Information is Energy: Definition of a physically based concept of information 3658408618, 9783658408619

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Lienhard Pagel

Information is Energy Definition of a physically based concept of information

Information is Energy

Lienhard Pagel

Information is Energy Definition of a physically based concept of information

Lienhard Pagel Fakultät für Informatik und Elektrotechnik Universität Rostock Rostock, Germany

ISBN 978-3-658-40861-9 ISBN 978-3-658-40862-6  (eBook) https://doi.org/10.1007/978-3-658-40862-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 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 Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface

In this book, a concept of information is founded and introduced, which is characterized by objectivity and dynamics and which leads to an information conservation theorem. First thoughts on this subject go back a long way. During my studies in 1967, I started to investigate the relation between entropy and information. The occasion was a lecture on thermodynamics which was obligatory for electrical engineers at that time. This was a wonderful piece of luck. The concept of this new notion of information was introduced in my book “Microsystems” [70] in 2001. Now the concept of information is more comprehensively justified and verified. This book is written for electrical engineers, computer scientists, and natural scientists. Therefore, the quantity of entropy, used in all three disciplines but interpreted somewhat differently in each, is once again described in a fundamental way. These descriptions may be considered redundant. Entropy is and remains a central quantity in information theory, and thus also in this book. Also in the field of quantum mechanics, fundamental relations are described which can be found in textbooks. This is especially true of the introductory descriptions of the representation of quantum bits using the formalism of quantum mechanics and the theory of abstract automata. Physicists may find these explanations superfluous and may skim over them. However, this book should be readable by readers from all the disciplines addressed. The book is not a pure textbook. Exercises are therefore not given. For students of higher semesters, the reading is suitable for deepening and for dealing with the concept of information. In essence, it introduces the concept of a physically based and objective notion of information, shows its compatibility with thermodynamics, and demonstrates its usefulness. The book certainly raises more questions than can be answered. Also, little evidence is provided. The argument follows plausible assumptions, and not without risks. The argumentation leads to contradiction with common views on some points, and will therefore challenge dissent. If this enriches the dispute of opinion on the subject of information, at least one goal of the book would be achieved.

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Preface

The physical foundations on which the argument is based have been known since about 1930 or well before. These are NEWTON mechanics, quantum mechanics based on the SCHRÖDINGER equation, electrodynamics, and thermodynamics. Opinions about the notion of information from this period are also treated, because they are still partly present at least among engineers today. In the last chapter, knowledge gained in the last decades is considered. In particular, the view is extended to the field of cosmology. Although a number of physical, mathematical, cosmological, and also philosophical aspects are addressed in this book, it is and remains a book written by an engineer. I do not want to and cannot leave the point of view and the approach of an engineer, even if the argumentation is often based on physical facts. Engineers are explicitly invited to read the book. The term engineer may also be used here in the extended sense of the German Engineering Laws (IngG), according to which all natural scientists may call themselves engineers. The formulations in this book are purposely worded in a definite and assertive way. I want to challenge the professional community and stimulate discussion by clearly formulated statements, which occasionally are to be regarded rather as hypotheses. Even the title of the book, “Information is Energy”, is consciously worded in a challenging way. The response to the first edition of this book has been encouraging. However, it has not dispelled my impression that there is rather a lack of interest in the subject of information among many computer scientists. The second edition has been extended in some points, revised and formulated more clearly. In the new chapter “Consciousness”, the crown of information processing systems, systems with consciousness, is discussed. Some basic properties of logic up to GÖDEL’s theorems have a high relevance to the topic of information. I believe that the notion of dynamic information can contribute to the understanding of consciousness. Finally, I would like to convince all scientists and especially engineers that information always exists dynamically and objectively. I hope that the charm of the defined dynamic and objective concept of information will convince and lead to its application and further development. Klockenhagen October 2022

Lienhard Pagel

Acknowledgement

I feel the need to thank all the discussion partners. Special thanks go to Dr. Ingrid Hartmann for the critical discussion on thermodynamics and structure formation. I would like to thank Prof. Dr. Hans Röck and Prof. Dr. Clemens Cap for the stimulating and critical discussions that contributed to sharpening the concept. As an author, I am of course responsible for the content, especially for mistakes and misinterpretations.

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Contents

1 The Concept of Information—An Introduction. . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation and Goals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Computer Science and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Information Technology and Information. . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Physics and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Information in Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Philosophy and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6.1 Structure of the Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6.2 Information About Information. . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6.3 Categorization of Information. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Information—Physically and Dynamically Based. . . . . . . . . . . . . . . . . . . . . . 19 2.1 Objectivity of the Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Subject and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Information Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Dynamics of the Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Quasistatic Information in Computer Technology. . . . . . . . . . . . 22 2.3.3 Time and Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.4 Energy Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Definition of Dynamic Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Quantum Mechanical Limits of Information Transmission. . . . . 24 2.4.2 Heisenberg’s Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.3 Phenomenological Foundation of Dynamic Information . . . . . . 28 2.4.4 Parallel Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.5 Transfer 1 Out of N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Representation of Quantum Bits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 Qubits—Description with the Formalism of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5.2 Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5.3 Hadamard Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ix

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2.5.4 2.5.5 2.5.6

2.6

Polarization of Photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Systems of Two Quantum Bits . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Qbits—Description with the Formalism of Automata Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.7 No-cloning Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.8 Quantum Computing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.9 Physical Realizations of Quantum Bits. . . . . . . . . . . . . . . . . . . . 49 Properties of Dynamic Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.6.1 Dynamic Information and Quantum Bits . . . . . . . . . . . . . . . . . . 51 2.6.2 Signal and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6.3 Valuation of Entropy by the Transaction Time . . . . . . . . . . . . . . 53 2.6.4 Valuation of Entropy by Energy . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6.5 Objectivity of Energy and Information. . . . . . . . . . . . . . . . . . . . 55 2.6.6 Conservation of Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.6.7 Destruction and Creation of Physical States and Particles . . . . . 61 2.6.8 Hypothesis: Variable Relationships Between Information to Energy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6.9 Redundancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 A Comparative View on the Concept of Information . . . . . . . . . . . . . . . . . . . 67 3.1 Shannon Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Definition by Jaglom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 Algorithmic Information Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Information and Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 Potential and Actual Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.6 Syntactic and Semantic Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.7 Interpretations by Carl friedrich von weizsäcker . . . . . . . . . . . . . . . . . . 72 3.8 Theory of ur Alternatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.9 Pragmatic Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.10 Transinformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.11 Information in Business Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.12 Relational Information Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.13 Entropy and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.14 Equivalence Information and Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.15 Further Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.16 What is Information?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Entropy and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Entropy in Information Technology—Basics . . . . . . . . . . . . . . . . . . . . . . 83 4.1.1 The Concept of Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.2 Renyi Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.3 Entropy of a Probability Field. . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.4 Entropy of a Probability Field of Words. . . . . . . . . . . . . . . . . . . 88

Contents

4.2

4.3

4.4

4.5

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4.1.5 Enhancements of the Entropy Concept. . . . . . . . . . . . . . . . . . . . 88 4.1.6 Applicability of the Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1.7 Negentropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Entropy in Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.1 Basics of Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.2 Entropy, Energy, and Temperature. . . . . . . . . . . . . . . . . . . . . . . . 93 4.2.3 Components of Entropy: Energy and Volume. . . . . . . . . . . . . . . 96 Entropy in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.1 Entropy of the Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.2 Density Matrix and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.3 Brukner-zeilinger Information. . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3.4 Information About a State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Computers and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4.1 Transmission in the Presence of Thermal Noise. . . . . . . . . . . . . 103 4.4.2 The Landauer principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4.3 Reversible Computing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.4 Energy Conversion in Computers . . . . . . . . . . . . . . . . . . . . . . . . 105 Entropy Flow in Information Technology. . . . . . . . . . . . . . . . . . . . . . . . . 107 4.5.1 Stationary Information in Classical Mechanics. . . . . . . . . . . . . . 107 4.5.2 Bits and Quantum Bits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.5.3 Entropy Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.4 Entropy Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.5 Degrees of Freedom and Freedom. . . . . . . . . . . . . . . . . . . . . . . . 113

5 Dynamic Information and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1 Elementary Information Transfer in Thermodynamic Systems. . . . . . . . . 117 5.2 Asynchronous Energy and Entropy Transfer. . . . . . . . . . . . . . . . . . . . . . . 117 5.3 First Law of Thermodynamics and Dynamic Information . . . . . . . . . . . . 118 5.4 Adiabatic Processes—Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.4.1 Classical Ideal Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.4.2 Real Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.4.3 Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.4.4 The Flow of Dynamic Information in Adiabatic Processes. . . . . 124 5.4.5 Parallelization of Data Streams. . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.6 Serialization of Data Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.5 Isothermal Processes—Phase Transformation in Solids. . . . . . . . . . . . . . 132 5.6 Temperature Dependence of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.7 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.7.1 Noise as a Disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.7.2 Noise and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.7.3 Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.7.4 Quantum Mechanical Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

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6 Irreversible Processes and Structure Formation . . . . . . . . . . . . . . . . . . . . . . 141 6.1 Irreversible Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1.1 Time Symmetry and Irreversibility. . . . . . . . . . . . . . . . . . . . . . . 141 6.1.2 Thermodynamic Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.1.3 Irreversibility and Objectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.1.4 Dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.1.5 Information Conservation and Irreversibility . . . . . . . . . . . . . . . 157 6.2 Formation of Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2.1 Chaos Theory—Interpretation of Information. . . . . . . . . . . . . . . 158 6.2.2 Structure Formation and Entropy . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2.3 Order, Order Parameters and Entropy. . . . . . . . . . . . . . . . . . . . . 165 7 Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.1 Consciousness and the Strong AI Hypothesis. . . . . . . . . . . . . . . . . . . . . . 167 7.2 Consciousness: Overview and Introduction. . . . . . . . . . . . . . . . . . . . . . . . 169 7.2.1 Consciousness in Language Use and Delimitations . . . . . . . . . . 169 7.2.2 Consciousness in Philosophy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.2.3 Consciousness and Information. . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.2.4 Limited Self-Awareness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.2.5 Is Consciousness Definable?. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.3 Components of Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.4 Formal Logical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.4.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.4.2 Contradictions in Formal Logical Systems. . . . . . . . . . . . . . . . . 178 7.4.3 Contradictions in Systems with Consciousness. . . . . . . . . . . . . . 179 7.4.4 Attractiveness of Contradictory Systems. . . . . . . . . . . . . . . . . . . 180 7.5 The Self-Reference in Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.6 Limits of Self-Knowledge Due to Self-Reference. . . . . . . . . . . . . . . . . . . 185 7.7 Self-Reference in Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.8 Algorithmic Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.8.1 Self-Reference in Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.8.2 Loops in Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.8.3 Algorithmic Requirements for Consciousness . . . . . . . . . . . . . . 190 7.8.4 Consciousness and Life. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.9 Consciousness and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.10 Juridical and Political Aspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8 Astronomy and Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.1 Relativistic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.2 Light Cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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8.3 8.4 8.5

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Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Entropy of Black Holes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 The Universe and Its Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

9 Resume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Acronyms

Bit neologism from binary digit CBit classical bit in the representation of abstract automata Qbit quantum bit in the representation of abstract automata Qubit Quantumbit EPR Einstein-Podolsky-Rosen paradox AI Artificial Intelligence AC Artificial Consciousness min… the function: minimum of … max… the function: maximum of … NAND logical function of the negated ‘AND’, word combination from NOT AND Negentropy abbreviation for: negative entropy Pixel neologism from picture element Voxel neologism from Volumetric pixel or Volumetric picture element Ur a “Yes/No” decision in C. F. V. Weizsäcker’s ur theory

xv

Nomenclature

δ angle vector space C2 φ angle ϑ angle bra-representation of a wave function �. . . | ∆ Laplace operator ∆E energy difference ∆t time of transmission, general: time interval  Planck’s quantum of action /2π ket-representation of a wave function | . . .� EPR wave function |ΨEPR � λ wavelength µ chemical potential µ expected return, business sciences Ω base set ω angular frequency, elementary event Ψ wave function Σ sum symbol, subset quantity σ volatility of a rate, business sciences τ time constant, relaxation time a, b, c, d constants, can be complex a*, b*, c*, d* constants, conjugate complex C capacitor, capacitance c speed of light E energy G gravitational constant g gravitational acceleration H Hadamard transform h Planck’s quantum of action

xvii

xviii

I information √ i −1 J inertial momentum kB Boltzmann constant m mass of particle N number of objects P probability measure p momentum, probability, investment vector p probability R electrical resistor S entropy S sum of moments, business sciences SH Bekenstein-Hawking entropy SF entropy flow T temperature, time interval t time TH Hawking temperature U potential V volume v velocity w probability Z number of possible states

Nomenclature

1

The Concept of Information—An Introduction

1.1

Motivation and Goals

Our era is often termed the “Information Age”. Information is constantly flowing almost everywhere1 . It seems that never before more information has been exchanged than today. A major reason for this development is the technical progress in the field of electronics and in particular microelectronics. The manufacturing processes of microelectronics allow the manufacture of millions of transistor functions at the price of a few cents. This makes even complex information processing and transmission affordable and is therefore ubiquitous, at least in industrialized countries. The effects of computers, especially their networking on our daily lives, are obvious. Our prosperity is essentially shaped by information technology and the exchange of information. Most households have more than 10 microcontrollers or computers in operation. Their performance does not have to hide behind the mainframe computers of the fifties of the last century. This shows that bold predictions of past decades have been far exceeded. Whether washing machine, television, or car, computers are usually integrated everywhere as socalled embedded systems, i.e. microcontrollers and complex circuits. The ubiquitous processing and transmission of information have not only changed our material life but also our thinking and feeling. People have internalized the fact that technical systems in their immediate environment exchange information with each other, often process complex algorithms and communicate with them. Our time is characterized in particular by the rapid availability of information, even over great distances and in complex contexts. Whether individuals receive more information

1 The term “information” comes from the Latin word “informatio” and is called “imagination” or

“explanation”. The verb “informare” means “shape, shape; teach someone, educate through instruction; describe something; think something” [26].

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 L. Pagel, Information is Energy, https://doi.org/10.1007/978-3-658-40862-6_1

1

2

1 The Concept of Information—An Introduction

under these circumstances and whether they understand more contexts is questionable. The amount of information that a person can receive per unit of time is biologically limited. Possibly it is different information that is received by people today, possibly it is interpreted more superficially. The hunter-gatherers of earlier millennia certainly had more detailed knowledge about their immediate environment than people today. Information and knowledge had an immediate importance for the existence of humans, which is why evolution had to equip us humans with more and more extensive information processing abilities. Evolution is taking place on a different level today. Advances in the field of artificial intelligence enable communication directly with technical systems. The interfaces between humans and machines are adapting more and more to humans. The power of algorithms and computers is becoming more visible where humans long believed they were superior to computers. Computers have long been winning at chess and go against world champions made of flesh and blood. Whether artificial systems can have consciousness and how far we are from artificial systems with consciousness are exciting questions of our time. They will be discussed in this book, they are directly related to information. Information technology has become an essential part of our culture. The flow of personal information, which can no longer be controlled by the individual, also seems to have become an integral part of our culture. This loss of control and its influence on our individuality and freedom has found its way only incompletely into our personal and social consciousness. It is all the more astonishing that the concept of information has so far been insufficiently defined. The increasingly successful handling of information has obviously blocked the view for the essence of information in many areas. The purpose of this book is to discuss the essence of information and to sharpen the concept of information. The approach is less technically oriented. The source of inspiration is the natural sciences. Approaches from the humanities are supportive, but not causal. Secondary effects of information in society and economy are not considered. The extent to which the concept of information has implications for the humanities and economics is not discussed in detail here. In this book, information is defined objectively and dynamically. This concept of information is called dynamic information. This concept is not intended to represent a particular type of information or a particular interpretation of the information. It does not stand beside the other concepts of information, it should replace them. It defines information in its essence. Starting from the central concept of entropy, a relation of information technology to thermodynamics and quantum mechanics is established. The relationship between information and energy is the key to understanding information. Information is regarded as a dynamic and measurable quantity in the physical sense, which is defined independently of man and can exist independently.

1.2

Computer Science and Information

1.2

3

Computer Science and Information

Computer science is a discipline that emerged from mathematics and whose subject is information. However, this subject of computer science is still insufficiently defined within computer science. Computer scientists sometimes complain about this fact. However, it is not clear that remedy has been found. In a slightly joking way Karl Hantzschmann2 said that information is what computer scientists deal with. The computer scientists are probably in good company. C. F. v. Weizsäcker writes under the heading “The sense of cybernetics” in [95]: One of the methodological principles of science is that certain fundamental questions are not asked. It is characteristic of physics ... that it does not really ask what matter is, of biology that it does not really ask what life is, and of psychology that it does not really ask what soul is ... . This fact is probably methodically fundamental to the success of science. If we wanted to ask these most difficult questions at the same time as we do science, we would lose all time and energy to solve the solvable questions.

The success of computer science and information technology is undisputed. This may be the reason why there is still a multitude of very different definitions for information next to each other. Klemm has comprehensively described the situation in an article [51]. The situation of the computer science is also characterized by Rozenberg [77] very aptly by writing about Carl Adam Petri: According to Petri, it is one of the shortcomings of computer science as a science that it does not follow an appropriate law, like the laws of conservation of energy in physics. As a law of nature, it could enable the discovery of new forms of information. Independently of this, such a law would lead to the concept of the information balance and its measurability. One should be able to present one’s balance after a piece of information processing. For Petri it is quite clear that in the world of information processing the consideration of the information balance is a quite normal one, ... An important reason why this fact is not generally accepted in the world of computer science is that it is not clear what should be balanced, the question is what information actually is. As Petri sees things, it is something that could be explained by terms of physics, but not necessarily by terms of already existing physics.

Carl Adam Petri obviously thought about the measurement of information and also about a conservation law for the information. It is obvious to explore the relation of energy to information more exactly, if there is already an energy conservation law. It is unmistakable, but nevertheless not generally accepted, that energy and information are inseparably connected with each other.

2 Prof. for Algorithms and Theory of Programming at the University of Rostock.

4

1.3

1 The Concept of Information—An Introduction

Information Technology and Information

In contrast to computer science, energy plays an increasing role in information technology. The ratio of transmitted information to energy consumption has long been a crucial issue for transmission. The performance of information-transmitting and information-processing systems is decisively dependent on energy consumption or, better still, on energy efficiency. Information processing structures are increasingly being adapted to energy conditions. The ever-increasing degree of integration forces developers to accommodate more and more components that convert electrical energy into heat in the smallest possible volume. Heat dissipation by heat conduction and convection is the limiting factor for further integration in many highly integrated systems. The ratio of processed information to energy consumption becomes the decisive factor for the further integration and design of complex electronic systems with high functionality. In information technology, researchers and developers have been keeping a close eye on energy efficiency, i.e. the relationship between transmitting information and the energy it requires. In view of the fact that the internet alone consumes more electrical energy worldwide than Germany, a connection between information and energy is not only of theoretical interest but also a significant factor in the global energy economy. In electronics, the so-called power delay product is often used to assess energy efficiency. The electrical power (energy per time unit) required by a gate for the transmission or elementary linking of elementary information units is multiplied by the delay time. The delay time is caused by various physical effects in the gates. In electronic semiconductor systems, these are often charging processes at parasitic capacitances and simply the transit time of charge carriers from input to output of a gate. The delay time contains the speed of the transmission and ultimately the amount of data transmitted per unit of time. A rather clear interpretation of the power delay product is the energy E = P · Δt = (ΔE/Δt) · Δt, which is used per transmitted information unit. This is not only about the energy required to represent an information unit. So, it is not only about the energy that is transported from one gate to the next but also about the energy that is necessary to technically realize this transport. Most of this energy is dissipated into heat. From a technical point of view, this is mostly energy loss. In recent decades, scaling has progressed faster than reducing the power per gate. As a result, chip cooling has become increasingly critical. The transmission of an information unit in electronic systems can be modeled and explained in a simple approximation by the charging of a capacitor. In Fig. 1.1, U0 is the operating voltage of the gate, R symbolizes the channel resistance of the driver transistor, and C is the input capacitance of the next gate, mainly the gate capacitance of the input transistors. A calculation at this very simplified model shows that 50% of the energy spent at

1.3

Information Technology and Information

5

Fig. 1.1 Simplified representation of a gate as RC circuit

 the gate U0 ∗ I (t)dt is converted to heat at the resistor R and only 50% is transferred to the next gate. Unfortunately, the energy transferred and stored in C is not reused in conventional gates, so that it is ultimately converted into heat. The remedy could be a so-called “adiabatic” circuit technique. In principle, electrical energy can be transferred from one gate to the next without loss. This would be possible, for example, by using inductors. A principle example can be found in [70]. Power electronics has had to find solutions to this problem because here 50% loss in power transfer from one unit to the next is not acceptable. However, these solutions are not applicable to highintegrity circuitry because of inductances that cannot be easily integrated. Other solutions are available in the field of optoelectronics. In principle, the transfer of photons from one gate to the next does not require any additional energy. Another interpretation of the power delay product is the quotient of electrical power (ΔE/Δt) and channel capacity (Bit/Δt). A “bit” represents a voltage pulse or better a transmitted charge that occurs with a certain probability. In digital technology, it is often intended to transmit a bit. In most cases, the coding is done so that the state “no charge” is coded with a “0” and “charge” with a “1”. With optimal coding, both states are equally distributed. Therefore, the average energy per bit is only half of the energy needed to transfer the charge. However, the polarity can also be changed and the information can be coded in the pulse length. Figure 1.2 shows the power delay product for some technical and natural information processing systems. The figure shows the power required to operate a gate over the delay time. It shows the relation to channel capacity. The energy, which represents an information unit itself, is included in the power. In lossless transmission, therefore, the power is not zero; it is reduced to the transmitted power. At the quantum limit, this is the minimum quantum mechanical power required. The power delay image does not directly show the power dissipation. In order to obtain this, the further used portion would have to be subtracted from the transmitted energy. With an adiabatic circuit technique, the power loss would be zero, even if much more than the quantum-mechanically necessary energy was transferred. Another interesting quantity is the energy delay product (EPD). If the energy of a switching process is meant, it cannot be smaller than the quantum of action according to the

6

1 The Concept of Information—An Introduction

Fig. 1.2 The efficiency of different information technology and electronics technologies in relation to the quantum of action

Heisenberg uncertainty relation for energy and time (see Sect. 2.4.2 “Heisenberg’s uncertainty relation”). Not only energy efficiency in the gates of integrated circuits is important. The interaction between software and hardware design at the component development level is an essential

1.4

Physics and Information

7

basis for the success of information technology. This refers to the shift of functionality from hardware to software or vice versa. Here, a certain equivalence between software and hardware regarding functionality is recognized and used. Among information technicians, it is hardly imaginable that energy is not involved in the handling of information. Some levels of abstraction are higher; the potentials are not yet exhausted. The interactions between information and physical structures are still insufficiently developed. An example is provided to illustrate this: In the field of quantum computing, the dependence of information structures on physical laws is obvious and dominant. Here, even the logic is influenced and determined by the physical processes. This concerns reversible logic, which is discussed in Sect. 2.5.6 “Qbits—description with the formalism of automata theory”. A closer conjunction of information theory with the physics of information processing will reveal new possibilities. Just as in information technology, the energy required to transfer a bit in a given time is of importance, the elucidation of the relationship between energy and information will increasingly clarify the relationship between computer science and physics. Information technology will have to translate the results—ultimately into affordable and applicable systems of general utility.

