Quantum Bio-informatics IV: From Quantum Information to Bio-informatics (Qp-Pq: Quantum Probability and White Noise Analysis) [1 ed.] 9814343757, 9789814343756

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. Quantum Bio-Informatics IV From Quantum Information to Bio-Informatics

QP-PQ: Quantum Probability and White Noise Analysis*

Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy

QP-PQ: Quantum Probability and White Noise Analysis Vol. 28:

Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 27:

Quantum Probability and Related Topics eds. R. Reballeda and M. Orszag

Vol. 26:

Quantum Bio-Informatics III From Quantum Information to Bio-Informatics eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 25:

Quantum Probability and Infinite Dimensional Analysis Proceedings of the 29th Conference eds. H. Ouerdiane and A. Barhaumi

Vol. 24:

Quantum Bio-Informatics II From Quantum Information to Bio-informatics eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 23:

Quantum Probability and Related Topics eds. J. C. Garcia, R. Quezada and S. B. Santz

Vol. 22:

Infinite Dimensional Stochastic Analysis eds. A. N. Sengupta and P. Sundar

Vol. 21:

Quantum Bio-Informatics From Quantum Information to Bio-Informatics eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 20:

Quantum Probability and Infinite Dimensional Analysis eds. L. Accardi, W. Freudenberg and M. Schilrmann

Vol. 19:

Quantum Information and Computing eds. L. Accardi, M. Ohya and N. Watanabe

Vol. 18:

Quantum Probability and Infinite-Dimensional Analysis From Foundations to Applications eds. M. Schilrmann and U. Franz

Vol. 17:

Fundamental Aspects of Quantum Physics eds. L. Accardi and S. Tasaki

Vol. 16:

Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsui and A. Hara

Vol. 15:

Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg

*For the complete list of the published titles in this series, please visit: www.worldscibooks.com/series/qqpwna_series.shtml

QP-PQ Quantum Probability and White Noise Analysis Volume XXVIII

Juantum

Bio-liljormatics IV From Quantum Information to Bio-Informatics Tokyo University of Science, Japan

10 - 13 March 2010

Editors

Luigi Accardi Universita di Roma "Tor Vergata ", Italy

Wolfgang Freudenberg Brandenburgische Technische Universitat Cottbus, Germany

Masanori Obya Tokyo University o/Science, Japan

Ｇｾｷｯｲｬ､＠

Scientific

NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CH ENNAI

Published by

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QUANTUM BIO-INFORMATICS IV QP-PQ: Quantum Probability and White Noise Analysis - Vol. 28 Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This bo.o.k, o.r parts thereo.f, may no.t be repro.duced in any fo.rm o.r by any means, electronic o.r mechanical, including pho.to.co.pying, reco.rding o.r any info.rmatio.n sto.rage and retrieval system no.w kno.wn o.r to. be invented. witho.ut written permissio.n fro.m the Publisher.

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (p. v)

PREFACE

This volume is based on the fourth international conference of quantum bio-informatics held at the QBI Center of Tokyo University of Sciences. The purpose of the conference is towards new stage making interdisciplinary bridges in mathematics, physics, information and life sciences, in particular, research for new paradigm for information science and life science on the basis of quantum theory. More than 100 researchers in various fields such as mathematics, physics, information and biology come from all over the world. The conference was held for nearly one week, and we had a lot of fruitful discussion. In this fourth conference, particular attention is come up on quantum entanglement, simulation of bio-systems, brain function, quantum like dynamics and adaptive systems. Most of speakers gave care to the relation between their own topics and the mystery of life. The papers submitted in this volume are all referred, whose contents are related to one of the following subjects: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Mathematics of Cryptography and its related topics Quantum algorithm and computation Quantum entanglement Quantum entropy and information dynamics Quantum dynamics and time operator Stochastic dynamics and white noise analysis Brain activity Quantum like models and PD game Quantum physics and superconductivity Quantum tomography and sufficiency Adaptation in Plants Alignment of sequences

Luigi Accardi Wolfgang Freudenberg Masanori Ohya

v

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. vii- x)

