Open Quantum Systems in Biology, Cognitive and Social Sciences 3031290232, 9783031290237

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Open Quantum Systems in Biology, Cognitive and Social Sciences

Andrei Y. Khrennikov

Open Quantum Systems in Biology, Cognitive and Social Sciences

Andrei Y. Khrennikov International Center for Mathematical Modeling Linnaeus University Växjö, Sweden

ISBN 978-3-031-29023-7 ISBN 978-3-031-29024-4 (eBook) https://doi.org/10.1007/978-3-031-29024-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to Masanori Ohya with whom I cooperated on quantum-like modeling in cognition and molecular biology for 25 years. He passed away too early and not all his ideas were realized. I shall always remember his hospitality at Noda-city (near Tokyo) and our daylong discussions.

Preface

This book is about quantum-like modeling and its applications. Such modeling is built on the methodology and the mathematical apparatus of quantum theory and it is directed to applications outside of physics, namely to biology, cognition, psychology, decision-making, economics, finances, social and political sciences, and artificial intelligence. It is of essential importance to signify that this approach can be explored for macroscopic systems and the system’s size is not significant. The quantum-like framework is applicable on all scales, that is to say from proteins and genes to animals, humans, and ecological and social systems. The crucial role is played by the character of information processing by a system and matching with the laws of quantum information theory. Systems are treated as information processors. Metaphorically one may say that system’s “hardware”, its physical and biological structures, are not so significant, but the system’s “software” plays the central role. We can speak about quantum bioinformatics [25] which should not be mixed with quantum biophysics [16]. The latter studies the genuine quantum physical processes in biosystems, e.g., in cells. It is important to point out the immense influence of mathematics in physics, emphasized by many scientists, and in particular, by E. Wigner [464]. However, mathematical tools commonly used in theoretical and mathematical biology, cognition, and psychology are not as efficient as in theoretical and mathematical physics. In his usual provocative manner, I. Gelfand, one of the famous Soviet mathematicians, contrasted Wigner’s thesis by pointing to “ineffectiveness of mathematics in biology”—this remark was mentioned by Arnold [17] with reference to Gelfand. From my point of view, Gelfand’s statement has to be reformulated and one would speak about ineffectiveness of mathematics that is commonly used in biology, cognition, and psychology. I presume someone has the intention to model the microsystems behavior, say electrons, atoms, photons, within the classical analysis of functions defined on phase space, A = A(q, p). In this case, one would confront difficulties and soon would notice either the impossibility of such description of quantum

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phenomena or at least its ineffectiveness.1 Physicists explored a new branch of mathematics, the theory of operators in complex Hilbert space in order to describe the quantum phenomena in an effective way. And in quantum physics, the noncommutative operator calculus works very well. Similarly, one should search for novel mathematics which is proper for biological and mental phenomena. This book advertises the same mathematics that was employed in quantum physics, noncommutative operator calculus in complex Hilbert space. Why is it so attractive for discussed applications? Personally, I was mainly driven by specialties of Quantum Probability (QP) calculus which matches mental phenomena very well [230, 231, 254]. This point will be discussed later in very detail. But one can look even at the deeper level. It is useful to extract the basic problems in mathematical modeling of mental phenomena highlighted by the experts in the field. I can recommend two handbooks [29, 195], and especially article [273] in the first one and the preface of the second one. Article [273] can be considered as the seed of this book. In it, I argue that living systems should be modeled within open quantum systems theory (see also [268]). The preface [29] is started with a brilliant citation from a story written by Edgar Allan Poe (1845) entitled “The Purloined Letter”. In this story, a protagonist, Mr. C. Auguste Dupin discussed the limits of mathematics applicability: “Mathematical axioms are not axioms of general truth. What is true of relation— of form and quantity—is often grossly false in regard to morals, for example. In this latter science it is very usually untrue that the aggregated parts are equal to the whole. [...] two motives, each of a given value, have not, necessarily, a value when united, equal to the sum of their values apart.” One can be surprised by Poe’s doubts about the applicability of mathematics (of 19s) century to moral phenomena (cf. with attempts of say Freud to proceed with “classical mathematics”). He also expressed doubts about the validity of the valueadditivity law. This is a very deep statement, and in quantum mathematics, it is formulated as “eigenvalues of the sum of operators C = A + B are not equal to the sums of the eigenvalues of the summands”, i.e., generally ci = ai + bi . In fact, the violation of the value-additivity law is the key point of von Neumann’s no-go theorem [451]; the first statement on the impossibility of classical reduction of quantum theory. Then, the authors of [29] also pointed out the noncommutativity effect in conjunctions, A&B = B& A. This order effect is also naturally formalized in the quantum framework. In fact, these two effects, the value-nonadditivity and the order ones, are closely connected. In probabilistic terms, they jointly expressed the violation of the formula of total probability and interference of probabilities (Sects. 1.3 and 1.4). 1

I stress effectiveness of the quantum description and do not highlight various no-go statements concerning the impossibility of the classical description (cf. [88, 89]).

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The essential part of the book is devoted to the order effect. This is a good place to mention that its QP-realization in decision-making has been done in the article [452]. We remark that in [29] the discussion is not coupled to quantum-like modeling: the authors searched for novel mathematical tools for psychology, but their considerations really cry for the appeal to quantum formalism. The main message [29] was that a variety of mathematical methods could be explored to solve the problems mentioned by Edgar Allan Poe which I completely agree with. Quantum formalism should not be treated as pretending to be the unique mathematical tool for modeling mental phenomena. A while ago, in response to the developments of using the quantum formalism outside of quantum mechanics, the eminent quantum physicist (the Nobel Prize 2023) Anton Zeilinger told me, “Why should it be precisely the quantum mechanics’ formalism? Maybe its generalization would be more adequate…” And he is right, for the moment, despite its tremendous success, quantum-like modeling is still at the testing stage. May be one day new, more advanced mathematical formalism will be suggested for modeling in cognition, psychology, and decision-making. However, from my viewpoint, quantum formalism is the most successful due to its simplicity. The reader may be surprised: “Simplicity? But the quantum theory is mysterious and very complicated!” One would immediately recall the famous statement commonly assigned to Richard Feynman “I think I can safely say that nobody understands quantum mechanics.” But here “understanding” is related to the interpretation problem of quantum mechanics; its formalism is very simple; it is linear algebra. And in quantum information theory, which is the most useful for applications, including quantum engineering, linear state spaces are finite dimensional. So, this is the matrix calculus in H = Cn . Linear evolution is very rapid and this is the advantage of the quantum-like like representation of mental states and the corresponding linear processing of them. Although I put so much effort into justification of quantum-like modeling through QP analysis and especially its contextual nature, slowly I started to understand that the seed of cognition quantumness (not only of humans, but also other biosystems), is in the logic structure of information processing. Quantum logic corresponds to the linear representation of information. The basic law distinguishing classical (Boolean) and quantum logic is the distributivity law, it is violated in quantum logic (see article [371] for the details). In this book, the logical aspects of quantum-like information processing are only briefly mentioned. After this motivational block (see also Sect. 1.1), we continue to discuss the book’s content in more detail. We emphasize that quantum-like modeling of cognition should be sharply distinguished from the quantum brain studies (see, e.g., [66, 67, 182, 208, 376, 443, 447, 448]) attempting to reduce information processing by cognitive systems including “generation of consciousness” to the quantum physical effects in the brain. However, we do not criticize the quantum brain project, although its difficulties are well known: e.g., the brain is too hot and big, and the scales of neurons operating are too rough to be compared with the quantum physical scales.

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In quantum-like modeling, it is simply not important whether the genuine quantum physical processes in the brain’s cells contribute to cognition or not. Generally, quantum-like modeling is performed on the meta-level of cognition; it does not concern the biophysical processes in neurons (however, cf. Chap. 6). In this framework, a biosystem, in particular, the brain is treated as a black box in which information processing cannot be described by the Classical Probability [318] (CP), and hence, the classical information theory. Non-classical probability and information theories are in demand. In particular, in decision-making, exploring CP leads to various paradoxes which are typically coupled to the irrational behavior of humans. My suggestion [230] was to employ Quantum Probability (QP) and quantum information theories, instead of the classical ones. Why should especially quantum theory be involved? This is a complex problem. There exist plenty of other models different from CP and QP. For example, the use of QP in decision-making was not derived from some basic principles for cognition and psychology. Commonly, QP is used pragmatically—to resolve paradoxes and to have a general probabilistic framework applicable to decision-making, in all areas of humanities and economics, as well as in biology. There is no priory reason to hope that QP would cover all problems which arise in decision-making. One might find paradoxes even in QP-based decision theory. May be other probabilistic models different both from CP and QP should be employed. Surprisingly, physicists have the same problem. In contrast to relativity theory, QM was not derived from natural physical principles (see Zeilinger [438] for the discussion on this problem). There is no reason to expect that all experiments in micro-world would match QP constraints. In physics, one typically debates CP versus QP, and classical versus quantum physics. However, one can even test whether physics of microsystems can violate the QP laws, i.e., whether electrons and photons can behave exotically even from the QP viewpoint. The corresponding test is given by the Sorkin equality [417] for the three slit experiment (see Chap. 15). This is really surprising that two- and threeslit experiments have so different probabilistic structures. The three-slit experiment was done by the Weihs group (Austria). They did not find deviations from QP, the Sorkin inequality was not violated [413, 414]. Similar experiments can be done for decision-making by humans (Chap. 15). So, my recommendation to the reader: “Apply the quantum-like models, but be cautious!” The essence of this book is the application of the special part of quantum theory— the open quantum systems theory. Any live biosystem is an open system. To analyze its behavior, it is natural to apply this theory [273] as the most general (known for the moment) theory describing the interaction of a system S and the surrounding environment E. As was already emphasized, biosystems are considered as information processors, and open quantum systems theory is treated as a part of the quantum information theory. The interaction between S and E is not classical force-like, but based on the information exchange. Generally, the open quantum systems theory is used as a theory of open information systems.

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In particular, the treatment of biosystems as open information systems can explain order stability in them, i.e., to present the quantum-like formalization of Schrödinger’s speculations in his famous book “What is life?" [406]. Quantum entropy (say von Neumann or linear entropy) is employed as a quantitative measure of order. Its dynamics in the process of information exchange between S and E crucially differs for the dynamics of classical entropy (Chaps. 4 and 5). In this book, the basic quantum master equation is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. This equation describes Markovian evolution. The latter posseses a strong constraint on the class of the mental state dynamics. By portraying cognition in the open quantum system framework, we suggest a quantum-like model of the brain’s functioning (Chap. 6). Now the brain is not treated as simply a black box. The starting point is consideration of the electrochemical states of neurons encoded in action potentials. Such states generate the brain’s mental states which are mathematically formalized as quantum states processed with the open quantum systems dynamics. This approach leads to the theory known as quantum dynamical decision-making or decoherence decision-making [18, 28]. Mathematically the decoherence process is modeled with the GKSL-equation [18] or more generally with the state-differentiation dynamics which was suggested in articles [26, 28]. We remark that in such a framework, the mental state collapse can be excluded from consideration. Decoherence is a deep foundational notion. Heuristically, it can be interpreted as the loss of quantumness, transition from QP to CP, and washing out of interference of probabilities (see also [442]). It is quantified with linear entropy, the measure of state’s purity. In quantum information theory, typically, decoherence is considered as a negative factor disturbing information processing. In quantum dynamical decisionmaking, decoherence plays the constructive role as decisions’ generator. The open quantum systems theory is also used for the mathematical formalization of the consciousness–unconsciousness interaction, the information exchange between them (Chap. 7). Consciousness plays the role of a measurement device, it performs observations over the states of unconsciousness. These observations can be interpreted as the brain’s self-observations. So, human’s thoughts and decisions are generated in the complex process of interaction between unconscious and conscious states. From the viewpoint of quantum foundations, we use Bohr’s interpretation of the outcomes of quantum measurements as generated in the complex process of interaction between a system and measurement apparatus (Sect. 17.3). In particular, these outcomes are not objective properties of a system that could be associated with it before measurement. In the same way, the mind is not objective. Bohr’s ideology structured within open quantum systems theory matches the Higher Order Theory of Consciousness [326, 328]. In this framework, we model the emotional coloring of conscious experiences. Such coloring is framed as contextualization. So, the theory of emotions is coupled to such a hot topic of quantum foundations as contextuality and the Bell inequalities. One of the specialties of this book is exploring of quantum instruments in applications to psychology and decision-making. Quantum instruments are the basic tools of the modern theory of quantum measurements. They describe the quantum state

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transformations generated by measurements’ feedback. Such transformations are not reduced to ones based on the projections. More general state space transformations are also needed. They cause a representation of quantum observable by Positive Operator Valued Measures (POVMs), generalized observables. Typically, the latter is considered as the basic entities of the modern theory of quantum measurements, especially in quantum information theory. However, POVMs are just byproducts of quantum instruments. POVM does not determine uniquely a state transformation coupled to measurement’s feedback on the system’s state. In physics, quantum instruments, and in particular, POVMs associated with them were introduced at the advanced stage of quantum theory’s development. However, modeling of cognition and decision-making should be based on quantum instruments even for the basic psychological effects; for example, the combination of the question order and response replicability effects [369, 370]. The von Neumann measurement theory has a restricted domain of applications (see Chap. 14 on the impossibility statement). The quantum instrument formalism is derived from open quantum systems theory through the indirect measurement scheme employing the unitary operator realization of the interaction between a system and a measurement apparatus. We hope that this book would serve as an introduction to quantum instrument theory for scientists working in mathematical modeling in cognition, psychology, and decision-making. Generally, we advertise exploring open quantum systems in modeling behavior of biosystems, from genes, proteins, and cells, to animals, humans, and ecological and social systems, from simple interference effects as lactose–glucose metabolism in cells to cognition, unconscious–conscious interaction, and collective social behavior. Globally, the book content is divided into six parts devoted to different areas of quantum-like modeling. Part I is introductory and it motivates the use of quantum theory. Part II is on applications of open quantum systems theory to general biological phenomena with an emphasis on the problem of stability in biosystems, here the main mathematical tool is the quantum master equation generating decoherence; the latter is treated positively as the source of state’s stabilization. Part III is an introduction to the quantum instruments theory with applications to psychology and decision-making. In Part IV, we move to the social systems and modeling of their behavior with open quantum systems and field theory, social laser and social Fröhlich condensation. Part V contains a discussion on the possible generalizations of the quantum-like modeling, by employing more exotic theories; in particular, it contains a no-go theorem for proceeding solely with von Neumann measurements. Detailed presentation of the mathematical formalism and quantum foundations is postponed to Part VI (the last part of the book), with an emphasis on the role of incompatibility and contextuality and the differences in the views of Bohr and Bell; the appendix contains mathematical technicalities related to some statements in the book. Växjö, Sweden December 2022

Andrei Y. Khrennikov

Acknowledgments

I discussed the book’s content with practically all leading experts in quantum-like modeling and I am very grateful to all who gave me advice or stimulated my thoughts with the inquiries. I am especially thankful to Luigi Accardi, Alexander Alodjants, Massanari Asano, Harald Atmanspacher, Fabio Bagarello, Peter Bruza, Jerome Busmeyer, Acacio de Barros, Ehtibar Dzhafarov, Andrei Grib, Emmanuel Haven, Polina Khrennikova, Masanao Ozawa, Arkady Plotnitsky, Emmanuel Pothos, Zeno Toffano, Noboru Watanabe, and Alexander Wendt. On some occasions, I debated with Stuart Hameroff, Roger Penrose, and Giuseppe Vitiello, such debates clarified the commonalities and differences between quantum physical and quantum-like modeling. During 22 years of Växjö conferences on the quantum foundations, I had numerous discussions with the world’s leading experts in the field, and I am especially thankful to Alain Aspect, Mauro D’ Ariano, Avashlom Elitzur, Christopher Fuchs, Philippe Grangier, Karl Hess, Anton Zeilinger, Gregg Jaeger, Arkady Plotnitsky, Karl Svozil, Lev Vaidman, and Igor Volovich. And I am grateful to Giorgio Parisi, Sergey Kozyrev, Kazuyuki Kuchitsu, and Ichiro Yamato for conversations on the foundational issues of modeling complex biological systems. This book would never be written without the everyday support of Anja Nertyk, especially during the turbulent period of spring 2022. Växjö, Sweden December 2022

Andrei Y. Khrennikov

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Contents

Part I 1

Quantum-Like Modeling

Interplay Between Classical and Quantum Probability . . . . . . . . . . . . 1.1 Quantum-Like Models: Motivation . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Briefly About Classical and Quantum Probabilities . . . . . . . . . . . 1.3 Interference of Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Bayesian Versus Non-Bayesian Inference . . . . . . . . . . . . . . . . . . . 1.5 Quantum-Like Paradigm: Contextual Information Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 6 7

2

Quantum Formalism for Decision-Making . . . . . . . . . . . . . . . . . . . . . . . 2.1 Elementary Quantum Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quantum-Like Model for Decision-Making . . . . . . . . . . . . . . . . . 2.3 Decision-Making via Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Decision Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 15 16 18

3

Classical Versus Quantum Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Savage Sure Thing Principle and Interference of Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Quantum-Like Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Coupling of Social Lasing to Deprivation of Classical Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Question Order and Response Replicability Effects . . . . . . . . . . 3.5 State-Dependent Incompatibility . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 4

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21 22 24 25 27

Biosystems as Open Quantum-Like Systems

What is Life? Open Quantum Systems Approach . . . . . . . . . . . . . . . . 4.1 Schrödinger About Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Theory of Open Quantum Systems and Order in Biosystems . . . 4.3 Supplement to Elementary Quantum Vocabulary . . . . . . . . . . . . . 4.3.1 Superoperators and Quantum Channels . . . . . . . . . . . . . 4.3.2 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.3

4.4 4.5

4.6 4.7 4.8 4.9 4.10 5

6

Isolated System: Schrödinger and Von Neumann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gorini-Kossakowski-Sudarshan-Lindblad Equation . . . . . . . . . . 4.5.1 Stabilization to Steady State . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Quantum Markov Dynamics . . . . . . . . . . . . . . . . . . . . . . Camel-Like Dynamics of Quantum Entropy . . . . . . . . . . . . . . . . . Open Systems Generating Disorder . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Linear Entropy (Decoherence) . . . . . . . . . . . . . . . . . Specialty of Biosystem’s States and Dynamics . . . . . . . . . . . . . . . Adaptation to the Environment: Illustrative Examples . . . . . . . .

37 38 39 40 40 41 44 46 48 50

Order Stability in Complex Biosystems . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Biosystem: Global Order from Local Disorder . . . . . . . . . . . . . . . 5.2 Compound Classical System: Local and Global Orders . . . . . . . 5.3 Compound Quantum System: Global and Local Orders . . . . . . . 5.4 Global Order from Local Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Complex Biosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Concluding Discussion: Order Stability in Bio and AI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 57 59 60

Brain Functioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Psychic Versus Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . 6.2 From Electrochemical Uncertainty in Action Potentials to Quantum Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Electrochemical and Quantum Information States: Nonlinear Versus Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Quantum Physics of Brain’s Functioning: Impossibility of Superposition of Neuron’s States . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Mental Function as Decoherence Machine . . . . . . . . . . . . . . . . . . 6.6 Model 1: Collapse of Mental Wave . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Model 2: Open Quantum System Dynamics . . . . . . . . . . . . . . . . . 6.7.1 General Quantum Dynamics of Mental State . . . . . . . . . 6.7.2 Critical Analysis of Open Quantum System Approach to Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Model 3: Differentiation of Mental State . . . . . . . . . . . . . . . . . . . . 6.8.1 Signatures of Environments in a Density Operator . . . . 6.8.2 Mathematical Scheme of Differentiation Process . . . . . 6.9 Autopoiesis: Quantum Information Representation . . . . . . . . . . . 6.10 Entanglement: Physics Versus Cognition . . . . . . . . . . . . . . . . . . . 6.11 Ontic and Epistemic Portrayals of Mental Processes . . . . . . . . . . 6.12 Concluding Discussion on Quantum-Like Modeling of Brain’s Functioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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66 70 71 72 75 76 77 79 80 80 81 83 84 86 89

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Emotional Coloring of Conscious Experiences . . . . . . . . . . . . . . . . . . . 91 7.1 Quantum Formalization of Emotional Coloring of Conscious Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 The First and Higher Order Theories of Consciousness . . . . . . . 92 7.3 Contextuality of The Higher Order Theory of Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.4 Perceptions and Emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.5 Unconscious and Conscious Information Processing . . . . . . . . . . 96 7.5.1 Unconsciousness as System . . . . . . . . . . . . . . . . . . . . . . . 96 7.5.2 Consciousness as Observer . . . . . . . . . . . . . . . . . . . . . . . . 96 7.5.3 Unconscious and Conscious Generation of Perceptions and Emotions . . . . . . . . . . . . . . . . . . . . . . 96 7.5.4 Conscious Experiences: Basic and Supplementary . . . . 97 7.6 Incompatible Conscious Observables . . . . . . . . . . . . . . . . . . . . . . . 97 7.7 Degeneration Resolution of Conscious Experiences via Contextual Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.8 Tensor Product Decomposition of Unconscious State Space . . . 100 7.9 CHSH Inequality: Test of Emotional Contextuality . . . . . . . . . . . 102 7.10 Concluding Remarks on Emotional Coloring . . . . . . . . . . . . . . . . 103

Part III Quantum Instruments in Psychology and Decision-Making 8

9

Quantum Instruments and Positive Operator Valued Measures . . . . 8.1 Von Neumann Observables and von Neumann-Lüders Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Davis–Lewis–Ozawa Quantum Instruments . . . . . . . . . . . . . . . . . 8.3 Positive Operator Valued Measures . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Quantum Instruments from Indirect Measurements . . . . . . . . . . . 8.5 Naimark Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 110 113 115 118

Question Order and Response Replicability Effects . . . . . . . . . . . . . . . 9.1 Quantum Instruments for Questions . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Questions as Indirect Measurements . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Von Neumann Observables for Questions . . . . . . . . . . . 9.2.2 Quantum a-Instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Quantum Algorithm for Decision-Making . . . . . . . . . . . 9.2.4 Quantum b-Instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Combination of Order and Response Replicability Effects . . . . . 9.3.1 Stability of Question Order Effect . . . . . . . . . . . . . . . . . . 9.3.2 Non-atomicity of Instruments . . . . . . . . . . . . . . . . . . . . .

119 119 121 121 122 125 126 127 129 130

10 Psychological Effects and QQ-equality . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.1 Question Order and Response Replicability Effects and QQ-equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.1.1 Observables, Belief and Personality States . . . . . . . . . . . 134

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10.1.2 Instrument Measuring a-Observable . . . . . . . . . . . . . . . . 10.1.3 Instrument Measuring b-Observable . . . . . . . . . . . . . . . . 10.2 Generalization of Wang–Busemeyer Postulates . . . . . . . . . . . . . . 10.3 Mind State Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Response Replicability Effect: Personality States . . . . . . . . . . . . 10.5 Question Order Effect: Personality States . . . . . . . . . . . . . . . . . . . 10.6 Matching with QQ-equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Linking Experimental and Theoretical Data . . . . . . . . . . . . . . . . . 10.8 Independence of Belief and Personality States . . . . . . . . . . . . . . . 10.9 Modeling Statistical Data from Clinton–Gore Poll . . . . . . . . . . . 10.10 On Postulate 5QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 137 138 141 142 144 145 146 149 152 155

Part IV Analysis of Social Systems within Open Quantum System Theory 11 Social Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Social-Information Waves Shaking the World . . . . . . . . . . . . . . . 11.2 Social Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Social Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Social-Information Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Absorption and Emission of Infons by Social Atom . . . . . . . . . . 11.6 Social and Physical Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Echo Chamber: Social Coherence Reinforcing . . . . . . . . . . . . . . . 11.8 Illustrating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10 Social Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.11 Concluding Remarks On Social Laser . . . . . . . . . . . . . . . . . . . . . . 12 Stability in Biological, Ecological, and Social Systems via Fröhlich Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Modeling of Fröhlich Condensation . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Coherent Vibrations in Biomolecules and Cells . . . . . . 12.1.2 Long-Range Nonlinear Interactions . . . . . . . . . . . . . . . . 12.1.3 Quantum-Like Modeling of Fröhlich Condensation . . . 12.1.4 Fröhlich Condensation of Information Excitations . . . . 12.2 Review on Fröhlich’s Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Conditions for Fröhlich Condensation . . . . . . . . . . . . . . . . . . . . . . 12.4 Quantum Formalism for Fröhlich Condensation . . . . . . . . . . . . . 12.5 Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Condensation of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Information Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Stability of Complex Information Societies . . . . . . . . . . . . . . . . . 12.9 Order in Pack of Wolfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Concluding Remarks on Physical and Social Fröhlich Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 162 163 164 169 171 176 178 184 186 188 191 191 192 192 193 193 194 197 197 198 199 201 202 204 208

Contents

xix

13 Social Laser and Networks Within Mean Field Theory . . . . . . . . . . . . 13.1 Social Networks: Laser Physics, Phase Transition, and Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Ising Model for Complex Networks: Equilibrium Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Ising-Type Interaction ín Networks . . . . . . . . . . . . . . . . . 13.2.2 Phase Transition and Network’s Structure . . . . . . . . . . . 13.3 Non-equilibrium Phase Transition in Social Laser . . . . . . . . . . . . 13.3.1 Primary Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Social Laser: Mean Field Framework . . . . . . . . . . . . . . . 13.3.3 Social Laser: Phase Transition . . . . . . . . . . . . . . . . . . . . . 13.4 Social Laser Dynamics and Information Spread . . . . . . . . . . . . . . 13.4.1 Viral Information Cascades in Social Laser . . . . . . . . . . 13.4.2 Velocity of Information Reinforced . . . . . . . . . . . . . . . . . 13.5 Concluding Discussion on Social Laser and Networks . . . . . . . . Part V

211 211 213 213 217 219 219 222 226 229 229 231 233

Boundaries of Applicability of Quantum-Like Modeling

14 No-Go Theorem for Modeling with Von Neumann Observables . . . . 14.1 Sequential Measurements in Physics and Psychology . . . . . . . . . 14.2 Von Neumann Observables and Unitary Inter-measurement Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Measurement Sequences: Evolution (In)Effectiveness and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Measurement Sequences a → a . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Measurement Sequences a → b → a . . . . . . . . . . . . . . . . . . . . .

241 244 246

15 Probabilistic Structure of Cognition: May Be Even Worse than Quantum? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Comparing Foundations of Quantum Physics and Cognition . . . 15.2 Abstract Presentation for Sorkin’s Equality . . . . . . . . . . . . . . . . . 15.3 Derivation of Sorkin’s Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Emigration Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Triple-Store Experiment in Economics . . . . . . . . . . . . . . . . . . . . .

251 251 252 255 257 258

Part VI

237 237 240

Foundations and Mathematics

16 Formalism of Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Mathematical Structure of Quantum Theory . . . . . . . . . . . . . . . . . 16.2 Quantum Mechanics as Axiomatic Theory . . . . . . . . . . . . . . . . . . 16.3 Projection Postulate: Von Neumann Versus Lüders Forms . . . . . 16.4 Classical Probability: Kolmogorov Axiomatics . . . . . . . . . . . . . . 16.5 Quantum Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Derivation of Interference of Probabilities . . . . . . . . . . . . . . . . . . 16.7 Compatible Versus Incompatible Observables . . . . . . . . . . . . . . . 16.8 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

261 261 264 266 267 269 271 272 273

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Contents

16.9 16.10 16.11 16.12

Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symbolism of Ket- and Bra-Vectors . . . . . . . . . . . . . . . . . . . . . . . . Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

274 276 276 277

17 Contextuality, Complementarity, and Bell Tests . . . . . . . . . . . . . . . . . . 17.1 Preliminary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Växjö Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Thinking over Bohr’s Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Bohr Contextuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Bohr’s Principle of Contextuality-Complementarity . . . 17.4 Probabilistic Viewpoint on Contextuality-Complementarity . . . 17.5 Clauser, Horne, Shimony, and Holt (CHSH) Inequality . . . . . . . 17.6 CHSH-Inequality for Quantum Observables . . . . . . . . . . . . . . . . . 17.7 Signaling in Physical Versus Psychological Experiments . . . . . . 17.8 Contextuality-by-Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9 Mental Signaling: Fundamental or Technical? . . . . . . . . . . . . . . . 17.10 Sources of Signaling Compatible with Quantum Formalism . . . 17.11 Meal Choice Experiment: Possible Source of Signaling . . . . . . . 17.12 Concluding Remarks on Cognitive Tests with Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.13 Joint Measurement Contextuality . . . . . . . . . . . . . . . . . . . . . . . . . . 17.14 Contextual Resolution of Degeneration of Eigenvalues of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 279 282 284 285 285 287 290 292 294 295 299 299 304

18 Quantum Statistics from Indistinguishability . . . . . . . . . . . . . . . . . . . . 18.1 Thermodynamics from Gibbs Ideal Ensembles . . . . . . . . . . . . . . 18.2 Distinguishable Systems: Classical Statistics . . . . . . . . . . . . . . . . 18.3 Indistinguishable Systems: Quantum Statistics . . . . . . . . . . . . . . . 18.4 Classical and Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311 312 313 315

Part VII

307 308 309

Supplement on Decision-Making

19 Classical Expected Utility Theory and Its Paradoxes . . . . . . . . . . . . . . 19.1 Von Neumann and Morgenstern: Expected Utility Theory . . . . . 19.2 Allais Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Savage: Subjective Expected Utility Theory . . . . . . . . . . . . . . . . . 19.4 Ellsberg Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Quantum-Like Modeling of Subjective Expected Utility . . . . . .

319 319 320 321 322 324

20 Belief State Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 The Spirit of Copenhagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Interpretations: Statistical Versus Individual . . . . . . . . . . . . . . . . . 20.3 QBism: Subjective Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Växjö Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327 327 328 329 330

Contents

21 God as Decision Maker and Quantum Bayesianism . . . . . . . . . . . . . . . 21.1 Bohr Versus Bell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Supplement on Quantum Bayesianism (QBism) . . . . . . . . . . . . . 21.3 QBism Versus Copenhagen Interpretation . . . . . . . . . . . . . . . . . . . 21.4 Free Will Given by God to Adam is the Source of Irreducible Uncertainty in the Universe . . . . . . . . . . . . . . . . . . 21.5 God as Decision Maker Operating with Subjective Probability Assigned to His Personal Experiences . . . . . . . . . . . .

xxi

333 333 334 335 336 337

Appendix A: Technicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Part I

Quantum-Like Modeling

The central objective of this part is to illuminate the major directions of quantum-like modeling, especially in cognitive psychology, decision-making, and social science (see Part II for general quantum-like modeling for biosystems). First, we introduce the motivation for operating with quantum probability outside of physics. Then we recount the basics of classical and quantum probability (CP and QP) theories and set forth the principles of quantum-like modeling of decision-making. In particular, consideration of a special quantum-like model, “decision-making via decoherence”, leads to coupling with the theory of open quantum systems. The Savage Sure Thing principle is the starting point of the discussion on CP vs. QP-based notions of rationality. The violation of the total probability formula is widely used in the description of the classical rationality violation. The classical total probability formula is modified by adding the interference term, the magnitude of which can be treated as the degree of deviation from classical rationality. The problem of rationality is coupled with the disjunction and question order effects. Attempts to combine the latter with another psychological effect, the response replicability effect, conduct us to inquire about the possibility to proceed using the standard quantum measurement theory (with the representation of observables by Hermitian operators and the quantum state update via the projection postulate). We demonstrate that this psychological effects combination can be modeled with quantum instruments. The social laser theory briefly mentioned in this part contains a comprehensive description of the classical rationality violation by humans overloaded with information. This theory describes, in particular, Stimulated Amplification of Social Actions (SASA), e.g., in the form of color revolutions and other social tsunamis. This part is written schematically with the minimal introduction into the quantum methodology and mathematical apparatus. The detailed presentation of quantum axiomatics and the mathematical structure of quantum mechanics is postponed to Part VI. We expect that the majority of the readers have some quantum experience; Sect. 2.1 serves as a short recollection. Readers who have no idea about the quantum theory may try to proceed by just reading Sect. 2.1. If it would not work, they can jump to Part VI and follow Chap. 16; Chap. 17 is devoted to the deep foundational problems, contextuality, complementarity, and Bell inequalities.

Chapter 1

Interplay Between Classical and Quantum Probability

1.1 Quantum-Like Models: Motivation In quantum-like modeling, a biosystem is a black box which information processing can’t be described by the classical probability [318] (CP) and, hence, the classical information theory. Biosystem’s size is not critical. The key point is that information processing in such a system follows the laws of quantum probability [271] (QP) and information. How and why do biosystems use the quantum-like representation of information? This question is rather perplexing and the answers are presented in the following chapters (see, especially Chap. 6). It is important to denote that it can be found plenty of probabilistic data which doesn’t match CP, for example, in biology on all scales, from molecular biology to ecology, cognitive psychology, and decision-making. In decision-making, such data is typically coupled to probability fallacies and irrational behavior of agents (see Chap. 19 for paradoxes). The existence of this data is the primary reason for appealing to QP instead of CP [87, 386, 387]. The situation is similar to quantum physics derived not from natural physical principles, as say special relativity, but created “by hands” to describe the statistics of outputs of experiments in atomic physics, cf. Zeilinger [438]. One may say that the appeal to QP and, consequently, to quantum information theory to model, e.g., humans’ behavior and decision-making is too exotic. The following natural question arises: Why do we apply QP to humans, or, generally, biosystems? Now we discuss the motivation for quantum-like modeling of decision-making in more detail. We recall that as early as the 1970s, Tversky, one of the most cited psychologists of all time, and Kahneman, who took the Nobel prize in economics in 2002, for prospect theory, which he co-developed with Tversky, have been demonstrating cases where classical probability prescription and human behavior persis-

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_1

3

4

1 Interplay Between Classical and Quantum Probability

tently diverge [221–225]. Kahneman points out that, today, we are at the theoretical cross-roads, with huge divisions across conflicting, entrenched theoretical positions: Should we continue relying on CP as the basis for descriptive and normative predictions in decision-making (and perhaps ascribe inconsistencies to methodological idiosyncrasies)? Should we abandon probability theory completely and instead pursue explanations based on heuristics, as Tversky and Kahneman proposed? However, the use of probabilistic and statistical methods is really the cornerstone of the modern scientific methodology, both in natural and social sciences. Thus, although the heuristic approach to decision-making can’t be discarded completely, it seems more natural not to reject the probabilistic approach to decision-making, but to search for novel probabilistic models. And QP [26] is a good candidate for an alternative to CP. We continue the discussion on basics of quantum-like modeling in Chap. 4 by concentrating on its quantum information aspects and coupling with theory of open quantum systems. Generally, the quantum-like modeling project is very successful. It is shown that such models can be applied to modeling the behavior of biosystems on all spatial scales. Genes, proteins, cells, animals, humans, and social systems can be treated as information processors and decision makers within the same mathematical model [25]. From the information perspective, we can speak about the creation of a new theory—quantum bioinformatics—which is a part of quantum information theory, but not of quantum biophysics. One of the advantages of this theory is its generality, its framework covers all possible biological systems. Another advantage of the quantum-like representation is its linearity. The quantum state space is a complex Hilbert space and dynamical equations are linear differential equations. The classical biophysical dynamics beyond quantum information representation are typically nonlinear and very complicated. The use of linear space representation simplifies the processing structure. The quantum information representation means that generally large clusters of classical biophysical states are encoded by a few quantum states: the quantum structure arises as the result of coarse graining (Chap. 6). It leads to essential information compressing. It also implies an increase of stability in state processing (Chap. 4). This is a rather unusual viewpoint on the use of the quantum information representation as lowering the complexity and instability of information encoding and processing. Since the essential part of this book is devoted to psychology and decision-making, it is important to highlight the contribution of the quantum-like approach to mathematical modeling of psychological effects, e.g., the order effect, conjunction and disjunction effects, and response replicability effect (see, e.g., [87, 89, 90, 190, 191, 254, 369, 370, 386–388, 452, 453]).

1.2 Briefly About Classical and Quantum Probabilities

5

1.2 Briefly About Classical and Quantum Probabilities In this section, we do not concern the complex problem of the interpretation of probability. We briefly discuss this problem in Chap. 20. CP was mathematically formalized by Kolmogorov [318] (1933). This is the calculus of probability measures, nonnegative weight P(A) is assigned to any event A. Events are represented by a family of sets forming a special set-theoretical structure (Chap. 16). The main property of classical probability P is its additivity: if two events A1 , A2 are disjoint, i.e., in the set-representation, A1 ∩ A2 = ∅, then the probability of disjunction of these events equals the sum of probabilities P(A1 ∪ A2 ) = P(A1 ) + P(A2 ). In the mathematical theory, the condition of additivity is extended to countableadditivity, i.e., for a sequence of pairwise disjoint events A1 , . . . , An , . . . , P(A1 ∪ A2 ∪ · · · ∪ An ∪ . . .) = P(A1 ) + P(A2 ) + · · · + P(An ) + . . . . However, as was emphasized by Kolmogorov [318], countable-additivity can’t be checked experimentally, so this is the pure mathematical condition which is needed for establishing the theory of Lebesque integration. QP [26] is the calculus of complex probability amplitudes (wave functions) or in the abstract formalism complex vectors—quantum states. Thus, instead of the operations with probability measures, one operates with vectors belonging to a complex Hilbert space. In the L 2 -space, each complex amplitude ψ = ψ(x), x ∈ R3 is normalized by one, i.e.,  |ψ(x)|2 d x = 1, R3

generates the probability by the Born rule: Probability to find a quantum system at the point x is given by P(x|ψ) = |ψ(x)|2 .

(1.1)

Then, for any (Borel) subset A of R3 ,  P(x ∈ A|ψ) =

P(x|ψ)d x. A

If we consider just one observable, then the possibility to represent its probability distribution with the complex amplitude does not imply foundational consequences. The essence of QP is that the same amplitude (quantum state) can be used to generate the probability distributions for all possible quantum observables; some of them are incompatible, i.e., they can’t be represented as random variables on the same prob-

6

1 Interplay Between Classical and Quantum Probability

ability space. A probability measure P which can be used for all these observables does not exist, but a common complex amplitude ψ (quantum state) does exist. What is its meaning? This is the most complicated foundational problem of QM, the problem of the interpretation of a quantum state. This problem is discussed by many authors, see also my books [231, 250, 251], and it is characterized by the diversity of viewpoints. For example, I suggested the Växjö interpretation [251, 256] (Chaps. 17 and 20). By this interpretation, a quantum state represents the complete experimental context, in other words, the combination of the preparation and measurement procedures. In cognitive applications, we can interpret ψ as a mental state or a belief state—we will use both terms equivalently. Of course, one can interpret a mental state. In Chap. 20, we discuss the problem of interpretations in connection with decision theory.

1.3 Interference of Probabilities By operating with complex probability amplitudes, instead of the direct operation with probabilities, one can violate the basic laws of CP, in particular, additivity of probability. One can get that, for disjoint events, the probability of disjunction is strictly smaller or larger than the sum of probabilities P(A1 ∪ A2 ) < P(A1 ) + P(A2 ) or P(A1 ∪ A2 ) > P(A1 ) + P(A2 ), since QP calculus leads to the formula [231–233, 250, 251]  P(A1 ∪ A2 ) = P(A1 ) + P(A2 ) + 2 cos θ P(A1 )P(A2 ).

(1.2)

The additional term is known as the interference term. To derive this formula, we represent probabilities with the Born rule P(Ai ) = |ψi |2 , i = 1, 2, P(A1 ∪ A2 ) = |ψ1 + ψ2 |2 .

(1.3)

Then algebra of complex numbers implies that |ψ1 + ψ2 |2 = |ψ1 |2 + |ψ2 |2 + 2 cos θ |ψ1 ||ψ2 |,

(1.4)

where θ is formed of arguments of complex numbers ψi = |ψi |eiθi as θ = θ1 − θ2 . Thus, interference of probabilities has a simple origin: Born’s rule + complex algebra. We recall that interference is the basic feature of waves, so often one speaks about the probability waves. These are not usual waves in physical space, these are

1.4 Bayesian Versus Non-Bayesian Inference

7

fluctuations of probability that can lead to its amplification, namely, constructive interference for cos θ > 0) or diminishing (destructive interference for cos θ < 0). Surprisingly, physicist Richard Feynman (and not a mathematician) was the first who derived the formula for interference of probabilities (1.2) by analyzing the probabilistic structure of the two slit experiment [149]. This is the good place to present Feynman’s views on the role of probability in quantum physics [148] (italic was added by me): From about the beginning of the twentieth century experimental physics amassed an impressive array of strange phenomena which demonstrated the inadequacy of classical physics. The attempts to discover a theoretical structure for the new phenomena led at first to a confusion in which it appeared that light,and electrons, sometimes behaved like waves and sometimes like particles. This apparent inconsistency was completely resolved in 1926 and 1927 in the theory called quantum mechanics. The new theory asserts that there are experiments for which the exact outcome is fundamentally unpredictable, and that in these cases one has to be satisfied with computing probabilities of various outcomes. But far more fundamental was the discovery that in nature the laws of combining probabilities were not those of the classical probability theory of Laplace.

I also worked a lot on the interference of probabilities (see monographs [251, 254] for the complete account of my studies in this direction). Although both Feynman and I couple the appearance of the interference term to the contextuality of quantum theory,1 My approach differs from Feynman’s one. The right probabilistic meaning to the interference of probabilities can be assigned not via the violation of additivity of probability (this was Feynman’s interpretation), but via perturbation of the classical formula of total probability (Sect. 1.11). The latter assumes the conditional interpretation of QP (see also Ballentine [42–44]).

1.4 Bayesian Versus Non-Bayesian Inference We recall the Bayes formula for conditional probability in the CP-framework P(B|A) =

P(B ∩ A) , P(A) > 0. P(A)

(1.5)

This definition of conditional probability implies that the probability of joint occurrence of events A and B can be expressed via the conditional probability as P(A ∩ B) = P(A)P(B|A),

(1.6)

or 1

In the example of the two slit experiment considered by Feynman [149], there are involved three different contexts: C12 = both slits are open, C1 = only slit 1 is open, and C2 = only slit 2 is open [251].

8

1 Interplay Between Classical and Quantum Probability

P(A ∩ B) = P(B)P(A|B).

(1.7)

In CP, the LHSs of these two formulas give the same answer. However, in QP this is not the case. The formula of total probability (FTP) is a simple consequence of the Bayes formula. Consider a pair a and b of discrete random variables CP-representing observables. Then  P(a = α)P(b = β|a = α). (1.8) P(b = β) = α

Bayesian inference within CP is used to determine the posterior probability from two given probabilities: 1. prior probability, 2. likelihood function. By using Bayesian inference, one can find the posterior probability with the aid of the Bayes theorem (a trivial consequence of the Bayes formula for conditional probability): P(E | H ) · P(H ) P(H | E) = (1.9) P(E) where H stands for a hypothesis whose validity is checked. Typically, decision maker considers a few competing hypotheses (Hi ), and the task is to determine which is the most probable; P(H ) is the prior probability, it gives the estimate of the hypothesis’ probability before the observational data E is obtained. The output of inference is the posterior probability P(H | E); this is the probability after data E is observed. P(E | H ) is the probability of observing E under the assumption of that the hypothesis H is true. This quantity is called the likelihood. As a function of E with H fixed, it indicates the compatibility of the evidence with the given hypothesis. The quantity P(E) is known as model evidence for the realization of data E. This factor is the same for all possible hypotheses being considered within this model. It is determined by FTP  P(H j )P(E|H j ), P(E) = j

where (H j ) is a complete set of hypothesis under consideration. Hence P(E | Hi ) · P(Hi ) P(Hi | E) =  j P(H j )P(E|H j )

(1.10)

To make a decision, one compares posterior probabilities (P(Hi | E)) and finds the maximal of them.

1.5 Quantum-Like Paradigm: Contextual Information Processing

9

We remark that the Bayesian approach to conditional probability plays a crucial role in CP modeling of cognition, machine learning, and artificial intelligence, especially applications of Bayesian networks. QP provides the possibility to relax some constraints on statistical data posed by CP. One such constraint is classical FTP which is a consequence of additivity of CP and the Bayes formula in the definition of conditional probability. More generally, QP extends the calculus of conditional probabilities and the operation of the probability update is generalized via the quantum operation of the state update. The latter is based on the projection postulate (Chap. 16) or quantum instrument theory (Chap. 8). Thus, the QP decision-making project can be considered as extension of boundaries of Bayesian probability inference. In particular, classical FTP (1.8) is modified; the quantum analog of FTP has the additional term representing interference between the complex probability amplitudes representing probabilities via the Born rule. For dichotomous observable a, it has the form P(b = β|ψ) = P(a = α1 |ψ)P(b = β|a = α1 , ψ) + P(a = α2 |ψ)P(b = β|a = α2 , ψ)+

(1.11) 2 cos θ



P(a = α1 |ψ)P(b = β|a = α1 , ψ)P(a = α2 |ψ)P(b = β|a = α2 , ψ).

This FTP modifies the probability-interference formula (1.2) (see also Sect. 16.6); the QP-conditional probabilities are defined in Sect. 16.5. FTP with interference term was derived in my works [231–233, 250, 251]. I was inspired by Feynman’s analysis of the probabilistic structure of the two slit experiment [149] that led him to the conclusion on non-additivity of QP and violation of the CP-additivity axiom. Since quantum probabilities are conditional (or better to say contextual) probabilities, it is more natural to express Feynman’s conclusion as the FTP violation. QP inference is framed within the complex Hilbert space formalism via state update generated by measurement feedback on a quantum state. In the simplest case, this state update is mathematically described by the projection postulate and more generally by quantum instruments (Chap. 8). By using quantum conditional probabilities and FTP with the interference term, this quantum scheme can be mathematically formed similarly to the Bayes theorem. In this book, we don’t describe this CP-like representation of the probability update via state update (see articles [192, 193]).

1.5 Quantum-Like Paradigm: Contextual Information Processing The following paradigm can be used to motivate the applications of QP outside of physics.

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1 Interplay Between Classical and Quantum Probability

Quantum-like paradigm (Khrennikov 1999): The mathematical formalism of quantum information and probability theories can be used to model behavior not only of genuine quantum physical systems, but all context-sensitive systems, e.g., humans, ecological, social, political, and spiritual systems. Contextual information processing can’t be based on the complete resolution of ambiguity. It is meaningless to do this for the concrete context, if tomorrow, or in a few seconds or even milliseconds, the context will be totally different. Therefore, such systems process ambiguities represented by superpositions of alternatives. Mathematically, ambiguity-superpositions are described as linear combinations of quantum states. For cognitive systems, they are interpreted as the belief states. We illustrate such information processing with the following toy example. Ivan has a girl friend named Sara; he is not sure whether he loves her or not. His belief state can be mathematically expressed as a superposition ψ(t) = c0 (t)|not love + c1 (t)|love,

(1.12)

where |c0 (t)|2 + |c1 (t)|2 = 1. These complex amplitudes determine intensities of his feeling. They fluctuate in the process of his belief-state evolution. He can process information in two ways: • classically: at each instant of time t, Ivan resolves uncertainty and determines whether he loves Sara and then he updates his behavior on the basis of this information; • quantum-likely: Ivan does not try to resolve uncertainty and proceeds with superposition |ψ(t). The latter processing saves a lot of computational and psychic resources. Ivan being in love superposition can use these resources to solve other life problems. Even “quantum-like Ivan” may soon or later resolve this ambiguity—to perform the self measurement. But he may as well proceed with this love superposition for years and even for his whole life. Self measurement can happen as the result of some external circumstances since Ivan is an open quantum information system.

Chapter 2

Quantum Formalism for Decision-Making

2.1 Elementary Quantum Vocabulary Denote by the symbol H a complex Hilbert space, the scalar product of two vectors ψ and φ is denoted as ψ|φ. We consider only finite dimensional state spaces, as typically one does in quantum information theory. Real physics is based on infinite dimensional Hilbert spaces, such as the space L 2 of square integrable functions. Pure States A pure quantum state is mathematically represented by a normalized vector in H, ||ψ||2 = ψ|ψ = 1. Two vectors ψ1 and ψ2 such that ψ2 = eiθ ψ1 correspond to the same pure quantum state. To be rigorous, a pure state is a class of normalized vectors which differ by phase factors. Thus, one should be careful with geometric illustration for operations on pure states (cf. [89]). Since H is a linear space, it is possible to form linear combinations of its vectors; in the quantum formalism linear combinations (with normalization) are known as state superpositions. By choosing in H some orthonormal basis, it can be represented as the space of vectors with complex coordinates, ψ = (z 1 , z 2 , . . . , z n ), z j ∈ C, where C is the set of complex numbers. The scalar product is defined as u|v =



u¯ i vi , u = (u 1 , . . . , u n ), v = (v1 , . . . , vn ),

(2.1)

i

where, for a complex number z = x + i y, x, y ∈ R, its conjugate is denoted by z¯ , here z¯ = x − i y. The absolute value of z is given by |z|2 = z z¯ = x 2 + y 2 . Von Neumann Observables - Hermitian Operators In the standard quantum measurement theory (due to von Neumann [451]), an observable a is mathematically represented by a Hermitian operator A (thus, A = A ). Its

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_2

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eigenvalues (spectrum)1 encode the possible outcomes, α = α1 , . . . , αk . This is the spectral postulate of quantum mechanics. By choosing an orthonormal basis H can be represented as H = Cn and an observable as a Hermitian matrix, i.e., A = (ai j ), where ai j = a¯ ji . Matrix’s eigenvalues encode the outputs of observations. Thus, the quantum measurement model is mathematically formalized as the linear algebraic calculus. Due to linearity quantum theory is mathematically very simple. Simplicity is one of the attractive sides of this theory. Consider an observable a with Hermitian operator A. Let E(α) be projection on the eigenspace of vectors corresponding to the eigenvalue α. So, E(α) is projection on the subspace of H consisting of all vectors satisfying the linear equation Aψ = αψ. Denote this subspace Hα . The operator A can be represented as the weighted some of its eigen-projections (the spectral decomposition): A=



α E(α).

(2.2)

α

If, for some eigenvalue α, the subspace Hα is one dimensional, then it is generated by one eigenvector. Such eigenvalue is called nondegenerate. If all eigenvalues of A are nondegenerate, then A has nondegenerate spectrum. In Dirac’s notation (Sect. 16.10), the eigenstate for nondegenerate eigenvalue is denoted by the symbol |α, i.e., A|α = α|α. To represent quantum states, eigenvectors are normalized, i.e., α|α = 1. Often Dirac notations are extended to all vectors, i.e., ψ ∈ H, is denoted by the symbol |ψ. Projection E on the vector |ψ is denoted as |ψψ|. Such notations generate a kind of algebra, e.g., E acts on the vector |φ as the formal multiplication E|φ = |ψψ||φ = ψ|φ|ψ.

(2.3)

In Dirac’s notation the spectral decomposition of a Hermitian operator with nondegenerate spectrum can be written as A=



α|αα|.

(2.4)

α

We don’t not use Dirac’s notation consistently, but operate with both symbols ψ and |ψ. For the state ψ, the probability of the outcome α is given by the Born rule: P(a = α|ψ) = E(α)ψ|ψ = E(α)ψ2 . 1

Since H is finite dimensional spectrum is reduced to the set of operator’s eigenvalues.

(2.5)

2.1 Elementary Quantum Vocabulary

13

If eigenvalue α is nondegenerate, then since E(α)|ψ = α|ψ|α, the Born rule can be written as (2.6) P(a = α|ψ) = |α|ψ|2 , cf. (1.1). For an observable a represented by Hermitian operator A, a measurement with the outcome a = α generates back-action onto system’s state: ψ → ψα = E(α)ψ/E A (α)ψ.

(2.7)

This is the projection postulate in the Lüders form [341] (see Sect. 16.3). If the eigenvalue α is nondegenerate, then ψ is projected on the eigenvector |α (with the corresponding normalization): ψ → ψα = |α.

(2.8)

Representation of Mixed States by Density Operators Án operator B is positively defined, B ≥ 0, if the corresponding quadratic form is non-negative: for any vector ψ ∈ H, Aψ|ψ ≥ 0. Generally quantum theory works with the quantum states given by density operators. Typically, such quantum state is interpreted as describing a statistical mixture of pure states.2 A density operator ρ is determined by the conditions: • ρ = ρ  (so, it is a Hermitian operator) • ρ ≥ 0, • Trρ = 1. The space of density operators is denoted by the symbol D ≡ D(H). We remark that each pure state can be represented by a density operator, projection on the state vector, see (2.3). If a quantum state is mathematically described as a density operator ρ, then the Born’s rule has the form: P(a = α|ρ) = TrρE(α). (2.9) (see Chap. 16 for detail). The state update is given by the formula ρ → ρα = E(α)ρ E(α)/TrE(α)ρE(α).

(2.10)

In Sect. 2.3 we will discuss the comparative interpretation of the states represented by normalized vectors and density operators. Compatible Versus Incompatible Observables In contrast to classical physics, in quantum physics some observables can’t be jointly measured. They are called incompatible. In the mathematical formalism incompatible 2

Although this interpretation is widely used in physics and it will be also often used in this book, it is ambiguous, since the same density operator ρ can represent different statistical mixtures.

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2 Quantum Formalism for Decision-Making

observables a and b are represented by noncommutative operators (matrices) A and B, i.e., [A, B] = 0. Compatible observables a and b can be jointly measurable and, hence, their joint probability distribution (JPD) is well defined. They are represented by commuting Hermitian operators A and B. Commutativity of operators implies commutativity of their projections onto eigensubspaces, [E a (α), E b (β)] = 0, for all eigenvalues α and β. In quantum formalism JPD for compatible observables (for joint measurements on systems in state ψ) is defined by the equality: P(a = α, b = β|ψ) = E b (β)E a (α)ψ2 = E a (α)E b (β)ψ2 ,

(2.11)

How can JPD be defined in the case of incompatible observables? They are not jointly measurable and formally JPD, as the probability of the pairwise outcomes (a = α, b = β) is not defined. However, conditional probability is well defined even for incompatible quantum observables (Sect. 16.5). Hence, one can use the quantum analog of the CP-formula (1.6) connecting JPD and conditional probability. Of course, for incompatible observables this is not JPD in the usual sense, but so to say sequential JPD: P(a = α, b = β|ψ) ≡ P(a = α; ψ)P(b = β|a = α; ψ) == E b (β)E a (α)ψ2 . (2.12) In CP the same JPD can be defined either via formula (1.6) or formula (1.7). The QP analogs of these formulas lead to the different answers that is not surprising: formula (2.12) determines the sequential a → b JPD, but the the sequential b → a JPD is given by P(b = β, a = α|ψ) ≡ P(b = β; ψ)P(a = α|b = β; ψ) == E a (α)E b (β)ψ2 . (2.13) and if [A, B] = 0, then the right hand sides of (2.12) and (2.13) do not need coincide. Projection Valued Measures - PVMs In physics, the values of observables are real numbers, the eigenvalues of Hermitian operators. We point out that the right hand side of the Born rule contains only orthogonal projectors and association of them with real numbers is not crucial. Instead of eigenvalues, we can consider just some index labeling orthogonal projections. This label can belong to any set X and then the probability distribution given by the Born rule is defined on the set X. For example, X can be the set of all possible words in Russian or Chinese language, set of possible conscious experiences, decisions, questions, emotions. This possibility to operate with an arbitrary set of symbols X is

2.2 Quantum-Like Model for Decision-Making

15

very important for applications outside of physics. For simplicity, we consider finite set of outcomes (2.14) X = {x1 , . . . , xm }. We recall that the spectral family of orthogonal projections for a Hermitian operator A satisfies the normalization condition  E(x) = I. (2.15) x

where I is the unit operator, and the mutual orthogonality condition E(x) ⊥ E(y), x = y.

(2.16)

Now, let us represent an observable a with the range of values X by the family of orthogonal projectors, E = (E(x))x∈X satisfying two constraints (2.15), (2.16). They guarantee that, for any state ψ, the quantity determined by the Born rule can be interpreted as probability, i.e., it is normalized by one 

P(a = x|ψ) = 1.

(2.17)

x∈X

Such family of projectors (E(x))x∈X determines a projector-valued measure (PVM), μ(G) =



E(x),

(2.18)

x∈G

where G is a subset of X. Sometimes (especially in Chap. 6), we use PVMs for mathematical representation of observables. This is very natural generalization of the Hermitian operator representation. Later (Chap. 8) we invent a larger class of observables given by positive operator valued measures (POVMs). Observables given by PVM will be called von Neumann observables. (We remark that typically von Neumann observables are identified with Hermitian operators.)

2.2 Quantum-Like Model for Decision-Making The belief (mental) state of a decision maker, say Alice, is represented as a quantum state, typically by a pure state ψ belonging to H (space of belief states). Generally belief states are represented by density operators.3 3

The idea to represent mental states as quantum states was elaborated by a few authors, see, e.g., my works [230, 231, 241–243], see also [87, 89, 90, 386–388].

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2 Quantum Formalism for Decision-Making

The problem of the belief state’s interpretation is the cognitive counterpart of the problem of the quantum state interpretation - the “wave function interpretation problem”. The latter is one the main foundational problems of QM and it is characterized by the diversity of the viewpoints (Chap. 20). A decision problem is a question a which is asked to Alice or a task which she must perform (to select some output). In the quantum-like framework a is mathematically described by a quantum observable. General mathematical formalization of quantum observables is given within the theory of quantum instruments (Chap. 8). For the beginning, we start with the simplest quantum observables given by Hermitian operators. So, say a question (or task) a is represented by a Hermitian operator A, its eigenvalues encode possible answers to this question, (task). For Alice in the belief state ψ, the probability P(a = α j |ψ) of decision α is given by the Born rule (2.5); if the belief state is given by density operator ρ, then probability is calculated with formula (2.9). In decision modeling, it is important to determine not only the probabilities of various decisions, but also the transformation of belief states resulting from decisionmaking. In the simplest model, the decision with the outcome a = α induces projection of the belief state ψ onto the α-eigenspace Hα of the Hermitian operator A, see (2.7); if state is mixed then the state update is given by (2.10). We remark that in the case of nondegenerate eigenvalue α the formulas for calculation of probability and the state update are simplified (at least for the pure initial state ψ), see (2.6), (2.8). And a newcomer in the field of quantum-like modeling might be attracted by this simplicity and the illustrative power of drawing vectors and their projections. However, by some reason mental observables can’t be represented by Hermitian operators with nondegenerate spectra (Sect. 16.5). In fact, the situation is even more complex. In contrast to physics, where one can proceed rather far with Hermitian operators and the state update of the projection type, in psychology and cognition even the simplest psychological effects can’t be modeled in this way, more general belief state updates have to be used, updates described by quantum instruments (Chap. 8).

2.3 Decision-Making via Decoherence In Sect. 4.5, we consider another quantum-like framework which is known as quantum dynamical decision-making or decoherence decision-making. Here the decision states are steady states of the open quantum system dynamics describing the belief (mental) state’s evolution in the process of decision-making. Such modeling of decision state generation found many applications, from genetics and molecular biology to psychology and cognition, ecology and sociology [18–21, 24, 25].4 4

In Chap. 4 the decoherence framework is used to model approaching of stability in biosystems, from proteins and cells to ecological systems.

2.3 Decision-Making via Decoherence

17

The belief state is mathematically described by a density operator. Its evolution t → ρ(t) in the process of decision-making is modeled by the open quantum system dynamics (Sect. 4.4). Its Markovean (i.e., memoryless) approximation is given by Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation; in this book we mainly operate with this approximate dynamical equation, the simplest quantum master equation. One of the distinguishing features of the open quantum system dynamics is that it immediately transfers a pure state given by a vector |ψ into a mixture of pure states described by a density operator ρ. In the quantum-like modeling of decisionmaking, this mathematical property is interpreted as destruction of the purity of Alice’s belief state resulting from interaction with surrounding psycho-physical environment. Thus, if Alice initially was in the state ρt0 = |ψψ|, then typically at any t > t0 her state is a statistical mixture of pure states which is mathematically represented by a density operator ρ(t). This transformation of a pure state into statistical mixture is known as decoherence. Decoherence is a complex foundational notion of quantum mechanics [442]. We proceed with the operational definition, “pure state → statistical mixture”. A pure state |ψ can be represented as superposition of eigenstates of the Hermitian operator A representing the question a asked to Alice: |ψ =



cα |α,

(2.19)

α

  where α p(a = α|ψ) = α |cα |2 = 1. The open quantum system dynamics destroys this superposition; so decoherence can be treated as destruction of superpositions. In probabilistic terms, superposition corresponds to interference of probabilities and “quantumness of probability” (Sect. 16.6). Hence decoherence destroys interference and transfer QPs into CPs; so to say quantumness is destroyed by decoherence. Quantitatively the degree of quantumness can be expressed through state’s purity (Sect. 4.8). Here we are not able to go deeper into the discussion on the decoherence interpretation. Thus, before starting the process of decision-making, e.g., preparing an answer to the question a, Alice forms superposition (2.19) of belief states (|α) corresponding possible answers. This is the state of deep quantum uncertainty. For example, Alice can assign the equal weights to all possible answers, p(a = α|ψ) = 1/N , where N is the number of possible answers. However, even such superposition is probabilistically different from the classical uniform probability distribution. This difference is not visible in operating with just a single question a. But consideration of another question b would demonstrate this difference via the interference of probabilities, given by FTP with the interference term (1.11). To see interference, question b should be incompatible with question a, i.e., their operators do not commute, [A, B] = 0. Quantumness of superposition is encoded in phases of the complex coefficients cα = α|ψ = |cα |eiθα .

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Even for equal amplitudes |cα |, the pure states with different phases react to measurements in different way. And these states differ crucially from the state given by the density operator  |cα |2 |αα|. ρ= α

So, initially Alice is in the state of deep uncertainty (2.19); her aim is to resolve this uncertainty. We describe her process of decision-making by the open quantum system dynamics. It transfers superposition into the classical statistical mixture ρ(t). In our model of decision-making via decoherence, a decision state is given by a steady state of this dynamics. (2.20) ρdecision = lim ρ(t) t→∞

(if this limit exists). Of course, this limit procedure is just the mathematical idealization, it encodes damping of state’s fluctuations, and damping is due to the interaction with system’s environment. Hence, ρdecision ≈ ρ(τ ),

(2.21)

for sufficiently large τ (w.r.t. decision maker’s temporal scale). How does Alice extract the concrete answer from the classical probability distribution given by the density operator (2.20)? One of the solutions is that at the final stage of the decision process, i.e., after approaching the steady state, Alice uses the classical random generator to select the answer a = α; probabilities are encoded in ρdecision . We discuss this complex problem in more detail in Sect. 6.5.

2.4 Decision Paradoxes The use of the quantum-like models can resolve some paradoxes of classical theory of decision-making. Such paradoxes started to be suggested immediately after publication of the book of von Neumann and Morgenstern [450] on expected utility theory, e.g., the Allais [7] (1953), Ellsberg [144] (1961), and Machina [342] (1987) paradoxes. See Chap. 19 for brief presentation of the Allais and Ellsberg paradoxes. The first one is the paradox of expected utility theory which basics were set von Neumann and Morgenstern [450] and the second one is the paradox of subjective expected utility theory which basics were set by Savage [400]. Typically a paradox (as a sign of irrational behavior) is probabilistically expressed as the violation of the law of additivity of probability (or FTP), i.e., that P(A1 ∪ A2 ) = P(A1 ) + P(A2 ), for A1 ∩ A2 = ∅.

2.4 Decision Paradoxes

19

In turn the latter can be explained by contextuality of probability: one cannot proceed with a single probability measure P, decision contexts are portrayed with context-dependent probability spaces and random variables representing decisionobservables (e.g., questions). The paradoxes of decision theories can be mathematically described by the Växjö model for contextual probability theory [251] (Sect. 17.2) or by Contextuality by Default theory (see, e.g., [131] and Sect. 17.8). The quantum-like approach provides the general and consistent mathematical representation of contextual probabilistic theories (see, e.g., [282] for subjective expected utility). The number of paradoxes generated by the classical decision-making theory is really amazing. The authors of the review [145] counted 35 basic paradoxes. During many years DM-theory was developed through creation of paradoxes and resolving them through modifications of the theory, e.g., from expected utility theory to the prospect theory. But any modified theory suffered of new paradoxes. The use of QP can resolve all such paradoxes, at least this is claimed in papers [27, 282].