1.4

Physics and Information

It seems that from physics there is a greater affinity for the clarification of the concept of information than from computer science. Hans- Joachim Mascheck sees information and energy lying closely together. He writes [65]: The role of the concept of information in physics is quite comparable to the role of the concept of energy. Energy and information are two quantities, which penetrate all fields of physics, because all physical processes are connected with energy transformations and at the same time with information transfer.

A statement of Norbert Wiener, the founder of cybernetics, stands in clear contradiction to this. He thinks that information is basically also conceivable without energy. From him comes the much-quoted sentence [97]: Information is information, neither matter nor energy. No materialism that does not take this into account can survive today.

With reference to this statement, it should be expressly stated in the sense of a delimitation that in this book only verified statements of physics are to be used. Any processes assumed outside of today’s physics are not considered and by the way are not necessary. For the understanding of information processing processes in technology and also in biology up to the human brain, the known physical bases are sufficient. This does not mean that everything

8

1 The Concept of Information—An Introduction

is understood. This only means that it is not due to the fundamental laws of physics, but to their application to very complex systems and their interpretation. In physics, thermodynamics is mainly concerned with entropy and information. Werner Ebeling makes a direct reference to thermodynamics and writes [15]: Information entropy is the average amount of information required to describe the special realization of a random event if the coding is optimal. According to this, the information stored in bit converted entropy of a thermodynamic system, is the amount of information required with optimal coding to indicate all details of an instantaneous state—e.g., in the case of a gas-filled volumes the marking of the positions and velocities of all gas molecules with the quantum theoretically possible accuracy.

This approach is based on the system and an observer who is informed about the state of the system. The entropy of a system at a given moment is therefore the amount of information the observer must receive in order to be informed about the current state. Because in the classical point of view measurements of location and velocity are possible with unlimited high accuracy, the observer has a certain arbitrariness here. The value of entropy or the amount of information depends on how exactly the observer wants to have the measurements. Thus, the value of entropy is defined with the exception of a constant amount. Actually, in classical mechanics, only entropy differences are defined. This point of view is used in the following explanation of the information concept. However, it also becomes problematic if quantum mechanical systems are considered. Then a hypothetical observer is to be assumed, who should have no interaction with the observed system, who is physically not realizable, because he would disturb the system by the collapse of the wave function. This observer facilitates the explanations, but is not needed later. Also, Ebeling sees a close connection of information with energy and entropy. In addition he writes [16]: Consciously simplifying by neglecting semantic aspects, the physicist understands information as an exchangeable quantity that is closely linked to energy and entropy and reduces the uncertainty of the state of a system.

Ebeling also makes statements about the essence of information. He sees it with energy and entropy on one level. Unusual in the sense of the usual view of information technology is the negation of the difference between the carrier of the information and the information itself. He means [16]: Although the physical carrier and the carried form an inseparable dialectical unit, we consider it more favorable not to define the information as carrier and carried, because the usual physical definitions of the basic quantities energy and entropy do not include the carrier either.

It seems that the concept of signal (as carrier of information) is questioned here. In general, “the signal is called the carrier of information” [103].

1.4

Physics and Information

9

Hans- Joachim Mascheck [65] sees parallels between information and energy as physical quantities. He writes [65]: Two important variables are involved in all natural processes: energy and information. In thermodynamics this finds its expression in the existence of two main sentences. For special problems the energy or the information can be in the foreground such as power electronics and information electronics. But always the other partners are also present.

But Mascheck sees the information as a process that takes place between man and environment. He nevertheless sees the information as a physical quantity [65]. The relationship between human thinking and the environment appear to us at first as pure information relations. However, the human brain is a part of the reality, in which the laws of physics are also valid.

In this book, the information is defined without the need of a subject. The term “subject” is closely related to the term consciousness or is used synonymously. The introduced concept of information now allows an unconstrained contemplation of a subject, i.e. a system with consciousness. The consideration of consciousness as a property of a physical system in Chap. 7 “Consciousness” closes the circle. In quantum theory, a pragmatic view of information seems to prevail. Many physicists regard the bit as a unit of information and equate entropy with information ([37, 94]). Unless expressly defined otherwise, it seems physicists understand entropy as information. It seems that from the point of view of quantum mechanics the preservation of the probability3 of the wave function (see Sect. 2.6.6) “Flow density of probability in the quantum mechanics” is regarded as the reason for the preservation of the information. However, this probability density (the square of the wave function) cannot be directly identified with the information. This does not mean entropy either. It is probably also about the preservation of structure, if, for example, a particle falls into a black hole. In Chap. 8, “Astronomy and cosmology” black holes are treated. Scores are problematic. However, it can be stated that, in physics, a justified access to the concept of information is sought. From the direction of computer science, this is not yet recognizable. This may be due to the fact that information is historical rather a physical term. Finally, physics knows the term entropy since about 1867. It was introduced by Rudolf Clausius. Ludwig Boltzmann and Willard Gibbs gave entropy a statistical meaning in 1887. Only three quarters of a century later Claude Channon found in the environment of information theory 1948 a formula for the average information content of a sign, which is identical with the “thermodynamic entropy”. The Shannon entropy is also called information entropy. In view of the longer and more comprehensive experience with the concept 3 Here, the preservation of the integral over the probability density of the wave function is meant.

10

1 The Concept of Information—An Introduction

of entropy, the question of whether the physicists are the more experienced information theoreticians in comparison to the computer scientists is, of course, slightly polemical. Physics does not know a quantity called “information”, but, in statistical physics, there is the concept of entropy. Entropy in physics is often directly associated with the concept of information. Entropy, like energy and temperature, is a fundamental quantity for characterization of a thermodynamic system. They are probably the most important quantities in statistical physics. It’s curious that physics doesn’t know a quantity of information, but no less than Leonard Susskind means [89]: The minus-first law of physics: Bits are not destructible.4 .

Susskind thus places the preservation of information before the preservation of energy. He speaks of bits as information and also means quantum bits, for example, photons of a certain polarization. Here not the abstract bit is meant, but a physical object, which also represents energy and also has a time behavior. Ben- Naim [5] contradicts the account that Shannon’s entropy is information. He formulates Information is an abstract concept which could be subjective, important, exciting, meaningful, etc. On the other hand, SMI5 is a measure of information. It is not information. ... SMI is an attribute assigned to an objective probability distribution of a system. It is not a substantial quantity. It does not have a mass, a charge or location.

It is reported that John von Neumann has pointed out to Shannon that his formula (4.6) agrees with the thermodynamic definition of entropy. Ben- Naim agrees that von Neumann thus did science a disservice. The opinions about the relation of entropy to information are quite different in physics.

1.5

Information in Biology

The storage and transmission of information are fundamental processes in biology. They will be briefly examined here. On the one hand, the storage of information in DNA is important for the transmission of information from one generation to the next. In these processes, evolution takes place through mutations and selection. These processes take place at the atomic and molecular levels. The information is stored in molecular structures and the processing of the information, i.e. 4 Leonard Susskind puts this sentence before all considerations in a lecture titled “The World as

Hologram”. The formulation is: “The minus-first law of physics: Bits are indestructible”. He adds: “Information is forever.” 5 Shannon’s Measure of Information.

1.5

Information in Biology

11

logical linkages and copying, are also molecular processes. These processes also include electronic processes because the interactions between atoms and molecules are dominated by processes in the atomic shell. As a general rule, it can be stated that these processes are abundantly endowed with energy and are also therefore sufficiently resistant to thermal fluctuations. On the other hand, in the neurons, the information transfer takes place in an electrical way, more precisely in an ionic way. Ions are the carriers of information. These processes are also abundant with energy, the Fig. 1.2 shows the relations. From the author’s point of view, the concept of information is not yet a dominant concept in biology. Nevertheless, information of gene sequences, structural information of tissues, and information content of biological signals are spoken of. The concept of “biocommunication” is obviously interpreted as “information change” and regarded as fundamental for life. Knappitsch and Wessel [53] give a comprehensive overview and formulate the proposition: The process of communication permanently surrounds every living being and is essential to its existence and continued existence. However, what exactly does this term refer to?

To what extent the concept of information plays an essential role at all is questionable. Taking the concept of “genetic information” as an example, Michael Kray and Martin Mahner in the “Lexikon der Biologie” [56]: In fact, coding DNA (coding sequence) cannot be distinguished from non-coding DNA by means of Shannon’s information measure H6 ... . It is therefore not surprising if the molecular biological processes in protein synthesis (proteins, translation) can be described and explained just as well without the aid of this concept of information, so that the use of information theory may be didactically helpful (analogy), but ultimately contributes nothing to the explanation in the methodological sense.

Ben- Naim describes in [5] the main information transfer processes in living cells. Aptly, he quotes Gatlin from her book Information Theory and the Living System [29]: Life may be defined operationally as an information processing system—a structural hierarchy of functional units—that has acquired through evolution the ability to store and process the information necessary for its own accurate reproduction. The key word in the definition is information.

Schrödinger also makes a connection between entropy and life in his famous book “What is Life” [79]:

6 formula (4.6)

12

1 The Concept of Information—An Introduction Living organism ... delay the decay into thermodynamic equilibrium (death), by feeding upon negative entropy, attracting a stream of negative entropy upon itself ... and to maintain itself on a stationary and fairly low entropy level.

In Sect. 3.1 “Shannon Information”, Sect. 4.1.7 “Negentropy”, Chap. 4 “Entropie and Information”, and Sect. 6.2.2 “Structure formation in open systems by entropy export” I will address these not unproblematic views. However, quite particularly the concept of consciousness is tangential to biology and life sciences in general. It is closely connected with the concept of information. This topic is discussed in the Chap. 7 “Consciousness” in more detail. It may well be summed up that the life sciences have a high natural and occasionally strongly differentiated affinity to the concept of information.

1.6

Philosophy and Information

Also, from the point of view of the more general sciences, the situation around information is deplored. A quotation from Michael Symonds [90] clarifies the situation: The question of the meaning and the object of the concept of information is the central blind spot in the field of vision of science and philosophy. This fundamental problem is seldom recognized as such and is only addressed by a few. But the fact is, we simply do not know: What the hell is information?

1.6.1

Structure of the Information

In order to gain access to the concept of information from the point of view of philosophy, the tripartite nature of the concept of information [26] can be entered into. It is in the sense of a semiotic7 , i.e. in the sense of sign theory, three-dimensionality: • The syntax that refers to relationships between signs. Essential are rules, which allow the creation of valid character strings, which can be understood on a semantic level, by combining single characters. Syntactics is the theory of relations between the signs of a semiotic system. • The semantics, which deals with relations between signs and concepts. Here the meaning of the signs stands in the foreground. Semantics thus concerns the meaning of the information units and their relations with each other. • The pragmatics, which deals with the relations between signs and their users. The individual understanding of signs, for example, plays a role here. It is about how signs work, 7 The semiotics (Greek σ ημιoν = signal, sign) is the general theory of linguistic signs [49]. It is

part of epistemology.

1.6

Philosophy and Information

13

in which context of action they are integrated, and how users can and react to it. Pragmatics therefore concerns the effect of information units and their relationships with each other. This three-dimensionality is also very useful for engineers for practical work. It is widely used in the field of programming languages in electrical engineering and computer science. However, it does not answer the question of what information is. How does the concept of information fit into the problem areas of philosophy? After Lyre [64] the questions of theoretical philosophy allow an allocation of the information to three problem areas: • Ontology (in short: the doctrine of the end or being) – What kind of object is the information? – How does information exist? – Does the information concern the kind of knowledge about an object area or the things themselves? – Does a consistent information ontology need a new understanding of the previous concept of substance in physics regarding energy and space-time? • Epistemology (cognitive science, theory of knowledge) – Can information be thought without an observer? – Is a subject needed? – Is information ultimately an objective or subjective quantity? – Does the difference between the ontological and the epistemic character of information disappear because the information represents both the building material of the world and the knowledge of this world? That would be the original theory of C. F. von Weizsäcker quite close. • Semantics (science of meaning, science of the meaning of signs) – Can aspects of semantics be formalized? – Is meaning measurable? In philosophy there should be agreement that one can basically speak of information processes in three areas [26]: 1. in the field of cognition and the production of ideas by social subjects (cognition), 2. in the field of the exchange of knowledge and transport of social subjects via ideas (communication), and 3. in the field of joint actions, the implementation of which requires social subjects to reconcile knowledge and ideas (cooperation). But how the concept of information is to be interpreted concerning these three areas leads to the so-called “Capurro trilemma” [12], that Fuchs [26] worded as follows:

14

1 The Concept of Information—An Introduction

Either the term information means in all areas of knowledge: 1. exactly the same: If the terms of information used in the different sciences were synonymous, then what is called “information” should apply to the world of stones (physics) in the same sense as in the world of people (psychology, etc.). However, there are good reasons why the qualitative differences between these worlds should not be mentioned. This possibility is out of the question. 2. or only something similar: Suppose the terms were analogous. Which of the various information terms should then provide the PRIMUM ANALOGATUM, the standard of comparison for the others, and with what justification? If, for example, it were the concept of information of human science, it would have to be accepted to anthropomorphize if non-human phenomena were to be treated, i.e. erroneously to transfer concepts from one area—here the human area—to another where they do not fit, for example, to have to claim that atoms talk to each other when they combine to form molecules, and so on. A consequence to be discarded. For this reason, this possibility is also out of the question. 3. or something completely different in each case: If the terms were equivok, i.e. identical words for incomparable designates, science would be in a bad position. It resembled the Tower of Babel, the subjects could not communicate with each other, just as Kuhn assumes that paradigms also detach from each other; the objects of knowledge would be disparate, if at all delimitable. So also the last possibility is unsatisfactory. Nevertheless, the variants of synonymy, analogism, and equivocation enjoy a large following. From the author’s point of view, the first possibility, synonymy, has been all too recklessly abandoned. Physics applies its concepts, such as energy, to all areas and is not afraid to apply them to the human world and the world of stones. Why shouldn’t this also succeed with a physically justified concept of information? However, this does not automatically solve the questions of interpretation in these areas. The listed questions should clarify the breadth of the questions that are connected with the concept of information from a philosophical point of view. Information is also a central concept of philosophy. It is itself a part of philosophy. A basic question in connection with information is the assignment of information to matter, substance, or energy. If information is neither material nor ideal (Norbert Wiener [97]), then a “third realm of being” would have to be constituted. But this question now goes into the marrow of philosophy, it concerns the basic questions of philosophy. Does the information really deserve such a special position beside the material and the ideal? In [49], no reason is seen to see the information as a ‘third realm of being’, it is in the sense of logic neither object nor the property of an object, but property of properties (predicate predicate). The concept of information is placed next to the basic categories “Substance” and “Energy” as the third basic category, i.e. “Information”.

1.6

Philosophy and Information

15

Philosophical questions should be addressed only selectively in this book. However, a statement is absolutely necessary for some questions. After all, an objectively defined concept of information is presented.

1.6.2

Information About Information

So what is the reason that the information is the “central blind spot” of philosophy? Probably the reason for this situation lies in the fact that science, to put it simply, collects and processes information about objects, but in the case of information, the information itself becomes an object. More comprehensively this means: The basic statement of analytical philosophy is that “all knowledge can only be gained through experience” [64]. This statement is closely connected with the tradition of logical empiricism. Experience is, according to Carl Friedrich von Weizsäcker, quite generally characterizable as “learning from the facts of the past for the possibilities of the future” [64]. Facts can only be obtained by collecting information about objects. Facts are information. The question whether information about objects is possible without disturbing the objects can be answered in classical mechanics with “yes”. The effect of the information can be arbitrarily small and negligible in comparison to the effect of the object. If quantum mechanics is used as a basis, information about a system can only be obtained if the system exchanges effects with the observer, which cannot be smaller than the Planck quantum of action. The observed system is changed or even destroyed. To receive information about information does not seem to succeed anymore, especially if the effects of the object are close to the quantum of action. The smallest unit of information, the quantum bit, is not divisible in the classical sense, because an effect quantum is not divisible. According to the no-cloning theorem [75] (see Chap. 2.5.7 “No-cloning theorem”), it is also not possible to copy quantum-mechanical states, e.g. a quantum bit, as a quantum object.8 The information seems to be unique. If an observer has information about an object, then obviously this information can no longer be present in the object. Not only the information about information is problematic but also the information about objects. At least this applies if one goes to the level of quantum bits. If information is to be considered in pure form, then the quantum bit is very suitable as an indivisible element of information. It is pure information, has no further properties than to be true or false, and is by the way the simplest non-trivial quantum object. With the question of information about information, the human sciences may have a greater problem than the natural sciences. In particular, epistemology must find an answer to the question of how to separate body and mind. Insights and knowledge can only be composed of quantum bits. Even if these concepts are very complex in nature, epistemology 8 This concerns quantum bits in general. After dereferencing and the breakdown of the wave function

after a measurement, the result, which is more of a classic bit, can be copied.

16

1 The Concept of Information—An Introduction

will not be able to avoid recognizing the uniqueness of information in the sense of being basically indivisible and basically not copyable. In contrast to philosophy, practical information technology today still has no real problem obtaining information about information. The energies used to transfer information are in most cases many times greater than necessary. In many cases, the information can be understood as a macroscopic object and can therefore be observed almost without disturbance. This situation will soon change when the effect for the transfer of a bit comes close to the quantum of action. In some cases, especially in optical information transfer, technical processes are at or near the quantum of action. There one will have to ask oneself whether one can obtain information about information. In the field of quantum cryptography, it is assumed that this is practically impossible. That would be pure security against eavesdropping. Quantum information technology is close to its first applications.

1.6.3

Categorization of Information

According to Mascheck [65], there are different approaches to classify the information in the structure of the basic categories: • between the terms – Information—Movement—Energy—Mass resp. – Information—Form (in Aristoteles- Platon sense)—Motion—Mass relationships in the sense of an equivalence or unity of nature (von Weizsäcker) can be established. • information is neither movement nor matter, but a third (Kerschner). • information is the mediator between mind and matter (in the sense of overcoming the contradiction materialism—idealism). These approaches show how far at least some philosophers are from an objective approach to the concept of information. An experimentally measurable and verifiable concept of information is probably not to be expected from philosophy for the time being. It should be emphatically recalled that information can be transmitted and processed with material devices and appliances. The known physical laws are sufficient for their construction and explanation. The mind could sit in the brain. But this consists of switching neurons, which work together in very complex and massively parallel and not yet fully understood ways. It is not recognizable that a new physics is needed for this. There is no reason to assume that the brain could not be equivalent to a Turing machine. Thus, all algorithms, if they need finite resources, would be realizable in our brain. This is essentially the Church-Turing hypothesis, which states that the class of Turing calculable functions is exactly the class of intuitively calculable functions. An intuitively calculable function is a function that can be executed by the human brain. Conversely, the

1.6

Philosophy and Information

17

hypothesis says that everything that cannot be calculated with a Turing machine cannot be calculated at all. The hypothesis is not refuted, it is generally accepted; it is also not proved. The reason lies in the not exactly comprehensible definition of intuitively calculable functions. This is the field of artificial intelligence (AI). It is reasonable to assume that there is a Turing machine equivalent to our brain. This would mean that there may also be computers that are equivalent to our brain. This is the so-called “strong AI thesis”. This is to be understood as a statement in principle, because the complexity of today’s computers does not yet come close to that of our brain. Practically, there are still open questions, such as the programming of learning complex processes. In principle, however, there is nothing recognizable against it. The coupling of information to the mind or consciousness is not necessary. Information can also exist without our mind and outside of our consciousness. After all, data from geological epochs in which no human being existed are accepted as information. Nevertheless, there would be information from those times, even if evolution had never produced man. On the other hand, consciousness or the subject is inconceivable without information, more precisely without any information processing. In Chap. 7, “Consciousness” the topic is deepened. Information does not need consciousness, but consciousness needs information and information processing.

2

Information—Physically and Dynamically Based

2.1

Objectivity of the Information

The variety of information terms proposed so far demonstrates the enormous relation of the term to almost all fields of knowledge and activities of man. This also makes clear that there can be many attributes to information. The difference between measurable quantities and subjective interpretations seems to be blurred. For scientists and engineers, the emphasis on the objectivity of information seems unnecessary. In view of the discussions whether information should be ascribed a separate mode of existence of matter or whether information can only be grasped by concepts which lie outside our known physics, a clear statement is necessary: in this book a dynamic concept of information is introduced which stands on the secure foundations of physics, especially quantum physics. Thus, in this book a concept of information is founded, which is physically defined and allows further interpretations in a great variety. This dynamic and objective concept of information leads to a measurable quantity, which exists and works like impulse or energy independently of human observations.

2.2

Subject and Information

The question arises as to how subjects might relate to information. What is actually a subject? In philosophy, a subject is understood as the “human spirit, the soul, the self-determined and self-determining I-consciousness” [105]. Does information need a subject or a soul? In [96] Wenzel cites Norbert Wiener in context with the “formal equal treatment of living beings and machines as operative systems for information processing”:

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 L. Pagel, Information is Energy, https://doi.org/10.1007/978-3-658-40862-6_2

19

20

2 Information—Physically and Dynamically Based It is clear that our brain, our nervous system form a message switching system. I mean, the analogy between our nerves and telephone lines is plausible.

Wenzel means that cybernetics has accomplished the identification of concretely representational entities (something existing which has an internal connection) with information. If our brain is considered as a message-switching system, the actual seat of the human mind can be considered as a technical system. Thus, the subject is “disenchanted”1 . As little as energy needs a subject, so little is a subject or a human being necessary for the transmission of information or for the reception of information. The subject is quite recognizable as structure and dynamics of special complex objects, but unnecessary. If a subject is the recipient of a message or information, there is almost always a context in which the message is interpreted. The characteristics of the recipient significantly determine the effect of the information. These are not inconsequential characteristics; they concern the fundamental characteristics of the recipient. George Steiner [88] writes that information or messages are related to associations in the receiver: This associative faculty of ours is so extensive and detailed that it probably equals in its uniqueness the sum of our personal identity, our personality.

Information processing in the receiver is a part of information transmission and processing. The “disenchanted” receiver or subject forms, as it were, the context of the information reception, while the information transmission and the information itself are context-free. In quantum mechanics, the observer has a special meaning and is not a subject in the philosophical sense. Heisenberg [34] has clarified: Of course, the introduction of the observer must not be misunderstood to mean, for instance, that subjectivist features should be brought into the description of nature. Rather, the observer has only the function of registering decisions, i.e. processes in space and time, whereby it does not matter whether the observer is an apparatus or a living being; but the registration, i.e. the transition from the possible to the factual,2 is absolutely necessary here and cannot be omitted from the interpretation of QT.

It is similar with communication. Originally it had the meaning of the exchange of information between people. Today, the term is also used for information transmission or signal exchange between devices or technical systems in general.

1 See Sect. 7.2 “Consciousness: Overview and introduction”. 2 The possible and the factual define the potential and the actual information, see on this Sect. 3.5

“Potential and actual information”.

2.3

Information Transfer

2.3

21

Information Transfer

What are the indispensable minimal properties that information must have? How can these properties be found? Of course, the starting point must be the use of the term information in our time, the information age. The result should also be compatible with this usage. The central property of information is transferability. Information that is not transmissible, i.e. immovable, cannot be received by any other structure and therefore can be considered irrelevant or nonexistent. The question whether everything that can be transmitted is also information is answered here with “Yes”. One reason for this lies in the objective approach, i.e. in the independence of information from the opinion of people. Whatever is transmitted, the receiver always receives at least the information that something specific has arrived or has been transmitted. Other physical quantities have similar properties, for example, energy. Energy can be useful for us, unnoticed or unrecognized, it exists independently of us and is measurable. It also acts without our intervention and without our knowledge about it. An energy which is present at one place and in principle can never be transferred to another place and otherwise has no effect on other places is not relevant. It cannot develop any effect in the physical sense of product of energy and time.