CONTENTS Preface

v

The QP-DYN Algorithms

1

L. Accardi, M. Regoli and M. Ohya

Study of Transcriptional Regulatory Network Based on Cis Module Database

17

S. Akasaka, T. Urushibam, T. Suzuki and S. Miyazaki On Lie Group-Lie Algebra Correspondences of Unitary Groups in Finite von Neumann Algebras

29

H. Ando, 1. Ojima and Y. Matsuzawa On a General Form of Time Operators of a Hmiltonian with Purely Discrete Spectrum

41

A. Ami Quantum Uncertainty and Decision-making in Game Theory

51

M. Asano, M. Ohya, Y. Tanaka, A. Khrennikov and 1. Basieva New Types of Quantum Entropies and Additive Information Capacities

61

V. P. B elavkin Non-Markovian Dynamics of Quantum Systems

91

D. Chrusciriski and A . Kossakowski Self-collapses of Quantum Systems and Brain Activities

K.-H. Fichtner, L. Fichtner, W. Freudenberg and M. Ohya

vii

101

viii

Statistical Analysis of Random Number Generators

117

L. Accardi and M. Gabler Entangled Effects of Two Consecutive Pairs in Residues and Its Use in Alignment

129

T. Ham, K. Sato and M. Ohya The Passage from Digital to Analogue in White Noise Analysis and Applications

137

T. Hida Remarks on the Degree of Entanglement

145

D. Chrusciriski, Y. Hirota, T. Matsuoka and M. Ohya A Completely Discrete Particle Model Derived from a Stochastic Partial Differential Equation by Point Systems

157

K.-H. Fichtner, K. Inoue and M. Ohya On Quantum Algorithm for Exptime Problem

173

S. Iriyama and M. Ohya On Sufficient Algebraic Conditions for Identification of Quantum States

185

A. J amiolkowski Concurrence and Its Estimations by Entanglement Witnesses

199

J. Jurkowski Classical Wave Model of Quantum-like Processing in Brain

209

A. Khrennikov Entanglement Mapping vs. Quantum Conditional Probability Operator

D. Chrusciriski, A. Kossakowski, T. Matsuoka and M. Ohya

223

ix

Constructing Multipartite Entanglement Witnesses

237

M. Michalski On Kadison- Schwarz Property of Quantum Quadratic Operators on M 2 (C)

255

F. Mukhamedov and A . Abduganiev On Phase Transitions in Quantum Markov Chains on Cayley Tree

267

L. Accardi, F. Mukhamedov and M. Saburov Space( -Time) Emergence as Symmetry Breaking Effect 1. Ojima Use of Cryptographic Ideas to Interpret Biological Phenomena (and Vice Versa)

279

291

M. R egoli Discrete Approximation to Operators in White Noise Analysis

311

Si Si Bogoliubov Type Equations via Infinite-dimensional Equations for Measures

321

V. V. Kozlov and O. G. Smolyanov Analysis of Several Categorical Data Using Measure of Proportional Reduction in Variation

339

K. Yamamoto, K. Tahata, N. Miyamoto and S. Tomizawa The Electron Reservoir Hypothesis for Two-dimensional Electron Systems

355

K. Yamada, T. Uchida, M. Fujita, H. Koizumi and T. Toyoda On the Correspondence between Newtonian and Functional Mechanics E. V. Piskovskiy and 1. V. Volovich

363

x

Quantile-Quantile Plots: An Approach for the Inter-species Comparison of Promoter Architecture in Eukaryotes

373

K. Feldmeier, J. Kilian, K. Harter, D. Wanke and K. W. Berendzen Entropy Type Complexities in Quantum Dynamical Processes

387

N. Watanabe

A Fair Sampling Test for Ekert Protocol

403

G. Adenier, A. Yu. Khrennikov and N. Watanabe Brownian Dynamics Simulation of Macromolecule Diffusion in a Protocell

413

T. A ndo and J. Skolnick Signaling Network of Envitonmental Sensing and Adaptation in Plants: Key Roles of Calcium Ion

427

K. K uchitsu and T. K urusu NetzCope: A Tool for Displaying and Analyzing Complex Networks

437

M. J. Barber, L. Streit and O. Strogan

Study of HIV-1 Evolution by Coding Theory and Entropic Chaos Degree

451

K. Sato The Prediction of Botulinum Toxin Structure Based on in Silico and in Vitro Analysis

461

T. Suzuki and S. Miyazaki On the Mechanism of D-wave High Tc Superconductivity by the Interplay of Jahn-Teller Physics and Mott Physics H. Ushio, S. Matsuno and H. Kamimura