Chapter 3

Classical Versus Quantum Rationality

3.1 Savage Sure Thing Principle and Interference of Probabilities In classical theory of decision-making, the rational behavior of agents is formalized with the Savage Sure Thing Principle (STP) [400]: If you prefer prospect b+ to prospect b− if a possible future event A happens (a = +1); and you prefer prospect b+ still if future event A does not happen (a = −1); then you should prefer prospect b+ , despite having no knowledge of whether or not event A will happen. Savage’s illustration refers to a person deciding whether or not to buy a certain property shortly before a presidential election, the outcome of which could radically affect the property market [400]: “A businessman contemplates buying a certain piece of property. He considers the outcome of the next presidential election relevant. So, to clarify the matter to himself, he asks whether he would buy if he knew that the Democratic candidate was going to win, and decides that he would. Similarly, he considers whether he would buy if he knew that the Republican candidate was going to win, and again finds that he would. Seeing that he would buy in either event, he decides that he should buy, even though he does not know which event obtains, or will obtain, as we would ordinarily say. It is all too seldom that a decision can be arrived at on the basis of this principle, but except possibly for the assumption of simple ordering, I know of no other extralogical principle governing decisions that find such ready acceptance.” STP is considered as the axiom of rationality of decision makers [400]. It plays the important role in decision-making and economics in the framework of Savage’s subjective utility theory. In the latter, the probability is formalized in the CP-framework [318] and it is endowed with the subjective interpretation. Generally, subjective expected utility theory was challenged by the Ellsberg paradox (see Chap. 19). The quantum-like models of subjective expected utility resolving this paradox were developed in articles [27, 282]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_3

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One of the first violations of the Savage Sure Thing Principle was found by Tversky and Shafir [401, 434]. This study was motivated by the behavior of their own students in San Francisco who had the custom to take their after-exam vacation in Hawaii. Tversky and Shafir showed that significantly more students report that they would purchase a nonrefundable Hawaii vacation if they knew that they had passed or failed an important exam as opposed to the case when they did not know the outcome of the exam. (See articles [248, 282] for quantum-like modeling.) We stress that STP is a simple consequence of FTP P(b = β) =



P(a = α)P(b = β|a = α).

(3.1)

α

Here the b-variable determine the decision variable, say b = +1, “buy house”, b = −1, “not buy house”, and the a-variable represents conditions, say outcomes of elections. Violation of FTP implies violation of STP. Thus, the degree of satisfaction of FTP can be used as a statistical test of classical (STP-type) rationality. In cognitive psychology, violation of STP is known as the disjunction effect. Plenty of statistical data was collected in cognitive psychology in experiments demonstrating disjunction effect. For example, in experiments of the Prisoners’ Dilemma type [221– 225]. Such data violate FTP. The latter implies irrationality (from classical viewpoint) of agents participating in experiments (mainly students). We recall that FTP is derived from two assumptions that are firmly incorporated into the Kolmogorov axiomatics: 1. Additive law for probability. 2. Baeys formula for conditional probability. Therefore, violation of FTP and, hence, of STP (and classical rationality) is generated either by violation of additivity of probability or the Bayes law for conditional probability or by the combination of these factors. Generally, this leads to the impossibility to use in decision-making Bayesian inference. Quantum(-like) agents proceed with more general inference machinery based on the quantum state update. Hence, classical rationality is Bayesian inference rationality and quantum rationality is non-Bayesian inference rationality.1

3.2 Quantum-Like Rationality In the light of above considerations, one can ask Are quantum agents irrational?

1

Of course, non-Bayesian probability updates are not reduced to quantum ones, given by state transformations in the complex Hilbert space. One may expect that human decision-making violates not only classical, but even quantum rationality.

3.2 Quantum-Like Rationality

23

As was discussed, by using QP, it is possible to violate FTP and hence STP. Therefore, generally quantum-like agents are (classically) irrational. However, we can question the CP approach to the mathematical formalization of decision-making and, consequently, the corresponding notion of rationality. We define quantum(like) rationality as respecting the quantum calculus of probabilities and the quantum formula for interference of probabilities, FTP with the interference term. In the framework of quantum-like modeling, violation of the CP laws including the Bayes formula for conditional probability or even additivity of probability are not exotic at all. Moreover, the situation in which the probabilistic data satisfies FTP seems to be rather an exception than the norm. We can speculate that QP processing of information was resulted from evolution of biological systems, not only humans, but even animals and simple bio-organisms. The question whether the “genuine human behavior” should be characterized by classical rationality taken as the normative theory for the rational decision-making is very complicated. Violation of FTP implies violation of STP. Thus, satisfaction of FTP can be used as a statistical test of rationality. Plenty of statistical data was collected in cognitive psychology in experiments demonstrating disjunction effect (works of Shafir, Tversky, Kahneman [401, 434], [221–225]). Such data violate FTP. The latter implies irrationality of agents participating in experiments (mainly students). For comparison of classical and quantumlike rationalities, see also [386, 428]. Generally, the violation of classical rationality is the violation of Bayesian inference employed for the probability update. The quantum probability update based on the state update as observation’s feedback is more general and it can lead to the probability transformations that are impossible within CP and its Bayesian inference counterpart. This question was highlighted in the article [261] on violation of Aumann’s theorem by quantum agents. Logic of Mind CP is based on classical Boolean logic and QP is based on quantum non-Boolean logic (Chap. 16). The latter leads to the relaxation of the basic rules of classical logic, in particular, the distributivity law for conjunction and disjunction. Roughly speaking, information processing based on the laws of quantum logic is simpler and needs less computational resources: one can operate in partial Boolean subalgebra of quantum logic, and one does not need to construct conjunctions, disjunctions, and negations for all possible events. Such information processing is superior in the situation of information overload. It saves a lot of information resources. However, this kind of information processing can lead to inconsistencies between conclusions which can be derived on the basis of different subalgebras. These inconsistencies are behavioral irrationalities (from the viewpoint of Boolean logic) and probability fallacies—the signs that people can’t adequately operate with probabilities. We claim that people operate very well with probabilities, but they use non-classical probability calculus, e.g., QP. In fact, QP is the calculus of probability amplitudes or complex vector state calculus. Thus, by making decisions in the state of uncertainty, humans operate with such amplitudes (states) by using the linear

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3 Classical Versus Quantum Rationality

algebra calculus in complex linear space. Finally, to assign probabilities for possible outcomes of decision, they transfer amplitudes into probabilities by using the Born rule. Finally, we point to article [371] on the quantum logic of human mind. It contains new and interesting approaches to analyze the human mind logical structure, especially nondistributivity role.

3.3 Coupling of Social Lasing to Deprivation of Classical Rationality One of the consequences of information overload is that information loses its content. A human has no possibility to analyze deeply the content of communications delivered by mass media and social networks. People process information without even attempting to construct an extended Boolean algebra of events. They operate with labels such as Covid-19, vaccination, pandemic, etc., without trying to go deeper beyond these labels. Contentless information behaves as a bosonic quantum field which is similar to the quantum electromagnetic field. Interaction of humans with such quantum information fields can generate a variety of quantum-like behavioral effects. One of them is social lasing, stimulated amplification of social actions (SASA) [12, 269, 276, 277, 287, 289, 290] (Chap. 11). In social laser theory, humans play the role of atoms, social atoms (s-atoms). Interaction of the information field composed of indistinguishable—up to some parameters, as say social energy—excitations with a gain medium composed of s-atoms generate the cascade type process of emission of social actions. SASA describes well, e.g., color revolutions and other types of mass protests. Over the past years, our society has been constantly shaken by high-amplitude information waves. These are waves of enormous social energy. They are often destructive and are a kind of information tsunami. The main distinguishing features of these waves are their high amplitude, coherence—the homogeneous nature of the social actions they generate—and the short time required for their generation and relaxation. These are huge singular spikes. As shown in the mentioned works, such waves can be modeled by using the social laser, which describes SASA. “Actions” are interpreted very broadly, from mass protests, in particular, leading to color revolutions such as the Orange or Maidan revolutions in Ukraine or the recent mass protests in USA, for example, anti-Baiden protests for fair votes, Belarus -anti-Lukashenko protests for fair votes, anti-Putin protests for the liberation of Naval’nii in Russia, Germany, UK, Australia, Canada, and Sweden, for instance, protests against corona-fascism and violation of the basic human rights with pandemic-justification as well as generating of “right voting” and other collective decisions such as acceptance of lockdown and support of the total vaccination against Covid-19 by the majority of population.

3.4 Question Order and Response Replicability Effects

25

3.4 Question Order and Response Replicability Effects This section serves as a brief introduction to Chaps. 9, 10, and 14. The question order effect (QOE) is dependence of the (sequential) joint probability distribution on questions’ order. In cognitive and social science, the following opinion pool is known as the basic example of the order effect. This is the Clinton–Gore opinion pool. In this experiment, American citizens were asked one question at a time, e.g. • a = “Is Bill Clinton honest and trustworthy?” • b = “Is Al Gore honest and trustworthy?” Two sequential probability distributions were calculated on the basis of the experimental statistical data, Pab and Pba —first question a and then question b and vice verse). And it was found that Pab = Pba This is surprising from the CP viewpoint, where aandb are represented as random variables taking values ±1. In CP, for any pair of values α, β = ±1, Pab (α, β) = P((a = α) ∧ (b = β)) = P((b = β) ∧ (a = α)) = Pba (β, α). Here the key-point is commutativity of the operation of Boolean conjunction which is CP realized as the set-intersection operation. We stress that QOE was one of the basic motivations for applying QM-formalism to psychological phenomena, see article of Wang and Busemeyer [452]. Wang and Busemeyer [452] reasoned that QOE can be modeled by appealing to incompatible observables which are mathematically represented by noncommutative Hermitian operators A, B. One observable is the question about Clinton and another about Gore. From this viewpoint. QOE is a consequence of the existence of incompatible observables. So, in this framework, Pab = Pba is equivalent to noncommutativity of Hermitian operators A and B representing the questions a, b, i.e., [A, B] = 0. However, the life is not so simple. For incompatible observables, one operates with sequential JPD given by (2.12). So, Pab (α, β|ψ) ≡ P(a = α; ψ)P(b = β|a = α; ψ) == E b (β)E a (α)ψ2 , where E a (α) and E b (β) are projection of the Hermitian operators A and B corresponding to eigenvalues α and β. In the same way Pba (β, α|ψ) ≡ P(b = β|ψ)P(a = α|b = β; ψ) = E a (α)E b (β)ψ2 .

26

3 Classical Versus Quantum Rationality

If [A, B] = 0, then, for some α, β, [E a (α), E b (β)] = 0, and sequential JPDs do not need to coincide, i.e., generally E b (β)E a (α)ψ2 = E a (α)E b (β)ψ2 .

(3.2)

A strong objection to the Wang-Busemeyer strategy w.r.t. QOE, i.e., using Hermitian operators to represent mental observables, e.g., questions asked to people, was presented in paper [262] (see Chap. 14). This objection is based on the response replicability effect (RRE) which also plays the important role in QM. Suppose that Alice is asked some question a and she replies, e.g., “yes”. If immediately after answering she is asked this question again, then she replies “yes” with probability one. We call this property a − a response replicability. All quantum measurements represented by Hermitian operators and the resulting state update given by the projection postulate satisfy a − a response replicability, because any projection is idempotent, E 2 = E. The Clinton–Gore opinion poll as well as any decision-making experiment satisfies a − a response replicability. If Alice has answered “yes” to question a, then she will repeat this answer (with probability 1), if this question will be repeated. Human decision-making has another distinguishing feature which can be called a − b − a response replicability. Suppose that after answering the a-question with say the “yes” answer, Alice is asked another question b. She replied to it with some answer. And then she is asked the question a again. In the social opinion pools and other natural decision-making experiments, Alice definitely repeats her original answer to a, “yes”. This is a − b − a response replicability. Combination of a − a with a − b − a and b − a − b response replicability is called the response replicability effect. Is it possible to combine the order and response replicability effects by using the quantum formalism? As was pointed out in paper [262] (see Chap. 14), by using the calculus of Hermitian observables and the projection postulate it is impossible to combine RRE with QOE. In short, to generate QOE Hermitian operators A, B should be noncommutative, but the latter destroys a − b − a response replicability of A. This was the very strong objection to the use of the quantum formalism in psychology. The impossibility of QP-modeling RRE+QOE was considered as the first paradox of QP decision theory. After publication of article [262], even the main actors of the quantum-like modeling, including the book author, were in doubts whether the quantum formalism is adequate to human behavior. However, recently the problem of combination of RRE and QOE has been solved withing the quantum formalism, but by using a more advanced mathematical formalism—quantum instrument theory describing generalized quantum observables. Psychological data can be described by quantum theory, but by its modern and advanced formalism, the instrument theory [369, 370] (Chaps. 8, 9, and 10). We remark that representation of quantum observables by Hermitian operators and realization of the state update with orthogonal projectors is too restrictive even in quantum physics, especially in quantum information theory. Generally, quantum

3.5 State-Dependent Incompatibility

27

observables are represented by positive operator valued measures (POVMs) and the process of measurement by quantum instruments. However, the basic quantum observables, position, momentum, spin, energy, can be mathematically described by Hermitian operators and the state update by the projection postulate. For cognitive systems, the situation is worse, even such simple cognitive effects as QOE and RRE can’t be modeled without appealing to generalized observables. Theory of quantum instruments is mathematically quite complicated and we postpone its presentation and application to modeling combinations of psychological effects to Chaps. 8, 9, and 10. The readers who are interested merely in psychology and cognition can jump directly to these chapters. In fact, I am surprized that even the basic mental observables, in contrast to the physical observables such as say position, momentum, and energy, can’t be mathematically described by Hermitian operators and from the very beginning one has to use the generalized quantum observables.

3.5 State-Dependent Incompatibility In connection with QOE, it is natural to point to the notion of state-dependent incompatibility. This notion is not so widely used in quantum physics, because there the basic incompatible physical observables are represented by noncommutative operators (cf., however, Ozawa [364–367] and especially his article [368]). However, it seems that in applications to cognition and decision-making belief-state dependence plays the important role. Let us look at inequality (3.2). To be sure in the presence of QOE, it is sufficient to find such a belief-state ψ that this inequality holds for this concrete ψ. The necessary condition for this is the ψ-state noncommutativity, i.e., that (3.3) [E a (α), E b (β)]ψ = 0. This notion has not been yet widely used in quantum-like modeling (see only article [370]). We also remark that noncommutativity of two operators, [A, B] = 0, does not imply state noncommutativity for an arbitrary state ψ. For some pairs of noncommutative operators, there can exist the states such that [A, B]ψ = 0. For such states, QOE is impossible, in spite of noncommutativity of the operators representing observables. Thus, it is more useful to operate with state-based compatibility and incompatibility of observables.

Part II

Biosystems as Open Quantum-Like Systems

Any alive biosystem is an open system and to analyze its behavior is reasonable to take advantage of the open quantum systems theory, whether the biosystems are acknowledged as information processors and the open quantum systems theory is treated as a part of the quantum information theory. The latter is the most general information theory comprising the classical information theory as a particular case, thus this part of the book concerns information processing in complex biosystems. From the information viewpoint, even a cell or a protein are very complex systems. We emphasize again that this book is not on the genuine quantum physical processes in biosystems and we apply the quantum-like theory of open systems. In particular, its application is not constrained by the system’s size, but is applicable on all scales of space, time, and complexity, from genomes, proteins, and cells, to animals, humans, and ecological and social systems. Treating of biosystems as quantum-like information processors can expound order stability in them, i.e., present the quantum-like formalization of Schrödinger’s speculations in his notorious book “What is life?”. The process of biosystem’s adaptation to the surrounding environment is described by the Gorini-Kossakowski-SudarshanLindblad Equation, where the Von Neumann and linear quantum entropies are employed as measures of the disorder degree. We highlight the role of quantum dynamics class in generating the camel-like shape for quantum entropies. Camel’s hump represents: (a) the entropy increase in the process of the initial adaptation to the environment; (b) the entropy decrease at the post-adaptation stage of the dynamics. Our analysis is based on numerical simulation, and to describe such a class of quantum dynamics analytically is a must-have. The quantum information treatment of order stability in compound biosystems leads to an interesting result. Global stability can be preserved in spite of an increase of disorder in subsystems. Such behavior is impossible in the classical information framework. As an application of the open quantum systems theory to cognition, we suggest a quantum-like model of the brain’s functioning. In this model, the general approach of this book—to start directly with the quantum information representation of byosystems’ states—is broken. We start with the consideration of electrochemical states of

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Part II: Biosystems as Open Quantum-Like Systems

neurons encoded in action potentials. Such states generate the brain’s mental states which are processed with the open quantum systems dynamics. The complex problem of mathematical formalization of consciousness and unconsciousness and their interaction can also be handled within the open quantum systems theory. Consciousness plays the role of an apparatus performing measurements over unconsciousness. This formalism matches well with the Higher Order Theory of Consciousness. It is applied to model mathematically the emotional coloring of conscious experiences. Such coloring is framed as contextualization. So, the theory of emotions is coupled to such a hot topic in quantum foundations as contextuality. Finally, we discuss the Bell type experiments for emotional coloring.

Chapter 4

What is Life? Open Quantum Systems Approach

4.1 Schrödinger About Life Schrödinger pointed out to order stability as one of the main distinguishing features of biosystems. Entropy is the basic quantitative measure of order. In physical systems, entropy has the tendency to increase (Second Law of Thermodynamics for isolated classical systems and dissipation in open classical and quantum systems). Schrödinger emphasized the ability of biosystems to beat this tendency. As was shown in article [19], a biosystem S which processes information in the quantumlike way can preserve or even improve its order structure in the process of information exchange with environment E. This order stability is of the dynamical nature, i.e., not simply constancy of entropy, but its stabilization generally to the level not exceeding entropy of the initial state. We emphasize the role of the special class of quantum dynamics and initial states generating the camel-like shape for entropy-evolution in the process of interaction with a new environment1 E: • (1) entropy (disorder) increasing in the process of adaptation to the specific features of surrounding environment E; • (2) entropy decreasing (order increasing) resulting from adaptation; • (3) the restoration of order or even its increase for limiting steady state. In the latter case the entropy of the steady state can become even lower than the entropy of the initial state. The results of this chapter on the camel-like behavior of entropy are based merely on numerical simulation and creation of the corresponding theory is an interesting and important problem. We recall that Schrödinger’s book [406] (1944) had the big influence on the future interplay between physics and biology and development of biophysics. The book’s title poses one of the most fundamental questions of the modern science “What 1

Here we follow paper [295].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_4

31

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4 What is Life? Open Quantum Systems Approach

is life?”. We are still far from obtaining an answer to it, may be no closer than Schrödinger was in 1944. The main distinguishing feature of biosystems (from a physicist’s viewpoint) was presented by Schrödinger [406] as follows: “What is the characteristic feature of life? When is a piece of matter said to be alive? When it goes on ‘doing something’, moving, exchanging material with its environment, and so forth, and that for a much longer period than we would expect of an inanimate piece of matter to ‘keep going’ under similar circumstances.” In his book [406], Schrödinger compared order stability in biosystems with the state evolution for non-living system: “When a system that is not alive is isolated or placed in a uniform environment, all motion usually comes to a standstill very soon as a result of various kinds of friction; differences of electric or chemical potential are equalized, substances which tend to form a chemical compound do so, temperature becomes uniform by heat conduction. After that the whole system fades away into a dead, inert lump of matter. A permanent state is reached, in which no observable events occur. The physicist calls this the state of thermodynamical equilibrium, or of maximum entropy’. Practically, a state of this kind is usually reached very rapidly. Theoretically, it is very often not yet an absolute equilibrium, not yet the true maximum of entropy. But then the final approach to‘ equilibrium is very slow.” See also widely cited paper of Gatenby and Frieden [172]: “Living systems are distinguished in nature by their ability to maintain stable, ordered states far from equilibrium. This is despite constant buffeting by thermodynamic forces that, if unopposed, will inevitably increase disorder.” Schrödinger pointed out that biosystems do not follow the laws of classical physics. Their behavior looks strange especially from the thermodynamical viewpoint. They are able (in an unclear way) to violate the Second Law of Thermodynamics. In principle, this is not surprising, since biosystems are fundamentally open systems. Schrödinger emphasized that a completely isolated biosystem is simply dead. And he pointed out that a biosystem is able to escape following the Second Law of Thermodynamics only via exchange with its environment. But, then he pointed to the main question related to this process: What kind of exchange characterizes stability of a living system? Matter? Energy? Of course, biosystems are continuously performing such exchanges and, in this way, they preserve their material and energy order. But, Schrödinger remarks that the same is done by non-alive systems. So, neither matter nor energy exchange can lead to bio-violation of Second Law of Thermodynamics and guarantee stability of biosystems. Schrödinger tried to resolve the mystery of life by using the approach that nowadays is known as quantum biophysics (so, he was one of its fathers). He compared quantum physical processes with biological ones; in particular, he compared the role of a physical molecular with gene, as carriers of information. However, he did not succeed in resolution of life’s mystery. Then, in the last part of the book he considered

4.1 Schrödinger About Life

33

the phenomenon of life from the viewpoint of entropy exchange between a biosystem S and environment E. This was one of the first steps toward information treatment of biological processes. He noted that formally stability and order’s preservation within a biosystem S can be modeled by means of negative entropy. So, S consumes negative entropy from the environment E and, in this way, it compensates its own entropy increase, and consequently preserve order inside it. But, appealing to such a notion as negative entropy is really speculative, albeit philosophically attractive. Schrödinger’s speculations on peculiarities of entropy transfer as the basis for preservation of alive-states can be interpreted as the first step toward study of the phenomenon of life from the positions of information theory. We stress that by explaining the basis of order preserving in living systems, Schrödinger did not have in mind explanation of homeostasis: the state of steady internal, physical, and chemical conditions maintained by living systems. (Homeostasis is mathematically described by the feedback loop models with balance equations. Nowadays, with sensors and computing blocks, it is easy to construct a physical system in the state of energy, material, and chemical homeostasis.) In [406], biostability was stability of interconnected regulation mechanisms including processing of mental information by the brain. Finally, he concluded that it would be impossible to explain bio-behavior within the known laws of physics, neither classical nor quantum. May be new physical laws will be formulated to explain the phenomenon of life. Although quantum information theory is the result of the natural evolution of quantum theory, its creation is often called the second quantum revolution - to highlight the novelty of the applications of the quantum concepts, methodology, and mathematical formalism, to foundations, theory, experiment, and engineering. The main message of Schrödinger’s book [406] was that biosystems are subject not only to material or energy constraints imposed by the physical environment, but also to the information constraints imposed by the information environment. Biosystems are considered as open systems interacting with their physico-informational environments. Nowadays, this viewpoint is well accommodated in biology, see, e.g., Gatenby and Frieden [172]: “In a 1970 review [217], Johnson characterized information theory (IT) as a “general calculus for biology. It is clear that life without matter and energy is impossible. Johnson’s manuscript emphasized that life without information is likewise impossible. Since the article, remarkable progress has been made toward understanding the informational basis for life...” However, after a few years of successful applications of classical information theory to biology, the bio-community started to recognize its limitations [172]: “it seems clear that, in the 35 years since Johnson’s original article, IT using traditional Shannon methods has not become, as predicted, the “general calculus” of biology. Although the lack of wide spread application is probably the result of multiple limitations...” In [172], Gatenby and Frieden stressed that in further applications of information theory in biology, “the focus will be on new methods... .” In particular, researchers “address limitations of the IT methodology by applying new

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statistical and modeling approaches to information dynamics, including bio-informatics, dynamical systems, game theory, graph theory, and measurement theory.” We remark that the process of IT-reconsideration of biology was closely connected with the similar process in physics. Modern physics differs essentially from physics of the middle of the last century. Highlighting the role of information is one of the main outputs of development of physical theory, starting with Wheeler’s “It from bit” [463] to the recent quantum information revolution (known as the second quantum revolution in physics).

4.2 Theory of Open Quantum Systems and Order in Biosystems We present a quantum information reformulation of Schrödinger’s discussion on biostability as based on entropy exchange with biosystem’s environment. Our suggestion is to treat entropy on the basis of quantum information theory (as the von Neumann entropy [451] or linear entropy). By using this theory, we follow the aforementioned recommendation of Gatenby and Frieden to focus on new methods in the information approach in biology. The quantum entropy represents uncertainty in distribution of quantum information states. And the open quantum systems theory provides such new methods. For open quantum systems, the problem of escaping transition to disorder can be formalized with quantum Markov dynamics. Within this theory, we demonstrated (with the concrete quantum master equations) that a biosystem S with quantumlike information processing can preserve or even improve its order structure in the process of information exchange with its environment E. This order stability is of the dynamical nature, i.e., not simply constancy of entropy, but its stabilization to the level not exceeding the entropy of the initial state. In introduction to this chapter we highlighted the role of the camel-like behavior of the quantum entropy (Sect. 4.6) in the process of interaction of the system with the surrounding environment and proposed a bio-information interpretation of such behavior, as three stage dynamics. In the framework of quantum Markov dynamics, we present an illustrative example of such behavior, see (4.21), (4.22) and Figs. 4.1 and 4.3; in Sect. 4.7 we describe the class of quantum Markov dynamics for which the camel-like behavior is impossible; they are well known in physics - the unital dynamics. Such dynamics were highlighted in studies on the quantum version of the Second Law of Thermodynamics [334] (see also [203]). Apart from the basic quantum entropy, the von Neumann entropy [451], we analyze behavior of the linear entropy reflecting decoherence of the system’s quantum state (Sect. 4.8). As we see (Fig. 4.3), quantum Markov dynamics can prevent decoherence and even increase the state’s coherence. This feature of quantum state dynamics is very important for modeling information processes in (open) biosystems.

4.3 Supplement to Elementary Quantum Vocabulary

35

We conclude by highlighting the problem of the mathematical description of the possible states of a living system S and speculate that such states should generate the camel-like dynamics of the quantum entropies for all possible surrounding (physicoinformational) environments (Sect. 4.10). The essence of this chapter is the biological interpretation of the well known mathematical results and highlighting the special class of quantum master equations which phenomenologically describe biologically-natural adaptive behavior. The question “What is life?” is very complex and this book does not pretend to give the final answer. This is just a step toward quantum information formalization of Schrödinger’s ideas.

4.3 Supplement to Elementary Quantum Vocabulary Here we deliver a new portion of the quantum formalism (see also Sect. 2.1 and Chap. 16). As always we work with finite dimensional state space, Hilbert space H with the scalar product ψ1 |ψ2 . Denote the space of linear operators in H by the symbol L(H). We recall again that we work in finite dimensional linear space. In the infinite-dimensional case one should use the space of bounded (continuous) linear operators. An operator U ∈ L(H) is called unitary if it preserves the scalar product on H, i.e., U ψ|U ψ = ψ|ψ. One important consequence of unitarity is that U −1 = U  ,

(4.1)

the inverse operator coincides with the adjoint operator. An operator Q ∈ L(H) is called positive, Q ≥ 0, if its quadratic form is positively semi-definite, i.e., ψ|Q|ψ ≥ 0 for any vector ψ ∈ H.

4.3.1 Superoperators and Quantum Channels We remark that the space of linear operators L(H) is by itself a complex linear space. It is equipped with the scalar product A|B = TrA B, A, B, ∈ L(H).

(4.2)

Thus, L(H) has the Hilbert space structure. We consider linear operators acting in this Hilbert space, T : L(H) → L(H). Such operators are called superoperators. A superoperator T : L(H) → L(H) is called positive, T ≥ 0, if it maps the set of positive operators into itself, i.e., T Q ≥ 0, for any Q ≥ 0; in particular, for a density operator ρ, T ρ ≥ 0.

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We remark that the space of density operators D ≡ D(H) is a convex subset of L(H). In future studies we will be interested in superoperators for which D is an invariant subset, i.e., T (D) ⊂ D. To define a quantum channel, we have to introduce the notion of completely positivity for a superoperator. The heuristic meaning of this notion is not straightforward. In Chap. 8 (Sect. 8.2) it will be discussed in very detail. For the moment, one can treat completely positivity just as some generalization of the notion of positivity. A superoperator T : L(H) → L(H) is called completely positive if its natural extension T ⊗ I to the tensor product L(H) ⊗ L(M) = L(H ⊗ M), where M is an arbitrary finite dimensional complex Hilbert space, is again a positive superoperator on L(H) ⊗ L(M). Let S be a physical system with state space H. The extension T ⊗ I means that one takes into account the presence of another system S (with state space M) which does not interact with S. But I does not disturb the state of S . So, it is natural to expect that T ⊗ I should map the states of the compound system (S, S ) into its states. Definition A quantum channel is any trace-preserving completely positive superoperator T : L(H) → L(H). In particular, for any ρ ∈ D, T ρ ∈ D.

4.3.2 Von Neumann Entropy The von Neumann entropy is defined as S(ρ) = −T r ρ ln ρ,

(4.3)

where ρ is a density operator. There exists an orthonormal basis | j consisting of eigenvectors of ρ, i.e., ρ| j = p j | j  (where p j ≥ 0 and j p j = 1). In this basis, the matrix of the operator ρ ln ρ has the form diag(pj ln pj ; ) hence S(ρ) = −



p j ln p j .

(4.4)

j

This is the classical Shannon entropy for the probability distribution ( p j ). However, the von Neumann entropy has the classical form, but only w.r.t. this to special basis. We present three basic properties of the von Neumann entropy.

4.3 Supplement to Elementary Quantum Vocabulary

37

1. S(ρ) = 0 if and only if ρ is a pure quantum state, i.e., ρ = |ψψ|. 2. For a unitary operator U, S(U ρU  ) = S(ρ). 3. The maximum of entropy is approached on the state ρdisorder = I /N and S(ρdisorder ) = ln N , where N is the dimension of the state space. It is natural to call ρdisorder = I /N the state of maximal disorder.

4.3.3 Isolated System: Schrödinger and Von Neumann Equations For an isolated quantum system, the evolution of its pure state is described by the Schrödinger equation: d i ψ(t) = H ψ(t), ψ(0) = ψ. (4.5) dt This equation implies that the pure state ψ(t) evolves unitarily:

where

ψ(t) = U (t)ψ0 ,

(4.6)

U (t) = e−it H .

(4.7)

The map t → U (t) is one parametric group of unitary operators. • U (t + s) = U (t)U (s), • U (−t) = U −1 (t), • U (0) = I. Since we work with the finite dimensional Hilbert spaces,  then exponent of any linear operator is determined as the power series e V = ∞ n=0 V /n! converging in the operator norm. In quantum physics, Hamiltonian H is associated with the energy-observable. However, in quantum-like modeling describing information processing in bio or AI systems, the operator H has no direct coupling with physical energy. This is the evolution-generator describing information interactions inside a system. In some applications H is associated with mental (psychic) energy (Chap. 11). Schrödinger’s dynamics for a pure state implies that the dynamics of a mixed state (represented by a density operator) is described by the von Neumann equation: dρ (t) = −i[H, ρ(t)], ρ(0) = ρ0 . dt

(4.8)

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Here

ρ(t) = U (t)ρ0 U (t) .

(4.9)

This formula implies that, for an isolated system, the von Neumann entropy is preserved.

4.4 Open Quantum Systems An isolated system is an idealization of the real situation; generally all systems are open, i.e., their interact with surrounding environment. For an open system S, the dynamics of its state has to be modeled via the interaction with surrounding environment E. The states of S and E are represented in the Hilbert spaces H and K. The compound system S + E is represented in the tensor product H ⊗ K (for the notion of tensor product see Sect. 16.9). This system is treated as an isolated system and in accordance with quantum theory,the dynamics of its pure state can be described by the Schrödinger equation: i

d (t) = H (t)(t), (0) = 0 , dt

(4.10)

where (t) is the pure state of the system S + E and H is its Hamiltonian. This equation implies that the pure state (t) evolves unitary : (t) = U (t)0 , where U (t) = e−it H . Here Hamiltonian, the evolution-generator, describing information interactions has the form H = HS + HE + HS,E , where HS , HE are Hamiltonians of the systems and HS,E is the interaction Hamiltonian. This equation implies that evolution of the density operator R(t) of the system S + E is described by von Neumann equation (Chap. 4, Sect. 4.3.3): dR (t) = −i[H, R(t)], R(0) = R0 , dt

(4.11)

However, the state R(t) is too complex: the environment includes too many degrees of freedom. Therefore, we are interested only the state of S; its dynamics is obtained via tracing of the state of S + E w.r.t. the degrees of freedom of E : ρ(t) = Tr K R(t), where Tr K denotes the partial trace w.r.t. Hilbert space K.

(4.12)

4.5 Gorini-Kossakowski-Sudarshan-Lindblad Equation

39

4.5 Gorini-Kossakowski-Sudarshan-Lindblad Equation As a rule, the open quantum system dynamics (4.12) is very complex and its mathematical analysis is difficult. Its simplest approximation is the quantum Markovian dynamics given by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation [209]. This is one of the quantum master equations. (In physics, it is often called the Lindblad equation.) dρ (t) = −i[H, ρ(t)] + L[ρ(t)], ρ(0) = ρ0 . dt

(4.13)

Here Hermitian operator (Hamiltonian) H ∈ L(H) typically describes the internal dynamics of system S. In some situations H can also contain counterparts related to E. But in this book such situations will not be considered. Hence, here Hamiltonian H generates the internal dynamics of the system S. Superoperator L : L(H) → L(H) transferring the set of density operators D(H) into itself describes the interaction with environment E. In physics L is known as the dissipation operator. This terminology is justified by the fact that typically the interaction with the environment leads to entropy increasing (Sect. 4.7). In other words the order in S decreases. However, in biological and cognitive applications we are interested in the GKSL-dynamics which should generate entropy decreasing and order increasing (Chap. 4). Therefore we shall not use the terminology “dissipation operator” which is common in physics and call L the interaction operator. If L = 0, no interaction, then GKSL equation (4.13) is reduced von Neumann equation (4.8) and, for pure states, to Schrödinger equation (4.5). The superoperator L can be represented in the form: Lρ =

 j

  1 γ j C j ρC j − {C j C j , ρ} , 2

(4.14)

where operators C j ∈ L(H) and, for a pair of operators F1 , F2 , {F1 , F2 } = F1 F2 + F2 F1 , is their anticommutator. The operators C j are called “collapse operators” or “quantum jump operators” (or Lindblad operators). Note that the solution ρ(t) remains a positive matrix with trace 1 if started with such ρ0 (so this general form of dynamics preserves trace in the same way as Schrödinger dynamics does). The coupling constants γ j ≥ 0 describe the strength of interaction between the system S and its environment E. They can be interpreted as inverse relaxation time (the higher the value of this constant, the faster the extinction of oscillatory behavior). Generally each jump operator C j describes the special type of interaction of the system S with the environment E and it is characterized by its own relaxation time τ j and interaction constant γ j = 1/τ j .

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4.5.1 Stabilization to Steady State Here we continue consideration started in Sect. 2.3. Under some conditions on the initial state ρ0 = ρ(0) and the interaction between a system S and the environment E, state evolution t → ρ(t) can lead to a steady state: lim ρ(t) = ρsteady .

t→∞

(4.15)

In this book, we restrict considerations to the evolution driven by the GKSL equation. We note that in this case a steady state satisfies the stationary GKSL equation: i[H, ρsteady ] = L[ρsteady ].

(4.16)

It is also important to point out that generally a steady state of the quantum master equation is not unique, it depends on the class of initial conditions. Limiting state ρsteady expresses the stability with respect to the influence of concrete environment E. Of course, in the real world the limit-state would be never approached. The mathematical formula (4.15) describes the process of stabilization and damping of fluctuations. But, they would be never disappear completely with time. The key aspect of the GKSL equation is that, whereas the standard Schrödinger equation produces incessant periodic oscillations in all probabilities for all observables (unless the state is an eigenvector of Hamiltonian H ), the GKSL equation eventually produces stabilization of its solution to a steady state [64]. This state represents approaching of the stationary regime of reacting to the influence of the environment.2 General conditions of existence and uniqueness of steady state are complicated. Later we shall point to a sufficient conditions. In this chapter approaching a steady state by a biosystem interacting with the surrounding environment expresses establishing equilibrium in information exchange between them. In the model of decision-making via decoherence (Sect. 2.3), a steady state has the meaning of the decision state. In the present chapter, we consider a more general problem of biosystem’s adaptation to the surrounding environment. Decision-making via decoherence will be again considered in Sect. 6.7 as a part of modeling brain’s functioning.

4.5.2 Quantum Markov Dynamics The GKSL equation is a quantum master equation for Markov dynamics. To formulate the Markov property, consider the evolution superoperator Tt : L(H) → L(H), 2

We recall that in this book state spaces are finite dimensional. In the infinite-dimensional case, even Schrödinger dynamics can lead to stabilization [38].

4.6 Camel-Like Dynamics of Quantum Entropy

41

ρ(t) = Tt ρ0 .

(4.17)

Tt = et , where ρ = −i[H, ρ] + L[ρ].

(4.18)

It has the form This representation implies that the map t → Tt , [0, +∞) → L(H), is the one parametric (super)operator semigroup, i.e., T0 = I, Tt1 +t2 = Tt2 ◦ Tt1 , t1 , t2 ≥ 0.

(4.19)

The latter equality represents Markov property: ρ(t + s) = Tt+s ρ0 = Ts ρt , for any s ≥ 0,

(4.20)

to determine system’s state at any instant of time t ≥ t, it is sufficient to know its state at time t.

4.6 Camel-Like Dynamics of Quantum Entropy We stress that the GKSL equation (Sect. 4.5) has been widely applied outside of quantum physics to a variety of problems in cognition, psychology, decision-making, and economics [18–21, 24, 25]. We consider the following simple GKSL equation illustrating the basic features of interaction of a biosystem S with its environment E: 1 ρ (t) = −i[H, ρ(t)] + γ(Cρ(t)C  − {C  C, ρ(t)}). 2

(4.21)

Here γ is the coupling constant representing the strength of interaction between S and its environment E. We want to present the model example of the order-stable dynamics that matches the biological order preserving behavior. We select the Hamiltonian of S and the interaction operator C as follows:  0 H = σx =  1

   0 1 1 , C =   ,  00 0

(4.22)

(In Sect. 4.7, one can find a hint why we selected the operators in this way.) System’s Hamiltonian is the Pauli matrix σx . We select the initial state as the density matrix   1  0.1 −0.1i  . (4.23) ρ(0) =  2 0.1i 0.9 

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4 What is Life? Open Quantum Systems Approach

Fig. 4.1 Order-stable (camel-like) dynamics of the quantum (von Neumann) entropy

The curves in Fig. 4.1 correspond to the values of the coupling constant γ = 0, 1, 2, 3, 4, 5. For γ = 0, the system is isolated and it preserves its entropy (straight line). The environment E that is described by the interaction operator (Lindbladian L determined by one operator C) has the strong ordering effect on the system. The increase of the coupling constant γ implies quicker stabilization and lower entropy of the steady state. As we can see from Fig. 4.1, entropy’s dynamics has four stages (so the dynamics is more complicated than was presented in the introduction to this chapter): • 1. Increasing and approaching the maximum value: the interaction with a new environment E and learning its features increases disorder in the system S. • 2. Decreasing and approaching the minimum value: by using the results of learning during the first stage, S recovers order. • 3. Slight increasing: the process of further interaction with E can bring a new portion of disorder into S. • 4. Stabilizing: S approaches the steady state ρsteady = limt→∞ ρ(t). We call graphs of such shape, Fig. 4.1, camel-like graphs. For small interaction constant γ (long relaxation time τ = 1/γ) there are two humps, with the second hump essentially dumped; increase of γ leads to stronger dumping of the second hump and then to its disappearance; for say γ = 5, the “camel” becomes one humped. The presence of the second hump and its dependence on γ is a finer characteristics of dynamics and we shall concentrate on the presence of the first (relatively big) hump. The entropic camel is characterized by the inequality: max S(ρ(t)) >> S(ρsteady ). t

(4.24)

4.6 Camel-Like Dynamics of Quantum Entropy

43

For large γ, we have even the inequality: S(ρ0 ) > S(ρsteady ).

(4.25)

The main distinguishing feature of such dynamics is that its steady state is not characterized by entropy’s maximum value corresponding to the first hump of the camel-like graph (cf., with the typical behavior of a physical system, Sect. 4.7). The entropy of the steady state can be essentially lower than the entropy approached at the first stage of dynamics - adaptation to the features of the environment E and search for adequate reaction to it. After this stage, S begins to improve its functioning and to increase the degree of internal order, consequently the entropy decreases. In Schrödinger’s terms [406], after the stage of adaptation to the environment’s features the system S starts to “absorb order” from E (to absorb negative entropy, in Schrödinger’s words). Illustrative biological examples are presented in Sect. 4.10. Of course, not any environment would deliver order. A typical quantum physical environment delivers disorder and generates the dissipation process leading to entropy’s increase. In such situation, the steady state is characterized by the maximum of entropy (Sect. 4.7, Fig. 4.2). To be more precise, we have to speak not about order-stable or disorder-generating environments, but about interactions which generate order or disorder. We emphasize that, for the camel-like dynamics, the entropy of the steady state ρsteady can be essentially lower than the entropy of the initial state ρ0 . For very strong coupling with the environment (graph for γ = 5), the final entropy is practically zero. Hence, adaptation via interaction with E can improve the level of order in S. The system absorbs from the environment the information useful for internal ordering. Thus, a special class of open quantum systems exemplified by dynamics (4.21), (4.22) is a very good candidate for mathematical modeling of order stability in biosystems. Schrödinger described the aforementioned dynamics with the notion of negative entropy [406]: “Every process, event, happening -call it what you will; in a word, everything that is going on in Nature means an increase of the entropy of the part of the world where it is going on. Thus a living organism continually increases its entropy - or, as you may say, produces positive entropy -and thus tends to approach the dangerous state of maximum entropy, which is of death. It can only keep aloof from it, i.e. alive, by continually drawing from its environment negative entropy - which is something very positive as we shall immediately see. What an organism feeds upon is negative entropy. Or, to put it less paradoxically, the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive.” We now discuss applicability of the Markov approximation given by the GKSL equation to modeling of interaction of a biosystem S with its environment E (see also article [310]). The basic condition for derivation of the GKSL equation is weak coupling of S with E (see, e.g., [64, 209]). This condition is very natural in the biological framework. While any biosystem S is fundamentally open, its coupling

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4 What is Life? Open Quantum Systems Approach

to the surrounding environment should be weak; otherwise functioning of S can be destroyed by E. For example, strong interaction with electromagnetic radiation may cause immediate death. In the same way, intensive absorption of information coming, e.g., from mass-media and other sources can lead to information overload, to a state of stress, and even to death. Schrödinger pointed out that isolation mechanisms play the crucial role in survivability of a biosystem [406]).

4.7 Open Systems Generating Disorder In this section we describe a typical behavior of quantum physical systems. The results presented here illustrate how biosystems should not behave, to stay alive. The unitality of quantum dynamics is discussed as a condition for non-camel graphic behavior. A quantum channel is called unital if T (I ) = I, where I denotes the unit operator. The dynamics of a typical physical system is characterized by unitality. We formulate the following simple theorem which is important for our further reasoning. This theorem is well known (see, e.g., [64, 334]); for reader’s convenience, we present its proof in Appendix A (Sect. A.1). Theorem 4.7.1 For a quantum channel T , S(T ρ) ≥ S(ρ).

(4.26)

for all quantum states if and only if T is unital.



Consider open quantum systems dynamics and master equation (4.13) for the density operator of the system S, ρ0 → ρ(t) = Tt ρ0 , where Tt = et , where ρ = −i[H, ρ] + L[ρ].

(4.27)

Suppose that, for any t ≥ 0, the dynamical quantum channel Tt is unital, i.e., Tt I = I, t ≥ 0.

(4.28)

Inequality (4.26) implies that, for any t, S(Tt ρ0 ) ≥ S(ρ0 ).

(4.29)

Markov property (Sect. 4.5) implies that, for any t > 0, S(Ts ρ(t)) ≥ S(ρ(t)), for any s ≥ 0.

(4.30)

4.7 Open Systems Generating Disorder

45

Thus, entropy does not decrease at any instant of time t. We recall that we consider finite dimensional state spaces; in the infinite-dimensional case, the situation is essentially more complicated [203, 334]. We obtained a kind of the Second Law of Thermodynamics, but for open (quantum) systems (see paper [334] for more precise formulation of this law): If the system’s evolution can be described by unital dynamics, then the von Neumann entropy gain during evolution is nonnegative. If a quantum dynamics is non-unital, then quantum entropy can decrease and fluctuate (Sect. 4.6, Fig. 4.1). Theorem 4.7.2 Dynamical channel Tt is unital, for some time-interval [0, δ], δ > 0, if and only if the unit operator satisfies the equation: LI = 0

(4.31)

Proof (1). Since Tt (I ) = I, we have dtd Tt (I ) = 0, and, hence, Tt (I ) I = 0. The dynamical channel is invertible, since Tt = et , where  : L(H) → L(H) is a bounded linear (super)operator (we consider a finite dimensional case). Hence,  I = i[H, I ] + L I = 0 and (4.31) holds. (2). Now, let (4.31) hold, then  I = 0 and Tt I = et I = I. In fact, we found that unitality on any (in principle, an arbitrary small timeinterval) implies global unitality, on [0, +∞), because condition (4.31) implies the global unitality. Proposition 4.7.3 If, for each state, the entropy gain is nonnegative on an arbitrary small time-interval, then the dynamical channel Tt is unital for all t ∈ [0, +∞), and, hence, the entropy gain is nonnegative on [0, +∞).  Theorem 4.7.2 and Proposition 4.7.3 imply the following form of the the Second Law of Thermodynamics for open quantum systems: Theorem 4.7.4 For GKSL-dynamics, the von Neumann entropy gain during evolution is nonnegative, iff (equivalent) conditions (4.28), (4.31) hold.  Now we rewrite Eq. (4.31) in terms of the operator representation of the Lindbladian L .   γ j (C j C j − C j C j ) = γ j [C j , C j ] 0 = LI = j

j

We recall that in functional analysis a linear operator is called normal, if it commutes with its adjoint operator: [C, C  ] = 0. (4.32) Thus normality of all operators C j is sufficient for unitality of dynamics and, hence, the entropy gain is nonnegative (globally). In the simplest case (see Eq. (4.21)), the

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4 What is Life? Open Quantum Systems Approach

Fig. 4.2 Disorder-generating dynamics of the von Neumann entropy

sum is reduced to one term, and the condition (4.32) is necessary and sufficient for unitality and entropy increase. (One can easily check that the operator C in (4.21)) given by (4.22) is not normal.) For example, consider Eq. (4.21) with some Hermitain interaction operator C, say C = σz (Pauli z-matrix) with the same Hamiltonian and initial condition as before (see (4.22), (4.23)); see Fig. 4.2 for graphs (for the same value of γ = 0, 1, ..., 5). We see that entropy monotonically increases and approaches its maximal value ln 2.

4.8

Dynamics of Linear Entropy (Decoherence)

As we know, the von Neumann entropy approaches its minimal value, S = 0, for pure states. This points to a connection between the degrees of purity and order in a system. How can one define purity of a quantum state? Pure states are represented by density operators of the form ρ = |ψψ|. As any projector, such a state is an idempotent operator, i .e., ρ2 = ρ. Since this projector is one dimensional, Tr ρ = 1. Hence, if a quantum state is pure, then (4.33) P(ρ) ≡ Tr ρ2 = 1, and vice verse. The quantity P(ρ) is called purity of ρ. Purity equals one iff the state is pure.

4.8 Dynamics of Linear Entropy (Decoherence)

47

Fig. 4.3 Order-stable (camel-like) behavior of the linear entropy (decoherence)

On the basis of purity, one introduces a new sort of entropy, linear entropy (see, e.g., [468]) given by (4.34) S L (ρ) = 1 − P(ρ). This entropy behaves similarly to the von Neumann entropy: • 1. S L (ρ) = 0 if and only if ρ is a pure quantum state. • 2. For a unitary operator U, S L (U ρU  ) = S L (ρ). • 3. The maximum of entropy is approached on the state ρdisorder = I /N and S(ρdisorder ) = 1 − 1/N , where N is the dimension of the state space. In particular, the linear entropy of an isolated quantum system is constant. Open quantum system dynamics can generate camel-like shapes of the linear entropy, cf. Figs. 4.3 and 4.1 (a camel-like shape) as well Figs. 4.4 and 4.2 (monotonic increase). One of its advantages is that it is easier to calculate than the von Neumann entropy (To calculate the von Neumann entropy, one has to find the eigenbasis of ρ and this can be computationally difficult.) The main distinguishing feature of the linear entropy is straightforward coupling with purity of the quantum state. This sort of entropy is also used as a measure of decoherence (see, e.g., [47, 48] for applications for analysis of experimental data). The latter can be treated as the loss of purity. For a quantum physical system, typically open system dynamics leads to decoherence, loss of purity; linear entropy approaches one. The basic feature of biosystems is their ability to beat decoherence and increase states’ purity - the camel-like behavior of the linear entropy. For linear entropy we can formulate the analog of Theorem 4.7.4:

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Fig. 4.4 Disorder-generating dynamics of the linear entropy (decoherence)

Theorem 4.8.1 For GKSL-dynamics, the linear entropy gain during evolution is nonnegative, iff (equivalent) conditions (4.28), (4.31) hold.  For example, consider curves of unital dynamics in Fig. 4.2. Its evolution of the linear entropy is presented in Fig. 4.4.

4.9 Specialty of Biosystem’s States and Dynamics It is useful to try to assign some biological meaning to unitality, Tt I = I. In the N -dimensional case, we can consider the state ρdisorder = I /N . This is the state of maximal disorder (with respect to the von Neumann and linear entropies), S(ρdisorder ) = ln N and S L (ρdisorder ) = 1 − 1/N . The unital dynamics is characterized by transformation of the maximal disorder state into itself, i.e., Tt cannot decrease disorder and generate order of some degree. However, it is not clear whether a biosystem can be alive in the state of total disorder, ρdisorder . Generally, we consider the unitality as a condition excluding the possibility of the camel-like dynamics for all possible states. The camel-like dynamics of a quantum system represented by the curves in Figs. 4.1 and 4.3 is a good candidate for modeling of living systems. The main problem is the absence of mathematical formalization of the camel-like behavior. Violation of the unitality condition implies only that, for some states, entropy has the camel-like form. But, generally non-unitality does not imply the camel-like dynamics for all states, as one would like to have for biological systems.

4.9 Specialty of Biosystem’s States and Dynamics

49

This interplay between the form of dynamics and states’ features is a complex issue, since not only the description of possible interactions with environments (encoded in the coefficients of the Lindbladian), but even the description of the class of possible states D S (H) of a biosystem S is a difficult problem. It is natural to assume that D S (H) is a proper subspace of the space of all density operators D(H). A similar problem is well known even in quantum physics. Textbooks may claim that all density operators correspond to physically realizable states, but experts know that only special quantum states can be prepared experimentally. The states of our interest are limited to those compatible with life. Denote the space of such states by the symbol Dlife (H). Subclasses of the states correspond to the health (including aging) conditions of the life. The description of the mathematical structure of the space Dlife is a complex problem. We mention one concrete question: Is Dlife (H) a convex set? We recall that the set D(H) is convex and this fact has very important consequence for the basic probabilistic features of the quantum information processing [115, 364]. The environmental states describe the conditions of system’s environment, but they are manifested only in the L-operator. In view of the above consideration, maybe attempting to describe a class of dynamics that generate, for all states belonging to D(H), the camel-like behavior of the von Neumann entropy is not at all a fruitful research strategy. Instead, we can define for each GKSL-dynamics the class of states D(H|E) generating the camel-like behavior; so D(H|E) is the union of the all such trajectories of the entropy of S for the environment E. Then it is natural to suppose that Dlife (H) ⊂ ∪E D(H|E), with union w.r.t. all possible environments supporting alive-states. We remark that for the unital dynamical systems the set of states Dlife (H) is empty. We have the following picture of the life-state evolution. Let ρ0 be the initial state at the moment when S meets new environment E and let ρ0 ∈ D(H|E). Then a camel-like trajectory of the system’s entropy is generated and the steady state ρstead is approached. If E stable, then S is comfortable in it and the entropy is constant (with small fluctuations). Let now S meet a new environment E (that can be generated, e.g., by change of some parameters describing the state of E). Then if ρstead ∈ D(H|E ), then a new camel-like trajectory is generated and so on. In principle, we can consider a life-evolution trajectory, starting with ρ0 = ρbirth ∈ D(H|Ebirth ). A life-trajectory consists of camel-like blocks, and the last block violates the camel-like structure of entropy: here entropy behaves in accordance with behavior of physical dissipating systems and entropy grows to the maximal value.

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4.10 Adaptation to the Environment: Illustrative Examples Human Adaptation to Surrounding Environment Suppose that somebody, say, Ivan suddenly finds himself in an unknown forest; he shall find a possibility to feed himself, adapt to weather conditions, secure his life from animals and so on. At the first stage, he investigates features of the forestenvironment and adapts his behavior to these features: disorder in his information representation of this context increases. Then he starts to explore the results of adaptation and his life becomes more ordered, up to stabilization to a steady state corresponding to this forest-environment (see [19]). This is an example for discussion of the meaning of the second hump of our entropy camel. From the curves in Fig. 4.1, we see that with the increase of the coupling constant the second hump disappears. It seems that it corresponds to the following situation: a biosystem S has adapted to the environment E and started to explore the features of E. However, if γ is relatively small, S was not able to adapt to all basic features of the environment and in the process of exploration, S meets some new features of E. So, the process of adaptation continues, but on a smaller scale. This generates the second hump and only then the state’s stabilization. Of course, there can be many humps with decreasing amplitude. Cell’s Adaptation to Lactose-Glucose Environment Another example is the lactose-glucose metabolism in a cell [20]. If the proportion of glucose-lactose in cell’s environment is changed, e.g., the concentration of glucose goes down and the concentration of lactose goes up, a cell S should change the regime of glucose-lactose consumption: at this stage entropy increases, since S cannot immediately update the information on the concentrations. Then when S “becomes sure” that the glucose concentration became low, it starts to consume lactose and entropy goes to its minimum value. Then S approaches the steady state corresponding to lactose consumption. Epimutations In contrast to the previous examples, epimutation changes not only bio-information processing (change in gene expression), but even the physical structure of the cell as, e.g., promoter methylation and histone modifications. At the stage of, e.g., promoter methylation (as the reaction to a new environment) entropy increases, then it goes down and finally stabilizes [21]. Neuron Interacting with Electrochemical Environment This example [284] will be presented in very detail in Chap. 6 starting with quantum information representation of a single neuron’s state. Of course, a neuron s is a not an isolated biosystem; it interacts with other neurons via electric signals as well as with the surrounding chemical environment, including a variety of hormones; denote this environment by E. Thus we are again in the framework of the theory of open quantum systems. In our model, by being involved in some cognitive task the neuron

4.10 Adaptation to the Environment: Illustrative Examples

51

s receives signals from E. The entropy of the quantum information state increases. The neuron is “in doubts” and it tries to adapt its state to the signals coming from other neurons as well as to the chemical context. After this period of adaptation, s starts to fire synchronically with other neurons involved in the same cognitive task. The quantum entropy decreases to its minimum. The electric state of synchronic firing corresponds to the steady state in the quantum information representation. In reality, a cognitive task is performed not by a single neuron, but by a neural network. In particular, synchronicity is achieved inside this network. Hence, a biosystem S is a neural network, its quantum information states are (generally entangled) states of neurons inside S. Synchronous firing corresponds to a steady state of S. The second hump of the entropy curves can be generated according to the general scheme. If S is too weakly connected with other neural networks that are relevant to performance of the cognitive task, then, after the first stage of interaction and adaptation with E, S can get signals which do not match the previous adaptation. They generate a new state update and a new hump that is essentially lower than the first one. This is the good place to mention articles [116–118] on generation of quantum-like behavior by neural network modeling.

Chapter 5

Order Stability in Complex Biosystems

5.1 Biosystem: Global Order from Local Disorder This chapter (essentially based on article [303]) is also motivated by Schrödinger’s book [406] in that he considered one of the most fundamental and intriguing problems of modern science: “What is life?” (so we continue the study started in Chap. 4). In this chapter, we model the order stability inside of a complex biosystem S that is composed of a few subsystems Si , i = 1, 2 . . . , N . We study the following problem of big complexity: Can a composed system S = (Si ) preserve the “global order” in itself, in spite of increase of local disorder (i.e., in its subsystems)? In the mathematical framework, this question is formulated as follows: Can S = (Si ) preserve its entropy while some of its subsystems Si (or even all) increase their entropies? We show that within quantum information theory the answer is positive. The key point is that in quantum theory the state of a compound system is not reduced to the states of its subsystems. The entropy balance in S is not based on summation of the entropies generated by subsystems. Here, the significant role is played by entanglement, nonclassical correlations between the states of subsystems Si of S. In the absence of entanglement, entropy behaves classically: the entropy of S equals the sum of entropies of Si . We explore the following feature of quantum channels (dynamical maps describing the state evolution): they can transfer non-entangled states into entangled. By using this feature we present the scheme of the concrete quantum channels construction preserving the global entropy and increasing all local entropies (see Appendix A, Sect. A.2). The construction is technically quite complicated. We restrict considerations to the case of two subsystems. We start with qubit state spaces of the subsystems and then consider the general case of N -dimensional state spaces. Our construction is explicitly based on representation of channels through orthonormal bases in the subsystems state spaces. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_5

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5 Order Stability in Complex Biosystems

This construction of desired quantum channels is restricted to the unitary evolution of the S-state. Generalization to non-unitary channels and complex systems with a large number of subsystems is also possible.

5.2 Compound Classical System: Local and Global Orders Micro and Macrostates Suppose that at some instant of time, a system S can be in one of states labeled by symbols x j , j = 1, 2, . . . , n. We call them microstates of S; set A S = {x j }. Let ρ be  a probability distribution on A S , that is, ρ = ( p(x) : x ∈ A S ), where p( x) ≥ 0, x p(x) = 1. We call ρ a macrostate, or simply state. For state ρ, entropy is defined as S(ρ) = −



p(x)ln p(x).

(5.1)

In bio applications (see Sect. 5.2), this quantity can be interpreted in the following way. Suppose that microstates of S can be “scanned” by other biosystems. Thus the microstate dynamics can be treated as signaling to other biosystems. For simplicity, we consider the discrete time dynamics; it generates the sequence of symbols X = x(τ1 ), . . . , x(τ N ), . . . ,

(5.2)

where τ is the time parameter of the microstate dynamics. Mathematically, this dynamic (signaling) is modeled as a random process. Under some conditions, the probability p(x) can be interpreted as the frequency probability - the limiting frequency of occurrence of the symbol x in the process’s trajectory X : p(x) = lim N (x)/N , N →∞

where N (x) is the number of occurrences of x in the sequence (5.2) (see [231, 250] for the frequency interpretation of probability). If system S is able to preserve its microstate, say one concrete y ∈ A S , then p(y) = 1 and p(x) = 0, x = y, and entropy S = 0. (The microstate can fluctuate visiting x-states different from y, but not so often, as 0 = lim N (x)/N ). In contrast, if the microstate of system S fluctuates covering uniformly A S , then p(x) = 1/n and entropy S = ln n. Thus entropy can be used as the measure of state-stability, order preservation in S. The increase of entropy implies the decrease of information, the diminishing of order, and death, or at least decay. On the contrary, the decrease of entropy means the increase of information, the rise of order, and life or at least the improvement of self-organization.

5.2 Compound Classical System: Local and Global Orders

55

We are interested in compound systems S = (S1 , S2 ). The (statistical) states of  S are represented by probability distributions ρ = ( p(x, y)), where p(x, y) ≥ 0, x y p(x, y) = 1. The entropy of S is given by S(ρ) = −



p(x, y) ln p(x, y).

(5.3)

xy

For a compound system, the states of its subsystems are given by the marginal probability distributions: ρ1 = ( p1 (x) =



p(x, y), ρ2 = ( p2 (y) =

y



p(x, y)),

(5.4)

x

and the corresponding entropies are S(ρ1 ) = −



p1 (x) ln p1 (x), S(ρ2 ) = −



x

p2 (y) ln p2 (y).

(5.5)

j

We can consider two biosystems, say two cells, that communicate with each other: S2 “feels” x-states of S1 and vice verse: cell-signaling. Systems Si can represent as well neural networks in the brain, social systems, or AI-systems [327]. If ρ = ρ1 ⊗ ρ2 (the direct product of probability measures), that is, probability p(x, y) = p1 (x) p2 (y), then S(ρ) = S(ρ1 ) + S(ρ2 ).

(5.6)

Generally, additivity is violated and only the subadditivity inequality holds S(ρ) ≤ S(ρ1 ) + S(ρ2 ).

(5.7)

In the quantum case, the situation is the same. We now point out the specific classical constraint between the entropy of a compound system and subsystems’ entropies: (5.8) S(ρ) ≥ S(ρi ). Quantum information processing relaxes this constraint; in such processing the global order in a compound biosystem S can be preserved, in spite generating of local disorders in its subsystems Si . Consider a model of signaling between biosystems based on recognition not of microstates, but macrostates. So, S1 and S2 communicate by recognition of the macrostates of each other (the probability distributions). There are two time scales, the fine time scale parameter τ and the rough time scale parameter t corresponding to the micro and macro state dynamics, respectively. The τ -scale dynamics determines macrostates evolving with the t-scale dynamics.

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Suppose that state of S evolves in accordance with some dynamics t → ρ(t). It generates dynamics of subsystems’ states t → ρi (t). Suppose that initially S had very low entropy, S(ρ0 ) =  1, and find the frequency ν M (1) = n M (1)/M, where n M (1) is the number of -intervals such that neuron N produces a spike. (Here  is a parameter of our model.) Then p1 ≈ ν M (1). We repeat that steady states play the exceptional role in our model as decision states (see Chaps. 1, 4 and Sects. 6.7.1, 6.8.2 of this chapter as well as works [18, 19, 24, 26, 27, 284, 291]). But, typically steady states are not pure, these are mixed states represented by density operators. In modeling the process of decision-making by the brain, we proceed with the frequency approach to probability. (In CP this approach, i.e., coupling probability with the frequency of occurrences, is based on the law of large numbers.) We note that observer can count spikes and operate with the frequency probability. However, immediately the following question arises: Who is an observer? In our model the brain (more concretely, each mental function) is a system that is able to perform self-observations, detection of information states of neurons and neural networks (but not electrochemical states). Consider now a group of neurons G (say m neurons) connected into a neural network. Quantum information state space of G is given by the tensor product of the state spaces for individual neurons. If the neurons do not interact the G-state is factorized into the tensor product of the states of individual neurons,

6.2 From Electrochemical Uncertainty in Action Potentials to Quantum Superposition

|ψG = |ψ1  ⊗ · · · ⊗ |ψm  ≡ |ψ1 . . . ψm .

69

(6.3)

If the neurons interact and these interactions generate correlations, their state is given by a non-factorisable vector belonging to HG , an entangled state, e.g., for two neurons, √ (6.4) |ψG = (|00 + |11)/ 2. This √ state is generated by a pair of neurons firing synchronically. The coefficient 1/ 2 gives the amplitude of probabilities p(00) = p(11) = 1/2 (hence, for a sufficiently long time interval T, the proportion of the periods of synchronized firing and relaxation). Consider also the state √ |ψG = (|01 + |10)/ 2.

(6.5)

In this state the neurons fire in anti-phase, if N1 fires, then N2 does not and vice verse. Generally, entangled states give the quantum information representation of perfect correlations (see [55]). So, we do not associate any sort of quantum magic with entangled states. Since entangled states play so important role in quantum information theory, we can speculate that approaching of the perfect correlations between neurons or groups of neurons also plays the important role brain’s functioning; if our quantum information model has some degree of adequacy with real information processing in the brain. In this model, pure states are typically used as the initial states for mental functions starting the process of decision-making. Say, an electromagnetic external signal (e.g., coming from a visual image) generates action potentials of neurons which are quantum informationally represented as superpositions. The latter are either correlated or anti-correlated via frequencies of firings. This state of neurons is represented as an entangled state. Consider observables a = ±1 given by the operators A such that Ai |0 = −|0, Ai |1 = |1. Consider now observables ai on the state spaces of two neurons given by the operators A1 = A ⊗ I and A2 = I ⊗ A. Their correlation has the form (6.6) A1 A2 ψ ≡ A1 A2 ψ|ψ = A ⊗ Aψ|ψ. For state (6.4), A1 A2 ψ = 1; for state (6.5), A1 A2 ψ = −1. So, these states correspond to the perfect correlations. Now consider a factorisable state, say |ψG =

1 1 (|0 + |1) ⊗ (|0 + |1) = (|00 + |01 + |10 + |11). 2 2

(6.7)

This state represents functioning of a network of two neurons such that all possible combinations of firing and relaxation are equally possible. There are no correlations in the firings of N1 and N2 .