2.3.1

Dynamics of the Information

In order to clarify the dynamic character of the information, a static, i.e. in principle immobile information is to be considered hypothetically. Immobile should mean here that the information cannot be transferred, that the information has no interaction with other systems. If such information should exist, nobody will know that it exists, because it is isolated. A claim that static information exists here or there can neither be proven nor disproven. In terms of simplicity of theories, it should not be considered. In physics, there is a similar situation with the energy E = mc2 . Who considers in everyday life that a human being represents in the relativistic term, an amount of energy of 2.5 · 1012 kWh? Because this energy cannot be converted in the vast majority of situations, it is also mostly meaningless. Only in the sun, in nuclear power stations and nuclear weapons this kind of energy can work. Information is therefore always dynamic. It must be physically transferable. Here a clear border is to be drawn to esoteric (in the sense of spiritual) conceptions. Only processes are considered which are physically describable. The concept of dynamics is to be defined more precisely. On the one hand, dynamics means movement. Movement is connected with change in time. In this sense, it means that information is always in motion. From a quantum mechanical point of view, objects are also

22

2 Information—Physically and Dynamically Based

always in motion. Even when information is stored in the form of quantum states, it is in motion. For example, a particle “sitting” in a potential well at the lowest level has a matter wavelength and a natural frequency. It can be considered as a vibrating and thus moving particle. On the other hand, dynamics can mean that systems are moving, but are just not moving at the moment under consideration. The dynamics of information discussed here includes both aspects.

2.3.2

Quasistatic Information in Computer Technology

Information must move in order to be received, processed or stored. But isn’t the information on the hard disk static? So does static information make sense after all? In today’s computer technology it does, because about 10,000 or more quantum bits are used to transmit or store one bit. This is the state of the art in 2020, which means that such a classical bit can be divided and copied almost any number of times. It becomes observable and it can also be considered quasistatic. So in computers and also in our brains, the transmission of bits is “shooting at sparrows with cannons”. In the end, much more information is processed than is perceptible to us in the result. The many “by the way” transmitted information contains data about the state of the electrons in the transistor, the state of the neurons and much more. Ultimately, they leave the computer via the waste heat of the circuits and are released to the environment as entropy in the fan. The fan realizes an entropy flow—that is thermodynamically indisputable. But this is also an information channel with a quite considerable channel capacity. This is objectively present, it is just not interpreted and perceived by us in general as “information flow”. It is similar with our brain. The uninterpreted information is emitted as heat, i.e. as an entropy current. Unlike the bits stored on hard disks and in neurons, quantum bits (qubits) are not divisible3 . This means that the information content is not divisible. If one has knowledge of the existence of a qubit4 , then you have it, then it is nowhere else.

2.3.3

Time and Space

The transmission of information is linked to time. There is at least a time before the transmission and a time after the transmission. This also implies a direction of the time. The 3 Photons can be divided by semipermeable mirrors (beam splitter), but this does not divide the

information. Only one receiver can receive the bit. The theorem of Holevo forbids the multiplication and also the annihilation of bits in quantum registers, see Sect. 2.4.3 “Phenomenological foundation of dynamic information”. 4 The properties of quantum bits are discussed in 2.5 “Representation of quantum bits”.

2.3

Information Transfer

23

reversibility of processes and also of transmission processes is basically given in the Newton mechanics and in the quantum mechanics. To be able to define a transfer process at all, a time direction must be given. The time is considered like an independent variable. However, it is useful to consider the symmetry of time in the context of the second law of thermodynamics (entropy theorem). For further considerations see the topics irreversibility and structure formation in Chap. 6 “Irreversible processes and structure formation”. In the consideration of times and instants, a non-relativistic point of view shall be taken for the time being. Delays by the finite transfer speed of energy and information are neglected. Thus also the conservation law of energy remains valid. The conservation of energy requires simultaneity and this is not absolutely definable in relativistic systems. One has to be aware of the importance of this limitation; after all, information transfer and information processing takes place in electronic systems with almost, and in photonic systems, with the speed of light. This problem is discussed in Sect. 8.1 “Relativistic effects”. The transfer process needs a boundary between systems, it is after all transferred from one place to another place. However, this boundary is subject to uncertainty for quantum mechanical reasons, resulting from the Heisenberg uncertainty relation for location and momentum. A difference in location (or boundary) and a difference in time are to be regarded as essential elements of a transmission—and the transmission as an essential element of information. Both quantities can be mediated by velocity or momentum and cannot be arbitrarily small and are subject to the following quantum mechanical limit: Δpx · Δx  .

(2.1)

For example, if the momentum in the x-direction px is known with an accuracy of Δpx , then the location x of the particle is defined with an accuracy of Δx only. Consequently, the dynamics of a transfer process affects the sharpness of the local boundary of the process. In general, there is therefore a relationship between the speed of transfer and the volume required for it. Fast, high-energy processes, require less space and are more sharply delimited. The physical quantities time and space (location) should not be considered as absolute quantities. In the quantum mechanical representation of objects by wave functions, there is a mathematical symmetry between momentum space and position space. Both representations are equivalent without restrictions. In this sense, information transfer can also be considered in momentum space. Because the time behavior correlates with the impulse, this symmetry can also be seen between time and position. Whether this results in advantages or other aspects depends on the concrete question.

24

2.3.4

2 Information—Physically and Dynamically Based

Energy Transfer

What is the role of energy? Is a transfer of information possible without energy transfer? If esoteric processes are excluded, this is not possible according to general opinion (see also Sect. 1.4 “Physics and information”). This consideration refers explicitly to elementary events. Independently, of course, the energy balance between sender and receiver can be balanced if two transmission processes occur simultaneously at different locations in opposite directions. Without energy transfer no information transfer can take place, because without energy in the receiver no effect can be generated. Thus an information transmission is always associated with an energy transmission. The energy portion could be regarded as a carrier of the information. What about the information content of a transmitted energy portion? Does every energy portion also have an information content? As it was already stated: Yes, at least the information is transferred that the energy was transferred. The opinion is contestable, if in the question, what is information, the receiver is asked as subject. Objectively, every transferred energy portion is carrier of at least one information, namely the information about its presence. In summary, it can be stated that an information transfer is bound to energy transfer and energy transfer is bound to information transfer. Both are not separable. Information is not even imagineable without energy, because even in the imagination, which usually takes place in the brain of a human being, for each switching operation of a neuron at least 10−11 W s are converted.

2.4

Definition of Dynamic Information

2.4.1

Quantum Mechanical Limits of Information Transmission

In order to clarify the relationship between energy and information more precisely, the basic question is to be answered, how much energy is necessary to transmit a certain amount of information. The elementary information unit is a yes/no decision, i.e. a bit. For the further considerations, a sender (the system will be briefly referred to as Alice) and a receiver (Bert) are assumed in an environment in which no interfering influences act. Alice wants to communicate a decision to Bert with the least possible energy expenditure. Bert observes the energetic state E of his input. To keep other impacts away during the transmission, it is assumed for the time being that noises, especially thermal noise, have been made ineffective by cooling. Then Bert will measure at his input for the time being only the quantum fluctuation of the energy. Their cause lies in the statistical character of the wave function. A wave function describes the state and dynamics of a particle in quantum mechanics and is generally a complex function of location and time. The square of the wave function is interpreted as the presence probability of the particle. Due to this statistical character of

2.4

Definition of Dynamic Information

25

Energy

ΔE

Δt

Time

Fig. 2.1 Detection of an energy change during quantum fluctuation

the wave function, the quantum mechanical fluctuation of a measured quantity cannot be eliminated in principle. Now, if Alice wants to change the energy state at Bert’s input for the purpose of transmitting the yes/no decision, Bert must be able to filter out this change from the quantum noise (or quantum fluctuation). Bert will need a time Δt to detect an energy change ΔE (see Fig. 2.1). This time Δt shall be called the transmission time. Clearly, the smaller ΔE is, the more time Bert needs to detect the energy change, for example, by averaging. The relation between ΔE and Δt is given by Heisenberg’s uncertainty relation5 ΔE · Δt  h

(2.2)

The transmitted yes/no decision has been transmitted with the lowest possible energy. This minimum-energy bit is generally called quantum bit or qubit. The minimum energy required for a bit thus depends on the time allowed for the transmission. The relation (2.2) is a fundamental relation for information transmission. If an effect should be considered which is larger than the quantum of action, this can and should be quantized, i.e. “decomposed” into individual actions. This decomposition is always possible, because in principle every quantum object can be decomposed into quantum bits. The energy described here as minimal can therefore also be regarded as the energy corresponding to the bit, it is nevertheless minimal.

5 There are various statements and justifications concerning the size of the indeterminacy. Here we

refer to Sect. 2.4.2 “HEISENBERG’s uncertainty relation”.

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2 Information—Physically and Dynamically Based

The question arises, with which certainty in the sense of an error rate Bert can detect the bit. More time means more transmission certainty. This question is treated in the Sect. 2.4.2 “HEISENBERG’s uncertainty relation”. What is the relationship between energy and action in information transfer? An action h “realizes” a transport of energy in a certain time. For the existence of a bit not the energy is the decisive quantity, but the action. If Alice has a bit, then that bit is an energy that is constantly moving to a next state in a time Δt—even within Alice. When transferring to Bert, energy is transferred from Alice to Bert’s system. But, this must happen in a time Δt. Thus, an action is transferred from Alice to Bert. The associated energy is not static. It cannot be readily “used” within Bert at any rate; it is linked to Δt because of the conservation of energy and the relation (2.2). The receiver receives not only a bit but also the time behavior, i.e. the dynamics of the bit. In this consideration it is better to speak of a quantum bit. Thus, the information transmission is not only the transmission of an energy but also a process is transmitted, which has its own dynamics and can basically exist independently. Thus, a quantum bit has an existence characterized by ΔE and Δt before and after transmission. The dynamical processes within Alice and Bert are internal information transfer processes. They are, of course, also subject to the laws of information transfer and, in particular, to Heisenberg’s uncertainty relation (2.2). Interaction processes with other quantum bits within Alice and Bert can change the energy and transmission time of quantum bits. Parallelization (see Sect. 5.4.5), serialization (see Sect. 5.4.6) and more generally dissipation processes (see Sect. 6.1.4) provide further dynamics. It can be seen that the velocity of a bit in terms of a carry-over velocity cannot be considered without taking into account the conservation of energy and Eq. (2.2). Basically, it must be noted that here in the transmission and especially in the detection of a quantum bit, the existence or non-existence of an energy portion was considered, that is, two energetic states. In an analogous way, a quantum bit could also be encoded in other energy states—in the spin of an electron or in the phase of a photon. The two quantum mechanical states of the quantum bit generally form a basis, and their scalar product is zero6 .

2.4.2

HEISENBERG’s Uncertainty Relation

Fundamental for the information transfer is the Heisenberg uncertainty relation (2.2) with respect to energy and time. It deserves a closer look, if only because different versions can be found in the literature. It is about the size of the uncertainty. How sure can Bert be of having received a bit? Bert could stop the observation if there is a change of energy at his input and he can count on a certain probability to have received a bit. Then the transaction time would be smaller than required by the relation (2.2). But: Bert could interpret a random spike in the quantum noise as a change in energy. The probability of a correct transmission increases with the observation time. Where is the limit? When does it make sense for Bert to stop the 6 Measurement bases are discussed in the Sect. 2.5.2 “Measurement” discussed.

2.4

Definition of Dynamic Information

27

observation? Exactly this question must be answered with respect to the interpretation of the Heisenberg uncertainty relation (2.2) and probably cannot be answered conclusively today. It concerns the justification of the uncertainty relation (2.2). This is a very fundamental question which is discussed, among other things, in relation to the interpretation of quantum mechanics. The process of measurement is a critical point for quantum mechanics. A look at the literature reveals the following variety of rationales: ΔE · Δt ≈ : A derivation of this relation can be found in Landau [58], p. 44. A system with two weakly interacting parts is considered. A perturbation-theoretic consideration yields, for two measurements of the energy between which time Δt has elapsed, the energy difference ΔE. It has the most probable value /Δt. This relation is valid regardless of the strength of the interaction. It is criticized that this is only an example. This result is also obtained by Niels Bohr, in which he calculates the time required for a wave packet to cross a given region. The approach is heuristic and based on properties of the Fourier transformation. ΔE · Δt  /2: In [66] reference is made to an elaboration of Mandelstam and Tamm. Here, for the time Δt, which a particle needs to traverse the distance Δx, Δt = Δx/v is taken quite classically. Quantum mechanically, Δt = Δx/ |x| ˙ inserted into the general indeterminacy relation and one obtains the indeterminacy relation ΔE · Δt  /2. A heuristic trace back to the location-momentum relation Δx · Δp =  also uses Δt = Δx/v. For the energy uncertainty, an energy uncertainty ΔE = vΔp/2 is calculated from E kin = p 2 /(2m) = vp/2. Substituting into the location-momentum relation, we get ΔE · Δt  /2. ΔE · Δt ≈ h: This relation can be derived from the lifetime τ of an excited state and the natural width of spectral lines ΔE [66]. The width of spectral lines is inverse to the lifetime: ΔE = /τ , where τ = 1/Δω. Here, the excited atom is thought of as an oscillator whose amplitude decreases exponentially due to radiation. The amplitude of the electromagnetic radiation is then exponentially damped with the decay time τ . Then the spectrum has the form of a Lorenz distribution with the width Δω = 1/τ . An essential problem with the different relations is the mostly not exact definition what actually ΔE and Δt is in the statistical sense. The energy-time uncertainty relation has the peculiarity that, in contrast to the locationmomentum uncertainty relation Δx · Δpx ≈ , the location x and the momentum px have no exact value, but the energy is defined at every instant. The quantum fluctuation is the cause of the scattering. A quotation from [66], which goes back to Landau, brings this very clearly to the point: The time [necessary for measurement] is limited by the relation ΔE · Δt > , which has been established very often, but has been interpreted correctly only by Bohr. This relation obviously does not mean that the energy cannot be known exactly at a certain time (otherwise

28

2 Information—Physically and Dynamically Based the concept of energy would have no sense at all), but it also does not mean that the energy cannot be measured with arbitrary accuracy within a short time.

The relation ΔE · Δt ≈ h is chosen for further consideration of the information transfer because it gives more time to the measurement and thus reduces the probability of error. The relation ΔE · Δt ≈  obviously provides the most probable error. It takes a little more time to fully acquire a bit. This is not an exact reasoning, so the principal uncertainty with the uncertainty remains. Later corrections are not excluded.

2.4.3

Phenomenological Foundation of Dynamic Information

So far, the term information has been used in the context of the transmission of yes/no decisions. Quantum mechanical constraints enforce a dynamic in the transmission of the yes/no decision. Energy and time must be included in the consideration. This is the reason for introducing a new dynamic concept of information. If the relation (2.2) is resolved to ΔE, one obtains the energy necessary for the transmission of a bit h (2.3) Δt The consideration is now extended to a sequence of transmissions, i.e. to a sequence of bits. The bits are supposed to be statistically independent of each other. Then the energy portions ΔE add up to the total energy E. ΔE =

E=

N  i=1

N  h ΔE i = Δti

(2.4)

i=1

If we assume for simplicity that the transaction times Δt are the same for each bit, then we get for this special case E=

N 

ΔE i =

i=1

Nh Δt

(2.5)

The N stands for the number of bits transmitted. The bit is the unit of measurement of entropy and is dimensionless. One can consider a yes/no decision as an entropy of one bit. Because entropy is an additive quantity, the entropies of individual quantum bits can be summed to the entropy of all bits S7 . Then the following relation is valid E =h

S . Δt

(2.6)

7 N bits represent 2 N possibilities. Each possibility occurs with probability p = 1/2 N . According to Eq. (4.6), we obtain for entropy S = log2 2 N = N .

2.4

Definition of Dynamic Information

29

So the energy is directly connected with a quantity which has the dimension of a channel capacity. The information is a subsystem with its own energy and entropy, passing from the sender to the receiver in time Δt. If the consideration is extended to the transmission of bits within systems, this quantity corresponds to the channel capacity that an observer needed to be constantly informed about the state of a system with certain energy. This is the dynamic information that the system has. If the information is understood as a dynamic quantity, the quantity entropy per time unit S/Δt offers itself as a new information term. This information is called dynamic information and denoted by I. The transaction time Δt plays a central role. From (2.6) follows E = hI

(2.7)

This relation (2.7) defines an objective and dynamic notion of information and is for the time being a postulate for the general relation between information and energy. This dynamic concept of information implies because of the energy conservation theorem also an information conservation theorem, more precisely a theorem for the conservation of dynamic information. It is examined in the further chapters for its usefulness and usability. The relation (2.7) with respect to (2.5) is of course only valid for synchronous transmissions, i.e. with invariant Δt. However, it shall also apply and be examined for transfers with different Δt, i.e. asynchronous transfers. For this case, that the transaction times for the individual bits are not equal, the energy and information portions must be summed up separately according to the Eq. (2.4). Then per bit 1Bit = ΔI (2.8) Δt The quantity “1 bit” stands for the transmitted entropy difference. It is the smallest unit that can be transmitted. The dynamic information consists of information units which are linked with classical information bits. A bit is the unit of measurement of the entropy. For the sake of shorter notation, a unit of dynamic information linked to a bit shall be called a dynamic bit. It is connected via 1 bit · /Δt or 1 bit · ΔE to the classical bit. It must be emphasized that a dynamic bit is not a uniform quantity and does not have a uniform value. The energies can be arbitrary. Concerning the range of validity of the relations (2.7) and (2.8) it should be noted that for the time being it is limited to non-relativistic quantum mechanical systems. This means that in the overall system considered, the energy and entropy do not change (because of the reversibility of quantum processes). This must be so because the Schrödinger equation ΔE = 

30

2 Information—Physically and Dynamically Based

is invariant to time reversal. However, energy and entropy transport between subsystems is possible. If N transmitted quantum bits are considered as quantum registers, the states can be mixed with infinite probability amplitudes. It looks like one could also store infinite information. A fundamental theorem of Holevo8 shows that a quantum register of length N can also store only N classical bits. Also only N classical bits can be read out again [94]. Thus the bridge between quantum bits and classical bits is established. The previous considerations have referred to the transmission time Δt. However, the energy portion ΔE has also an expansion. For quantum mechanical reasons, the location of the energy portion is subject to an uncertainty resulting from the Heisenberg uncertainty relation for location x and momentum p, Δpx Δx  . For photons, the uncertainty of the location difference is of the order of the wavelength λ = c/ΔE (c is the speed of light). For electrons √ or other particles not moving at the speed of light, the De- Broglie wavelength λ = / 2mΔE is governing. Consequently, the space requirement of an energy portion or an information depends on the energy and thus also on the time behavior. These relationships also explain why quantum systems (and quantum bits) must be very small and very fast at room temperature to “survive”. A somewhat more detailed discussion can be found in [70]. The number of bits, i.e. the entropy of a system, need not be preserved in an open system. For example, when the volume for the propagation of a system is increased, the number of bits can be increased. Thus, bits can also be lost from a system, For example, if two photons produce a higher energy photon by frequency doubling or multiphoton excitation. This does not contradict the entropy theorem. It is very important to distinguish between open and closed systems in these processes. In summary, it should be stated that information can be meaningfully understood only by taking quantum mechanics into account. Information is quantum mechanical.

2.4.4

Parallel Channels

The above consideration assumes a single “message channel” between Alice and Bert. What if such a transfer process can take place at several positions of the interface between A and B (see Fig. 2.2). For example, if a photon with energy ΔE is detected by a CCD chip with 2n pixels. In this case, the photon carries n bits with it. The energy of the photon is of course ΔE and per bit an energy of ΔE/n is dropped. It should be noted that the photon has to go through n decision processes before it is detected by a pixel. Where and how these processes take place shall not be discussed here. The only certainty is that they must have taken place somewhere and at some time before detection. These decision processes 8 See Sect. 2.6.1 “Dynamic information and quantum bits”.

2.4

Definition of Dynamic Information

31

Fig. 2.2 Distribution and transmission on two channels

are physical processes in which energy (ΔE) is transferred and each process takes a time Δt = /ΔE. So the “selection process” takes a total of n ∗ Δt. This process compares well with the expansion of a gas as shown in Fig. 5.2. The entropy multiplies while the energy remains unchanged. The dynamic information is conserved, as is the energy. The consequent application (2.2) means also that selection processes must be included. In addition one more example: To find out in which of 8 pots a particle or electron is, three yes/no questions have to be answered. These three measuring processes are of course also subject to (2.2). This is imaginable in such a way that each decision process spans one dimension. Then a selection from 8 possibilities would be representable in three dimensions, each dimension representing one bit. When the selection has been made in each dimension, the result is fixed. In summary, if the number of possibilities is increased alone, i.e. if more volume is available, the energy of the particle(s) is conserved, but the energy is diluted and distributed among the bits. The lower energy leads to a temporal prolongation of the processes, which can be explained by necessary selection processes.

2.4.5

Transfer 1 Out of N

For the further investigations the relation of information and energy is interesting for the practically important case that a pulse is sent as 1 and no energy is transmitted in the other time elements (One Hot Coding, Fig. 2.3). So, only one energy portion is sent during N time elements. What is the relation between energy and information here? So far, it has been assumed that a permanent stream of data bits flows during data transmission. If, in addition, the information is coded in the properties of the particle9 , quasi stationary processes prevail. However, when the information is encoded in the presence of particles, a different situation arises. For practical applications, for example, in 1-out-of-Nencoding, this case is of interest. What is the energy required when the bit is distributed to

9 This could be the spin or the polarization.

32

2 Information—Physically and Dynamically Based

Fig. 2.3 Transmission of a pulse in N time units, each having a duration of τ . Only one pulse is detected, no pulse may be registered in the other N-1 time units

multiple receivers. In the following, a rough estimation about the behavior of the dynamic information is made on the basis of a quasiclassical consideration. It is modeled that in N time elements of duration τ a pulse of duration τ with a detectable energy ΔE T is transmitted. Thus, the total time available for this transmission is T = N τ (see Fig. 2.3). A 0 is to be detected in each of the other N − 1 time elements. The number of bits transmitted is obtained by counting the number of yes/no decisions required to determine the position of the pulse. The number B of bits transmitted is B = log2 (N ). This estimation applies accordingly to the case where, for example, a photon is to be detected by N detectors, such as a photon on a CCD chip. For the transmission of one bit in the time element τ an energy of ΔE τ = /τ would be required according to the uncertainty relation. Now there is the added requirement that no 1 should be detected in the remaining N − 1 time elements. Because quantum fluctuation occurs in these time elements, the detection of a 1 cannot be excluded in principle. However, the probability should be sufficiently small. Now it should be estimated by which amount the energy of the one pulse ΔE T must be higher in comparison to the energy ΔE τ , which is necessary for the transmission of only one bit. The pulse should be detected with a certain high probability as a single one pulse. After the transmission time T the receiver has the knowledge at which time the pulse was sent. Because the pulse, depending on the size of N , transmits several bits, more precisely log2 N bits, its energy should also be correspondingly higher. In order to estimate the energy ΔE T , information about the probability distribution of the amplitudes is needed. For the following estimation it is only assumed that the probability for the occurrence of high amplitudes decreases exponentially with the energy. A general approach with 2 still free parameters A and E 0 shall be chosen, where for practical reasons the base 2 is chosen for the exponential function: −E

w0 (E) = A · 2 E0

(2.9)

The procedure “1 out of N” (One Hot Coding) means that only in one time slot or in only one detector a pulse is received. The sum of all individual probabilities should then be 1.