469

Quantum BiD-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 1-15)

THE QP-DYN ALGORITHMS LUIGI ACCARDI, MASSIMO REGOLI and MASANORI OHYA* Centro Vito Volterra, Universitii di Roma Tor Vergata, *Quantum Bio-informatic Center, Tokyo Universty of Science

1. QP-DYN algorithms: general scheme

The common denomination QP-DYN refers to a family of symmetric cryptographic algorithms based on a single mathematical structure, but differing in a potentially infinite class of specific realizations. We will see in the following how this flexibility in the choice of the realization can be used to arbitrarily increase security. The construction of these algorithms is inspired to a special class of deterministic dynamical systems (Anosov systems) whose chaotic properties are well known in the mathematical literature (see [5]). The theoretical bases of QPK-DYN algorithms are in the theory of chaotic dynamical systems and are described in the paper [2] which develops the previous [1] in which the main idea of the method was applied to the construction of sequences of pseudo-random vectors .. This paper also explains why the transition from the pseudo-random generation method to the cryptographic algorithm is non trivial: the point is that almost all results on this class of dynamical systems (in particular the proof of the of the caoticity proprerties) are based on techniques of measure theory and therefore are valid up to sets of zero measure. On the other hand it can be proved that the set of rational numbers, which are the only ones computers can deal with, is in the excluded zero measure set. Therefore the standard mathematical theory cannot guarantee the required chaotic properties. More refined mathematical arguments (dealing with the ergodic properties of periodic orbits, see discussion in [2]) can be of help, but this direction of the theory is much less developed than the standard one. For this reason the mathematical theory, even if providing a useful intuition of the direction to pursue, cannot guarantee a priori satisfactory statistical properties of the generated sequences

2

and these have to be verified a posteriori using the standard batteries of statistical tests. Moreover a straightforward transposition of the pseudo-random generation program produces an easily breakable cryptographic protocol (for more details see section (8)). Therefore the simulation of the chaotic dynamic system must be integrated with tricks of specifically cryptographic nature. After the above mentioned proposal in [1], other authors have developed the idea to use the hyperbolic automorphism of tori (Anosov systems) for the generation of pseudo-random sequences (e.g. [12], [9], [4], [13], [11], [6]), however we have not found evidence, in the literature, of cryptographic applications of such algorithms. In the analogy with dynamical systems the secret key is the dynamical law and the potentially public part of the key (seed) is the initial state of the system. Given these data, the secret shared key (SSK) is easily constructed from the orbit of the system corresponding to the given initial state. The general idea of the algorithm can thus be summarized as follows: A (Alice) and B (Bob) share the (secret) dynamical law of a dynamical system whose state space is public. In order to produce an SSK they publicly exchange an initial state of the system and each of them constructs the SSK by applying a pubicly known procedure to the orbit. By increasing the complexity of the dynamical law one can increase the security beyond any limit, but also the constructive complexity of the system increases and this decreases its speed. The balance between these two competitive requirements characterizes the good cryptographic algorithms. A posteriori one can recognize that, by appropriately choosing the state spce, any pseudo-random generator can be fitted into this scheme. Thus what makes the difference are the specific features of the state spce and of the dynamical law. The goal of every algorithm of the QP-DYN family is the following: given in input a text T of binary length IT E N U {oo}, produce a key of length equal to IT. Such algorithms are used in two situations: (i) the length of the text is known a priori: IT < 00 (ii) the length of the text is not known a priori: IT = 00 Case (i) is typical in storage applications, Case (ii) in streaming applications. The QP-DYN protocols are separated in three independent subalgorithms:

3

(I) Initialization algorithm (II) Secret shared key (SSK) generation algorithm (III) Codification and decodification algorithms (i.e. use of the SSK to exchange messages). The role of the initialization algorithms is to quickly loose memory of the public initial state (seed). This is equivalent to introduce a modification of the dynamical law in the first steps of the algorithm and can be done in a number of ways. The codification and decodification algorithms can be chosen arbitrarily however, since in the present case the SSK has the same length as the text and has very good statistical properties, while the potentially public part of the key can be changed for every text at zero cost, as codification/decodification algorithm it is sufficient to use the XOR of the SSK with the clear text in one time pad modality which, because of Shannon's theorem, is the one of maximum security. We will concentrate our attention on the core of the algorithm, i.e. the dynamical law which produces the SSK. All the specific protocols described in the following have been realized in software programs that have been tested in a variety of applications. The paper [8], based on part of of Markus Gabler's PhD thesis, discusses the results of a statistical analysis of the QP-DYN pseudo-random generators. This analysis has been repeated many times by several independent groups confirming the good statistical properties of these generators. The report [10] has been realized by the research group directed by Prof. Giuseppe Italiano of the Department of Computer science, Systems and Production, of the University of Rome Tor Vergata and compares the performances of the QP-DYN algorithms on cellular telephones with several known suites of cryptographic algorithms realizing public key exchange (RSA, Diffie-Hellmann, Elliptic curves) and subsequent encoding/decoding (AES, RC5). The result is that the QP-DYN suite produces longer SSK in shorter times. The content of the present paper is the following: - section (2) describes the dynamical systems underlying the QP-DYN algorithms - section (3) introduces the notion of key generating function (KGF) and describes how the secret key shared (SSK) is constructed from the orbits of the dynamical systems discussed in section (2) - section (4) explains why the dynamical systems described in section (2) are not adequate for cryptographic purposes and outlines the modifications

4

of the dynamical law introduced inm order to achieve this goal - section (5.1) describes the orbit jump function - section (6) describes the mechanism of machine truncation - section (7) introduces the use of multiple dynamical systems and explains how the KGF is modified in this framework - section (8) shows how the introduction of the cut function alone is sufficient to increase enormously the complexity of attacks even to the single matrix algorithm - section (9) shows how the introduction of a second dynamical system changes qualitatively the situation with respect to possible attacks in the sense that the attacker now faces an indeterminate rather than a difficult problem. Finally let us notice that the full control on the mathematical structure, in particular the heavy use of modular multiplications, has a price in terms of speed of the algorithm (about 80 machine cycles per byte): this is quite fast for most purposes, but not enough to rank the present algorithm among the fastest presently available stream ciphers. A faster version (by a factor of about 8) ofthe QP-DYN algorithm (QPDYN-S) has been implemented in software and submitted to all the tests of the evaluation program of the Lausanne SASC Conference (13-14 February 2008) (see [14]), available in the web page of the conference and consisting of 8 measures of speed and agility. We compared the performances of QP-DYN-S with the 8 finalist algorithms in the software profile selected by the conference. The results of these tests proved that QP-DYN-S was among the most performing 4 finalist algorithms. No algorithm, among the 8 finalists (plus QP-DYN-S), turned out better than the other ones in all these 8 parameters. For example our QP-DYN-S was about twice slower than the fastest one (10,25 machine cycles per byte against 4,48) but better in agility (21,44 against 29,50) and definitively faster than some popular algorithms, such as Salsa 20. A detailed description of the P-DYN-S algorithm will be discussed elsewhere.

2. Dynamical systems underlying the QP-DYN algorithms The QPK-DYN cryptographic algorithms are realized using variations of the class of (discrete time) dynamical systems described in the present section.