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In quantum information representation, correlations in neuronal networks are mathematically described by entangled states. For example, for a neural network composed of m neurons, the synchronized firing can generate an entangled state of the form: (6.8) |ψG = c0 |00 . . . 0 + c1 |11 . . . 1, where |c0 |2 + |c1 |2 = 1. These coefficients give the complex amplitudes of probabilities p(00 . . . 0) = |c0 |2 , p(11 . . . 1) = |c1 |2 (hence, for a sufficiently long time interval T, the proportion of periods of synchronized firing and relaxation). Synchronized firing of neurons performing realization of some mental function seems to be the easiest way to generate correlations at the level of quantum information representation via entangled states of the form (6.8). We stress that entangled states generated by (anti-)synchronized neuronal firings are the important mental computational resource. We also remark that the states considered above, (6.4), (6.5), are symmetric: neurons’ permutations do not change them. They belong to the symmetric subspace sym HG of the state space HG . Neurons in such states are indistinguishable. The latter seems to play the important role in computational stability. The question whether the quantum-like model of brains functioning should be based on symmetric state sym space HG (reflecting indistinguishability of neurons) or on the complete quantum information state space HG is complex. We speculate that some mental functions sym use indistinguishability of neurons and they process states belonging to HG ; others work in HG (see also [284, 291]).

6.3 Electrochemical and Quantum Information States: Nonlinear Versus Linear Dynamics The Hodgkin–Huxley equation [206] is the ordinary differential equation describing the dynamics of the action potential of a neuron. This dynamics is nonlinear. Nonlinearity is crucial to represent non-periodic (chaotic) pattern of activation appearing at some physical conditions (see, for example, [4, 5, 207]. (Nonlinearity is a necessary condition of chaos [427].) However, although the chaotic patterns are typical for electrochemical processes in the brain, they are not characteristic for cognitive processes. Chaotic behavior is never detected at the cognition level (at least for psychically healthy people). Rather, as pointed out in our model, the frequency of activations in a long interval T seems to be truly important to capture cognition process. The quantity of frequency is determined regardless of chaotic or periodic behavior at the electrochemical level. Corresponding to the frequency probability, the superposition state can be assigned for a neuron. Generally, this is the superposition (6.1). Mathematically such states can be described by the quantum mechanical formalism based on the complex linear space representation. Quantum dynamics is linear. Thus, transforming the electro-

6.4 Quantum Physics of Brain’s Functioning: Impossibility of Superposition …

71

chemical states to the quantum information states (superpositions) the brain transfers the nonlinear dynamics of the Hodgkin–Huxley type into the linear quantum dynamics. The quantum-like representation is free form chaotic patterns characteristic for underlyning electrochemical representation. Thus, elimination of chaotic patters is a principal feature of quantum information representation of the electrochemical dynamics of the action potentials of neurons.

6.4 Quantum Physics of Brain’s Functioning: Impossibility of Superposition of Neuron’s States This is the good place to make a few remarks on models reducing cognition and even consciousness to genuine quantum physical processes in the brain (e.g., [182, 376, 393, 443], see also [66, 67, 208, 397, 447, 448]). These attempts are rooted in the strong belief that the quantum theory is the ultimate physical theory. Supporters of the ultimate character of quantum theory assume that all processes in nature can be reduced to quantum physical processes. In particular, brain’s functioning and its “products”, cognition and consciousness, are generated by quantum physical processes in the brain. We point to the three main streams in the quantum brain project: 1. Quantum Mechanics: consciousness from entanglement in microtubules (Hameroff [182]). 2. Quantum Field Theory: consciousness as the states of the quantum field generated by the brain (started by Umezawa [443], see also [393], and nowadays this QFTbrain model is actively developed by Vitiello [397, 447, 448]). 3. Quantum Gravity: consciousness as the chain of collapses of superpositions of masses in the brain (Penrose [376]) These streams have their own advantages and disadvantages. We are not interested in the physical problems related to the quantum brain approach. For us it is crucial that all quantum brain models suffer of the impossibility to couple them to the paradigm of the neuronal processing of cognitive information. The neurons are not the basic entities of these models, but just some supplementary physical structures. And this attitude to diminish the role of neurons in cognition has the clear explanation. All authors developing the genuine quantum physical models of brain’s functioning pointed to the impossibility of considering neurons in a state of superposition. For example, Ricciardi and Umezawa [393] said: We do not intend to consider necessarily the neurons as the fundamental units of the brain.

Similar statements can be found in the works of Penrose [376], Hameroff [182], Vitiello [397, 447, 448], and Bernroider [67]. As a consequence, in such models neurons are not considered as the basic units of information processing. And this

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viewpoint on neurons’ role diminished the interest of the experts in neurophysiology and cognitive science to the quantum brain models. In contrast to quantum physical brain models, we present a natural electrochemical basis for superposition states of a neuron or a group of neurons - in the latter case, states are generally non-separable - entangled. We stress that genuine quantum physical models of brain’s functioning also appeal to superposition of states. However, in contrast to our quantum-like model, the latter is the superposition of quantum physical states, not the superposition of information states. The electrochemical processes in neural networks generating quantum potentials of neurons are macroscopic and globally described by classical physics, but, of course, at the molecular level the quantum physics also plays the important role.

6.5 Mental Function as Decoherence Machine This section is based on Sect. 2.3 on decoherence decision-making. In accordance with the previous consideration, a neural network G generates quantum-like states, linear superpositions of the basic states (|α) corresponding to the neural code used by G. Here α = (α1 , . . . , α N ), α j = 0, . . . , m − 1, where N is the number of neurons in G and m gives the number of symbols in this neural code. Hence, |α = |α1 . . . αn . Denote this state space by the symbol HG . Consider a mental function F that is physically based on the neural network G. We suppose that F has discrete outputs ( f j ) (possible “decisions”). Mathematically a mental function F based on G is represented by some orthonormal basis (|γ ) in HG . These are basic mental states associated with this mental function. We emphasize that the basis (|γ ) does not need to coincide with the basis (|α) corresponding to the neural code used by the mental function F. Various mental functions can use the same neural code (may be they all use the same code) and a few mental functions can be physically based on the same neural network, but be represented by different bases (|γ  ≡ |γ F ). For example, consider the quiescent-firing neural code, i.e., m = 2, and G consisting of a single neuron, the neural basis is written as |0, |1. For example, F can be determined by the basis √ √ |γ 1 = (|0 + |1)/ 2, |γ 2 = (|0 − |1)/ 2. Consider now F which is physically based on G consisting of two neurons; let F use the dichotomous neural code, i.e., the α-basis has the form: |00, |10, |01, |11, For example, F can be determined by the Bell basis √ √ |γ 1 = (|00 + |11)/ 2, |γ 2 = (|00 − |11)/ 2,

6.5 Mental Function as Decoherence Machine

73

√ √ |γ 3 = (|01 + |10)/ 2, |γ 4 = (|01 − |10)/ 2. Each basis vector |γ  corresponds to some value f of F. However, generally this correspondence does not need be one-to-one: a few basis vectors can correspond to the same value f of mental function F. In the simplest case, F has only two outputs, f = ±1 (decisions “yes”/“no”). Thus HG = H−;G ⊕ H+;G , where the subspaces are generated by the vectors corresponding to outputs f = ±1. Generally H = ⊕ f H f ;G . Suppose that, as in the above example, F is determined by the Bell basis and subspaces H±;G are generated by vectors, |γ 1 , |γ 2 and |γ 3 , |γ 4 , respectively. Then, F generates the outputs F = ±1 for, respectively, correlated and anti-correlated states. Generally a state contains both correlated and anti-correlated components. Let |ψ0 be the initial mental state, superposition of the neural code basis states, |ψ0 =



cα |α,

α



|cα |2 = 1.

(6.9)

α

This representation is of the electrochemical origin, but of the quantum information nature. By our mathematical model of decoherence decision-making (Sect. 2.3), the mental function F works to transfer this superposition into a density operator ρ F that is diagonal with respect to the F-basis, i.e., ρF =



pγ |γ γ |,

(6.10)

γ

 where pγ ≥ 0, γ pγ = Tr ρ F = 1. In quantum physics, this transition is known as decoherence. So, in our model [18, 19, 24, 27] a mental function generates its outputs via decoherence of the initial mental state which is typically assumed to be a pure state. We recall that decoherence process diminish states quantumness. Quantitatively the latter can be expressed through state’s purity (Sect. 4.8). In terms of density operators, F generates transformation ρ0 ≡ |ψ0 ψ|0 → ρ F .

(6.11)

This output state can be considered as a classical state - classical with respect to mental function F (the basis (|γ )). The ρ F is the classical statistical mixture of pure states (|γ ) with probabilistic weights pγ . Thus, in our model a mental function resolves uncertainty presented in the initial mental state |ψ0 by “differentiating” it (see Sect. 6.8.2) into weighted mixture of states (|γ ) associated with F. The probabilities of F-outputs are obtained by summation of probabilities

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pf =

pγ .

(6.12)

|γ ∈H f ;G

The final step of F-functioning, selection of the concrete output f, can be based on a classical random generator with the probability distribution ( p f ). Another possibility is that F selects the output f corresponding to the maximal probability p f . A mental function F can be considered as a self-observable. In coming sections we present a few quantum measurement models for the process of resolution of initial uncertainty encoded in mental state |ψ0 , i.e., for transition (6.11). Ontic and Epistemic Theories We remark that the outputs of a mental function F are determined by subspaces H f ;G : F = f for |ψ ∈ H f ;G (with probability one). In the operator terms, F is described by projections (Q f ) on the subspaces (H f ;G ). If F-outputs are encoded by real numbers, then these projections compose the Hermitian operator denoted by the same symbol F,  f Qf, (6.13) F= f

The mental function F can be identified with this operator (Sect. 6.6).1 In this formalism decomposition of projections Q f into one dimensional projections corresponding to the basis (|γ ), Qf =



|γ γ |,

(6.14)

|γ ∈H f ;G

does not play any role; it is hidden even from F which cannot approach its own internal structure; a mental function self-observes only the outputs F = f. The following part of this section is devoted to the discussion on the onticepistemic structure of functioning of a mental function. This discussion is merely of philosophical and methodological value and the reader who is not interested in such issues can jump to the next sections. We point out that, in contrast to the von Neumann based scheme, the special scheme of decision-making by decoherence, so- called differentiation (Sect. 6.8), is based on the use of the basis (|γ ). This basis determines the hidden structure of F which is not visible for F by itself, for F as a self-observer. We can interpret vectors (|γ ) as hidden variables and the description of functioning of F in terms of these vectors as ontic. The description of F in terms of projections (or operator (6.13)) can be interpreted as empistemic, it is about knowledge which F extracts from (quantum information) states of the neural network G.

1

In principle, there is no need to label the F-outputs by real numbers. They can be symbols of any origin. Hence, the mental function F can be mathematically viewed as PVM, F = (Q f ).

6.6 Model 1: Collapse of Mental Wave

75

We emphasize that both these descriptions are quantum. This is interesting, since typically the quantum description is epistemic and the ontic one is classical (see Atmanspacher [32], Khrennikov [255]). Moreover, we should not forget about the primary level of onticity given by action potentials, onticity of the electrochemical processes in the brain (and generally in human’s body). The γ -basis description is epistemic w.r.t. the electrochemical one. Thus, our framework has the multilevel ontic-epistemic structure. Each lower level is treated as ontic w.r.t. the next higher level which in turn is treated as ontic for the next higher level. To call the γ -description epistemic, we should point out to some observer who “sees” the γ -structure of the (quantum information) states of G. Who is it? A mental function F operates (as an observer) in the conscious domain of brain’s mental processing. We can speculate that γ -representation corresponds to operation in the unconscious domain. Transformation of action potentials into the neural code basis (|α) and then into (|γ )-basis happens without coupling to consciousness. And the process of state differentiation (Sect. 6.8) w.r.t. (|γ )-basis also happens unconsciously. We can imagine that each F (treated as a self-observer) is based on an unconscious quantum processing device described as a quantum channel (Chap. 4, Sect. 4.7). And the latter is determined by the γ -basis. As is typical in ontic-epistemic structuring of scientific theories, the same epistemic model may have a few different ontic models beyond it. In the above quantum ontic-epistemic framework, the selections of different bases in subspaces H f ;G correspond to the use of different hidden variables (ontic structures). As we shall see, these structures correspond to the different signatures of the environments (Sect. 6.8.1) of the neural network G – the neural basis of F = FG .

6.6 Model 1: Collapse of Mental Wave We start with the presentation of the canonical quantum model for transition (6.11). It is convenient to proceed with the initial state given by an arbitrary density operator ρ0 . In this model a mental function F is represented by the Hermitian operator (6.13). The value f is observed with probability given by the Born rule (Sect. 2.1): p f = Trρ0 Q f

(6.15)

and the post-observation state with specified value f is given by ρf =

Q f ρ0 Q f Tr Q f ρ0 Q f

and without output specification by the state

(6.16)

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ρF =



pf ρf.

(6.17)

In particular, for pure initial state |ψ0 (the density operator ρ0 = |ψ0 ψ0 |), the post-measurement state is again the pure state: |ψ f  = Q f |ψ0 / Q f |ψ0  .

(6.18)

This is the projection postulate [341, 451]. This mathematical model leads to jump-like state transformation, |ψ0  → |ψ f ; this transformation is often called “wave function collapse.” A plenty of quantum (-like) models of cognition refer to “mental state collapse” [66, 67, 182, 208, 376]. However, even in genuine quantum physics the notion of wave function collapse (and generally appealing to the projection postulate [341, 451]) is the most doubtful notion of quantum theory. Its straightforward use in modeling of cognition mystifies brain’s functioning.

6.7 Model 2: Open Quantum System Dynamics In modeling of brain’s functioning, our aim is to proceed without mental state collapse, without referring to the projection postulate. Here we appeal to theory of open quantum systems and corresponding treatment of the measurement process [115, 209, 367]. In this theory, observation (measurement) on a system S is considered as interaction of S with its environment E. System’s state dynamics generated by this interaction is described by the quantum dynamical equation (Sects. 2.3 and 4.4). Here we briefly repeat the basics of quantum Markovean modeling of the state dynamics of a system interacting with the surrounding environment. Under the assumption that quantum state dynamics is Markovean, the quantum dynamical equation is the analog of the classical master equation for probabilities. This is the GKSL equation, dρ (t) = −[H, ρ(t)] + Lρ(t), ρ(0) = ρ0 , dt

(6.19)

where H is Hamiltonian of S and L is a super-operator. Commonly operator H represents the state dynamics in the absence of outer environment; L describes interaction with the environment (Chap. 4). For “natural” systems, environments, and interactions (encoded in operators H and L) the state ρ(t) asymptotically approaches some steady state ρ F , (6.20) ρ F = lim ρ(t), t→∞

this is a solution of the stationary equation

6.7 Model 2: Open Quantum System Dynamics

[H, ρ F ] = Lρ F .

77

(6.21)

This state is considered as the post-measurement state, but without determination of the concrete output. In our model of quantum dynamical (decoherence) decision-making (Sect. 2.3), the state ρ F should be diagonal with respect to the F-basis (|γ ). Thus, it has the form (6.10)  pγ |γ γ |. (6.22) ρF = γ

Probabilities p f are determined via summation, see (6.12). In this scheme which is based on the quantum master equation, the constraint of diagonalization in the (|γ )-basis looks as ad hoc.2 The determining role of (|γ )-basis becomes clear in the differentiation scheme (Sect. 6.8). The role of system S is played by neural network G (that can be reduced even to a single neuron) and E is its electrochemical environment, including the electrical and chemical signals from other brain’s networks, working with other mental functions, as well as from other body’s parts.

6.7.1 General Quantum Dynamics of Mental State For a mental function F, a system in our case neural network G and environment E, are unified in a compound quantum(-like) system, (S, E). Its state space is mathematically represented as the tensor product of state spaces of subsystems, i.e., as H = HG ⊗ KE . For a mental function F, the state dynamics of the compound system is described by the Schrödinger equation, i

d = HF |(t), |(0) = |0 , dt

(6.23)

i.e., |(t) evolves unitary, |(t) = U F (t)|0 , where U F (t) is one parametric group of unitary operators, U F (t) : H → H. The state of ρ(t) of the neural network G (its quantum information state) is obtained via averaging the state R(t) = |(t)(t)| of the compound system with respect to the degrees of freedom of environment E: ρ(t) = Tr K R(t).

2

(6.24)

Coupling of this basis with the coefficients of the equation is not straightforward. In applications, we typically start with an equation and then by finding its steady state we associate its eigenbasis with some mental function F. If ρ F depends on ρ0 , then the situation is even more complex.

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This is the general quantum dynamical equation; the GKSL equation is derived as its approximation under some conditions [209]. The mental analogs of these conditions and analysis of their applicability to cognitive system were presented in article [310]. We remark that the partial trace is again a pure state only for factorisable states of a compound system, i.e., generally it is a mixed state represented by a density operator. And this dynamics can be non-Markov with the memory effects. In fact, the real dynamics is surely non-Markovean, but consideration of non-Markovean quantum dynamics makes the model essentially more complicated; in particular w.r.t. the problem of existence of steady states. As in the Markovean case, the outputs of F (decisions) are generated via approaching  of a steady state, see (6.20), that is diagonal in the F-basis (|γ ) : ρ F = γ pγ |γ γ |. This state, or more precisely, this decomposition of the density operator, is the classical statistical mixture of the basic mental states determining the mental function. The probabilities of F-outputs are given by (6.12). Finally, we make a remark on the limit-procedure (6.20) (see also Chap. 5). Of course, to approach the decision state ρ F , the network works only a finite period of time - until state’s fluctuations become small with respect to stabilization parameter > 0. This parameter is one of characteristics of the mental function ≡ F ; in principle, the model can be more complicated with depending on the initial state. This determines the degree of fluctuations. The clearer picture corresponds to the discrete dynamics which can be obtained via time-discretization of the quantum dynamical equation (6.24). Here the state evolution is given by the sequence of density operators, ρ(t0 ) = ρ0 , . . . , ρ(tn ), . . . where tn+1 = tn + δ and δ > 0 is the step of discretization. Then the approximate decision state is determined by the condition (6.25) ||ρ(tn ) − ρ(tn−1 )|| ≤ . We note that in this book we proceed in the finite dimensional case and the choice of the concrete norm on the space operators does not play any role. In the infinite dimensional case, one can use either sup or Tr norm. In Sect. 6.5 the following difficult question has been briefly discussed: How does F select the concrete value F = f on the basis of the steady state ρ F ? For the moment, one can only speculate on this issue. I speculate in the following way. The state ρ F is the classical statistical mixture of the mental states (|γ ) determining the quantum information representation of self-observable F. Thus, in the process of approaching ρ F quantumness was washed out from the state. In physical terms, this is the process of decoherence w.r.t. the concrete basis (|γ ). It seems that the selection of the concrete value f from this statistical mixture can be described in purely classical probabilistic terms, as, e.g., a random generator producing the outputs F = f with probabilities p f given by (6.12). As was mentioned in Sect. 6.5, one can refer to other models of the decision-making on the basis of a classical probability measure.

6.7 Model 2: Open Quantum System Dynamics

79

6.7.2 Critical Analysis of Open Quantum System Approach to Cognition Appealing to the theory of open quantum systems and the use of the quantum dynamical equation (see (6.24) and its Markovean version (6.19)) provide the possibility to proceed without “mental state collapse”, as resulting from decision-making. However, this measurement scheme is too abstract. We do not take into account the internal structure of the process of “differentiation” of the initial state into mixture of F-basic states (|γ ). We also point that stabilization to steady state ρ F deforms probabilities encoded in the initial state ρ0 . Generally the output probability pγ = γ |ρ F |γ  is not equal to the input probability p0γ = γ |ρ0 |γ . Consequently, even the probabilities for the outputs of mental function F can be modified; generally p0 f =





p0,γ = p f =

|γ ∈H f ;G

pγ .

(6.26)

|γ ∈H f ;G

To be more concrete, consider a pure initial state ρ0 = |ψ0 ψ0 |. This state encodes potentiality of realization of state |γ  with the probability p0γ = |ψ0 |γ |2 . These potentialities correspond to electrochemical uncertainty generated by action potentials of neurons. It would be natural to expect that these potentialities would coincide with potentialities encoded in the output state ρ F that are given by probabilities pγ . However, generally pγ = |ψ0 |γ |2 . Roughly speaking the stabilization scheme based on the quantum dynamical equation is too general, it generates too wide class of the output distributions. One can desire a quantum measurement scheme without the projection postulate and with preservation of the probabilities. Such a scheme will be presented in the next section. We remark that the canonical quantum measurement scheme based on the projection postulate [341, 451] (Sect. 6.6) reproduces the probability distribution of F-outputs ( p f ) encoded in the initial state ρ0 , see (6.15): p f = Trρ0 Q f =



p0γ .

(6.27)

|γ ∈H f ;G

However, the projection postulate scheme [341, 451] suffers of the collapse-like state-transition. We want to open the “collapse back box” and to preserve probability distribution. We also repeat that the projection scheme does not describe the internal structure of projectors Q f , coupling to the collection of mental states (|γ ) determining the mental function.

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6.8 Model 3: Differentiation of Mental State In this section we follow the article [28] in which the general quantum dynamical scheme was concreted based on the interpretation of a density operator as a signature of the environment surrounding system S.

6.8.1 Signatures of Environments in a Density Operator A density operator is commonly interpreted as a mixed state: statistical mixture of pure states. However, this interpretation is really inconsistent: density operator ρ permits different decompositions into mixtures, statistical ensembles of pure states (|πi ):   ρ= qi |πi πi |, where qi ≥ 0, qi = 1. (6.28) i

Theory of open quantum systems matches well with the following interpretation of states given by density operators. By the Naimark’s dilation theorem, the density operator ρ representing the state of system S can be generated as a partial trace of a pure state of the compound system (S, E). By this interpretation it is meaningless to interpret ρ intrinsically, i.e., without coupling to environment E. Different decompositions of ρ into the statistical mixtures of pure states correspond to variety of the interactions and environments. According to D’Ariano [110], only a pure state of system S is informationally complete. A state given by a density operator that does not correspond to a pure state of S is informationally incomplete: it carries the impact of environment E; different environments can generate the same ρ-state. In this book (Sect. 6.8.2), we consider the most simple case of orthogonal states determining a mental function. And in this case the difference between considering the operator (the left-hand side of (6.28)) and the decomposition into one dimensional projections (in the right-hand side of of (6.28))) is essentially washed out’. In fact, we are interested not in the density operator, but in its concrete decomposition and the corresponding classical probability distribution (qi )). In [28], the general differentiation scheme was elaborated for non-orthogonal decompositions. In this case, different decompositions correspond to the signatures of different environments. They all are encoded in the same operator and they are invisible in the standard operator representation of a quantum state. In the formalism of article [28], quantum states are given not by operators, but by families of one dimensional projectors and probabilities (|πi  πi | , qi ), where generally πi |π j  = 0, for i = j.

6.8 Model 3: Differentiation of Mental State

81

6.8.2 Mathematical Scheme of Differentiation Process We now present the model of differentiation [28] by which a system S experiences step by step state transitions under the influence of environmental factors. This approach can be used for modeling various natural and mental phenomena: cell’s differentiation, evolution of biological populations, decision-making. In this book, we apply the state-differentiation scheme to model generation of outputs of mental function F as differentiation of the initial state |ψ0  into a classical mixture of the basic states (|γ -states determining F (see also [28]). We recall that the mental state |ψ0  is generated by a neural network G from action potentials. Let us consider a typical state transition caused by a quantum measurement (in our model, functioning of a mental function F): |ψ0  → (|γ , pγ ).

(6.29)

Here, as in the above consideration, |ψ0  denotes the initial state of F represented by density operator ρo ≡ |ψ0  ψ0 | ; (|γ , pγ ) denotes a classical statistical mixture of the basis states for F. The initial state |ψ0  can be expanded with respect to the F-basis as |ψ0  =

√

pγ |γ ,

γ

√ √ where pγ denotes a complex number satisfying | pγ |2 = pγ , i.e., it contains the phase as well. Hence, the density operator corresponding to the initial state can be expressed as the sum of two terms, diagonal and off-diagonal, |ψ0  ψ0 | =



pγ |γ  γ | +

γ

√

pγ

√

  pγ  ∗ |γ  γ   .

(6.30)

γ =γ 

 The first term γ pγ |γ  γ | corresponds to the classical probability distribution {|γ , pγ }. In physics, the process of vanishing of the second term is known as “decoherence.” It represents accomplishment of the measurement of observable with γ -basis. The relation of |ψ0  and {γ , pγ } is represented as  γ

Mγ |ψ0  ψ0 | Mγ =

 γ

| γ |ψ0  |2 |γ  γ | =



pγ |γ  γ | ,

(6.31)

γ

with the use of projection operator Mγ = |γ  γ |. The transition probability | γ |ψ0  |2 is equal to pγ . Thus in this model the initial probability distribution encoded in |ψ0  is not deformed. We want to realize projection transformation (6.31) as a process, i.e., to exclude the collapse-like state transformation. We consider such a process - differentiation, the state of mental function F is differentiated to {γ , pγ } step by step through a large number of state transitions.

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Each iteration is represented by a map in the space of density operators, a quantum channel. State iterations are given by ρ(0) = |ψ0  ψ0 | → ρ(1) = ρ(0) → ρ(2) = ρ(1) → · · · → ρ(n) = ρ(n − 1).

This process of iterations is called differentiation [28], if lim ρ(n) =



n→∞

pγ |γ  γ | ,

(6.32)

γ

is satisfied. The concrete construction of quantum channel was described in [28] and now we represent it briefly by coupling with the quantum-like model of brain’s functioning. The electrochemical environment of neural network G (on that mental function F is based) has many components. The initial quantum information states of these components do not depend on the initial state |ψ0 . Let |   | be the initial state of one of components of the environment; it belongs to environment’s state space denoted by K(= C N ). The initial state of the compound system belongs to state space H ⊗ K. We assume that it is factorized in the tensor product of the states of the subsystems: |ψ0  ψ0 | ⊗ |   | ≡ |ψ0  ψ0 | . In our model, already the first iteration through the quantum channel destroys separability and generates the entangled state. Canonically a quantum channel is realized with the aid of a unitary transformation of the state space of a compound system, U : H ⊗ K → H ⊗ K, U |ψ0  ψ0 | ⊗ |   | U ∗ . The operator U describes the interaction between the state of the mental function and environment. In [28], the following concrete construction was proposed: U=



|γ  γ | ⊗ u γ .

(6.33)

γ

Here, each operator u γ : K → K is unitary. And more concretely, U |ψ0  ⊗ |  =

√

  pγ |γ  ⊗  γ ,

γ

where

   γ = u γ |  .

(6.34)

6.9 Autopoiesis: Quantum Information Representation

It can be shown that if

83

     γ  =  γ 

for some γ = γ  , then the output state is entangled, i.e., it cannot be factorized into the state of the mental function and surrounding environment. Take now in (6.34) the spatial trace with respect to K. We select the orthonormal basis in K, say (|ψ j ) Nj=1 , and obtain that N             (|γ  γ  ) =

γ |ψ j ψ j | γ  |γ  γ   = γ | γ  |γ  γ   . j=1

In [28], it was shown that (6.32) holds, i.e., quantum channel is a differentiation channel. It was also shown that the off-diagonal terms in density operators ρ(n) approach zero with factors  γ | γ  n . (We remark that, since operators u γ : K → K are unitary, vector | γ  has the unit norm, so | γ | γ  | < 1).

6.9 Autopoiesis: Quantum Information Representation In fact, the presented model reflects only a part of brain’s functioning structure. The brain is not feed-forward computer handling solely stimuli coming from environment, its functioning is characterized by autopoiesis of excitation patterns. (This is bio-information version of biological [347] and social [340] autopoiesis, see further consideration.) Even in the absence of external stimuli the brain is spontaneously activated by transiting stochastically among various internal states. Each state represents activation patterns of a large number of neurons. It is a sort of framework (package) of patterns that is optimized to receive a specific external stimulus. In other words, the brain preliminarily prepares various packages in order to react quickly to various stimuli. Importantly, these packages of activation patterns are expected to be formed (and memorized) gradually through multiple interactions between neurons. This memorization is also can be modeled as a differentiation process. We model autopoiesis by using quantum information theory, so activation patters are associated with quantum-like states. The same state |ψ1  can correspond to a package of electrochemical activation patterns. Thus, learning to possible stimuli is performed at the quantum information level. And autopoiesis under consideration is (quantum) bio-information autopoiesis, as reproduction quantum information states. However, the situation is even more complicated, as we see below. There are two stages of differentiation. The first differentiation mechanism creates a set of packages for the neural network connected to one concrete neuron. The second differentiation mechanism leads on/off of activation for this neuron. The second stage of differentiation can be seen as the feedback of quantumlike dynamics to the electrochemical dynamics (that in turn generates quantum-like states). And autopoiesis under consideration also has two levels, electrochemical and

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quantum information. This is a complex issue and we shall continue its study in a future work.

6.10 Entanglement: Physics Versus Cognition We repeat that a pure quantum state belonging to tensor product H of state spaces Hi of systems Si , i = 1, 2, is called entangled, if it is not separable, i.e., it cannot be factorized into product of states belonging to spaces Hi , i = 1, 2. Entangled states play the crucial role in quantum computing. The standard explanation of their role can be presented in the following way. First consider some separable state |ψ = |ψ1  ⊗ |ψ2 .

(6.35)

Let U1 be some unitary transformation in H1 , a local operation. By transforming |ψ with the aid of this operation, |ψU1  = |U1 ψ1  ⊗ |ψ2 , we do not change the state of S2 . Of course, the state of compound system S = (S1 , S2 ) is changed. However, the reactions of states |ψ and |ψU1  to measurements on S2 are statistically the same. By performing measurements on S2 , an observer would never get to know that the state of S1 was changed. Now consider an entangled state, say (6.4). Then algebra of tensor product implies that transformation U1 changes state |ψ in such a way that (statistically) reactions on measurements on S2 are changed as well. How can it happen? Theoretically S2 can be far away from S1 , say S1 on the Earth and S2 on the Moon… In quantum physics, this phenomenon is commonly explained by so-called quantum nonlocality, the mystical action at a distance [62]. This explanation is based on Bell’s inequality. Its violation is considered as the confirmation of quantum nonlocality. The latter is still actively debated (see, e.g., my recent paper [272, 286, 288, 292]). However, commonly the quantum information community feels it comfortably with instantaneous action at a distance as the computational resource justifying quantum computational superiority. In fact, quantum nonlocality is really a mess. One typically mixes possible nonlocality of probable subquantum models, as e.g., Bohmian mechanics, and “genuine quantum nonlocality” presented in the projection postulate [286, 288, 292]. By this postulate a measurement on S1 with output x implies state’s projection in H1 . This instantaneously changes the total state of S and this change has a nontrivial impact even on possible measurements on S2 . We have criticized the use of the projection postulate [252] (that is just a mathematical symbolic encoding of the complex process of the interaction with a measurement device). Nevertheless, by using the individual interpretation of a quantum state (i.e., by associating a quantum state with the concrete quantum system), one confronts the really terrible problem of explanation of “state change at a distance”. To escape this problem, a part of the quantum community appeal to the statistical interpretation of the quantum state (Einstein-Ballentine). By this interpretation a quantum state

6.10 Entanglement: Physics Versus Cognition

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represents statistical features of a large ensemble of identically prepared quantum systems. Thus a measurement on S1 does not change the real physical state of S2 . However, it really changes the statistics of predictions for possible observations on S2 . In probability theory and statistics, such a procedure is well known and it is called probability inference. The main difference is in the rules for probability update. The quantum update is mathematically expressed by the projection postulate [341, 451] or more generally via differentiation process [28]. After this rather long intercourse into quantum foundations, we turn to neurons and our model. We are not ready to assume that neurons cooperate via the instantaneous action at a distance (well, compare with studies on the “really quantum brain” [66, 67, 182, 208, 376, 393, 397, 443, 447, 448]). One of the problems with opening the door to the action at a distance for mental states is that by saying “A” one should also say “B” - to assume inter-brains nonlocality with all anomalous phenomena. As was already pointed out, the quantum-like brain project does not reject completely that genuine quantum processes plays a role in the brain functioning (e.g., for our quantum information model, they can contribute into generation of action potentials and transmission of signals between neurons). Following to the Cromwell principle, I cannot even reject completely the hypothesis on mental nonlocality (e.g., [243, 430]). But, let us try to find a more simple explanation of the quantum-like state update in our model of the brain operating with quantum information states. (We recall that these states correspond to distribution of action potentials and frequency of neurons’ firings.) We proceed in the framework of theory of open quantum (information) systems. In a neural network G (serving for accomplishment of some mental function F), each neuron N plays the role of the environment for other neurons. Any change of the electrochemical state of N (that is at the quantum information level is expressed as differentiation [28]3 ) can be considered as a change of the environment for other neurons. In the presence of correlations, i.e., for an entangled state, this local environmental change induces modification of the firing pattern of other neurons. Yes, neurons can be sufficiently far from each other, in different domains of the brain. But, by taking into account the contribution of electromagnetic field as a component of the electrochemical interactions, we can assume that in the brain (which is very small comparing with light’s velocity) neuronal electrochemical states can interact very quickly. In this framework, entanglement is reduced to correlations. (So, there is nothing mystical in it.) These correlations play the role of the statistical interaction between neurons, by modifying the rates of their firings. The crucial point is that these are correlations at the level of quantum information states. They can be stable even for chaotic patterns in underlying electrochemical dynamics (described by the Hodgkin–Huxley equation and its generalizations).

3

We note that differentiation process unifies both unitary state transformation and measurement. There is no such sharp distinguishing between them as in the projection measurement scheme.

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6.11 Ontic and Epistemic Portrayals of Mental Processes This section is devoted to the discussion on the ontic-epistemic structure of functioning of a mental function. This discussion is merely of the philosophical and methodological value and the reader who is not interested in such issues can jump to the last section of this chapter. Physical and Informational Representations Interrelation of the electrochemical and quantum information states of the brain is the complex problem. The evolving (via quantum dynamics) state ρ(t) is the quantum information state encoding uncertainty in the action potentials of the neurons composing the network G. The signals coming from the environment E are electrochemical. And they determine the operator coefficients of the GKSL equation or the unitary operators encoding differentiation. In this way, our model connects physical and information states. Discreteness, the Notion of Phenomenon One of the distinguishing features of this encoding is its discreteness. The classical action potential, see Fig. 6.1, is continuous. But, the quantum information state is superposition of the basis states corresponding to the discrete alphabet of a neural code. Discreteness (quantization of observable’s outcomes) plays the crucial role in quantum mechanics. Bohr emphasized this within formalization based on the notion of phenomenon - a discrete event of observation, say a dot on the registration photoemulsion screen in the interference experiments or the click of a detector, see Bohr [74] (vol. 2, p. 64): ... in actual experiments, all observations are expressed by unambiguous statements referring, for instance, to the registration of the point at which an electron arrives at a photographic plate. ... the appropriate physical interpretation of the symbolic quantum mechanical formalism amounts only to predictions, of determinate or statistical character, pertaining to individual phenomena ... .

Thus, although quantum theory produces statistical predictions, its observables generate individual phenomena. Discreteness of detection events is the fundamental feature of quantum physics justifying existence of quantum systems, carriers of quanta. Hence, quantum mechanics provides the discrete representation of nature which is constructed via discrete events of observations. This issue is not so trivial and in modern foundational literature it is often completely ignored. Operating with phenomena, discrete detection events, instead of continuous signals, is in fact one of the main foundational issues of quantum mechanics. This theme is important, e.g., in distinguishing “classical” and “genuine quantum” entanglements [288, 292]. The former is entanglement of the degrees of freedom of the classical electromagnetic field [320, 418, 419], the latter is entanglement of the degrees of freedom of quantum systems, e.g., atoms or photons.4 Otherwise 4

May be the expression “degrees of freedom” is misleading; may be it is better to speak about entanglement of observables.

6.11 Ontic and Epistemic Portrayals of Mental Processes

87

to explain this difference one should appeal to the ambiguous notion of quantum nonlocality (spooky action at a distrance) [418, 419] This issue is closely related to distinguishing of quantum and quasi-classical theories, see Grangier et al. [180] for such experiments which are known as the “photon existence experiments”. Ground and Upper Levels of Brain’s Functioning As was already mentioned, we proceed with the two-level structuring of scientific theories, the ontic-epistemic one. From Bohr’s viewpoint, quantum mechanics is an epistemic theory, and it is important to add “of discrete (quantized) observations.” The electrochemical processes belong to the ontic level – physical reality of brain’s functioning. Quantum information processing belongs to the epistemic level, i.e., brain’s mental functions represent their knowledge about the surrounding environment within the quantum information framework. This is the good place to repeat once again that the quantum information representation does not need to be related to quantum physics in the brain. The quantum information states (superpositions of the classical states of a neural code) can be generated by macroscopic electrochemical waves propagating in the brain. In fact, we use the quantum information counterpart of quantum theory. However, by following the writings of the fathers of quantum theory, especially Bohr [74] and Schrödinger [407], we immediately understand that they also used the information interpretation (well, without using the notion “information). This interpretation was further developed in the works of Zeilinger and Brukner [86], Plotnitsky [379], D’Ariano [110], Fuchs and Schack [165]. In fact, QBism is so to say the extreme version of the information interpretation (the epistemic viewpoint on quantum theory), QBists highlighted the individual agent perspective on measurement’s outcomes. As was noted in Chap. 1, in quantum-like modeling QBism’s viewpoint is very attractive [192]. In this chapter, the role of an individual agent is played by a mental function. Our model suggested the two-level description of brain’s functioning: • The ground level: electrochemical processes in the brain and body. • The upper level: quantum information processing of superpositions of neural code’s states. Genuine Quantum Physical Processes in the Brain In principle, we do not reject the possibility that quantum physical processes in the brain, say at cells’ membranes, can contribute to generation of the quantum information states. Here, one should distinguish two different sorts of quantum theoretical description: • quantum physical theory at the micro-scale (the ontic level), • quantum information theory at the macro-scale (the epistemic level). Both portrayals are based on the same mathematical formalism, the quantum formalism of complex Hilbert state space. I see one crucial difference between these two quantum portrayals. Quantum physical theory is based on infinite dimensional Hilbert spaces of the L 2 -type; quantum information theory is based on finite dimensional n-qubit spaces.

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Ontic and Epistemic Levels of Quantum Information Representation The outputs of a mental function F are determined by subspaces H f ;G : F = f for |ψ ∈ H f ;G (with probability one). In the operator terms, F is described by projections (Q f ) on the subspaces (H f ;G ). These projections determine the outputs’ probabilities In the von Neumann formalism, decomposition of projections Q f into one dimensional projections (|γ γ |) corresponding to the F-basis (|γ ), Qf =



|γ γ |,

(6.36)

|γ ∈H f ;G

does not play any role; it is hidden even from F which cannot approach its own internal structure; a mental function self-observes only the outputs F = f. However, decision-making via differentiation (Sect. 6.8) explores the basis (|γ ). This basis determines the hidden structure of F which is not visible for F by itself, for F as a self-observer. We can interpret vectors (|γ ) as hidden variables and modeling of functioning of F in terms of these vectors as ontic. The portrayal of F in terms of projections (Q f ) is epistemic, it is about knowledge which F extracts from (quantum information) states of the neural network G. We emphasize that both these descriptions are quantum. This is interesting, since typically the quantum description is epistemic and the ontic one is classical (see Atmanspacher [32], Khrennikov [255]. The primary level of onticity given by action potentials, onticity of the electrochemical processes in the brain (and generally in human’s body). The γ -basis description is epistemic w.r.t. the electrochemical one. Thus, our framework has the multilevel ontic-epistemic structure. Each lower level is treated as ontic w.r.t. the next higher level which in turn is treated as ontic for the next higher level. To call the γ -portrayal epistemic, we should point out to some observer who “sees” the γ -structure of the (quantum information) states of G. Who is it? A mental function F operates (as an observer) in the conscious domain of brain’s mental processing. We can speculate that γ -representation corresponds to operation in the unconscious domain. Transformation of action potentials into the neural code basis (|α) and then into (|γ )-basis happens without coupling to consciousness. And the process of state differentiation (Sect. 6.8) w.r.t. (|γ )-basis also happens unconsciously. We can imagine that each F (treated as a self-observer) is based on an unconscious quantum processing device described as a quantum channel. And the latter is determined by the γ -basis. As is typical in ontic-epistemic structuring of scientific theories, the same epistemic model may have a few different ontic models beyond it. In the above quantum ontic-epistemic framework, the selections of different bases in the subspaces H f ;G correspond to the use of different hidden variables (ontic structures). These structures correspond to the different signatures of the environments (Sect. 6.8.1) of the neural network G – the neural basis of F = FG .

6.12 Concluding Discussion on Quantum-Like Modeling of Brain’s Functioning

89

6.12 Concluding Discussion on Quantum-Like Modeling of Brain’s Functioning Quantum information revolution stimulated essentially applications of the quantum formalism to model cognition and decision-making. Generally such modeling is not based on real quantum physics; the brain is considered as a black box processing information in accordance of the laws of quantum information and probability. The natural problem of coupling of the quantum-like models with the electrochemical processes in the brain arises. Following [284, 291], we proceed toward solving this problem with the two-level model, with classical electrochemical (ontic) level and quantum-like (epistemic) level. In this model, uncertainty in generation of spikes is transformed into quantumlike superposition. The main idea is that the brain is able to transfer the nonlinear dynamics of the Hodgkin–Huxley type [206] into linear quantum-like dynamics. At the level of coding, this means that a classical neural code is extended to include superposition states. By moving from the nonlinear dynamics of the electrochemical processes in neural networks, the brain is able to escape chaotic behavior. The latter is characteristic for nonlinear dynamical systems, including the Hodgkin–Huxley differential equation [4, 5, 207]. We realize a mental function as a quantum-like observable. Here we put the emphasis to the evident fact that the brain is an open system. Since the theory of open quantum systems is the most general mathematical theory of open systems, it is natural to apply it for modeling of brain’s functioning (see [18, 19, 24, 26, 27, 284, 291] and [397, 447, 448]). We analyze advantages and disadvantages of the canonical quantum measurement scheme based on the projection postulate as well as the open system model based on the quantum dynamical equation (with Markovean and non-Markovean dynamics). Finally, we presented a very general scheme (differentiation [28]) for transformation of quantum-like superpositions corresponding to action potentials’ uncertainty into classical statistical mixtures of decision states. The use of the quantum measurement theory to model brain’s functioning leads to the fundamental foundational problem of external versus internal (self-) observations. We briefly discussed this problem in Sect. 6.9 in connection with information autopoiesis in brain’s functioning. This problem is far from clarification.

Chapter 7

Emotional Coloring of Conscious Experiences

7.1 Quantum Formalization of Emotional Coloring of Conscious Experiences We start with the following citation from [328]: Although emotions, or feelings, are the most significant events in our lives, there has been relatively little contact between theories of emotion and emerging theories of consciousness in cognitive science.

We want to formalize this contact with the aid of the quantum-like theory of consciousness [268].1 We apply the mathematical model of cooperative work of unconsciousness and consciousness to describe the process of emotional coloring of perceptions and more general conscious experiences including decision-making. The brain performs self-measurements. To model such self-measurements, we split the brain, as an information processor, into two subsystems, unconsciousness UC and consciousness C. The former plays the role of a system under observation and the latter of an observer [268].2 To model the cooperation of perceptions and emotions, the state space of UC is decomposed into the tensor product of corresponding state spaces, (7.1) HUC = Hper ⊗ Hem . Two classes of observables Oper and Oem are invented: for conscious experiencing of perceptions and emotions, respectively. These are observables acting on the state spaces Hper and Hem . In this sense these are “psycho-local observables”. However, the tensor product of a perception-observable and an emotion-observable acts on 1

Here we follow article [302]. See [6, 141, 229, 243] for mathematical modeling of join functioning of unconsciousness and unconsciousness based on treelike geometry of the brain.

2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_7

91

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7 Emotional Coloring of Conscious Experiences

HUC and the entanglement effect of the unconscious states coupled to perceptions and emotions plays the important role. In principle, it can be checked experimentally (see Sect. 7.9). The domain of applicability of our quantum-like model is not restricted to emotional coloring of perceptions; we present the very general scheme of coloring of one class of conscious experiences with another, “basic experiences” are colored with “supplementary experiences”. The aim and the origin of such coloring will be discussed below. Generally, we operate with the tensor decomposition HUC = Hbas ⊗ Hsup .

(7.2)

And the two classes of observables Obas and Osup are invented. Emotional coloring is coupled to quantum contextuality—emotion observables determine contexts for perceptions and other basic conscious experiences which are also treated as observables. We highlight that contextualization reduces degeneration of spectra for observables belonging to the set Obas and acting in the space of unconscious states HUC . Such contextual reduction is very important, since the unconsciousness state space HUC has high dimension. Each conscious experience a = x is based on multidimensional eigensubspace Hx of HUC . We follow Chap. 17 and treat contextuality very generally, as Bohr contextuality. We recall that the Bohr contextuality-complementarity principle leads to rejection of ”naive realism”. Observable’s outcomes cannot be interpreted as the objective properties of systems. The observed value cannot be assigned to a system before measurement; it is created in the complex process of interaction between a system and a measurement apparatus.

7.2 The First and Higher Order Theories of Consciousness There are two basic and competing theories of consciousness: • the First Order Theory of Consciousness [70, 71, 129, 324, 432]; • the Higher Order Theory of Consciousness [97, 326, 396]. We characterize these theories with the following citation from [328]: First-order theorists, such as Block, argue that processing related to a stimulus is all that is needed for there to be phenomenal consciousness of that stimulus [70, 71, 129, 324, 432]. Conscious states, on these kinds of views, are states that make us aware of the external environment. Additional processes, such as attention, working memory, and metacognition, simply allow cognitive access to and introspection about the first-order state. In the case of visual stimuli, the first-order representation underlying phenomenal consciousness is usually said to involve the visual cortex, especially the secondary rather than primary visual cortex. Cortical circuits, especially involving the prefrontal and parietal cortex, simply make possible cognitive or introspective access to the phenomenal experience occurring in the visual cortex.

7.2 The First and Higher Order Theories of Consciousness

93

In contrast, David Rosenthal and other higher-order theorists argue that a first-order state resulting from stimulus-processing alone is not enough to make possible the conscious experience of a stimulus. In addition to having a representation of the external stimulus one also must be aware of this stimulus representation. This is made possible by a HOR, which makes the first-order state conscious. In other words, consciousness exists by virtue of the relation between the first- and higher-order states. Cognitive processes, such as attention, working memory, and metacognition are key to the conscious experience of the first-order state. In neural terms, the areas of the GNC, such as the prefrontal and parietal cortex, make conscious the sensory information represented in the secondary visual cortex.

The Higher Order Theory distinguishes between unconscious and conscious processing of mental information in the brain. By this theory, what makes cognition conscious is a higher-order observation of the first-order processing. And in quantum theory observation is not simply inspection of the system’s state. This is a complex process of interaction between a system and a measurement device. This is the good place to cite Bohr again [74] (see Chap. 17): This crucial point ... implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled.

This viewpoint matches better with the Higher Order Theory of Consciousness. A conscious experience is not simply introspection of the UC-state. In this chapter, we proceed without describing the process of UC − C interaction. In particular, we do not operate with the states of C and we do not appeal to the scheme of indirect quantum measurements (Sect. 8.4). We neither apply the general theory of quantum instruments, but use only the observables given by PVMs (Sect. 2.1). We remark that, although Bohr’s viewpoint on the outcomes of quantum measurements dominates in QM, a few respectable scientists claimed that these outcomes can be considered as the objective properties of physical systems. They treat quantum measurements as just approaching the premeasurement values of observables, the objective properties of the systems under observation. This position was presented in the paper of Einstein, Podolsky, and Rosen (EPR) [142]. Later Bell elaborated (and modified) EPR’s argument [60, 62], but he confronted the problem of nonlocality (Chap. 17). This line of thought matches better to the First Order Theory of Consciousness. We will follow Bohr’s line of thought [73] and hence couple quantum measurement theory with the Higher Order Theory of Consciousness. In application to emotions, the First Order Theory of Consciousness matches the somatic theories of emotions rooted to James [215]; the first of them is James-Lange theory [96]. Nowadays this viewpoint on emotions is advertised by some prominent neuro-physiologists, e.g., Damasio [33].

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7.3 Contextuality of The Higher Order Theory of Consciousness The quantum-like model of this paper formalizes the general scheme of the Higher Order Theory of Consciousness, as the mathematical framework for generation of conscious experiences as the outcomes of measurements performed by consciousness over unconsciousness and based on interaction between them. Besides formalization of the general scheme, the model describes the concrete feature of the Higher Order Theory of Consciousness—emotional contextualization of conscious experiences such as perceptions and decisions. This model highlighted contextuality of information processing by the brain in the concrete framework of emotional coloring. Contextuality can be tested experimentally, as was done in both in physics and cognition with tests based on the Bell-type inequalities (see Chap. 17). Contextuality is a delicate property of cognition, and it is one of the signatures of indeterminacy of conscious experiences generation process. Indeterminacy mathematical characterization is the impossibility to define JPD for some observables. In quantum physics such observables are called incompatible ones. Their existence is a consequence of Bohr’s contextuality-complementarity principle. Mathematically this principle is supported by the rules of quantization: classical variables with nontrivial Poisson brackets are transformed into quantum observables with nontrivial commutators. Indeterminacy of conscious experiences is well accepted in cognitive studies. Therefore, one may ask why contextuality has not been widely tested in experimental research? The recent interest to contextual studies, both theoretical and experimental, diffused to cognition from quantum theory. The possible answer is that the basic experimental protocols in cognitive studies are grounded on one fixed complex of experimental conditions, but to check contextuality, an experiment has to involve at least three, but better even four, different experimental complexes. The majority of experiments were not done in such multi-contextual framework.

7.4 Perceptions and Emotions Perception Representation of Sensations We follow to von Helmholtz [449] theory of sensation-perception. Perceptions are not simply copies of sensations, not “impressions like the imprint of a key on wax”, but the results of complex signal processing including unconscious cognitive processing. In the modern science formulation, the process of perception creation can be described as follows [354]: Sensory information undergoes extensive associative elaboration and attentional modulation as it becomes incorporated into the texture of cognition. This process occurs along a core

7.4 Perceptions and Emotions

95

synaptic hierarchy which includes the primary sensory, upstream unimodal, downstream unimodal, heteromodal, paralimbic and limbic zones of the cerebral cortex.

Contexts Representation by Emotions As is emphasized in [328], Emotion schema are learned in childhood and used to categorize situations as one goes through life. As one becomes more emotionally experienced, the states become more differentiated: fright comes to be distinguished from startle, panic, dread, and anxiety.

In our terminology, each emotion-generation scheme is crystallized on the basic life-contexts. Context-labeling is the basic function of emotions. Surrounding environment contextualization was one of the first biosystems cognitive tasks, and this ability was developed in parallel with the sensation-perception system’s establishment. Memory is heavily involved in emotional activity, both for memorizing the contexts’ features and for comparison of new perceptions with these contexts. (See, e.g., [329] for the memory’s role in cognition). Thus, evolutionary a mental information processing system representing the basic life-contexts was designed. This system is fixed at the level of the brain, and more generally, the nervous system, hardware. But, memorizing a contextual experiences’ variety is done on the basis of the experiencing various situations (see above citation from [328]). This context-refection system was the root of the present emotion-system in humans. The latter has complex cognitive functions, not only contextual. However, in this paper we concern mainly contextuality. Emotions represent adaptive reactions to environmental challenges; they are a result of human evolution; from the viewpoint of computational resources they provided optimal solutions to ancient and recurring problems that faced our ancestors [143].3 We emphasize that in our model emotions are conscious, cf. [328]: One of our view’s implication is that emotions can never be unconscious. Responses controlled by subcortical survival circuits that operate unconsciously sometimes occur in conjunction with emotional feelings but are not emotions. An emotion is the conscious experience that occurs when you are aware that you are in particular kind of situation that you have come, through your experiences, to think of as a fearful situation. If you are not aware that you are afraid, you are not afraid; if you are not afraid, you aren’t feeling fear.

3

Although we do not follow the James-Lange theory of emotions [96], this is the good place to mention that James [215] pointed out that “feeling of the same changes as they occur is the emotion. Here, for us the key words are “the same changes as they occur”, i.e., the complex of repeatable bodily changes—physiological encoding of a context.

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7.5 Unconscious and Conscious Information Processing 7.5.1 Unconsciousness as System An essential part of information processing in the brain is performed unconsciously; the information system responsible for such processing, call it unconsciousness, is denoted by the symbol UC. The space of its states is denoted by H ≡ HUC . In the quantum-like model, this is a complex Hilbert space. The reader does not need couple the notion of unconsciousness with the names of James [216], Freud [155], and Jung [219] (although the author of this paper was strongly influenced by them, cf. with the previous works [6]–[229]). In this book, UC denotes a special collection of information processors of the brain performing pre-observational mental state processing.

7.5.2 Consciousness as Observer Perceptions and emotions are commonly treated as conscious entities. So, in our model the brain contains another information processing system generating conscious experiences; denote it by the symbol C. In our quantum-like framework, its functioning is modeled as performing measurements on the system UC. Introduction of two systems UC and C matches the quantum measurement scheme, UC is the analog of a physical system exposed to measurements and C is the analog of a complex of measurement apparatuses. We note that unconscious-conscious modeling of the brain’s functioning matches well to the philosophic paradigm of ontic-epistemic structuring of scientific theories. The ontic level is about both physical and mental reality as it is if nobody observes it, the epistemic level describes observations (see Atmanspacher [32]). In contrast to quantum physics, in our model the epistemic level is related not to external observers, but to brain’s self-observations. In the brain, unconscious and conscious processes are closely coupled and the sharp separation between them is impossible (cf. Brenner [80] and Brenner and Igamberdiev [81].

7.5.3 Unconscious and Conscious Generation of Perceptions and Emotions In this paper, we shall be mainly concentrated on functioning of two information processors transforming • sensations → perceptions, • contexts → emotions.

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97

Both processors have conscious outputs. Their functioning is strongly correlated; in the formalism of quantum theory correlations are represented by entangled states. We denote unconscious counterparts of these processors by the symbols UCper and UCem , respectively. In modeling of the emotional coloring of perception (its contextualization), we shall consider the compound information system (UCper , UCem ). This is the good place to mention the first theory of emotions, the James-Lange theory [96]. James claimed that “I am trembling. Therefore I am afraid.” He stated: “My thesis ... is that the bodily changes follow directly the perception of the exciting fact and that our feeling of the same changes as they occur is the emotion.” This way of thinking matches with the First Order Theory of Consciousness and the EPR-Bell viewpoint on quantum measurements. Following LeDoux [329] (see also [328]), we treat emotions within the Higher Order Theory of Consciousness coupled to Bohr’s interpretation of quantum measurements.

7.5.4 Conscious Experiences: Basic and Supplementary Although we are mainly interested in perceptions’ emotional coloring, the formalism under consideration can be applied to the very general class of compound information processing systems, (UCbas , UCsup ). The latter is used for determining stable repeatable and evolutionary fixed contexts for the former. Simplest generalization of the perception-emotion scheme is emotional contextualization of decision-making which modeling is based the compound system (UCdm , UCem ). Generation of conscious experiences (basic and supplementary) is modeled with quantum observables; denote the corresponding classes by the symbols Obas and Osup . In particular, we shall consider the pairs (Oper , Oem ) and (Odm , Oem ). An expert in cognition may suggest other pairs of basic and supplementary conscious experiences. One of the properties of the supplementary mental experiences is their rapid processing. They wouldn’t inhibit the basic mental experiences processing. We remark that emotion’s generation is characterized by high speed (see [329]).

7.6 Incompatible Conscious Observables We stress that the space of observables Oper can contain incompatible perceptions as well as Oem can contain incompatible emotions. In physics incompatibility is often seen as the exotic property of quantum theory—comparing with classical physical theory. Philosophically, this notion is formalized in the Bohr’s contextualitycomplementarity principle. In the mental framework, the notion of incompatibility

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can be interpreted very naturally. There exist incompatible emotions which cannot be experienced simultaneously; say happiness and sadness, pride and shame. And it is also evident that there exist incompatible, i.e., jointly unobservable, perceptions and other conscious experiences, say hot and cold. We stress once again that mental observations are brain’s self-observations. This self-observational structure clarifies the incompatibility issue. The necessity to operate with various incompatible conscious entities, or mental observables, is the main root of the quantum-like information representation. In the absence of incompatibility, i.e., if, for the same mental state, the brain was able to construct the consistent probabilistic representation in the form of JPD of all possible combinations of say emotions, the quantum state formalism would be unnecessary.

7.7 Degeneration Resolution of Conscious Experiences via Contextual Coloring We start with presentation of the general scheme for the resolution of degeneration of the C-outcomes. In this scheme C operates with two classes of observables Obas and Osup representing the “basic and supplementary conscious experiences”, respectively. General Scheme In the quantum-like model of conscious experience generation [268], consciousness C is modeled as a system performing observations over unconsciousness UC. As in Sect. 7.5.1, the symbol H denotes the space of UC-states. A conscious observable a is mathematically described as a PVM-observable (Sect. 2.1) E a = (E a (x), x ∈ X ), where X is the set of conscious experiences. The latter can be, for example, the set of language expression or visual images. The values of an observable are determined by the subspaces Hxa (see Postulate 2QL, Chap. 10, Sect. 10.2). If consciousness C detects a state belonging to Hxa , then it feels the conscious experience x ∈ X. The space of unconscious states H has high dimension. In reality, it is infinite dimensional, since this is quantum information representation of the electrochemical waves in the brain (see [284] for details). We restrict modeling to the finitedimensional case; state spaces have very high dimension, dim H >> 1. If the “conscious-experience vocabulary” X (for observable a) is not so large, i.e., number of points in set X p(By) ≥ p(Bn) ≥ p(An), and satisfy QQE (10.71). Then, there exist input joint probabilities q(0), q(1), q(2) and p(0, 0), p(0, 1), p(1, 0), p(1, 1) uniquely such that the model outputs the joint probabilities { p(Ab, Ba), p(Bb, Ab) | a, b = y, n} if and only if the following conditions hold: (i) p(Ay By) ≥ p(By Ay), (ii) p(Ay Bn) ≥ p(Bn Ay), p(Ay Bn) − p(By An) p(Ay Bn) − p(Bn Ay) (iii) ≥ , p(Ay) − p(By) p(Ay) − p(Bn) p(Ay By) p(By Ay) ≥ , (iv) p(By) p(Ay) p(Ay Bn) p(Bn Ay) ≥ , (v) p(Bn) p(Ay) p(An By) p(By An) (vi) ≥ , p(An) p(By)

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(vii)

p(Bn An) p(An Bn) ≥ . p(An) p(Bn)

Proof Necessity: Relations (i) and (ii) follow from (10.108) and (10.109) with the assumption p(Ay) ≥ p(By) ≥ p(Bn). Relation (iii) follows from relations (10.108), (10.109) and 1 − q(1) ≥ q(2); note that the assumption p(Ay) > p(By) implies the relation p(Bn) > p(An). From (10.100) and (10.104), we have q(0) p(1, 1) p(Ay By) = + q(2), p(Ay) p(Ay) p(By Ay) q(0) p(1, 1) = + q(2), p(By) p(By) and hence relation (iv) follows from the assumption p(Ay) > p(By) and p(1, 1) ≥ 0. Relations (v), (vi), (vii) follow similarly from conditions q(0) p(1, 0) ≥ 0, q(0) p(1, 0) ≥ 0, and q(0) p(0, 0) ≥ 0. Sufficiency: Suppose that q(0), q(1), q(2), p(0, 0), . . . , p(1, 1) satisfy (10.100), …, (10.107), and q(0) + q(1) + q(2) = 1. It suffices to prove that they are nonnegative. We have p(2) ≥ 0 by relation (i) and (10.108). Similarly, p(1) ≥ 0 follows from relation (ii) and (10.109), and q(0) ≥ 0 follows from relation (iii) and q(0) + q(1) + q(2) = 1. Then, p(0, 0) ≥ 0 follows from relation (iv) and (10.100). The non-negativity of p(1, 0), p(0, 1), p(0, 0) follows similarly. 

10.9 Modeling Statistical Data from Clinton–Gore Poll In this section, we shall show that the well-known data from Clinton–Gore Poll [359] can be reproduced within ±0.75% of errors from our model. Consider the following data from Clinton–Gore experiment [359, 452, 453]. p(Ay By) = 0.4899,

(10.120)

p(Ay Bn) = 0.0447, p(An By) = 0.1767, p(An Bn) = 0.2887,

(10.121) (10.122) (10.123)

p(By Ay) = 0.5625, p(By An) = 0.1991,

(10.124) (10.125)

p(Bn Ay) = 0.0255, p(Bn An) = 0.2129.

(10.126) (10.127)

QQE is approximately satisfied with good accuracy.

10.9 Modeling Statistical Data from Clinton–Gore Poll

153

q = [ p(By Ay) + p(Bn An)] − [ p(Ay By) + p(An Bn)] = [ p(Ay Bn) + p(An By)] − [ p(By An) + p(Bn Ay)] = −0.0032.

(10.128)

Thus, their QQE-renormalization ar p(Aa, Bb), ar p(Bb, Aa) are expected to approximate the original data p(Aa, Bb), p(Bb, Aa) with good accuracy. For the Clinton–Gore poll, we have p(Ay By) + p(An Bn) + p(By Ay) + p(Bn An) = 0.7770, 2 p(Ay Bn) + p(An By) + p(By An) + p(Bn Ay) S2 = = 0.2230, 2 S1 =

and we obtain their QQE-renormalizations as follows. p(Ay By) = 0.4889, p(Ay By) + p(An Bn) ar p(Ay By) − 1 = −0.0021. (−0.21%), p(Ay By) p(An Bn) = 0.2881, S1 × p(Ay By) + p(An Bn) ar p(Ay Bn) − 1 = −0.0021. (−0.21%), p(Ay Bn) p(By Ay) = 0.5637, S1 × p(By Ay) + p(Bn An) ar p(By Ay) − 1 = 0.0021. (+0.21%), p(By Ay) p(Bn An) = 0.2133, S1 × p(By Ay) + p(Bn An) ar p(Bn An) − 1 = 0.0020. (+0.20%), p(Bn An) p(Ay Bn) = 0.0450, S2 × p(Ay Bn) + p(An By) ar p(Ay Bn) − 1 = 0.0072. (+0.72%), p(Ay Bn) p(An By) S2 × = 0.1780, p(Ay Bn) + p(An By) ar p(An By) − 1 = 0.0072. (+0.72%), p(An By) p(By An) S2 × = 0.1977, p(By An) + p(Bn Ay) ar p(An By) − 1 = −0.0071. (−0.71%), p(An By)

ar p(Ay By) = S1 × Error = ar p(An Bn) = Error = ar p(By Ay) = Error = ar p(Bn An) = Error = ar p(Ay Bn) = Error = ar p(An By) = Error = ar p(By An) = Error =

(10.129) (10.130)

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p(Bn Ay) = 0.0253, p(By An) + p(Bn Ay) ar p(Bn Ay) Error = − 1 = −0.0075.(−0.75%). p(Bn Ay)

ar p(Bn Ay) = S2 ×

Under the assumption of the independence of the personality state, from Eq. (10.108) and the relation q(0) + q(1) + q(2) = 1, we can determine q(0), q(1), q(2) by the QQE-normalized data as follows. q(2) = 0.3288,

(10.131)

q(1) = 0.0668, q(0) = 0.6045.

(10.132) (10.133)

We can determine p(α, β) for α, β = 0, 1 by Eqs. (10.100)–(10.107) as follows. p(1, 1) = 0.5184, p(1, 0) = 0.0155, p(0, 1) = 0.2429,

(10.134) (10.135) (10.136)

p(0, 0) = 0.2231.