2.5

Representation of Quantum Bits

33

In the simple case the probabilities w0 (E τ ) can be added. To estimate by how much ΔE T must be higher, we set the probabilities equal: With this condition and (2.9) we get N · w(ΔE τ ) ≈ 1 N · A·2

ΔE τ /E 0

≈1

(2.10) (2.11)

After logarithmizing, one gets for energy to be expended: ΔE T ≈ E 0 log2 (N ) − log2 (A).

(2.12)

E 0 and A can be made plausible if one assumes that exactly one bit is transmitted for N = 2. If one is only interested in the ratio between the energy for one bit and B bits, w0 (E) must be equal to 0.5, because then only two possibilities exist. Then E 0 can be set equal to ΔE τ and A = 1. (2.13) ΔE T ≈ ΔE τ · log2 (N ) It can be seen that the energy expended is proportional log2 N , i.e. the number of bits transmitted. This is the entropy of the pulse for a 1–out–N –transmission. For the special case N = 2 (one bit), ΔE T = ΔE τ is obtained. In summary, it can be stated that even in the case of information transmission with longer pauses, the energy expenditure grows with increasing entropy, as required by the relations (2.6) and (2.7).

2.5

Representation of Quantum Bits

In the Sect. 2.4.1 some basic properties of a quantum bit have already been explained and used. However, for further considerations on quantum mechanics, a treatment of the representation of quantum bits is necessary. Before discussing the representation on quantum bits, a clarification of concepts is necessary. If a quantum bit is to be understood as quantum object, then one speaks of a Quantum bit or short Qubit. If quantum bits are represented in a simplified way as a generalization of simple, classical abstract automata (in the context of automata theory), then one speaks of a Qbit.

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Qbits are suitable to represent finite and discrete quantum mechanical systems. They are not suitable to describe continuous systems [111]. A classical bit in the representation of abstract automata is called Cbit. First the complete description of a quantum bit shall be treated10 .

2.5.1

Qubits—Description with the Formalism of Quantum Mechanics

The quantum bit is the smallest and simplest nontrivial object in quantum mechanics. Basic and clear descriptions of quantum bits are given by Franz Embacher in [19] and de Vries in [94]. The state of a qubit can be represented in terms of a two-dimensional vector with complex components.   a |Ψ  ≡ . b

(2.14)

As usual, the Dirac’s bra-ket notation is used here. A state vector is represented as “ket” √ |.... a and b can be complex variables. It could be a = r + i · m, where i = −1. r is called real part and m imaginary part of the complex number a. Let the set of all vectors |... form a complex two-dimensional vector space denoted by C2 . On this set is an inner scalar product for 2 elements     a c |Ψ  = , |Φ = (2.15) b d is defined as follows: Ψ |Φ = a ∗ c + b∗ d

(2.16)

Here the * means that it is the conjugate complex number to a: If a = r + i · m, then , a ∗ = r − i · m. The object Ψ | can be written as.   Ψ | = a ∗ b∗

(2.17)

10 In principle, the following explanations of the properties of quantum bits are not new. This book

is also written for engineers who are not familiar with the formalism of quantum mechanics.

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Representation of Quantum Bits

35

From a mathematical point of view, Ψ | is the hermitian conjugate11 row vector to |Ψ . The scalar product can then be written as a matrix multiplication. The matrix multiplication then delivers the scalar product:    ∗ ∗ c = a ∗ c + b∗ d Ψ |Φ = a b d

(2.18)

In other words, ψ| can be taken as a call to form the scalar product with |Φ. The scalar product has four important properties: 1. The scalar product is zero if two vectors are orthogonal to each other, φ|ψ = 0 . 2. The scalar product of an object with itself is always real and positive, φ|φ ≥ 0 . 3. A vector is called normalized if φ|φ = 1. The normalization of a non-normalized vector can be achieved if the vector is divided by its magnitude. 4. The scalar product is generally not symmetric, it is φ|ψ = ψ|φ∗ . In addition to these properties of the scalar product, there are two more simple calculation rules: 5. Vectors of thesame  dimension   can beadded by adding the components one by one: a c a+c |Ψ  + |Φ = + = b d b+d 6. Vectors can be multiplied by complex numbers by multiplying the components one by one:     a A·a A · |Ψ  = A · = b A·b By defining the scalar product, the two-dimensional vector space C2 becomes a vector space with scalar product or a Hilbert space. In this example, we are dealing with a twodimensional Hilbert space. The dimension of a Hilbert space corresponds to the number of classical states of the system. For the representation of quantum bits C2 is accordingly suitable. A normalized element of the Hilbert space can be called a state vector or a wave function of a quantum mechanical system. State vectors that differ only in phase (multiplication by a complex number with magnitude 1) describe the same state. 11 Hermitian conjugate means that a matrix is equal to its transposed and complex conjugate matrix.

The transposed matrix of A = (ai j ) is A = (a ji ). The conjugate complex matrix of A is A∗ = (ai∗j ).

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Accordingly, a linear combination of state vectors is also a state vector of the system after normalization. This property is called the superposition principle. If |Ψ1  and |Ψ2  are state vectors, then the following is true |Ψ  = c1 |Ψ1  + c2 |Ψ2 ,

(2.19)

where c1 and c2 can be complex numbers. |Ψ  is also a possible state of the system after normalization. In the case of quantum bits this means that a linear combination of a “Yes” bit with a “No” bit is also a possible state. In classical logic, a bit can only take the state “Yes” or “No”. Nevertheless, one can determine the state of a quantum bit by a measurement. Then the system must “decide”.

2.5.2

Measurement

In quantum mechanics, the concrete determination of a state is of special importance. Simplifying we assume that from a quantum bit is to be measured whether a “Yes” or a “No” occurs. This measurement represents a measurement basis. The two base states can be represented with   0 |Y es = 1

  1 und |N o = 0

(2.20)

According to the superposition principle, both states can be mixed according to (2.19): |Ψ  = a0 |N o + a1 |Y es =

  a0 . a1

(2.21)

This means that any element of the Hilbert space can be expressed by these two basis vectors. Geometrically, this situation can be simplistically represented in two dimensions as in Fig. 2.4.

Fig. 2.4 Representation of a quantum bit in the standard base

2.5

Representation of Quantum Bits

37

Because a1 and a2 can be complex, the vector |Ψ  can move on a complex unit sphere. The representation is a special case for real a1 and a2 . More generally, a quantum bit can be represented as follows:  |Ψ  = e

iδ

 e−iφ/2 cos(ϑ/2) . eiφ/2 sin(ϑ/2)

(2.22)

For the case that φ = δ = 0 we get the representation |Ψ  =

  cos(ϑ/2) , sin(ϑ/2)

(2.23)

as shown in Fig. 2.4. The representation of a quantum bit by (2.22) suggests that there is a lot of information in a quantum bit due to the three independent variables δ, φ and ϑ. This is not so. The variables δ and φ have no influence on a measurement result. They influence the interaction of quantum bits with each other before the measurement. A bit more precisely, the absolute phase of quantum bits is not relevant, but only their phase difference. Only ϑ shifts the probability between |Y es and |N o. The normalization condition for Eq. (2.21) is a12 + a02 = 1. a12 and a02 are the probabilities for determining a “Yes” or a “No”. In the standard basis, the two basis vectors are orthogonal to each other. It is valid N o|Y es = 0. Both basis vectors are normalized, of course, Y es|Y es = 1 and N o|N o = 1. It can be shown that only one classical bit can be extracted from a quantum bit by measurement. This statement can be extended to a quantum register. A quantum register of size n is a system of n quantum bits. The theorem of Holevo states that only n classical bits can be stored in a quantum register of size n. Also only these n classical bits can be read out again. This theorem is of fundamental importance for the interface between classical information technology and quantum systems. Now, if a measurement is made on the quantum object |Ψ , the probability that • a “Yes” is determined, equal to P(“Yes”) = a12 , • a “No” equals P(“No”) = a02 . The measurement changes the quantum object itself. A state vector is assigned to each of the possible measurement results, and these state vectors are orthogonal to each other. They form a measurement basis. It is said that during a measurement the wave function collapses. The process is also called decoherence. After the measurement, the quantum object is in the “Yes” state (|Y es) or in the “No” state (|N o). Figure 2.5 illustrates the process of measurement.

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Fig. 2.5 Representation of a measurement on the quantum bit |Ψ . The result is a quantum bit |Ψm . The measurement basis here is the so-called standard basis

2.5.3

HADAMARD Transform

Another possible basis can be generated by the Hadamard transformation. The transformation H 12 can be applied to quantum bits as follows:  1  H (|0) ≡ H |0 = √ |0 + |1 2  1  H (|1) ≡ H |1 = √ |0 − |1 . 2

(2.24)

This new basis can be defined by applying the Hadamard transformation to the basis vectors of the standard basis. The new basis vectors H1 and H2 are defined by |H1  = H |N ein |H2  = H |J a

 1  = √ |N ein + |J a 2  1  = √ |N ein − |J a 2

(2.25)

Both vectors are orthogonal to each other; the scalar product H1 |H2  is equal to zero. The basis vectors are rotated with respect to the standard basis by 45◦ , as shown in Fig. 2.6. The Hadamard basis is a measurement basis and is in principle equivalent to the standard basis. It has a physical interpretation. The Hadamard transformation is suitable to describe a beam splitter. A beam of light is split at a semi-transparent mirror. For simplicity, let it be a lossless 50/50 mirror. One part is reflected, the other part passes through the mirror, as 12 H is also used here as operator.

2.5

Representation of Quantum Bits

39

Fig. 2.6 Representation of the Hadamard basis versus the standard one. |H1  and |H2  form an orthogonal basis

shown in Fig. 2.7. Note that there is a phase jump of 180◦ when reflecting from the optically denser medium. The transformation is easily recognizable. If only one |0in is taken as input, the result is two partial beams with the same phase. However, if |1in is taken, the result is a |0out and a |1out that is shifted in phase by 180◦ and is therefore negative. If the Hadamard transformation is performed twice in succession, it cancels out, it is |Qout = H H |Qin = |Qin.

(2.26)

The series arrangement of two transformations according to Eq. (2.26) is shown in Fig. 2.8 and is physically a Mach- Zehnder interferometer [8]. If a phase shifter Φ is additionally inserted into the beam path, it can be used to shift the distribution of probabilities for the output bits. Thus a hardware is created, with which quantum bits can be manipulated. This is a possible basis for future optical quantum computers. By the help of the Mach- Zehnder interferometer the nature of the quantum bit as a wave becomes visible. The phase position at the second mirror influences the result significantly. Interesting is the fact that a bit or photon hitting a mirror is split. More precisely, the wave

Fig. 2.7 Illustrative representation of the Hadamard transform on a beam splitter

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Fig. 2.8 Two Hadamard transformations form a Mach- Zehnder interferometer

function splits into two parts. Both halves can interfere again at the second mirror. One photon does not travel along the upper light path and another along the lower. That would mean a doubling of a photon, which would already contradict the conservation of energy. There is also not half a photon running on top and the other half on the bottom. If one wanted to know where the photon is, a measurement would have to be carried out. If, for example, it is determined at the lower light path that the photon is there, then it will not be in the upper light path at this moment. The wave function is destroyed by the measurement. Of course, interference at the second mirror can no longer take place. It is remarkable that the measurement of the photon takes place after the division of the photon in the first mirror. The upper half is already on its way during the measurement in the upper path. It has already passed a part of its way. Nevertheless, when the bit is identified as being in the lower path, the upper half is destroyed. The lengths of the paths, which the two parts have already traveled, do not play a role. Practically a long-distance action takes place, which is experimentally proved. The two parts of the wave function belong to a wave function which exists as a whole or not. The quantum mechanics is not a local theory. The basis vectors of the Hadamard basis can define a measurement. The basis vectors are, as it were, pure states which the system can take after a measurement. The systems |Y es and |N o are abstract quantum systems. Which physical systems and which measurements can be hidden behind them?

2.5.4

Polarization of Photons

First, an example shall be considered. Let the information, i.e. the quantum bit, be “encoded” in the polarization of a photon. It is practical to rename the states: • |Y es =⇒ |V  (vertical polarized) • |N o =⇒ |H  (horizontal polarized)

2.5

Representation of Quantum Bits

41

Fig. 2.9 Analysis of the polarization state of a photon

A photon to be measured is now sent through polarizing filters. The polarization angle of the filter θ is 0◦ , if the polarization plane is horizontal. Behind it is a photon detector. In the arrangement in Fig. 2.9, a rotation of the polarization filter by θ = 45◦ is shown. The polarizing filter can be rotated by the angle θ . Thus, if θ = 0◦ , all horizontally polarized photons will pass through the filter. For any angle θ , the probability for horizontally polarized photons to pass the filter is equal to cos2 θ , for vertically polarized photons it is equal to sin2 θ . For θ = 45◦ , both probabilities are 50%. This case would correspond to the basis according to (2.24). If a bit is to be analyzed without loss, i.e. without absorption at the polarization filter, a Glan- Taylor polarization beam splitter can be used. Then a second detector can be used to determine whether a photon with opposite polarization has also arrived. What are relations concerning energy and time at the detection of the photon? In the Sect. 2.4.3 the relation of energy and time for the detection has been described. In the case of a photon, the energy depends on the frequency ν of the photon. The energy is E = hν =

h . τ

(2.27)

The absorption of a photon, for example, by excitation of an electron in a silicon atom of the detector, takes place in a time τ which is about τ = 1/ν. τ is to be regarded here as transaction time for the bit. The similarity of Eq. (2.27) to Heisenberg’s uncertainty relation (2.2) in Chap. 2.4 is evident.

2.5.5

Systems of Two Quantum Bits

As an example a system is explained, which consists of a system A, it is the system of Alice. The second system B is the system of Bert. Both systems have their own Hilbert space and both systems can take the states |0 and |1 and their superpositions. Let Alice’s qubits be called |0 A and |1 A , and Bert’s be called |0 B and |1 B correspondingly. The state of the overall system is expressed by writing the states side by side. A possible state can be |0 A and

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|1 B . There are 4 possible states of the total system. Based on the superposition principle, any linear combination of these 4 possibilities can also be a state of the total system |Ψ : |Ψ  = a · |0 A |0 B + b · |0 A |1 B + c · |1 A |0 B + d · |1 A |1 B

(2.28)

The normalization condition is |a|2 + |b|2 + |c|2 + |d|2 = 1.

(2.29)

States with mixed states become interesting. On the one hand, Alice, for example, can take a mixed state that mixes her two basis vectors: |Ψ  = √1 (|0 A + |1 A )|1 B . So far Alice 2 and Bert can determine their states themselves. However, mixed states of type 1 |Ψ E P R  = √ (|0 A |1 B − |1 A |0 B ) 2

(2.30)

are also possible. This is the famous EPR or spin singlet state13 . Because Alice and Bert cannot generate this state by their own operations, it is called “entangled”. Thus, a state is entangled if it cannot be written as a product of a state vector of Alice and a state vector of Bert. The states |0 A |1 B or √1 (|0 A + |1 A )|1 B are thus not entangled. 2 The computational rules for handling two-qubit systems are simple: Linearity A linear combination in a subsystem can be transferred to the overall system. Brackets can be multiplied out. For example 1 1 1 √ (|0 A + |1 A )|1 B = √ |0 A |1 B + √ |1 A |1 B 2 2 2

(2.31)

Independence of operations in the subsystems. The procedure for forming inner products is as follows: A 0| B 1| × |0 A |1 B

= A 0|0 A × B 1|1 B = 1 × 1 = 1

(2.32)

When performing measurements on the overall system, Alice and Bert, for example, can use their standard base. Then the following measurement results are possible: |0 A |0 B |0 A |1 B |1 A |0 B |1 A |1 B

13 Einstein-Podolsky-Rosen effect.

(2.33)

2.5

Representation of Quantum Bits

43

Now, when the system is in the EPR state, the following probabilities for the measurement results will occur: | A 0| B 0|Ψ E P R |2 = 0 1 | A 0| B 1|Ψ E P R |2 = 2 (2.34) 1 2 | A 1| B 0|Ψ E P R | = 2 | A 1| B 1|Ψ E P R |2 = 0 The result is characteristic for the EPR state. Alice and Bert can never measure the same bit. If Alice measures |0, Bert can only measure |1 and vice versa. As an application, 2 spin systems can be considered. Both systems can have spin 1 or −1. But both systems cannot take the same spin. The sum of the spins must always be 0. This is exactly what the EPR state realizes. Finally, it should be mentioned that the measurement basis can also consist of four entangled states. In this case, the measurement cannot be performed by local measurements in the subsystems, but only on the whole system. An example would be the so-called Bell basis [19].

2.5.6

Qbits—Description with the Formalism of Automata Theory

Quantum bits as objects of quantum mechanics can be represented as abstract classical automata. Thereby all properties relevant for quantum bits are represented. The quantum bits are represented as generalizations of simple classical automata [111]. The formalism of quantum mechanics can be dispensed with. Thus, it is not the behavior of particles that is represented, but the behavior of abstract qbits. The disadvantage is that continuous quantum objects cannot be represented. For quantum information this disadvantage is not essential for the time being. The main advantage is the simpler description of the properties of quantum bits. At the same time, the fundamental interpretation problems of quantum mechanics are not lost. The procedure is to be outlined briefly. Basically, finite and deterministic automata are assumed, which transform a given initial state into a defined final state by a transformation. Several transformations can be executed successively and finite automata can be combined to complex systems in order to execute more complex operations. One can think of this as interconnecting gates to form a computer. The automaton should be able to take N states. The set of all possible N states is called state space Z n . N is often written in powers of two: N = 2n . This is convenient when dealing with systems with n qubits. The states should be written in a redundant way as column vectors.

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⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 1 ⎜0⎟ ⎜1⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎟ |0 = ⎜ ⎝...⎠ , |1 = ⎝...⎠ , ..., |N − 1 = ⎝...⎠ 1 0 0

(2.35)

Again the ket notation was introduced. The N -vectors can form basis of a vector space C N with the elements |α or |β. Functions are to be represented as matrices. An input vector |α is transformed into the output vector |β by matrix multiplication: |β = M |α

(2.36)

A system that can take only 2 states (state space Z 2 ) shall be called Cbit (classical bit). The representation is analogous to Eq. (2.20):   0 |1 = 1

  1 |0 = 0

und

(2.37)

An AND gate can be defined as follows AN D|00 = |00 AN D|01 = |00

(2.38)

AN D|10 = |00 AN D|11 = |01 The associated matrix is then: ⎛

1 ⎜0 |β = AN D |α = ⎜ ⎝0 0

1 0 0 0

1 0 0 0

⎞ 0 1⎟ ⎟ |α 0⎠

(2.39)

0

Here it is to be noted that because of symmetry the result was supplemented with a bit (see 2.38), the first bit of the result is always 0. The AND gate is a transformation. The matrix realizes a transformation function. Like NAND and OR, this gate is not reversible. There are also reversible logical operations, like negation. But in the AND operation, the input is not reproducible if the output is known. This contradicts the reversibility of quantum mechanical processes, but is not a problem for classical logic. However, it is possible to build all logic from reversible operations. Logical reversibility can be shown by means of the Toffoli gate T . The gate is a three-bit gate and realizes the following function

2.5

Representation of Quantum Bits

45

Fig. 2.10 Schematic representation of the Toffoli gate and its use to realize a reversible NAND

T |000 = |000 T |001 = |001 T |010 = |010 T |011 = |011

(2.40)

T |101 = |101 T |110 = |111 T |111 = |110 The gate realizes a reversible NAND for the special case c = 1. This is immediately obvious from (2.40). Since any logic can be built from NAND gates, it is also shown that any logic can be built reversibly using the Toffoli gate. However, an additional and redundant bit is required. This reversibility of logic is important because until the 1980 s, it was assumed that input data could not be reproduced from the result of logical operations, that logic was in principle irreversible. It is interesting that quantum properties have not been directly included in the previous considerations of abstract automata (see Fig. 2.10). Automata can be composed. In classical logic, for example, arbitrary systems can be built only by combining NAND gates. When systems are combined, states are also combined. Two systems with state spaces Z N and Z M can be combined into a state space Z N ,M that has N · M states. The transformations can be performed sequentially. This would correspond to a series connection of systems or gates. The transform functions of combined systems can also be formed as a product of the partial functions. Up to now only classical bits were represented. How are Qbits represented now? For this the following extensions must be made [111]: 1. The states of a Qbit are represented by vectors of a vector space C2 . If n Qbits are n combined, they are represented in the vector space C2 . The basis vectors of the standard basis |0 and |1 are the possible states of a Qbit. 2. The transformations are reversible linear transformations that preserve the scalar product (unitary transformations or matrices). All reversible transformations that are possible on Cbits are also possible on Qbits.

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3. All unitary transformations are accepted as operators. These are all linear transformations which transform orthogonal bases back into orthogonal bases. Any unit vectors can occur as states. The points 1. and 2. describe also Cbits, the third point is new for Qbits. The following points are added: 4. Only unitary transformations are accepted. 5. As a consequence, arbitrary unit vectors can appear as states. The state vectors are linear combinations of the basis vectors. This is the expression of the superposition principle for states. Which transformations are possible and useful with Qbits? The classical Cbit transformations of the gates • • • • •

I -gate: leaves the state unchanged, X -gate: inverts the state, C X -gate (CNOT): controlled inversion, T (Toffoli) gate: can be used as a NAND gate, U f -gate: realized functions

are unitary like their products. For Qbits further transformations are possible, like the already known Hadamard transformation. So with Qbits arbitrary discrete quantum systems can be composed. They are abstract automata. The representation is formally similar to the representation for quantum bits, but simpler. The state space corresponds to the Hilbert space for functions.

2.5.7

No-cloning Theorem

For handling with quantum bits it is important to know that quantum bits in general cannot be copied or “cloned”. It can be shown that cloning leads to contradiction. To show this, let a qubit |ψ = c0 0| + c1 1| be copied by a cloning function C. The cloning function has the following effect: C|ψ|β = |ψ|ψ

(2.41)

The cloning operator must return the following result for the base states: C|0|β = |0|0 C|1|β = |1|1

(2.42)

2.5

Representation of Quantum Bits

47

These relations are now substituted into the left-hand side of (2.41). First (2.41) is multiplied out: (2.43) C|ψ|β = C(c0 0| + c1 1|)|β) = C(c0 |0|β + c1 |1|β) and now (2.42) is inserted: C|ψ|β = c0 |0|0 + c1 |1|1

(2.44)

But the right side is expanded:

=

C|ψ|β = (c0 |0 + c1 |1)(c0 |0 + c1 |1) 2 c0 |0|0 + c0 c1 |0|1 + c1 c0 |1|0 + c12 |1|1

(2.45)

Now left side (2.44) should be equal to right side (2.45), but in general this is not true for any c02 + c12 = 1. This means that the cloning operator in general leads to the contradiction. Any qubit cannot be cloned. However, for the case where c0 = 0 or c1 = 0, (2.44) is equal to (2.45). This means that cloning is possible for classical basis states |0 or |1 (corresponding to (2.42)).