5

A dynamical system of this class is determined by: (i1) a natural integer dEN, called dimension of the algorithm (i2) an invertible d x d matrix M with coefficients M[i, j] in the natural integers (we will write simply M E M (d; N)) representing the law of motion of the dynamical system (i3) a natural integer PEN, called module (typically it is a large prime number) Therefore a such dynamical system is determined by the triple: (1)

{d, M,p}

As usual we often identify a natural integer with the string of O's and 1 's defined from its binary expansion. We will use integers of m bits, typically m=32

or m is a multiple of 32. Denoting 71,p the finite field with p elements, identified to the set of natural integers {O, 1, 2, ... ,p - I} , the state space of the dynamical system is by definition the vector space ＷＱＬｾＮ＠ The system is supposed to be reversible, i.e. denoting det(M): the determinant of the matrix M: det(M)

i- 0

mod p

Definition 2.1. The orbit of the vector Va E ＷＱＬｾ＠ O(M,va):= {va} U {Vi E 71,;

Vi+l

is by definition the set

= MVi

1 ｾ＠

i EN}

and Va is called the initial vector of the orbit. Since ＷＱＬｾ＠ contains exactly pd different vectors any orbit 0 is a finite set and therefore for each vector Va there exists TEN such that VT = Va. The smallest T with this property is called the period of Va. Since the dynamical system is deterministic and reversible, any vector in O(M, va) has the same period as Va and an orbit can intersect itself only if it comes back to the initial vector Va.

3. From sequences of vectors to sequences of bits The orbits described in the previous section generate pseudo-random binary sequences of arbitrary length through the use of a sequence of key generating functions (KGF) nEN

6

As explained above we identify an integer with its binary expansion therecan be thought to transform n d-dimensional vectors fore each function with integer components into a single binary string. Each step of the algorithm produces a d-dimensional vector with integer components (more precisely in {O, 1, ... ,p - I}). Each of these numbers is represented in the basis 2 with b binary digits (typically b = 32). Therefore every step of the algorithm produces a string of d . b bit. Consequently, after n steps of the algorithm a string of n . d . b bit will be obtained. The sequence of key generating functions ("'n) uses these strings to produce iteratively the SSK as follows: starting from the initial vector Va at step 0, after n steps the algorithm either has halted or has produced the vectors Va, V1, ... ,Vn . The (n + 1)-th step is the following. The algorithm: (i) compares "'n (V1' ... ,vn ) with IT (ii-a) stops if

"'n

(2) (ii-b) otherwise computes the (n V n +1

+ 1)-th vector

=

MV n

(mod p)

(3)

(iii) computes the (n + 1)-step key "'n+1 (V1' ... , vn+d (iv) goes to the next step. If the iteration is stopped at step N the pseudo-random sequence produced is "'N(Va, ... , VN). 3.1. Recursive construction of the sequence (K,n)

A computationally efficient way to construct the sequence ("'n) consists in computing recursively each by fixing a function:

"'n

"': N d x N

and defining the first step KGF "'1 : N d "'1 :

----t

----t

N

N by means of the prescription

x E N d ----t "'l(X) := ",(x, 0) EN

The sequence ("'n) is then defined inductively for n way: "'n+1: (x,y) ENd x N

----t

ｾ＠

",(0, n)

:=

2 in the following

"'n+1(X,y):= ",(x, "'n(Y)) EN

Definition 3.1. A function", : N d

",(x, n)

ｾ＠

n

X

N

----t

N that satisfies the condition

(4)

7

will be called a binary d-dimensional KGF. Remark 3.1. There are many interesting classes of binary d-dimensional KGF which are computationally easy to handle. The choice of such functions can be used: (i) in order to create personalizations of the algorithm (ii) in order to increase its robustness by keeping secret such choices In the following section we will describe the choice made in the present implementation of the QP-DYN algorithm. 3.2. KGF by left concatenation Definition 3.2. Given a function CXJ

A : 1'1

--7

1'1 ==

U {O, l}m m=l

the A-left concatenation function K,)., :

1'1 d

X

1'1

--7

1'1

is defined by

K,).,((nl, ... ,nd);n):= [A(nd), ... ,A(nd,n] where the right hand side denotes the binary string obtained by left concatenation of the binary strings A(nd), ... , A(nd, n in the given order. Remark 3.2. The A-left concatenation function K,)." defined by (5), depends on the choice of the function A. The two choices that we have currently implemented are: (i) the random truncation: A(n) removes all the bits of n from the left up to the first 1 included (if the first 1 is not removed, then each component of a vector produces a string of bits whose left extreme is always 1, thus decreasing the chaoticity of the procedure) (ii) the deterministic truncation of order c: A(n) removes the first T bits of n from left, where T is a pre-defined number (this choice is more convenient in hardware implementations). Example 3.1. If the components of the vectors are b-bit numbers, then in case (ii) each component produces b - T bits so that n vectors produce a string of (b - T)dn bits.