(10.137)

Note that the unity of the total probability is satisfied. p(1, 1) + p(1, 0) + p(0, 1) + p(0, 0) = 1.0000. Thus, the belief state p(α, β) and the personality state q(γ ) are determined from the experimental data. Then, our quantum model with the belief state p(α, β) and the personality state q(γ ) accurately reconstructs the QQE renormalized data ar p(Aa, Bb), ar p(Bb, Aa) for a, b = y, n as follows. ar p(Ay By) = p(1, 1)q(0) + [ p(1, 1) + p(1, 0)]q(2) = 0.4889, ar p(Ay Bn) = p(1, 0)q(0) + [ p(1, 1) + p(1, 0)]q(1) = 0.0450, ar p(An By) = p(0, 1)q(0) + [ p(0, 1) + p(0, 0)]q(1) = 0.1780, ar p(An Bn) = p(0, 0)q(0) + [ p(0, 1) + p(0, 0)]q(2) = 0.2881, ar p(By Ay) = p(1, 1)q(0) + [ p(0, 1) + p(1, 1)]q(2) = 0.5637, ar p(By An) = p(0, 1)q(0) + [ p(0, 1) + p(1, 1)]q(1) = 0.1977, ar p(Bn Ay) = p(1, 0)q(0) + [ p(0, 0) + p(1, 0)]q(1) = 0.0253, ar p(Bn An) = p(0, 0)q(0) + [ p(0, 0) + p(1, 0)]q(2) = 0.2133. Therefore, all data of the QQR-renormalizations are accurately reproduced, and we conclude that our quantum model reproduces the statistics of the Clinton– Gore Poll data almost faithfully (within ±0.75% of error) with a prior belief state

10.10 On Postulate 5QL

155

{ p(0, 0), . . . , p(1, 1)} independent of the question order. Thus, this model successfully corrects for the order effect in the data to determine what in the model is the genuine distribution of the opinions.

10.10 On Postulate 5QL As we know, a density operator ρ can be decomposed into probabilistic mixtures of pure states in various ways. We show that Postulate 5QL is invariant with respect to such decompositions. Suppose we conduct an experiment on 100 people. Then, we can suppose the subject j = 1, ...100 is in a pure state ψ j and the ensemble is described by the mixed state ρ1 = (1/100)  j |ψ j ψ j |. We conduct another experiment on 200 people and let and ρ2 = (1/200)  k |φk φk |. The third experiment is conducted on 300 people, and ρ3 = (1/300) l |ξl ξl |. Then, P(Aa Bb|ρ1 ) = (1/100)



P(Aa Bb|ψ j ), P(Bb Aa|ρ1 ) = (1/100)

j

P(Aa Bb|ρ2 ) = (1/200)

P(Bb Aa|ψ j ),

j



P(Aa Bb|φ j ), P(Bb Aa|ρ2 ) = (1/200)

j

P(Aa Bb|ρ3 ) = (1/300)







P(Bb Aa|φ j ),

j

P(Aa Bb|ξ j ), P(Bb Aa|ρ3 ) = (1/300)

j



P(Bb Aa|ξ j ).

j

It follows that if pρ1 + p  ρ2 = ρ3 , where p, p  > 0, p  = 1 − p. Then p P(Aa Bb|ρ1 ) + p  P(Aa Bb|ρ2 ) = P(Aa Bb|ρ3 ). This is consistent with P(Aa Bb|ρ) = Tr[I B (b)I A (a)ρ]. This also holds for any longer sequences and the joint probability distribution depends only on ρ, it is independent of its decomposition into pure states, orthogonal or not.

Part IV

Analysis of Social Systems within Open Quantum System Theory

This part is devoted to the modeling of social behavior with the methodology and the mathematical apparatus of the quantum field theory. The basic field under consideration is the social information field. Its excitations, infons, are analogs of photons. They carry socially valuable information emitted by mass media and social networks. Humans are treated as social analogs of atoms. They interact with infons and absorb quantized social energy contained in the information. This framework is applied for the development of social laser theory. This theory describes the coherent emission of spikes of social energy—social tsunamis. Nowadays, social lasers are actively explored in social engineering, e.g., in color revolutions, mass protests, and the generation of coherent collective decisions. We also consider the social analog of Fröhlich condensation. Here, societal stability is approached at a high social temperature with the aid of a continuous energy supply and its redistribution between people, from highly energetic individuals to less energetic ones. The crucial role is played by the presence of the social bath absorbing a part of energy in the process of its redistribution. The same model can be applied to stability modeling not only in human societies, but even in groups of animals, i.e., wolf packs. Social networks contribute both to social lasing and Fröhlich condensation. Internet-based Echo Chambers are the basic components of social laser resonators. It is amazing that the network-based social lasers are more effective than lasers based on disconnected elements, e.g., physical or social atoms.

Chapter 11

Social Laser

11.1 Social-Information Waves Shaking the World During the last years, the grounds of the modern world were shaken by the coherent information waves of very high amplitude. The basic distinguishing property of such waves is that they carry huge amount of social energy. Thus, they are not just the waves, widely distributing some special information content throughout the human society. In contrast, their information content is surprisingly small. Typically, it is reduced to one or a few labels, we can say “quasi-colors”. At the same time information waves carry very big emotional charge, a lot of social energy. So, they can have strong destructive as well as constructive impact on the human society. I suggested [267] to model generation of very powerful and coherent socialinformation waves within social laser theory which was later developed in collaboration with a few coauthors (see [12, 269, 276, 277, 287, 289, 290]). At the beginning of its creation, this theory was oriented toward modeling Stimulated Amplification of Social Actions (SASA) such as color revolutions and other mass protests around the world (cf. with socio-political studies, e.g., [79, 147, 321, 345, 358, 403, 429, 433]). However, later it became clear that the social laser theory has essentially wider domain of applications. And we will demonstrate this in the present chapter. We remark that one of the creators of the laser theory, Haken, considered laserequations to illustrate the mathematical analogy with self-organization processes in complex physical, biological, and social systems [184–186]. Our aim is different. We want to formalize quantum-like properties of social-information systems which can lead to SASA. The basic entities of social laser theory are social energy, atoms, and fields [425], [267], , [65, 336, 377, 389] (cf. also with [155, 216, 219, 220]). Social atoms represent humans. They exchange quanta of social energy with the social-information field which is composed of excitations carried by communications massively emitted by mass media and social networks. The lasing scheme can be formulated with these entities as the process of social energy pumping in a human gain medium and then stimulated emission of a cascade of social actions. The latter is understood very gen© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_11

159

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erally as actions both in physical and social-information spaces: mass protests, color revolutions, wars as well as collective decisions on the important societal problems. Article [307] was the important methodological step toward approaching closer similarity with quantum physics. There were invented the social-information analogs of photons which are called infons. These are excitations of the quantum socialinformation field carrying social energy and coarse-grained content of communications (their color and quasi-color). We also analyze the dynamics of iterations of the cascades of infons in the social resonators. The latter is based on social networks coupled to laser’s gain medium composed of social atoms. The special attention is paid to the role of Echo Chambers. They increase color and quasi-color coherence of the social-information field. Our theory formalizes the basic conditions for generation successful social lasing [277]: • Indistinguishability of social atoms. The human gain medium, population exposed to the social-information radiation, should be composed of social atoms— “creatures without tribe”: the role of national, cultural, religious, and even gender differences should be reduced as much as possible (at least during the lasing period). • Indistinguishability of infons (content-ignorance). Social atoms process information communications without deep analyzing of their contents; they extract only the basic labels-“quasi-colors”, encoding the communications’ contents (“clip thinking”, “popcorn brain”). In formal terms, infons carrying the same color (social energy) and quasi-color (rough content) are indistinguishable. Of course, humans are still humans, not social atoms; thus, in contrast to quantum physics, it is impossible to create human gain mediums composed of completely indistinguishable creatures. People still have say names, gender, nationality, but such characteristics are ignored in the regime of social lasing.1 In the same way, information communications still can be distinguished if one analyzes them deeply. However, social atoms who are overloaded by information cannot do this. The indistinguishability condition leads to quantum statistics, Bose–Einstein, or Fermi–Dirac, or parastatistics (Chap. 18). Quantum Bose–Einstein statistics of infons is the basis for generation of social-information cascades. The important components of lasers, both physical and social, are resonators [277], optical cavities versus social networks. Such devices play the double role: • amplification of the beam of physical vs. social-information radiation; • improving coherence of this beam. Social laser resonators play the crucial role in generation of coherent information waves of high amplitude. They are based on the Internet Echo Chambers associated with social networks, blogs, and YouTube channels. Their functioning is based on 1

Indistinguishability is definitely context dependent, i.e., we discuss creation of contexts which are stimulating for social lasing. In such social, psychological, political, and economic conditions, people behave as social atoms.

11.1 Social-Information Waves Shaking the World

161

the feedback process of posting and commenting, the process that exponentially amplifies the information waves that are initially induced by mass media (see also Chap. 12). Echo Chambers improve the coherence of the information flow through the statistical elimination of communications that do not match the main stream. This statistical elimination is a consequence of the bosonic nature of the quantum information field. We compare functioning of the optical and information mirrors. The latter represents the feedback process in the internet systems such as, e.g., YouTube. In contrast to the optical mirror, the information mirror not only reflects excitations of the quantum information field, but also multiplies them. Thus, this is a kind of reflector-multiplier. As the result of this multiplication effect, social resonators are more effective than physical ones. However, as in physics, resonator’s efficiency depends on a variety of parameters. One of such parameters is the coefficient of reflection-multiplication (Sect. 11.9). We analyze the multilayer structure of an information mirror and dependence of this coefficient on the layer (Sect. 11.9). In the previous studies, social laser theory was mainly applied for modeling destructive processes, social tsunamis, e.g., in the form of mass protests. Collective decisions were typically societal responses to emergency situations and negative emotions as fear, angst, and often aggression were basic for the process of social energy pumping preceding collective decision-making in the form of stimulated emission. So, even for lasing focused to a good goal, social energy pumping preceding emission was not based on love between human beings. Now we analyze the possibility to create “a societal benefit laser”, i.e., directed for societal good and not involving negative emotions, instincts, and feelings (Sect. 11.8). In particular, we discuss the possibility of creation of anti-war laser based not on angst of mutual destruction or nuclear apocalypses, but on care, compassion, attention, cordiality, forgiveness, friendliness, humanism, kindness, mutual understanding, cordiality, and brotherhood. Unfortunately, we came to the pessimistic conclusions. The role of social networks as lasing resonators is illustrated by the massive protests during COVID-19 pandemic (Sect. 11.8; cf.,e.g., [104, 445]). We discuss creation of the competing beams of social-information radiation and the conditions for their creation and coexistence, e.g., pro-democratic and prorepublican beams during election campaigns in USA or the pro-war and pro-peace beams generated since February 2022 in mass media and Internet resources. The role of social networks is highlighted once again (cf. [41, 290]). Social laser theory predicts that a human gain medium can approach the state of population inversion with infons of one sort (quasi-color), but the stimulated emission can be done by injection of a batch of infons of a different quasi-color, memorylessness of social atoms. This is very important property of social laser which can be widely used in social engineering. We illustrate this theoretical property with a few examples from the modern social-political life. In quantum physics the process of the energy exchange between an atom and the electromagnetic field depends not only on atom’s energy levels, but also on the spins of electrons distributed on these levels as well as spins of photons in the surrounding field [353]. This part of quantum theory was missed in its previous applications to

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social laser (with exception of [290]). It is natural to assume that as well as in physical atoms in social atoms energy is stored in special blocks, “social electrons”. Such a block is characterized not only by the amount of social energy stored in it, but also by the additional characteristics—a social spin. Heuristically, it can be imagined as an axis in the mental space. It is stable and the distribution of such social spins in a social atom plays the important role in energy exchange between s-atoms and the surrounding social-information field (Sect. 11.10). In this paper we shall widely use the abbreviation s-for “social”, say s-atom, s-energy.

11.2 Social Atom A human is the minimal indivisible entity of society, a social atom (s-atom). The atomic viewpoint on the human being has very long history; see Thims’ book [425], in this book the reader can find discussions and references on the basic human-atomistic (or molecular) models: • chemical entity (Johann Goethe, 1809), • point atom (Humphry Davy, 1813), • human molecule (Hippolyte Taine, 1869, Emile Boutmy, 1904, Henry Adams, 1910), Vilfredo Pareto, 1916, Pierre Teilhard, 1947), • social molecule (Thomas Huxley, 1871), • economic molecule (Leon Walras, c. 1870s), • human atom (Ferninand Schiller, 1891, Erich Fromm, 1956), • human chemical and human chemical element (William Fairburn, 1914), • acquaintanceship atom, collective atom, individual atom, psychological atom (Jacob Moreno, 1951), • dissipative structure (Ilya Prigogine, 1971), • human atomism (Arthur Iberall, 1987), • social atom (Mark Buchanan, 2007). Although these authors suggested different definitions, generally they follow the same paradigm: operating with human beings as individual information processors described by just a few parameters characterizing information interaction. Thus, practically infinite complexity of a human being was reduced to these basic parameters, in the simplest case to social energy (Sect. 11.3). This reduction of complexity made humans treatable thermodynamically (Chap. 18). On the other hand, ignoring human complexity diminishes the explanatory power of such models; typically, they can describe statistical behavior of humans, but not explain why they behave in one or another way. The distinguished property of our approach is the quantum-like treatment of variables, as representing observations. Another distinguished property is invention of the quantum social-information field, i.e., s-atoms can interact not only with each

11.3 Social Energy

163

other, as in aforementioned theories, but also with the social-information field. The latter is also interpreted and modeled in the quantum-like framework. We have to make the following terminological remark. Generally we shall operate with the words “it, its” by speaking about s-atoms. This is motivated by their indistinguishability in the process of social lasing and losing the basic human characteristics (Sect. 11.4). However, sometimes we would operate with the words “human, she, he, her, him”, when the human nature of s-atoms should be noted.

11.3 Social Energy As was already highlighted, Bohr denied the objectivity of quantum variables. Measurements’ outcomes are generated in the process of complex interaction between a system and a measurement device. This approach is fruitful for introduction of social energy (s-energy). We do not need to create a deep neurophysiological or psycho-social theory to justify this notion (cf. [155, 216, 219, 220]). S-energy is an observable measuring the degree of social excitement of a person. It can be done with a variety of measurement devices. They can be calibrated with different scales, the simplest scale is dichotomous, E = E a0 , E a1 , where these values are assigned to relaxation and excitement, respectively. The simplest measurement procedure is done with question: “Do you feel your socially excited or not?” From quantum operational viewpoint, such invention of the s-energy observable seems to be justified. For more advanced and psychologically justified measurement procedures for s-energy, see [65, 377]. This operational invention of s-energy as well as other mental observables can be illustrated by the citation of Bohr [75]: Indeed, the necessity of considering the interaction between the measuring instruments and the object under investigation in atomic mechanics corresponds closely to peculiar difficulties met in psychological analysis which arise from the fact that the mental content is invaiably altered when the attention is concentrated on it.

In future quantum-like modeling the crucial role will be played not by the absolute values of the energy levels, but by their difference. For two-level s-atom (with relaxed and excited states), (11.1) E a = E a1 − E a0 . If both levels are high, but the energy gap is small, then such s-atom would not be able to perform a strong s-action. Say, she would never participate in demonstrations leading to brutal clashed with police. Throughout this book we consider only two level s-atoms (see [289] for social laser based on the multilevel s-atoms).

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The energy levels determine the corresponding mental states of an s-atom which are denoted as |E a0 , |E a1 . The main feature of quantum representation of states is the existence of superpositions, e.g., s-atom can be not only in the mental states |E a0 , |E a1 , corresponding to the concrete values of s-energy, but also in superposition states of the form: |ψ = ca0 |E a0  + ca1 |E a1 ,

(11.2)

where |ca0 |2 + |ca1 |2 = 1, ca j ∈ C. The complex coefficients ca j , j = 0, 1, encode the probabilities, (11.3) p j = P(E = E j |ψ) = |c j |2 is the probability that s-atom in the mental state |ψ would answer that his senergy equals E j . (Here we consider the introspective measurement procedure of s-energy when s-atom is asked to report his energetic feeling and the set of answers is restricted to • “I feel me relaxed”, • “I feel me excited.” This probability depends on the state |ψ of s-atom, this fact is reflected in the symbol P(E = E j |ψ). This formula is the Born’s rule. The mental state space of s-atom is a complex Hilbert space H. For two level s-atom, H is two dimensional with the orthonormal basis |E a0 , |E a1  (qubit space). As is typical in physics, the scalar product of two vectors from H is denoted as ψ1 |ψ2 . In terms of the scalar product of s-atom’s states, the Born rule is written as p j = |E a j |ψ|2 .

(11.4)

The existence of superposition states is the mathematical expression of non-objectivity of s-energy. Before s-atom is asked to estimate his-her s-energy, he-she does not know its value. Of course, one can design other methodologies for measurement of s-energy which are not based on self-observations.

11.4 Social-Information Field In accordance with QFT, a field represents an ensemble of its energetic excitations. Mathematically, this excitation structure of a field is described in Fock space. Quantum fields are described with the operators of creation and annihilation of excitations.

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Quantum field Excitations—Photons, Phonons, Infons Quantum field excitations are treated on equal grounds with “real systems” such as atoms or electrons. Say excitations of the electromagnetic fields are photons. Moreover, excitations corresponding vibrations, e.g., of atoms in a crystal or dipoles in a molecular, also treated as systems, phonons. In social studies, we proceed in the same way. The social-information (s-information) quantum field which is generated by mass media and Internet-based s-networks is an ensemble of energetic excitations, each excitation is determined by the portion, “quantum”, of s-energy. The field excitations are generated by the sources of socially relevant information, mainly by mass media and s-networks. Each communication emitted by them carries a quantum of s-energy. We call such quanta infons. We remark that the s-information quantum field should not be imagined as a continuous wave propagating in physical space. Mathematically any quantum field is a structure of the very high degree of abstractness: operator valued distribution (generalized function). Connection of a quantum field with real world is via detection of its excitations. In physics photon’s energy can be connected with light’s frequency and hence the color of the corresponding spectral line. In the same way we can color infons, depending on the amount of s-energy: • low—red, • intermediate—yellow, green, blue, • high—violet. For example, during the pandemic the majority of communication on COVID-19 was of the violet color; during the spring of year 2022 news about the war were also violet, but the communications about COVID-19 were colored in red. News about sexual affairs of politicians and stars can be colored in yellow. We hope that this s-energy/color terminology would not be misleading. In ordinary life red means danger and attracts more attention than violet. But, in physics red photons are low energetic and violet photons are highly energetic. We keep the physical picture. So, the red colored infons carry small amounts of s-energy and violet infons are highly energetic. In fact, in physics characterization of QFT excitations is not reduced to energy. For example, a photon also has polarization. Generally, photon’s state |Eα is characterized by the parameters E = energy and α = (polarization, temporal, and spatial extensions). S-information field can also have some characteristics additional to s-energy and related to communication’s content. We call such characteristics the quasi-color of infon; its state can be encoded as |Eα, where E and α are s-energy and the quasi-color respectively.

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Quantum Versus Quantum-like Indistinguishability Introduction of the quasi-color is a delicate process related to such foundational issue as indistinguishability of quantum systems (see, e.g., [43]). Quantum theory assumes that two photons in the state |Eα are indistinguishable. Moreover, it is claimed that there are no hidden variables, additional photon’s characteristics which would provide a possibility to distinguish two photons in the state |Eα. So, in quantum physics indistinguishability has the fundamental character. Indistinguishability plays the crucial role in derivation of quantum statistics for energy distribution in the framework of statistical thermodynamics [405]. This is the good place to mention once again phonons, the quanta of vibrations, e.g., in a crystal, excitations of the field of mechanical fluctuations. To describe the special quantum effects, phonons have to be indistinguishable, up to some parameters, the energy is the basic one. While photons still match with our image of particles (at least up to some degree), phonons are really immaterial entities carrying information about the relative fluctuations of atoms. So, phonons are not so different from infons. In quantum-like modeling of the excitations of the s-information field infons are indistinguishable, up to s-energy E and the quasi-color α. This is the important assumption beyond social laser theory. However, there is one important difference between quantum and quantum-like indistinguishabilities. The former is genuine and irreducible and the latter is relative to context. In one context some social variables are important and they should be included in the quasi-color α, in another context they do not play any role, so they are not included in α. But, we cannot deny their existence. For example, humans have names, but their names do not play any role in the process of social lasing in the form of mass protests. So, humans are indistinguishable w.r.t. to the name variable. The slogan “Black Lives Matter” is the integral quasi-color which was crucial in the protests in USA. The concrete names of black people who experienced racism, discrimination, and racial inequality were hidden in this quasicolor. So, indistinguishability in social laser theory is quantum-like. It is up to the characteristics determining the process of lasing. These characteristics form the quasi-color α. And infons’ indistinguishability is up to s-energy and this quasi-color. For example, α can be “corruption”, then infon’s state is characterized by s-energy (color) and corruption label (quasi-color). So, whole content of a communication is reduced to the corruption label. Of course, a communication can carry a lot of additional information, but in the process of this concrete lasing it would be ignored.

Social-Information Field as Bosonic Quantum Field We start with consideration of quantum physics. We are interested in absorption and emission of photons by atoms, especially in the balance of these processes.

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Bosonic creation operator a ∗ and annihilation operator a verify the canonical commutation relation in the form [a ∗ , a] = a ∗ · a − a · a ∗ = I, where I is the unit operator. These operators are not Hermitian, so they do not represent quantum observables. The quantum observable for this system is the photon number operator given by the composition of the operators, n = a∗a

(11.5)

giving for a laser the number of photons in the considered mode u. Here u encodes both photon’s momentum (in particular its energy) and polarization. So, all operators are labeled by the mode-parameter u as au∗ , au .n u . The quantum field description of the stimulated emission is a collective effect, i.e., an atom interacts with a batch of photons and not just with an individual photon. The crucial role is played by the Bose–Einstein statistics of the photons. Consider the n-photon state |n, u, for a fixed mode u. This state can be represented in the form of the action of the creation operator aα† on the vacuum state |0, u >: |n, u =

 n √  au† / n! |0, u

(11.6)

This representation gives the possibility to find (with simple calculations in the QFT state space—the Fock space) that the transition probability amplitude from the state |n, u > to the state |n + 1, u > equals Pu (n → n + 1) =

√

n + 1.

(11.7)

On the other hand, it is well known that the reverse process of absorption is characterized by the transition probability amplitude from the state |n, u > to the state |n − 1, u > given by √ (11.8) Pu (n → n − 1) = n . Thus, in a quantum bosonic field the increase in the number of photons leads to an increase of the probability of generation of one more photon in the same state. This constitutes one of the “quantum advantages” of laser-stimulated emission showing that the emission of a coherent photon is more probable than the absorption. In social science and finance, the bosonic and fermionic algebras of operators were used in works [24, 36]. It is interesting that applications outside of physics lead to the use of the operator algebra mixing bosonic and fermionic features in the so-called qubit operator algebra [312] for applications to the theory of decision-making. Bosonic statistics of infons was derived in article [267] (see Chap. 18) on the basis of their indistinguishability up to the parameter u = (E, α), where E is infon’s

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color (its s-energy) and α is infon’s quasi-color (content-label). Generally indistinguishability implies one of quantum statistics, Bose–Einstein, Fermi–Diract, or parastatistics. The first one is characterized by an arbitrary number of systems for the same mode u, so the states |n, u are defined for any n = 0, 1, 2, ... . The second one reflects the Pauli exclusion principle, here n = 0, 1. The parastatistics assumes the concrete upper bound q for n, i.e., n = 0, 1, ..., q, for the q-statistics. We selected for infons the Bose–Einstein statistics, since mass media and Internet can generate (practically) any number of communications, messages, news, posts, comments on the same mode u. Parastatistics are not present in quantum physics. But they may be found in cognitive and social phenomena. For example, one can estimate the number of possible infons on the same mode u which can be generated in human civilization. But, for very large q, the difference between the q-statistics and the Bose–Einstein statistics is practically invisible. For infons (under the assumption of the Bose–Einstein statistics), we can use the formalism of operators of creation and annihilation. In combination with Born’s rule for the transition probabilities we obtain the formulas (11.7) and (11.8). Hence, in the social framework we arrive at the same conclusion as in the physical one:

Interpretation of Infon’s State In above consideration we operated with the word “to treat”. But, who does treat (analyze) infon’s content and the level of its energizing? This is s-atom which plays not only the role of infon’s receiver, but also of its analyzer. Thus, the state of infon is not objective. This state is assigned to infon by s-atom receiving it. This is a subjective state. In principle, it depends on the receiver-analyzer: s-atom. Even “to be or not to be entangled” is subjective. In contrast, in physics a state is considered as the state of a system, say the state of photon. However, this state’s objectivity is questioned by Schrödinger [406]. In this article, he considered a quantum state as mathematical representation of potentiality to obtain outcomes of observables. This ideology was structured within Quantum Bayesianism (QBism) developed by Fuchs and coauthors (e.g., [163–168]). In QBism quantum probabilities given by the Born rule are interpreted as subjective probabilities of individual agents dealing with quantum theory and experiment. We also interpret probabilities as subjective probabilities assigned to infons by s-atoms. Generally variability of such probabilities and quantum-like states beyond them can be high. However, to be able to apply quantum-like models for collective phenomena, including social laser, we assume that populations of s-atoms under consideration are characterized by homogeneous assignment of states and, hence, subjective probabilities. This homogeneity results from social rules, traditions (cultural, historical), opinion-structuring with the aid of mass media and social networks.

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So, within such a population a quantum-state can be considered as infon’s state. One can speak about state’s objectivity w.r.t. to some population of s-atoms. Social lasers work within such populations.

11.5 Absorption and Emission of Infons by Social Atom Here we consider processes of absorption and emission of quanta of s-energy by s-atoms interacting with the excitations of the s-information field—infons.

Absorption Consider physical atoms with two levels of energy, excited and relaxed, E 1 and E 0 . The difference between these levels E a = E a1 − E a0

(11.9)

is the basic energetic parameter of an atom, its spectral line.2 Two level atom reacts only to photons carrying energy E matching with atom’s spectral line (Bohr’s rule): (11.10) E a = E. In quantum-like modeling, we apply Bohr’s rule to s-atoms and infons. So, two level s-atom reacts only to infons carrying energy E matching atom’s spectral line, see (11.9) and (11.10). If infon carries too high energy which is larger than the spectral line, E > E a , then this s-atom would not be able to absorb this infon. For example, infon carrying s-energy E is a call for upraise against the government. And s-atom is a bank clerk (say Elena) in Moscow. Elena has the liberal views and hates Putin’s regime, but her spectral line is too small to absorb s-energy carried by such infon and to move from the ground state to the excited state. She simply ignores such highly energetic communication, news, or internet-post. Similarly, if infon’s s-energy E is less than spectral line E a , then Elena would not be excited by such infon. As well as a physical atom cannot collect energy from a few low energy photons, with E < E a , s-atom cannot collect s-energy from a few infons carrying small portions of s-energy. S-atom either absorbs infon (if their colors match each other), or she does not react to it. In the same way, s-atom cannot “eat” just a portion of s-energy carried by highly energetic infon with E > E a . 2

In the case of two level atom, it has just one spectral line. Generally there are a few spectral lines corresponding to differences between energy levels, E a;ij = E ai − E aj , i > j. This is atom’s spectrum.

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Spontaneous and Stimulated Emissions In the quantum-like theory, the process of infon emission by excited s-atom is also characterized by its spectral lines. In the case of two level s-atom this is just the number E a . S-atom can emit only infon satisfying (11.10). As in physics, emission can be spontaneous when s-atom suddenly emits infon— at random instance of time and with a random quasi-color. Another sort of emission is known as stimulated; s-atom emits infon as the result of interaction with infons of the surrounding s-information field. Ideally a single infon in the state |Eα, where E = E a , stimulates excited s-atom to emit infon in the precisely the same state. So, emitted infon has not only the same s-energy as stimulating infon, but also the same quasi-color α. However, since this process is probabilistic (as all quantumlike processes), the real stimulation of emission is possible only with fields of high density. So, s-atom should interact with a strong s-information field, with a cloud of excitations which have the same s-energy (equal to E a ) and quasi-color. Inside such field the probability of emission is high. In quantum physics spontaneous emission of photon by atom is considered as exhibition of irreducible quantum randomness. However, such picture might be adequate only for completely isolated atom. But real atom is never completely isolated. Background radiation is everywhere. Therefore, it may be that even the spontaneous emission is not a totally random process. It can be stimulated by fluctuations of the background electromagnetic field and interactions with other atoms. In the same way, spontaneous emission of social excitation might be generated by fluctuations in the surrounding social environment: occasional news, a scandal with a partner, a problem at work, and so on. The quasi-color which emitting s-atom assigns to infon (or action in physical space generated by this infon) may reflect the environment’s quasi-color. In physics the photon absorption-emission condition (11.10) is satisfied only approximately (11.11) E ≈ E a . The spectral line broadening is always present. In an ensemble of atoms E a = E a (ω) is the Gaussian random variable. This is a bell centered at the mean average ¯ a . The dispersion of the Gaussian distribution depends on an ensemble value E of atoms. Ensembles with small dispersion are better as gain mediums for physical lasing, but deviations from exact law (11.10) are possible. It is natural to assume Gaussian distribution realization of exact laws even for social systems; in particular, absorption of excitations of the information field by satoms. Thus, deviations from (11.10) are possible. But a good human gain medium (an ensemble of s-atoms selected for social lasing) should be energetically homogeneous. Therefore, the corresponding Gaussian distribution should have very small dispersion. The latter is also the important necessary condition for functioning of physical laser.

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Memorylessness of Atoms Finally, we discuss one interesting feature of interrelation of absorption and emission: Consider quantum physics. Suppose that an atom absorbed a photon with momentum vector p. This vector determines the direction of photon’s propagation and its length |p| determines photon’s energy. The process of absorption is characterized by matching of energies (11.10), so the direction of photon’s propagation given by α ≡ p/|p|

(11.12)

does not play any role in the process of absorption. The most interesting for us is that an atom “forgets” the direction α of incoming photon. In the process of spontaneous emission, an atom emits a photon in an arbitrary direction. Moreover, in the process of stimulated emission an atom emits a photon with momentum which is identical to momentum of stimulating photons, the stimulating electromagnetic field. The same “memory washing” is a feature of quantum-like model, since its mathematical formalism is identical to quantum physical theory. So, s-atom does not remember the quasi-color α of infon, say a news, which it has absorbed. It can emit spontaneously infon, say a post in a social network, of an arbitrary quasi-color. When s-atom is stimulated for emission, it emits infon of the same quasi-color as stimulating infons, say news. This memorylessness of s-atoms is very important in social engineering, including social lasing (see also Sect. 11.10).

11.6 Social and Physical Lasers For simplicity, we continue to consider two level atoms, both physical and social. We shall present the social lasing scheme parallelly to the scheme of physical lasing.

Laser’s Components and Stages of Lasing Physical laser has three main components: • A gain medium composed of atoms. • A source of energy (the electromagnetic field). • A resonator (an optical cavity).

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Physical lasing has two main stages: • (A). Energy pumping. Energy is pumped into a gain medium; the aim is to approach the population inversion—more than 50% of atoms should be transferred into the excited state. • (B). Stimulated emission. A batch of photons propagating in the same direction α given by (11.12) is injected into the gain medium. They stimulate the cascade process of the emission of photons by atoms. At the both stages, the colors of photons and atoms’ spectral line match each other at least approximately (for two level atoms there is just one spectral line). The gain medium should be color-homogeneous. Photons produced during the B-stage copy the direction of propagation from stimulating photons. The latter was injected along the main axis of the optical cavity—laser’s resonator. As was pointed out in Sect. 11.5, the directions of photons’ propagation at the A and B stages can be totally different (memorylessness of atoms). Some important details will be mentioned below to illustrate the corresponding details of social lasing. The social laser also has three components: • A gain medium composed of s-atoms (humans). • A source of s-energy (the s-information field). • A social resonator (Internet-based social networks). Social lasing also has two main stages: • (A). Energy pumping. S-energy is pumped into a human gain medium; the aim is to approach the population inversion—more than 50% of s-atoms should be transferred into the excited state. • (B). Stimulated emission. A batch of infons of the same quasi-color α is injected into the gain medium. They stimulate the cascade process of the emission of infons by s-atoms. Now we describe these stages in more detail: The mass media and Internet pump s-energy into a human gain medium aiming to approach the population inversion—to excite the majority of people. The gain medium should be homogeneous w.r.t. its spectral structure, ideally E a = Const. In reality E a is a Gaussian random variable with very small standard deviation. The s-energies of infons (communications, news, messages, internet-videos) used for energy pumping do not need be so sharply concentrated around average E a of E a . S-atoms would simply ignore infons essentially deviating from E a . And such s-energy losses are compensated by the powerful flows of information generated by modern mass media and Internet. After achievement of the population inversion, the stimulated emission is started. A batch of infons (say communications, news) is injected into the human gain medium. The first constraint is that s-energy of these stimulating infons should match the spectral line of s-atoms, see (11.10) (in the ideal case). In reality it is sufficient to control the approximate matching condition (11.11). So, s-energy of injected infons should not deviate essentially from E a . Another constaint on injected infons is that

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they all should carry the same quasi-color α, say α = COVID-19, or α = vaccination, or α = Russian aggression against Ukraine (depending on the socio-political context and the aims of social lasing). This injected beam of social-information radiation generates the cascade process in the gain medium. Quasi-color homogeneity of the stimulating information injection is the basis of quasi-color coherence of the laser beam of infons. Later this social-information beam is transferred into the social action matching infons’ color (s-energy amplitude) and quasi-color (information content). Infons’ homogeneity should be very high. Here statistical deviations are not acceptable, since infons of other quasi-colors would also generate their own cascades. Such “noise-cascades” would destroy quasi-color coherence of the output beam of social radiation. They should be then eliminated with the aid of social resonators.

Cascade’s Dynamics In the simplified picture, each infon stimulates s-atom to emit infon having the same color and quasi-color with its stimulator. Resulting two infons stimulate two s-atoms to emit two new infons, so one stimulating infon resulted in four infons which interact with four s-atoms and so on. After say 20 steps there are 220 , approximately one million of infons (the excitations of the social-information field) of the same color and quasi-color. In reality, the process is probabilistic: s-atom reacts to stimulating infon only with some probability. The latter rapidly increases with the increase of the density of the social-information field. And field’s bosonic nature is crucial (Sect. 11.4). In physics, the beam induced in the gain medium by the stimulating injection of coherently colored photons is not the laser’s output beam. Laser has an additional component which plays the crucial role in increasing both the amplitude and coherence of the output beam. This is a laser resonator. For lasers emitting photons— excitations of the quantum electromagnetic field, this is an optical cavity. Its mirrors reflect beams generated inside the gain medium and send them back to this medium. In this way, the cascade process in the gain medium is repeated many times.

Coherence Increasing Within Resonator The process of reflection from the mirrors also increases spatial coherence of the beam. The photons propagating not precisely along the main axis of the cavity are reflected outside of the cavity and disappear. We remark that the stimulating beam is sent along this axis. The cascade photons copy the direction of propagation in space given by momentum vector (11.12) of initially injected photons. The photon dynamics inside the cavity also increases temporal coherence. The first beam of cascade photons (induced by the initial injection) has very small temporal

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dispersion, since produced photons (moving with the light velocity) pass very quickly through the atom gain medium. This temporally sharp wave of energy approaches the mirror and it is reflected back—toward the gain medium. Of course, some arriving to the mirror photons can deviate from the main spike; they are also reflected toward the gain medium and induce cascades. But the latter are negligibly small compared with the cascades induced by iterations of the main spike, because the probability of photon’s emission by an atom depends on the density of the surrounding electromagnetic field. We remark that during the beam iterations (through reflections) energy is continued to be pumped into the gain medium from outside. So, atoms which emitted photons in the preceding iterations absorb newly incoming photons and move to the excited state. Intensity of pumping of energy quanta into the gain medium should be high enough, higher than some threshold depending on laser’s parameters (see [287] for details; cf. [184–186]). This threshold is called the lasing threshold. If the intensity of energy pumping is lower than the lasing threshold, then too many atoms would spontaneously relax between two reflection-iterations of the basic wave of photons. On one hand, this is the energy loss; on the other hand, the mini-cascades created in the excited gain medium by spontaneous emissions would lower coherence of the radiation beam. If the intensity is higher than the lasing threshold, then practically all energy pumped into the gain medium is transferred into the basic radiation wave propagating along the cavity’s axis. We remark that in physics the coherence of the laser beam is typically discussed in the classical wave framework [187] by operating with frequencies and phases. In social applications, I was not able to invent the proper analogs of these notions. We proceed solely with s-energy and quasi-color. And coherence is defined in these terms. However, even information beams have the temporal structure. And we shall discuss corresponding coherence. In our quantum-like model the social laser also should have a resonator, a kind of two mirrors which reflect infons and send them back into the human gain medium— to interact again with s-atoms in the human gain medium and to stimulate them to emit infons. The role of such social resonators is played by Internet-based information systems, such as YouTube, Facebook, Instagram, Bastyon, Telegram, Life Journal, VK, and so on. The main distinguishing feature of these systems is the possibility of the rapid feedback to communications, news, and videos in the form of comments, comments on comments and so on. The beam of infons created from the initial stimulating injection (typically by mass media’s giants such as say BBC and CNN, Washington Post, New York Times, and Guardian) is distributed over internet channels and create new posts (in the form of articles and videos), each of them is actively commented. Each comment plays the role of a mirror. But this is a kind of an active mirror, not only reflecting infons but creating them. By reading a comment say a YouTube user absorbs infon matching the color and quasi-color of the post and transits to the excited state. In physics, this is a part laser theory—laser with active mirrors. Some of its social counterparts were developed in article []. So, social resonators are more complicated devices than simply passive reflecting optical mirrors used in physical laser.

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As well as the physical resonator, the social resonator not only amplifies the beam of social radiation inside the human gain medium, but also increases its coherence w.r.t. the quasi-color α of the stimulating injection. Posts quasi-colored differently from α disappear in the massive flow α-infons. Besides quasi-color coherence, a social resonator increases temporal coherence. Since each stimulating news, as say with α = “the start of the war at Ukraine”, rises up to a wave of the α-posts, practically immediately. We should not forget that communications propagate via optical fibers and wireless with the light velocity. Comments appearing in the s-networks with a delay from the main wave are typically ignored.

Summarizing Social Laser Theory • Our quantum-like model is of the quantum field type, the social-information field. Its excitations are called infons. Each infon transports quantum of s-energy. The latter determines infons’ color, red infons are low energetic and violet infons are highly energetic. • Each s-atom is characterized by the s-energy spectrum; in the simplest case of two levels, this is the difference between the energies of the excitation and relaxation states, E a = E 1 − E 0 . • Beside s-energy (color), infons (the excitations of the information field) are characterized by other labels, quasi-colors, carrying content of information communications. • Coherence corresponds to quasi-color sharpness; ideal social laser emits a single quasi-color mode, denoted say by the symbol α. • Excited s-atoms by interacting with α-colored infons also emit α-colored infons. • The amount of s-energy carried by stimulating infons (communications, news, comments, Internet-posts) should match the color of s-atoms in the gain medium. • To approach the population inversion, s-energy is pumped into the gain medium. Pumping should be intensive, since s-atoms have the tendency spontaneously relax and emit infons with randomly distributed quasi-colors. • This energy pumping is driven by the mass media and the Internet sources. • The gain medium should be homogeneous with respect to s-energy spectrum. Ideally (for the two level case), all s-atoms should have the same color E a . However, in reality, it is impossible to create such human gain medium. As in physics, the spectral line broadening has to be taken into account. • The quasi-colors of infons in energy pumping have no direct connection with the quasi-color of infons generated by stimulating emission (memorylessness of s-atoms). • Infons follow the Bose–Einstein statistics. • This statistics matches with the bandwagon effect in psychology [107] (see article [290] for details).

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• The probability of emission of the α-colored infon by s-atom in a human gain medium increases very quickly with the increase of the intensity of the socialinformation field on the α-colored mode. • The stimulating injection of homogeneously quasi-colored infons gives rise to the cascade of coherent (w.r.t. the color and quasi-color) infons. • The created social radiation beam is amplified in the social resonators based on Internet information systems, say YouTube, Facebook. • The social resonators, especially in the form of Internet-based Echo Chambers, also improve quasi-color coherence (Sect. 11.7). • When the power of the beam of coherent infons becomes very high, infons are transformed into s-actions, either in physical or information spaces. Actions are understood very generally from collective coherent decision-making to physical actions, as mass protests, demonstrations, color revolutions For example, a gain medium consisting of humans in the excited state and stimulated by the anti-corruption quasi-colored information field would “radiate” a wave of anti-corruption protests. The same gain medium stimulated by an information field carrying another quasi-color would generate the wave of actions corresponding to this last quasi-color. For social laser engineering, it is very important that the quasicolors of s-energy supply and stimulation of emission do not need to coincide— memorylessness.

11.7 Echo Chamber: Social Coherence Reinforcing The detailed presentation of social resonators theory can be found in article [290]. In the latter we highlighted the differences between the physical resonators of the cavity type, so to say “passive reflectors”, and the social resonators which are based on the “s-mirrors. Such mirrors can be treated as active reflectors producing on demand of users new infons. Here we shall consider in more detail special, but at the same time very important type of social resonators, namely, Internet-based Echo Chambers (see also [290]). In our notations, it can be defined as following: Echo Chamber is a system in that some beams of infons carrying (as their quasicolors) news, communications, ideas, and behavioral patterns are amplified and sharped through their feedback propagation inside this system. In parallel to such amplification, infons carrying quasi-colors different from those determined by the concrete Echo Chamber are suppressed. In our terms, an Echo Chamber is a device for transmission and active re-emission (not simply reflection) of infons—the excitations of the quantum social-information field. Its main purpose is amplification of this field and increasing its quasi-color coherence via distilling from “s-noise”, i.e., infons colored and quasi-colored differently from Echo Chamber’s basic color and quasi-color.

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We underline that an Echo Chamber is considered as a component of the social laser, as its resonator. The coherent output of an Echo Chamber, the quasi-color of this output, is determined not only by the internal characteristics of the Echo Chamber, but also by the quasi-color of stimulated emission in laser. The same Echo Chamber may be turned in the accordance with the aim of the stimulated emission in progress. Of course, such turning is not possible for every Echo Chamber. Some of them are stable w.r.t. to their basic quasi-colors. Let us consider functioning of some Internet-based Echo Chamber; for example, one that is based on some s-group in Facebook and composed of s-atoms. The degree of their indistinguishability can vary depending on the concrete Echo Chamber. Say, names are still present in Facebook, but they have some meaning only for the restricted circle of friends; in Instagram or Snapchat, even names disappear and s-atoms operate just with nicknames. By a s-group we understand some sub-network of say Facebook, for example, s-group “Quantum Physics”. The main feature of a s-group is that all posts and comments are visible for all members of this s-group. Thus, if I put the post “Getting rid of nonlocality from quantum physics”, then it would be visible for all members of this s-group, and they would be able to put their own comments or posts related to my initiation post. This is simplification of the general structure of posting in Facebook, with constraints that are set by clustering into “friends” and “followers”. We assume that the ensemble of s-atoms of this Echo Chamber approached population inversion, so the majority of them are already excited. A batch of communications of the same quasi-color α and carrying quanta of s-energy E c = E a is injected in the Echo Chamber. Excited s-atoms interact with the stimulating communications and, with some probability, emit information excitations of the same quasi-color as the injected stimulators. These emitted quanta of s-energy are represented in the form of new posts in Echo Chamber’s s-group. Each post plays the role of a mirror, it reflects the information excitation that has generated this post. However, the analogy with the optics may be misleading. In classical optics, each light ray is reflected by the mirror again as one ray. In quantum optics, each photon reflected by the mirror is again just one photon. An ideal mirror reflects all photons, the real one absorbs some of them. In contrast, “the mirror of an Echo Chamber”, the information mirror, is s-energy multiplier. A physical analog of such a multiplier works as follows. Each light ray is reflected as a batch of rays or in the quantum picture, matching the situation better, each photon by interacting with such a mirror generates a batch of photons. Of course, the usual physical mirror cannot reflect more photons than the number of incoming ones, due to the energy conservation law. Hence, the discussed device is hypothetical. The Internet-based system of posting news and communications works as a multiplier. Each posted news or communication emits a batch of “information rays” directed to all possible receivers—the s-atoms of Echo Chamber’s s-group who are active at this time. In the quantum model, each post works as an information analog of the photon’s emitter. It emits quanta of s-energy, the power of the information field increases. Consequently excited s-atoms emit their own posts and comments

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with higher probability. We repeat that new posts have the same quasi-color as the initiating information excitations that were injected in the Echo Chamber. By reaction we understood emission of a new message, say a comment. If satom simply reads a posted communication, but does not emit its own information excitation, then we do not consider such reading as a reaction. For the moment, we consider only the process of stimulated emission. Later we shall consider absorption as well. In the latter, reaction means transition from the ground state to the excited state; so, not simply reading. In fact, a relaxed s-atom can read a post or a comment without absorbing a quantum of s-energy sufficient for approaching the state of excitement.

11.8 Illustrating Examples In this section, we would like to illustrate previous theoretical considerations by the additional examples of social laser’s use at the modern socio-political arena, potential and real uses.

A Societal Benefit Laser In the previous works on the social laser [267–290], we main attention was paid to its destructive power. This apparatus for social engineering was considered merely from SASA-viewpoint. A social-information tsunami can be not less destructive than a physical tsunami. And nowadays the destructive power of social-information tsunamis is essentially higher than the power of ocean’s waves. After my lecture at the event “Quantum Social Science Bootcamp’s 22”,3 graduate and post-graduate students were curious about the social laser potential for generation of something good for people, so to say “the waves of love” between people, e.g., to stop the wars in Syria or Ukraine, to struggle against poverty in Africa, Latin America, India, or to intensify the struggle against destruction of planet’s ecological system. During the discussion we quickly came to the common conclusion that although theoretically it is possible to generate societal benefit lasing, practical realization is difficult. As was already pointed out, the energy gap E a in typical s-atom of the modern society is very big. We are living in the highly energetic society. Hence, the energy pumping has to be done with highly s-energetic infons (say communications, news). The main source of human psychic (mental) energy are our basic instincts. Messages carrying horror, anger, fear, hate deliver huge amounts of the psychic energy which can be later (at the stage of stimulating emission) transferred into the s-energy. It

3

see https://u.osu.edu/quantumbootcamp/recordings-and-resources/.

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is not easy to create highly s-energetic messages with quasi-color α = “societal benefit”. During the aforementioned discussion after my lecture, one master student remarked that why not to use highly energetic communications with “terrible content” to approach the population inversion and then to start stimulating emission with a batch (quasi-) colored with α = “societal benefit”? Theoretically it can be done (see Sect. 11.5 on the possibility to pump s-energy with α-quasi-colored infons and perform the stimulated emission with beta-quasi-colored infons, α = β. The main obstacle to such s-engineering “transferring bad into good” is that the stimulating infons must carry big amounts of s-energy matching E a which is assumed to be big in the modern society. If “excitation’s color is not purple enough”, then s-atoms would simply ignore such infons, even if mass media and s-networks would start to generate them intensively. At the same time, we can point to such social lasing campaigns for “social benefits” as say “Me Too” , or “Black Lives Matter”, or“Critical Race Theory”. So, social lasing tsunamis in social-information space can serve for social benefits. The sexual instinct is also one of the basic instincts, and it can serve as a source of both s-energy pumping to approach the population inversion in human gain medium and to start the stimulated emission. This instinct played the important role in the hippie motion in 1960. This motion dynamics had some features of social lasing, although in the absence of internet s-resonators were not advanced. However, instead of the internet based Echo Chambers, youngsters used physical meetings with joint camping, festivals, creation of hippie communities. And we repeat once again that the sexual instinct was heavily involved in s-energy generation and transmission.

COVID-19 Protests During the COVID-19 pandemic, the human gain medium was overheated by the shock news about the spread of this terrible disease, by its deadly consequences, by life during lockdowns, and numerous rigid restrictions on social life (e.g., masks in public places and somewhere, e.g., OAE, even at the streets), by the QR-codes and obligatory vaccination for some professions, e.g., the personal of hospitals. Such communications were often repeated a few times, their content could vary, but not the basic quasi-color, α = COVID-19. In some countries, even the most democratic ones as in Sweden, the laws were changed by restricting the basic freedoms, including the basic constitutional right for meetings and demonstrations. Scientists also actively contributed in generation COVID-19 fear. For example, some mathematical models of disease spread predicted millions of deaths from COVID-19 in UK and hundreds of thousands in Sweden, if the rigid restrictions, including lockdowns and masks would not be invented [146, 151]. My personal opinion is that such the mathematical models were really primitive, basically the very old SIR-dynamics (may be disturbed by a stochastic term for noise). This is the good place to mention the new model of disease spread [293, 294, 296]; it

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took into account the social cluster structure of population. This model predicted the opposite effect, comparing with the majority of models, of lockdowns and other rigid restrictions. In contrast to, e.g., [146, 151], such restrictions slowdown approaching of natural immunity in human population (cf. with conventional predictions [82–84]. By summarizing we can say that at the end of the year 2020 and the beginning of the year 2021, the state of population inversion was approached in European countries, Australia, Canada, USA, and Russia. The human gain medium was ready for radiating a huge spike of s-energy, a spike which could destroy the basics of society. Various s-groups and individuals started to generate information excitations against WHO’s COVID-19 policy and their governments following this policy. Social networks resonated these excitations. This led to generation of local spikes of protests, world-wide and especially in Australia, Germany, the Netherlands, France, UK, Canada, and even Sweden (see, e.g., [104, 445]). (In Sweden, the COVID-19 restrictions were really mild by comparing with majority of countries: no lockdowns, no masks.) Powerful SASA was about to start world-wide. However, extended suppression of functioning of the social resonators, especially by YouTube and Twitter, prevented creation of the global wave of protests. At the same time, the local spikes relaxed some portions of s-energy and in this way s-temperature was lowed.

Pro-war and Pro-peace Beaming: Competitions of Stimulated Emissions Coming back to societal benefit lasing theme, consider a war between two countries or blocks of countries. And suppose that some group of policy makers wants to use the social laser technology to generate the wave of peaceful thoughts and actions. Assume that this group is powerful enough to generate a strong injection of communications for peace. In principle, it is possible to connect with a message having quasi-color α = “peace” big amount of s-energy. Unfortunately, to generate such social lasing for peace the war should be going for sufficiently long time. And it should lead to big casualties from both sides or at least from one of them. Otherwise even spontaneously emitted hate cascade would destroy the processes of stimulation of a cascade of thoughts and actions for peace. Here we come to the problem of competing stimulation in human gain medium approached the state of population inversion. The main problem in starting such process is that another group at the political arena might not be interested to end this war. By using their information resources they could continue social lasing in favor of the war, α = “war”. And the aggression instinct is so powerful that there is a big chance that such the α = “war” beam of social laser would be essentially stronger than the α = “peace” beam. In such competition of s-energy beams of two quasi-colors, the crucial role is played by social resonators, in the form of s-networks based on YouTube, Facebook,

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Live Journal, Bastion, Twitter, Telegram, Instagram, Yandex, Vkontakte, and so on. Therefore, it is so important to control such information resources (e.g., one can understand the motivation of Elon Musk to buy Twitter). Without the control of social resonators, it is practically impossible to start stimulating emission. The initial (stimulating) batch of information excitations which is not supported by social resonators would pass through information space in a flash and disappear. As was stressed in [335], The dissemination and control of information are indispensable ingredients of violent conflict, with all parties involved in a conflict or at war seeking to frame the discussion on their own terms. Those attempts at information control often involve the dissemination of misinformation or disinformation (i.e., information that is incorrect by accident or intent, respectively).

Therefore, it is not surprising that Elon Mask was ready to spend 43 billion dollars to buy Twitter.

Russian-Ukrainian War and Relaxation of COVID-19 Generated Social Energy Now we turn again to the COVID-19 pandemic. During the years 2020–21 human society collected a lot of s-energy and approached the state of population inversion. Of course, the state of population inversion was not approached in whole world; say in Egypt and other African countries COVID-19 did not lead to massive transition of people into the excited state. We speak about this transition in European countries (both West and East Europe), USA, Canada, Australia. It is interesting that in China, in spite of very high degree of COVID-19 related restrictions, generally the mental state of population could not be characterized as excited. Chinese population took these restrictions rather calmly by following automatically COVID-19 state recommendations, as people here would do in any other case. On the other hand, the mass protests of Canadian track-drivers demonstrated that the degree of social tensions in Canadian society was very high. These protests can be considered as a test of COVID-19 generated instability in Western society. West society’s stability was saved only due to diminishing the activity of social resonators via the direct censorship as well as mobilization of police forces to terminate demonstrations and protests that happened in physical space. The social lasing in the form of the COVID-19 protests was put on hole. But the society was in the unstable state; any stimulated or even spontaneous emission could start tsunami in information space with transmission of its social power into physical space. One can speculate that only the war between Russia and Ukraine relaxed the huge amount of s-energy collected during the pandemic in Europe, USA, Canada, Russia, and Ukraine. The COVID-19 energy was transferred into the war energy. Here we discussed mainly the processes in social-information space, i.e., not the real war

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battles in physical space. In February, March, and April of the year 2022, the war ignited the real information tsunami in mass media and s-networks, the massive flows of s-energy were radiated wold-wide. This is the excellent example of the possibility to relax s-energy collected within α1 quasi-colored information campaign via a new stimulating emission which is quasi-colored with totally different quasi-color α2 . But, as always in socio-political studies, it is impossible to exclude that the sequence COVID-19→war was just the occasional flow of events.

Russian-Ukrainian Information War: Competing Beams of Social Radiation It is interesting that the war generated two powerful beams of radiation—two rays of social lasing. The rays had the opposite quasi-colors, anti-and pro-Russia quasicolors, αa R , α p R . These social lasing rays were stimulated by mass media of Western countries, Ukraine and Russia and they were amplified by the corresponding Echo Chambers. Echo Chambers were Internet-based. Russian government tried to block some pro-Western social resonators. It was not able to block concrete Echo Chambers or individual bloggers, since the Internet resources were controlled by the Western corporations. So, the Russian government chosen the strategy of the total blockade of some resources, say Facebook, Instagram, Linkedin. It even planned to block YouTube, but finally decided not to do this and to use this internet system for own purposes. Surprisingly similar strategy was applied by Western governments to some Russian resources, e.g., to “Russia Today” TV channel. European and American bloggers and, e.g., YouTube channels delivering information excitations of α R quasi-color (or excitations which might be interpreted as having such quasicolor) were accused in supporting the Russian war propaganda. In particular, we can mention American conservative TV and YouTube channels as say FOX News. And YouTube and Facebook also massively block videos of Russian (typically state coupled) bloggers, especially delivering the news of military operations at Ukraine. At the same time YouTube was actively used by Russia, the state employed bloggers delivered pro-Russia colored news, comments, event-interpretations about the war. Pro-Western resources were definitely more powerful and covered wider population throughout the world, including some shifts of Russian youngsters. Some Arabic, Indian, Latin American, and Chinese channels generated α R −communications. The latter was important for the active part of Russian patriotic population, since it made the feeling that Russia is not isolated at the world’s political arena. Especially important was functioning of American conservative channels, as say Fox News. Such channels by criticizing the Biden administration contributed to the α p R −beam of war-radiation. The war-lasing also illustrated the possibility to generate two laser beams from different population clusters. Such multi-beam social lasing is an interesting problem which deserves the special study. One of the sources of multi-beam lasing is sharp

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clustering of modern population into big clusters coupled to distinct Echo Chambers, used for amplification of different laser beams. For example, the majority of Russians and Ukrainians are still rigidly coupled to the state owned or supported information resources (including bloggers and “private channels”). It is interesting that Russians ignore practically completely the Ukrainian internet channels, even the Russian speaking ones; the same can be said about Ukrainians who generally understand well Russian and in principle may follow pro-Russian bloggers and channels. But mostly there is no interest for creation of the international picture of the situation. Why? One of the reasons is that Internet became the very dangerous place. Nowadays the information space is not less dangerous than physical space. For a “wrong” video, comment, and even like one can be heavily punished and in Russia and Ukraine “wrong” Internet-message can lead to prison and for many years. However, fear of punishment is not crucial in generation of information clustering of population.

Reddit Against Wall Street For those who didn’t know or forgot the story about a s-network uprising against Wall Street, we recall some details by following [344]. GameStop is a US video game retailer that has lost much of its market share to online trade and whose stock plummeted from $56 a share in 2013 to about $5 in 2019. Some big hedge funds decided that they would cash in on GameStop’s misery by shorting its shares. A short is a bet that an asset, such as a share, will decline in price. It’s a manoeuvre that can generate huge profits. But if the asset price doesn’t fall, investors can also lose a lot of money. A bunch of Reddit geeks on the online forum r/wallstreetbets, an investment discussion group that boasts more than 6 million users, decided to buy GameStop shares en masse. Perhaps they saw it as an investment, perhaps they were bored, perhaps they wanted to inflict pain on Wall Street. Whatever the reason, the consequence was to push GameStop’s share price up. And up. Once it became a global story, others piled in too, boosting the share price from about 40 to almost 400 in a matter of days. As a result, big investors lost big... . The story, however, is not just about traders getting their comeuppance, but also about the absurdity of the stock market.

To analyze this event, we shall appeal to social laser theory with its application to s-atoms operating at the financial market. In this framework this “global story” demonstrated not the absurdity of the stock market, but rather the possibility to use new financial technology for generation of short squeezing. And as usually, this GameStop short squeezing generated huge profits for those who designed and ignited the process of stimulated amplification of coherent s-actions. In this case “sactions” were posting comments at Reddit expressing believes (hopes, instructions) that GameStopp shares will go up in price. These were actions in the information space. They led to actions in the financial space—buying of GameStopp shares. We finalize this short consideration of Reddit “uprising” by a few citations from media-sources [411]:

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And though the share price dipped on Monday, Feb. 1, by more than 30%, many Reddit users say they’re buying more GameStop stock, convinced it’ll rocket even higher. Jaime Rogozinski, the apparent founder of the Reddit community at the heart of all this, told The Wall Street Journal it’s like ‘a train wreck happening in real time.’ Keith Gill, the trader in the Reddit community who helped kick off the battle, told the paper he ‘didn’t expect this.’ There might be something cathartic in watching the wolves of Wall Street themselves being savaged, but we should not romanticise the Reddit geeks. This was not an ‘uprising’ or ‘the French Revolution of finance’, as Donald Trump’s former communications director Anthony Scaramucci absurdly described it, but a scheme to play professional investors at their own game. [Guardian]

11.9 Technical Details Losses, Coefficient of Reflection-multiplication As in physical lasing, the above ideal scheme is complicated by a few factors related to losses of s-energy in the Echo Chamber. As is known, not all photons are reflected by mirrors of the optical cavity, a part of them is absorbed by the mirrors. The coefficient of reflection plays the fundamental role. The same problem arises in social lasing. An essential part of posts is absorbed by the information mirror of the Echo Chamber: for some posts, the probability that they would be read by members of the s-group is practically zero. Additional (essential) loss of s-energy resulted from getting rid of communications carrying quasi-colors different from the quasi-color α of the bunch of the communications initiating the feedback dynamics in the Echo Chamber. Such communications are generated by spontaneous emission of atoms in the s-group. The real model is even more complex. The information mirror is not homogeneous, “areas of its surface” differ by the degree of readability and reaction. The areas can be either rigidly incorporated in the structure of the s-group or be formed in the process of its functioning. As an example of functionally created information layers, we can point to ones which are coupled to the names of some members of the s-group, say “area” related to the posts of a Nobel Prize Laureate has a high degree of readability and reaction. But, of course, one does not need to be such a big name to approach a high level of readability and reaction. For example, even in science the strategy of active following to the main stream speculations can have a very good effect. Top bloggers and YouTubers create areas of the information mirror with high coefficients of reflectionmultiplication (see below (11.14)) through collecting subscriptions to their blogs and YouTube channels. It is clear that the probability of readability and reaction to a post depends heavily on the area of its location in the information space of a s-group or generally Facebook, YouTube, or Instagram. The reflection-multiplication coefficient of the information mirror varies essentially.

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Consider first the physical mirror and photons reflected by it. From the very beginning, it is convenient to consider an inhomogeneous mirror with the reflection coefficient depending on mirror’s layers. Suppose that k-photons are emitted to area x and n of them were reflected, i.e., (k − n) were absorbed. Then the probability of reflection by this area P(x) ≈ n/k, for large k. (11.13) Now, for the information mirror, consider a sequence of posts, j = 1, 2, ..., k, that were put in its area x. Let n j denotes the number of group’s members who reacts to post j. Each n j varies between 0 and N , where N is the total number of group’s members. Then coefficient of reflection-multiplication P(x) ≈ (

k 

n j )/k N , for large k, N .

(11.14)

j

If practically all posts generate reactions of practically all members of the group, then n j ≈ N and P(x) ≈ 1.

Improvement of Coherence: Direct and Indirect Filtering We have already discussed in detail the multilayer structure of the information mirror of an Echo Chamber. This is one of the basic information structures giving the possibility to generate inside it the information field of the very high degree of coherence: a very big wave of information excitations of the same quasi-color, the quasi-color of stimulating communications. It is sufficient to stimulate atoms having the potential of posting in the areas of the information surface having the high coefficients of reflection-multiplication. These areas would generate a huge information wave directed to the rest of the s-group. Spontaneously emitted communications would be directed to areas with the low coefficients of reflection-multiplication. How is this process directed by the Internet engines? It is described by the model of the dynamical evaluation of the readability history of a post. We shall turn to this model in Sect. 11.9. Although the dynamical evaluation plays the crucial role in generating the coherent information waves, one has not to ignore the impact of straightforward filtering. We again use the analogy with physics. In the process of lasing, the dynamical feedback process in the cavity excludes the excitation of the electromagnetic field propagating in the wrong directions. In this way, laser generates the sharply directed beam of light. However, one may want some additional specification for excitations in the light beam. For example, one wants that all photons would be of the same polarization. It can be easily done by putting the additional filter, the polarization filter, that eliminates from the beam all photons with “wrong polarization”. Of course, the use of an additional filter would weaker the power of the output beam. The latter

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is the price for coherence increasing. In social lasing, the role of such polarization filters is played by say Google, Facebook, Instagram, or Yandex control filtering, e.g., with respect to the political correctness constraints. Besides numerous moderators, this filtering system uses the keywords search engines as well as the rigid system of “self-control”. In the latter, users report on “wrongly (quasi-)colored posts and comments” of each other; the reports are directed both to the provider and to s-groups—to attract the attention to such posts and comments.

Dynamical Evaluation System The dynamical evaluation system used, e.g., by YouTube, increases post’s visibility on the basis of its reading history, more readings imply higher visibility (at least theoretically). However, the multilayer structure of the information mirror of YouTube should also be taken into account. The main internet-platforms assign high visibility to biggest actors of the mass media, say BBC, EuroNews, RT, that started to use actively these platforms. Then, and this may be even more important, these internet-platforms assign high visibility to the most popular topics, say nowadays the coronavirus epidemy, videos, posts, and comments carrying this quasi-color are elevated automatically in the information mirrors of Google, YouTube, or Yandex. Of course, the real evaluation system of the main internet actors is more complicated and the aforementioned dynamical evaluation system is only one of its components, may be very important. We would never get the answer to the question so widely discussed in communities of bloggers and YouTubers: How are the claims on unfair policy of the internet-platforms justified? By unfair policy they understand assigning additional readings and likes to some internet-communications or withdraw some of them from other communications.