2.5.8

Quantum Computing

Quantum computers are handling quantum bits and quantum information. In general, quantum information is the information present in quantum mechanical systems. It cannot be described by the laws of classical information theory. The theory of quantum information must build the foundations for quantum computers, quantum cryptography, and other quantum information technologies. To take advantage of quantum bits in computers, computers must be built to work with quantum bits. The hardware for quantum computers is fundamentally different from today’s classical computers. In 2000, D. DiVincenzo [14] formulated requirements that a universal quantum computer should fulfill: 1. There must be a scalable physical system with well-characterized quantum bits. 2. The system must have the ability to initialize states of quantum bits before the computer starts. 3. The decoherence time should be much longer than the operation time of the gates. Decoherence should be negligible. 4. The system must contain a sufficient set of universal quantum gates that can process a sequence of unitary transformations. 5. Measuring devices for determining the states of quantum bits must be available.

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The next two requirements were added later: 6. The system must have the ability to convert stationary and movable quantum bits into each other. 7. Moving quantum bits must be able to be transferred between different locations without falsification. From a technical point of view, the requirements are rather fundamental and not sufficient. For example, requirements regarding the error probability in the transmission of quantum bits are missing. In this context, a quantification of the requirement against the decoherence time is necessary. This quantitative information will be essential for the type of error handling, the fault tolerance of algorithms and measures to implement redundancy in the hardware. Only one thing is likely to be certain: The result of a calculation must be correct. Many of these requirements can be realized in the laboratory, but it will probably be decades before practicable universal computers are realized. In the field of quantum cryptography, practicable solutions seem more feasible. So where are the advantages of quantum computing? First, some technical advantages shall be mentioned: 1. Handling quantum bits will significantly increase energy efficiency. Instead of using thousands of quantum bits to represent one bit, only energies of one order of magnitude given by the uncertainty relation (2.2) will be used. In the illustration of Fig. 1.2 and Fig. 4.4 one will be on the quantum limit or at least close to it. 2. The processing speeds will be close to what is quantum mechanically possible. An example would be photonic systems. Photons travel at the speed of light. 3. Quantum systems operating at room temperature must prevail over the thermal noise of the environment. As can be seen from Fig. 4.4, only relatively high-energy quanta can be considered at room temperature because they would otherwise be destroyed by the thermal energy of the environment. These systems can only be fast then because of ΔEΔt ≈ h, even as electronic systems. 4. Quantum systems can handle mixed states. Thus the wave function becomes the object of the quantum computer. By means of the so-called Deutsch algorithm14 it can be exemplarily shown how quantum algorithms take advantage of the specific properties of quantum bits (see [111]). In classical computing, it is “only” the basis vectors that are used. Practically every gate realizes a decoherence. The properties of the wave function are not used in general. In quantum computing, in addition to the technical advantages, considerable speed advantages are expected in connection with point 4, because quantum bits can be processed as 14 The algorithm named after the English physicist David Deutsch.

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Representation of Quantum Bits

49

mixed states and thus massive parallel computing can be used for certain tasks. Simplified, it is practically possible to compute with probabilities that would have to be determined by repeating the algorithms in classical computing. The road to usable, manageable and, above all, universal quantum computers is still very long for engineers.

2.5.9

Physical Realizations of Quantum Bits

In the Sect. 2.4.1 the dynamic information has been introduced heuristically on the basis of a rather abstract example. The quantum bit and its properties played a significant role. The quantum bit has not been associated with a concrete physical object so far. Depending on the further conditions, even simple quantum objects can be “composed” of several or very many quantum bits. Four examples of an encoding of a quantum bit into a physical object shall be mentioned, which are also of practical importance: Polarization of a photon The information is encoded in the polarization direction of an electromagnetic wave, a photon (see also Sect. 2.5.4). For example, a horizontal polarization could be denoted by |H  and a vertical polarization by |V . For the measurement this means: At a certain time a photon comes unconditionally on a measuring device, the measuring device detects the polarization direction. As measuring device could be used a polarization filter with following photon detector (see Fig. 2.9). Presence of a photon Information can be transmitted, for example, by detecting or not detecting a horizontally polarized photon at a given time. State of an electron in an atom or ion A bit can be encoded in two special states of an electron in an atom or ion. Usually the ground state is one state of the quantum bit, for example, |0 and the excited state is the other state |1. As an example, let us mention the encoding of a quantum bit in electron states of the 40 Ca + ion. The calcium ion is useful in many ways for experiments in quantum optics. The term scheme is shown in Fig. 2.11. For encoding, the states 42 P1/2 with |0 and 32 D5/2 with |1 of the 40 Ca + ion can be called [40]. These states are suitable because they have a natural lifetime of about 1 s. This is a very long lifetime compared to the other states shown in Fig. 2.11, which have a natural lifetime around 10−8 s. As shown in Fig. 2.12, the readout of the quantum bit can be done via the fluorescence of the states 42 S1/2 − 42 P1/2 . If the electron is in the state |0 at the level 42 P1/2 , fluorescence is possible, in the state |1 at the level 32 D5/2 fluorescence is not possible, the ion remains dark. Spin of an electron A quantum bit can be encoded into the spin of an electron. The measurement of the spin could in principle be done on atoms using a Stern- Gerlach arrangement.

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Fig. 2.11 Term diagram of a 40 Ca + ion

Fig. 2.12 Term diagram for the encoding of a quantum bit into the states 42 P1/2 with |0 and 32 D5/2 with |1 of a 40 Ca + ion

Fig. 2.13 Term scheme for the encoding of a quantum bit into spin states

In the 40 Ca + ion, the Zeemann substates of level 42 P1/2 can be used. The term scheme and a possible encoding is shown in Fig. 2.13. In the detection of a quantum bit, the measurement basis plays a critical role. When measured, the quantum bit splits into a component of the measurement basis. Of special interest are measurements which are not performed with the standard base. By rotation of the measuring apparatus, rotation of the polarizer at the photon measurement or of the magnetic field at the Stern- Gerlach measurement, the

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measuring base can be changed. With this rotation, the measurement base also rotates in Hilbert space. An example would be the Hadamard measurement base.

2.6

Properties of Dynamic Information

2.6.1

Dynamic Information and Quantum Bits

In the Sect. 2.4 the concept of dynamic information was justified on the basis of a phenomenological description of a transmission process. Now a more general justification for the dynamic information concept shall be given. If one recognizes the quantum bit as the smallest unit of information, then the energy and time behavior is thereby described by the Eq. (2.3). The quantum bit and thus the information is to be considered accordingly always in the context with energy and time. The quantum bit can be considered as an elementary component of all finite objects. Quantum mechanical objects can be described by wave functions. The wave functions can be represented in a Hilbert space. A Hilbert space is a vector space spanned by basis vectors. According to a fundamental theorem of quantum theory, which is essentially based on the spectral theorem of functional analysis and goes back to Feynman, every wave function is representable as a superposition of quantum bits. That is, given a suitable choice of the basis of Hilbert space, any wave function can be represented as a sum or superposition of quantum bits. Vice versa, any wave function can be decomposed into quantum bits or evolved according to quantum bits. So it is allowed to imagine the world as a sum of quantum bits. There “energetic” objects are explicitly included. This also means that every system can be considered as an information system. In this context we should refer to the ur hypothesis or ur theory of Carl Friedrich von Weizsäcker [95], which is quite close to this idea. Von Weizsäcker assumes that all objects of this world are built up of so-called “ur objects”. These primordial objects are elementary decisions, which are answerable with “yes” or “no”. The information of an event is the number of its undecided primal alternatives. The ur alternatives express the form of an event. A short description of the facts can be found in [26]. Elementary particles need not be identical with Weizsäckers “ur objects”. Similarly, a classical elementary particle or a classical bit need not be identical to a quantum bit. A quantum bit is often a rather abstract part of a wave function. In Eq. (2.6) the entropy S gives the number of transferred quantum bits, the transferred energy is determined by Δt. The number of quantum bits can change in an open system. The annihilation and creation of particles is possible. For example, photons can be created and annihilated in a photon gas. In closed systems, the annihilation and generation processes of quantum bits should balance when the system is in equilibrium or is close to equilibrium.

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In this more general description of dynamical information, the restriction of synchronous transmission that led to (2.3) can be abandoned. The “decomposition” of a wave function into quantum bits is not linked to temporal conditions. Some basic questions seem to lead to a contradiction. An explanation of the irreversible entropy increase in classical, thermodynamic and closed systems is not possible in this context or not possible without additional assumptions. New quantum bits would have to be created. The energy for this would certainly be available. It could be provided by slowing down other quantum bits. Are such processes possible? Where should a new quantum object come from? If the entropy should increase, then this must be independent of other quantum bits. The theorem of Holevo also rules out the creation of new bits in any quantum mechanical processes in a quantum register. Extending the notion of quantum registers to arbitrary finite objects, the creation of new bits should also be impossible in other subsystems. An irreversible increase of entropy in a closed system is not imaginable with elementary processes, because all processes which can be described by the Schrödinger equation are reversible, also the logical operations. The world of quantum computing is reversible, at least if one imagines a domain (of any size) that is finite and closed. Measurement on quantum mechanical objects could bring irreversibility, but it means an intervention in the system. Then the system is no longer closed. Some questions remain to be answered, especially for the practical handling of quantum bits. What is the meaning of the seemingly “informationless” extra bits in parallelization and serialization? Is there any similarity with the extra bits in reversible logic? Certainly, the notion of a “bit” needs to be looked at more exactly. A bit is an “open” yes/no decision. It can also be considered as an elementary unit of entropy.

2.6.2

Signal and Information

A classical approach to information transmission distinguishes between signal and information. So far, the explanations have not spoken of a separable, informationless signal and the actual information. In many considerations a signal is regarded as a carrier of the information. As already mentioned, Ebeling sees [16] the carrier and the carried as a dialectical unity. In light of quantum mechanics, this separation must be questioned. If one transmits a portion of energy with a minimal effect (a quantum bit), nothing else is transmitted except this energy. The fact remains that only an energy portion was transferred. Physically there is no energy for any additions or additional information. These additions would have to be placed in a new quantum bit. A quantum bit has no attributes, because these would be additional information. There is not even any information about the information. The signal is the information. If this is so, the concept of the signal is not needed. However, the conceptual separation of signal and information is useful in classical mechanics. Here an observer can in principle follow the motion of signals without restric-

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tions and have knowledge about information and information flows. This is based on the condition that the energies necessary for the observation or measurement of the signal or information flows are very small compared to the energies of the signal transmission processes and therefore the observation does not influence the signal transmission process. The relationship between signal and information is often defined by the receiver, who extracts information from the signal through interpretation. The meaning of the signal is determined by the receiver. If the information processes are objectified, only signal or information transmission takes place. The transmitted energies change the receiver. Interpretation takes place as an action of the energy portion on the receiver. The energy portion will have influences on the state and further course of the processes in the receiver. In unstable receiver systems very small energies can also cause large effects. This case corresponds to the everyday understanding of information. A statement containing only a few bits can change the world, even start wars. Very little energy can trigger many energy transformations. Even if the energetic relation between the occurring effect and the triggering information can be very large, this is no reason to consider the information as energy-less. It is basically irrelevant whether the receiver is a digital electronic system, a human brain consisting of billions of neurons or any physical system. In this objective approach, the distinction between signal and information is unnecessary. Therefore, only the concept of information will be used in the following text.

2.6.3

Valuation of Entropy by the Transaction Time

Information has been defined as a dynamic quantity by its very nature. This is also compatible with the everyday understanding of information. This says namely that fast information is more valuable than slow information. Each bit also has its own time in the sense of a relaxation time in which it can move or, more precisely, in which it can be transmitted. What is the use of a bit today, which needs 10,000 years for its transmission? It may very well exist, but it is of only minor use. The faster a bit is, the more valuable the information. The importance of transaction time for information is illustrated by the example of data transmission between stock exchanges in London and New York. In order to have time advantages over the usual transmission routes via satellite or public cables across the Atlantic, a direct fiber optic cable link was installed between the two exchanges at a substantial cost. The time advantage of parts of a second is obviously worth a lot of money. Now, it is a charm of the concept of information introduced above that the bit is divided by its own transaction time. The factor 1/τ could also be taken as a valuation factor for the usefulness of a bit.

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2.6.4

2 Information—Physically and Dynamically Based

Valuation of Entropy by Energy

With the help of the uncertainty relation between energy and time, the valuation of a bit by the transaction time can also be converted into a valuation by energy. In the Eq. (2.8), (2.3) can be inserted. 1Bit = 1BitΔE (2.46) ΔI =  Δt A system can consist of a number of dynamic information units, or dynamic bits for short. Like the energies, the dynamic bits are added: I = Bit1 · ΔE 1 + Bit2 · ΔE 2 + Bit3 · ΔE 3 + ... =

N 

Bitn · ΔE n .

(2.47)

n=1

This equation describes the composition of a complex system of dynamic information units with non-unitary energy and without interaction among themselves. In thermodynamic systems, after sufficient time, the energies would follow a probability distribution corresponding to the character of the interaction. How does the interaction between dynamical bits present itself? To clarify this question, it is necessary to consider the dynamical bits as a component of physical systems. Basically, energy, momentum and angular momentum can be exchanged. Thereby the conservation laws of these quantities must be considered. Provided that the dynamic bits are conserved, the energies can redistribute among themselves if only the sum of the energies is conserved. A conservation of the number of dynamic bits cannot be postulated without further ado in general and in particular in open systems. The reason is that elementary particles can be created and annihilated. To illustrate the interaction, consider a quantum system, such as a harmonic oscillator, which is excited by photons, for example. The initial state has the energy E 0 . As shown in Fig. 2.14, a photon shall excite the oscillator. The photon has the energy E 1 . The energy of the excited state is E 0 + E 1 . After excitation, the photon no longer exists. The bit of the photon goes to the oscillator and takes a new higher state there. The energy difference must of course fit to the energy of the photon. The balance looks now as follows: 1Bit Photon · E 1 + 1Bit Oszill0 · E 2 = 1Bit Oszill1 (E 1 + E 2 )  1Bit Oszill0  Bit Ozsill1  1Bit Photon + = Δt Photon Δt Oszill0 Δt Oszill1

(2.48)

This could look like the annihilation of a bit. For the simple case that E 1 = E 2 = E would result:  1Bit Photon  1Bit Oszill0  Bit Oszill1 + = Δt Δt Δt/2 (2.49) 1 Bit Photon · E + 1 Bit Oszill0 · E = 2 · 1 Bit Oszill1 · E

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55

Fig. 2.14 Conversion of a dynamic bit by excitation of an oscillator. The photon is destroyed

However, it must always be kept in mind that a bit is defined only on a probability distribution. In a statistical system, the additional energy in the quantum system will create entropy. The new high energy bit has more possibilities to take states, and therefore more entropy. From this point of view, the Eqs. 2.47 are not to be taken for the single process of an excitation, but for a statistical process where the quantum system should have several particles so that energy exchange becomes possible. Another example would be the two-photon excitation of a state in an atom or molecule. After recombination, two photons of lower energy would net to one photon of higher energy. If no further losses occur in the process, the resulting photon carries away the sum of the energies of the two annihilated photons. For the momentum p the same is valid because of p = E/c. Thereby two “low-valued” photons are made into one “higher-valued” photon. Since the “higher-order” photon has more energy, thus would correspond to a higher temperature, it also has more entropy. Of course, in such a case one cannot speak of temperature. The Wien displacement law could be consulted for understanding. Because of the reversibility of quantum mechanical processes, the inverse process is also possible. In general, particles can be divided to create new dynamical bits. Thereby the correlation of the resulting bits has to be discussed, because these are not completely independent, but can be randomized is statistically described systems. In summary, it can be stated that the introduction of dynamic bits results in a valuation of classical information bits. Faster bits have a higher value, which can be measured in energy units. The price for the higher valuation is just the additional expense of energy.

2.6.5

Objectivity of Energy and Information

By means of two examples the objective character of the defined concept of information shall be explained illustratively. It is about the question, what is considered as information and what as energy.

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Fig. 2.15 Inner photoelectric effect in pn junction shown in band model; conversion of light into pair of electric charges

Example: Inner photoelectric effect In information and energy transfer, the Inner Photoelectric Effect plays an important role in a pn junction. A photon impacts a pn junction, an electron-hole pair is generated and a momentum is created in the potential gradient of the pn junction. Is this an information transfer? In a CCD camera, an image is generated from the light; this is generally information. In a photovoltaic system, the same process is used to convert energy (see Fig. 2.15). Neither the process flow nor its characteristics provide information about whether it is information that is flowing or energy. Only the further technical environment allows such an evaluation of the process. The evaluation is subjective. Example: Gas bubble in Greenland ice If the concept of information is decoupled from the subject, then everything that is material is information. At first sight this sounds unusual. An arbitrary example shall illustrate this. Some gas bubble in Greenland ice is to be considered. Is the chemical composition of the trapped gas information? If, as defined by C. F. von Weizsaecker15 , information is only what is understood, the composition of a gas bubble would not be information. Objectively, this is information. For, when a drill brings the environment of the gas bubble to the surface and ultimately into a mass spectrometer of a laboratory, this can produce a mass spectrum that is generally regarded as information and whose content is processed in a computer. Ultimately, information-processing processes are constantly taking place around us and also within us, regardless of whether we take note of them. Information of a system The indeterminacy of a bit sequence can be specified as its entropy and it can be calculated and measured how much energy has to be transferred for its transmission, if the time allowed for it is known. A certain energy does not realize a certain entropy in itself, but its transport

15 See Sect. 3.7 “Interpretations by Carl Friedrich von Weizsäcker”.

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or its movement. An energy is confronted with a quantity which has the unit of measurement bit per time unit. This quantity can describe the entropy flow in a channel (channel capacity) or the entropy flow within a system. The state of the system can be described as a point in phase space. The entropy flow within the system is the amount of information an observer needed to know about the state of the system as a trajectory in phase space. The transaction time Δt is then a unit of time during which the state does not change significantly. More precisely, in terms of the sampling theorem, it is the maximum time between measurements of the state that may elapse in order to describe the trajectory “lossless” by sampling. The entropy per unit time thus defined is the amount of information that an observer would have to receive in order to be constantly informed about the state of the system. In analogous systems it is the entropy of the course of the curve which describes the system in phase space. In clocked systems, the transaction time τ is the clock time and the entropy is the measure of the indeterminacy of a state. After each cycle time the state changes. The quantity entropy per time τ describes the information flow within the system. The following point of view suggests itself: The system sends itself the new state from one clock time to the next, where (2.7) holds. The unit of time is the sampling time. Physically it can also be defined by (2.2). It is important to note that entropy per unit time can be the information transferred from one system to another, i.e. across the system boundary. Similarly, transmission processes take place within systems, which must also be regarded as information transmission. However, here the sender and receiver are in the same place, they are identical. In clocked technical systems there is a cycle time. In natural thermodynamic systems the processes run asynchronously. For a better understanding, one could think of the system under consideration as being composed of many subsystems communicating with each other. In the case of the internal information transfer of a system, one should speak of the information of the system. Just as one can speak of the energy of a system. Accordingly, the information is like the energy a state variable of a thermodynamic system. In the end it is the same. In this view, a single crystal has little information (it is easy to describe), while an amorphous body or a gas has a lot of information. There is chaos. That is, the amorphous solid has a lot of information and therefore can export information (see Sect. 6.2 “Formation of structure”). This would only be possible if order is established within the exporting system. But what is order?

2.6.6

Conservation of Information

Principle of conservation of energy and conservation of information The definition of information as a dynamic quantity has the charm to be coupled with energy in the closest way and allows the formulation of an information conservation law based on the energy conservation law. It is essential that no restrictions are made with respect to

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the sender and receiver. Although in Sect. 2.4.1 Alice and Bert were considered, both can in principle be replaced by technical or other natural systems. Therefore the difference in principle between information and energy transmission disappears. If there are no differences, both quantities, energy and information, can also be regarded as equivalent or identical. Here lies a high relevance for philosophical and especially epistemological considerations. The conservation of dynamical information and its illustration is the subject of the following sections. Of course, in all investigations it must be taken care that the conservation laws are valid only for closed systems, i.e. for systems which have no interaction with other systems or the environment. Flow density of probability in the quantum mechanics The fundamental laws of quantum mechanics contain a very essential conservation law. It is about the conservation of the probability density of the wave function during the time evolution of a quantum object. The property of conservation of probability density is called “unitarity”. By unitarity is understood in quantum physics the “conservation of the normalization” of states. Simplifying one can speak of the “conservation of probability”. Unitary time evolution means: As long as there is no measurement, states orthogonal to each other always change into orthogonal states. To characterize a quantum object, the wave function Ψ (x, t), whose value depends on the location x and the time t, can be used. For the location x is written here; the following considerations can be extended to three-dimensional functions. According to the Copenhagen interpretation, the square of the wave function |Ψ (r , t)|2 can be interpreted as the probability density P(x, t). Because the object must be somewhere, and exactly once, the sum of all probabilities over the complete space must be 1. The particle must be somewhere with 100% probability. The normalization condition must hold:

|Ψ (x, t)|2 d x = 1 (2.50) This normalization condition can be easily achieved for an initial state by multiplication with a constant. What about the normalization condition in future points in time? The system evolves with time, taking on other states. It can be shown that the normalization condition holds for all times if it is satisfied only for the initial state t = 0 [28]. To demonstrate this, the basic equation of quantum mechanics is used, the Schrödinger equation. ∂Ψ (x, t) 2 ∂ 2 Ψ (x, t) =− ∂t 2m ∂x2 For the first derivative of the probability density P(x, f ) we get i

(2.51)

2.6

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59

∂ ∂Ψ ∗ ∂ P(x, t) ∂Ψ ∂ |Ψ |2 = (Ψ ∗ Ψ ) = = Ψ + Ψ∗ (2.52) ∂t ∂t ∂t ∂t ∂t If the right-hand expressions are converted into derivatives with respect to x by using the Schrödinger equation and the conjugate complex equation, we get:     ∂Ψ ∂ P(x, t) ∂ ∂Ψ ∗ Ψ∗ =− − Ψ (2.53) ∂t ∂x 2 i m ∂x ∂x If now a probability current density j(x, t) is defined as follows:    ∂Ψ ∂Ψ ∗ j(x, t) = Ψ∗ − Ψ , 2i m ∂t ∂t

(2.54)

so is valid ∂ ∂ P(x, t) + j(x, t) = 0 (2.55) ∂t ∂x This is the continuity equation for the probability current density. If P(x, t) decreases in time at location x, a corresponding current must leave point x. The conservation law (2.55) is preserved if a term for a real potential V (x) in the form V (x)ψ(x, t) is added to the Eq. (2.51). This shows that the probability density is conserved as a whole. This continuity equation for the probability current density states that quantum objects or even parts of them cannot disappear. It also says that in the process of the development of a quantum system no new components can be added, if parts do not disappear at the same time in the same measure. For information, this means that the number of quantum bits in a system is conserved. This statement is extendable to systems consisting of several particles. If a system consists of N particles, these are described in the one-dimensional case by a wave function of the shape Ψ (x1 , x2 , ..., x N ). The quantity |Ψ (x1 , x2 , ..., x N )|2 gives the probability density for the particle 1 to be encountered at x1 , particle 2 at x2 , and the N -th particle at x N . For particles that do not interact, the wave functions can be multiplied. It is simply Ψ (x1 , x2 , ..., x N ) = Ψ1 (x1 )Ψ2 (x2 )...Ψ N (x N ). Analogous to Eq. (2.50), the normalization condition must also apply here:

|Ψ (x1 , x2 , ..., x N )|2 d x1 d x2 ...d x N = 1. (2.56) In analogous way also the Schrödinger equation can be written. According to the Eqs. (2.52) to (2.54) a conservation law can be derived analogously (2.55). Also for N -particle systems the probability density is conserved. A normalized initial state is preserved even as time progresses and the system develops. This is a fundamental property of quantum mechanical systems and follows directly and only from the basic equation of quantum mechanics, the Schrödinger equation.