8

4. Modifications of the dynamical law The cryptographic robustness of the SSK, constructed in section (3) above, is based on the fact that the reconstruction of the dynamical law of a complex deterministic system from its orbits, is a very difficult problem even if these orbits are relatively simple. For example the reconstruction of the gravitational law from the elliptic orbits of the planets in the solar system has requided nearly one century of hard work of the best mathematicians, physicists and astronomers. If the reconstruction of the dynamical law of the system from its orbits is easy, then the cryptographic scheme outlined in the previous sections is weak under clear text attacks in the sense that, if an attacker E can obtain a pair of the form (clear text , encrypted text) then she can easily reconstruct the secret key. The dynamical system described in section (2) has two main defects: (i) it is not enough chaotic, i.e. does not pass some statistical tests (ii) it is not enough complex, i.e. the matrix M can be easily reconstructed once one knows a segment of orbit including a number of vectors of order d (this attack is described in the first lines of section (8)). The reason of this weakness is the linearity of the dynamical law described in section (2). One can remedy to these drawbacks by introducing additional nonlinearities which are strong enough to destroy any attempt to reconstruct, in an efficient way, the dynamical law from an arbitrary number of its orbits, but simple enough to implement, in order not to reduce the speed of the algorithm. The additional operations, introduced to hide the initial algebraic structure of the dynamical law are the following: (i) the cut (already described at the end of section (3.2)) (ii) the jump of orbit (see section (5)) (iii) the machine truncation (see section (6)) (iv) the XOR with another sequence produced by another dynamical system (see section (7)) Section (7.1) is dedicated to estimate how much part of the algebraic structure can be recovered after the single operation of cut and at which computational cost. The estimate is done in the case of fixed cut. In the case of random cut, corresponding to the currently implemented software version of the algorithm the complexity grows.

9

5. Orbit jump function We have already seen that, since all operations are taken modulo p, the space of the possible vectors contains a finite number of points, hence every orbit of every dynamical system is periodic and this can create problems even with simple statisical tests. It is usually assumed that a desirable condition for good cryptographic sequences is to have good statistical properties, i.e. to be able to pass some strongly demanding batteries of statistical tests (see however the considerations in [16] which show that this dogma has to to be taken with great caution). In order to achieve such a good statistical behavior it is necessary to make so that the periods of such orbits are at least very long (which is a necessary but not sufficient condition for chaoticity). To achieve this goal the original dynamical system is modified as follows. The algorithm confronts, at every step, the last vector produced, say V n , with the initial vector of the orbit Va (since the system is reversible only Va has to be memorized). If the two coincide, Vn is replaced by J(vn-d where J : N d --+ N d is a function, called jump function. This is equivalent to begin a new orbit from the vector J(vn-d, that therefore becomes the new initial point. Thus the protocol described in section (3) becomes modified as follows: (ii-c) after step (3) the algorithm verifies if

(6) (ii-d) if this happens, goes to step (iii) (ii-e) if (6) does not hold, defines V n+l

:=

J(vn )

(7)

and then goes to step (iii). It is clear from the above description that the role of the jump function J is to prevent, as long as possible, the occurrence of a periodic orbit, improving in this way the chaoticity of the generated sequences. Clearly this empirical prescription is not sufficient to guarantee the absence of short periods, in fact it is not difficult to build examples of systems that, even in presence of a jump function, are locked for ever in two short orbits. It is however an empirical fact that, with the introduction of a jump function, the occurrence of periodical orbits becomes very rare: in fact after several years of trials and millions of terabytes produced, such an occurrence has never shown up.

10

5.1. A choice of the orbit jump function It is clear that the jump function is largely arbitrary and its inclusion into the secret parameters of the algorithm improves its security. The algorithm actually implemented uses the following orbit jump function

J: v E ｚｾ＠

(2.....0·'. .. °0).. v+ (0).. 01 . ..