11.10 Social Spin Electrons in an atom have spin and photons of the electromagnetic field have polarization—the analog of spin. For simplicity, we will speak about spins of electrons and photons. Spin is characterized by the direction of some axis in threedimensional space. The notion of spin is the quantum mechanical counterpart of the notion of the angular momentum of a classical particle. The latter has not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Spin represents an axis via its projections. The crucial point is that spin’s projections on coordinate axes can only take the discrete values: {si : si ∈ {−s, −(s − 1), . . . , s − 1, s}},

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where i = x, y, z (the coordinate axes) and s is the quantity known as the principal spin quantum number [353, pp. 372–373]. Another distinguished feature of quantum spin is that this quantity, as any quantum quantity, can be treated only in terms of observations. Each spin’s projection is represented by an operator Si . These operators do not commute. Hence they cannot be jointly measurable; in particular, the joint probability distribution for their outcomes is not well defined. So, the classical image as the angular momentum vector can be misleading, but our imagination cannot produce anything better. Physicists often speak about spin vector. It is more correct to speak about spin in some direction: to select some direction in the three-dimensional space v = (v1 , v2 , v3 ) and consider the operator of spin projection on this axis Sv . The energy exchange between the electromagnetic field and an atom depends on spins’ structure of the electrons in an atom and photons in the field. Their spins should match each another, otherwise neither absorption nor emission can happen. In social laser model developed up to now, this spin structure is not taken into account, although it is natural to assume that both s-atoms and information excitations are “oriented along some social axes” and they interact only when their orientation axes are close. It is natural (by following the socio-physical methodology) to call such axes social spins. Of course, the quantum-like model is more complicated, since, as we know from quantum physics, spin cannot be mathematically described as a vector. We can operate only with spin operators, the spin projection operators. So, we propose the following heuristic picture. People have their personal social axes as well as information excitations (carried by communications, news, messages, posts in s-networks, comments to these posts). The degree of interaction between s-atoms and information excitations depends on matching of their social axes. As in physics, the processes are probabilistic, small angles between axes imply higher probabilities of absorption and emission, if directions are opposite, then probability of interaction is practically zero. Once again, as in physics, the picture of axes is too naive, one has to translate it to the operator language and work with projections. The social-political preferences of the majority of American population can be described by one axis; say spin up and down encode the liberal and conservative views respectively. (I am sorry by encoding the conservative preferences as “social spin down”. We can encode spins another way around.) If a s-atom (say from California) has the spin up preference, then she would not react to a communication carrying the conservative preference; similarly a s-atom (say from Arizona) with spin down preferences would not react to an information excitation about, e.g., climate change. In the human gain medium consisting of liberal s-atoms, one cannot start stimulated emission by sending a batch of information excitations with social spin down. Such liberal atoms would neither react to radiation emitted by conservative Echo Chambers. Of course, the majority of spin up s-atoms are simply disconnected from such Echo Chambers But even by meeting occasionally an information excitation with spin down, a liberal s-atom would not interact with it; moreover even high density flow of such information excitations would not stimulate spin up s-atoms to emit or absorb quanta of s-energy.

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The natural question about the origin of spin in a human arises. We stress that social spin is a stable quantity. Typically a person is keeping with liberal (or conservative) values during many years, or even during the whole life. As always, there are exceptions from any rule. For example, Donald Trump changed his views completely and moved from the Democratic Party to the Republican Party. However, this case is so unusual that some Republicans claimed that Trump played the role of Trojan Horse; he infiltrated into the Republican Party to destroy its image, in particular, by his eclectic speeches and posts in Twitter. The basic question is whether human’s spin has the biological or social basis? or their combination? We do not know the answer. Heuristically human spin is determined by the s-environment (therefore, we called it social spin). But it is even possible that human spin is determined by her genome and physiology of organism’s functioning. In any event, it is clear that the present theory of social laser must be developed to take into account the social analog of quantum spin.

11.11 Concluding Remarks On Social Laser We hope that this chapter would be useful for researchers working both in humanitarian and natural sciences. In this book, the methodology of social laser theory was enriched through the invention of infon. This is an analog of photon. Photons are the excitations of the quantum electromagnetic field, and infons are the excitations of the quantum social-information field. We emphasize the bosonic nature of these fields which is the basic factor leading to the generation of the cascade process of the stimulated emission in the lasers’ gain media, both physical and human. The infons-language is convenient to describe the dynamics of cascades’ iterations within social laser-the gain medium and resonator. Social resonators are implemented by coupling laser’s gain medium to s-networks. We emphasize the role of resonators in, e.g., lasing for the competing candidates in the presidential elections. Generally, we are interested in the process of creation of two competing beams of s-actions, as the mentioned elections or war and peace beams in contemporary information space. As well as in article [290], we highlight the role of Internet-based Echo Chambers in increase of the amplitude as well as color and quasi-color coherence of the beams of social radiation. Echo Chambers are also used to increase temporal coherence, to make the spike of social radiation sharply concentrated in the time domain. It is the good place to point out that Internet-based resonator is a kind of active mirror, in contrast to the optical cavities with the reflecting mirrors. Once again (cf. with [267, 290]) we highlighted the possibility to supply s-energy to the gain medium with infons of the quasi-color different from the quasi-color of infons in the stimulating injection (and hence the output beam of social radiation). This property of s-atoms is called memeorylessness. It plays the important role in social engineering. The real aim of social lasing can be deemed at least at the stage of

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s-energy pumping in the gain medium. Moreover, s-energy produced by one social laser can be used at the stage of approaching population inversion in another social laser. In turn the latter can be used for new social laser and so on ... ; say the “climate change beam” of social radiation was used for the A-state of COVD-19 social lasing, in terms of the s-energy emitted by the latter was directed to create the “war beam”. The natural question arises: What kind of social lasing may be expected after the war?

Chapter 12

Stability in Biological, Ecological, and Social Systems via Fröhlich Condensation

12.1 Modeling of Fröhlich Condensation As was demonstrated by Fröhlich [157–159, 161, 162], some biosystems can approach the stationary energy state through the process of phonons’ condensation on the lowest vibrational mode—the Fröhlich condensation(see also for review Vasconcellos et al., 2012, and references in it). This stabilization is approached not through system’s isolation, but through sufficiently intensive energy pumping and an energy exchange between different vibration modes (energy levels) of a biosystem and with the surrounding heat reservoir E. Such a biosystem can preserve its stability, despite permanent energy supply, and it neither totally relaxes to the state of thermodynamic equilibrium with E: . . . under appropriate conditions a phenomenon quite similar to a Bose condensation may occur in substances which possess longitudinal electric modes. If energy is fed into these modes and thence transferred to other degrees of freedom of the substance, then a stationary state will be reached in which the energy content of the electric modes is larger than in the thermal equilibrium. This excess energy is found to be channelled into a single mode–exactly as in Bose condensation–provided the energy supply exceeds a critical value. Under these circumstances a random supply of energy is thus not completely thermalized but partly used in maintaining a coherent electric wave in the substance.

Fröhlich [158]. This process has some similarity with the process of the Bose–Einstein condensation. The crucial difference between them is that the Fröhlich condensation is reached by systems in the heat reservoirs with sufficiently high temperature which are exposed to sufficiently intensive energy supply. One can say that Bose–Einstein and Fröhlich condensations are the passages to stability through freezing and high temperature energy exchange, respectively. Since biosystems frozen to very low temperature are dead, the phenomenon of the Fröhlich condensation is more interesting from the biological viewpoint.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_12

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12.1.1 Coherent Vibrations in Biomolecules and Cells The basic example of the systems with Fröhlich condensation are giant dipole oscillators in macromolecules inside a cell, e.g., proteins or genomes; solution in a cell is their heat reservoir E. Energy supply to a cell is electromagnetic and chemical. The giant dipoles approach synchronized oscillations at the lowest energy mode, which is nevertheless essentially higher than the average energy of excitations in E . And behavior of the whole macromolecule is characterized by coherence of oscillations or in terms of energy, by coherence of the energy distribution. Starting with such systems, Fröhlich [157–159, 161, 162] extended the domain of applicability of his theory to the level of cells composing body’s organs, e.g., lever. In all cells of this organ, all dipoles in macromolecules are involved in the coherent fluctuations. This regime leads to synchronization of all biological processes in an organ. In [162], it was emphasized that Many other types of physical order may be imposed on certain systems when they are lifted from their thermal equilibrium state even though they are spatially disordered. This does not hold, however, for all systems, and no general criteria has been found, so far, which when satisfied permits particular systems to exhibit some type of order when energy is supplied to them. Most systems, however, simply raise their temperature. Biological materials, when they exhibit typical biological properties, must be ‘fed’, i.e., energy passes through them. This energy may arise from chemical processes, or it may be supplied by the Sun.

So, biosystems have some specialties which generally are not characteristic of physical systems. The aim of this paper is to formulate such specialties in the framework of information biology.

12.1.2 Long-Range Nonlinear Interactions Fröhlich used the methods of nonequilibrium thermodynamics, the condensation corresponds to the stationary state which is far from equilibrium; cf. [446]. Such states are generated by nonlinear dynamics for energy exchange between systems coupled via long-range nonlinear interactions, e.g., of Coulomb’s type. Straightforward mathematical modeling of such dynamics is the complex problem. Therefore, the statistical methods are useful, and their application led Fröhlich to his discovery. Quantum theory provides one of the most powerful tools for statistical modeling, with the quantum master equation for the density operator. For Fröhlich condensation, the quantum model was developed, for example, in articles [440, 470]. In particular, the quantum approach rigorously formalized the original thermodynamic considerations of Fröhlich. The phenomenon of coherent vibrations in biomolecules was studied with open quantum systems theory and its quantum modeling has a lot in common with laser theory, see article [440] for exploration of this commonality.

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12.1.3 Quantum-Like Modeling of Fröhlich Condensation Our aim is to show that the range of applicability of Fröhlich’s theory can be extended to macroscopic biosystems, including the ecological and social systems. The possibility of such extension was indirectly mentioned in the original works of Fröhlich. The analysis of cancer’s generation [162] was done in the abstract framework by discussing the collective modes regulating behavior of cells in an organ. Originally such modes were identified with frequencies or vibration energies in dipoles in biomolecules, but the condensation model was not rigidly coupled to vibrations [162]: it will be noticed that the term ‘collective mode’ might be given a different physical interpretation from the one suggested here, provided individual cells have collective properties which enable them to react to collective model and which could be destroyed by external influences.

See also [184]. Quantum physical modeling [440, 470] of coherent synchronization of dipoles in say DNA molecule belongs to the domain of the standard applications of quantum theory. Its extension to macroscopic biosystems is not straightforward, a variety of methodological and interpretational problems must be resolved. This can be done in the framework of quantum-like modeling. As was already pointed out a few times, in this approach the physical structure of a system is not crucial; only the way of information processing is important. In some situations, the mathematical formalism of quantum theory matches better the structure of processed information. Then, it is pragmatically natural to use the quantum operator description. We do not claim that such situations cannot be described classically. The quantum-like formalism gives the possibility to model the Fröhlich condensation for arbitrary collective modes from proteins and cells to ecological and social systems, by using the calculus of creation and annihilation operators. This calculus is basic in quantum theory of the Fröhlich condensate and laser physics [440, 470]. We apply it in the general information framework, with the operators of creation and annihilation of the social and behavioral excitations.

12.1.4 Fröhlich Condensation of Information Excitations As in the previous chapters, biosystems are considered as information processors and quantum-like modeling is used to describe information processing within the quantum mathematical formalism. Instead of photons and phonons, we consider quanta of information, each carries a few attributes. The latter determines the collective modes for condensation. As is standard in quantum theory, at least with the Copenhagen interpretation, these modes are invented as observables, and not as objective properties of information processors. For simplicity, we consider just a single mode; say A with the spectrum (the range of values)

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0 < a1 < a2 < · · · < an .

(12.1)

Biosystems, information processors (I -processors), absorb and emit information excitations with amplitudes given by the differences between A-lines, ai j = (ai − a j ). And the information Fröhlich condensation is creation of the stationary state corresponding to the minimal nonzero value, A = a1 .

12.2 Review on Fröhlich’s Works Fröhlich [157–159, 161, 162] considered a biosystem containing the following components: 1. Oscillating segments of giant dipoles in macromolecules. 2. A heat bath; say protein molecular in a cell filled with solution. The system is open, and it is exposed by an external energy pump directed to the oscillating units. They are characterized by the frequency band: ω1 < · · · < ωn .

(12.2)

For the further considerations, it is convenient to give the description solely in terms of energy. Frequency modes (12.2) can be associated with the corresponding energy band: (12.3) E 1a < · · · < E na . Oscillations of single units can be described by appropriate superposition of these waves. In Fröhlich’s papers, the waves were treated as classical vibration waves. In quantum and quantum-like models, one considers superposition of the energy states |E ja  :   c j |E ja , |c j |2 = 1. (12.4) |ψ == j

j

(3) Fröhlich [162] did not emphasize the dipole nature of oscillating subsystems. He called them simply ‘units’: Consider a suspension of a large number of units in a substance which will be treated as a heat bath.

And this terminology is convenient for quantum-like modeling. The external energy pump is characterized by supplies Ji to the corresponding energy modes; it was supposed that supply is uniform w.r.t. to the energy band, i.e., Ji = J (the mode-homogeneous supply). The heat reservoir is assumed to be in the state of the thermodynamical equilibrium characterized by the Planck formula for the average number Nib of excitations corresponding to each energy mode—reservoir’s spectrum is discrete:

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195

E 1b < · · · < E nb . Hence, Nib =

1 e

E ib λ

−1

,

(12.5)

(12.6)

where λ is the energy scale parameter. In thermodynamics, the energy scaling parameter λ is represented as λ = K T, where T is bath temperature, K is the Boltzmann constant. Each unit exchanges energy with the heat bath; this is an exchange of energy in quanta (discreteness of energy transfer). The net rate of loss L 1i of the mode with energy E ia (containing n i quanta) is written as L 1i = φ[n i e

E ia λ

− (1 + n i )],

(12.7)

where the coefficient φ > 0 encodes interaction of (bio-)units with the bath (e.g., solution in a cell). Of course, besides this straightforward interaction with the heat bath, the units are involved into the energy exchange between different modes and the bath: emission and absorption of E ia -energy excitation. The net range of loss L 2i of the mode E ia due to such processes can be written in the form: L 2i = χ

 E ia −E ja [n i (1 + n j )e λ − n j (1 + n i )],

(12.8)

j

where χ is the transition probability, it generally depends on the indexes i, k, but, for simplicity, Fröhlich neglected by this dependence. He remarked that this assumption essentially simplifies calculations, but it does not influence the result. The quantities L 1i , L 2i were phenomenologically selected to match the requirement that in the absence of energy supply, J = 0, thermal equilibrium L 1i = 0, L 2i = 0

(12.9)

requires the Planck distribution for n i , ni =

1 e

E ia λ

−1

.(5)

(12.10)

Here the Bose–Einstein statistics of the equilibrium state is assumed (see Chap. 18). The condition of stationarity in the presence of external energy supply and heat bath in the state of thermodynamical equilibrium can be written in the form of equation: J = L 1i + L 2i .

(12.11)

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This equation gives us (see [] for the derivation) the following formula: ni = where

D Ve

 D = 1+

E ia λ

−1

,

(12.12)

 J , φ + χN

 and N = j n j is the total number of the energy excitations in the system. We recall that J is the intensity of energy pumping which is assumed to be the same for all modes. It can be proven that the coefficient V ≤ 1 and, hence, it can be represented in the form: (12.13) V = e−μ/λ , where in thermodynamics the quantity 0 ≤ μ ≤ E 1a is the chemical potential . Thus, n i has the form D n i = Eia −μ , (12.14) e λ −1 Up to now we worked with the standard balance equation and its stationary state under condition of the Bose–Einstein distribution of energy in the heat bath. The condensation to the E 1a -mode takes place when chemical potential μ approaches E 1a very closely. This can be approached in the high temperature regime K T >> E 1a .

(12.15)

Thus, the temperature of the heat bath—solution in a cell—must be essentially higher than the energy of the ground level of oscillations in the units (electric dipoles) of a biosystem (protein or genome). In this regime, ni ≈

λD E ia − μ

(12.16)

and this approximation is good for the pumping level J exceeding some threshold J0 . Thus, for (12.17) J ≥ J0 , n 1 crucially dominates over the rest of n j . This is the phenomenon of Fröhlich condensation.

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197

12.3 Conditions for Fröhlich Condensation We now list the conditions which combination leads to this phenomenon: • D Discrete character of the energy exchange between a system, the heat bath, and external energy source. • BE Bose–Einstein statistics of the thermal equilibrium state of the heat reservoir. • BV Big heat capacity of reservoir comparing with capacity of a bio-component. • HT High temperature of the reservoir. • ES High intensity of external energy supply. Often condition BE is coupled with mystical features of quantum systems. However, in fact, it is based on heuristically very simple and natural condition of indistinguishability [405]; see Chap. 18.

12.4 Quantum Formalism for Fröhlich Condensation My interest for the Fröhlich condensation was stimulated by paper [440]. It was written by the group of experts in quantum modeling of laser processes, and it is readable only by such experts. The paper contains many technical details which are not important for the majority of readers of this book. It is better to follow earlier paper [470] which presentation is simpler and more useful for our further transition to the quantum-like framework. Consider biounits with discrete energy spectrum (12.3) or vibrations (12.2). To each energy mode E ia , the creation and annihilation operators ai and ai are assigned. These modes interact with the heat bath filled by the energy excitations with spectrum (see (12.5)): E 1b < · · · < E nb The corresponding creation and annihilation operators are denoted by the symbols bi and bi . The external energy supply has the energy spectrum E 1c < · · · < E nc ,

(12.18)

which is associated with creation and annihilation operators denoted by the symbols ci and ci . Hamiltonian of compound system: dipole, its heat bath, and external field of energy pumping, is written in the terms of the creation and annihilation operators in the standard way: (12.19) H = H0 + Hab1 + Hab2 + Hac .(14) The operator H0 represents the free energies of the corresponding subsystems: H0 =

 i

E ia ai ai +

 j

E jb bj b j +

 k

E kc ck ck

(12.20)

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Later we use abbreviation ‘h.t.’ for ‘Hermitian conjugate’ and write other parts of Hamiltonian in the form:  [αai bj bk + h.c.], (12.21) Hab1 = ij

Hab2 =

 [βai a j bk + h.c.],

(12.22)

i jk

Hac =

 [γ c j ai + h.c].

(12.23)

ij

Here α and γ are the coefficients for the interaction of dipole with the heat bath and the external flow of energy, respectively, leading to creation and annihilation of single energy excitation in the dipole. The coefficient β is the interaction constant for the dipole-bath energy exchange leading to creation of one energy excitation in the dipole and annihilation of another. Higher order processes, i.e., creation of annihilation of two or more energy excitations in the dipole at the same time are not considered. Such processes are not typical for biosystems. By using this Hamiltonian and the trace operation, one can write the dynamical equation for the average number n i (t) of energy excitations for the dipole system. As well as in the Fröhlich considerations, Wu and Austin [470] assume that the state of the heat bath can be modeled as the thermal equilibrium state. The Planck expression for the number of E-energy excitations is substituted into the dynamical equation for n i (t). For stationary state w.r.t. the distribution of energy excitations in dipoles, it is required that the average value of the change of the phonon number with any energy be zero. This equation leads to Fröhlich’s expression (12.14). To derive the existence of the supply threshold J0 for which the chemical potential μ approaches E 1a , Wu and Austin [470] referred to article [162].1

12.5 Cancer Fröhlich [162] applied his theory to modeling of cancer initiation process in a tissue or an organ of an organism. He described this process in biophysical terms: The problem of cancer is one of the controls of cell division. While bacterial cells under appropriate conditions keep on dividing, cells in a fully grown differentiated tissue or organ divide relatively rarely, though they divide more rapidly in growing tissues and organs. We shall assume that a controlling agency belonging to the whole organ, i.e., a collective 1 Surprisingly, I was not able to find a consistent and rigorous presentation of the Fröhlich formalism, neither in framework of classical nonequilibrium thermodynamics nor in the quantum framework (article [] is difficult for reading).

12.6 Condensation of Information

199

property, exists and that individual cells respond to it. …This then implies that a coherent electric vibration carried collectively by the individual nuclei extends through the whole tissue (organ), and that the vibrations of each individual cell respond to it and are thus held in the appropriate phase, such that the total electric energy is a minimum. In general, this vibrating polarization field will form a complicated spacial pattern.

By Fröhlich this polarization field controls the rate of cell’s division. Perturbation of the field destroys the control mechanism and leads to unregulated division of some cells. This can become a self-stimulating process: increase in the number of cells vibrating out of the common phase makes the polarization field weaker and the latter in turn leads to weaker control of coherence in cells’ behavior.

12.6 Condensation of Information In a long series of works, see, e.g., [243, 247, 253] I have been working on the purely informational model of biological social, and financial processes. In this project, the appeal to quantum information is crucial since the latter is about observational and not objective features of systems. Here the previous studies are structured to formulate the purely informational version of the Fröhlich condensation. Information Excitations Following the paradigm of information biology (see Chap. 4), we consider biosystems as information processors (I -systems). They absorb and emit portions of information. Portions are discrete, otherwise it is difficult to interpret and use them. So, we can speak about information quanta or excitations. Each excitation has a few attributes (A1 , . . . , A N ), as in physics photon (light excitation) has say energy A1 , polarization A2 , and momentum A3 . For simplicity, we mainly proceed with excitations carrying single attribute A. So, we can speak about A-excitations. Their magnitude is quantified with discrete spectrum. For example, A can be s-energy. By following the quantum-like approach, we model Fröhlich condensation at all scales of physical, biological, ecological, and social organization. Information Formulation of Condensation Conditions We consider a complex I -system S and its subsystems Sk . Similarly, to condition D (Sect. 12.3), it is assumed that A-spectrum is discrete, see (12.3). In the previous considerations, S was a cell, and its units were dipoles in some biomolecular, say protein. Now a complex I -system S can be a human organ and its units are cells; for example, the brain and neurons in it. The S can be an ecosystem and Sk be different organisms living in it. In Chap. 11, S was a social group or just a virtual social system based on a social network. The complex I -system S is compound of units Sk and information reservoir E. The latter is filled by a variety of I -excitations in the state of equilibrium with the Bose–Einstein statistics given by the Planck formula—condition BE (Sect. 12.3). Here the coefficient λ gives the scale of exchange of A-excitations in a compound system S. One can couple λ with a kind of temperature for ‘a gas of A-excitations’;

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see Sect. 12.6. We also assume that the information capacity of Sk is essentially less than the capacity of E (condition BV). The system S is exposed to supply of external information carrying A-excitations. The model assumes discrete A-spectra both for the external information supply and for the reservoir–condition D (Sect. 12.3). We consider I -systems exposed for information supply of high intensity, condition ES. Bose–Einstein Statistics of Information Excitations What are the sufficient conditions for Bose–Einstein statistics of information excitations in the bath? Why did we transfer condition BE to the domain of information? Here one can follow book [405] on the statistical mechanical derivation of the classical and quantum statistics of the equilibrium states of the bath. In fact, the only difference between systems leading to classical versus quantum statistics is their distinguishability versus indistinguishability (Chap. 18). And we recall that Schrödinger treated the attributes assigned to systems in the Copenhagen like manner, i.e., as w.r.t. observables. In article 267 this distinguishability versus indistinguishability reasoning was applied to information excitations (see Chap. 18). Thus, the BE-condition can be transformed into the condition: ID Indistinguishability. We remark that indistinguishability is understood in the following way (Chap. 18): Two I -excitations (quanta of information) having the same A-attributes are considered as indistinguishable. So, it does not mean that two concrete communications C1 and C2 carrying I -excitations coincide. They can essentially differ in their content. But, if the Aattributes of these communications coincide, A(C1 ) = A(C2 ) then C1 and C2 are indistinguishable, but only for this concrete information attribute A. For example, let A be s-energy coupled to the war in Ukraine; A is quantified as (12.18). If two communications (generated by mass-media) carry the same amount of s-energy coupled to the war, A(C1 ) = E ic = A(C2 ), then C1 and C2 are indistinguishable—in the context of the war in Ukraine. We can consider a model with a few attributes of information A1 , . . . , A N . If Ak (C1 ) = Ak (C2 ), k = 1, 2 . . . N , then communications C1 and C2 are indistinguishable (w.r.t. these system of attributes). By proceeding solely with ID, one derives (Chap. 18) all possible quantum statistics, cf. [405]. In the Fröhlich I -context, the Fermi–Dirac statistics and parastatistics seems to be improper. However, who does know? We cannot exclude the possibility that, in biology, cognitive social sciences, such statistics also would be useful. High Information Temperature We recall that A-excitations are quantified with discrete spectrum (12.3). Condition HT, see (12.15), can be rewritten as λ >> E 1a .

(12.24)

12.7 Information Temperature

201

The ground level of A-excitation E 1a of components Sk of compound I -system S is essentially less than the scale of exchange of A-quanta with information reservoir E in the compound system S. This is the most delicate condition. A consistent introduction of a kind of I -temperature (Sect. 12.7) would clarify it essentially. Information Energy For further considerations, it is convenient to assume that information excitations carry a kind of energy representing potentiality to perform actions of the information nature—information energy (I -energy).2 We will introduce different types of I energy, e.g., social, private, and behavioral I -energies. I -temperature (Sect. 12.7) represents the intensity of I -energy exchange. Redistribution of Information Excitations Suppose that each I -excitation carries the A-attribute. For simplicity, we operate with just one attribute, e.g., I -energy. I -processors have the A-spectrum (E ia ), the range of possible A-values, with associated creation and annihilation operators ai and ai . The I -reservoir is characterized by its A-spectrum (E ib ) with associated operators bi and bi , and external information supply by spectrum (E ic ) and operators ci and ci . The description with the creation and annihilation operators is formal; it is insensitive to the kind of excitations which generation and annihilation is described— the energy excitations or other types. The corresponding Hamiltonians and the master equation can be used (Sect. 12.4). The quantum-like model can also be considered for I -excitations carrying multiple attributes A = (A1 . . . , A N ); the corresponding creation and annihilation operators will be labeled by multi-indexes.

12.7 Information Temperature We introduce I -temperature thermodynamically by consideration of the state of equilibrium for the information reservoir E. This is equilibrium for exchange by Aexcitations between E and subsystems emitting these excitations to E and absorbing them from it. To make closer the analogy with physics, we select A = E as I -energy. For the equilibrium state, the exchange parameter λ for the I -energy excitations can be determined from the Planck formula (12.6): λ=

E ib . ln(1 + 1/Nib )

(12.25)

For the equilibrium state, RHS of (12.6) does not depend on i. This feature can serve as a test for approaching I -equilibrium. I -temperature (w.r.t. its fixed attribute 2

For cognitive systems, information actions can (but generally do not need) generate actions in physical space. So, information energy can be transferred into physical energy. For the moment, we do not have a mathematical model of such transition. In turn, generation of information excitations consumes physical energy; the following picture arises: …→physical energy→information energy→physical energy→…

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A) can be introduced via calibration with some parameter k with dimension “Adimension/temperature”: E ib . (12.26) T = k ln(1 + 1/Nib ) (We stress that in physics selection k = K , where K is the Boltzmann constant is also the subject for the agreement.) The Planck formula can be written as in physics: Nib =

1 e

E ib kT

−1

,

(12.27)

where k is an information analogue of the Boltzmann constant. By the agreement we select it as independent from an information reservoir. (In principle, the parameter k can depend on information reservoirs.) The I -version of the high temperature condition HT has the form: (12.28) kT >> E 1a . Hence, E must be hot (in the sense of the above definition of information temperature).

12.8 Stability of Complex Information Societies We now apply the I -reformulation of Fröhlich condensation theory to model order stability in complex social systems. Social Energy and Atoms Following Chap. 11, humans are considered as social analogues of atoms, s-atoms, discrete elementary components of human society. We are interested only in I processing performed by s-atoms, so these are I -processors. Consider one of the basic attributes of I -excitations—social energy (s-energy) quantifying human’s potential to perform social actions (see Chap. 11). Generally, actions are of the information nature, they are related to information exchange, e.g., decision-making, problem-solving, but some of them have the potential to be transformed into actions in physical space. s-energy is the special form of I -energy; its specialty is in the class of I -processors, humans. It is supposed that s-energy spectrum of s-atom can be discretely quantified (see D and (12.2)). This is how human cognition works. It calibrates the degree of excitement carried by I -excitations. In the simplest scale, these are just two values, E 1a and E 2a : weak and strong excitement. Generally, the scale is more complex, see (12.2). External supply of s-energy is composed of I -excitations which are generated by a variety of information sources, mainly mass-media. Each communication, news, comment, article, video, carries a discrete portion of s-energy. The quantification of s-energy depends on an individual; in complete agreement with the Copenhagen ideology s-energy is not the objective property of a communication. Its concrete

12.8 Stability of Complex Information Societies

203

value is assigned by observers and humans. In principle, people can assign, often unconsciously, different values of s-energy to the same communication, say news. For social lasing as well as for Fröhlich condensation, homogeneity of a social group is important. In laser physics, impurities in atom population—active medium—disturb generation of the coherent laser pulses. Coherent condensation of DNA-vibrations in all cells in the concrete organ is due to homogeneity of spectra of dipole vibrations in all DNA-molecules. The latter, in turn, is due to the specific geometry of DNAmolecules in this organ. Social Conditions for Condensate • BE (Bose–Einstein statistics for excitations in social reservoir; Chap. 18): This is a consequence of indistinguishability condition ID. Of course, information stored in reservoir E is not totally contentless. But up to some degree, it can be characterized by just one observable—s-energy (single attribute of I -excitations, A = E). s-atoms are attracted by the degree of excitement (s-energy) carried by communication: comment, post, news, article. The model can be generalized by completing the s-energy with additional attributes B, C, D, . . . (quasicolors in theory of social lasing)3 of I -excitations. Indistinguishability up to s-energy and these attributes also leads to the Planck formula . • BV (Large social reservoir): The s-energy reservoir E is one of the basic components of the compound socio-informational system S. The informational environment of modern humans includes the variety of internet platforms as YOUTUBE, Live Journal, Facebook, Telegram, Twitter, and so on. They present the big collection of videos, articles, photos, posts, comments, comments on comments. It is natural to assume that in terms of the s-energy capacity E is very large comparing with s-atom. • HT (High social temperature regime): Social temperature T can be introduced in the general information framework (see Sect. 12.7). But, as in physics, this thermodynamical introduction via consideration of the state of information equilibrium can be completed by the operational approach, by considering so to say social thermometers and calibration procedures for them. In principle, one can proceed without appealing to social temperature, by operating with λ as model’s parameter, characterizing the scale of s-energy exchange in a society; see (12.24). • ES (High intensity of social supply): In the modern society, information supply has very high intensity, mass-media, and social networks pump huge flows of information carrying huge social energy. Moreover, these information flows are densely populated by highly energetic news: wars, pandemics, terror-attacks, scandals. It should be stressed that people are overloaded by variety of information coming from TV, newspapers, and various gadgets. This leads to contentless information processing, up to s-energy and some additional attributes. People do not have the computational and time resources for detailed cognitive analysis of information (clip thinking, popcorn brain, . . .). 3

In this theory, s-energy plays the special role. Therefore it is useful to separate it from the rest of attributes.

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The basis of the Fröhlich social condensation process is s-energy redistribution between possible levels of s-energy of s-atoms. Fröhlich condensation for social excitations leads to a stable society which is characterized by the homogeneous structure of the s-energy distribution. Open Information Societies as Social Fröhlich Condensate In contrast to the authoritarian societies, open democracies like USA, EU-countries, Canada, Australia, and so on preserve order stability practically without censorship and restriction of information production and its flows. Moreover, despite the powerful information flows of all kinds, the open societies are characterized by concentration of the most population on the approximately the same mode of s-energy. In particular, the slice of hyperactive people those who can disturb societal order is not so wide. Such people quickly “radiate” their energy. Where does it go? Its essential part is absorbed by the reservoir of social energy E. These highly charged guys emit a part of their energy into social networks, by posting comments, photos, videos, clips, articles at say Live-journal and responding to posts of others. Nowadays the main part of energy exchange between people goes through E. So, creation of the huge information (s-energy) reservoir is one of the basic conditions of social stability of open societies. Another basic condition is continuous pumping of s-energy via very powerful information flows. Stability is possible if s-energy supply overcomes some threshold. Social temperature (Sect. 12.7) has to be sufficiently high (so hot and shock news are really the necessary element of the stabilization structure). Fröhlich condensation has not only stabilization function, but it is also the powerful mechanism of societal energy accumulation on the basic active mode. Most members of such society are in the state of sufficiently high energy, so they are functionally active. The latter is also the important function of the social Fröhlich condensate. In contrast to the Fröhlich condensate model for the open information societies, the authoritarian societies can be quantum-like modeled with the Bose–Einstein condensate. However, the latter model is not so exciting, since approaching order stability via a kind of social freezing can be expected from the very beginning.

12.9 Order in Pack of Wolfs Consider a pack of wolfs, it reminds a macromolecule; wolfs are analogues of dipoles (or s-atoms in the model of the social condensate). As well as dipoles in a macromolecule, or humans in a social group, wolfs in a pack are coupled via nonlinear interactions and behavioral interactions. The nature and structure of these interactions is complex, and their mathematical formalization is difficult, if possible, at all. Their main impact is generation of order stability in a wolf pack and approaching the stable level of sufficiently high behavioral activity. The latter will be expressed in terms of behavioral energy. The wolfs in a pack behave coherently. And we want to model this behavior with Fröhlich’s theory. The latter is simple mathematically,

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205

one does not need to go deeply into the description of the individual behavioral interactions; statistical considerations lead to clearly formulated conditions of coherent behavior of the wolfs in a pack. Natural Information Reservoir Wolfs live in the natural environment E which reminds the heat bath in a cell or the reservoir of s-energy in human society; E includes forest, animals living in it, and natural (e.g., weather) conditions. This environment is treated as an information environment. Hence, e.g., animals are considered not as physical bodies, but as I processors: the sources and receivers of information signals, visual, audio, smell. This is the ocean of information and wolfs interact with it. They absorb information from E and emit their own signals (e.g., wolf’s howl and smell) into E. Exchange of Behavioral Energy We are interested in the special sort of I -energy (the attribute of information)— behavioral energy (b-energy). b-energy is expressed as potentiality to a variety of behavioral actions inside a wolf pack as well as with the components of E. So, wolfs are considered as discrete systems (so to say behavioral atoms) emitting and absorbing b-energy. The latter is like s-energy of humans. However, it is more natural to speak about animals’ behavior and not socialization. Inside the pack, wolfs exchange B-excitations, either straightforwardly, as struggle for does, for the place in pack’s hierarchy, or through more complicated chains of b-energy transitions. For example, in hunting the most energetic wolfs, the beater wolfs, spend essential part of their b-energy resources to catch a deer. But, then the meat is distributed through the whole wolf pack and those which, before hunting, were in low B-state transit to higher energy state. This transition is based on the physical energy contained in meet, chemical energy collected in organic molecules. Behavioral and Physical Energies Now we come to the complicated issue of interrelation of the physical and information energies. (We already had similar discussion for s-energy.) The process of transition of the physical energy into the psychic energy has been discussed a lot in psychology, cognition, behavioral, and social studies. However, in spite of the tremendous efforts, the notion of psychical, mental, or social energy as the objective property of say a human being was not firmly established. We defined information energy as measurable quantity, as an observable over information processors. In this approach, the physical energy can be treated as a kind of hidden variable behind the information energy. In quantum theory, the hidden variables conjecture has been intensively discussed: hidden variables can be found behind observable quantities. Such hidden variables can be determined in the classical manner, as objectively assigned to systems. The common consensus, based on the interpretation of violations of the Bell inequalities, is that hidden variables cannot be introduced straightforwardly, as context independent quantities. However, the context-dependent hidden variables can naturally be invented. And biosystem’s behavior is fundamentally contextual. In this framework, coupling of the physical and behavioral energies is contextual, i.e., it depends on

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12 Stability in Biological, Ecological, and Social Systems via Fröhlich Condensation

context. It means that depending on context an animal can get different amounts of b-energy from the same amount of food. Bio-contexts are very complex. And it is practically impossible to describe them mathematically, in such a detail that it would be possible to model the process of transformation of calories into b-energy. However, as was pointed out, in this book we do not try to model the transfer between the physical and behavioral energies. External Supply of Behavioral Energy Let us turn to the original Fröhlich model. External energy supply comes to a cell in the form of electromagnetic and chemical energies: photons and ions arriving into a cell from outside. A cell by itself is the environment for macromolecules, e.g., proteins. A molecule absorbs quantum of external energy and then it is redistributed inside this molecule. Redistribution is performed via transfer of vibration quanta—phonons. Fröhlich pointed out that the energy forms involved in the process of coherent condensation are not important. The crucial role is assigned to interrelation between the energy scales of external supply, vibrations of dipoles, and the heat bath. They are hierarchically ordered, from highly energetic external quanta to less energetic internal vibrations, and, finally, to the heat bath excitations having lower magnitudes. And coming back to wolf’s behavior, we can relate to external supply Bexcitations which have higher magnitudes than b-energy redistribution in a pack. The latter in turn have higher magnitudes than reservoir’s B-excitations. Such highly energetic B-supply includes arrival to E of animals or humans which serve for wolfs as new high B-excitations, in the form of nutrition and danger. It also includes extreme changes of weather conditions as well as forest fire. Conditions for Behavioral Fröhlich Condensate We now go through conditions for generation of the Fröhlich condensate (Sect. 12.3). • BE (Bose–Einstein statistics for excitations in the environmental I -reservoir; Chap. 18): This condition is a consequence of indistinguishability behavioral excitations. Of course, such excitations are not completely indistinguishable. Besides b-energy, they carry some attributes, e.g., sounds, smells, etc. And animals react not only to the b-energy magnitude, but also to the magnitudes of other attributes. Each attribute contributes to animal’s behavior. For simplicity, in the model under consideration I -excitations are distinguished only w.r.t. the B—energy magnitude. • BV (Large information environment): I -capacity of the environmental reservoir E must be essentially bigger than single wolf’s I -capacity. This is really the case if a wolf lives in the natural environment and not in a cell in Zoo; cf. [349, 402] and Sect. 12.9. • HT (High temperature regime): Behavioral temperature of the environment must be sufficiently high, so exchange of behavioral excitations in E should have the high degree of activity. This happens only in the environment populated by actively interacting animals or (and) with varying natural conditions. If one does not want to appeal to a behavioral analogue of temperature, then it is possible to operate with the parameter λ quantifying the scale of energy exchange.

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207

• ES (High intensity of external energy supply): In the framework of alpha-wolfs theory this condition is a consequence of active behavioral regulation driven in a pack by alpha-male and female. In some sense, this is the straightforward way to approach order stability. The case of self-regulation without strict and violent control from the side of alpha-wolfs is more interesting from the viewpoint of theory of Fröhlich condensation. Here coherent behavior of wolfs is conditioned by active external behavioral stimulation. Hence, the wolf’s pack would behave as the Fröhlich behavioral condensate in the forest or another environment E which is sufficiently densely populated by actively interacting animals (BV, HT) and with sufficiently intensive external supply (ES) of newly arriving to E animals, humans, and variability of external, for example, weather conditions. In other words, order stability in a wolf’s pack is approached via redistribution of b-energy from highly behaviorally active wolfs to less active pack’s members in the environment E with the aforementioned properties. The same model can be used for the herds of other animals, say elephants. Savanna which is densely populated by behaviorally active animals is the good example of the environment generating behavioral order in animals’ herds. Savanna is characterized by high behavioral temperature and intensive supply of behavioral energy, in particular, via transformation of physical energy of nutrition. This is high-energy stability, savanna’s Fröhlich condensate. Packs Controlled by Alpha-Wolfs Up to now, we did not consider the role of alpha-wolfs, males and females, in creation of behavioral stability in a wolf pack. Although highlighting the alpha-wolfs role is common in research on wolf’s behavior, in the light of the recent studies one should take this concept with caution. The concept goes back to the article written by animal behaviorist Schenkel [402] and it spread widely in research on wolf’s collective behavior. However, recently the concept of a wolf pack rigid alpha-control was criticized starting with Mech [349]. Schenkel’s picture of a supreme pack leader who dominates through permanent fight and reigns superior to the other wolves in his pack corresponds to artificially created packs in Zoo which he used for his studies, but the naturally created packs of wild wolfs are primarily based on family relations and the alpha-dominance is not at all the main source of order stability in a pack (Mech, 1999). A pack which is ruled by alpha-wolfs is an animal analogue of the authoritarian society. Order stability is such a pack, based on brutal control from the alpha-wolfs side. As was pointed out, such packs were formed in ZOOs. We note that in such a situation creation of the behavioral Fröhlich condensate is in principle impossible, because conditions listed in Sect. 12.9 are violated: capacity of the I -reservoir is very low (cf. BV), external I -supply is neither intensive (cf. BS), new I -excitations are so rare that they cannot be treated as indistinguishable (cf. BE).

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12.10 Concluding Remarks on Physical and Social Fröhlich Condensation In this chapter, we reviewed the Fröhlich condensate theory without going deeply into technical details. We hope that this presentation will be useful for experts in biosystems, cognition, psychology, decision-making, behavioral and social sciences. Physicists who are interested in aforementioned applications can also earn something from this review, since, in contrast to the Bose–Einstein condensation, the Fröhlich condensation was not highly advertised in physics. We presented two approaches: • Nonequilibrium thermodynamics. • Formalism of open quantum systems. Although the quantum approach is relatively formally simple and rather rigorous, the thermodynamical approach which was originally suggested by Fröhlich should not be neglected, because it provides the clear heuristic picture of statistical processes. The quantum picture is given in terms of creation and annihilation operators, Hamiltonian dynamics of the compound system, a system + its environment (reservoir). These leads via averaging w.r.t. the environment degrees of freedom to a quantum master equation, and analysis of conditions leading to approaching a far from equilibrium steady state. The thermodynamical approach emphasizes the role of the equilibrium state of the reservoir and its high temperature. Both approaches highlight the role of high intensity of external energy supply. As an illustrative example, we briefly present the Fröhlich model of cancer’s initiation as resulting from the coherent vibrations of the dipoles in cell’s macromolecules destruction of DNA and proteins. The coherent vibrations condensate can be destroyed by resonant microwave radiation; condensate’s perturbation leads to uncontrollable cells’ division. In this book, the conditions sufficient for creation of the coherent condensate far from equilibrium were reformulated without straightforward coupling to the physical energy notion. These conditions match the information approach to biology, physics, behavioral and social sciences. Starting with these conditions, we presented the model of the Fröhlich-like information condensate with invention of the information energy notion. As its special forms, we consider the social and behavioral energies. In the terms of the s-energy exchange between s-atoms, representing humans, and the large information reservoir, of Internet-based social networks and information exchange platforms, we formulated conditions for far from equilibrium order stability in the modern open society—democracies of USA, EU, Canada, Australia, Japan, South Korea…. The model emphasizes that the external supply of s-energy has to be high enough, higher than so to say the stability threshold. This supply is based on the information excitations flow generated by a variety of information sources, mainly, mass-media. The social Fröhlich condensate model spotlights the role of powerful information supply generation for order stability in the modern open society. Essential decrease in the information stream would lead to social decoherence and might be even to soci-

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209

ety’s collapse. As in the biosystems, in the social systems Fröhlich-like condensation serves not only for preservation of order stability, but also for s-energy redistribution. Highly energetic members of society redistribute their energy to weakly energetic ones. The latter does not fall to the state of equilibrium with I -reservoir, so they are in the sustainable state corresponding to the minimal energy characteristic for the modern open societies. Mathematically the model is based on the quantum formalism for creation and annihilation operators for social excitations and the quantum master equation, which leads to a stationary state under the aforementioned constraints for exchange by quanta of s-energy. In Sect. 12.9, similarly of societal stability in a far from equilibrium state, we model coherent collective behavior of animals with the illustrative example of a wolf pack. In this framework, it is natural to operate with the notion of the behavioral energy and its excitations. The exchange of such excitations between wolfs and with surrounding information environment (I -reservoir) is mathematically modeled with the operators of creation and annihilation and the quantum master equation. The ‘wolf’s Fröhlich condensate’ can exist only in a rich information-environment and a wolf pack has to be exposed to intensive supply of external information carrying novel behavioral excitations. We hope that this book as well as the recent articles [305, 306] would attract attention of biologists, psychologists, and social and behavioral studies experts to the Fröhlich condensate model for order stability preservation via distribution of social and behavioral energies and coherent behavior. We conclude this chapter with the statement of F. Fröhlich (the son of H. Fröhlich) presented in [156] (see also [446]): If one thinks without preconceptions of collective phenomena in which the behaviour of discrete constitutive individuals is modified, and constituting a large collective group where the whole is more than and different from a simple addition of its parts, living organisms would indeed seem to be the ideal example.

Chapter 13

Social Laser and Networks Within Mean Field Theory

13.1 Social Networks: Laser Physics, Phase Transition, and Critical Phenomena This chapter extends the social laser theory presented in Chap. 11; this extension is due to the paper of Alodjants, Bazhenov, Khrennikov, and Bukhanovsky [12]. We emphasize that the current life-style may be characterized as a sequence of online representations in various networks. The rapid growth of information (network) resources in terms of information exchange and processing leads to an exponential growth in the information being processed. Sometimes, enormous (social) enhancement effect can result from unimportant, at first sight, information spread. Contrary, in some cases, socially actual information and knowledge rapidly attenuate. • How fast does socially actual (social) information spread in social networks and communities? • How are these communities transformed under some perturbations? These questions are becoming critical for a well-functioning society. To find answers to these questions, we suggest to use the novel paradigm of social laser. As was stressed in this chapter, the social laser model can be explored to describe real-life socially actual information processing and reinforcement in the presence of strong mass media action, which is determined as a mass media pump. Understanding the physics of laser generation and coherence has always been a key task for laser physics. Main (coherent) statistical features of laser irradiation were explained within quantum and nonlinear optics theory [15, 176, 183, 392, 399]. Another theory of phase transitions and critical phenomena based on statistical and thermodynamic approach was able to explain coherent phase ordering in condensed matter [99], and macroscopic coherent effects of superconductivity [337]. The relationship between these two theories has always been the subject of heated debates that beneficially affected laser science as a whole. The main difference between the two fundamental theoretical approaches is that initially phase transitions were © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_13

211

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13 Social Laser and Networks Within Mean Field Theory

considered only for thermodynamic equilibrium systems [337]. However, laser systems are not thermodynamically equilibrium (cf. with Fröhlich condensation). To proceed to social applications, we shall not go deeply into interrelation of laser physics with phase transition and critical phenomena theories. We restrict our consideration to the discussion about some analogies occurring in open (driven— dissipative) and thermodynamically equilibrium laser-like systems exhibiting phase transitions [119, 178, 227]. The thermodynamic approach is widely applied for condensed or solid state systems, which are ordered in a certain way, or not ordered at all, like gases. The study of phase transitions in complex structured systems has recently been of great interest [68, 126, 373]. Although systems, such as spin glasses, have been studied for a long time, these are various complex networks, including the social networks, that are in the focus now [372]. The main reason for the growing interest in such systems is the large-scale Internet network development and global digitalization of the human society, which cover all aspects of humans life [403]. Providing subscribers with an opportunity to socialize globally, the Internet itself has become a social phenomenon that unites people in various social communities and on business and educational platforms (as well as contribute to social clustering and polarization). By examining phase transitions in such networks, we are able to make some conclusions (within certain limits) about social, information, communication, and emotional processes occurring in cyberspace [45, 205, 362]. In particular, public (social) networks promote a coherent macroscopic social impact involving a large number of people online. They provide information amplification and cascading, which may be elucidated by computer-mediated social studies [102]. In real life, coherent processes occurring in the networks provide social cohesion, which in some cases leads to cascading social processes (Arab spring, Orange and Maindan revolutions in Ukraine, Occupy Wall Street movements, mass protests against Lucashenko in Belarus, vaccination vs. anti-vaccination campaigns, etc.) in different countries [198, 321, 345]. One of intriguing problems at the nexus of modern quantum physics and statistical physics, social sciences, cognitive and computer-oriented studies is how close various statistical (laser-like) physical models may describe “online” social events occurring due to current Internet communication facilities. From physical point of view, the coherent processes discussed are similar to laser field generation, which results in huge energy release in ensemble of two- (or multi-) level oscillators under population inversion conditions. Notably, current studies of processes occurring in social communities are based on fundamentals of statistical physics and phase transitions theory. Traditionally, the approach of statistical physics is applied to complex systems description, which can include various social networks exhibiting social communities, cf. [374]. In particular, statistical Ising model is at the basis of many socially oriented statistical models, which are related to the exchange of information between people [126, 332, 420, 422, 457]. A simplified two-level (spin-like) decision-making (DM) agents model may be used in this case [171]; DM agents can be treated as s-atoms (Chap. 11). In particular, Ising-like models suppose collective opinion formation and social impact

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213

[204, 319], epidemic, information, and rumor spreading [351, 363, 375]. However, the major part of the models examined is thermodynamically equilibrium, whereas real-life social processes require a non-equilibrium approach. In this sense, the problem of non-equilibrium phase transition is of primary interest, as it may appear in laser-like systems possessing Ising-like Hamiltonians, cf. [57, 227]. Such models become crucial for feature modeling of distributed intelligence systems, which presume artificial and natural intelligence agents of DM behavior and their interaction capacity in the framework of a socially oriented network [181]. In this chapter, we extend the social laser theory by establishing a simple (generic) model of a social laser as a system possessing the second-order (non-equilibrium) phase transition, which presumes a complex network topology inherent to real network social communities. To be more specific, we examine networks, which may be characterized by means of power-law degree distribution function. Such networks allow for a relative simple analytical description [45]. Here it is the good place to mention the quantum field approach to modeling of cognition and consciousness (Vitiello [447, 448]) explaining the long range correlations in the brain. We start with the brief description of a socially oriented Ising model, which exhibits the second-order ferromagnetic (ordering state)—paramagnetic (disordering state) phase transition at equilibrium. Then we proceed to social laser mean field theory relevant to the interaction of two-level DM agents (s-atoms) with classical (pump) and quantum-like information fields. At the heart of these models there are basic statistical properties of complex networks, which create the link between the non-equilibrium social laser paradigm and existing socially oriented Ising-type models. To be more specific, we examine networks with power-law degree distribution which possess relatively simple analytical treatments for key parameters inherent to the models established in this work, cf. [441]. Finally, the social laser non-equilibrium behavior is examined. Social lasing phenomena are represented as an information cascade process, which relates to the application of the SIS (susceptible-infectedsusceptible) model for characterizing information spread in social networks.

13.2 Ising Model for Complex Networks: Equilibrium Phase Transition 13.2.1 Ising-Type Interaction ín Networks We start with the Ising model defined on annealed networks. We assume that twolevel (spin-1/2) quantum systems randomly occupy N nodes of a complex network— see Fig. 13.1. In the framework of social community analysis we model decisionmaking agents with these simple quantum systems and assume that they possess information exchange action. Thus, we represent this community as a graph with nontrivial (specific) properties. The Ising Hamiltonian of the model reads as

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13 Social Laser and Networks Within Mean Field Theory

Fig. 13.1 Power-law degree distribution networks for a γ = 1.5, b γ = 2.5 and c γ = 4.5, which correspond to anomalous, scale-free and random regimes, respectively, d—power-law degree distributions in a logarithmic scale for the networks given in a–c. The number of nodes is N = 1000

H=−

 ij

Ji j σiz σ jz −

1 h i σiz , 2 i

(13.1)

where the spin matrices σiz , i = 1, ..., N , characterize the i-th agent DM spin component. The first term in (13.1) characterizes interaction between DM agents. In real life such an interaction is implemented by some information exchange, which is still hidden for an external “observer” within the Ising model. Notably, for this model we cannot predict directly how information is relevant to DM agents inhomogeneously distributed over the network. However, we can conclude how this information spreads within the network because of the first term in (13.1), which depends on the network topology. The sum in (13.1) is performed over the graph vertices with certain adjacency matrix Ai j proportional to Ji j , which stores the information about the graph structure: matrix element Ai j = 1 if two vertices are linked and Ai j = 0 otherwise. Parameter h i in (13.1) characterizes the coupling of a DM agent with external (information) field. We examine the case when this information is the same for all DM agents and represents classical (strong) information pumping. In other words, we set h i = h for arbitrary i (hereafter for simplicity we put the Planck and Boltzmann constants  = 1, k B = 1). We are interested in the annealed network approach that presumes a weighted, fully connected graph model. The network dynamically rewires. We recast parameter Ji j that indicates the coupling between the nodes in Eq. (13.1) through probability pi j as Ji j = J pi j , where J is a constant, and pi j is the probability for two nodes i and j to be connected:

pi j = P(Ai j = 1) =

ki k j , N k

(13.2)

where Ai j is an element of the adjacency matrix, ki is i-th node degree, which indicates an expected number of the node neighbors and is taken from distribution p(k). In (13.2), the quantity

13.2 Ising Model for Complex Networks: Equilibrium Phase Transition

k =

215

1  ki N i

is an average degree. Noteworthy, the annealed network approach is valid for pi j  1 and large enough N , cf. [45]. Thus, the strength of two spins interaction Ji j is a variable parameter and depends on particular network characteristics; it is greater for two pairs of nodes with the highest k coefficient. In practice, different approaches provide an explanation for a real-world network topology [126]. Such networks may exhibit the power-law degree distribution. Since the number of nodes is large enough, N  1, we are interested in network structures, which admit continuous degree distribution p(k). To be more specific, in this work we examine networks with distribution function p(k) defined as γ −1

p(k) =

(γ − 1)kmin , kγ

(13.3)

where γ is a degree exponent that covers anomalous (1 < γ < 2)—Fig. 13.1a, scalefree (2 < γ < 3)—Fig. 13.1b, and random (γ > 3)—Fig. 13.1c regimes, cf. [45]. The properties of scale-free networks possessing distribution (13.3) for γ = 2 and γ = 3 should be calculated separately. The normalization condition for p(k) is represented as +∞ p(k)dk = 1.

(13.4)

kmin

The scale-free networks possess preferential attachment phenomena, which result in hubs appearance. The largest hub is described by degree kmax called a natural cutoff. The condition +∞ p(k)dk =

1 N

(13.5)

kmax

can be used if the network with N nodes possesses more than one node with k > kmax . From (13.4) and (13.5), we immediately obtain 1

kmax = kmin N γ −1 .

(13.6)

Figure 13.1d demonstrates probability distribution function (13.3) plotted in a logarithm scale for the networks shown in Fig. 13.1a–c. The node degree fluctuations grow at γ ≤ 3, cf. [57]. Hubs in Fig. 13.1d appear as dots in the right corner of the distribution function. The number of hubs and their size dramatically grow with vanishing γ in the anomalous regime where kmax /kmin > N , cf. Fig. 13.1a and the green line in Fig. 13.1d.

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13 Social Laser and Networks Within Mean Field Theory

Real-world networks mostly possess degree exponent γ > 2. The BianconiBarabási (BB) network appropriately models such a network. In particular, the BB network is based on a fitness model, which accounts that the probability of the newly attached node link is proportional to its fitness parameter [45]. This parameter may be interpreted in the framework of statistical mechanics approach and leads to γ = 2.255, [68]. In particular, it is possible to map the BB network to gas particles assuming that energy of each node is determined by its fitness parameter. As a result, we can clearly distinguish the new BEC phase of the network. The network condensation phenomena presumes super-hub formation, which accumulates a huge number of links; strictly speaking, such a network is not a scale-free any more. The statistical properties of networks may be characterized by means of the n-th moment for the degree distribution defined as:

K

(n)

kmax ≡ k  = n

k n p(k)dk,

(13.7)

kmin

where n is a positive integer. In this work, we are interested in the first and normalized nth order (n = 2, 3, 4) degree correlation functions, which are defined as ζn ≡

K (n) , n = 2, 3, 4. K (1)

(13.8)

In Fig. 13.2, we represent the main properties of scale-free network statistical characteristics as a function of degree exponent γ . Remarkably, the nth order correlation functions defined in (13.7) diverge at γ = 1. On the contrary, their combination ζ ≡ [K (1) ]2 K (4) /[K (2) ]3 remains finite.

Fig. 13.2 Dependence of a ζ , k, ζ2 , ζ3 and ζ4 on degree exponent γ for kmin = 1, N = 1000, and b ζ2 (dash line) and k (solid lines) on N for kmin = 1 and different values of γ . Both figures are plotted in logarithmic scale

13.2 Ising Model for Complex Networks: Equilibrium Phase Transition

217

13.2.2 Phase Transition and Network’s Structure Here, we use the Hamiltonian formalism, which is common in physics and widely used in the framework of DM agents interaction, see, e.g., [126, 204, 319, 420, 422]. It allows to speak about some social energy stored in (13.1). In this sense, the interpretation of social spin is straightforward: different spin components correspond to agents preferences with the same spin energy. For example, with no external information in elections people as DM agents can prefer to vote a candidate or reject them without any social energy changing. The external (information) field or interaction with other agents influences our potential decisions. External information may be enough to flip our decision or not. In this case, the temperature, T , represents some vital macroscopic parameter, which manifests about microscopic processes occurring in the system . In physics temperature indicates energy exchange for the canonical ensemble of fixed number of particles with the thermal bath. In social science, it is also possible to introduce social temperature; it stands for some sophisticated (fitting) parameter, which may be discussed in the framework of various models in economy, finance, etc., cf. [100, 128, 357]. In this work, we assume that social temperature indicates abilities of DM agents for information exchange with other agents and with an environment, respectively. In other words, thermalization, which is a keystone problem for any physical system, may be achieved in social networks through information exchange. In this case, the system approaches thermal equilibrium at time periods long enough. Comparing our model with the models, which presume money exchange financial markets, we can identify the temperature parameter as an average number of messages per agent in the network, cf. [100, 128]. In this work, we consider the mean-field approach to the system described by the Ising Hamiltonian (13.1). At social level of DM, the system is totally characterized by the order parameter Sz defined as Sz =

1  ki σiz , N k i

(13.9)

and represents a weighted average spin component, cf. [431]. Collective variable Sz obeys self-consistent equation 1 Sz = k



kmax kp(k)tanh

 β (4J Sz k + h) dk, 2

kmin

where β ≡ 1/T is reciprocal temperature; in (13.10) substitution k max kmin

... p(k)dk is performed.

(13.10)

1 N

 i

... →

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13 Social Laser and Networks Within Mean Field Theory

At high enough temperatures and external field h = 0 Eq. (13.10) admits nonzero solution for collective spin component Sz . This solution corresponds to some ferromagnetic (FM) phase possessing Sz = 0 and indicating some certain DM at macroscopic level. Critical reciprocal temperature βc , which provides this solution, is determined from Eq. (13.10) and represented as βc =

1 , 2J ζ2

(13.11)

where parameter ζ2 defined in Eq. (13.8). The dependence given in (13.11) admits simple interpretation in the framework of information exchange. In particular, the strongest interaction between the DM agents evokes the highest critical temperature, Tc ≡ 1/βc . Simultaneously, a large number of hubs can manifest the activity of DM agents. In this limit, ζ2 is also large enough and corresponds to high temperature of phase transition Tc . Critical temperature Tc becomes very large (βc  0) in the vicinity of γ = 1 where ζ2 enormously increases, see Fig. 13.2a. In the opposite case, for large degree exponent γ the ζ2 approaches kmin and critical temperature becomes Tc ∝ kmin . Remarkably, similar arguments are still true in the mean-field approximation if we neglect degree correlations in the network and suppose in (13.11) ζ2  k, which leads to Tc  2J k.

(13.12)

Equations (13.11) and (13.12) play a crucial role in phase transition occurring in the finite size Ising model. The equations establish a clear connection between the system temperature and network statistical properties. For a given temperature, T , the critical value, ζ2,c , is determined from (13.11) as

ζ2,c =

T . 2J

(13.13)

Then, in Eq. (13.10) we expand the function tanh(x) ≈ x − 13 x 3 with x = + h). In this limit, we can represent Eq. (13.10) in the form of

β (4J Sz k 2

ASz − B Sz3 +

h = 0, 2T

(13.14)

where coefficients A and B are defined as A=

ζ2 − 1, ζ2,c

B=

ζ ζ23 . 3 3 ζ2,c

(13.15)

13.3 Non-equilibrium Phase Transition in Social Laser

219

From (13.14) in the vicinity of critical point ζ2 = ζ2,c at h = 0 we obtain 

3 Sz = ζc



1/2 ζ2 −1 . ζ2,c

(13.16)

The properties of parameter ζ may be inferred from Fig. 13.2a. The magnitude of ζ is finite at γ = 1 and reaches its maximum value at γmax  21 [ln(N ) + 2 − √ ln(N )[ln(N ) − 4]], which implies γmax ≈ 2.21 for the networks with N = 1000 nodes established in Figs.13.1 and 13.2a, respectively. Equation (13.16) establishes the second-order phase transition from paramagnetic (PM) state (Sz = 0) to ferromagnetic (FM) one (Sz = 0), which occurs if normalized degree correlation function obeys the ζ2 ≥ ζ2,c condition. For social network systems, such a phase transition means transformation from disorder to some ordering state with opinion formation or voting. Hence, phase transition for the analyzed Ising model appears only due to finite size effects, cf. [126, 332, 422, 457]. Remarkably, for power-law degree distribution networks parameter ζ2 is the function of number of nodes N . The dependence of ζ2 versus N for various γ is shown in Fig. 13.2b. As clearly seen, ζ2 grows significantly within the anomalous domain of γ . Therefore, social networks, which possess a growing number of hubs (decreasing γ ), promote the occurrence of some ordering state; it may be characterized in terms of definite social polarization, which is discussed in more detail below. In the presence of non-vanishing external (pump) field h for ζ2 → ζ2,c the order parameter may be obtained in the form  Sz 

3h 4J ζc ζ2,c

1/3

 

3h 4J kc

1/3 .

(13.17)

The right-hand part of Eq. (13.17) is valid for large degree exponent γ with the valid factorization of parameters ζ2 and ζ . Notably, the main features of average degree k in this limit resembles ζ2 , see Fig. 13.2.

13.3 Non-equilibrium Phase Transition in Social Laser 13.3.1 Primary Consideration In this section, we examine the social lasing effect as a non-equilibrium phase transition that occur in social systems possessing some specific features. The model we discuss in this work may be formulated in a general form in terms of the quantum field theory apparatus. For simplicity we consider N two-level systems (TLS), which comprise gain laser medium; each system is inherent to a well-distinguished ground (|g) and excited (|e)

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13 Social Laser and Networks Within Mean Field Theory

states possessing different energies E g and E e , (E e > E g ), respectively. We describe TLS at states |e and |g by bosonic annihilation and creation operators, aˆ j and aˆ †j for ground state and bˆ j and bˆ †j for excited state, respectively. We characterize the jth TLS by resonant frequency (energy) of transition ω j = E e, j − E g, j ( j = 1, 2, . . . , N ); note that we use units where the Plank constant equal to 1. We assume that interaction of TLSs ensemble with irradiation occurs in the resonator, which supports a multimode regime described by an annihilation (creation) † field operator fˆv , ( fˆv ), which freely evolves in time with common frequency ω, see Fig. 13.3a, cf. [421]. Within the framework of dipole and rotating-wave approximations, the generic Hamiltonian of the system is given by: 1 Hˆ = − 2

N 

ω j (bˆ †j bˆ j − aˆ †j aˆ j )

j=1

N kj  1  ˆ † ˆ † † ω f v f v + g j ( fˆv aˆ †j bˆ j + bˆ †j aˆ j fˆv ) + i P( fˆv − f v ) , + N j=1 v=1

(13.18)

where g j characterizes interaction strength (energy) of TLS with field fˆv . Thereafter, we assume that gi = g for all TLS. In (13.18) k j is the jth node degree, which can be understood as a mean number of modes occurring in a physical laser resonator. Physically, 2g represents so-called vacuum Rabi-splitting frequency, which indicates the rate, at which one quantum excitation may appear due to the interaction with a resonator field, cf. [313]. From quantum theory it is known that even in the absence of photons in a resonator (so-called vacuum state) the resonant probability of TLS transition from the ground state to the excited one within time t is W = sin 2 [gt], cf. [409]. Considering short time intervals such as gt 3, see Fig. 13.2. Dependence (13.36) for information field on social mass media pump P plays a crucial role in social laser features. In physics, gain medium pumped by strong classical field, which may be coherent or incoherent, see Fig. 13.3a. The latter possesses a broad spectrum [421]. In the framework of social studies, the role of coherent mass-media pump is straightforward. It strongly supports echo chamber transition frequency ω. The role of an incoherent social mass-media pump may be much more interesting, since in this case DM agents are influenced by strong and wide spectrum

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229

of different viewpoints. However, even in this case the social pumping effect may be strong enough due to so-called confirmation bias, which results from individuals’ feature to absorb information in a way that confirms one’s prior beliefs supported by an echo chamber. In social networks confirmation bias possess amplification by means of so-called filter bubbles; they display only the information that individuals are likely to agree with, while excluding opposing views [315].