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Information conservation as a physical conservation law Conservation laws result from fundamental invariances. According to the theorem of Emmy Noether, to every continuous symmetry of a physical system corresponds a conservation law. A symmetry is understood to be a transformation of the physical system that does not change the behavior of the system in any way. Such a transformation may be, for example, a displacement or a rotation of the system. If one considers information transfer and information processing processes as quantum mechanical physical processes, the invariances or symmetries for conservation laws are used up anyway: • From the homogeneity of time, i.e. the temporal invariance of the systems against time displacement, follows the law of conservation of energy. • From homogeneity of place, i.e. invariance to local displacement, follows the law of conservation of momentum. • From the isotropy of space, i.e. invariance to rotation, follows the conservation law of angular momentum. • From the symmetry of the phase of a charged particle, i.e. the invariance of the systems to phase shift, follows the conservation of charge. The conservation of the baryon and lepton number plays a role in nuclear physics. Such processes shall not be considered in detail here. If there should be a separate information conservation law, there should also be an additional corresponding symmetry. Such a “white spot” would probably have been recognized by the physicists long ago. Also this fact suggests the conclusion that the dynamic information defined here is identical with the energy. So the dynamic information is to be regarded as an interpretation of the energy. Consequently, information conservation is just the energy conservation and results from the homogeneity of the time. One could object with regard to Adam Petri (see citation in 1.2.) that the definition of information and the conservation of information would first require new physics. This is opposed by the fact that information transmission and information processing can be described very well with the known laws of physics. Here no open or unsolved problems are recognizable concerning the elementary physical processes. In this context it is to be referred again to Wiener’s statement that information is neither matter nor energy. Information would then be a quantity standing outside of the present physics. It would stand then also outside of thermodynamics and would have nothing to do with entropy. This is not appropriate. It is certainly possible to formulate a non-contradictory information theory which places the concept of information outside matter. However, since information has to be transmitted and processed and this is not possible without energy, many additional assumptions would be necessary, especially concerning the relation between energy or matter and information.

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61

Following the principle of simplicity, models with as less additional assumptions as possible should be preferred.

2.6.7

Destruction and Creation of Physical States and Particles

In the context of information conservation and the conservation of quantum bits, the annihilation and generation operators for states and particles should not go unmentioned. If in a system of particles the particles change their states or a system changes its state, then a mathematical description for the change is necessary. Such a change from state A to B can be described by the annihilation of a particle at A and the generation at B. Thus annihilation need not mean that conservation laws are violated in closed systems. What is important is the balance between annihilation and generation. In the classical thermodynamic representation of systems, the entropy can increase due to interventions in the system, resulting in the generation of additional bits. In particular, when a system is given more volume to exist in, the entropy and hence the number of bits can increase16 . The system is no longer closed when it is extended or spatially expanded. Also C. F. v. Weizsäcker17 assumes in his ur theory that in the course of time the number of “ur”s in the universe increases. The universe expands, it widens.

2.6.8

Hypothesis: Variable Relationships Between Information to Energy?

In the Sect. 2.4.1 “Quantum mechanical limits of information transmission” a fixed relation between energy and information was justified. This reasoning was done in the sense of a positive argumentation. Here the attempt shall be made to assume a variable relation of energy and information and to lead to the contradiction. Now the hypothetical case shall be considered that the relation between information and energy can change. The first question should be whether a bit can be transferred in a given time Δt with less than one action ? This question is to be answered with “No”. This would contradict the Heisenberg uncertainty relation for energy and time. So the energy of a bit could only increase. The second question, whether a bit could also be transmitted with more energy, is not trivial. However, two arguments can be made against this case: First, actions larger than  can be quantized and interpreted as multiple bits. Finally, because of their quantization, the action can only occur as an integer multiple of . More generally, any wave function can be represented as a superposition of quantum bits.

16 See Sect. 5.4.1“Classical ideal gas”. 17 See Sect. 3.8 “Theory of ur alternatives”.

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So any “big bit” or more generally any information that carries more energy than is necessary could always be decomposed into quantum bits. On the other hand, any wave function can be built by superposition of quantum bits. This means that every system, which is subjectively regarded as “energetic”, can be represented as a sum of quantum bits, i.e. as a sum of elementary information units. The linearity of the Schrödinger equation and the superposition principle following from it as well as the spectral theorem (development theorem) are the fundamental causes for this fact18 . Because these are different representations of the wave functions, an “energetic” representation of a wave function and a representation as a sum of quantum bits are only two different views of one and the same object. Ultimately, the question of whether an object represents information or energy is simply a matter of the frame of reference, or more precisely, the basis of the Hilbert space in which the wave function of the object is represented. Second: It would then be necessary to combine the energies of two existing bits into one bit when generating an energetically “overloaded” bit, which would correspond to the annihilation of entropy. This would violate the entropy theorem, but would be possible in the individual case19 . Also the copying of quantum bits is not possible according to the no-cloning theorem [75]. If this were possible, the information of a quantum bit could be distributed to several quantum bits and the ratio of information to energy could be shifted in favor of energy. Cloning would correspond to a reduction of entropy. Another independent argument is the accepted fact that in closed systems energy remains constant and entropy also remains constant or at most could increase20 . This means that the energy per bit cannot decrease on average. An increase of the energy per unit of information (measured in bits/Δt ) would then only be possible if the transaction times Δt would shorten on average, which is only possible if the average energy in the system increases, because of the indeterminacy relation (see (2.2)). This would contradict the assumption that the energy remains constant. But this is an argument with mean values. In single areas and limited time periods a violation of the fixed relation between information and energy cannot be excluded with this argumentation. Another line of argumentation would arise if one assumes that there are processes which would shift the relation between energy and information. If these processes were serializable, i.e. the output could also be input of a similar process, then the relation would in principle be arbitrarily shiftable, which would certainly collide with the indeterminacy relation or the no-cloning theorem. The arguments given are not proofs. They should be developed further. So far they are rather circumstantial. 18 Personal communication of Prof. Dieter Bauer University of Rostock. 19 But this means that the “individual cases” must balance out on the average. 20 An increase of entropy in closed systems is seen critically, see 4.1.3 “Entropy of a probability

field”.

2.6

Properties of Dynamic Information

2.6.9

63

Redundancy

What is the meaning of redundancy in the picture of dynamic information? Can it be argued that in the case of redundancy energy is transferred without the receiver receiving information? Redundancy means that signals are transmitted that are not actually information for the receiver. In [102] one finds: “A unit of information is redundant if it can be omitted without loss of information.” It is questionable whether this unit of information deserves its designation, since it is not information. Manfred Broy [11] writes more correctly in the context of encoding information: If ... more bits than necessary for the uniqueness of the coding are used ..., then the coding is always ... redundant.

Critically, it must be noted that the question of what actually constitutes information remains unclear here. Redundancy is not useless with regard to transmission security. One possible interpretation could be that the redundancy of information depends on the recipient. It then has a subjective component. On the other hand, it seems to be exactly measurable as a “deviation” from the optimal coding. But, according to Broy: Strictly speaking, it only makes sense to speak of redundancy if a probability distribution is given for the considered set of information [11].

Is redundancy an objective quantity? Can one speak of redundancy if no agreements have been made or can be made between sender and receiver? An example will clarify the question: Would a receiver receiving a bit sequence 010101010101 not want to predict that the next bit will be a 0? But if this sequence is now a bit sequence from the transmission of equally probable words (16 bits = 1 word), then there is no redundancy in this bit sequence at all. By chance, for example, the word 010101010101 was received, which has the same probability as the bit sequence 0110010110011101 or any other word. The regularity in the structure of the bit sequence leads to the assumption of redundancy. In another representation, e.g. decimal representation, this bit sequence appears as 21845. The representation of information is very dependent on the reference system used. Particularly simple or outstanding representations can be created or destroyed by a change of the reference system. If there are no agreements between the sender and the receiver, especially if the receiver does not know the properties of the information channel, especially about the future behavior with respect to the probability distribution, the redundancy cannot be defined unambiguously. The algorithmic information notion21 should be able to answer these questions. But algorithmic information is not clearly defined. In particular, it is not decidable whether there 21 See Sect. 3.3 “Algorithmic information theory”.

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Fig. 2.16 Periodic stimulation of a pendulum

is a law of formation in a sequence of numbers that appears random. This is a fundamental statement of theoretical computer science, which is related to the halting problem22 . But just such a formation law would define redundancy. In principle, then, the question whether redundancy exists cannot be decided. In the light of quantum mechanics, the question of redundancy can be answered quite clearly. A quantum mechanical system is to be considered, which comprises sender and receiver. Identical subsystems, that means subsystems which are in exactly the same state, are not possible. This means that redundancy in the sense of identical copies is not possible. Also the no-cloning theorem forbids the generation of identical quantum bits. In this context, it is not conceivable that redundancy is possible in the transmission of information in the context of quantum mechanics in the sense of conventional communications technology. Redundancy is a subjective phenomenon. What processes take place physically when apparently “useless information” is to be transmitted? It shall be shown in the following that these “information” are not “accepted” by the receiver at all. By means of the following examples the objectification shall be illustrated. Example: The pendulum To illustrate this, we will consider as a receiver a system which contains an oscillator as an essential component. For the classical case one could imagine a pendulum (Fig. 2.16). If the pendulum is excited monotonically in the rhythm of the resonant frequency by the sender, the first pulses will cause the pendulum to oscillate. The receiver absorbs energy. When the pendulum is activated, hardly any energy is transmitted, although the behavior of the sender has not changed. Only the compensation of the friction would still be transmitted. The “success” of an intended transmission is, of course, dependent on the structure and condition of the receiver. The transmission is effective at the beginning, but then, when the transmission becomes redundant in the classical sense, hardly any energy is transmitted. It could be interpreted that when redundancy occurs, the receiver no longer accepts the information. If the monotonicity were interrupted, for example, by changing the phase of the excitation, energy would immediately be absorbed again by the receiver. 22 The halting problem is discussed in Sect. 6“ The halting problem”.

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Properties of Dynamic Information

65

In the case of the oscillation stimulation also energy (thus an information) can be delivered back to the sender. This information could be understood as a response or reaction to the information transmission. Example: Saving a quantum bit For the quantum mechanical case, the saving of a bit by excitation of a state in a quantum system shall be explained as an example. Practically, it could be a color center in a solid or a 40 Ca + -ion (see 2.5.9) which is used to save a bit (see Fig. 2.17). The sender may now want to save a bit. He sends the bit in the form of a first photon, which stimulates the electron in the receiver. The transmission is successful. The bit has been saved by the sender in the receiver. The saving process is now repeated. So if a second, in the classical sense redundant bit, is sent in the form of a photon, a stimulation is no longer possible because the electron is already stimulated. The photon is not absorbed. The receiver has become transparent after the absorption of the first photon. Either the receiver has a reflection device for this case, then the photon would go back to the sender, or the photon passes the receiver and is absorbed in a third system behind it. One could interpret that the first photon was interesting for the memory (receiver). It did not accept the second one because it was redundant. The reaction of the receiver to a bit depends on the state of the receiver. If one would regard this as a subject, then one could also say that the receiver interprets a bit sent to him in dependence on his state. The concept of interpretation of information is, at least in the case under consideration, traced back to a physical process. In principle, this approach can be applied to more complex systems, including humans as recipients. Here, the interpretation processes are very complex, but always consist of ele-

Fig. 2.17 Saving a bit and a second save attempt

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mentary physical and logical processes. The description and research of these interpretation processes is the subject of neurology and psychology. Redundancy in the classical sense plays a role in the transmission of information, but is essentially determined by the interaction between sender and receiver. In this physical interpretation it is quite close to the common term. However, it must be emphasized that each transmitted portion of energy represents information for the receiver. A subjective concept of redundancy is not necessary. The discussion on redundancy clearly shows that information transmission is not only a transmission from a sender to a receiver but also quite essentially characterized by interaction between sender and receiver. The sender, however, only deserves its designation if more energy goes to the receiver than vice versa. The question arises whether a strict separation between sender and receiver is possible. It is probably the case that in certain technical arrangements a distinction is traditionally made between sender and receiver.

A Comparative View on the Concept of Information

3.1

SHANNON Information

Shannon1 . defined information in terms of entropy [13] as early as 1948. He wrote that information is the surprise that is in a transmitted sequence. It is the elimination of uncertainty by the transmission of signals [73]. Coding, channel capacity, and redundancy play a crucial role in the information transmission. Shannon assumes that there is uncertainty at the receiver. The uncertainty can be interpreted as entropy at the receiver. Accordingly, an effect of information is the removal of entropy at the receiver. However, this view is problematic because it does not fit thermodynamics. The reduction of entropy in a system as a result of information transfer into the system, which also results in energy transfer into the system, is thermodynamically very questionable. In the case of thermodynamic equilibrium or near equilibrium, it is not possible. The entropy of the transmitted message is in any case the central quantity of the information transmission. The central relation (2.7) expresses this also. However, the reduction of the Shannon notion of information to entropy does not take into account dynamics. Therefore, the notion of channel capacity (entropy per unit time) is better suited to describe information. In contrast to the Shannon view, the dynamic information defined by Eq. (2.7) recognizes that energy and entropy flow to the receiver through the information transfer and consequently energy and entropy must increase in the receiver. This consideration avoids therefore also the contradiction with the entropy decrease in the receiver. In the special case of systems with the same clock frequency, the information notions of Shannon and the concept of dynamical information introduced here are the same or at least quite similar with respect to the measurement of information. To a good approximation, this 1 Often spoken of Hartley-Shannon’s concept of information.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 L. Pagel, Information is Energy, https://doi.org/10.1007/978-3-658-40862-6_3

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3 A Comparative View on the Concept of Information

is also true for isothermal thermodynamic systems. Of central importance is the fact that Shannon information does not know a time behavior. Be that as it may, entropy is the central quantity in information theory. The different handling of entropy in information technology and in thermodynamics is striking. In information technology, senders and receivers are needed. The elimination of uncertainty at the receiver is discussed. In the logical building of thermodynamics, these concepts do not appear. There, entropy is a fundamental physical quantity. Thermodynamics also does not need negentropy or syntropy. In Chap. 4, the entropy is treated more exactly and also the negentropy is discussed.

3.2

Definition by JAGLOM

A. M. and I. M. Jaglom [45] define information about conditional entropy. They consider two experiments α and β, where the outcome of the experiment β may depend on the realization of the experiment α. The entropy of the composite experiment S(αβ) is S(αβ) = S(α)+ S(β) if both experiments are independent. If β depends on α, S(αβ) = S(α)+ Sα (β). Here, Sα (β) is called conditional entropy. The information contained in the trial α about the trial β is now defined as a difference as follows. (3.1) I (α, β) = S(β) − Sα (β) I (α, β) indicates to what extent the realization of the experiment α reduces the indeterminacy of the experiment β. Thus, the information is an entropy, more precisely an entropy difference.

3.3

Algorithmic Information Theory

The central concept of an information theory founded by Kolmogorov, Solomonoff, and Chaitin is the complexity of strings. This so-called Kolmogorov complexity K (x) of a word x is the length of a shortest program of a programming language M (in dual representation) which outputs x [108]. The measure of information is the minimum length of a program that a computer would need to realize the desired structure [73]. This definition eliminates problems of redundancy. Compared to the Shannon definition, it is immediately apparent here that repetitions or other regularities of a word should be recognized and lead to the optimal encoding. Interesting cases in this context are the software random generators. Here, long strings are generated with little program code, which seem to be very close to complete disorder, but are not. If one applies the Shannon concept of information to such a string, a large

3.3

Algorithmic Information Theory

69

value for the information certainly comes out, if only the probabilities for the occurrence of a character enter into the result. An example will show this. The string: 3, 21, 147, 5, 35, 245, 179, 229, 67, 213, 211, 197, 99, 181, 243, 169, 131, 149, 19, 133, 163, 117, 51, 101, 195, 85, 83, 69, 227, 53, 115, 37, 3, 21, 147, … is generated by the very simple instruction: z i = 7 · z i−1

(3.2)

A complete program for the calculation of 100000 numbers is written in the C programming language: char z[100000]; z[0]=3; for(int i = 1; i No (by definition of M*) No − > Yes (by definition of M*)

The Russel antinomy was an occasion to banish from the Logical System of the so-called “Principa Mathematica” any self-reference in order to avoid contradictions. But Gödel shows that this does not help either. The Barber paradox A similar paradox is the “barber paradox” by Berntrand Russell. The village barber is a villager who shaves all those villagers who do not shave themselves. Does the village barber shave himself?

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The EPIMENIDES or liar paradox Also the sentence “I lie” leads to a contradiction. If the sentence is a lie, it is also true. Hofstadter [41] writes to this: In its absolute purest version, Gödel’s discovery represents the translation of an ancient philosophical paradox into the language of mathematics. It is the so-called Epimenides or Liar paradox. Epimenides was a Cretan who spoke an immortal sentence: ‘All Cretans are liars.’

The use of the word “I” in the short form “I lie” is noteworthy. The “I” realizes self-reference. The realization of one’s own existence is an essential characteristic of consciousness. Strange loop formation Hofstadter also considers self-reference to be decisive. He writes in [41] There always seems to be the same catch in these paradoxes: Self-reference or “strange loop formation.” So if one’s goal is to eliminate all paradoxes, why not try to eliminate self-reference and anything that might lead to it? This is not as easy as it seems, because in some circumstances it is difficult to determine where self-reference occurs. It can spread over a whole Strange Loop with different steps as in the ‘extended’ version of the Epimenides, ...: The following sentence is false. The preceding sentence is correct.

The GÖDEL incompleteness theorems The first Gödel incompleteness theorem states that, in recursively enumerable systems of arithmetic, it is not possible to formally prove or disprove all statements. It states: Any sufficiently powerful, recursively enumerable formal system is either inconsistent or incomplete or something more prosaic: A formal system describing at least arithmetic cannot be both complete and noncontradictory. The proof is demanding, but the basic idea can be formulated understandably [84]: Gödel constructed a statement G in the formal system which says: “This statement G is unprovable.” If G is provable, then so is non-G and vice versa. So G is not provable (provided the system is free of contradictions so far). But that is just what G says: therefore G is true. More precisely, we understand that G is true, but in the formal system it cannot be deduced.

But, whether a string is provable or not can be formalized precisely, and even the seemingly suspect self-reference in the proposition can be circumvented. Gödel assigned a number

7.6

Limits of Self-Knowledge Due to Self-Reference

185

to each string, which was later called the Gödel number. The second Gödel incompleteness theorem states that a logically consistent system is unable to prove its own logical consistency. Every sufficiently powerful consistent system cannot prove its own consistency. Roger Penrose argues against the strong AI hypothesis in [72], arguing mainly with Gödel’s incompleteness theorems. Hofstadter elaborates in [41] (p. 503) which systems are affected by Gödel’s incompleteness theorem. He lists three conditions: 1. The system must be rich enough that all statements about numbers can be expressed in it. 2. All general recursive relations should be represented by formulas in this system. 3. The axioms and typographic patterns defined by rules must be determinable by a finite decision procedure. Hofstadter elaborates: The wealth of the system brings about its own fall. Essentially, the fall comes because the system is strong enough to contain self-referential statements.

Limits to the growth of logical systems become apparent here.

7.6

Limits of Self-Knowledge Due to Self-Reference

It seems obvious that systems have fundamental problems with themselves that amount to limitations on their self-knowledge. With other systems, however, this is not so. The sentence “I lie” is fundamentally a problem, “you lie” is not. This suggests that another system can get more or at least different knowledge about a system than the system can get from itself. Perhaps this fact is an explanation for the fact that people need the exchange of information with other people. Because an intensive and time-consuming exchange of information is required for the realization of fundamental characteristics, fundamental knowledge cannot be exchanged with too many partners. People need others’ opinions about themselves. This is certainly also one reason why people prefer to live in pairs. One can go one step further and assume that the genome of a human can be represented by a logical system. After all, the genome encodes processes in the cell and in the whole organism which can be described mathematically and logically. Then the limitations of self-knowledge and the effects of contradictions would also apply to a genome. This would also mean that such a system would self-destruct by the propagation of contradictions, if

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only it were sufficiently complex enough16 . This danger would not exist for simple living organisms, provided that their complexity allows non-contradiction. If the genome is not refreshed by components from outside, the system could become unstable due to the spread of contradictions. This would be inbreeding with the known consequences. A solution is the mixing of two genomes during the reproduction. That would be then from the point of view of formal logical systems a reason for the necessity of the mixing of the genes and for the generative reproduction. These thoughts do not only apply to biological systems, of course. They are the consequence of mathematical theorems and of course also apply to artificial intelligence and artificial systems with consciousness. Seen in the light, the limitation of self-knowledge by the self-reference creates a fundamentally unsatisfactory situation for systems with consciousness: The statement “I lie” leads to contradiction and the statement I “I do not lie” is not provable because of the second Gödel theorem. It is probably not possible to do worse.

7.7

Self-Reference in Electronics

Logic is not only a matter of mathematics or computer science. Logical functions are realized in electronics by mostly integrated electronic circuits. The properties of formal logic must therefore also be visible in logical circuits and can be grasped, i.e. touched. Mostly logical circuits can interpret “true” and “false” as voltage or current states. Intermediate states are mostly not possible and excluded by positive coupling. The negator An electronic equivalent to the sentence “I lie” is a back-coupled negator. The negator turns a true statement (true) into a false one (false). It is the electronic liar. If you now apply this liar or negator to itself, i.e. present its statement to itself, then we have a contradiction; TRUE is FALSE, low level is high level (see Fig. 7.1). This is not possible in terms of circuitry and physics. If the input of the negator accepts only 2 states, the delay time between the change of the input signal and the response on the output solves the problem. Then this simple circuit is an oscillator that is constantly trying to solve its contradiction, so to speak. The self-reference makes the system dynamic, it causes continuous changes. Prosaically: it animates the system. Such oscillators are important in electronics such as clock generators. The clock generator of a processor “animates” the computer. If it is switched off, the computer is “dead”. Can this function of the feedback negator also be transferred to “neuronal circuits”, in our brain? Thus, the self-reference generates an oscillator which possibly drives our brain the way the clock generator drives a microprocessor. Hofstadter [42] writes in the chapter “I cannot live without myself” (p. 241) the sentence: 16 The Gödel theorems apply to sufficiently complex systems; to put it simply, to systems powerful

enough to represent arithmetic.

7.7

Self-Reference in Electronics

187

Fig. 7.1 Loopback negator or inverter, oscillator

“‘I’ seems to be for us the root of all our actions and choices”.