-->

1

. '.

.

00··· 1

1

(8)

6. Machine truncation The fact that usual machines deal with numbers with a pre-definite number of bits, say m, can be exploited to introduce an additional nonlinearity which increases the randomness of the system. Let m be the number of binary digits (precision) available for the computation and let the modulus p be chosen so to satisfy 2m ｾ＠

2P

Suppose that the entries of the secret key matrix M are not taken modulo p but are large enough (the more of them have m or m - 1 bits the better) so that, when one constructs the orbit, the summation occurring in the matrix-vector product, may lead to vectors whose components exceed 2m bits. When this happens the machine truncates the result to m bits, before computing the modulo p, according to the scheme: [

ＨｾｍＧ｛ｩＬ＠

k]· V[k])

mod 22m

1 mod p

7. Use of multiple dynamical systems An additional nonlinearity to the dynamics described in section (2) can be introduced by replicating the original procedure. Two possible choices to achieve this goal are: (i) to introduce a new dynamical law M' leaving the environment, i.e. the field Zp, unaltered (ii) to introduce a new environment Zp" leaving the dynamical law M unaltered. Both choices lead to different dynamical systems, i.e. to different orbits.

11

Since the choice (ii) has the computational advantage that, at each step of the iteration, one saves a matrix-vector multiplication, we have implemented this choice. In what follows we will illustrate this implementation.

7.1. The 2 prime protocol Given a text T of length IT we consider two dynamical systems

{d,M,p',va'''''n,J}

(9)

with: (il) the same dimension dEN (i2) the same dynamical law M E M(d; N) (i3) the same initial vector Va E N d (i4) the same orbit jump function J : N d ---+ N d , given by (8) (i5) the same key generating functions (KGF) ""n : Ndn ---+ N (n E N) (i6) different moduli pi, p" E N (prime numbers). One then executes the algorithm described in section (3) with the only difference that step (iii) (computation of the (n + 1)-step key) is replaced by the following two steps: «iii)-2syst-a) having produced, without any cut, the two (n + I)-step keys

(remember that, in the present implementation, the KGF the algorithm computes

""n are the same) (10)

where, for any two binary strings x, y, x EB y denotes the string x XOR y and if necessary the two strings are made of the same length by adding zeros on the left. (v-2syst-b) then removes from the bits of (10) all the leading O's and the first leading 1 (or, in the case of fixed cut, the first T bits from left). The result is the (n + I)- step key of the modified algorithm: -

- ( ""n+l Vl ,

I ') .. · ,Vn+l; Vl, .. ·,vn+ l

The stopping rule is the same as in section (3). Remark 7.1. It may happen that the string (10) has some leading bits equal to zero because, depending on the choice of the initial vector, the modulo operations

12

with pi and p" may enter the game only after a certain number of orbit steps: in these steps the vectors produced by the two systems are identical. Therefore an initial part of the resulting sequence should not be considered: the length of the omitted part is a parameter (which can even be public with no harm for the security of the algorithm) included in the initialization procedure.

8. Attacks to the 1-matrix algorithm The robustness analysis that follows has been developed in the worst possible hypotheses for the defender. That is: - it is only considered the case of a single dynamical system (thus excluding the most important security factor) - the machine cut is excluded - the jump function is excluded - one supposes that the only secret key is the matrix M while the following information are considered public: - the prime number p (module), - the dimension d, - the initial data initial Vo, - the KGF sequence (h: n ): left concatenation without permutations Furthermore: - the bit cut (see the end of section (3.2)) is considered fixed and public. - the most favorable case for the attacker E is considered, i.e.: the clear text attack, in which the both original text T and the encrypted text are known to the attacker. Clearly the degree of security grows if, as it is always possible, some of these informations are part of the secret key shared a priori. The fact that, even under these extreme conditions, the breaking complexity of the algorithm can be very high helps to guess why up to now it has not been possible to find, even at theoretical level, attacks to the 2-matrix version of the algorithm. Suppose that: (i) E knows d + 1 consecutive (column) vectors of the orbit starting from some lEN: {VI,VI+l,'" ,Vl+d} (ii) the first d among these vectors are linearly independent and define the following (column) matrices:

v=

(VI,'"