13.4 Social Laser Dynamics and Information Spread 13.4.1 Viral Information Cascades in Social Laser Modern social media creates unprecedented opportunities for DM agents to influence each other. This is facilitated by various Internet-based resources sharing applications (e.g., YouTube), blogs and microblogs (e.g., LiveJournal, Twitter etc.), social networks (e.g., Facebook, Myspace, VKontakte). In this sense, information field E responsible for the influence of DM agents cannot be considered as a constant in time, cf. (13.36). For example, some information published by the user can almost immediately become “viral” and accessible to millions of people. It can provoke a rapid growth of so-called information cascades [101, 455]. In this case, dissemination of information in social media is the object for close attention of researchers who work in the field of mathematics, computer science, statistical physics, sociology, and psychology. Time dependent phenomena for the social laser may be inferred from Ginzburg– Landau equation (13.26). In particular, from (13.26) for average s-photon number variable n ph = E 2 (which corresponds to number of retweets, as example) we obtain √ n˙ ph = 2 An ph − 2Bn 2ph + 2P(t) n ph ,

(13.37)

where we recover slow time dependence of pump P → P(t) on time. Equation (13.37) may be recognized as a rate equation for the number of photons generated by laser at the output, cf. [188]. In the framework of social media studies, the similar equation (with P = 0) characterizes a so-called susceptibleinfected-susceptible (SIS) epidemic model, which describes information spread in social media based on the growth of infected agents population, cf. [45, 375]. Recent remarkable applications of such a model to rumor and misinformation spreading are given in [218, 343]. In laser theory, the pump field plays a crucial role. It is used to create required population inversion σz . Then, the pump may be switched-off, P = 0. In this limit, Eq. (13.37) admits a simple analytical solution n ph =

Ave2 At , 1 + Bve2 At

(13.38)

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13 Social Laser and Networks Within Mean Field Theory

Fig. 13.4 a Time dependence of average s-photon number n ph (t) for A = 1 and B = 0.25. Pump rates are: P = 0 (blue curve), P = 1 (red curve), P(t) = sin(0.37t) (green curve), P(t) = sin(1.3t) (magenta curve), and P(t) = e−0.12t (black curve), respectively. b Viral Information cascade in social laser network. The cascade starts with a socially stimulated repost (retweet) in the presence of spontaneously posted message, which together deliver information to two other agents in the network and initiate new “spontaneously” or socially stimulated messages. The messages occur in accordance with the rules explained in Fig. 13.3c

where we define v = n/(A ¯ − B n); ¯ n¯ is an average photon number at initial time t = 0. Figure 13.4a exhibits another important result of this work and explains the role of the mass media pump in the social laser. As seen in Fig. 13.4a, we plot solutions of Eq. (13.37) for s-photon average number n ph for various pump P(t). We can suppose that n ph describes reinforcement of some viral information supported by the echo chamber. Parameter g 2 kσz /  in this case simply describes message diffusion rate whereas κ characterizes recovery rate; the s-photons escape from the echo chamber, which amplifies this information with rate κ, cf. Eq. (13.28). The blue curve in Fig. 13.4a is relevant to typical S-shape behavior, which is relevant to the SIS model and characterizes the (exponential) growth of n ph above the threshold (A > 0). The pump field creates the social laser threshold and then switches off, P = 0. This limit corresponds to growing information cascade, which is schematically depicted in Fig. 13.4b. In the social laser, a cascade starts with a “spontaneously” posted message. Then, in the presence of required inversion, or a relevant average node degree this message evokes avalanche of messages (s-photons) in the network created by other DM agents as a result of socially stimulated emission phenomena, cf. Fig. 13.3b. In particular, such a cascade may be described in the framework of cascade generating function, which characterizes details of cascade microscopic dynamics, cf. patterns (3), (5) given in Fig. 2 of [173]. At the same time, the information cascade shown in Fig. 13.4b may be explained in the framework of structural virality paradigm, which considers viral information diffusion [177].

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231

In the presence of the pump of mass media an average s-photon number, as it is shown Fig. 13.4a, follows pump main features. In particular, a permanent pump possesses essential acceleration of information diffusion, see the red curve in Fig. 13.4a. For the black and magenta curves we suppose that P(t) = P0 sin[νt], where P0 is a pump field amplitude, and ν is frequency of the pump variation. In adiabatic approximation (ν  ω) DM agents follows mass media pump evolution, which is clearly seen for the black and magenta curves behavior shown in Fig. 13.4a, respectively. Within the established time window, the green curve may be associated with power-law decay of an information cascade. In contrast, the black line in Fig. 13.4a characterizes the exponential decay of the cascade demolished by mass media pump P(t) = P0 e−νe t . Within the large time window, it approaches the blue curve obtained at P = 0. Thus, the plots in Fig. 13.4a may reproduce a variety of information cascade properties obtained experimentally depending on a mass media pump field action P(t), cf. [105, 198, 218, 346, 390]. In particular, the SIS model in the framework of social laser possesses simple and very elegant interpretation. The DM agent is initially susceptible to s-photon. The s-photon absorption evokes infection of the agent with some probability, which corresponds to transition to the excited state. Then, s-photon emission (which practically implies posting a tweet, for example) changes the s-photon number (number of tweets) and simultaneously transfers DM agent to the ground state back. Notably, the same agent again becomes susceptible to absorb s-photon, which agrees with basic SIS model predictions, [218, 343]. Thus, the total number of emitted s-photons in the social laser model is accompanied with diminishing of population inversion. The lasing threshold in this case corresponds to the epidemic threshold. If the lasing threshold does not fulfill (A < 0), the properties of n ph strictly depend on the pump P. In particular, in this limit it is possible to obtain n ph

√ (e At (A n¯ + P) − P)2 = , A2

(13.39)

For A < 0 Eq. (13.39) provides critical pump field Pc = A2 n¯ when n ph = n. ¯ For P > Pc the n ph grows exponentially.

13.4.2 Velocity of Information Reinforced This is a keystone question that we can address to the social laser paradigm from the beginning. Up to date it is unclear how network peculiarities affect social laser gain medium, cf. Fig. 13.3b. As we have already pointed out, Fig. 13.4 and (13.38) may be understood in the framework of various models of disease spreading. To be more specific, even simple SI model can help to create a link between information growth presented in Fig. 13.4a and network peculiarities. To demonstrate that, let us examine the social laser model well above the threshold with a switched-off pump

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field, P = 0 (we can simply neglect by κ for the SI model in this limit). At any time moment we suppose that all DM agents in a social network belong to two large classes of “infected” or “susceptible” individuals. A DM agent belongs to set of “infected” population if he/she posts a message about some potentially viral content. Otherwise, he/she is “susceptible” (S). Notably, transfer from “infected” to “susceptible” agents within this model is not provided, which is clearly inherent to the SI approach. We suppose that an infected agent posts the message, which is passed to a susceptibleone (here we refer to so-called SI model for simplicity). Both of them are part of the social laser social network community, see Fig. 13.4b. Within the time interval dt a probability that corresponds to the message passage to a susceptible agent is dt (we suppose that is small enough, dt  1). Then, let us assume that susceptible agent j possesses node degree k j . In this case a “susceptible” agent receives the message (becomes infected) with probability k j dt. The enhancement of this probability k j times is obvious: people more intensively communicated with individuals who possess more (communicative) links in social media. Let us assume that k j approaches its average value k. Fraction of nodes with average degree k, which are “infected” in the network, is defined as I , and the fraction, which are not “infected”, is defined as 1 − I , respectively. A simple equation that governs I variable is represented as (cf. [45]) I˙ = (1 − I )kI.

(13.40)

The solution of Eq. (13.40) is I =

I0 e kt , 1 − I0 + I0 e kt

(13.41)

where I (t = 0) = I0 defines the initial condition. Comparing Eq. (1.12) with (13.38) we can conclude that  2g 2 σz / . Equation (1.12) provides a definition of characteristic time scale tk required to achieve 1/e fraction of all “susceptible” DM agents; from (1.12) we obtain tk =

1 1  ,

k 2A

(13.42)

where we omit the term with κ, which is relevant to losses. The obtained in (13.42) plays an important role for complex networks with the power-law degree distribution, which mimics a real-life social media. Actually, the characteristic time scale tk is inversely proportional to the rate, at which a viral message spreads through social media. As it is followed from the inset to Fig. 13.2, social networks with degree exponent γ > 3 may be recognized as socially passive in the framework of interaction exchange. The situation changes in the social laser limit. Echo chambers provide increasing of information exchange and social excitation of network agents. For the networks with power-law degree distribution it means vanishing degree exponent γ , which leads to clusterization and hubs appearing,

13.5 Concluding Discussion on Social Laser and Networks

233

cf. Fig. 13.1. The rate of information transfer enormously increases (tk vanishes) within anomalous network domain 1 < γ < 2 which corresponds to growing k. Notably, such an effect almost no longer depends on κ. This regime corresponds to a so-called superstrong coupling regime, when coupling between TLSs and resonator field behaves strongly enough during one round trip in the resonator, cf. [350]. Hence, we can conclude that in the framework of social laser paradigm we can expect huge acceleration of information spread and rapid information cascade, which result from a superstrong coupling regime. We can represent such a condition in the form G  , κ, ω F S R ,

(13.43)

where ω F S R is a free spectral range of the resonator, which is inversely proportional to the “size” of the resonator [350]. In quantum physics, a strong coupling regime provides observation of various coherent matter-field interaction effects, like exciton-polariton Bose–Einstein condensates [124]. Roughly speaking, in this limit we need no population imbalance at all. It may be also seen from definition (13.29); threshold value of σthr vanishes at g(k)  κ,  for large N , cf. Fig. 13.2b. We expect that it is possible to obtain condition (13.43) by choosing an appropriate network topology within 1 < γ < 2 domain. This result represents an important impact for current social science. It means that in the presence of a strong mass media pump and echo chambers occurring in social media we do not need to obtain a large population inversion, which leads to large number of emotionally excited people. It is enough to have some strong majority who can attract people, create echo chambers in social networks, and provide necessary network communicative topology (which characterized by some specific γ for power-law networks, in particular). It is worth noticing that huge information cascades in this situation appear due to a spontaneous DM process. Then, all population is likely to follow these cascades.

13.5 Concluding Discussion on Social Laser and Networks The social laser concept explains growing and fast-paced processes generated in current social media. The social laser, as the physical laser device, contains all keystone elements, which are necessary for reinforcement of social information. At the core of the social laser is social network media, which may be modeled (with some approximation) by means of networks whose degree distribution follows power law. The DM agents, which we established as simple quantum TLS, occupy the nodes of this network (we consider the case where each agent occupies only one node). An echo chamber, which mimics an optical resonator known from physical laser device, represents a key feature of social media. In particular, echo chamber may provide enhancement of one privileged s-information mode. We have demonstrated a clear link between the social laser and Ising model. The Ising model is used in sociophysics that considers networks and characterizes a social impact. Unlike the

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13 Social Laser and Networks Within Mean Field Theory

Ising model, the social laser accounts for elementary processes of s-photon absorption, spontaneous emission, and (socially) stimulated emission, which are relevant to on-line information exchange between DM agents in the network. As a result, social lasing represents a coherent non-equilibrium process that implies the formation of macroscopic social polarization, (viral) information cascade creation in the presence of population imbalance (social bias). In this chapter, we use by mean-field theory, which has a twofold meaning for a social laser system. In particular, we obtain a set of equations to describe the social laser in mean-field approximation, which deals with average values of relevant operators and neglect all quantum (Langevin) noises. We examine a network structure in the limit when fluctuations of node degree may be neglected. In this case, features of the socially oriented Ising model and social laser provide the existence of some critical average node degree, which defines the second-order phase transition threshold for the network topology and nodes. In this limit, it was shown that a coherent information field represents the order parameter and provides a huge social impact in social laser network media. An important difference between the social laser and currently actual models of echo chambers and related phenomena is the presence of a mass media pump, which plays a crucial role in social laser generic features. The mass media pump provides population inversion required for lasing occurrence in the presence of various decay mechanisms. In this work, we establish a clear link between the SIS/SI epidemiological models and the social laser rate equation for an average number of s-photons, which may be simply recognized as an average total number of messages posted or tweeted in a social network. We have shown that the social laser field immediately generates information cascades occurring in social media. Remarkably, the mass media pump enables to reinforce and accelerate cascade growth in different ways. Notably, DM agents follow an adiabatically time dependent mass media pump, which acts in a social network community and contributes to the reproduction of various reliable scenarios for information cascade evolution in time. We have achieved an important result, which relates to seminal features of parameter g(k) that characterizes coupling strength of DM agent with √ social laser s-information modes; we have shown that g(k) is proportional to k for the network structure possessing average node degree k. We have shown that the time of information spreading in a social network is inversely proportional to average node degree k, which may be large enough in social laser systems. As a result, we can obtain a huge information cascade and social impact at the “output” of the social laser, which is manipulated by the mass media pump.

Part V

Boundaries of Applicability of Quantum-Like Modeling

Application of the quantum methodology and formalism to decision-making was stimulated by the paradoxes and inconsistencies of the CP-based theory, probability fallacies, and visible irrationality of people’s behavior in some cognitive and social contexts (irrationality which was also framed in the CP framework). Appealing to quantum theory is only one possible way to proceed nonclassically. May be this theory does not cover the whole body of the cognitive phenomena. In this part, we analyze this complex problem. In Chap.14, we show that the description of questions by von Neumann observables has restricted applicability. In Chap.15, we discuss the possibility that the probabilistic structure of the cognitive phenomena is even more exotic than the structure of quantum probabilities. Here we follow Sorkin, who analyzed a similar problem in physics.

Chapter 14

No-Go Theorem for Modeling with Von Neumann Observables

14.1 Sequential Measurements in Physics and Psychology On a very general level, QM accounts for the probability distributions of measurement results using two kinds of entities, called observables a and states ψ (of the system on which the measurements are made). Let us assume that measurements are performed in a series of consecutive trials numbered 1, 2, . . .. In each trial t the experimenter decides what measurement to make (e.g., what question to ask), and this amounts to choosing an observable a. In a psychological experiment the a-values are the responses that a participant is allowed to give, such as Yes and No. The probabilities of these outcomes in trial t (conditioned on all the previous measurements and their outcomes) are computed as some function of the observable a and of the state ψ (t) in which the system (a particle in quantum physics, or a participant in psychology) is at the beginning of trial t,   P [a = v in trial t | measurements in trials 1, . . . , t − 1] = F ψ (t) , a, v . (14.1) This measurement changes the state of the system, so that at the end of trial t the state is ψ (t+1) , generally different from ψ (t) . The change ψ (t) → ψ (t+1) depends on the observable a, the state ψ (t) , and the value a = v observed in trial t,   ψ (t+1) = G ψ (t) , a, v .

(14.2)

On this level of generality, a psychologist will easily recognize in (14.1)–(14.2) a probabilistic version of the time-honored Stimulus-Organism-Response (S-O-R) scheme for explaining behavior [469]. This scheme involves stimuli (corresponding to a), responses (corresponding to v), and internal states (corresponding to ψ). It does not matter whether one simply identifies a with a stimulus, or interprets a as a kind of internal representation thereof, while interpreting the stimulus itself as part of the measurement procedure (together with the instructions and experimental setup, that are usually fixed for the entire sequence of trials). What is important is that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_14

237

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14 No-Go Theorem for Modeling with Von Neumann Observables

the stimulus determines the observable a uniquely, so that if the same stimulus is presented in two different trials t and t  , one can assume that a is the same in both of them. The state ψ (t+1) determined by (14.2) may remain unchanged between the response v terminating trial t and the presentation of (the stimulus corresponding to) the new observable that initiates trial t + 1. In some applications this interval can indeed be negligibly small or even zero, but if it is not, one has to allow for the evolution of ψ (t+1) within it. In QM, the “pure” evolution of the state (assuming no intervening inter-trial inputs) is described by some function   ψ(t+1) = H ψ (t+1) ,  ,

(14.3)

where  is the time interval between the recording of v in trial t and the observable in trial t + 1. This scheme is somewhat simplistic: one could allow H to depend, in addition to the time interval , on the observable a and the outcome v in trial t. We do not consider such complex inter-trial dynamics schemes in this chapter. The reason we single out opinion polling and compare it to psychophysics is that they exemplify two very different types of stimulus-response relations. In a typical opinion polling experiment, a group of participants is asked one question at a time, e.g., a = “Is Bill Clinton honest and trustworthy?” and b = “Is Al Gore honest and trustworthy?” [359]. The two questions, obviously, differ from each other in many respects, none of which has anything to do with their content: the words “Clinton” and “Gore” sound different, and the participants know many aspects in which Clinton and Gore differ, besides their honesty or dishonesty. Therefore, if a question, say, b, were presented to a participant more than once, she would normally recognize that it had already been asked, which in turn would compel her to repeat it, unless she wants to contradict herself. One can think of situations when the respondent can change her opinion, e.g., if another question posed between two replications of the question provides new information or reminds something forgotten. Thus, if the answer to the question a = “Do you want to eat this chocolate bar?” is Yes, and the second question is b = “Do you want to lose weight?,” the replications of a may very well elicit response No. It is even conceivable that if one simply repeats the chocolate question twice, the person will change her mind, as she may think the replication of the question is intended to make her “think again.” In a wide class of situations, however, changing one’s response would be highly unexpected and even bizarre (e.g., replace a in the example above with “Do you like chocolate?”). We assume that the pairs of questions asked, e.g., in Moore’s study [359] are of this type. In a typical psychophysical task, the stimuli used are identical in all respects except for the property that a participant is asked to judge. Consider a simple detection paradigm in which the observer is presented one stimulus at a time, the stimulus being either a (containing a signal to be detected) or b (the “empty” stimulus, in which the signal is absent). For instance, a may be a tilted line segment, and b the same line segment but vertical, the tilt (which is the signal to be detected) being too small for all answers to be correct. Clearly, the participant in such an experiment

14.1 Sequential Measurements in Physics and Psychology

239

cannot first decide that the stimulus being presented now has already been presented before, and that it has to be judged to be a because so it was before. With this distinction in mind, however, the formalism (14.1)–(14.3) can be equally applied to both types of situations. In both cases we can consider some observables a and b. The values of a and b are the possible responses one records. In the psychophysical example, a = va and b = vb each can attain one of two values: 1 = “I think the stimulus was tilted” or 0 = “I think the stimulus was vertical”. The psychophysical analysis consists in identifying the hit-rate and false-alarm-rate functions (conditioned on the previous stimuli and responses)   P [a = 1 in trial t | measurements in trials 1, . . . , t − 1] = F ψ (t) , a, 1 , (14.4) P [b = 1 in trial t | measurements in trials 1, . . . , t − 1] = F ψ (t) , b, 1 . The learning (or sequential-effect) aspect of such analysis consists in identifying the function   (14.5) ψ (t+1) = G ψ (t) , S, v , S ∈ {a, b} , v ∈ {0, 1} , combined with the “pure” inter-trial dynamics (14.3). In the opinion polling example (say, about Clinton’s and Gore’s honesty), there are two hypothetical observables: a, corresponding to the question a =“Is Bill Clinton honest?”, and b, corresponding to the question b =“Is Al Gore honest?”, each observable having two possible values, 0 =“Yes” and 1 =“No”. The analysis, formally, is precisely the same as above, except that one no longer uses the terms “hits” and “false alarms” (because “honesty” is not a signal objectively present in one of the two politicians and absent in another). It is worth noting that in the opinion polling the observables a, b are defined by the corresponding questions alone only because the allowable responses (Yes or No) and the instructions (“Respond to this question”) do not vary from one trial to another. If the allowable responses varied (e.g., if they were Yes and No in some trials, and Yes, No, and Not Sure in other trials), or if the instruction varied (say, in some trials “Respond as quickly as possible”, in other trials “Think carefully and respond”), they would have also contributed to the identification of the observables. Analogously, in our psychophysics example, the observables are defined by stimuli alone because the instruction to the participants (“Tell us whether the line you see is tilted or vertical”) and the responses allowed (“Tilted” and “Vertical”) remain fixed throughout the successive trials. In quantum physics, a classical example falling within the same formal scheme as the examples above is one involving measuring the spin of a particle in a given direction. Let the experimenter choose one of two possible directions, a or b (unit vectors in space along which the experimenter sets a spin detector). If the particle is a spin- 21 one, such as an electron, then the spin for each direction chosen can have one of two possible values, 1 =“up” or 0 =“down” (we need not discuss the physical meaning of these designations). These 1 and 0 are then the possible values of the

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14 No-Go Theorem for Modeling with Von Neumann Observables

observables a and b one associates with the two directions, and the analysis again consists in identifying the functions F, G, and H .

14.2 Von Neumann Observables and Unitary Inter-measurement Evolution For simplicity (and because all our examples involve binary outcomes), in this chapter we will only deal with the observables a that have two possible values, denoted 0 and 1. This means that all our observables can be presented as Hermitian operators of the projection type1 A = E1, (14.6) and E 02 = E 0 , E 12 = E 1 , E 0 E 1 = 0, E 0 + E 1 = I.

(14.7)

Each eigenvalue v (0 or 1) has its multiplicity 1 ≤ d < n. This is the dimensionality of the eigenspace V associated with v, which is the space spanning the d pairwise orthogonal eigenvectors associated with v (i.e., the space of all linear combinations of these eigenvectors). Multiplication of E v by any vector x is the orthogonal projection of this vector into V . If d = 1, the eigenspace V is the ray containing a unique unit length eigenvector of A corresponding to v. The eigenvalue 1 − v has the multiplicity n − d, the dimensionality of the eigenspace V ⊥ which is orthogonal to V . If both d = 1 and n − d = 1 (i.e., n = 2), then A is said to have a non-degenerate spectrum. We assume the spectra are generally degenerate (n ≥ 2). The eigenvalues 0, 1 of A in a given trial generally cannot be predicted, but one can predict the probabilities of their occurrence. To compute these probabilities, QM uses the notion of a state of the system. In any given trial the state is unique, and it is represented by a unit length state vector ψ.2 If the system is in a state ψ (t) in trial t, and the measurement is performed on the observable a given by operator A, the probabilities of the outcomes of this measurement are given by   F ψ (t) , a, v = P [a = v in trial t | measurements in trials 1, . . . , t − 1] (14.8)  2 = E v ψ (t) , ψ (t) =  E v ψ (t)  , 1

These are observables of the von Neumann type, see Chaps. 16 and 8. As we know from theory of quantum instruments, it is not enough to determine observables and, hence, the probabilities of their outcomes. We also have to determine the form of the state update resulting from measurement’s feedback action; in this chapter it is given by Lüders postulate. Refined said, we consider quantum instruments of the von Neumann-Lüders class (Chap. 8, Sect. 8.1). 2 For simplicity, we assume throughout the chapter that the system is always in a pure state. A more general mixed state is represented by a density matrix, which is essentially a set of up to n distinct pure states occurring with some probabilities. The same as with the assumption that n is finite, the restriction of our analysis to pure states is not critical.

14.3 Measurement Sequences: Evolution (In)Effectiveness and Stability

241

where v = 0, 1. Note that these probabilities are conditioned on the previous observables, in trials 1, . . . , t − 1, and their observed values. Given that the observed outcome in trial t is v, the state ψ (t) changes into ψ (t+1) according to   E v ψ (t) G ψ (t) , a, v = = ψ (t+1) . (14.9)

E v ψ (t)

This equation represents the von Neumann-Lüders projection postulate of QM. The denominator is nonzero because it is the square root of P [a = v in trialt], and (14.9)  is predicated on v having been observed. The geometric meaning of G ψ (t) , a, v is that ψ (t) is orthogonally projected by E v into the eigenspace V and then normalized to unit length. Finally, the inter-trial dynamics of the state vector in QM (between v and the next observable, separated by interval ) is represented by the unitary evolution formula   H ψ (t+1) ,  = U ψ (t+1) = ψ(t+1) ,

(14.10)

where U is a unitary matrix, defined by the property U−1 = U† .

(14.11)

Here, U−1 is the matrix inverse (U−1 U = U U−1 = I ), and U† is the conjugate transpose of U , obtained by transposing U and replacing each entry x + i y in it with its complex conjugate x − i y. The unitary matrix U should also be made a function of inter-trial variations in the environment (such as variations in overall noise level, or other participants’ responses) if they are non-negligible. The identity matrix I is a unitary matrix: if U = I , (14.10) describes no inter-trial dynamics, with the state remaining the same through the interval . Note that the eigenvalue v itself does not enter the computations. This justifies treating it as merely a label for the eigenprojectors and eigenspaces (so instead of 0, 1 we could use any other labels).

14.3 Measurement Sequences: Evolution (In)Effectiveness and Stability In this section we introduce terminology and preliminary considerations needed in the subsequent analysis. Throughout the chapter we will make use of the following way of describing measurements performed in successive trials: (a1 , v1 , p1 ) → · · · → (ar , vr , pr ) .

(14.12)

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14 No-Go Theorem for Modeling with Von Neumann Observables

We call this a measurement sequence. Each triple in the sequence consists of an observable a being measured, an outcome v recorded (0 or 1), and its conditional probability p. The probability is conditioned on the observables measured and the outcomes recorded in the previous trials of the same measurement sequence. Thus, p1 = P [a1 = v1 in trial 1] , p2 = P [a2 = v2 in trial 2 | a1 = v1 in trial 1] , p2 = P [a3 = v3 in trial 3 | a1 = v1 in trial 1, and a2 = v2 in trial 2] , ...

(14.13)

As we assume that the outcomes v1 , v2 , . . . in a measurement sequence have been recorded, all probabilities p1 , p2 , . . . are positive if the measurement sequence exists. Recall that the observables a1 , a2 , . . . in a sequence are uniquely determined by the measurement procedures applied, a1 , a2 , . . ., and that the outcomes (0 or 1) are eigenvalues of these observables. Consider now the two-trial measurement sequence (a, v, p) → (b, w, q) ,

(14.14)

where v, w ∈ {0, 1}. Let the quantum observable A (representing the question a) have the eigenprojectors E 0 , E 1 , and the quantum observable B (representing the question b) have the eigenprojectors Q 0 , Q 1 ; hence, A and B are represented by projection operators A = E 1 and B = Q 1 , respectively. If the initial state of the system is ψ = ψ (1) , we have

and ψ (1) transforms into

p = E v ψ 2 ,

(14.15)

Ev ψ = ψ (2) .

E v ψ

(14.16)

Assuming an interval  between the two trials, ψ (2) evolves into ψ(2) = U ψ (2) =

U E v ψ .

E v ψ

(14.17)

This is the state vector paired with B in the next measurement, yielding, with the help of some algebra,   2

U† Q w U E v ψ 2 U E ψ

Q w  v q = Q w ψ(2) 2 = = .

E v ψ 2

E v ψ 2

(14.18)

As a special case U can be the identity matrix (no inter-trial changes in the state vector), and then we have

14.3 Measurement Sequences: Evolution (In)Effectiveness and Stability

q= because in this case

Q w E v ψ 2 ,

E v ψ 2

  U† Q w U = Q w .

243

(14.19)

(14.20)

It is possible, however, that the latter equality holds even if U† is not the identity matrix. In fact it is easy to see that this happens if and only if U and B commute, i.e., (14.21) U B = BU . For the proof of this, see paper [262]. We will say that Definition 14.3.1 A unitary operator U is ineffective for a quantum observable B given by the operator B if the two operators commute.  The justification for this terminology should be transparent: due to (14.20), in the computation (14.18) of the probability q the evolution operator can be ignored, yielding (14.19). The notion of inefficiency of the evolution operator will play an important role in the analysis of repeated measurements below. Our next consideration regards the set of all possible values of the initial state vector ψ for a given measurement sequence. In the applications of QM in physics, this set is assumed to cover the entire Hilbert space in which they are defined. We are not justified to adopt this assumption in psychology, it would be too strong: one could argue that the initial states in a given experiment may be forbidden to attain values within certain areas of the Hilbert space. At the same time, it seems even less reasonable to allow for the possibility that the initial state for a given measurement sequence is always fixed at one particular value. The initial state vectors, as follows from both the QM principles and common sense, should depend on the system’s history prior to the given experiment, and this should create some variability from one replication of this experiment to another. This is important, because, given a set of observables, specially chosen initial state vectors may exhibit “atypical” behaviors, those that would disappear if the state vector were modified even slightly. This leads us to adopting the following Stability Principle: If ψ is a possible initial state vector for a given measurement sequence in an n-dimensional Hilbert space, then there is an open ball Br (ψ) centered at ψ with a sufficiently small radius r, such that any vector ψ + δ in this ball, normalized by its length ψ + δ , is also a possible initial state vector for this measurement sequence. We will say that Definition 14.3.2 A property of a measurement sequence is (or holds) stable for an initial vector ψ, if it holds for all state vectors within a sufficiently small Br (ψ).  Almost all our propositions below are proved under this stability clause (see open access article for the proofs [262]).

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14 No-Go Theorem for Modeling with Von Neumann Observables

14.4 Measurement Sequences a → a Using the definitions and the language just introduced, we will now focus on the consequences of (14.8)–(14.10) for repeated measurements with repeated responses,   (a, v, p) → a, v, p  .

(14.22)

Consider an opinion polling experiment, with questions like a =“Is Bill Clinton trustworthy?” [359]. As argued for in Sect. 14.1, if the same question is posed twice, a → a, a typical respondent, who perhaps hesitated when choosing the response the first time she was asked a, would now certainly be expected to repeat it, perhaps with some display of surprise at being asked the question she has just answered. This may not be true for all possible questions, but it is certainly true for a vast class thereof. Let us formulate this as Property 1 (a → a repeatability). For some nonempty class of questions, if a question is repeated twice in successive trials (separated by one of a broad range of inter-trial intervals), the response to it will also be repeated. If a question a within the scope of this property is represented by Hermitian operator (projection) A = E 1 , we are dealing with the measurement sequence (14.22) in which p  = 1. Such a measurement sequence does not disagree with the formulas (14.8)–(14.9)–(14.10), and in fact is even predicted by them if the intervening intertrial evolution of the state vector is assumed to be ineffective. Indeed, (14.18) for the measurement sequence (14.22) acquires the form p =

 

U† E v U E v ψ 2

E v ψ 2

,

(14.23)

and the inefficiency of U for a implies p =

E v2 ψ 2 = 1,

E v ψ 2

(14.24)

because E v2 = E v holds for all projection operators. This is easy to understand informally. An outcome v observed in the first measurement, (a, v, p), is associated with an eigenspace V . The measurement orthogonally projects the state vector ψ = ψ (1) into this eigenspace, and this projection is normalized to become the new state ψ (2) . The application of the same measurement to ψ (2) orthogonally projects it into V again, but since ψ (2) is already in V , it does not change. The squared length of the projection therefore is 1, and this is what the probability p  is.

14.4 Measurement Sequences a → a

245

As it turns out, under the stability principle, effective inter-trial evolution is in fact excluded for the observables representing the questions falling within the scope of Property 1. In other words, for all such questions, the unitary operators U can be ignored in all probability computations. The following definition formalizes Property 1 (a → a repeatability) within the quantum formalism: Let us say that Definition 14.4.1 A quantum observable A representing a question a has the Lüders property with respect to a state vector ψ if the existence of the measurement (a, v, p)  for this ψ and an outcome v ∈ {0, 1} implies that the property  p = 1 holds stable   for this ψ in the measurement sequence (a, v, p) → a, v, p . In other words, the Lüders property means that an answer to a question a (represented by operator A) is repeated if the question is repeated, and that this is true not just for one initial state vector ψ, but for all state vectors sufficiently close to it. We now can formulate our first proposition (see [262] for its proof). Proposition 14.4.2 (repeated measurements) An observable A has the Lüders prop erty if and only if U in (14.10) is ineffective for A. In the formulation of Property 1, the interval  and the question represented by A can vary within some broad limits, whence the inefficiency of U for a should also hold for each of these intervals combined with each of these questions. We have to be careful not to overgeneralize the Lüders property and the ensuing inefficiency property. As we discussed, one can think of situations where replications of a question may lead the respondent to “change her mind.” The most striking contrast, however, is provided by psychophysical applications of QM. Here, the inter-trial dynamics not only cannot be ignored, it must play a central role. Let us illustrate this on an old but very thorough study by Atkinson, Carterette, and Kinchla [31]. In the experiments they report, each stimulus consisted of two sideby-side identical fields of luminance L, to one of which a small luminance increment L could be added, serving as the signal to be detected. There were three stimuli: a = (L + L , L) , b = (L , L + L) , c = (L , L) .

(14.25)

In each trial the observer indicated which of the two fields, right one or left one, contained the signal. There were thus two possible responses: Left and Right. An application of QM analysis to these experiments requires a, b, c to be translated into observables (Hermitian operators) A, B, C, each with two eigenvalues, say, 0 = Left and 1 = Right. In the experiments we consider no feedback was given to the observers following a response. This is a desirable feature. It makes the sequence of trials we consider formally comparable to successive measurements of spins in quantum physics: measurements simply follow each other, with no interventions in between.3 3

However, this precaution seems unnecessary, as the results of the experiments with feedback in Ref. [31] do not qualitatively differ from the ones we discuss here.

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14 No-Go Theorem for Modeling with Von Neumann Observables

We are interested in measurement sequences     (a, 0, p1 ) → a, 0, p1  , (a, 1, p2 ) → a, 1, p2  , (b, 0, p3 ) → b, 0, p3  , (b, 1, p4 ) → b, 1, p4  , (c, 0, p5 ) → c, 0, p5 , (c, 1, p6 ) → c, 1, p6 .

(14.26)

Recall that the probabilities pi (i = 1, . . . , 6) are conditioned on previous measurements, so that, e.g., p1 + p2 = 1 while p1 + p2 = 1. For each observer, the probabilities were estimated from the last 400 trials out of 800 (to ensure an “asymptotic” level of performance). The results, averaged over 24 observers, were as follows: Experiment 1 index p p  1 0.65 0.73 2 0.35 0.38 3 0.36 0.39 4 0.64 0.71 5 0.50 0.53 6 0.50 0.60

Experiment 2 index p p  1 0.56 0.70 2 0.44 0.41 3 0.27 0.31 4 0.73 0.79 5 0.39 0.50 6 0.61 0.65

The leftmost column in each table corresponds to the index associated with p and p  in (14.26). Thus, the first row shows p1 and p1 , the last one shows p6 and p6 . The two experiments differed in one respect only: in Experiment 1 the stimuli a and b were presented with equal probabilities, while in Experiment 2 the stimulus b was three times more probable that a (the probability of c was 0.2 in both experiments). The results show, in accordance with conventional detection models, that this manipulation makes responses in Experiment 2 biased toward the correct response to b. This aspect of the data, however, is not of any significance for us. What is significant, is that, in accordance with Proposition 14.4.2, we should conclude that the inter-trial evolution (14.10) here intervenes always and significantly.

14.5 Measurement Sequences a → b → a Returning to the opinion polling experiments, consider the situation involving two questions, such as a =“Is Bill Clinton honest?” and b =“Is Al Gore honest?” The two questions are posed in one of the two orders, a → b or b → a, to a large group of people. The same as with asking the same question twice in a row, one would normally consider it unnecessary to extend these sequences by asking one of the two questions again, by repeating b or a after having asked a and b. A typical respondent, again, will be expected to repeat her first response. We find it “almost certain” (the “almost” being inserted here because we cannot refer to any systematic experimental

14.5 Measurement Sequences a → b → a

247

study of this obvious expectation) that from the nonempty (in reality, vast) class of questions falling within the scope of Property 1 one can always choose pairs of questions falling within the scope of the following extension of this property. Property 2 (RRE, see Chap. 1). Within a nonempty subclass of questions (and for the same set of inter-trial intervals) for which Property 1 holds, if a question a is asked following questions a and b (in either order), the response to it will necessarily be the same as that given to the question a the first time it was asked. As always, we represent questions a, b with quantum observables A, B (Hermitian operators and projection state update resulting from measurement’s feedback action). We use the following notation: the probability of obtaining a value v when measuring the observable a is denoted pva , qva , etc. (the letters p, q, etc. distinguishing different measurements); we use analogous notation for the probability of obtaining a value w when measuring the observable b.4 Consider the measurement sequence   (a, v, pva ) → (b, w, pwb ) → a, v, pv

(14.27)

  = 1 and qwa = 1. As it turns out, this RRE implies that in these sequences pva property has an important consequence (assuming the two inter-trial intervals in the measurement sequences belong to the same class as  in Proposition 14.4.2).

Proposition 14.5.1 (alternating measurements) Let observables a and b represented by operators A and B possess the Lüders property, and let the measurement sequences (14.28) (a, v, pva ) → (b, w, pwb ) exist for all v, w ∈ {0, 1}, and some initial state vector ψ. Then, in the measurement  = 1 holds stable for this ψ if and only if the sequences (14.27), the property pva operators A and B commute, AB = B A.  In other words, if the probabilities pva , pwb , qwb , qva are nonzero in (14.28) for   = 1 and qwa = 1 for all state vectors some ψ, the sequences (14.27) exist with pva in a small neighborhood of ψ if and only if AB = B A. (See [262] for Proposition’s proof.) To illustrate how this works, recall that the Hermitian operators A and B commute if and only if they have one and the same set of orthonormal eigenvectors e1 , . . . , en (generally, not unique). Since the operators A and B have two eigenvalues each, the difference between the two observables is in how these eigenvectors are grouped into two eigenspaces. Take one of the measurement sequences of the (14.27)-type, say,    . (a, 1, p1a ) → (b, 0, p0b ) → a, 1, p1a 4

(14.29)

Probabilities are either experimental or quantum-theoretical; the reader can assign to them the meaning depending on contexts.

248

14 No-Go Theorem for Modeling with Von Neumann Observables

Since A and B have the Lüders property, all the probabilities are the same as if there was no inter-trial dynamics involved. Proceeding under this assumption, the first measurement projects the initial vector ψ = ψ (1) into V1 that spans some of the vectors e1 , . . . , en . Let this projection (after its length was normalized to 1) be ψ (2) . The second measurement projects ψ (2) into the intersection V1 ∩ W0 that spans a smaller subset of these vectors. The third measurement then, since the second normalized projection ψ (3) is already in V1 , does not change it, ψ (4) = ψ (3) . This  , being the scalar product of ψ (3) and ψ (4) , must means that the third probability, p1A be unity. The commutativity of A and B is important because it has an experimentally testable consequence. Proposition 14.5.2 (no order effect) If observables a and b possessing the Lüders property commute, then in the measurement sequences (a, v, pva ) → (b, w, pwb ) , (b, w, qwb ) → (a, v, qva ) the joint probabilities of the two outcomes are the same, pva pwb = qwb qva .

(14.30)

P [a = v in trial 1] = P [a = v in trial 2]

(14.31)

P [b = w in trial 1] = P [b = w in trial 2] .

(14.32)

Consequently,

and

(See [262] for the proof.) To clearly understand what is being stated, recall that pwb is the conditional probability of observing the value w of b given that before this the outcome was the value v of a. So, the product of pva pwb is the overall probability of the first of the two sequences in the proposition. The value of qwb qva is understood analogously. Equation (14.30) therefore states that P [a = v in trial 1 and b = w in trial 2] = P [b = w in trial 1 and a = v in trial 2] .

Equations (14.30)–(14.32) are empirically testable predictions. Moreover, if we assume that the questions like “Is Clinton honest” and “Is Gore honest” fall within the scope of RRE (and it would be amazing if they did not), these predictions are known to be de facto falsified.

14.5 Measurement Sequences a → b → a

249

Property 3 (QOE+RRE). Within a nonempty subclass of questions for which (RRE) holds (and for the same set of inter-trial intervals), the joint probability of two successive responses depends on the order in which the questions were posed. This is question order effect in combination with “response replicability effect (QOE + RRE, Chaps. 1, 9 and 10). We recall that QOE has in fact been presented as one for whose understanding QM is especially useful: the empirical finding that pva pwb = qwb qva is explained in Ref. [452] by assuming that the operators A and B representing the observables a and b do not commute. In the survey reported by Moore [359], about 1,000 people were asked two questions, one half of them in one order, the other half in another. The probability estimates are presented for four pairs of questions: the first pair was about the honesty of Bill Clinton (a) and Al Gore (b), the second about the honesty of Newt Gingrich (a) and Bob Dole (b), etc. probability of Yes to a question pair in a → b in b → a 1 0.50 0.57 2 0.41 0.33 3 0.41 0.53 4 0.64 0.52

probability of Yes to b question pair in a → b in b → a 1 0.60 0.68 2 0.64 0.60 3 0.56 0.46 4 0.33 0.45

The results are presented in the form of P [a = 1 in trial i] and P [b = 1 in trial i], i = 1, 2, so the tested predictions are (14.31) and (14.32). As we can see, for all question pairs, the probability estimates of Yes to the same question differ depending on whether the question was asked first or second. Given the sample size (about 500 respondents per question pair in a given order) the differences are not attributable to chance variation. Properties 1, 2, and 3 turn out to be incompatible within the framework of the von Neumann part of quantum measurement theory (observables as Hermitian operators and state updates as projections). We should conclude therefore that QM cannot be applied to the questions that have these properties without modifications. These modifications have been done in the framework of the theory of quantum instruments [369, 370] (Chaps. 9 and 10). We remark that when paper [262] was written its authors were not sure that aforementioned modifications are possible at all. This paper contained a discussion on the (im)possibility to proceed with POVMs, but it suffers of mathematically nonrigorous considerations. The next paper in this direction [49] only increased doubts in possibility to describe QOE+RRE within QM: atomic quantum instruments (those most widely used in quantum information theory) neither work. So, the situation, before publication of article [369], was really critical for quantum-like modeling in cognition and psychology.

Chapter 15

Probabilistic Structure of Cognition: May Be Even Worse than Quantum?

15.1 Comparing Foundations of Quantum Physics and Cognition In this chapter, we follow the article of Basieva and Khrennikov [50]. QP calculus relaxes a few important classical probabilistic constraints. For example, one of the basic laws of classical probability theory, the formula of total probability (FTP) can be violated (Chap. 1). In physical terms violation of FTP for quantum observables is represented as interference of probabilities [231, 232, 251]. As was found by Khrennikov [254, 264], FTP is violated as well for statistical data collected in experimental research in cognitive psychology. Busemeyer et al. [87] found coupling of the violation of FTP with the disjunction effect in psychology. In principle, there are no reasons to expect that quantum probability matches perfectly cognition. Yes, now we are sure that cognition cannot be represented with the aid of classical probability (Kolmogorov’s set-theoretic axiomatics, 1933), e.g., because FTP is violated. However, it may happen that quantum probability, although so successful for modeling of some features of cognition, cannot cover completely cognitive phenomena. There might be some effects which (probabilistically) are neither classical nor quantum. There are no reasons to expect that quantum probability would cover all statistical experiments with micro-systems. Recently one of the world’s best experimental groups working in quantum foundations, the group of prof. Weihs from Innsbruck, put tremendous efforts to find violations of the basic rule of quantum probability theory, the Born rule, by testing Sorkin’s equality [417], see [413, 414].1 1

Another model leading to violation of this rule was proposed in the framework of so-called prequantum classical statistical field theory (PCSFT) [263]. In PCSFT, Born’s rule is perturbed, quantum probability is defined as p(x) = |ψ(x)|2 + α|ψ(x)|4 , α 0. P(A)

(16.22)

We stress that other axioms are independent of this definition. We also present the formula of total probability (FTP) which is a simple consequence of the Bayes formula. Consider the pair, a and b, of discrete random variables. Then  P(a = α)P(b = β|a = α). (16.23) P(b = β) = α

Thus the b-probability distribution can be calculated from the a-probability distribution and the conditional probabilities P(b = β|a = α). These conditional probabilities are also known as transition probabilities.

16.5 Quantum Conditional Probability In the classical Kolmogorov probabilistic model (Sect. 16.4), besides probabilities one operates with the conditional probabilities defined by the Bayes formula (see Sect. 16.4, formula (16.22)). The Born’s postulate defining quantum probability should also be completed by a definition of the conditional probability. We have remarked that, for one concrete observable, the probability given by Born’s rule can be treated classically. However, the definition of the conditional probability involves two observables. Such situations cannot be treated classically. Thus conditional probability is really a quantum probability. Let observables a and b be represented by Hermitian operators A and B with spectral decompositions A=

 m

αm E a (αm ), B =



βm E b (βm )

(16.24)

m

Let ψ be a pure state and let E(αm )ψ = 0. Then the probability to get the value b = βm under the condition that the value a = αk was observed in the preceding measurement of the observable a on the state ψ is given by the formula

270

16 Formalism of Quantum Theory

P(b = βm |a = αk ; ψ) ≡

E b (βm ) E a (αk )ψ2 E a (αk ) ψ2

(16.25)

One can motivate this definition by appealing to the projection postulate (Lüders’ version). After the a-measurement with output a = αk initially prepared state ψ is projected onto the state E a (αk )ψ ψαk = . E a (αk ) ψ Then one applies the Born rule to the b-measurement for this state. Thus, quantum conditional probability is based on the operation of the state update resulting from observation and the calculus of quantum conditional probabilities is a “probabilistic shadow” of the calculus of quantum states. Let the operator A have nondegenerate spectrum, i.e., for any eigenvalue α the corresponding eigenspace is one dimensional. Then P(b = βm |a = αk ; ψ) = E b (βm )eka 2

(16.26)

(here Aeka = αk eka ). Thus, in this case the conditional probability does not depend on the original state ψ. We can say that the memory about the original state was destroyed. If also the operator B has nondegenerate spectrum then we have: P(b = βm |a = αk ; ψ) = |emb |eka |2 and P(a = αk |b = βm ; ψ) = |eka |emb |2 . Symmetry of the scalar product implies the following important result: Let both operators A and B have purely discrete nondegenerate spectra and let Ek a ψ = 0 and Em b ψ = 0. Then conditional probability is symmetric and it does not depend on the original state ψ: P(b = βm |a = αk ; ψ) = P(a = αk |b = βm ; ψ) = |emb |eka |2 .

(16.27)

So, the state-label can be omitted and we can simply write P(b = βm |a = αk ), P(a = αk |b = βm ). We remark that, classical (Kolmogorov-Bayes) conditional probability is not symmetric, besides very special situations. Thus QM is described by a very specific probabilistic model. Consider two nondegenerate observables. Set pβ|α = P(b = β|a = α). The matrix of transition probabilities P b|a = ( pβ|α ) is not only stochastic, i.e.,  β

pβ|α = 1

16.6 Derivation of Interference of Probabilities

271

but it is even doubly stochastic.:  α

pβ|α =

 α

|eβb |eαa |2 = eβb |eβb  = 1.

In Kolmogorov’s model, stochasticity is the general property of matrices of transition probabilities. However, in general classical matrices of transition probabilities are not doubly stochastic. Hence, double stochasticity is a very specific property of quantum probability. We remark [254] that statistical data collected outside quantum physics, e.g., in decision-making by humans and psychology, violates the quantum law of double stochasticity. Such data cannot be mathematically represented with the aid of Hermitian operators with nondegenerate spectra. One has to consider either Hermitian operators with degenerate spectra or positive operator valued measures (POVMs).

16.6 Derivation of Interference of Probabilities Let H2 = C × C be the two dimensional complex Hilbert space and let ψ ∈ H2 be a quantum state. Let us consider two dichotomous observables b = β1 , β2 and a = α1 , α2 represented by Hermitian operators B and A, respectively (one may consider simply Hermitian matrices). Let eb = {eβb } and ea = {eαa } be two orthonormal bases consisting of eigenvectors of the operators. The state ψ can be represented in the two ways (16.28) ψ = c1 e1a + c2 e2a , cα = ψ|eαa ; ψ = d1 e1b + d2 e2b , dβ = ψ|eβb .

(16.29)

P(a = α) ≡ P(a = α|ψ) = |cα |2 .

(16.30)

P(b = β) ≡ P(b = β|ψ) = |dβ |2 .

(16.31)

By Postulate 4 we have

The possibility to expand one basis with respect to another basis induces connection between the probabilities P(a = α) and P(b = β). Let us expand the vectors eαa with respect to the basis eb e1a = u 11 e1b + u 12 e2b (16.32) e2a = u 21 e1b + u 22 e2b

(16.33)

where u αβ = eαa , eβb . Thus d1 = c1 u 11 + c2 u 21 , d2 = c1 u 12 + c1 u 22 . We obtain the quantum rule for transformation of probabilities:

272

16 Formalism of Quantum Theory

P(b = β) = |c1 u 1β + c2 u 2β |2 .

(16.34)

On the other hand, by the definition of quantum conditional probability, see (16.25), we obtain: (16.35) P(b = β|a = α) = |eαa |eβb |2 . By combining (16.30), (16.31) and (16.34), (16.35) we obtain the quantum formula of total probability – the formula of the interference of probabilities: P(b = β) =



P(a = α)P(b = β|a = α)

(16.36)

α

+2 cos θ



P(a = α1 )P(b = β|a = α1 )P(a = α2 )P(b = β|a = α2 )

In general cos θ = 0. Thus the quantum FTP does not coincide with the classical FTP (16.23) which is based on the Bayes’ formula.

16.7 Compatible Versus Incompatible Observables In quantum physics, observables a and b are called compatible if they can be jointly measurable and the joint probability distribution (JPD) P(a = x1 , b = x2 |ψ) is well defined; observables which cannot be jointly measurable and, hence, their JPD cannot be defined are called incompatible. In the mathematical formalism, compatibility and incompatibility are formalized through commutativity and noncommutativity, respectively. If observables are described as Hermitian operators A and B, compatibility is encoded as commutativity of operators, [A, B] = 0;

(16.37)

if they are observables given by PVMs a = (E a (x1 )) and b = (E b (x2 )), then compatibility is mathematically formalized as [E a (x1 ), E b (x2 )] = 0;

(16.38)

for all x1 ∈ X a , x2 ∈ X b , where X a and X b are the ranges of values of these observables. Incompatibility of observables given by Hermitian operators A and B is described the noncommutativity condition [A, B] = 0 or, for POV-observables, as violation of (16.38) at least for one pair (x1 , x2 ). For compatible observables, JPD is given by the following extension of the Born’s rule: P(a = x1 , b = x2 |ψ) = E a (x1 )E b (x2 )ψ2 = E b (x2 )E a (x1 )ψ2 .

(16.39)

16.8 Quantum Logic

273

By using JPD compatible observables can be modeled in the classical probabilistic formalism. The formula (16.39) can be generalized for an arbitrary number of compatible observables a1 , . . . , am as P(a1 = x1 . . . am = xm |ψ) = E a1 (x1 ) . . . E am b(xm )ψ2

(16.40)

and this expression is invariant w.r.t. permutations of observables. We point out that compatibility of quantum observables is identical to pairwise compatibility, if observables are pairwise compatible, i.e., each pair of them is jointly measurable and JPD for each pair ai , a j is well defined, then they are jointly measurable and their JPD is well defined by formula (16.40). This is the specific property of quantum observables. Generally there are no reasons to expect that pairwise measurement should imply joint measurement of the vector of observables a1 , . . . , am or in the probabilistic terms - the existence of the pairwise JPD would imply the existence of JPD for this vector. We now make the following remark on quantum conditional probability for compatible observables, by combining formulas (16.25) and (16.39), we obtain: P(b = βm |a = αk ; ψ) =

P(a = x1 , b = x2 |ψ) , P(a = x1 |ψ)

(16.41)

hence the classical Bayes formula for conditional probability (16.22) is recovered. Thus, for compatible observables, the calculus of conditional probabilities coincides with the classical one. We stress that compatibility vs. incompatibility are coupled to joint measurability vs. non-measurability; in probabilistic terms: existence vs. nonexistence of JPD; in operator terms: commutativity vs. noncommutativity. The latter algebraic characterization is often coupled straightforwardly with the order effect: measurement of observable a and then b generates statistics which differs from statistics in measurement of observable b and then b. In fact, this is the misleading interpretation of noncommutativity of operators.

16.8 Quantum Logic Birkhoff and von Neumann [69] suggested to represent events (propositions) by orthogonal projectors in complex Hilbert space H. For an orthogonal projector E, we set H E = EH, its image, and vice versa, for subspace L of H, the corresponding orthogonal projector is denoted by the symbol E L . The set of orthogonal projectors is a lattice with the order structure: E ≤ Q iff H E ⊂ H Q or equivalently, for any ψ ∈ H, ψ|Eψ ≤ ψ|Qψ.

274

16 Formalism of Quantum Theory

We recall that the lattice of projectors is endowed with operations “and” (∧) and “or” (∨). For two projectors E1, E2, the projector R = E1 ∧ E2 is defined as the projector onto the subspace H R = H E1 ∩ H E2 and the projector S = E1 ∨ E2 is defined as the projector onto the subspace H R defined as the minimal linear subspace containing the set-theoretic union H E1 ∪ H E2 of subspaces H E1 , H E2 : this is the space of all linear combinations of vectors belonging these subspaces. The operation of negation is defined as the orthogonal complement: E ⊥ = {y ∈ H : y|x = 0 for all x ∈ H E }. In the language of subspaces the operation “and” coincides with the usual settheoretic intersection, but the operations “or” and “not” are nontrivial deformations of the corresponding set-theoretic operations. It is natural to expect that such deformations can induce deviations from classical Boolean logic. Consider the following simple example. Let H be two dimensional Hilbert space √ with the orthonormal basis (e1 , e2 ) and let v = (e1 + e2 )/ 2. Then E v ∧ E e1 = 0 and E v ∧ E e2 = 0, but E v ∧ (E e1 ∨ E e2 ) = E v . Hence, for quantum events, in general the distributivity law is violated: E ∧ (E1 ∨ E2) = (E ∧ E1) ∨ (E ∧ E2)

(16.42)

The lattice of orthogonal projectors is called quantum logic. It is considered as a (very special) generalization of classical Boolean logic.2 Any sub-lattice consisting of commuting projectors can be treated as classical Boolean logic.

16.9 Tensor Product Consider two arbitrary (finite) dimensional Hilbert spaces, H1 , H2 . For each pair of vectors ψ ∈ H1 , φ ∈ H2 , form a new formal entity denoted by ψ ⊗ φ. Then consider the sums =



ψi ⊗ φi .

i

On the set of such formal sums we can introduce the linear space structure. This is the tensor product H = H1 ⊗ H2 . In particular, if we take orthonormal bases in (1) (2) Hk , (e(k) j ), k = 1, 2, then (ei j = ei ⊗ e j ) form an orthonormal basis in H, any  ∈ H, can be represented as = 2



ci j ei j ≡



ci j ei(1) ⊗ e(2) j ,

(16.43)

See article [371] for detailed analysis of the impact of nondistributivity of quantum logic for cognition and decision-making.

16.9 Tensor Product

275

We also introduce the notion of the tensor product of operators. Consider two linear operators Ai : Hi → Hi , i = 1, 2. Their tensor product A ≡ A1 ⊗ A2 : H → H is defined starting with the tensor products of two vectors: A ψ ⊗ φ = (A1 ψ) ⊗ (A2 φ). Then it is extended by linearity. By using the coordinate representation (16.43) the tensor product of operators can be represented as A=



ci j Aei j ≡



ci j A1 ei(1) ⊗ A2 e(2) j ,

(16.44)

If operators Ai , i = 1, 2, are represented by matrices (akl(i) ), with respect to the fixed bases, then the matrix (akl.nm ) of the operator A with respect to the tensor product of these bases can be easily calculated. In the same way one defines the tensor product of Hilbert spaces, H1 , . . . , Hn , denoted by the symbol H = H1 ⊗ · · · ⊗ Hn . We start with forming the formal entities ψ1 ⊗ · · · ⊗ ψn , where ψ j ∈ H j , j = 1, . . . , n. Tensor product space is defined as the set of all sums 

ψ1 j ⊗ · · · ⊗ ψn j

j

(which has to be constrained by some algebraic relations, but we omit such details). Take orthonormal bases in Hk , (e(k) j ), k = 1, . . . , n. Then any  ∈ H can be represented as =

 α

cα eα ≡

 α=( j1 ... jn )

(n) c j1 ... jn e(1) j1 ⊗ · · · ⊗ e jn .

(16.45)

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16 Formalism of Quantum Theory

16.10 Symbolism of Ket- and Bra-Vectors Dirac’s notations [125] are widely used in quantum information theory. Vectors of Hilbert space H are called ket-vectors, they are denoted as |ψ. The elements of the dual space H of H, the space of linear continuous functionals on H, are called bravectors, they are denoted as ψ|. Dirac used the expression ψ|φ for the duality form between H and H, i.e., ψ|φ is the result of application of the linear functional f ψ ≡ ψ| to the vector |φ. In the Hilbert case one can identify H with its dual space H and the duality form with the scalar product. Consider an observable a given by the Hermitian operator A with nondegenerate spectrum. Thus, the normalized eigenvectors ei of A form the orthonormal basis in H (we recall that our considerations are restricted to the case of finite dimensional H). Let Aei = αi ei . In Dirac’s notation ei is written as |αi  and, hence, any pure state can be written as   ci |αi , |ci |2 = 1; (16.46) |ψ = i

i

projection onto |αi  is written as E αi = |αi αi |, and it acts formally as E αi |ψ = (|αi αi |)|ψ ≡ αi |ψ|αi , and the operator A is written as A=



αi |αi αi |.

(16.47)

i

Now consider two Hilbert spaces H1 and H2 and their tensor product H = H1 ⊗ H2 . Let (|αi ) and (|βi ) be orthonormal bases in H1 and H2 corresponding to the eigenvalues of two observables A and B. Then vectors |αi  ⊗ |β j  form the orthonormal basis in H. Typically in physics the sign of the tensor product is omitted and these vectors are written as |αi |β j  or even as |αi β j . Thus any vector ψ ∈ H = H1 ⊗ H2 can be represented as ψ=



ci j |αi β j ,

(16.48)

ij

where ci j ∈ C.

16.11 Qubit In quantum information theory typically qubit states (quantum analogs of bits) are represented with the aid of observables having the eigenvalues 0, 1. Qubit space is two dimensional, its vectors have the form |ψ = c0 |0 + c1 |1, |c0 |2 + |c1 |2 = 1.

(16.49)

16.12 Entanglement

277

A pair of qubits is represented in the tensor product of single qubit spaces, here pure states can be represented as superpositions: |ψ = c00 |00 + c01 |01 + c10 |10 + c11 |00.

(16.50)

 where i j |ci j |2 = 1. In the same way the n-qubit state is represented in the tensor product of n one-qubit state spaces (it has the dimension 2n ): |ψ =



cx1 ...xn |x1 . . . xn ,

(16.51)

x j =0,1



where

|cx1 ...xn |2 = 1.

x j =0,1

We remark that the dimension of the n qubit state space grows exponentially with the growth of n. The natural question about possible physical realizations of such multI -dimensional state spaces arises. The answer to it is not completely clear; it depends very much on the used interpretation of the wave function.

16.12 Entanglement Consider the tensor product H = H1 ⊗ H 2 ⊗ · · · ⊗ Hn of Hilbert spaces Hk , k = 1, 2, . . . , n. The states of the space H can be separable and non-separable (entangled). We start by considering pure states. The (pure) states from the first class, separable pure states, can be represented in the form: |ψ = ⊗nk=1 |ψk  = |ψ1 . . . ψn ,

(16.52)

where |ψk  ∈ Hk . The states which cannot be represented in this way are called nonseparable, or entangled. Thus mathematically the notion of entanglement is very simple, it means impossibility of tensor factorization. For example, let us consider the tensor product of two one-qubit spaces. Select in each of them an orthonormal basis denoted as |0, |1. The corresponding orthonormal basis in the tensor product has the form |00, |01, |10, |11.

(16.53)

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16 Formalism of Quantum Theory

Then, so called Bell’s states √ √ |+  = (|00 + |11)/ 2, |−  = (|00 − |11)/ 2;

(16.54)

√ √ | +  = (|01 + |10)/ 2, | −  = (|01 − |10)/ 2,

(16.55)

are entangled. They also form an orthogonal basis in two qubit space. Entanglement is typically considered as one of the most distinguished features of quantum information theory. Its interpretation is a difficult foundational problem. Entanglement is typically coupled to quantum nonlocality and spooky action at a distance, i.e., to one of quantum exotics. We proceed pragmatically and couple entanglement to correlations of observables under conditional measurements (see article [55] for details). The definition of entanglement can be generalized to mixed states. A state ρ ∈ D (H1 ⊗ H2 ) is called separable if it can be represented in the form: ρ=



ckm ρ1(k) ⊗ ρ2(m) ,

(16.56)

km ( j)

where ρi ∈ D (Hi ) , i = 1, 2. A compound state that cannot be represented in this form is called entangled.

Chapter 17

Contextuality, Complementarity, and Bell Tests

17.1 Preliminary Discussion Forgotten Contribution of Bohr to Contextuality Theory Contextuality is one of the hottest topics of modern quantum physics, both theoretical and experimental. During the recent 20 years, it was discussed in numerous papers published in top physical journals. Unfortunate of these discussions is that from the very beginning contextuality (in fact, its special form which we call “join measurement contextuality”(JMC), Sect. 17.13) was coupled to the issue of nonlocality. It was Bell’s intention in his analysis of the possible seeds of the violation of the Bell type inequalities [61]. Surprisingly, Bell had never mentioned [62] general contextuality which we call “Bohr contextuality”. The latter has no straightforward connection to the Bell inequalities; it is coupled to the notion of incompatibility of observables - the Bohr principle of complementarity. One of the explanations for this astonishing situation in quantum foundations is that Bohr presented his ideas in a vague way; moreover, he often changed his vague formulations a few times at different occasions. In this chapter we briefly present Bohr’s ideas about contextuality of quantum measurements and their role in his formulation of the complementarity principle. Then, we move to the Bell inequalities [60, 62]. This pathway towards these inequalities (i.e., via Bohr’s contextuality-complementarity) highlights the role of incompatibility of quantum observables in the Bell framework and gives the possibility to operate with the Bell inequalities without mentioning the ambiguous notion of quantum nonlocality (spooky action at a distance).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_17

279

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17 Contextuality, Complementarity, and Bell Tests

What Does Contextuality Mean? In this situation when so many researchers write and speak about quantum contextuality, one should be sure that this notion is well defined and its physical interpretation is clear and well known. In fact, before started to think about the meaning of contextuality, I was completely sure in this. Strangely enough, I was not able to create a consistent picture. And I was really shocked when by visiting the institute of Atom Physics in Vienna and having conversation with Rauch and Hasegawa, I found that they are also disappointed. They asked me about the contextuality meaning. And they performed the brilliant experiments [46, 95] to test contextuality in the framework of neuron interferometry! They had a vague picture of what was tested. They had a vague picture of what was tested and what is the physical meaning of their experimental results! Then, in Stockholm by being in the Ph.D. defense jury of one student who was supervised by prof. Bengtsson (let call her Alice), I asked Alice about the physical meaning of contextuality. (Her thesis was about it.) Alice answered that she has no idea about the physical interpretation of advanced mathematical results obtained in her thesis. Generally I like discussions. To stimulate a debate, I told that Rauch and Hasegawa had the strange idea that contextuality is just noncommutativity, a sort of the order effect in the sequential measurements (this was the final output of our discussions in Vienna). Unfortunately, in Stockholm the discussion (with comments of a few top experts, including Bengtsson and Lindblad) quickly finished with the conclusion that the question is interesting, but not for the Ph.D.-defense.

Jump from Contextuality to Bell Inequalities Typically by writing a paper about contextuality in QM one starts by referring to this notion as JMC: dependence of the outcomes of some observable a on its joint measurement with another observable b (see Sect. 17.13). However, this definition is countefactual and cannot be used in the experimental framework. Nevertheless, the “universal contextuality writer” is not disappointed by this situation and he immediately jumps to the Bell inequalities [60, 62] which are treated as noncontextual inequalities [14]. Moreover, often contextuality is identified with the violation of the Bell inequalities. This identification shadows the problem of the physical meaning of contextuality. One jumps from the problem of understanding to calculation of a numerical quantity, the degree of the violation of some Bell inequality. Such inequalities are numerous. And they can be tested in different experimental situations and generate the permanent flow of highly recognized papers. I suggested the following critical illustration to this strategy (contextuality = violation of the Bell inequalities) [308]. in the CP-framework consider the notion of a random sequence. Theory of randomness is the result of the intensive research (Mises, Church, Kolmogorov, Solomovov, Chatin, Martin-Löf; see [271]). This the-

17.1 Preliminary Discussion

281

oretical basis led to elaboration of the variety of randomness tests which are used to check whether some sequence of outputs of physical or digital random generator is random. The NIST test (a batch of tests for randomness) is the most widely used. So, here we also use tests, but beyond them there is the well developed theory of randomness. In particular, this leads to understanding that even if a sequence x passed the NIST test, this does not imply that it is random. In principle, there can be found another test such that x would not pass it. The latter would not be a surprise. In contrast to the above illustration by randomness theory, in quantum information theory contextuality is per definition the violation of some noncontextual (Bell) inequality.1 So, the theoretical notion is identified with a test. This is really bad! Not only from the theoretical viewpoint, but even from the practical one. As was mentioned, by working with randomness people understand well that even passing the NIST test does not guarantee randomness. In QM, passing the Bell test (in the sense of the violation of some Bell type inequality [60, 62]) is per definition is equivalent to contextuality. This is wrong strategy which led to skews in handling quantum contextuality.