A self-reference across two negators (Fig. 7.2) is a stable arrangement that can store information. So self-reference does not always have to cause oscillations. But also the storing is essential for the consciousness. Feedback can occur in more complex circuitry via multiple gates and need not be immediate and short. The behavior is analogous to loops in programs, which often arise over many program lines and do not always reveal themselves. Artificial neural networks Artificial neural networks can have the cardinality of Turing machines. This is a consequence of the “physical Church- Turing hypothesis”. Therefore, any computable physical system can be simulated by suitable artificial neural networks [52]. Artificial neural networks are realized by electronic circuits. Feedforward networks, even multilayer ones, have no feedback. No information is fed towards inputs. Loops are not provided. In contrast to this are recurrent neural networks (RNN), in which feedback loops are intentionally built in, in which signals are sent in the direction of the input (Fig. 7.3). Here, time delays are usually built in to prevent the system from oscillating too quickly. These feedback loops generate dynamic behavior and realize a memory. Feedbacks in our natural neural network are considered important. The title of a paper by H.- C. Pape [71] is “The thalamus, gateway to consciousness and rhythm generator in the brain”. The work gives an insight into an extraordinarily interesting and complex subject, which will not be discussed in depth here.

Fig. 7.2 Two feedback negators, memory

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Fig. 7.3 Recurrent neural network

7.8

Algorithmic Considerations

7.8.1

Self-Reference in Algorithms

The self-reference is quite certainly realizable if the algorithm has access to its own code. In lower programming languages like assembler, the access to the program code is possible, at least if a von Neumann structure is present. In order to be able to discuss the self-reference, a simple function is to be considered, which is to determine a property of the program by analysis (“observation”) of the program, in which it is called. An example shall clarify the self-reference: In programming practice occasionally very simple functions are used, which have the task to determine the byte sum over their own program. This value is unique. With such a function, the program itself can determine whether its code has been changed. Such a simple protection is not perfect, because only all changes are detected, which change the byte sum. After all, such a simple function allows the algorithm to determine with a high degree of probability whether it itself is still what it once was after its completion and installation or whether it is now someone else. In the field of security-relevant applications, such protection is important. The algorithm is also able to change its byte sum (property) itself. If this happens accidentally, it is usually catastrophic and is often prevented by computer architectures or operating systems by write protection. The simple example also makes it clear: To change an algorithm, the program code does not necessarily have to be changed, it is sufficient if parameters that control the behavior of the program and are stored in the working memory are changed. Thus, it becomes recognizable that the access to the program code can be only a part of the self-reference. The working memory must be included. The program has access to it anyway. However, a function, which “looks” at its own algorithm, then also “sees” its own activities, namely the “self-reference”. This is logical, because they are part of the algorithm. At the latest when the function stores its result of the self-consideration, it has changed its own result, probably already before.

7.8

Algorithmic Considerations

189

Here, the problem of self-reference becomes apparent, especially when a program wants to determine its own properties. The theorem of Rice states that. Whether the mentioned deviation in the self-assessment is relevant depends on the relevance of the self-analysis function for the program. If this self-observation is insignificant for the purpose of the program, the change of the algorithm by the self-observation is also not relevant. The situation can become critical if the identification of the own properties is used to change these own properties. Then the self-judgement is essential for the program. Another question is whether there is enough memory available to store the result of selfknowledge. If all memory resources are used, memory for the result is missing. Of course, the analyzing part of the system is also included in the result of the self-knowledge, which generally does not cause a problem. In case the system does not use its memory fully, i.e. essential parts are unused, there is no limitation of self-knowledge concerning memory. If stability is present for a long time, parts of the memory must of course be cleared occasionally, i.e. be forgotten. Otherwise, the system will reach its limit in regard to memory. The logical problem that the result of self-knowledge changes the object of knowledge remains. This “error” of self-knowledge could be reduced by renewed self-knowledge, which would eventually become a never-ending iterative process. So this loop should be interrupted, either by external events or by the “program structure”. These problems become manageable when a function analyzes another program, not itself. At least the storage of the result does not change the object of the analysis. The problem of the investigation of a changing program could be solved, in which one stops the program which can be examined temporarily. This possibility does not exist with the self-analysis. Problems with self-reference are also apparent in the definition of Kolmogorov complexity. They have been investigated and lead to the conclusion that the Kolmogorov complexity is not computable (see Sect. 3.3).

7.8.2

Loops in Algorithms

The example of the sentence “I lie”, shows in a simple way that its interpretation leads into a loop without end. It is important that this loop runs over two metalinguistic levels. If meta-linguistic abilities are important for systems with consciousness, then loops over multiple levels should be an important necessary criterion for consciousness. However, loops are found in almost every complex algorithm, so they are not a special feature. It is difficult to imagine an algorithm working in our brains for 70 years without internal loops. With the title statement of Hofstadter in [42] “I am a strange loop” the emphasis must then be on “strange”.

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Algorithms without inner loops would quickly terminate without inputs. As neural networks, they would quickly become inactive. However, they could be involved in outer loops. An example would be control algorithms.

7.8.3

Algorithmic Requirements for Consciousness

What criteria must an algorithm have that realizes consciousness? First, internal properties of the algorithm are to be considered in the sense of a “white box testing”. It is about the analysis of the instructions of the algorithm. The central property of a system with consciousness is self-reference and thus the ability to recognize itself and to interact with itself. Consciousness is closely related to self-reference and the notion of identity, especially one’s own identity. In [1] a more philosophical overview of identity and the “I-development” is given. Here is a first approach to formulate this requirement algorithmically: 1. A part of the system has a structure which corresponds to a special class in the sense of C++. This class is said to be called “Human”17 . The instances of these classes represent systems, especially systems with consciousness, which may be humans. The class describes the properties of individuals of the class human. It contains data describing properties and functions operating on these properties and the instances. The definition could look like this: class Human //definition of the class { /*parameter set of properties*/; /*set of functions*/; } Human M1, M2, ...; //definition of instances

2. The class Human contains exactly one instance which is called “I”. This is an instance that describes the properties of the system under consideration. //Definition of instances with "I". Human I, M1, M2, ...;

Here, an explicit class definition is shown. What is important for the system under consideration, however, is only that structures corresponding to it in the sense of an isomorphism can be identified. Regardless of whether it is a computer program or a neural network. Thus, we have the self-reference in the system. A comparison with the Cantor set definition18 suggests itself, where a set itself can be a member of this very set. This 17 “Human” here is to be understood only as a designation or name of the class. 18 See “Russels antinomy”.

7.8

Algorithmic Considerations

191

leads to the contradiction. The class human contains a set of instances, including the “I” and the system “I” contains (describes) all instances. An important question is whether the instances M1, M2 ... are necessary. Can an “I” be defined without another human being? Is it similar to a maximal number, which can exist only in comparison with a smaller number? The following points of view speak for the necessity of the connection with other systems with consciousness: – identity is based on distinction [1]. Distinction conditions other individuals. This strongly suggests that an individual, including the “I”, cannot exist without other individuals. Hoftstadter wrote: “I cannot live without myself”. Symbolically, it should be added: “I cannot live without you.”

This phrase is quite common. Usually, a certain person is meant. Here, any human being is meant. – Evolutionarily, consciousness most certainly originated in a social system of several people. Hermits can live, but originally originated from a group of people. But can a human being originate consciousness who has never seen a human being? One is inclined to answer the question with “No”. Against this necessity speaks: – self-reference can function as an “internal matter” of a system and it is not logically recognizably attached to contact with other systems. – a feedback negator can exist without other negators. However, it also has no consciousness. So far, however, it has not been defined in detail which properties the class Human should have. The instances are supposed to describe properties of systems. That is: – The class definition contains a set of algorithms that determine parameters of subsystems. – Objects are subsystems that can be described by a set of time-invariant parameters. – The information about the objects may have come into the system through inputs or may have been generated by logical reasoning. – There exists an isomorphism between the instance and the object that describes the instance. Thus, parameters are used for isomorphism which shows a temporal invariance, i.e. which change only within narrow limits. The recognition of objects as largely temporally invariant structures is of central importance.

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3.

Technically, this could mean looking for variables that are often read and rarely written to. A bundle of such variables, which lie close together in the memory, could define an object. These considerations are vague and should be specified with the goal of identifying objects in systems, i.e. data, with algorithms that are as simple as possible. An important role can be played by data described directly by input and output processes. They can be the memory of the system, by their nature they are not or rarely changed. The data set should be called “InOut”. A memory is important because it is an essential part of the identity of the system. It defines a part of the “I”. Crucial for systems with consciousness is the ability of the system to recognize itself as an object. Thus, the system would have to find a set I of parameters that defines its own behavior. The system must be able to change the parameters I that determine its own behavior (i.e. parameters of the instance “I”). For the change of the own behavior, a goal function is implemented, which is a component of the characteristics of the system. If this objective function ensures the survival of the system, the system is dynamically stable. The algorithm can be divided into two parts,

4. 5.

6.

– a basic part, which controls the running processes and whose behavior is mainly determined by the set of parameters I. – and a second top part that uses the parameters (I) to change them19 . This establishes two levels of language. The top part of the algorithm uses an objective function F(I,InOut) to change the parameter set I using I and InOut. With In = F(In−1 , I n Out), self-reference is realized, a central property of systems with consciousness. Because of the unavoidable delay, a finite time has elapsed between In and In−1 , which avoids a direct contradiction. 7. The algorithm must have sufficient time for the top part, in which it deals with itself, i.e. for self-reference. This means that its processing capacity is not substantially consumed by higher-priority and simple reactions of the base part, caused from the environment. 8. The algorithm needs sufficient inputs from the outside. The base part of the algorithm should predominate in terms of processing capacity. An exclusive preoccupation with itself would increasingly generate contradictions that spread like wildfire throughout the entire system. 9. The system must never return to a state it has already entered, regardless of whether and which inputs act on the system. The number of possible states must be greater than the number of states passed through in the lifetime of the system. This condition is computable and ensures complexity and evolution possibilities of the system in a simple and measurable way. If a deterministic system were to take a state a second time, a loop would be the result, representing an exact repetition of the sequences. This condition 19 Motivation research has found that the ability to change one’s own behavior is an essential prereq-

uisite for motivation.

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Algorithmic Considerations

193

also ensures that the system has sufficient memory available, because, otherwise, the high number of possible states would not be feasible. 10. The algorithm must be sufficiently complex. The property of the ability to self-reference, which is essential for systems with consciousness, is connected with the first Gödel incompleteness theorem. The complexity of the logical system is stated as a prerequisite for this ability. Simplifying, it is required that the system must include at least arithmetic. This condition has been specified over time. With self-reference, contradictions become part of the system. 11. The program path responsible for decisions runs serially. Subprocesses can run in parallel (unconsciously) (inner speech). This is independent of the realization of the algorithm, i.e. whether it runs on a neural network or a von Neumann architecture. 12. The algorithm must not terminate by itself. This requirement corresponds to experience. It is not logically mandatory. These requirements have a good chance of being verified. Whether they are sufficient for awareness is not yet clear. Requirements 1–4 are essential, from point 5 on the requirements could be plausible but not mandatory. These requirements are also a description of a system with consciousness. Whether it is possible in principle that a system with consciousness (the author) can describe a system with consciousness is very questionable. The theorem of Rice does not allow this. In particular, three postulates that a system can determine its own properties. Because systems with consciousness are inconsistent anyway, this is no wonder. The point here is a process of trying to find criteria for consciousness that are as close as possible to the essence of consciousness. The goal will possibly never be reached. Nevertheless, engineers will design information-processing systems that acquire or can acquire consciousness. It seems that the hardware requirements for this are already in place, the software is probably the challenge. Programming such a system can be done in several ways. As with humans, who are not born with consciousness, a comparatively simple program that enables complex processes (for example, the simulation of a neural network) can be learned. When applying the notion of algorithms to systems with consciousness, it should be noted that known systems with consciousness, i.e. humans, do not have a defined starting point for their “program.” Consciousness develops slowly after neurons in the embryonic stage fire each other through spontaneous activity and begin to structure and learn. The influence of initial conditions (input) on later behavior is likely to be extremely small. Moreover, human consciousness as an algorithm has no special destination in the sense of a result of an algorithm, because it is not intended to be terminated. The direct applicability of the notion of algorithm in theoretical computer science to conscious systems should be critically questioned. The author takes a critical view of the beginning and end of the algorithm. However, its realizability by an Turing machine or equivalent machines is not criticized.

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Fig. 7.4 The relationship between consciousness and life

It seems equally possible, however, that the system receives consciousness during its initial programming.

7.8.4

Consciousness and Life

Superficially, consciousness, and life already belong together because the only system we know that has consciousness is a living system, our brain. Because there is obviously life without consciousness and consciousness can exist without life, this relation is not trivial after all20 . If one assumes that the question about “living” and “conscious” can be answered with “Yes” or “No”, respectively, there would be four possibilities (Fig. 7.4). The deliberately somewhat flippant labeling makes contradictions clear. Undisputed is only the position “consciously living”, the human brain. The position “consciously not living” would correspond to a computer which realizes consciousness. That would be artificial consciousness (AC). Artificial intelligence (AI) could then be drawn in halfway to AC. This position becomes more explosive when life not based on carbon compounds is considered. Extraterrestrial life comes to mind. Would one then accept systems that would be considered technological and that have consciousness as life forms? If so, life and consciousness would move closer together or might even be identical. ... The position “dead without consciousness” seems clear. However, in view of Konrad Zese’s “computing universe”21 one has to ask whether very complex information-processing systems, just matter, may simply be denied consciousness. It must be taken into account that there are forms of consciousness which are not easily recognizable. This could be the case 20 The concept of autopoiesis explores the self-creation and self-preservation of systems. It assumes

that the product of a living system is itself, which means that there is no separation between producer and product (self-reference!). It is tangential to the topic of consciousness, but goes beyond it. 21 See [112] and Sect. 4.4 “Computers and thermodynamics”.

7.9

Consciousness and Time

195

if the inner and outer communication of these systems has not been grasped by us yet or is not understood as such. The position “unconsciously living” suggests that life can also exist without consciousness. It is questionable here whether the sharpness with which consciousness has been defined (also in this book) is sufficient to generally deny consciousness to life outside our brain. It is to be pointed out here to an aspect of the life, the self-reproduction. It is a prerequisite for the self-preservation of a living system. In this chapter, self-reference is often emphasized as an essential feature of consciousness. Do not self-reproduction and self-maintenance include self-reference? Can self-reference be demonstrated in the algorithms realized by the chemical processes in a living cell? This approach argues for a more graded property of consciousness. Consciousness should perhaps be a measurable quantity after all. But could consciousness be measured? There is still much to be done here. Without falling into pessimism it should be noted that according to the theorem of Rice possibly generally valid procedures for the “measurement” of the property “consciousness” cannot exist at all.

7.9

Consciousness and Time

In Sect. 6.1.3 “The flow of time”, it is pointed out that the running of time can be perceived by systems which are able to distinguish states, i.e. which can distinguish in particular the states “Yesterday” and “Today”. For this, an interaction with the environment is necessary, at least inputs are important. If a system has consciousness, then this ability is present with high probability. The flow of time can then be perceived and realized in the system. The question arises whether the perception of time is a general and compelling property of consciousness. The more precise question is whether a conscious system perceives time consciously. The unconscious perception of time seems to be necessary for actions to proceed in a coordinated way. And ultimately, serial internal speech realizes a temporal sequence of logical steps or thoughts. No reason is visible why consciousness is bound to the conscious perception of time. On the other hand, it was already justified that the time does not exist without consciousness. However, if the flow of time is closely related to consciousness, the question arises whether time runs objectively at all if consciousness is not involved. For a system with consciousness, it continues to run when it detects the death of another system with consciousness. Nevertheless, time remains bound to consciousness because the “surviving” system necessarily needs consciousness. The view that the flow of time is a subjective phenomenon is not new. The debate about time has always had a place in popular science articles. “A Brief History of Time” by

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Stephen Hawking and Leonard Mlodinow ([33]) and “The Order of Time” by Carlo Rovelli [76] are worth mentioning. In a contribution in [106] Carlo Rovelli is quoted: “It was a surprise when in the 60’s the fathers of quantum gravity Bryce DeWitt and John Wheeler set up their equation. It contained no time at all! This triggered a huge debate. How can you describe the world without time? Well, it can. Our normal conception of time works just fine in everyday life. But we have to accept that it is only an approximation. An approximation of an approximation. Time is a structure that shows up at some level, but it’s of little use when we think very fundamentally about the world.”

The time is an essential object of the relativity theory. However, the relation between time and consciousness can be described with high probability without the consideration of relativistic effects. Time and consciousness should be consistently explainable even in daily life and without the direct involvement of relativistic effects.

7.10

Juridical and Political Aspects

Consciousness is arguably the most important quality to attribute to a natural legal person. Under what circumstances can consciousness be attributed to them? Under what circumstances are statements or signatures legally relevant in the face of the influence of alcohol, other drugs, or disease? According to [24], “The proper work of consciousness is linked to the functioning of the sensory organs and to the intact functioning of neurophysiological processes between brain stem, diencephalon, and cerebral cortex.” In German civil law, a declaration of intent is considered void if it was made in a state of unconsciousness or temporary disturbance of mental activity. A disturbance of consciousness still leads to incapacity to commit an offense. A profound disturbance of consciousness may lead to incapacity for guilt. A more precise definition of consciousness would be helpful here. In politics, there is no need to worry about consciousness and self-awareness on the part of the actors, the politicians. Here, artificial consciousness matters. A first political question is whether technical systems with consciousness should have a say in political decisions, in particular, whether they should be given the right to vote. A second political question is whether the principle that all humans are equivalent still applies if technical systems with consciousness have significantly more capabilities than humans. If the ice of equivalence is broken, political systems could give more political weight to artificial systems with consciousness than to humans. One possible justification could be the superior capabilities (speed, memory capacity) of artificial systems. That would then be a management of society by “expertise”. Then the system would no longer be democratic. However, a dictatorial political system could also only claim

7.11

Summary

197

this “expertise”. Exactly then a precise definition of consciousness has a special political meaning. However, even without artificial consciousness, a regime of “experts” can be installed. This is a very topical question.

7.11

Summary

Formal logical systems without contradiction are self-contained and all derivable statements are trapped in the system of axioms and rules of derivation. Nevertheless, they can be very complex and powerful. To get out of this captivity, logical systems with contradictions are a solution. The way to contradiction is self-reference. This is a justification for the necessity of self-reference and information processing at several meta-linguistic levels. With it, the possibility is created to partially recognize own properties and also to change them. If a logical system contains even one contradiction, this contradiction spreads in complex systems like a wildfire immediately through the whole system and makes it logically useless. Due to the always-existing time delay (dynamic information), a contradiction cannot spread instantaneously, but in a system-dependent time scale. The more complex the system is, the slower the wildfire runs. This delay and the need for the system to respond simultaneously to external inputs ensures the internal stability of the system while maintaining dynamism and unconstraint. Inner speech ensures decision-making ability in systems with consciousness and likely avoids chaotic processes that would occur due to a very large number of individual processes and conflicting statements within the system. The following summarizes criteria that systems with awareness should meet. Criteria for Consciousness: The previous considerations suggest the following criteria for a system with awareness. They correspond to the requirements for algorithms22 , but are somewhat more general. 1. self-reference: a) passive: The system must be able to recognize, albeit incompletely, its own properties23 . b) active: The system must be able to change own properties24 . 2. complexity: The system must be sufficiently complex, it must at least be able to represent arithmetic.

22 See Sect. 7.8.3 “Algorithmic requirements for consciousness”. 23 From a technical point of view: read access to program or/and data. 24 From a technical point of view: write access to program or/and data memory.

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3. Contradiction: self-reference generates contradictions in a complex logical system25 . 4. creativity: the system is not consistent, which weakens the meaning of axioms/ propositions. 5. inputs: Dynamic stability is achieved by a balance of “destructive” processes, caused by in principle unavoidable contradictions, and the following processes, which prevent a “logical collapse”: a) Inertia of reasoning: logical operations take time, slowing the propagation of contradictions. b) Serialization by inner speech, this effectively slows down the propagation of contradictions even in massively parallel systems. c) Sufficient inputs that constantly feed the system new information that is not yet invalidated by the contradictions. Self-reference, complexity, and inputs are the “hard” criteria. Contradiction and creativity follow from self-reference and complexity. Inputs are necessary for the stability of the system and imply a meaningfulness of the system consciousness, outputs of course too. To phrase it very briefly: A complex logical system fulfills necessary conditions for the formation of consciousness. Here is meant a complex logical system in the sense of Gödels theorems. Thus, selfreference and contradictoriness are included. What is the relation of consciousness to dynamic information? The property of information to be dynamic and ultimately to convert energy and consume time provides the stability of the conscious system. Logical systems without time delay would be “instantly” “destroyed” by a contradiction.

25 This results in instability: logical contradictions propagate in the system and “destroy” existing

propositions/knowledge. This is a basis for creativity.

8

Astronomy and Cosmology

8.1

Relativistic Effects

In large systems, the speed of light must be taken into consideration. Time delays caused by the speed of light are then no longer negligible. This is the field of astronomy, especially of cosmology. The speed of light is the greatest possible speed with which energy and thus also information can be transferred. It is to be expected that with it principal limits are set to the information exchange. The essential idea of the Einstein theory of relativity is the consideration of the finiteness and the independence of the speed of light from the reference frame. Thus, the synchronization of the clocks is no longer possible as in the classical case. By relativistic effects, the simultaneity is lost and with it also the basis for the law of conservation of energy. This is also true for all other conservation laws. Without the notion of simultaneity, the conservation of energy cannot be verified. This is because conservation of energy states that the total energy of a system does not change from one point in time to another point in time. Consequently, this restriction must also apply to the conservation of information. The information conservation law founded in the Sect. 2.6.6 is therefore no longer valid over large distances. Nevertheless, like the law of conservation of energy, it is valid in smaller ranges, i.e. within ranges where time delays can be neglected due to the finiteness of the speed of light. The size of this range depends also on the kind of processes which are considered. It must be clarified that relativistic effects of the information transfer only apply if the information is bound to energy. If information would be, as Norbert Wiener says, “thinkable without energy”, it would not be bound to the speed of light as maximum speed of transmission. It would then be able to stand outside of the relativity theory. The question of super-speed-of-light information transfer in the transmission of quantum objects entangled with each other has been raised by Albert Einstein and led to discussions [98]. This issue is called the Einstein- Podolsky- Rosen paradox or EPR paradox for short. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 L. Pagel, Information is Energy, https://doi.org/10.1007/978-3-658-40862-6_8

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Fig. 8.1 EPR paradoxon

? H

-x

Emission of two Photons with opposite polarization 0

V

x

The issue is whether quantum mechanics as a theory satisfies the principles of locality and reality. Through the Bell inequalities, this question has become experimentally decidable. The experiments show that the Bell inequalities are violated. This shows that the locality of a physical theory need not be further required. Quantum mechanics is a nonlocal theory. Moreover, it was shown that there are no hidden variables in quantum mechanics [18]. Figure 8.1 shows the EPR situation. From one point, two photons are emitted, which are entangled with each other. The emission process may produce photons with opposite polarization in each case. A measurement of the polarization is now made on the photon flying to the right. At this moment, the left photon takes the opposite polarization. The question is how the left photon received the information from the result of the measurement on the right photon. The solution is that quantum mechanics is not local. The wave function of both photons exists as a whole until the measurement is made; regardless of how far apart the two photons already are. The effect from one photon to the other is instantaneous, without delay. Is here information transfer faster than light? It is not at that because the information exchange about the result between both sides can take place only with at most light speed. It does not come to the contradiction. During the discussion of the EPR paradox, it has become clear that quantum mechanics is not a local theory and thus information transfer at faster-than-light speeds cannot take place. The dynamic concept of information coined in this book is coupled to the energy and thus arranges itself casually into the theory of relativity. This is an additional argument to couple the information to the energy.