Bell Tests in Physics: Signaling and Other Anomalies in Data The first signs that addiction to one concrete test of contextuality (Bell inequalities [60, 62]) may lead to the wrong conclusions were observed by Adenier and Khrennikov [8]. Adenier was working on the translation of the Ph.D. thesis of Alain Aspect [30] (due to the joint agreement with prof. Aspect and Springer) and he pointed out to me that he found some strange anomalies in Aspect’s data. One of them was signaling. i.e., dependence of detection probability on one side (Bob’s lab) on the selection of an experimental setting on another side (Alice’s lab). Then, we found [8] signaling in the data from the famous experiment of Weihs closing the nonlocality loophole [459] (see also [458]); we point out to Weihs’ reply [460] to our critical analysis of his data. (At that time we understood that it is very difficult if possible at all to get the rough statistical data from experimenters, i.e., not the beautiful graphs illustrating articles, but click by click data. We were really surprised by experimenters unwillingness to share their data with researchers willing to analyze their experiments and in this way to attract further attention to their results. Even my wide contants in quantum community did not help us so much. Only Gregor Weihs sent his data, as a batch of CDs, to Guillaume Adenier. And, as any good will, this data sharing was punished by our finding of signaling in it.) Our publications [8] attracted attention to the problem of signaling in data collected in quantum experiments. Slowly people started to understand that experimenter cannot be happy by just getting higher degree of the violation of say the CHSHinequality, with higher confidence. Often this implied the increase of the degree 1

We point out to the variety of the Bell type inequalities; see article [259] on the analysis of equivalence of the statistical tests based on some inequalities.

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17 Contextuality, Complementarity, and Bell Tests

of signaling. Experimenters started to check the hypothesis of signaling in data. Unfortunately, our message was ignored by some experimenters, e.g., the data from the “the first loophole free experiment” [199] demonstrated statistically significant signaling. Any Bell test should be combined with the test of experimental statistical data on signaling. We pointed out that signaling was not the only problem in Aspect’s data. As he noted in his thesis [30], the data contains “anomalies” of the following type. Although the CHSH-combination of correlations violates the CHSH-inequality, the correlation for the concrete pair of angles θ1 , φ, as the function of these angles, does not match the theoretical prediction of QM, the graph of the experimental data differs essentially from the theoretical cos-graph. Our attempts to discuss this problem with other experimenters generated only replies that “we do not have such anomalies in our data”.

17.2 Växjö Model In the probabilistic terms, complementarity (incompatibility of observables) means that their joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with context-dependent family of probability spaces—the Växjö probability model (see [232–234, 236, 238, 239, 245, 249, 251, 256]): MZ = (PC , C ∈ Z), where Z is a family of contexts and, for each C ∈ Z, PC = (C , FC , PC ) is Kolmogorov probability space [318] (Sect. 16.4). Here C is a sample space, FC is a σ -algebra of subsets of C (events), and PC is a probability measure on FC . All these structures depend on context C. To develop a fruitful theory, Z must satisfy to some conditions on inter-relation between contexts [232–234, 238, 239, 249, 251, 256]. In CP the points of C represent elementary events, the most simple events which can happen within context C. Although these events are elementary, their structure can be complex and include the events corresponding to appearance of some parameters (“hidden variables”) for a system under observation and measurement devices, times of detection and so on. We remark that the Växjö probability model serves as the mathematical basis of the the Växjö interpretation of quantum mechanics, the contextual statistical interpretation [236, 245]. Observables are given by random variables on contextually-labeled probability spaces, measurable functions, aC : C → R. The same semantically defined observ-

17.2 Växjö Model

283

able a is represented by a family of random variables (aC , C ∈ Za ), where Za is the family of contexts for which the a-observable can be measured. In MZ averages and correlations are also labeled by contexts,  aC = E[aC |PC ] =

C

 aC (ω)d PC (ω) =

 abC = E[aC bC |PC ] =

R

xd Pa|C (x),

(17.1)

 C

aC (ω)bC (ω)d PC (ω) =

R

x yd Pa,b|C (x, y), (17.2)

where Pa|C is the probability distribution of ac and Pa,b|C is the JPD of the pair of random variables (aC , bC ). In (17.1) C ∈ Za and in (17.2) C ∈ Za,b = Za ∩ Zb . Since in context Za,b both observables a and b are represented by random variables, namely, by aC and bC , it is natural to assume that in this context both observables can be measured and the measure-theoretic JPD Pa,b|C represents mathematically the JPD for joint measurements of the pair of observables (a, b). In further sections, we analyze the probabilistic structure of QM by considering the Bell inequalities [60, 62] and concentrating on the CHSH-inequality [106]2 and the Fine theorem [150]. The Växjö probability model serves as the mathematical basis of the Växjö interpretation of quantum theory (Sect. 20.4).

Mental Contextuality In this book we are interested in applications of quantum theory outside of physics, e.g., to cognition, decision-making, and social science. The following questions naturally arise: 1. What are lessons from QM? 2. What are cognitive and social specialties? The main lesson (from my viewpoint) is that the Bohr contextualitycomplementarity principle is the basic principle of quantum theory. Everything else is its consequence (including JMC and the statements concerning the Bell inequalities). The main commonality between quantum physical and mental measurements is the irreducible character of contribution of the interaction between a system and measurement device (including self-measurements performed by the brain, its mental functions, Chap. 6). Here “system” is understood as an information system (see further chapters of the book). Our decisions, emotions, thoughts are outputs of such complex interactions. This viewpoint matches especially well to the model of unconsciousconscious interaction. In this model, a mental system and a self-measuring apparatus

2

See article [259] on using different Bell inequalities in experimental studies.

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17 Contextuality, Complementarity, and Bell Tests

(a mental function) are informationally located in some regions of the unconscious and consciousness, respectively (Chap. 7). As in physics, this viewpoint on measurements outputs leads to understanding that some mental phenomena are complementarity and that there exist incompatible mental (self-)observations. As was pointed out by Dzhafarov et al. [133] all known experiments with mental observables (say questions or tasks) demonstrate statistically significant signaling. We point to two alternatives: 1. Careful analysis of the cognitive experiments would point out to the sources of signaling which are compatible with the quantum formalism. These sources can be either technicalities, i.e., non-clean data or sampling of insufficient size, or the violations of some conditions assumed in the Bell test, the conditions on the state preparation and measurement procedures. 2. Signaling in mental data is irreducible, mental contextuality cannot be modeled within the quantum formalism. We shall discuss these issues in further sections of this chapter.

Summary of Preliminary Discussion We can conclude the discussion with a few statements: • The theoretical definition of contextuality as JMC suffers of appealing to conterfactuals. • Identification of contextuality with the violation of the Bell inequalities is not justified, neither physically nor mathematically (in the last case such an approach does not match the mathematical tradition). • The Bell tests have to accompanied with test on signaling. • “Unuploaded to internet experiments have no results” [265]. • Probabilistically contextuality-complementarity is described by contextual probability (as by the Växjö model, [232–234, 238, 239, 249, 251, 256]).

17.3 Thinking over Bohr’s Ideas This section is devoted to thinking over Bohr’s foundational works in terms of contextuality.

17.3 Thinking over Bohr’s Ideas

285

17.3.1 Bohr Contextuality The crucial question is about the physical meaning of contextuality; without answering to it, JMC (even by ignoring counterfactuality) is mystical, especially for spatially separated systems. Even spooky action at a distance is welcome—to resolve this mystery. In series of papers [281, 286] the physical meaning of contextuality was clarified through referring to the Bohr’s complementarity principle. Typically this principle is reduced to wave-particle duality. (In fact, Bohr had never used the latter terminology.) However, Bohr’s formulation of the complementarity principle is essentially deeper. Complementarity is not postulated; for Bohr, it is the natural consequence of the irreducible dependence of observable’s outcome on the experimental context. Thus, the outcomes of quantum observables are generated in the complex process of the interaction of a system and a measurement device [74] (see also [300, 383]). This dependence on the complex of experimental conditions is nothing else than a form of contextuality, Bohr-contextuality (Sect. 17.3.2). We remark that JMC is its special case. But, in contrast to JMC, the physical interpretation of Bohr-contextuality is transparent—dependence of results of measurements on experimental contexts. And it does not involve the use of conterfactuals. Such contextuality is the seed of complementarity, the existence of incompatible observables. (We recall that observables are incompatible if they cannot be measured jointly.) Moreover, contextuality without incompatibility loses its value. If all observables were compatible, then they might be jointly measured in a single experimental context and multicontextual consideration would be meaningless. One can go in deeper foundations of QM and ask: Why is dependence on experimental context (system-apparatus interaction) is irreducible? Bohr’s answer is that irreducibility is due to the existence of indivisible quantum of action given by the Planck constant (see articles [299, 301] for discussion and references). We stress that the interactions considered in quantum theory are not the classical force-like interactions. QM-interactions are represented by unitary operators and they follow the “physical laws” of quantum theory. This is the operational viewpoint on interaction which is widely used in theory of open quantum system. It is mathematically formalized in the indirect measurement scheme and the mathematical construction of quantum instruments from unitary interaction between a system and apparatus [114, 115, 364–366].

17.3.2 Bohr’s Principle of Contextuality-Complementarity The Bohr principle of complementarity [74] is typically presented as wave-particle duality, incompatibility of the position and momentum observables. The latter means the impossibility of their joint measurement. We remark that Bohr started with the

286

17 Contextuality, Complementarity, and Bell Tests

problem of incompatibility of these observables by discussing the two slit experiment. In this experiment position represented by “which slit?” observable and momentum is determined the detection dot on the registration screen. (This screen is covered by photo-emulsion and placed on some distance beyond the screen with two slits.) Later Bohr extended the wave-particle duality to arbitrary observables which cannot be jointly measured and formulated the principle of complementarity. He justified this principle by emphasizing contextuality of quantum measurements. The Bohr’s viewpoint on contextuality was wider than in the modern discussion on quantum contextuality related to the Bell inequality. The later is contextuality of joint measurement with a compatible observable (Sect. 17.13). In 1949, Bohr [74] presented the essence of complementarity in the following widely cited statement: This crucial point ... implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.

In short, Bohr’s way to the complementarity principle, the claim on the existence of incompatible quantum observables, can be presented as the following chain of reasoning [281, 286]: • CONT1 An outcome of any observable is composed of the contributions of a system and a measurement device. • CONT2 The whole experimental context has to be taken into account. • INCOMP1 There is no reason to expect that all experimental contexts can be combined with each other and all observables can be measured jointly; thus some observables can be incompatible. • INCOMP2 The Heisenberg uncertainty principle implies that the position and momentum observables are incompatible. The statements CONT1 + CONT2 and INCOMP1+ INCOMP12 compose the contextual and incompatibility parts of Bohr’s reasoning. Bohr considered the two slit experiment as the experimental confirmation of the Heisenberg uncertainty principle. Hence, for him INCOMP2 was experimentally verified. In the light of such structuring of Bohr’s thinking, it might be more natural to speak about two Bohr’s principles: • Contextuality Principle. • Complementarity Principle.

17.4 Probabilistic Viewpoint on Contextuality-Complementarity

287

We can unify these two principles and speak about the ContextualityComplementarity Principle, instead of simply the complementarity principle. Unfortunately, the contextual dimension of the complementarity is typically missing in the discussions on quantum foundations. Of course, the fact that outcomes of observables depend irreducibly on experimental conditions, does not imply the existence of other experimental conditions which are incompatible. However, if the outcomes were the objective properties of physical systems, then at least in principle any two experimental contexts would be combinable. By speaking that the wave and particle properties cannot be merged in a single experimental framework (the wave-particle duality) we have to remember that the seed of their dispersing is contextuality: they are determined within two different experimental contexts. This is the good place to remark that it is possible to establish wave-particle correlations in the conditional experiments as in article [152], where amplitude of the electromagnetic field was conditioned to photon detection. This experiment supports treatment of quantum probabilities as conditional ones. In the light of Bohr-contextuality, the following natural question arises: How can one prove that the concrete observables a and b cannot be jointly measured (i.e., that they are incompatible)? From the viewpoint of experimental verification, the notion of incompatibility is difficult. How can one show that the joint measurement of a and b is impossible? One can refer to the mathematical formalism of quantum theory and say that the observables a and b cannot be jointly measurable if the corresponding Hermitian operators A and B do not commute. But, another debater can say that may be this is just the artifact of the quantum formalism: yes, the operators do not commute, but observables still can be jointly measured. The latter argument was used by some experts in quantum foundations against the appeal to the Heisenberg uncertainty principle as justification of the existence of incompatible observables—INCOMP2.

17.4 Probabilistic Viewpoint on Contextuality-Complementarity The basic analysis on the (in)compatibility problem is done in the probabilistic terms. Suppose that observables a, b, c, . . . can be in principle jointly measured, but we are not able to design the corresponding measurement procedure. Nevertheless, the assumption of joint measurability, even hypothetical, implies the existence of JPD. What are consequences of JPD’s existence? We shall comeback to this question in Sect. 17.4. Now we remark that the principle of contextuality-complementarity can be reformulated in probabilistic terms. In short, we can say that the measurement part of QM is a (special) calculus of context-dependent probabilities. This viewpoint was presented in a series of works

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17 Contextuality, Complementarity, and Bell Tests

summarized in monograph [251] devoted to the calculus of context dependent probability measures (PC ), C ∈ Z, where Z is a family of contexts constrained by some consistency conditions. We emphasize that QP is a special contextual probabilistic calculus. Its specialty consists in the possibility to use a quantum state (the wave function) |ψ to unify generally incompatible contexts. This is the important feature of QP playing the crucial role in quantum foundations. In classical statistical physics the contextuality of observations is not emphasized. Here it is assumed that it is possible to proceed in the CP-framework: to introduce a single context-independent probability measure P and reproduce the probability distributions of all physical observables on the basis of P. This is really possible. However, the careful analysis of interplay of probability measures appearing in classical physics shows that even here contexuality cannot be ignored. In articles [240, 246], there are considered models, e.g., in theory of complex disordered systems (spin glasses), such that it is impossible to operate with just one fixed probability measure P. A variety of context dependent probabilities have to be explored. We especially emphasize the paper on classical probabilistic entanglement [11].

Existence of Joint Probability Distribution Let P = (, F , P) be a Kolmogorov probability space [318]. Each random variable a :  → R determines the probability distribution Pa . The crucial point is that all these distributions are encoded in the same probability measure P : Pa (α) = P(ω ∈  : a(ω) = α). (We consider only discrete random variables.) In CP, the probability distributions of all observables (represented by random variables) can be consistently unified on the basis of P. For any pair of random variables a, b, their JPD Pa,b is defined and the following condition of marginal consistency holds: Pa (α) =



Pa,b (α, β)

(17.3)

β

This condition means that observation of a jointly with b does not change the probability distribution of a. Equality (17.3) implies that, for any two observables b and c,  β

Pa,b (α, β) =



Pa,c (α, γ ).

(17.4)

γ

In fact, condition (17.4) is equivalent to (17.3): by selecting the random variable c such that c(ω) = 1 almost everywhere, we see that (17.4) implies (17.3). These considerations are easily generalized to a system of k random variables a1 , . . . , ak .

17.4 Probabilistic Viewpoint on Contextuality-Complementarity

289

Their JPD is well defined, Pa1 ,...,ak (α1 , . . . , αk ) = P(ω ∈  : a1 (ω) = α1 , · · · , ak (ω) = αk ). And marginal consistency conditions holds for all subsets of random variables (ai1 , . . . , aim ), m < k). Consider now some system of experimental observables a1 , . . . , ak . If the experimental design for their joint measurement exists, then it is possible to define their JPD Pa1 ,...,ak (α1 , . . . , αk ) (as the relative frequency of their joint outcomes). This probability measure P ≡ Pa1 ,...,ak can be used to define the Kolmogorov probability space, i.e., the case of joint measurement can be described by CP. Now consider the general situation: only some groups of observables can be jointly measured. For example, there are three observables a, b, c and only the pairs (a, b) and (a, c) can be measurable, i.e., only JPDs Pa,b and Pa,c can be defined and associated with the experimental data. There is no reason to assume the existence of JPD Pa,b,c . In this situation equality (17.4) may be violated. In the terminology of QM, this violation is called signaling. Typically one considers two labs, Alice’s and Bob’s labs. Alice measures the a-observable and Bob can choose whether to measure the b- or c-observable. If   Pa,b (α, β) = Pa,c (α, γ ), (17.5) β

γ

one says that the a-measurement procedure is disturbed (in some typically unknown way) by the selection of a measurement procedure by Bob, some signal from Bob’s lab approaches Alice’s lab and changes the probability distribution. This terminology, signaling versus no-signaling, is adapted to measurements on spatially separated systems and related to the issue of nonlocality. In quantum-like models, one typically works with spatially localized systems and is interested in contextuality (what ever it means). Therefore we called condition (17.4) marginal consistency (consistency of marginal probabilities) and (17.5) is marginal inconsistency. In the further presentation we shall use changeably both terminologies, marginal consistency versus inconsistency and no-signaling versus signaling. In future we shall be mainly interested in the CHSH inequality. In this framework, we shall work with four observables a1 , a2 and b1 , b2 ; experimenters are able to design measurement procedures only for some pairs of them, say (ai , b j ), i, j = 1, 2. In this situation, there is no reason to expect that one can define (even mathematically) the JPD Pa1 ,a2 ,b1 ,b2 (α1 , α2 , β1 , β2 ). This situation is typical for QM. This is a complex interplay of theory and experiment. Only probability distributions Pai ,b j can be experimentally verified. However, in theoretical speculation, we can consider JPD Pa1 ,a2 ,b1 ,b2 as a mathematical quantity. If it existed, we might expect that there would be some experimental design for joint measurement of the quadruple of observables (a1 , a2 , b1 , b2 ). On the other hand, if it does not exist, then it is meaningless even to try to design an experiment for their joint measurement.

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Now we turn back to marginal consistency; in general (if Pa1 ,a2 ,b1 ,b2 does not exist), it may be violated. However, in QM it is not violated: there is no signaling. This is the miracle feature of QM. Often it is coupled to spatial separation of systems: a1 or a2 are measured on S1 and b1 or b2 on S2 . And these systems are so far from each other that the light signal emitted from Bob’s lab cannot approach Alice’s lab during the time of the measurement and manipulation with the selection of experimental settings. However, as we shall see no-signaling is the general feature of the quantum formalism which has nothing to do with spatial separability nor even with consideration of the compound systems.

17.5

Clauser, Horne, Shimony, and Holt (CHSH) Inequality

We restrict further considerations to the CHSH-framework, i.e., we shall not consider other types of Bell inequalities. How can one get to know whether JPD exists? The answer to this question is given by a theorem of Fine [150] concerning the CHSH inequality. Consider dichotomous observables ai and b j (i, j = 1, 2) taking values ±1. In each pair (ai , b j ) observables are compatible, i.e., they can be jointly measurable and pairwise JPDs Pai ,b j are well defined. Consider correlation  ai b j  = E[ai b j ] =

αβ d Pai ,b j (α, β);

in the discrete case, ai b j  = E[ai b j ] =



αβ Pai ,b j (α, β).

αβ

By Fine’s theorem JPD Pa1 ,a2 ,b1 ,b2 exists if and only if the CHSH-inequality for these correlations is satisfied: |a1 b1  + a1 b2  + a2 b1  − a2 b2 | ≤ 2.

(17.6)

and the three other inequalities corresponding to all possible permutations of indexes i, j = 1, 2.

Derivation of CHSH Inequality Within Kolmogorov Theory The crucial assumption for derivation of the CHSH-inequality is that all correlations are w.r.t. the same Kolmogorov probability space P = (, F , P) and that all observ-

17.5 Clauser, Horne, Shimony, and Holt (CHSH) Inequality

291

ables ai , b j , i, j = 1, 2, can be mathematically represented as random variables on this space. Under the assumption of the JPD existence, one can select the sample space  = {−1, +1}4 and the probability measure P = Pa1 ,a2 ,b1 ,b2 . Thus, the CHSH inequality has the form,      [a1 (ω)b1 (ω) + a1 (ω)b2 (ω) + a2 (ω)b1 (ω) − a2 (ω)b2 (ω)]d P(ω) ≤ 2. (17.7) 

The variable ω can include hidden variables of a system, measurement devices, detection times, and so on. It is only important the possibility to use the same probability space to model all correlations. The latter is equivalent to the existence of JPD Pa1 ,a2 ,b1 ,b2 . This is the trivial part of Fine’s theorem, JPD implies the CHSH inequality. The other way around is more difficult [150]. This inequality can be proven by integration of the inequality −2 ≤ a1 (ω)b1 (ω) + a1 (ω)b2 (ω) + a2 (ω)b1 (ω) − a2 (ω)b2 (ω) ≤ 2 which is the consequence of the inequality −2 ≤ a1 b1 + a1 b2 + a2 b1 − a2 b2 ≤ 2 which holds for any quadrupole of real numbers belonging [−1, +1].

Role of No-signaling in Fine Theorem The above presentation of Fine’s result is common for physics’ folklore. However, Fine did not consider explicitly the CHSH inequalities presented above, see (17.6). He introduced four inequalities that are necessary and sufficient for the JPD to exist, but these inequalities are expressed differently to the CHSH inequalities. The CHSH inequalities are derivable from Fine’s four inequalities stated in Theorem 3 of his paper. We remark that the existence of the quadruple JPD implies marginal consistency (no-signaling). And the Fine theorem presupposed that marginal consistency. This is the good place to make the following remark. In quantum physics this very clear and simple meaning of violation of the CHSH-inequality (non-existence of JPD) is obscured by the issue of nonlocality. However, in this article we are not aimed to criticize the nonlocal interpretation of QM. If somebody speaks about spooky action at a distance and other mysteries of QM, we have no quarrel with this, since we only use the quantum formalism, not its special interpretation. Finally, we point out that the Bell type inequalities were considered already by Boole (1862) [76, 77] as necessary conditions for existence of a JPD.

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Violation of CHSH Inequality for Växjö Model If it is impossible to proceed with the same probability space for all correlations, one has to use the Växjö model (Sect. 17.2), and there is no reason to expect that the following inequality (and the corresponding permutations) would hold,    ’

 C11

aC11 (ω)bC11 (ω)d PC11 (ω) +

 C21

C12

 aC21 (ω)bC21 (ω)d PC21 (ω) −

C22

aC12 (ω)bC12 (ω)d PC12 (ω)+

(17.8)

  aC22 (ω)bC22 (ω)d PC22 (ω) ≤ 2,

where Ci j is the context for the joint measurement of the observables ai and b j . Here ai -observable is represented by random variables (aCi1 , aCi2 ) and bi -observable by random variables (bC1i , bC2i ). In the Växjö model the condition of no-signaling may be violated; for discrete variables, signaling means that 

PaC1 11,b1 (x, y) =

y



PaC1 12,b2 (x, y).

y

17.6 CHSH-Inequality for Quantum Observables In this section we present the purely quantum treatment of the CHSH inequality and highlight the role of incompatibility in its violation (we follow article [286]).

Non-compound Systems Although in QM the CHSH inequality is typically studied for compound systems with the emphasis to the role of the tensor product structure of the state space, in this section we shall not emphasize the latter and proceed for an arbitrary state space and operators. Consequences and simplifications for the tensor product case will be presented in Sect. 17.6. Observables ai , b j are described by (Hermitian) operators Ai , B j , i, j = 1, 2, [Ai , B j ] = 0, i, j = 1, 2.

(17.9)

17.6 CHSH-Inequality for Quantum Observables

293

We remark that generally [A1 , A2 ] = 0, [B1 , B2 ] = 0, i.e., the observables in the pairs a1 , a2 and b1 , b2 do not need to be compatible. Observables under consideration are dichotomous with values ±1. Hence, the corresponding operators are such that Ai2 = B 2j = I. The latter plays the crucial role in derivation of the Landau identity (17.13). Consider the CHSH correlation represented in the quantum formalism and normalized by 1/2, B =

1 [A1 B1  + A1 B2  + A2 B1  − A2 B2 ]. 2

(17.10)

This correlation is expressed via the Bell-operator: B=

1 [A1 (B1 + B2 ) + A2 (B1 − B2 )] 2

(17.11)

as B = ψ|B|ψ.

(17.12)

Simple calculations lead to the Landau identity: B2 = I − (1/4)[A1 , A2 ][B1 , B2 ].

(17.13)

If at least one commutator equals to zero, i.e., [A1 , A2 ] = 0,

(17.14)

[B1 , B2 ] = 0,

(17.15)

or

then, for quantum observables, we obtain the inequality |B| ≤ 1.

(17.16)

Derivation of (17.16) was based solely on quantum theory. This inequality is the consequence of compatibility for at least one pair of observables, A1 , A2 or B1 , B2 . Symbolically Eq. (17.16) is the usual CHSH-inequality, but its meaning is different. Equation (17.16) can be called the quantum CHSH inequality.

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Compound Systems Here, H = Ha ⊗ Hb and A j = A j ⊗ I, B j = I ⊗ B j , where Hermitian operators A j and B j act in Ha and Hb , respectively. Incompatibility of observables in factors of the tensor product is called local incompatibility. Theorem 17.6.1 Conjunction of local incompatibilities in H A and H B is the necessary and sufficient condition for the violation of the quantum CHSH-inequality.  Thus, the CHSH-inequality for observables which can be mathematically modelled within the quantum formalism is violated iff [A1 , A2 ] = 0 and [B1 , B2 ] = 0. This result is very supporting for coupling of the violation of the Bell type inequalities with incompatibility of observables.

Tsirelson Bound √ By using Landau identity (7.6) we can derive the Tsirelson bound 2 2 for the CHSH correlation of quantum observables, i.e., observables which are represented by Hermitian operators Ai , B j , i, j = 1, 2, with spectrum ±1, so Ai2 = B 2j = I. For such operators, for any state |ψ, we have: √ |A1 B1  + A1 B2  + A2 B1  − A2 B2 | ≤ 2 2.

(17.17)

On the other hand, if observables are not described by QM, then this bound can be exceeded. For the Växjö contextual probability model, the CHSH correlation may approach the value 4; the same is true for CbD model.

17.7 Signaling in Physical Versus Psychological Experiments By using the quantum calculus of probabilities, it is easy to check whether the nosignaling condition holds for quantum observables. However, experimenters were focused on observing as high violation of (17.6) as possible and they ignored the no-signaling condition. However, if the latter is violated, then a JPD automatically does not exist, and there is no reason to expect that (17.6) would be satisfied. The first paper highlighting the signaling problem in quantum experimental research was the article of Adenier and Khrennikov [8]. They demonstrated that statistical data collected in the basic experiments (for that time) performed by Aspect [30] and Weihs [459] violate the no-signaling condition. Nowadays no signaling

17.8 Contextuality-by-Default

295

condition is widely discussed in quantum information theory, but without referring to the pioneer work of Adenier and Khrennikov [8]. The experiments to check CHSH and other Bell-type inequalities were also performed for mental observables in the form of questions asked to people [23, 24], [10, 53, 91, 108, 133, 136]. The first such experiment was done in 2008 [108] and was based on the theoretical paper of Khrennikov [244]. As was pointed out by Dzhafarov et al. [133], all known experiments of this type suffer of signaling. Moreover, in contrast to physics, in psychology there are no theoretical reasons to expect nosignaling. In this situation Fine’s theorem is not applicable. And Dzhafarov and his coauthors were the first who understood the need of adapting the Bell-type inequalities to experimental data exhibiting signaling. Obviously, the interplay of whether or not a JPD exists for the quadruple of observables s = (a1 , a2 , b1 , b2 )

(17.18)

can’t be considered for signaling data.

17.8 Contextuality-by-Default Contextual Indexing Dzhafarov and his coauthors [131–135] proposed to consider, instead of quadruple of observables s, the corresponding octuple s which is generated by doubling each observable and associating s with four contexts of measurements of pairs, C11 = (a1 , b1 ), C12 = (a1 , b2 ), C21 = (a2 , b1 ), C22 = (a2 , b2 ).

(17.19)

Thus, the basic object of CbD-theory (for the CHSH inequality) is octuple of observables   (17.20) s = (a11 , b11 ), (a12 , b21 ), (a21 , b12 ), (a22 , b22 ) , so, e.g., observable a1 measured jointly with observable b j is denoted a1 j . The essence of CbD is coupling of observables to contexts. This coupling generates double-indexing of observables. The mathematical representations of the observable ai measured jointly with the observables b j , j = 1, 2 cannot be identified. This is the basic foundational counterpart of CbD. CbD uses the random variables language to present observables, so it operates with four pairs of random variables:   S = (A11 , B11 ), (A12 , B21 ), (A21 , B12 ), (A22 , B22 ) ,

(17.21)

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17 Contextuality, Complementarity, and Bell Tests

generally each pair of random variables is defined on its own probability space, Pi j = (i j , Fi j , Pi j ). So, CbD and the Växjö contextual probability model have the same foundational basis and operate with the families of Kolmogorov probability spaces corresponding to measurement contexts. But technically they diverge.

Coupling Method The main mathematical apparatus of CbD is the coupling method which is widely used in CP [338, 426]. The Växjö model differs from CbD by the calculus of transition probabilities. Let Pi = (i , Fi , Pi ), i = 1, 2, be two Kolmogorov probability spaces and let X i be random variables on these spaces. Then a coupling of X 1 and X 2 is a new probability space P = (, F , P) over which there are two random variables Y1 and Y2 such that Y1 has the same distribution as X 1 while Y2 has the same distribution as X 2. In the CHSH-framework the coupling method is applied as follows. Let Pi j = (i j , Fi j , Pi j ) be probability spaces with vector random variables (Ai j , B ji ), then their coupling is a probability space P = (, F , P) over which there are defined four vector random variables (Ai j , B ji ) having the same probability distributions as the vectors (Ai j , B ji ). The coupling method can be considered as a mechanism of embedding of contextual probability model into CP. This embedding gives the possibility to apply the mathematical apparatus of CP for generally non-Kolmogorovian model. Such embedding is not unique and in CbD one searches for an optimal (in some sense, see below) coupling. We remark that the Växjö model can also be embedded in some Kolmogorov probability space and with such embedding the contextual probabilities are realized as the classical conditional probabilities [266]. Thus, coupling technique gives the possibility to operate with variety of octuples of random variables S = (A11 , B11 , A12 , B21 , A21 , B12 , A22 , B22 ),

(17.22)

defined on the same probability space and giving couplings for the system of contextual random variables S.

17.8 Contextuality-by-Default

297

Measure of Contextuality By moving from quadruple S to octuple S, one confronts the problem of identity of an observable which is now represented by two different random variables, e.g., the observable ai is represented by the random variables Ai j (ω), j = 1, 2. In the presence of signaling one cannot expect the equality of two such random variables almost everywhere. Dzhafarov et al. came up with a novel treatment of the observableidentity problem. It is assumed that averages m a;i j = Ai j , m b;i j = Bi j 

(17.23)

Ci j = Ai j B ji 

(17.24)

and covariation

are fixed. These are measurable quantities. They can be statistically verified by experiment. Set (17.25) δ(ai ) = m a;i1 − m a;i2 , δ(b j ) = m b; j1 − m b; j2 , and 0 =

  1  δ(ai ) + δ(b j ) . 2 i j

(17.26)

This is the experimentally verifiable measure of signaling. We remark that in the coupling representation the joint satisfaction of the CHSH inequalities, i.e., (17.6) and other inequalities obtained from it via permutations, can be written in the form: max |A11 B11  + A12 B21  + A21 B12  + A21 B22  − 2Ai j B ji | ≤ 2. ij

(17.27)

In the signaling-free situation, e.g., in quantum physics, the difference between the left-hand and right-hand sides is considered as the measure of contextuality. Denote (1/2 times) this quantity by CHSH . It is also experimentally verifiable. Then Dzhafarov and coauthors introduced quantity (P) =



ai (P) +



b j (P),

(17.28)

where ai (P) = P(ω : Ai1 (ω) = Ai2 (ω)), b j (P) = P(ω : B j1 (ω) = B j2 (ω)). (17.29)

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17 Contextuality, Complementarity, and Bell Tests

Here ai (P) characterizes mismatching of representations of observable ai by random variables Ai1 and Ai2 with respect to probability measure P; b j (P) is interpreted in the same way. The problem of the identity of observables is formulated as the mismatching minimization or identity maximization problem (P) → min

(17.30)

with respect to all octuple probability distributions P satisfying constraints (17.23), (17.24). And it turns out, that min = min (P).

(17.31)

It is natural to consider the solutions of the identity maximization problem (17.30) as CP-representations for contextual system S. The corresponding random variables have the highest possible, in the presence of signaling, degree of identity. The quantity min − 0 is considered as the measure of “genuine contextuality”. This approach is very useful to study contextuality in the presence of signaling. The key point is the coupling of this measure of contextuality with the problem of the identity of observables measured in different contexts. As was pointed out in article [134] : ...contextuality means that random variables recorded under mutually incompatible conditions cannot be join together into a single system of jointly distributed random variables, provided one assumes that their identity across different conditions changes as little as possibly allowed by direct cross-influences (equivalently, by observed deviations from marginal selectivity).

Bell–Dzhafarov–Kujala Inequality This approach to contextuality due to Dzhafarov–Kujala can be reformulated in the CHSH-manner by using what we can call CHSH-BDK inequality: max |A11 B11  + A12 B21  + A21 B12  + A21 B22  − 2Ai j B ji | − 2 0 ≤ 2. ij

(17.32) It was proven that octuple-system S exhibits no genuine contextuality, i.e., min = 0 ,

(17.33)

if and only if the CHSH-BDK inequality is satisfied. The general formula for all cyclic systems was derived in [137], and a complete theory of cyclic systems is given in [140].

17.10 Sources of Signaling Compatible with Quantum Formalism

299

17.9 Mental Signaling: Fundamental or Technical? As was pointed out in article [133], data collected in psychological experiments contains statistically significant signaling patterns. One can wonder whether signaling is a fundamental feature of mental observations or a mere technicality, perhaps the consequence of badly designed or/and performed experiments. We recall that in physics signaling patterns were found in all Bell experiments during the first 30 years. Since quantum theory predicts the absence of signaling, signaling patterns in experimental statistical data were considered to be a technicality.3 Understanding the technical sources of signaling and finding ways to eliminate it required great efforts of the experimenters. Finally, Giustina et al. [175] and Shalm et al. [410] reported that the null hypothesis of signaling can be rejected for the data collected in these experiments. In psychology the situation is more complicated. There are no theoretical reasons to expect no signaling. And it is not obvious whether signaling is a technicality or a fundamental feature of cognition. For the moment, only a few experiments have been performed. One cannot exclude that in the future more advanced experiments would generate data without signaling. However, it may be that mental signaling is really a fundamental feature of cognition. In any event, it is interesting to attempt to find non-signaling contextual patterns in human behavior. As the first step towards such experiments, possible experimental sources of mental signaling should be analyzed. We shall do this in the next section.

17.10 Sources of Signaling Compatible with Quantum Formalism As was already emphasized, quantum measurement theory is free from signaling: marginals are consistent with JPDs. Now we prove this simple fact.

Quantum Theory: No-signaling Consider the quantum Hilbert space formalism, a state given by density operator ρ; three observables a, b, c represented by operators A, B, C (acting in H) with spectral families of projectors E a (x), E b (x), E c (x). It is assumed that in each pair (a, b) and (a, c) the observables are compatible, [A, B] = 0, [A, C] = 0. Then P(a = x, b = y|ρ) = TrρEa (x)Eb (y), P(a = x, c = y|ρ) = TrρEa (x)Ec (y) (17.34) 3

Here “technicality” refers to situations in which technical equipment, experimental design, improper calibration of detectors and so on, influence an experiment results.

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17 Contextuality, Complementarity, and Bell Tests

and hence 

P(a = x, b = y|ρ) = TrρEa (x)



y

Eb (y) = TrρEa (x)

(17.35)

y

= TrρEa (x)

 y

Ec (y) =



P(a = x, b = y|ρ).

y

and we point out that TrρEa (x) = P(a = x|ρ)

(17.36)

and, hence, both marginal probability distributions coincide with the probability of measurement of the a-observable alone. We remark that this proof of no-signaling can be easily extended to generalized quantum observables given by POVMs. So, in quantum measurement theory there is no place for signaling. We also recall that signaling (marginal inconsistency) is absent in classical (Kolmogorov) probability theory [318]. On the other hand, it is natural for contextual probability (as in the Växjö model [232–234, 238, 239, 249, 251, 256]).

No Signaling Even for Nonlocal Quantum Observables Now let H = H1 ⊗ H2 , where H1 , H2 be the state spaces of the subsystems S1 , S2 of the compound system S = (S1 , S2 ) and let the observables a, b, c are nonlocal, in the sense that their measurements are not localized to subsystems. The corresponding operators have the form A = A1 ⊗ A2 , B = B1 ⊗ B2 , C = C1 ⊗ C2 , where A2 , B1 , C1 do not need to be equal to I. Let us decompose say E a (x) into tensor product E 1a (x1 ) ⊗ E 2a (x2 ), where outcomes of a are labeled by pairs of numbers (x1 , x2 ) → x (the map from pairs to the a-outcomes is not one to one). However, the above general scheme based on (17.35) is still valid. The tensor product decomposition of projections does not play any role in summation in (17.35). Nonlocality of observables cannot generate signaling. This is unexpected fact, because typically signaling is associated with nonlocality. But, as we have seen, this is not nonlocality of observables. Now we turn to the quantum CHSH inequality. As we seen in Sect. 17.6, for quantum observables its violation is rigidly coupled only to their incompatibility. Even if Ai = Ai1 ⊗ Ai2 , i = 1, 2, but [A1 , A2 ] = 0, then the CHSH inequality is not violated. So, by quantum theory signaling is impossible. But, e.g., in decision-making, signaling patterns (expressing marginal inconsistency) were found in all known experiments. This is the contradiction between the quantum-like model for decision-

17.10 Sources of Signaling Compatible with Quantum Formalism

301

making and experiment. This situation questions the whole project on applications of the quantum formalism to modeling behavior of cognitive systems. However, there are some “loopholes” which can lead to marginal inconsistency.

Signaling on Selection of Experimental Settings Consider the Bohm-Bell experiment: a source of photons’ pairs S = (S1 , S2 ) and two polarization beam splitters (PBSs) in Alice’s and Bob’s labs; their output channels are coupled to the photo-detectors. Denote orientations of PBSs by θ and φ. Suppose now that the quantum observables representing measurements on S1 and S2 depend on both orientations, a = a(θ, φ), b = b(θ, φ). (17.37) They are represented by operators A = A(θ, φ), B = B(θ, φ).

(17.38)

Thus selection of setting φ for PBS in Bob’s lab changes the observable (measurement procedure) in Alice’s lab and vice verse. This is a kind of signaling between Bob’s lab and Alice’s lab, signaling carrying information about selection of experimental settings.4 In such a situation, P(a(θ, φ) = x, b(θ, φ) = y) = TrρEa(θ,φ) (x)Eb(θ,φ) (y) and hence 

P(a(θ, φ) = x, b(θ, φ) = y|ρ) = TrρEa(θ,φ) (x)



y

Eb(θ,φ)

(17.39)

(17.40)

y

= TrρEa(θ,φ) = P(a(θ, φ) = x|ρ),

P(a(θ, φ) = x, b(θ, φ ) = y|ρ) = TrρEa(θ,φ ) (x)





Ea(θ,φ ) (y) =

(17.41)

y

TrρEa(θ,φ ) (x) = P(a(θ, φ ) = x|ρ). Generally

4



TrρEa(θ,φ) (x) = TrρEa(θ,φ ) (x).

(17.42)

This can also be referred to the absence of free will of experimenters w.r.t. selection of experimental settings. But, we would not follow this line of thought (which is so natural for philosophy of superdeterminism).

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or, in the probabilistic terms, P(a(θ, φ) = x|ρ) = P(a(θ, φ ) = x|ρ).

(17.43)

We remark that decomposition of S into subsystems S1 and S2 and association of observables a and b with these subsystems did not play any role in quantum calculations. Such decomposition and coupling it with spatial locality is important only in the physics as the sufficient condition to prevent signaling on selection of experimental settings. In the probabilistic terms each pair of settings determines context C = (θ, φ) and the corresponding probability space. Thus, we are in the framework of the Växjö model for contextual probability. Here the possibility of signaling and violation of the Bell type inequalities is not surprising. In cognitive experiments, observables are typically questions asked to a system S (e.g., a human). As we have seen, dependence of questions a and b on the same set of parameters can generate signaling. This dependence is not surprising. Even if questions a and b are processed by different regions of the brain, the physical signaling between these regions cannot be neglected. If θ and φ are the contents of the a- and b-questions, then after a few milliseconds the area of the brain processing a = a(θ ) would get to “know” about the content of the b-question and thus a-processing would depend on both parameter, a = a(θ, φ). We remark that an essential part of information processing in the brain is performed via electromagnetic field; such signals propagate with the light velocity and the brain is very small as a physical body. On the other hand, some kind of mental localization must be taken into account; mental functions performing different tasks use their own information resources (may be partially overlapping). Without such mental localization, the brain5 would not be able to discriminate different mental tasks and their outputs. At least for some mental tasks (e.g., questions), dependence of a on the parameter φ (see (17.37)) can be weak. For such observables, signaling can be minimized. Are there other sources of signaling compatible with quantum formalism?

State Dependence on Experimental Settings Let us turn to quantum physics. Here “signaling” often has the form of real physical signaling and it can reflect the real experimental situation. We now discuss the first Bell-experiment in which the detection loophole was closed [174]. It was performed in Vienna by Zelinger’s group and it was characterized by statistically significant signaling. By being in Vienna directly after this experiment, I spoke with people who did it. They told the following story about the origin of signaling—marginal 5

The real situation is more complex; not only the brain, but the whole nervous system is involved in mental processing.

17.10 Sources of Signaling Compatible with Quantum Formalism

303

inconsistency. The photon source was based on laser generating emission of the pairs of entangled photons from the crystal. It happened (and it was recognized only afterwards) that the polarization beam splitters (PBSs) reflected some photons backward and by approaching the laser they changed its functioning and backward flow of photons depended on the orientations of PBSs. In this situation “signaling” was not from b-PBS to a-PBS, but both PBSs sent signals to the source. Selection of the concrete pair of PBSs changed functioning of the source; in the quantum terms this means modification of the state preparation procedure. In this case selection of a pair of orientations leads to generation of a quantum state depending on this pair, ρab . This state modification contributed into the signaling pattern in data. The above physical experimental illustration pointed out to state’s dependence on experimental context as a possible source of signaling. It is clear that, for ρ = ρa,b , generally (17.44) Trρab Ea (x) = Trρac Ea (x). This dependence also may lead to violation of the Bell inequalities. In the probabilistic terms this is again the area of application of the Växjö model with contexts associated with quantum states, the probability measures depend on the experimental settings. We remark that it seems that the state variability depending on experimental settings was the source of signaling in Weihs’ experiment which closed nonlocality loophole [459]. At least in this way we interpreted his reply [460] to our paper [8]. Since Weihs’ was able to separate two “labs” to a long distance [459], the signals from one lab could not approach another during the process of measurement. In quantum physics experimenters were able to block all possible sources of state’s dependence on the experimental settings. Thus, it is claimed that one can be sure that ρ does not depend on a and b. By using the orientations of PBSs θ, φ, i.e., ρ = ρ(θ, φ), the latter condition can be written as ∂ρ(θ, φ) ∂ρ(θ, φ) = 0, = 0. ∂θ ∂φ

(17.45)

Stability of state preparation is the delicate issue. As we have seen, the source by itself can be stable and generate approximately the same state ρ, but the presence of measurement devices can modify its functioning. Moreover, even if any feedback to the source from measuring devices is excluded, laser’s functioning can be disturbed by fluctuations. Typically violation of state statsbility cannot be observed directly and the appearance of a signaling pattern can be considered as a sign on state’s variation. In physics the signaling can be rigidly associated with fluctuations in state preparation. Spatial separation leads to local parameter dependence of observables, i.e., a = a(θ ) and b = b(φ). For cognitive systems, it seems to be impossible to distinguish two sources of signaling: • joint dependence on parameters θ, φ determining contents of questions, • state dependence on θ, φ.

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17.11 Meal Choice Experiment: Possible Source of Signaling Here we briefly present the“meal choice experiment” from article [53].6 Consider three courses of meal: starters, main course, and deserts. All dishes are assigned with high (H) or low (L) calorie levels. Each participant of experiment see on the screen only two courses, say starters and the main course. Each course contains two dishes, say for starters: two salads. One dish in each course is of the H -type and another is of the L-type. Before the experiment each participant (say Alice) is instructed that, in each of two courses presented on the screen she can select only one dish under the constraint that the combinations H H and L L are forbidden, e.g., she can select the high calorie salad and the low calorie dish from the main course or vice verse. From my viewpoint, this exclusion principle is the crucial part of the experimental design. Meal courses play the role of systems, i.e.. Sa , Sb , Sc . For these systems we define the observables a, b, c yielding the values ±1 corresponding to H/L choices. The experiment is based on three compound systems Sab = (Sa , Sb ), Sbc = (Sb , Sc ), Sac = (Sa , Sc ). On these systems one can perform joint measurements (a, b), (b, c), (a, c).7 By looking at the screen and by seeing the compound system (the two courses under consideration), e.g., Sab , Alice applies the H − L exclusive principle in the context of the (a, b)-measurement. At this stage of decision-making, she projects her mental state |ψ onto the H L − L H entangled state corresponding to this concrete pair of courses: |ψ → |ψab  = ≡

ψ|Ha L b |Ha L b  + ψ|L a Hb |L a Hb  |ψ|Ha L b |2 + |ψ|L a Hb |2

ψ| +a −b | +a −b  + ψ| −a +b | +a −b  . |ψ| +a −b |2 + |ψ| −a +b |2

And, for this state she performs the joint (a, b) measurement generating the correlation ab ≡ abψab = −1. Then, for the system Sbc , the (b, c)-measurement is preceded by the mental state preparation to adjust the H − L-exclusion principle to this measurement: |ψ → |ψbc  = ≡ 6

ψ|Hb L c |Hb L c  + ψ|L b Hc |L b Hc  |ψ|Hb L c |2 + |ψ|L b Hc |2

ψ| +b −c | +b −c  + ψ| −b +c | +b −c  . |ψ| +b −c |2 + |ψ| −b +c |2

See Sect. A.3 for more mathematical details. In contextuality theory, typically pairs are written to show cyclic structure of measurements: (a, b), (b, c), (c, a).

7

17.11 Meal Choice Experiment: Possible Source of Signaling

305

In the same way the (a, c)-measurement on Sac is preceded by the mental state adjustment: |ψ → |ψac  = ≡

ψ|Ha L c |Ha L c  + ψ|L a Hc |L a Hc  |ψ|Ha L c |2 + |ψ|L a Hc |2

ψ| +a −c | +a −c  + ψ| −a +c | +a −c  . |ψ| +a −c |2 + |ψ| −a +c |2

This scheme leads to correlations ab = bc = ac = −1. Thus, we reproduce the result from [53] within the quantum formalism (see Appendix A for detail): max[−ab + bc + ac, ab − bc + ac, ab + bc − ac, −ab − bc − ac] = 3

We remark that this preparation of the projection type leads to correlation ±1 independently of the initial mental state |ψ. This is the good feature of this experimental design: it is not easy (if possible at all) to prepare an ensemble of humans in the same mental state (at least a pure state |ψ). This preparation step can generate signaling in the form of marginal inconsistency, since the states |ψab  and |ψac  can lead to different marginal probability distributions for the a-observable (see Appendix A for detail). Thus, the experimental design and the statistical data collected in article [53] can be described in the quantum-like framework. The contradiction between the results of [53] and the quantum formalism is apparent. In fact, consideration of three observables is not so interesting for the problem of matching of the results of article [53] with quantum theory. The situation is delicate: for three observables, one has to distinguish the original Bell inequality [60, 62] and Suppes-Zanotti inequality (see [298] for details). Therefore, it is better to consider four different meal courses [53], systems S1 , S2 , S3 , S4 . (A gourmand can create a good gastronomic picture for this experiment; personally I am fine with just the main course, so my gastronomic creativity is very low.) Observables on these systems are denoted as a1 , b1 , a2 , b2 . (We use these notation to be closer to the quantum physics, the CHSH inequality. In the purely contextual framework, one typically uses notation x1 , x2 , x3 , x4 , i.e., without underlining the compound system structure.) We select the systems pairwise by creating the compound systems and selecting pairwise joint observations on them: S12 = (S1 , S2 ), S32 = (S3 , S2 ), S34 = (S3 , S4 ), S14 = (S1 , S4 ). By keeping close to quantum physics observables are symbolized as (a1 , b1 ), (a2 , b1 ), (a2 , b2 ), (a1 , b2 ).

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17 Contextuality, Complementarity, and Bell Tests

On these compound systems, there are performed joint measurements of the corresponding pairs of observables. The maximal possible value= 4 for the CHSH combination of correlations cannot be approached solely with the exclusion principle: the latter must be used for three pairs and, for the forth pair, Alice’s mental state must be adjusted by H − H, L − L matching principle, e.g., the matching principle is applied for the compound system S34 = (S3 , S4 ) and the (a2 , b2 )-measurement is preceded by state adjustment: |ψ → |ψa2 b2  = ≡

ψ|Ha2 Hb2 |Ha2 Hb2  + ψ|L a2 L b2 |L a2 L a2  |ψ|L a2 L b2 |2 + |ψ|L a2 Hb2 |2

ψ| +a2 +b2 | +a2 +b2  + ψ| −a2 −b2 | −a2 −b2  . |ψ| +a2 +b2 |2 + |ψ| −a2 −b2 |2

In article [53] the aforementioned pre-measurement state adjustments were not highlighted. The exclusion principle was formulated in the non-contextual form: Alice does not want both courses to be of high calorie or both to be low calorie.

Such pre-measurement preparation generates the entangled state |ψ = |H L + |L H , where the states |H , |L are context independent. With this interpretation of the exclusion principle, in the quantum framework sig√ naling is impossible; it neither possible to exceed the Tsirelson bound 2 2. Thus, the results of the experiment [53] might be interpreted as the confirmation of impossibility to represent some mental observables (in some mental contexts) as quantum observables. (This position was emphasized by Basieva.) One should say that the experiments [53] demonstrated that, not only the CP-model, but even the QP-model of decision-making has its limits of applicability. In principle, this would not be surprising. As was permanently repeated in this book, neither quantum physics nor quantum(-like) cognition were derived from some fundamental and heuristically attractive principles; the√quantum formalism is applied, since it works. Signaling and exceeding of the 2 2-bound are the strong arguments to claim that we cannot proceed with quantum observables and that more general contextual probability theories have to used (as, e.g., the √ Växjö model). Such models can generate both signaling and exceeding of the 2 2-bound. This is the position of Basieva. Personally, I do not supports this position. As was noted, the crucial role is played by the exclusion principle and the way of its using by Alice in the process of decision-making. Although this principle can be formulated in the context independent form(as was suggested by Basieva), it seems that in the real mental processing Alice adjusts this principle to the concrete measurement context, i.e., she performs pre-measurement projection of her mental state |ψ on context dependent state |ψab  (for the (a, b) measurement).

17.12 Concluding Remarks on Cognitive Tests with Bell Inequalities

307

Thus, we discussed two opposite viewpoints on signaling and exceeding of the Tsirelson bound in experiments presented in paper [53]: 1. They are signs of the restricted QP applicability in cognition, psychology, and decision-making. 2. They can be modeled within quantum measurement theory by taking into account the pre-measurement (context dependent) state projections. Can one design an experiment which would distinguish aforementioned two positions? The main question is whether Alice really performs the context dependent mental state projection.

17.12 Concluding Remarks on Cognitive Tests with Bell Inequalities Another coauthor of article [53] Ehtibar Dzhafarov claimed that by using the coupling method one does not need to appeal to QP at all (especially strongly this viewpoint is expressed in article [139]). All cognitive phenomena can be described by CP. Generally this is correct (cf. [266]). However, although coupling and conditioning constructions are mathematically rigorous, they are not so attractive as QP which is a linear space mental model. Another its advantageous property is the presence of the unifying structure—a quantum state representing a mental (belief) state (see [285] for an extended discussion). Paper [53] describes a variety of choice experiments, i.e., not only the meal choice, but also a shop choice and a space orientation experiments. All these experiments are based on the exclusion (or matching) constraints guarantying entangled generation (in the quantum framework). I interpret these constraints as mental state adjustments to the concrete experimental situation corresponding to concrete pairs of questions. All these experiments can be modeled within QP similarly to the meal choice experiment. It seems that even the first experiment [98] demonstrated the presence of “true contextuality”, i.e., the violation of the CHSH-BDK inequality, can also be modeled via quantum-like representation with context dependent state adjustment. However, in article [98] this adjustment is not so straightforward as in later article [53]. And more careful analysis is needed. The common feature of the experiments suggested in [53, 98] is that, in fact, people are asked the same question, say “High or low calorie?” But this question is asked in different contexts. In my opinion, this is precisely the framework which we have considered above, i.e., various state preparations. Other objections of applicability of the quantum formalism to psychology will be discussed in part 5.

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17 Contextuality, Complementarity, and Bell Tests

17.13 Joint Measurement Contextuality In quantum measurement theory, selection of observables co-measurable with an observable a is considered as specification of measurement context of a-measurements; the a-value in the b-context can differ from the a-value in the c-context, for the same premeasurment state ψ. This is the essence of special contextuality playing so important role in quantum information theory; in [298] we called it joint measurement contextuality (JMC): Definition 17.13.1 (JMC) If a, b, c are three quantum observables, such that a is compatible with b and c, a measurement of a might give different result depending upon whether a is measured with b or with c.  We note that contextual behavior corresponds to the case of incompatible quantum observables b and c, i.e., there exist indexes such that [E mb , E nc ] = 0. If all observables are pairwise commute, i.e., for all indexes [E ka , E mb ] = [E ka , E nc ] = [E mb , E nc ] = 0, then, for any state ψ, it is possible to construct the noncontextual model of measurement based on the JPD for the triples of outcomes, P(a = xk , b = ym , c = z n |ψ) = E ka E mb E nc ψ 2 = · · · = E nc E ka E mb ψ 2 . (17.46) If b and c are incompatible, such a model is impossible. This is the contextuality scenario. However, JMC-contextuality formalized via above Definition cannot be tested experimentally, since it involves counterfactual reasoning. The only possibility is to test contextuality indirectly with the aid of Bell-type inequalities [60, 62]. If the sets of observables’ outcomes coincide with subsets of the real line, then the above considerations can be essentially simplified—with the Hermitian linear operators A, B, C representing observables. The contextuality scenario is related to observables satisfying the commutation relations [A, B] = [A, C] = 0, [B, C] = 0.

(17.47)

However, the operator language is misleading and not only because representation of outcomes by real numbers is too special for coming cognitive applications. The main problem is that in the linear-operator approach the basic entities are outcomes, eigenvalues of the operator-observable. The subspace Hxa is constructed as the space of the eigenvectors of the operator A corresponding to the eigenvalue x. In our coming modeling, we proceed another way around. The basic structures are mutually orthogonal subspaces Hk such that H = ⊕k Hk .

(17.48)

Then each of these subspaces is identified with some value of the PVM-observable A given by projectors on these subspaces. Roughly speaking, first states then values.

17.14 Contextual Resolution of Degeneration of Eigenvalues of Observables

309

17.14 Contextual Resolution of Degeneration of Eigenvalues of Observables JMC can be used to reduce degeneration of eigenvalues of observables. We shall use this technique in Chap. 7, to reduce degeneration of observables representing mathematically conscious experiences. Consider an observable a which is mathematically described as PVM (E a (x))x∈X . For the discrete set of observance’s values X = (xk ), we use notation E ka . Suppose now that some of projections E ka are degenerate; dim Hka > 1, where Hka = E ka H. Moreover, for some outcomes, degeneration is very high, dim Hka >> 1. In this case, a large set of states corresponds to the same outcome xk . Observer may be unsatisfied by such a situation; observer wants to refine his observation and split (at least partially) the state space corresponding to the fixed outcome. How can it be done? The quantum measurement formalism presents the very natural and simple procedure for refinement of states’ structure. Consider another quantum PVM-observable (E mb ) compatible with the original observable a, i.e., [E ka , E mb ] = 0 for all indexes k, m (Chap. 16, Sect. 16.7). We remark that b-observable has its own set of outcomes, Y = (ym ), which need not coincide with X. For any (pure) quantum state ψ, the PVM-observables a and b are jointly measurable with outcomes (xk , ym ) and the join probability distribution P(a = xk , b = ym |ψ) = E ka E mb ψ 2 = E mb E ka ψ 2 .

(17.49)

The projection postulate and commutativity of projectors imply that the postmeasurement state is the same for the joint measurement with outcome (xk , ym ) and sequential measurements, first a = xk and then b = xm or vice verse. The postmeasurement state can be represented as follows ψ → ψa=xk ,b=xm =

E mb E ka ψ E ka E mb ψ = .

E ka E mb ψ

E mb E ka ψ

(17.50)

(a,b) The state space Hka is reduced to the space Hkm = E ka E mb H(= E mb E ks H); so (a,b) Hka = ⊕m Hkm

(17.51)

(direct sum decomposition). Thus, the observer can specify its post-observation state much better. The most fruitful refinement is based on an observable with nondegenerate spectrum. Let all projectors E mb be one dimensional, dim Hmb = 1, and let emb be the corresponding basis vector, i.e., E mb emb = δ( j − m)emb . In this case, the outcome (xk , ym ) completely determines the post-observation mental state: ψ → emb . The correspondence between labels (xk , ym ) and post-measurement states is one-to-one.

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17 Contextuality, Complementarity, and Bell Tests

Consider now another observable c which is also compatible with a, mathematically this is expressed as [E ka , E nc ] = 0 for all k, n. The state spaces corresponding to a-outcomes can also be refined w.r.t. c-outcomes, i.e., (a,c) , Hka = ⊕m Hkm

where

(17.52)

(a,c) = E ka E mc H = E mc E ks H. Hkm

In particular, if, for all n, dim Hnc = 1 with the basis vector emb , i.e., E cj enc = δ( j − m)enc , then the outcome (xk , z n ) completely determines the post-observation mental state: ψ → enc .

Chapter 18

Quantum Statistics from Indistinguishability

18.1

Thermodynamics from Gibbs Ideal Ensembles

As well as in our previous works [267, 269], here we follow Schrödinger’s book [405] presenting the abstract scheme of the derivation of the basic thermodynamical quantities from statistical mechanics for ensembles of distinguishable versus indistinguishable systems. Schrödinger presented Gibbs’ approach based on ideal ensembles of identical systems and extended it to quantum systems. In this way, he obtained quantum statistics, both the Bose–Einstein and Fermi–Dirac cases, as well as parastatistics. Gibbs’ approach to thermodynamics differs crucially from the physical approach of Maxwell and Boltzmann. Gibbs’ approach can be used in our mental modeling, in contrast to the Maxwell–Boltzmann approach. To proceed with the latter, one has to be sure that constraints imposed on physical gases can also be assumed for “information gases”, in particular, “social gases”. The validity of these constraints for nonphysical systems is not evident at all. The Maxwell–Boltzmann picture is based on N actually existing physical systems in actual physical interaction with each other, i.e., gas molecules or Planck oscillators in the corresponding quantum generalization. In Gibbs’ picture, N identical systems are mental copies of the one system under consideration. What could it mean to distribute a given amount of energy between these N mental copies? Schrödinger wrote [405], p. 3: The idea is, in my view, that you can, of course, imagine that you really had N copies of your system…Fixing your attention on one of them, you find it in a peculiar kind of “heat bath” which consists of the N − 1 others.

The crucial point is that both pictures, the physical and mental ones, lead to the same results, the same theory. Moreover, the mental picture is even easier adopted to the quantum case (the main argument from which Schrödinger’s analysis was initiated in [405]). By operating not with an ensemble of real systems, but with © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_18

311

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18 Quantum Statistics from Indistinguishability

an ideal ensemble in mental representation, we can apply the methods of statistical thermodynamics not only to physical systems, but to systems of any origin, biological, cognitive, social, political, and financial. The details of the output of the Schrödinger– Gibbs derivation of the basic thermodynamical laws can be found in book [405] which I strongly recommend to the reader.

18.2 Distinguishable Systems: Classical Statistics Consider an ensemble of N identical systems; each of them can be in one of the states ψ1 , . . . , ψk , . . . characterized by the corresponding energy levels E 1 , . . . , E k , . . . Here “energy” is any quantity with properties discussed in the previous section. The state of an ensemble is determined by the indication that • system no. 1 is in a state ψ j1 , • no. 2 in a state ψ j2 , . . . , • system no. N in a state, say ψ jN . Thus, a certain class of states of the ensemble will be indicated by saying that n 1 , . . . , n k , . . . of the N systems are in the states ψ1 , . . . , ψk , . . . It is important to remark that n 1 , . . . , n k , . . . determine not a single state, but a class of states, precisely because the systems in the ensemble are distinguishable. The number of states belonging to this class is equal to P = P(n 1 , . . . , n k , . . . |N , E) =

N! n 1 !n 2 ! . . . n k ! . . .

(18.1)

with the two constraints 



ni = N ,

i

n i E i = E,

(18.2)

i

where N is the number of systems in the ensemble and E is the total energy of this ensemble. Gibbs’ method is based on the following argument. If N is very large, ideally infinitely large, then practically all states will belong to the class of states corresponding to the maximal value of P. By taking into account the constraints (18.2) and using the Lagrange multipliers λ and β for these constraints we obtain 



eλ−β Ei = N ,

i

E i eλ−β Ei = E.

(18.3)

i

We form the partition function, the statistical sum over states: Z=

 i

e−β Ei .

(18.4)

18.3 Indistinguishable Systems: Quantum Statistics

313

Then the first multiplier can be easily found: λ = ln(N /Z ). This is just the normalization factor. The second multiplier plays more important role. It can be connected with average energy per a system in the ensemble: U = E/N =

∂ ln Z 1  . E i e−β Ei = − Z i ∂β

(18.5)

However, in general the functional dependence β = β(U ) is very complex. Therefore, in thermodynamics one assigns the direct meaning to β as inverse to temperature, β = 1/T, and then typically the average energy is expressed as a function of temperature, U = U (T ). Invention of temperature is based on considerations leading to a possibility to operate with thermometer. In physics, such considerations are justified. Proceeding pragmatically we just treat the parameter β as representing the average energy per system in the ensemble of N systems with the energy levels (E 1 , . . . , E k , . . .), see also Sect. 18.3. By using (18.3) we also obtain that the average number of systems per energy level is given by e−β Ek nk = . (18.6) n¯ k = N Z It is convenient to represent it in the form n¯ k = −

1 ∂ ln Z . β ∂ Ek

(18.7)

This set of equations indicates the distribution of the N systems over their energy level Schrödinger remarked [405], p. 8: It may be said to contain, in nutshell, the whole of thermodynamics.

18.3 Indistinguishable Systems: Quantum Statistics Consider a system which is composed of m indistinguishable subsystems. Compound system will be denoted by S and its subsystems by S with indexes. It is assumed that an observer can assign to all systems some quantity called energy and satisfying the

314

18 Quantum Statistics from Indistinguishability

additivity requirement with respect to its distribution over subsystems of a system. We shall denote the energy of S by E and the energy of S by E. These quantities should not be interpreted as objective properties of systems. Energy E = E(S) can be assigned to S as the output of observation performed by an observer on S. The same interpretation is used for energy E = E(S) assigned to S. As always derivation of thermodynamical quantities from ensemble statistics is started with construction of partition function Z . The possible energy levels of a subsystem S are denoted by E 1 , . . . , E j , . . .. An energy level Ek of m-particle compound system S is characterized by a sequence of natural numbers m 1 , . . . , m j , . . . of subsystems on the corresponding levels. The latter means “subsystems with measurement outputs” E 1 , . . . , E j , . . .. Here it is crucial that subsystems with the same value of energy E j are not distinguished one from another. This indistinguishability determines the form of Z and, hence, all thermodynamical quantities. In chapters on social lasing and Fröhlich condensate, we have always emphasized that indistinguishability of systems is w.r.t. to one or a few observables. Deeper inspection can show that in fact the systems under consideration are distinguishable w.r.t. to other parameters. But it is assumed that such supplementary parameters (“distinguishability parameters”) are not involved in the processes under consideration. We have  Es m s . (18.8) Ek = s

Thus, the partition function is given by the sum Z=



e−β

 s

Es m s

,

(18.9)

(m s )

where symbol (m s ) denotes an admissible set of numbers m s . For the moment, β > 0 is just a parameter of the model, a Lagrange multiplier. From ln Z , it is possible to deduce the basic thermodynamical quantities; in particular, the average value of m s m¯ s = −

1 ∂ ln Z . β ∂ Es

(18.10)

Now, different statistics for systems composed of indistinguishable subsystems can be obtained by consideration of different possible ranges of values of natural numbers m s : 1. m s = 0, 1, 2, . . . . (Bose–Einstein statistics), 2. m s = 0, . . . , q, where q ≥ 1 is a natural number (parastatistics). In physics, q = 1, i.e., m s = 0, 1; this is the case of the Fermi–Dirac statistics. We remark that in quantum physics selection of the Fermi–Dirac statistics from a bunch of parastistics is just postulated. Of course, it is confirmed by the experimental

18.4 Classical and Quantum statistics

315

situation. One cannot exclude that in social science parastistics with q > 1 can find applications. By restricting considerations to the Bose–Einstein and Fermi–Dirac statistics and following Schrödinger [405], we find that the corresponding partition functions can be expressed as Z = Z BE =

 s

 1 , Z = Z FD = (1 + e−β Es ). −β E s 1−e s

(18.11)

This leads to the following basic expression for the average value of m s m¯ s =

1 1 β Es e ξ

(18.12)

∓1

where 0 < ξ ≤ 1 is so called parameter of degeneration, ξ = ξ(m). We remark that, for photons, ξ = 1, with the Bose–Einstein statistics. Then one gets the average energy as U=

 s

αs 1 β Es e ∓ ξ

1

.