8.2

Light Cones

The catchment area for information is limited by the speed of light. In a space-time representation, this catchment area can be represented as a cone (see Fig. 8.2). Information can in principle be obtained from all points inside the cone. The areas outside are in principle inaccessible. Such a cone also exists for the future, only events within the light cone are accessible. The light cone could also be called an information cone to illustrate

8.3

Event Horizon

201

Fig. 8.2 Representation of light cones for one-dimensional places. The trajectory of an object (for example A) can only be continued within the light cone. Event B cannot be reached from A1 . The increase of the background lines corresponds to the speed of light, |d x/dt| = c

that information cannot be sent to all areas of space at any time. The transmissibility of information is limited here in principle. In this sense, an information theory is also always relativistic. Here, only the finiteness of the speed of light is considered. In the general relativity theory, the properties of the space must be considered additionally.

8.3

Event Horizon

An important concept for the information exchange is in the general relativity of the event horizon. If information is to be exchanged between systems, the speed of light and the curvature of space are taken into account, then information cannot be obtained from all points in space-time. There is another principal limit for the information transfer in the relativistic world. The cosmos is expanding. The event horizon indicates the maximum distance an object can be so that its light can reach us in the future. An object has crossed the event horizon when it is impossible at our location to receive any information from the object in the future. According to the standard model of cosmology, the event horizon is at a distance of about 16.2 billion light years. Events that take place beyond the event horizon are in principle inaccessible to a present-day observer for whom the event horizon is valid. Also here, we have a principal limit of the information exchange. In contrast, the observable horizon indicates whether information can have been received by an object at our location since the Big Bang. The observable universe is therefore the part of the universe which lies within our observation horizon. According to the Standard Model, the observable horizon lies at a distance of about 42 billion light years. The universe has an

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age of about 13.7 billion years. Here, it is to be considered that the universe has constantly enlarged since the big bang and the distances to be returned have constantly become larger. Event and observation horizons represent principal limits of information exchange.

8.4

Entropy of Black Holes

Black holes are “gravity traps”, areas from which light cannot escape, from which no energy or information can escape without taking quantum effects into account. They are characterized by their • mass, • angular momentum and • its electric charge So says the uniqueness theorem formulated by Werner Israel. John Wheeler put it more popularly: “Black holes have no hair1 .” The information paradox This looks like very little information is sufficient to describe the black hole. This also looks like the entropy of the matter that has fallen into the black hole disappears into the black hole and, at least to the outside observer, is annihilated and lost forever. This fact is often called information paradox. However, quantum effects are interesting. Thus, according to Hawking, the “tunneling” of the potential is possible, so that it seems possible in principle that a black hole does not have to exist forever, it can “evaporate”. The entropy of black holes can be calculated from the “area” of the event horizon. On closer inspection, mass or energy is not directly transported out of the black hole. In the immediate environment of the event horizon, pairs of virtual particles are created. These are virtual particles and antiparticles of the vacuum, which can be separated under certain circumstances, for example, in strong fields. Thereby, the case can occur that now a particle falls into the black hole and appears there as negative energy. In black holes, particles can have negative energies. This reduces the mass of the black hole. The other particle, which remained outside, must be supplied with energy in return. Conservation of energy must apply. It becomes real thereby. Consequently, a black hole radiates energy. Thus entropy is radiated. However, this radiation is not completely uniform. It carries information with itself (see Fig. 8.3). If quantum field theoretical aspects are included in the consideration, information can be stored as excitation of the event horizon. At the surface of the event horizon, quantum fluctuation takes place, which is influenced by the interior of the black hole. Thus, information 1 This statement is also known as the “no hair theorem”.

8.4

Entropy of Black Holes

203

Fig. 8.3 Representation of light cones near a black hole. The space is curved here. If the Schwarzschild radius (event horizon) is exceeded, no world line leads back

about the interior of the black hole is encoded on the “surface”. Thermodynamically, the temperature of the surface or event horizon is extremely high. After all, a lot of information or entropy is stored in it. Thus, the information about the matter which formed the black hole would not be lost and information would be indestructible. It can also not be destroyed by the fall of matter into a black hole. The expansion and the structure of the event horizon are related to the area A of the event horizon via the Bekenstein- Hawking entropy S H [37]: SH =

Ac3 4G

(8.1)

In it, A is the area of the event horizon2 , c is the speed of light,  is the quantum of action and G is the gravitational constant. S is the maximum entropy that can be stored in a spherical mass. It is interesting to note that the entropy content of a relativistic mass depends on its surface area, not its volume. The entropy is very high, also because the temperature is very high. The Hawking temperature is [101]: TH =

c3 8π G Mk B

(8.2)

The entropy of a black hole with solar mass has the unimaginably high value of 1077 k B [67]. This is much more than the sun has, which is paradoxical. It is called “entropy paradox”. The Bekenstein- Hawking entropy is a new kind of entropy and must be considered in the second law of thermodynamics (entropy theorem). In the so-called “extended entropy theorem”, the Bekenstein- Hawking entropy is thus added as a summand to the entropy. 2 This area is calculated from the Schwarzschild radius for the non-rotating case according to A = 4π Rs2 .

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It states that the sum of (classical thermodynamic) entropy and Bekenstein- Hawking entropy cannot become smaller. Remarkably, a system containing a black hole can be considered closed. According to this, the interior of the black hole is not “tied off” from the universe and is outside our world. With the Hawking temperature and the Bekenstein- Hawking entropy essential thermodynamic quantities for a black hole are defined. Following the four main laws of thermodynamics, four main laws are postulated for black holes: 0. Theorem: The temperature is constant in equilibrium. −→ In a stationary black hole, the surface gravity is constant over the entire event horizon. 1. law of thermodynamics: Energy conservation: d E = T d S + r ever sible wor k. −→ d M H = TH d S H +  H J H 2. law of thermodynamics: Entropy theorem: d S ≥ 0. −→ S H ≥ 0 3. law of thermodynamics: T = 0 is never reachable. −→ No state is attainable where the surface gravity is zero. M H is the mass or energy of the black hole. J is the angular momentum and  H = 4π/(M A). Whether or not information is destroyed in the black hole is a historically interesting question. Susskind describes in [89] his struggle with Hawking’s “information paradox”. He argues about the alleged loss of information when matter falls into black holes: But this would also mean that everything that had ever fallen into this black hole before would be lost, including any information contained in it. However, this conflicts with a fundamental mathematical property of quantum theory, whose equations are such that the history of every quantum object can in principle always be traced back, information can therefore never be completely lost.

Finally Hawking gave in: But modern physics has practice in getting used to the impossible, and in 2004 also Hawking gave in. Under great participation of the world press he explained that he was mistaken and had lost a corresponding bet with the American physicist John Preskill: Information is not lost in black holes, but remains on the level of string-theoretical structures at the event horizon and finally escapes with the—not completely uniform—Hawking radiation.

To explain the conservation of information, string theory must be invoked: In fact, however, the fact that the black hole information paradox is string-theoretically solvable is by no means proof that the string theory is correct. It is just as well possible that the quantum

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Entropy of Black Holes

205

theoretical principle of information conservation is violated on a fundamental level after all. This level is completely inaccessible to experiments for an unforeseeable time.

The string-theoretical approach explains that on the “surface”, the event horizon, there are very many degrees of freedom in the form of strings, which finally realize the high entropy. In addition, there is the extremely high temperature of this “surface”. This question concerns the fate of the information. In relativistic processes, however, the coordinate systems are of great importance for the description of place and time. For the conservation laws, the undefinable simultaneity of time points is a fundamental problem, which becomes extremely acute near the event horizon of a black hole. For a far outside observer, an object falling into a black hole is first accelerated, then, near the event horizon, the velocity goes to zero. The object falls infinitely long. In these coordinates, nothing disappears in the black hole, at least for the observer “watching” from a distance. For an observer in the falling object, the fall accelerates. No force acts on the central point of the object, but the object is torn apart by forces within the object (similar to tidal forces). In the center of the black hole, where the mass is concentrated in one point, there is a singularity. It is noticeable that, in considerations of this kind, the information is treated like energy. At least the argumentations are compatible with the assumption that information is an energy. Or argued negatively: If information is something which is not yet explainable or describable by present physics, then it would not be excluded that information could cross event horizons. Then the properties of information would not be known yet. An information, which would be thinkable without energy, could override all these barriers. Admittedly, intellectually this is tempting. The holographic universe Interesting is the idea that information about a volume can be stored on the bounding surface. In holography, views of three-dimensional objects are stored on two-dimensional photo plates. This is possible to a limited extent if the main purpose is to reproduce surfaces in a three-dimensional space. This is indeed mostly the case in our environment. True three-dimensionality also causes problems for our brain. It seems that our brain interprets the environment essentially “2.5-dimensional”. It is certainly also no coincidence that a holographic model of reality was developed by a quantum physicist, David Bohm, and a neuroscientist, Karl Pribram. The idea of mapping the information about a volume to the surface in terms of an isomorphism need not fail because all points in a surface can be mapped surjectively to a volume. The mapping is unique, but not possible in the opposite direction3 The idea is acceptable, however, because all objects in a finite volume are representable by a finite number of quantum bits. Geometrically, this means that space can be divided into “voxels” much like a surface can be divided into pixels. The term “voxel” is an abbreviation of “volume” and 3 By a Peano curve into a continuous surjective mapping from a curve (I 1 ) to a surface (I 2 ), this is analogously true for higher orders (I n → I n+1 ).

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“pixel” and describes a small finite contiguous volume in space that does not overlap with other volumes (voxels). Susskind formulates the holographic principle as follows [89]: The maximum amount of information in a region of space is proportional to the surface area of the region. He means “The world is an ‘pixilated’ world, not a ’voxilated’ world”. The thought has been transferred to a whole universe or whole universes. In holographic universes, the information content is largely determined by the surface. In such a universe also, another physics could prevail. If here is also talked about proofs, it should be clear that this book is written about information in our universe and the physics known to us. The processes in information transmission and processing are indeed determined by quantum mechanics and occasionally by relativistic effects, but they are quite terrestrial. There is no white spot recognizable in physics and information technology where processes could not be explained by the physics known to us. Even if not all processes in information technology are understood, this is primarily not due to the laws of physics, but to the scientists who simply have not yet found the corresponding models and solutions. This is especially true for complex processes. The human brain is probably one of the greatest challenges.

8.5

The Universe and Its Evolution

Essential in the development of the universe is its expansion since about 12.6 billion years. By the expansion, an ever more extended event horizon forms. The question arises, whether objects escape the observation by this in principle. From these objects, no information could come to us. The model of the expanding cosmos is a relativistic model. Therefore, the conservation of energy and information cannot be assumed, at least on a large scale. However, there has already been a phase of extremely fast expansion in the early phase of the development of the cosmos. In the so-called inflation between 10−33 s and 10−30 s, the universe could have expanded according to the standard model by the factor 1030 –1050 . Then many objects would have disappeared behind our event horizon. If these objects disappear behind the event horizon, we cannot receive any energy or information from them, but this does not mean that these objects no longer exist. The conservation of information gets a different meaning under cosmological aspects. It is no longer strictly valid as in classical non-relativistic systems. Similar to black holes, information behind an event horizon can escape our grasp, but is not lost, but could in principle reappear sometime within our event horizon. So the information is not destroyed, it remains. However, the expansion of the universe has an effect on space and the things that are in it. For example, photons are stretched with space. Their wavelength increases and the energy decreases.

8.5 The Universe and Its Evolution

207

Expansion of the universe The expansion of the universe is essentially adiabatic in the standard model. Thus, there should be no significant change in entropy. In the Sect. 5.4.4, it is reasoned that a system can give off information during adiabatic expansion. Thus, if the expansion of the universe were adiabatic, it could lose information. Where would the information go? Dark energy could solve the problem. Dark energy acts negatively and could absorb the information. Perhaps just therefore the dark energy is present. However, an expansion is possible also without energy loss, it would increase then only the entropy (see Fig. 5.2). Speculative predictions about dark matter and dark energy are certainly possible. Following the definition of information in Sect. 2.6.1, this dark matter should represent information as well. Currently, gravitational effects are the only way to obtain information about dark matter. In an analogous way could be speculated about the dark energy. Structure formation in the universe How does it look globally with the structure formation in the universe. In a very early phase, when the light radiation decoupled from the plasma, the universe was quite homogeneous. Today the background radiation has a temperature of 2.73 ◦ K and is distributed very homogeneously in the sky. The universe at that time consisted mainly of light elements, mainly about 75% hydrogen and about 25% helium. These elements were formed in primordial nucleosynthesis during the first 3 min of the universe’s existence. Only after that galaxies have formed during the cooling. Then stars have formed from the gases by gravitational collapses. A hierarchy has been formed, which has the structures: galaxy-clusters, galaxies, and stars. Heavy elements form during nuclear fusion within stars. Star explosions (supernovae) produce a variety of heavy elements that are emitted into the interstellar medium. In addition, many stars emit dust at the end of their lives. These clouds of gas and dust collapse back into stars. There is a cycle of star formation. It seems that the development of structures does not go in the direction of homogeneity, but in the other direction. It almost looks as if the entropy would decrease. However, in the cycle of stellar explosions and star formation, one direction of evolution is recognizable. In supernova explosions, heavy elements are blown into space. New star generations have a higher portion of heavy elements than previous ones. The trend in nuclear fusion is toward iron. Heavier nuclei would decay by nuclear fission. At the moment, the element frequency has not yet shifted significantly toward heavier nuclei. The nuclear fuel for the stars will last for many billions of years, but eventually only iron will remain. Probably, in this sense, the entropy will increase over wide areas. This would correspond to the second law of thermodynamics. However, it is not applicable to the universe without further assumptions, because the universe cannot be considered as closed. Moreover, it is questionable whether the concept of entropy can be applied to the universe. In smaller areas of the universe, one can assume the conservation of energy and information. But because of the not definable simultaneity in the universe and because of the expansion of the whole universe, this statement is problematic for the whole universe.

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Cosmological inflation Concerning the information transfer processes in the universe, there are oddities: The background radiation is isotropic except for very small fluctuations. It has very exactly the same temperature in all directions. If the speed of light is the highest speed at which information can be transmitted, how did areas that lie in opposite directions from us synchronize? How can the observed global isotropy of the universe be explained? The answer is a phase of inflation in an evolutionary stage that occurred before the decoupling of background radiation. According to the standard model of cosmology, this cosmological inflation started 10−35 s after the Big Bang and lasted until 10−30 s. During this time, the universe has expanded by a factor of 1030 –1050 . Today, manageable universe has had a size of about 1 m afterwards. About the causes of the cosmological inflation is still speculated. Undisputedly, however, it solves a number of problems [99]. 1. It explains that the cosmos has similar structures everywhere because all areas had temporary interaction before inflation. This explains also the isotropy of the background radiation. 2. It explains that the cosmos has no measurable space curvature. The rapid expansion during the inflationary phase made the cosmos flat. 3. It explains the density fluctuations as quantum fluctuations of the inflation field. From these density fluctuations later, galaxies and galaxy clusters have developed. 4. It explains that today no magnetic monopoles are observed. They have disappeared in the inflation phase. The cosmological inflation explains information processes which have influenced the structure of the universe essentially and fundamentally. There has been an information in the cosmos by interactions. Then the space has expanded inflationary. Thus, the interaction has been interrupted. Only much later, when the galaxies have originated, it comes again to appreciable interaction and information exchange.

9

Resume

Starting from the recognizable deficit of a physically founded concept of information which considers the dynamics as well as the objectivity of information, a new concept of information is founded in this book. The introduction of this concept of information is based on quantum mechanics and objectifies sender and receiver. As usual in information theory so far, entropy plays the central role. In extension of the previous approach, not the entropy itself, but the entropy transferred per time unit is defined as information. This information is called dynamic information. The distinction is necessary because entropy itself is often called information. This approach leads to a conservation law for this dynamic information. The close connection between the dynamic information and the transferred energy lets the difference in principle between information and energy fade into the background. In various application cases and examples, the conservation law is made plausible, its usefulness is shown and verified. Thermodynamics plays an important role because entropy and energy play a fundamental role in it. Any concept of information that uses entropy must be able to be embedded in the wellfounded building of thermodynamics. In information technology, dynamic information does not play a dominant role today, because technical processes still use too much energy for information transmission by many orders of magnitude. Or, speaking in the image of dynamic information, too much useless information is transmitted, which ultimately increases the entropy of the environment via the fans of electronic devices. This situation will change in the future when the energy efficiency of electronic circuits is substantially increased and more quantum optical methods are used in information technology. Quantum computing will then gain in importance. The concept of consciousness is directly related to the concept of information, because systems with consciousness are always information-processing systems. The concept of dynamic information is helpful for the understanding of consciousness and fits into this complex topic. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 L. Pagel, Information is Energy, https://doi.org/10.1007/978-3-658-40862-6_9

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9 Resume

Because the topic of information is very comprehensive, not all aspects could be covered, of course. Seen in this light, the book is more of a suggestion than an exhaustive treatise. Many aspects are still to be investigated. Mathematical proofs and generalizations are the next necessary steps. The concept of information is defined more sharply and the physical quantity of energy does not change its previous meaning. But one can see behind every energy and information and vice versa behind every information also energy. If one imagines the world consisting of quantum bits, then a fundamental difference between information and energy is no longer detectable. Both terms designate only different views on one and the same object. In view of the fact that information can often trigger major effects in the social sphere, the title “Information is energy” also takes on a more general meaning that goes beyond the purely physical interpretation of the terms information and energy. In summary, regardless of the reader’s point of view, information always has something to do with energy: social, physical, and informational.

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Index

A Adiabatic Logic, 105 Adiabatic processes, 119, 124 Algorithm, 188 α-entropy, 84 Alphabet, 86 Artificial consciousness (AC), 168 Artificial intelligence, 16, 167 Astronomy, 199 Automata theory, 43 Awareness, criteria, 190, 197

Classical ideal gas, 119 Collapse of the wave function, 40 Collapse of wave function, 37 Communication, 20 Computer, 102 Consciousness, 167 Conservation of information, 57 Contradictions, 178 Correlation entropy, 84 Cosmology, 199 Creativity, 176

B Boltzmann gas, 134 Ban-information unit, 109 Bekenstein-Hawking entropy, 203 Bit, 9, 86 Black holes, 202 Boltzmann constant, 95 Bongard information., 89 Bongard-Weiss entropy, 89 Bra-ket notation, 34 Brukner-Zeilinger entropy/information , 84 Business sciences, 76

D De-Broglie-wavelength, 106 Decoherence, 37, 40 Degree of freedom, 113 Density matrix, 100 Dirac’s bra-ket notation, 34 Dissipation, 150 Dit-information unit, 109 Dynamic bit, 29, 54 Dynamic information, 24

C Cbit, 34, 44 Chaos, 158 Chemical potential, 118 Church-Turing hypothese, 167 Church-Turing-hypothesis, 16

E Efference copy, 175 Electrons, 131 Energy and entropy, 96 Energy transfer, 24 Entanglement, 40, 42 Entropy, 8, 9, 53, 85, 88, 90, 100 Entropy and entropy, 96

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 L. Pagel, Information is Energy, https://doi.org/10.1007/978-3-658-40862-6

217

218 Entropy and volume, 96 Entropy flow, 107, 110, 111, 117 Entropy of a probability field, 88 Entropy of mixing, 153 Entropy transfer, 117 Entropy transmission, 110 Entropy, dimension, 95 Entropy, thermodynamic, 95 Epimenides paradox, 184 EPR state, 42 Event horizon, 201 F Flow density of probability, 58 Formation of structure, 158 G Gas bubble, 56 Gödel incompleteness theorems, 184 H Hadamard transform, 38 Halting problem, 182 Hartley entropy, 84 Hartley information unit, 109 Hawking radiation, 202 Heisenberg, 20 Hilbert space, 35 Holographic universe, 205 I Information, 1, 8, 29, 80 Information balance, 3 Information conservation, 60, 157 Information paradox, 202 Information source, 86, 87 Information Technology, 4 Information theory, algorithmic, 68 Information transfer, 21 Information transmission, 24 Information, C. F. von Weizsäcker, 72 Information, actual, 71 Information, potential, 71 Information, semantical, 72 Information, stationary, 107 Information, syntactical, 72

Index Internalization, 175 Interpretation of information, 66 Invariances, 60 Irreversibility, 141, 142, 157 Isothermal processes, 132 K Kolmogorov complexity, 93 Kolmogorov-Sinai entropy, 89 Kullback-Leibler information, 88 Knowledge, 71 Kolmogorov complexity, 68 Kuhlen, Rainer, 71 L Letter, 86 Liar paradox, 184 Liberty, 113 Life, 194 Light cones, 200 Logical systems, 176 Loops in algorithms, 189 M Maxwell’s demon, 146 Mach-Zehnder interferometer, 40 Machine consciousness(MC), 168 Markow chain, 112 Mix entropy, 153 N Nat-information unit, 109 Negentropy, 90 Nepit-information unit, 109 Neumann-Landauer-limit, 104 Neural networks, artificial, 187 Nit-information unit, 109 No-cloning theorem, 15, 46, 127 Noise, 136, 137, 139 Nyquist-Shannon sampling theorem, 96 O Objectivity, 142 Observer, 8, 20 One hot coding, 31 Open systems, 154

Index P Parallelization, 126 Phase transformation, 132 Photoelectric effect, 56 Photonics, 105 Photons, 127 Physical similarity, 122 Pn-junction, 56 Power-delay-produkt, 4 Pragmatic information, 75 Probability field, 85, 88 Probability measure, 85 Probability space, 85 Processual information, 77 Pseudorandom numbers, 69 Q Qbit, 33, 34, 43 Quantum bit, 15, 25, 33, 109 Quantum computing, 105 Quantum computing , 47 Quantum fluctuation, 24 Quantum limit, 4, 105 Quantum register, 37 Quantum systems, scaling, 122 Quasi-adiabatic electronics, 105 Qubit, 24, 33 Qubits, 34 R Random number generator, 69 Reafference principle, 175 Redundancy, 63, 66 Relational information theory, 77 Relativistic effects, 199 Renyi information, 83 Reversible computing, 105 Russel’s antinomy, 183 S Schrödinger equation, 124 Schrödinger equation, 98 Self-awareness, 173 Self-reference, 181, 184, 186, 188 Serialization of data streams, 132

219 Shannon entropy, 83, 84 Shannon information, 67 Shannon, Sn-Information unit, 109 Single particle system, 99 Space, 22 Space-time, 201 Speech, internal, 175 Spin-singlet state, 42 State space, 94 Stationary information source, 87 Strong AI hypothesis, 167 Structure formation, 159 Subject, 19 Synthetic consciousness, 168

T Theorem of Holevo, 30, 37, 52, 127, 139 Theory of ur alternatives, 74 Thermal noise, 103, 106 Thermodynamics, 8, 92, 102 Time, 22, 200 Time and consciousness, 195 Transaction time, 53, 71, 96, 111, 119, 129, 133 Transferred energy, 118, 119 Transferred entropy, 117 Transferred information, 119 Transinformation, 76 Transmission time, 24 Turing machine, 171 Turing-machine, 16, 70

U Universe, 206 Ur theory, 74

V Vector space, 34 Volume and entropy, 96 von Neumann entropy, 101

W Wave function, 99, 100 Word, 88