(18.13)

18.4 Classical and Quantum statistics The mathematical laws describing the average number of systems are presented on Fig. 18.1. These are very special mathematical laws. By considering gases composed of physical systems, by Boltzmann–Maxwell approach, one may think that these laws are specified by physics: the Maxwell–Boltzmann law for classical gases and the Bose–Einstein and Fermi–Dirac statistics for quantum gases. However, our derivation (in Gibbs’ framework) of formulas (18.6) and (18.12) has nothing to do with physical laws. It is based on consideration of distinguishable versus indistinguishable systems of any origin. Such a combinatorial derivation makes the origin of the forms of the graphs presented at Fig. 18.1 really mystical. Why is statistical behavior restricted only to such forms? Can one find some statistical behavior with a graph “in-between” the graphs corresponding to the Maxwell–Boltzmann, Bose–Einstein, and Fermi– Dirac statistics?1 What are additional constraints, different from distinguishability, which can induce new natural forms of statistical dependence on energy?

1

We remark that consideration of the parastatistics does not bring us really new mathematical laws. The parastatistics graphs are simple modifications of the graph for the Fermi–Dirac statistics.

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18 Quantum Statistics from Indistinguishability

Fig. 18.1 Functional dependence for Bose–Einstein, Maxwell–Boltzmann, and Fermi–Dirac statistics

In physics it is proven, both theoretically and experimentally, that only the Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics are realizable in nature. However, for social systems, other forms of statistical dependence f () = n() may appear. Of course, we are still mainly theorizing, because experimental verification of nonclassical statistics in social data, (say even the Bose–Einstein statistics for social energy in information flows) has not yet even started.

Part VII

Supplement on Decision-Making

This part contains three chapters supplementary to the previous chapters on decisionmaking. The first chapter is devoted to classical expected utility theory with contextual structuring of its paradoxes. The second chapter is about a few interpretations of quantum state and probability. If a reader is an atheist, then the third chapter can be either ignored or God can be treated as a kind of universal observer.

Chapter 19

Classical Expected Utility Theory and Its Paradoxes

19.1 Von Neumann and Morgenstern: Expected Utility Theory There are two lots, say A = (xi , Pi ) and B = (yi , Q i ), where (xi ) and (yi ) are outcomes and (Pi ) and (Q i ) are probabilities of these outcomes. Which lot do you select? An agent, say Alice, can simulate the experience that she draws the lot A (or B) and gets the outcome xi (or yi ). Let us represent such an event by (A, xi ) (or (B, yi )). As usual, Alice assigns the utilities u(xi ) and u(yi ) of (A, xi ) and (B, yi ), respectively. Here, u(x) is a utility function of outcome x.1 By using the utility function, the agent evaluates various comparisons for making the preference: A  B or B  A. The first mathematically consistent theory of decision-making was expected utility theory (von Neumann and Morgenstern [450]) based on the axioms: Completeness, Transitivity, Independence, Continuity. The axioms are given for the relation of utilities like u  v and the operation using probability like pu + (1 − p)v. = u(xi )Pi and This motivates an agent to operate with the expected utilities, E A  E B = u(yi )Q i , and to use their difference as the criterion for making the preference.

1

In the classical model the utility of an event depends on only its outcome. However, in the quantumlike model utility has the complex Hilbert space representation encoding all circumstances of realization of outcomes. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_19

319

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19 Classical Expected Utility Theory and Its Paradoxes

19.2 Allais Paradox We consider one of the versions of the Allais paradox [7]. Here we follow [467] with one modification, namely, using the word “context”, instead of the word “situation”. From the first view, this modification is innocent, but it gives the possibility to include this paradox in the general contextual structure of the foundational experiments of quantum physics [251] (Chap. 17). Let us consider two gambling contexts each is based on two lotteries. Let there be outcomes consisting of only losing money. Context C1 : • Lottery A1 : loss of $500 with probability 100%; • Lottery B1 : equal probabilities of losing $1000 or $0. Context C2 : • Lottery A2 : 10% chance of losing $500 and a 90% chance of losing $0; • Lottery B2 : 5% chance of losing $1000 and a 95% chance of losing $0. We calculate the expected utilities for all lotteries. Context C1 : • Lottery A1 : E A1 = u(−$500). • Lottery B1 : E B1 = 0.5u(−$1000) + 0.5u($0). Context C2 : • Lottery A2 : E A2 = 0.1u(−$500) + 0.9u($0). • Lottery B2 : E B2 = 0.05u(−$1000) + 0.95u($0). Often people prefer lottery A1 to lottery B1 in context C1 and lottery B2 to lottery A2 in context C2 .2 Hence, for these people, • Context C1 : E A1 = u(−$500) > E B1 = 0.5u(−$1000) + 0.5u($0). • Context C2 : E A2 = 0.1u(−$500) + 0.9u($0) < E B2 = 0.05u(−$1000) + 0.95u($0). After such calculations, one typically makes the following statement. “Two contexts have the same structure, which causes a paradox:” 0.1u(−$500) < 0.05u(−$1000) + 0.05u($0)

(19.1)

In literature on decision-making, this situation is typically treated as resulting from ambiguity aversion (uncertainty aversion): roughly speaking, people are afraid of uncertainty. However, I prefer the contextual explanation of the origin of this paradox. There are considered two different gambling contexts, C1 and C2 . It is natural to assume that they generate two different utility functions u C1 and u C2 . By operating with different functions one can proceed without paradoxes. However, operation with 2

Numerous experimental studies were performed in cognitive psychology and economics. In this book, we do not discuss the concrete experiments.

19.3 Savage: Subjective Expected Utility Theory

321

a context-dependent utility function is more natural for subjective expected utility theory which will be considered in the next section. Contextuality plays the crucial role in a few models described in this book (see, e.g., Chaps. 6, 7, 17). Contextuality matches well the foundations and formalism of quantum theory (Chap. 17). The quantum-like models of expected utility developed in articles [248, 282] can be treated as reflecting contextuality of the utility function and probability. Further paragraph can be difficult for readers unfamiliar with contextuality in quantum physics. It might be useful to start with reading Chap. 17. The “trick” leading to the paradox is the same as in quantum physics in theoretical and experimental studies of the violation of the Bell inequality [251]. What is wrong from the experimental viewpoint? As in the Bell experiments, statistical data collected in different experiments is put in the same probabilistic model, i.e., the same Kolmogorov probability space [318] P = (, F , P), where  is the space of elementary events and P is a probability measure on σ -algebra of events. In applications of this model (CP), observers are represented by random variables on P. However, both in physics and decision-making the multi-contextual experimental situation doesn’t guarantee that such single Kolmogorov probability space description is possible. Following the Växjö model [251] (Sect. 17.2), one should operate with a family of context-dependent probability spaces, PC = (C , FC , PC ). Observables are given by random variables on contextually-labeled probability spaces, measurable functions, aC : C → R. The same semantically defined observable a is represented by a family of random variables (aC ) the family of contexts for which the a-observable can be measured. In the Allais paradox, the role of observable is played by the utility function u. As was noted, it has to be context-labeled. Similar contextual description can be provided in the framework of Contextuality by Default (CBD) theory developed by Dzhafarov et al. (see, e.g., [131] and Sect. 17.8).

19.3 Savage: Subjective Expected Utility Theory As we have seen, the expected utility theory is not free of paradoxes, the most widely known is the Allais paradox [7]. This problem is fundamentally coupled to the interpretation of probability used in expected utility theory. The latter is based on the frequency (statistical) interpretation of probability. Therefore it is natural to test models of decision-making based on other interpretations. The most powerful alternative to the frequency probability is the subjective probability. The subjective probability is the measure of belief about whether a specific outcome is likely to occur. The subjective modification of expected utility theory was performed by Savage [400] who suggested to consider both utility and probability as subjectively determined by decision makers. However, as we see in the next section even the subjective expected utility theory is not free of paradoxes.

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19.4 Ellsberg Paradox The Ellsberg paradox [144] is a challenge for subjective expected utility theory [400]. In the simple decision-making framework—selection of a lottery—people behave in contrary with subjective expected utility theory.

Two Urns Paradox Let us consider the two urns version of this paradox. Take two urns, call them urn1 and urn2, containing the equal number of balls, say 100 balls. Decision makers are informed that urn1 contains 50 red and 50 black balls, but urn2 contains mixture of red and black balls in unknown proportion. Hence, for urn1, P(red) = P(black) = 1/2, but for urn2 these probabilities are unknown. In decision-making with subjective probabilities the latter two probabilities are assigned by decision maker subjectively. Any participant of the selection experiment is offered the following bets • Lottery A1 : get $1 if a red ball is drawn from urn urn1, $0 otherwise. • Lottery A2 : get $1 if a black ball is drawn from urn urn1, $0 otherwise. • Lottery B1 : get $1 if a red ball is drawn from urn urn2, $0 otherwise. • Lottery B2 : get $1 a if black ball is drawn from urn urn2, $0 otherwise. In such experiments typically people are indifferent between lotteries A1 and A2 and this is consistent with expected utility theory. But they strictly prefer A1 to B1 and A2 to B2 and this contradicts to subjective expected utility theory. This situation can be treated as resulting from ambiguity aversion (uncertainty aversion): people are afraid of uncertainty. Now let us perform contextual structuring of this gambling. Consider three gambling contexts. Context C1 : • Lottery A1 : get $1 if a red ball is drawn from urn urn1, $0 otherwise. • Lottery A2 : get $1 if a black ball is drawn from urn urn1, $0 otherwise. Context C2 : • Lottery A1 : get $1 if a red ball is drawn from urn urn1, $0 otherwise. • Lottery B1 : get $1 if a red ball is drawn from urn urn2, $0 otherwise. Context C3 : • Lottery A2 : get $1 if a black ball is drawn from urn urn1, $0 otherwise. • Lottery B2 : get $1 if a black ball is drawn from urn urn2, $0 otherwise. Alice should make the decisions and compare the lotteries in three different contexts and if she operates with contextual utility functions her decisions would not lead to a paradox. Moreover, subjective probabilities P(red from urn2|C2 ) and

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P(red from urn2|C3 ) = 1 − P(black from urn2|C3 ) do not need coincide. So, this is contextuality of probability (cf. [251]). However, subjective utility theory is not contextual in this sense. The quantum-like models of subjective expected utility [248, 282] reflect contextuality. We remark that quantum contextuality becomes visible only for entangled belief states.

One Urn Paradox Let us now consider the one-urn paradox. The urn contains 90 balls: 30 balls are red and 60 balls are either black or yellow in unknown proportions. Hence, P(red) = 1/3, but P(black) and P(yellow) are unknown. In decision-making with subjective probabilities, the latter two probabilities are assigned by decision maker subjectively. The participants then should select one of the following gambling scenarios: • Lottery A: get $100 if a red ball is drawn. • Lottery B: get $100 if black ball is drawn. Additionally, the participant may choose a separate gamble scenario within the same situational parameters: • Lottery C: get $100 if a red or yellow ball is drawn. • Lottery D: get $100 if a black or yellow ball is drawn. By subjective expected utility theory the choice between gambles is based on comparison of expected utilities between them. The latter depends on subjective assignments of P(black) and P(yellow). Alice does not need to assign exact values to them. It is sufficient to estimate them comparing with probability P(red). It is important to note that expected utility is calculated individually for each game. In this example the utility is the same in all lotteries, u($100), hence expected utility is determined by probabilities—their subjective determination. Alice strictly prefers lottery A to lottery B if and only if she assumes that P(red) > P(black). The latter implies that P(yellow) > P(black) and since P(black) + P(yellow) = 2/3, then P(yellow) > 1/3, Hence, P(red or yellow) > 2/3 = P(black or yellow), i.e., Alice should prefer lottery C to lottery D. However, ambiguity aversion would predict that people would strictly prefer lottery A to lottery B, and lottery D to lottery C. Numerous experimental studies confirmed this prediction. Once again I do not support the hypothesis that people suffer of ambiguity aversion. We can repeat contextuality story presented for previous paradoxes. Quantum-like models of subjective expected utility were developed in articles [27, 282]. In this book, we don’t want to go into technical details. We just point to the key point of the quantum-like models: to go beyond the classical subjective utility theory, Alice’s belief state ψ has to be entangled w.r.t. to lotteries’ choices. If ψ is separable, the quantum-like model reproduces the classical situation.

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19.5 Quantum-Like Modeling of Subjective Expected Utility Consider the space of belief states of an agent (Chap. 2). In accordance with the quantum-like modeling of cognition belief states are represented by normalized vectors of a complex Hilbert space H. These are so-called pure states. More generally, belief states are represented by density operators encoding classical probabilistic mixtures of pure states. Lotteries A and B are mathematically realized as two orthonormal bases in H : (|i a ) and (| jb ). Any vector |i a  represents the event (A, xi )—“selecting of the A-lottery which generates the outcome xi .” The same can be said about vectors of the B-basis. These events are not real, but imaginable. Alice plays with potential outcomes of the lotteries and compares them. We can also represent lotteries by Hermitian operators, the lotteries operators: A=

 i

xi |i a i a |, B =



y j | jb  jb |.

(19.2)

j

As in the classical theory, each outcome xi has some utility u i = u(xi ) (say amount of money). Thus our model is based on a mapping from eigenstates of the “lotteryoperators” to utilities (amounts of money), |i a  → u(xi ), | jb  → u(y j ). Thus, starting with two lotteries A and B with outcomes (xi ) and (y j ) with corresponding utilities u i = u(xi ) and u j = u(y j ), Alice represents these utilities (coupled to their lotteries) by two orthonormal bases in the belief-state space H : u i → |i a , u j → | jb .

(19.3)

We emphasize coupling of utilities to lotteries. Utility (derived from some monetary amount) has not only the value, but also so to say the “color” determined by circumstances surrounding the corresponding lottery—lottery’s context. Therefore even the same outcome z = xi = y j of two lotteries (having the same utility value u = u(z)) may be represented by two different vectors: |i a  = | jb . Moreover, outcomes inside each lottery are also coupled through selection of the fixed orthonormal basis. Finally, we remark that mapping (19.3) encodes the correlations between outcomes of lotteries (and their utilities). These correlations are mathematically expressed through quantum transition probabilities. The lotteries operators can be noncommuting, i.e., [A, B] = 0. In quantum theory noncommuting Hermitian operators represent complementary (or incompatible) observables: they cannot be measured jointly. In our quantum-like model of lottery selection, we can speak about complementary lotteries which are represented by noncommuting operators. In the DM-process for complementary lotteries, Alice does

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not create the joint image of outcomes of both of them. In mathematical terms the latter means the impossibility to determine the joint probability distribution for the pairs of outcomes (xi , y j ). Thus, instead of weighting probabilistically the pairs of outcomes, Alice analyzes the possibility of realization of an outcome say xi of the A-lottery and she accounts for its utility u(xi ). Then under the assumption of such realization she imagines possible realizations (y j ) of the B-lottery and compares the utilities u(y j ) and u(xi ). Suppose I have selected the A-lottery and its outcome xi was realized. What would be my earning (lost) if (instead) I were selected the B-lottery and its outcome y j were realized?

This kind of counterfactual reflections is mathematically described by the Hilbert space formalism and transition from the A-basis to the B-basis. Outputs of these comparisons are weighted through accounting Hilbert space coordinates. This accounting is described by the special comparison operator D. Since Alice cannot handle both lotteries simultaneously, she starts with imaging one of them say A, as in the above consideration. Then she performs similar counterfactual reasoning starting with the B-lottery. The comparison operator D has two counterparts representing the processes A → B and B → A comparisons. In the operator terms, transitions from one basis to another are represented by transition operators E ia → jb , E jb →ia . And the comparison operator D is compounded of these operators. See [282] for further development of theory.

Chapter 20

Belief State Interpretation

20.1 The Spirit of Copenhagen As was already pointed out, the problem of the quantum state (wave function) interpretation and more generally the interpretation of the mathematical formalism of quantum theory is one of the main foundational problems. The debates on it are characterized by controversy. As the organizer of the Växjö series of quantum foundations conferences (2000–2022), I can’t see any sign of opinion convergence to a kind of the universal interpretation (see, e.g., proceedings [3, 111, 235, 237, 257, 258] and my monographs [250, 271]). Typically an expert would say that he proceeds with the Copenhagen interpretation. But my experience based on hundreds of discussions with world’s leading experts show that practically everybody has his own picture of Copenhagen. Arkady Plotnitsky described this situation at the second Växjö conference in 2001 (see [378]); in his recent book [385] he wrote: - Bohr’s interpretation, in any of its versions, will be distinguished in this study from “the Copenhagen interpretation,” because there is no single such interpretation, as even Bohr has changed his a few times. It is more suitable to speak, as Heisenberg did [197], of “the Copenhagen spirit of quantum theory” or, as a handier shorthand, “the spirit of Copenhagen,” referring to certain common features of a group of interpretations, which may be different in their other features.

As well as formulas in CP [231, 250], the probability in the Born formula (2.5) can be interpreted either statistically or subjectively. However, in QP the interpretation problem is more complicated, because QP operates not solely with probabilities, but also with quantum states. Hence, not only probabilities, but also states have to be interpreted and the state-probability interpretation should be consistent.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_20

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20.2 Interpretations: Statistical Versus Individual By the statistical interpretation of probability, for a large ensemble of decision makers (N persons), the frequency of observation of the output a = α approaches (for N → ∞) the probability calculated theoretically with the Born rule: ν N (α) = n N (α)/N ≈ E(α)ψ2 ,

(20.1)

where n N (α) is the number of decision makers who answered a = α. This interpretation is the most useful to couple theory and experiment (see [231, 250] for the frequency interpretation of probability). In cognitive, psychological, social, and financial experiments statistical data is collected for an ensemble of humans (or animals). In quantum physics, this interpretation of probability is coupled to the statistical interpretation of the wave function by which it represents not the state of an individual quantum system, but of an ensemble of systems prepared by the same preparation procedure. This interpretation is typically associated with Einstein, latter it was highlighted by Ballentine [43]. In decision theory, by the statistical interpretation the belief state is associated with a large ensemble of decision makers which is selected under special conditions—a decision-making preparation procedure. Bohr and von Neumann assigned the wave function to an individual quantum system. However, for probability they used the statistical interpretation. This interplay of individual (for state) and statistical (for probability associated with this state by the Born rule) interpretations made the statistical structure of QM really mystical. This mystery would be resolved only via the solution of the quantum measurement problem: creation of a mathematical model for generation of the concrete outputs of the observable A on the basis of the initial state ψ. This is the complicated problem and maybe it could not be solved at all within quantum theory. Nevertheless, the individual interpretation of quantum state combined with the statistical interpretation probability is widely used in QM. It is reasonable to use this framework even in cognition, decision-making, and other applications outside of physics. By this interpretation the belief (mental) state ψ is assigned to the individual decision maker, but it determines the statistical probability for an ensemble of decision makers who have the same state ψ. Why does ψ generates the statistical probability which is experimentally verifiable (see (20.1)) with the frequency of observations ν N (α j )? This question would be replied if within quantum-like modeling one would solve the measurement problem: How does brain generate the concrete output a = x j starting with the mental state ψ?

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20.3 QBism: Subjective Interpretation By the subjective interpretation of probability Alice (whose belief state is ψ) assigns her own weight P(a = α) to the outcome a = α and in the quantum-like model of decision-making this subjective probability is given by the Born rule, i.e., P(a = α) = P(a = α|ψ). In general decision theory (i.e., not directly related to quantum experiments), the subjective interpretation of probability is widely applied to a variety of problems, including the framework of the subjective utility function. In the quantum-like reformulation, the Born rule gives subjective probabilities which decision makers assign to possible outcomes (see [27] for the quantum-like subjective utility model). In quantum physics the subjective interpretation of probability is associated with Quantum Baeysianism (QBism), see, e.g., [163–168] (see also [109, 279, 280] for discussions on QBism). We start with the following citation of Fuchs and Schack [166], pp. 3–4: The fundamental primitive of QBism is the concept of experience. According to QBism, quantum mechanics is a theory that any agent can use to evaluate her expectations for the content of her personal experience. QBism adopts the personalist Bayesian probability theory ... . This means that QBism interprets all probabilities, in particular those that occur in quantum mechanics, as an agent’s personal, subjective degrees of belief. This includes the case of certainty - even probabilities 0 or 1 are degrees of belief. ... In QBism, a measurement is an action an agent takes to elicit an experience. The measurement outcome is the experience so elicited. The measurement outcome is thus personal to the agent who takes the measurement action. In this sense, quantum mechanics, like probability theory, is a single user theory. A measurement does not reveal a pre-existing value. Rather, the measurement outcome is created in the measurement action. According to QBism, quantum mechanics can be applied to any physical system. QBism treats all physical systems in the same way, including atoms, beam splitters, Stern–Gerlach magnets, preparation devices, measurement apparatuses, all the way to living beings and other agents. In this, QBism differs crucially from various versions of the Copenhagen interpretation. ... An agent’s beliefs and experiences are necessarily local to that agent. This implies that the question of nonlocality simply does not arise in QBism.

What is about the QBism-interpretation of the belief state? This is the theory about structuring the individual’s experiences in the situation of uncertainty. The belief state ψ is individual’s state. Hence, ψ is not the state of say an electron involved in the process of measurement, but of an experimenter who creates the subjective probabilistic picture for possible outcomes of this experiment. In discussions on QBism, it is typically missed that it diminishes the role of the quantum state. QBists suggest to operate solely with probabilities. To realize this approach, they consider not standard quantum observables which are mathematically represented by Hermitian operators, but generalized observables which completely determine a quantum state—informationally complete POVMs. Such observables

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have n 2 outcomes, where n is the state space dimension. Another property is that the outcomes are not mutually exclusive, i.e., projections corresponding to different values are not mutually orthogonal. Any quantum state, pure or mixed, can be represented as the vector with n 2 coordinates given by probabilities for observation outcomes (for this state). Hence, QBists are able to eliminate the complex Hilbert state space from consideration and represent the quantum formalism as the purely probabilistic machinery that is used for probability update—quantum Bayesian inference. Of course, the price is very high: • model’s dimension is squared, • the basic observables are not von Neumann projection type observables, but generalized observables—symmetric informationally complete POVMs,1 • FTP derived within QBism is totally different from FTP of CP. My intention is that in quantum-like modeling one can use the QBism methodology, the personal experience viewpoint on outcomes of observations and on probability. However, the notion of the personal belief state has to be used as the seed for generation of subjective probabilities. The QBism’s program to reformulate the quantum formalism solely in terms of probability played the important role in quantum foundations, in particular, for demystification of quantum mechanics. However, in real applications the conventional quantum formalism with the Hilbert state space is more convenient, both in physics and decision-making. We continue the discussion on QBism in Sects. 21.2 and 21.3.

20.4 Växjö Interpretation By the Växjö interpretation [236, 245] the quantum formalism is the machinery for the probability update. The essence of this interpretation is representation of the QP-update similarly to the classical update. Originally in the quantum formalism the probability update is a derivative of the quantum state update. Reference to a complex vector to express the update of the real probability mystifies this procedure. By the Växjö interpretation, one can reformulate the QP-update scheme in purely probabilistic terms. The key element of this CP-like representation is the quantum FTP (with the interference term). In this sense, i.e., in attempting to demystify the QP-update by representing it in CP-like way, the Växjö interpretation is close to QBism. But the corresponding quantum versions of classical FTP are totally different. The FTP with the interference term (1.11) used in the Växjö interpretation is just an additive perturbation of CP-FTP. The interference term δ can be considered as a parameter describing correspondence between QP and CP; when δ → 0, the

1

Even in physics, to perform measurements of observables described by these POVMs is a difficult task. In biological and cognitive applications, nobody even tried to do measurements of such type.

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quantum FTP is smoothly transformed into the classical FTP. QBism’s analog of FTP cannot be related to the classical FTP in this way. QBism’s FTP is a kind of singulare perturbation of the classical FTP. Similarly to QBism the Växjö interpretation demystifies quantum theory, by portraying it as the special machinery for the probability update. In contrast to QBism, in physics the Växjö interpretation treats probabilities statistically. In quantum-like modeling they can be treated subjectively as well.

Chapter 21

God as Decision Maker and Quantum Bayesianism

21.1 Bohr Versus Bell The recent quantum information revolution (known as the second quantum revolution) had strong impact to foundations of quantum mechanics and natural science in general. It featured the role of information in physics; physical laws can be treated as describing consistent transformation of information corresponding to different experimental contexts. Thus the role of physical carriers of information was substantially diminished and the reductionist viewpoint to physics was faded. This information interpretation of quantum mechanics matches well with the views of Niels Bohr who repeatably pointed out that quantum mechanics is not about micro-reality as it is, but about measurements procedures giving the possibility to extract knowledge concerning these systems with the aid of measurement devices. Moreover, in his complementarity principle he stressed that measurements outputs are not objective properties of microsystems, but generated in the process complex interaction between an apparatus and a system. These Bohr’s views are known as the Copenhagen interpretation of quantum mechanics. One can consider this interpretation as highlighting the role of the observer, the human component of quantum theory. In particular, Wigner proceeded in this direction, but his views are not so widely accepted. In particular, the interpretations of the foundational experiments on quantum nonlocality, via demonstration of violation of the Bell inequality, highly depend on the problem of human’s free will. Denying the existence of the latter leads to the interpretation known as superdeteminism: everything is predetermined from the moment of the Big Bang, creation of the universe. Keeping to this interpretation is the easiest way to resolve the problem of nonlocality. Superdeterminism leads to the local viewpoint on quantum mechanics. By saying that humans have the free will, one confronts a number of complicated foundational and philosophic problems. Declaring that quantum mechanics is nonlocal is the simplest way to solve them. However, there is another way to interpret consistently the Bell-type experiments within the free will paradigm. Scientists who accept that observer has free will (at least to choose experimental settings) and suggest the probabilistic resolution of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4_21

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problem of nonlocality are known as the probability opposition to the nonlocal interpretation of quantum phenomena (see [288, 292] for the corresponding references). Careful treatment of probabilities involved in the Bell experiments gives the possibility to proceed without referring to mystical quantum nonlocality and spooky action at a distance [286]. The main issue is contextual dependence of probabilities and existence of incompatible observables corresponding to complementary experimental contexts—complexes of physical conditions (Chap. 17).

21.2 Supplement on Quantum Bayesianism (QBism) In the contextual playing with the quantum foundations, it can be relevant to go further and include even an observer into the experimental context. This approach lifts the role of human being and his mind in quantum theory. And this is the ideology of Quantum Bayesianism (QBism) displaying the role of the personal experience of an observer. Since such experiences are local, QBism easily resolves the problem of quantum nonlocality and this is one of its strong sides. QBism essentially demystifies quantum mechanics. On the other hand, for it is not so easy, if possible at all, for typical physicist, to recognize the need to involve the subjective element in the quantum foundations. I, personally, can be compared with two faced Janus: • In physics I stay on the position of aforementioned physicists. It is difficult to accept the need of the personal experiences of observers to make the quantum foundations consistent. QBists kill the monster of quantum nonlocality, but at the same time welcome human as the basic component of quantum theory. Even the treatment of quantum probabilities as subjective ones is in controversy with my intuition. For me these probabilities are statistical. Often Bayesianism—the probability update machinery including Bayesian inference—is rigidly coupled with the subjective interpretation of probability. However, this is not the case. In CP, the same mathematical formalism (Kolmogorov 1933 [318]) can be applied for probabilities interpreted both statistically and subjectively. Kolmogorov explored the statistical interpretation. In QP, genuine quantum Bayesianism is the machinery for the quantum state update and the latter generates the QP update. As well as in CP, this machinery can be endowed both with the statistical and subjective interpretations. I am in favor of the statistical one and according to my experience and intuition measurement is done by a physical apparatus and human’s involvement cannot play the crucial role in theory about quantum measurements, • In cognition and decision making I completely share the views of QBists— association of outcomes of observations with the personal experiences of observers. However, observations are here self-observations, as was described in Chaps. 6, 7. In Bohr’s manner, they can be represented as observations performed by consciousness over unconsciousness. Personal experiences of decision maker are conscious experiences generated via interaction of unconscious and conscious mental

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states. Probabilities are subjective probabilities of decision maker, typically they are assigned unconsciously. Moreover, these subjective probabilities have the statistical nature and generated by neural networks in the brain, as it was described in Chap. 6. Now we turn to QBism foundations. The essence of QBism is presented by one of its founders Christopher A. Fuchs [169] in the following words: QBism ...takes the idea of “the instruments of observation as a . . . prolongation of the sense organs of the observer’ deadly seriously and runs it to its logical conclusion. This is why QBists opt to say that the outcome of a quantum measurement is a personal experience for the agent gambling upon it. Whereas Bohr always had his classically describable measuring devices mediating between the registration of a measurement’s outcome and the individual agent’s experience, for QBism the outcome just is the experience.

Quantum theory is a statistical theory and it predicts probabilities of experimental outcomes. The aforementioned lifting of the role of observer is the basis of the subjective QP interpretation in QBism [169, 170]. A quantum state in QBism is just a state of belief of an observer on possible outcomes of experiment. Of course, an observer should be qualified, i.e., knowing the quantum theory. The authors of article [170] say mean: “but there is no sense in which the quantum state itself represents (, like pictures, copies, corresponds to, correlates with a part or a whole of the external world, much less a world that just is.” For this position. QBism is often criticized that the personal agent perspective leads to solipsism. This accusation in solipsism is analyzed and rejected by QBists (see article [169]). Generally QBism is strongly opposed to representationism, “attempts to directly represent—(map, picture, copy, correspond to, correlate with)—he universe—with “universe” here thought of in totality as a pre-existing, static system; an unchanging, monistic something that just is.” [170]. Philosophically, QBIsm is coupled to nonreductionism, empiricism, and meliorism. The latter coupling is especially important for this chapter and we consider it in more detail in Sect. 21.4. Following de Finetti [120, 121] QBists pointed out that consistent appealing to the subjective interpretation of probability should lead to reconsideration of the objective treatment of the scientific methodology. In fact, de Finetti was even more revolutionary than QBists, because his subjective treatment of scientific method was not restricted to physics.

21.3 QBism Versus Copenhagen Interpretation Although Bohr’s views and the Copenhagen interpretation stimulated development of QBism. QBists prefer not to be identified as just a special stream within the Copenhagen interpretation of quantum mechanics. As Fuchs wrote [169], “Without Niels Bohr, QBism would be nothing. But QBism is not Bohr. This paper attempts

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to show that, despite a popular misconception, QBism is no minor tweak to Bohr’s interpretation of quantum mechanics. It is something quite distinct.” D. Mermin claimed that QBism differs crucially from all interpretations in the spirit of Copenhagen, see [352], pp. 7–8: A fundamental difference between QBism and any flavor of Copenhagen, is that QBism explicitly introduces each user of quantum mechanics into the story, together with the world external to that user. Since every user is different, dividing the world differently into external and internal, every application of quantum mechanics to the world must ultimately refer, if only implicitly, to a particular user. But every version of Copenhagen takes a view of the world that makes no reference to the particular user who is trying to make sense of that world. Fuchs and Schack prefer the term “agent” to “user”. “Agent” serves to emphasize that the user takes actions on her world and experiences the consequences of her actions. I prefer the term “user” to emphasize Fuchs’ and Schack’s equally important point that science is a user’s manual. Its purpose is to help each of us make sense of our private experience induced in us by the world outside of us. It is crucial to note from the beginning that “user” does not mean a generic body of users. It means a particular individual person, who is making use of science to bring coherence to her own private perceptions. I can be a “user”. You can be a “user”. But we are not jointly a user, because my internal personal experience is inaccessible to you except insofar as I attempt to represent it to you verbally, and vice-versa. Science is about the interface between the experience of any particular person and the subset of the world that is external to that particular user. This is unlike anything in any version of Copenhagen. It is central to the QBist understanding of science.

21.4 Free Will Given by God to Adam is the Source of Irreducible Uncertainty in the Universe For QBists [170], “the world before the agent is malleable to some extent—that his actions really can change it.” Fuchs illustrates this position by the following allusion to Bible. Adam said to God, “I want the ability to write messages onto the world.” God replied, “You ask much of me. If you want to write upon the world, it cannot be so rigid a thing as I had originally intended. The world would have to have some malleability, with enough looseness for you to write upon its properties. It will make your world more unpredictable than it would have been – I may not be able to warn you about impending dangers like droughts and hurricanes as effectively as I could have – but I can make it such if you want.

So, Adam got the possibility to influence actively to the world, but the price was very high uncertainty. Since this chapter is based on paper [297], a part of the special volume devoted to the scientific enlightenment of the religious questions, it is interesting to discuss the probabilistic meaning of God’s reply to Adam. This reply clearly shows that free will of Adam restricts God’s ability to control events happening in the world. Thus human’s free will is the source of the irreducible uncertainty in the universe. It is natural to couple this free will generated uncertainty with irreducible quantum uncertainty and acausality which were first considered by von Neumann [451].

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This picture contradicts to the interpretation of God’s power as absolute, as the ability to control deterministically the stream of all events in the universe. So, if God did not give up and he is still trying to preserve order, his control machinery has to be probabilistic. In our model, God’s control is statistical, not individual. From this viewpoint, the religious would-picture is not deterministic at all. Statements that everything in human’s life is under divine providence do not match the above picture of the uncertain universe. In particular, catastrophic events in human’s life, say wars, are not punishments for sins. They are generated by humans’ free wills. In the uncertain universe paradigm, each person has big responsibility for his mental and physical actions.

21.5 God as Decision Maker Operating with Subjective Probability Assigned to His Personal Experiences As we see from the previous chapters of the book, quantum-like modeling was very successful in decision-making. The subjective interpretation of probability is very common in classical decision-making, e.g., as the basis of the subjective utility theory. In this book we focus to just briefly discussed the QP interpretation (Chap. 2) and, in particular, mentioned QBism and the subjective interpretation. This is the good place to mention now that probabilities considered in Chaps. 9, 10 can be interpreted as subjective ones and the answers to the questions as the personal experiences of humans. Thus, the mathematical formalism developed in these two chapters matches with QBism very well. Now consider God as decision maker who processes and updates the state of psycho-physical universe to preserve the order in the world, but only statistically. God also has his belief state and assigns subjective probability for possible outcomes—his possible personal experiences. God’s behavior can be described by the mathematical formalism presented in this book charged with the QBism interpretation. This interesting theme is beyond book’s topic, may be I shall continue quantum-like theological modeling in future works. Finally, we point out to the most intriguing issue of the quantum-like model of God’s order-control in the universe. Quantum theory is characterized by the presence of incompatible observables. In particular, they describe incompatible free wills. Such free wills cannot be embedded in a consistent mental picture. The following natural question arises: How can God manage such mutually complementary wills?

Appendix A

Technicalities

All‘s well that ends well

A.1

Proof of Theorem 4.7.1, Chap. 4

As was promised in Chap. 4, here we give the proof of Theorem 1. Proof We recall that in this paper we proceed with finite dimensional state spaces. Consider the space of dimension d. By the monotonicity of quantum Kullbaceibler information (relative entropy), S(T ρ|T ρ  ) ≤ S(ρ|ρ  ), where S(ρ|ρ  ) = Tr[ρlogρ] − Tr[ρ log ρ  ]. Then, S(ρ|I /d) = Tr[ρ log ρ] − Tr[ρ(− log d)] = log d − (−Tr[ρ log ρ]) = log d − S(ρ). If T is unital, i.e., T I = I, we have S(T ρ) − S(ρ) = [log d − S(T ρ||I /d)] − [log d − S(ρ||I /d)] = S(ρ|I /d) − S(T ρ|T (I /d)) ≥ 0. Thus, we conclude that S(T ρ) ≥ S(ρ),

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4

339

340

Appendix A: Technicalities

if T is unital. Conversely, if

S(T ρ) ≥ S(ρ)

for all states ρ, we have S(T (I /d)) ≥ S(I /d) = log d so that T (I /d) = I /d. Thus, we have proven that S(T ρ) ≥ S(ρ) for all states ρ if and only if T is unital.

A.2

Construction of Quantum Channels

Here we construct quantum channels which were discussed in Chap. 5. The constructions of the desired quantum channel for subsystem’s state spaces of dimensions N = 2 and N > 2 are different. In the latter case, the expressions for the von Neumann entropies of the subsystems Si , i = 1, 2, contain the factor log(N − 2). Therefore, we consider these cases separately.

A.2.1

Two Subsystems with Qubit State Spaces

      Let H1 and H2 be C2 and x0(i) , x1(i) be orthonormal bases in Hi (i = 1, 2). We define a completely positive channel  from D (H1 ⊗ H2 ) to D (H1 ⊗ H2 ) by  (•) ≡ V (•) V, where V is a linear map from H1 ⊗ H2 to H1 ⊗ H2 given by            1  (1)  (2)  (1)  (2)   V ≡ x0(1)  ⊗ x0(2)  x0 ⊗ x0 + x1 ⊗ x1 2            1  (1)  (2)  (1)  (2)   + x1(1)  ⊗ x1(2)  x0 ⊗ x0 − x1 ⊗ x1 2            1  (1)  (2)  (1)  (2)   + x0(1)  ⊗ x1(2)  x0 ⊗ x1 + x1 ⊗ x0 2            1  (1)  (2)  (1)  (2)   + x1(1)  ⊗ x0(2)  . x0 ⊗ x1 − x1 ⊗ x0 2 We remark that the operator V is unitary. Thus this channel is noiseless—it is given by the unitary dynamics. Let ρ0 be an initial compound state on H1 ⊗ H2 of the form:            x0(1)  ⊗ x0(2)  . ρ0 = x0(1) ⊗ x0(2)

Appendix A: Technicalities

341

We remark that this is the density operator corresponding to a pure state and the von Neumann entropy of ρ0 equals zero: S (ρ0 ) = 0. We point out that the pure state under consideration is separable (non-entangled) and hence the two marginal states of ρ0 are given by density operators corresponding to pure states:           ρ01 = x0(1) x0(1)  , ρ02 = x0(2) x0(2)  . The von Neumann entropy of two marginal states ρ01 and ρ02 are equal to zero: S (ρ01 ) = 0, S (ρ02 ) = 0. The final compound state  (ρ0 ) transmitted through the CP channel is    1  (1)   (2)   (1)   (2)  1  (1)  (2)  (1)  (2)  x0  ⊗ x0  + x1  ⊗ x1  .  (ρ0 ) = x0 ⊗ x0 + x1 ⊗ x1 2 2

We emphasize that this is the density operator corresponding to an entangled pure state. The entropy of transformed state  (ρ0 ) coincides with the entropy of the initial state: S ( (ρ0 )) = 0 = S (ρ0 ) . The two marginal states of  (ρ0 ) are 1 ρ01 =

1  (1)  (1)  1  (1)  (1)  1    1    x0  + x1 x1  , 2 ρ02 = x0(2) x0(2)  + x1(2) x1(2)  . x0 2 2 2 2

The von Neumann entropy of two marginal states 1 ρ01 and 2 ρ02 are S (i ρ0i ) = log 2 > S (ρ0i ) = S ( (ρ0 )) = 0. Thus the entropies of both subsystems increased for log 2-amount, but the entropy of S preserves its zero value.

A.2.2

Two Subsystems with N-dimensional State Spaces

We expand the above setting to N × N compound systems (N ≥ 3).   N −1  Let H1 and H2 be C N and xk(i) be orthonormal bases in Hi (i = 1, 2). k=0 We define a completely positive channel  from D (H1 ⊗ H2 ) to D (H1 ⊗ H2 ) by

342

Appendix A: Technicalities

 (•) ≡ V (•) V, where V is a linear map from H1 ⊗ H2 to H1 ⊗ H2 given by N −1

    ϕk, x (1)  ⊗ x (2)  , V = k  k,=0

where N −1    

    ϕk, = 2 ⊗ x (2) αk,, j x (1) j j+k mod N N j=0

αk,, j =

V =

− N 2−2 ( j = ) 1 ( j = )

(k = 0, 1, 2, · · · , N − 1)

       N − 2  (1)   (2)   (1)   (2)   (1)  (2) (1)  (2)  x0  ⊗ x0  x0 ⊗ x0 + x1 ⊗ x1 + · · · + x N −1 ⊗ x N −1 2



      2  (1)   (2)  N − 2  (1)   (2)   (1)  (2) (1)  (2)  + x0  ⊗ x1  x0 ⊗ x0 + − x1 ⊗ x1 + · · · + x N −1 ⊗ x N −1 N 2 +··· +



 2  (1)   (2)   (1)   (2)  N − 2  (1)   (2)  (1)  (2)  + x0  ⊗ x N −1  x0 ⊗ x0 + x1 ⊗ x1 + · · · + − x N −1 ⊗ x N −1 N 2

               2 N − 2  (1)  (2)  (1)  (2)  (1)  (2) (1)  (2)  + − x1  ⊗ x0  x0 ⊗ x1 + x1 ⊗ x2 + · · · + x N −1 ⊗ x0 N 2



      2  (1)   (2)  N − 2  (1)   (2)   (1)  (2) (1)  (2)  + x1  ⊗ x1  x0 ⊗ x1 + − x1 ⊗ x2 + · · · + x N −1 ⊗ x0 N 2 +··· +



 2  (1)   (2)   (1)   (2)  N − 2  (1)   (2)  (1)  (2)  + x1  ⊗ x N −1  x0 ⊗ x1 + x1 ⊗ x2 + · · · + − x N −1 ⊗ x0 N 2

               2 N − 2  (1)  (2)  (1)  (2)  (1)  (2) (1)  (2)  + − x2  ⊗ x0  x0 ⊗ x2 + x1 ⊗ x3 + · · · + x N −1 ⊗ x1 N 2



      2  (1)   (2)  N − 2  (1)   (2)   (1)  (2) (1)  (2)  + x2  ⊗ x1  x0 ⊗ x2 + − x1 ⊗ x3 + · · · + x N −1 ⊗ x1 N 2 +··· +

 

 2  (1)   (2)   (1)   (2)  N − 2  (1)   (2)  (1)  (2)  + x2  ⊗ x N −1  x0 ⊗ x2 + x1 ⊗ x3 + · · · + − x N −1 ⊗ x1 N 2 +········· +

       2 N − 2  (1)   (2)   (1)   (2)   (1)  (2) (1)  (2)  + − x N −1  ⊗ x0  x0 ⊗ x N −1 + x1 ⊗ x0 + · · · + x N −1 ⊗ x N −2 N 2

       2  (1)   (2)  N − 2  (1)   (2)   (1)  (2) (1)  (2)  + x N −1  ⊗ x1  x0 ⊗ x N −1 + − x1 ⊗ x0 + · · · + x N −1 ⊗ x N −2 N 2 +··· +



 2  (1)   (2)   (1)   (2)  N − 2  (1)   (2)  (1)  (2)  + x N −1  ⊗ x N −1  . x0 ⊗ x N −1 + x1 ⊗ x0 + · · · + − x N −1 ⊗ x N −2 N 2 2 N



−

Appendix A: Technicalities

343

The operator V is unitarity. Hence, this channel is noiseless. Let ρ0 be an initial compound state on H1 ⊗ H2 denoted by            x0(1)  ⊗ x0(2)  . ρ0 = x0(1) ⊗ x0(2) One finds the von Neumann entropy of ρ0 such that S (ρ0 ) = 0. The two marginal states of ρ0 are           ρ01 = x0(1) x0(1)  , ρ02 = x0(2) x0(2)  . The von Neumann entropy of the two marginal states ρ01 and ρ02 are S (ρ01 ) = 0, S (ρ02 ) = 0. The final compound state  (ρ0 ) transmitted through the CP channel is    N −1 N − 2  (1)   (2)   (1)   (2)  − x0 ⊗ x0 + xk ⊗ xk 2 k=1    N −1 N − 2 (1)  (2)  (1)  (2)  2 − x0  ⊗ x0  + xk  ⊗ xk  . N 2 k=1

2  (ρ0 ) = N

We also have the von Neumann entropy of  (ρ0 ) is S ( (ρ0 )) = 0 = S (ρ0 ) The two marginal states of  (ρ0 ) are 1 ρ01 =

N −1 4  (1)  (1)  (N − 2)2  (1)  (1)  + x xk  , x  x 0 0 N2 N 2 k=1 k

2 ρ02 =

N −1 4  (2)  (2).  (N − 2)2  (2)  (2)  x + x xk    x 0 0 N2 N 2 k=1 k

The von Neumann entropy of the two marginal states 1 ρ01 and 2 ρ02 are S (i ρ0i ) = 2 log N −

2 (N − 1)2 8 (N − 1) log (N − 2) − log 2 > S (ρ0i ) = S ( (ρ0 )) = 0. N2 N2

344

Appendix A: Technicalities

Consider the above general formulas for the case N = 3. Let ρ0 be an initial compound state on H1 ⊗ H2 denoted by            x0(1)  ⊗ x0(2)  ρ0 = x0(1) ⊗ x0(2) One finds the von Neumann entropy of ρ0 such that S (ρ0 ) = 0. The two marginal states of ρ0 are           ρ01 = x0(1) x0(1)  , ρ02 = x0(2) x0(2).  The von Neumann entropy of the two marginal states ρ01 and ρ02 are S (ρ01 ) = 0, S (ρ02 ) = 0. The final compound state  (ρ0 ) transmitted through the CP channel is

2  (1)   (2)   (1)   (2)   (ρ0 ) = x0 ⊗ x0 + x1 ⊗ x1 − 3

2 (1)  (2)  (1)  (2)  x0  ⊗ x0  + x1  ⊗ x1  − 3

 1  (1)   (2)  ⊗ x2 x 2 2  1 (1)  (2)  x  ⊗ x2  . 2 2

We also have the von Neumann entropy of  (ρ0 ) is S ( (ρ0 )) = 0 = S (ρ0 ) The two marginal states of  (ρ0 ) are 4  (1)  (1)  x0  + x 9 0 4    = x0(2) x0(2)  + 9

1 ρ01 = 2 ρ02

4  (1)  (1)  x1  + x 9 1 4  (2)  (2)  x1  + x 9 1

1  (1)  (1)  x2  , x 9 2 1  (2)  (2).  x2  x 9 2

The von Neumann entropy of two marginal states 1 ρ1 and 2 ρ02 are S (i ρ0i ) = 2 log 3 −

16 log 2 > S (ρ0i ) = S ( (ρ0 )) = 0. 9

Appendix A: Technicalities

A.3

345

Signaling from Contextual State Modification

In this section we present the framework of Sect. 17.11 (Chap. 17) in more detail and without direct connection to cognition. Consider the tensor product framework: state space H = Ha ⊗ Hb and operators A j = A j ⊗ I, B j = I ⊗ B j , j = 1, 2, where Hermitian operators A j and B j act in Ha and Hb , respectively, and have the eignenvectors |±ai , |±b j . Denote their tensor products by | +i + j , | +i − j , | −i + j , | −i − j ;

(A.1)

they form the basis in H. Here labeling by indexes of observables ai and b j is important. We shall consider states which are entangled w.r.t. to the concrete pairs of observables and not, as typically one does, rotationally invariant entangled states. For each pair of observables ai , b j , consider two (orthogonal) subspaces generated by the vectors in the brackets: L i j±∓ = {| +i − j , | −i + j }, L i j±± = {| +i + j , | −i − j }

(A.2)

and projections E i j±∓ , E i j±± onto these subspaces. We are interested in the quantum description of measurements of the pairs of observables (a1 , b1 ), (a1 , b2 ), (a2 , b1 ), (a2 , b2 ). They are represented by pairs of (Hermitian) operators (A1 , B1 ), (A1 , B2 ), (A2 , B1 ), (A2 , B2 ) acting in H. However, we modify the standard scheme for such measurements and suppose that measurement of each pair of observables (ai , b j ) is preceded by the projection of the initial state |ψ either onto L i j±± or onto L i j±∓ . Thus, by the Lüders postulate the initial state is transformed as |ψ → |ψi j±±  =

E i j±± |ψ , ||E i j±± |ψ||

(A.3)

|ψ → |ψi j±∓  =

E i j±∓ |ψ . ||E i j±∓ |ψ||

(A.4)

By using canonical bases in these subspaces we write |ψi j±± =

ψ| +i + j | +i + j  + ψ| −i − j | +i + j   , |ψ| +i + j |2 + |ψ| −i − j |2

(A.5)

|ψi j±∓ =

ψ| +i − j | +i − j  + ψ| −i + j | −i + j   , |ψ| +i − j |2 + |ψ| −i + j |2

(A.6)

346

Appendix A: Technicalities

Any projection represents additional part of the state preparation before the (ai , b j )-measurement. This state preparation is by itself based on the measurement (see later consideration). Now consider such a measurement scheme in which, for three pairs of indexes, projections are selected of the same type and the forth projection is of the different type. For example, for the pairs of observables (a1 , b1 ), (a1 , b2 ), (a2 , b1 ), projections are of the (±, ∓) type and, for the pair (a2 , b2 ), projection is of the (±, ±) type. Calculate correlations for this experimental scheme; for (a1 , b1 ), (a1 , b2 ), (a2 , b1 ), we have: (A.7) ai b j  = ψi j±∓ |Ai B j |ψi j±∓  = ψ| +i − j | +i − j  + ψ| −i + j | −i + j |Ai ⊗ B j |ψ| +i − j | +i − j  + ψ| −i + j | −i + j  = −1 |ψ| +i − j |2 + |ψ| −i + j |2

and a2 b2  = +1

(A.8)

|a1 b1  + a1 b2  + a2 b1  − a2 b2 | = −4

(A.9)

Therefore,

Thus, within the proposed quantum measurement scheme the CHSH correlation can exceed the Tsirelson bound and approach the maximal possible contextual value. Now we shall turn to the problem of signaling. The probabilities generated by the above experimental scheme can be theoretically represented as quantum conditional probabilities. Let us introduce observables di j represented by the operators Di j = E i j±± − E i j±∓ .

(A.10)

For example, for (a1 , b1 ), (a1 , b2 ), (a2 , b1 ), the probabilities under consideration can be written as q(ai = α, b j = β) ≡ p(ai = α, b j = β|di j = −; ψ),

(A.11)

q(a2 = α, b2 = β) ≡ p(a2 = α, b2 = β|d22 = +; ψ)

(A.12)

and

Then, e.g., in the first case we have: q(ai = +, b j = −) = ψi j±∓ |E ai (+)E b j (−)|ψi j±∓  =

|ψ| +i − j |2 |ψ| +i − j |2 + |ψ| −i + j |2

,

(A.13) q(ai = −, b j = +) =

|ψ| −i + j |2 , |ψ| +i − j |2 + |ψ| −i + j |2

(A.14)

Appendix A: Technicalities

347

and q(ai = +, b j = +) = q(ai = −, b j = −) = 0.

(A.15)

Hence, e.g., q j (ai = +) ≡



q(ai = +, b j = β) = q(ai = +, b j = −) =

β

|ψ| +i − j |2 |ψ| +i − j |2 + |ψ| −i + j |2

(A.16) Generally |ψ| +i − j |2 |ψ| + −ik |2  = , k = j. |ψ| +i − j |2 + |ψ| −i + j |2 |ψ| + −ik |2 + |ψ| − +ik |2 (A.17) Thus, signaling is the natural property of the presented measurement scheme within the quantum formalism. The source of signaling is the pre-measurement state preparation given by (A.3), (A.4). Now, for E i j±∓ projections, we have: ai  j = q j (ai = +) − q j (ai = −) = ψi j±∓ |E ai (+) − E ai (−)|ψi j±∓  = ψ|E i j±∓ Ai E i j±∓ |ψ =

(A.18)

|ψ| +i − j |2 − |ψ| −i + j |2 |ψ| +i − j |2 + |ψ| −i + j |2

Finally, for example, a1 1 − a1 2 = ψ|E 11±∓ A1 E 11±∓ |ψ − ψ|E 12±∓ A1 E 12±∓ |ψ.

(A.19)

or a1 1 − a1 2 =

|ψ| +1 −1 |2 − |ψ| −1 +1 |2 |ψ| +1 −2 |2 − |ψ| −1 +2 |2 − . 2 2 |ψ| +1 −1 | + |ψ| −1 +1 | |ψ| +1 −2 |2 + |ψ| −1 +2 |2

We can express the quantity 0 , (see (17.33), the measure of signaling) in the quantum terms: 0 =



1

(|ai 1 − ai 2 | + |bi 1 − bi 2 | 2 i=1,2 i=1,2

(A.20)

1 |ψ|E 11±∓ A1 E 11±∓ |ψ − ψ|E 12±∓ A1 E 12±∓ |ψ| + |ψ|E 21±∓ A2 E 21±∓ |ψ − ψ|E 22±± A2 E 22±± |ψ| 2  +|ψ|E 11±∓ B1 E 11±∓ |ψ − ψ|E 21±∓ B1 E 21±∓ |ψ| + |ψ|E 12±∓ B2 E 12±∓ |ψ − ψ|E 22±± B2 E 22±± |ψ| .

=

348

Appendix A: Technicalities

In the presented context the CHSH-DK inequality has the form: 4 − 0 > 2 or 0 < 2.

(A.21)

So, this is the upper bound for signaling; if signaling is too high it would shadow correlations. In the experiments of article [53] inequality (A.21) was satisfied. Hence, signaling (the direct influences in the terminology of Dzhafarov–Kujala) is rather low and one can find “true contextuality”. We also remark that projections commute with the spectral projections of operators Ai , B j . We show this E i j±∓ E ai (α)|ψ = E i j±∓ (ψ|α+ j |α+ j  + ψ|α− j |α− j ) = ψ|α− j | +i − j  and E ai (α)E i j±∓ |ψ = E ai (α)(ψ| +i − j | +i − j  + ψ| −i + j | −i + j ) = ψ|αi − j i j |αi − j .

Hence, [E i j±∓ , E ai (α)] = 0, α = ±, and in the same way [E i j±∓ , E b j (β)] = 0, β = ± as well as [E i j±± , E ai (α)] = 0, [E i j±∓ , E b j (β)] = 0. Observables (ai , b j , di j ) (see (A.10)) are compatible and, hence, each triple can be jointly measurable and we can find their JPDs: p(ai = +, b j = −, di j = −|ψ) = ||E b j (−)E ai (+)E i j |ψ||2 = |ψ| +i − j |2 ; p(ai = −, b j = +, di j = −|ψ) = ||E b j (+)E ai (−)E i j |ψ||2 = |ψ| −i + j |2 ; p(ai = +, b j = +, di j = −|ψ) = p(ai = −, b j = −, di j = −|ψ) = 0; p(ai = +, b j = +, di j = +|ψ) = ||E b j (+)E ai (+)ei j±± |ψ||2 = |ψ| +i + j |2 ; p(ai = −, b j = −, di j = +|ψ) = ||E b j (−)E ai (−)E i j |ψ||2 = |ψ| −i − j |2 ; p(ai = +, b j = −, di j = +|ψ) = p(ai = −, b j = +, ei j = −|ψ) = 0. We remark that conditional quantum probabilities can be represented as classical conditional probabilities, given by Bayes’ formula: q(ai = +, b j = −) ≡ p(ai = +, b j = −|di j = −; ψ) = p(ai = +, b j = −, di j = −|ψ)/ p(di j = −|ψ).

(A.22) However, for different i, j, generally we have to use different Kolmogorov probability spaces.

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Index

A Adjoint operator, 262 Algebra of sets, 267 Alpha-wolf, 207 Ambiguity aversion, 320, 322 Annealed network, 213 Annihilation operator, 167, 197 Atomic instrument, 130 Atomic quantum instruments, 114, 130 Autopoiesis, 65 Average, 268 Axiomatics of von Neumann and Morgenstern, 319 Axiom of completeness, 319 Axiom of continuity, 319 Axiom of independence, 319 Axiom of transitivity, 319 Axioms of Savage, 321

B Bayes formula, 7, 269 Bayesian inference, 8 Bayes theorem, 8 Behavioral energy, 205 Behavioral Fröhlich condensate, 206 Belief state, 149 Bell, 93 Bell–Dzhafarov–Kujala inequality, 298 Bell inequality, 280, 283 Black box, x, 3 Black Lives Matter, 179 Bohr contextuality, 92, 279, 285 Bohr contextuality-complementarity, 279, 283

Boltzmann constant, 202 Boolean logic, 23 Borel σ -algebra, 268 Born rule, 5, 68, 101 Bose–Einstein condensation, 191 Bose–Einstein statistics, 168, 200, 311, 314 Brain area, 100 Brain evolution, 100 Bra-vectors, 276

C Camel-like graph, 42 Camel-like shape, 31 Cancer, 198 Cascade, 230 Cell, 50, 192 Cerebral cortex, 95 CHSH-inequality, 283, 289–291 Classical probability, x, 3 Clinton-Gore Poll, 152–154, 238 Coherence, 160 Collapse, 65, 76 Commutator, 94 Complementarity, 282 Complementarity principle, 63 Completely positive map, 59 Completely positive operator, 36 Complex social system, 202 Compound system, 58 Conscious experience, 92, 98 Consciousness, 91 Contextuality, 92, 94 Contextuality-by-Default, 295

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Y. Khrennikov, Open Quantum Systems in Biology, Cognitive and Social Sciences, https://doi.org/10.1007/978-3-031-29024-4

369

370 Contextuality-complementarity 92, 94 Correlation, 85 Coupling method, 296 COVID-19, 179 Creation operator, 167, 197 Critical phenomena, 211 Critical Race Theory, 179

D Damasio, 93 Decoherence, 47, 73, 78 Decoherence decision-making, 16 Density operator, 263 Dipole oscillator, 192 Disjunction effect, 22 Dissipation operator, 39 Dissipation process, 43 Distributed processing, 100 Double stochasticity, 271

E Echo Chamber, 160 Einstein, 93 Electrochemical wave, 98 Emigration experiment, 257 Emotion observable, 92 Energy level, 313 Energy band, 194 Energy spectrum, 197 Energy supply, 192 Entangled state, 58, 69, 101, 277 Entanglement, 85 Entropy, 31 Epimutation, 50 Equilibrium state, 201 Event, 267, 273 Event, elementary, 267 Excitation, 192 Excited state, 163 Expected utility theory, 319

F Fabry-Pérot resonator, 221 Facebook, 224 Factorisable state, 58 Fermi–Dirac statistics, 311, 314 Field of sets, 268 Fine theorem, 283, 290, 291 Firing, 66

Index principle,

Formula of total probability, 8, 251 Fröhlich condensation, 191 G Genome, 192 Gibbs, 311 Gorini-Kossakowski-Sudarshan-Lindblad equation, 39 H Hamiltonian formalism, 217 Happiness, 98 Hardware, 99 Heat reservoir, 192 Hermitian operator, 262 Hilbert space, 261 Hodgkin–Huxley equation, 70 Homeostasis, 33 I Incompatible emotions, 97 Incompatible observables, 13 Incompatible of observables, 282 Incompatible perceptions, 97 Indistinguishability, 160, 200 Indistinguishable, 70 Indistinguishable systems, 311 Infon, 165 Information energy, 201 Information environment, 33, 60 Information excitation, 200 Information overload, 23 Information processor, 199 Information reservoir, 202 Information temperature, 201 Instrument, 141 Interaction-operator, 39 Interference of probabilities, 251, 271 Irrational behavior, 3 Ising model, 213 J James-Lange theory, 93 Joint measurement contextuality, 308 Joint probability distribution, 14, 147 K Ket-vectors, 276 Kolmogorov probability space, 267

Index L Lactose-glucose metabolism, 50 Lattice, 273 Lifetime, 224 Likelihood function, 8 Limbic zones, 95 Linearity, 4 Linear operator, 261 Lüders property, 245

M Macromolecule, 192 Macrostate, 54 Marginal inconsistency, 289 Markov dynamics, 40 Markovian dynamics, 39 Matrix doubly stochastic, 271 stochastic , 270 Me Too, 179 Mean-field theory, 222, 234 Mean value, 268 Measuring process, 115 Memorylessness, 171 Mental environment, 131 Mental function, 65, 73 Microtubules, 71 Mixed state, 263

N Naimark dilation theorem, 80 Naimark theorem, 118 Negative entropy, 33 Neural code, 66 Neural network, 51 Neuron, 66 Noncommutativity, 120 Noncommutativity of instruments, 120 Non-equilibrium, 221 Nonequilibrium thermodynamics, 208 Non-Markov, 78

O Observables compatible, 272 Observables incompatible, 272 Open information society, 204 Open quantum system, 131, 208 Open system, 33 Order effect, 119 Order parameter, 228 Order stability, 31

371 Oscillation, 192 Ozawa quantum instrument, 113, 115

P Parastatistics, 311, 314 Perception, 92 Personality state, 149 Phase transition, 211 Phenomenon, 86 Planck oscillators, 311 Positive operator, 35 Positive operator valued measures, 15, 27, 109, 271 Positive-semidefinite, 262 Pride, 98 Prior probability, 8 Probability fallacies, 3 Probe system, 116 Projection postulate, 262 Projection postulate, Lüders, 267 Projection postulate, von Neumann, 266 Projector-valued measure, 15 Proposition, 273 Prospect theory, 3 Protein, 192 Psychic energy, 63 Pure state, 263, 264

Q QBism, 329 Quantization, 94 Quantum gravity, 71 Quantum Baeysianism, 329 Quantum bioinformatics, vii, 4 Quantum biophysics, vii, 4 Quantum brain, ix Quantum channel, 36, 59 Quantum field theory, 219 Quantum information, 4 Quantum information revolution, 34 Quantum instrument, 26, 109, 112 Quantum logic, 23, 273 Quantum nonlocality, 84 Quantum probability, 3 Quasi-color, 175 Qubit, 276 Question order effect, 25, 133, 249 Quiescent, 66

372 R Random parameter, 267 variable, 268 Reddit, 183, 224 Relaxation time, 39 Relaxed state, 163 Response replicability effect, 26, 119, 133, 249 S Sadness, 98 Savage Sure Thing Principle, 21 Scalar product, 261 Schrödinger, 311 Second Law of Thermodynamics, 32 Self-measurements, 91 Self-observation, 68, 98 Self-observer, 66 Semi-classical equation, 222 Shame, 98 Signaling, 281, 284, 289 Simultaneous measurements, 120 Social atom, 159 Social condensation, 204 Social energy, 159 Social Fröhlich condensate, 204 Social information field, 159, 175 Social laser, 24, 159 Social resonator, 160 Social stability, 202 Societal benefit laser, 178 Software, 99 Somatic theory, 93 Sorkin equality, 251 Spectrum, nondegenerate, 265 Spectrum, purely discrete, 265 Spontaneous emission, 170, 223 Spooky action at a distance, 279 Stability Principle, 243 Statistical interpretation, 84

Index Steady state, 16, 18, 40 Stimulated amplification of social actions, 24 Stimulated emission, 170, 223 Stimulus-Organism-Response, 237 Subjective expected utility theory, 321 Subjective probability, 321 Superoperator, 59, 120 Superposition, 66, 266 Synaptic hierarchy, 95

T Telegram, 224 Tensor product, 275 Transition probability, 269, 270 Treelike geometry, 91 Triple-slit experiment, 252 Twitter, 224

U Uncertainty aversion, 320, 322 Unconsciousness, 91 Unital quantum channel, 44 Unitary operator, 35

V Växjö interpretation, 6, 282 Växjö model, 284, 292, 296 Växjö probability model, 282 VK, 224 Von Helmholtz, 94 Von Neumann equation, 37 Von Neumann-Lüders instrument, 110 Von Neumann observable, 15, 110, 113, 117

W Wolf, 204