,Vl+d-l) E

M(d;N)

Vi =

(Vl+l,'"

,V/+d ) E

M(d; N)

13

then MV = V' and this allows to obtain the secret key M = V ,- 1 hence to break the algorithm. However, since E only knows a binary string her problem is to recover from it the components of the vectors VI+i. This means that E has to discover which bit is the first bit of the first component of VI. Once she has this information, since she knows from the public structure of the algorithm that the bits are generated from the vectors by left concatenation without permutations and that the cut is constant and equal to T, E can determine each component of each of the vectors {VI+l, ... ,vI+d up to an ambiguity of T bits per component. This implies 2T possibilities for component and therefore 2dT possibilities per vector. Since E needs d + 1 vectors, she has to choose among 2d(d+ 1)T possibilities. For example, if d = 10 (a dimension that an usual personal computer can manage without any difficulty), then d(d + 1) = 110. Supposing, in order to further facilitate E's task, that T = 2, we see that E has to choose among 2220 possibilities. For each of these choices E must carry out one inversion and one multiplication of matrices of order 10 (each of these operations requires an order of 103 mUltiplications). Finally notice that an increment of d or T, e.g. 15 instead of 10 or 3 instead of 2, increases the construction complexity of the orbit by a factor that is at most quadratic in the increment, while the complexity of attack increases exponentially.

9. Attacks to the 2-matrix algorithm The attacks described in section (8) cannot be applied to the 2- matrix algorithm because in this case E can only recover the sequence

(where EB denotes the XOR operation) and it is impossible to know if, in this sequence, a 1 has been obtained from the combination of a 0 in "'N(Vl , . .. ,VN) (SSK of the first dynamical system) and of a 1 in '" N ( ｶｾ＠ , ... , ｶｾＩ＠ (SSK of the second dynamical system), or vice-versa. Similarly it is impossible to know if a 0 from two O's or two l's. In other words, and this is one of the main ideas of the new algorithm,

14

E is not facing a difficult problem, but an indeterminate one, namely: given a sum of two elements in a ring, reconstruct the value of the addends. Since, fixing arbitrarily one of the two elements, the knowledge of the sum determines the other one uniquely and since, given the information available to E, all the elements of the ring are equiprobable, it follows that the ambiguity is of the same order of the number of elements of the ring. In our case this means that, for every component of every vector, E has an ambiguity, of order p. For each vector the ambiguity will be therefore of order pd and, for d + 1 vectors, of order pd(d+l). Finally the simultaneous use of the three different fields, i.e. Zp" Zp'" Z2 (where the last one refers to the XOR operation), makes an algebraic attack, even at the statistical level, practically impossible. References 1. Accardi L., F. de Tisi, A. Di Libero: Sistemi dinamici instabili e generazione di sequencei pseudo-casuali, In: Rassegna di metodi statistici e applicazioni, W. Racugno (ed.) Pitagora Editrice, Bologna (1981) 1-32 2. Abundo M., Accardi L., Auricchio A.: Hyperbolic automorphisms of tori and pseudo-random sequences, Calcolo 29 (1992) 213-240 3. Accardi L., Regoli M.: Some simple algorithms for forms generations, L. Accardi (ed.) Fractals in nature and in mathematics, Acta Encyclopaedica, Istituto dell'Enciclopedia Italiana (1993) 109-116 4. L. Afferbach and H. Grothe, J. Comput. Appl. Math. 23, 127 (1988) 5. Arnold V.I., Avez A.: Ergodic problems in classical mechanics, New York: Benjamin (1968) 6. L. Barash, L.N. Shchur, Periodic orbits of the ensemble of Sinai-Arnold cat maps and pseudorandom number generation Physical Review E 73, 036701 (2006) The American Physical Society (2006) 7. M.Cugiani: Metodi Numerico statistici (1980) 8. Markus Gabler: Statistical Analysis of Random Number Generators October (2007); see also M. Gabler's paper in these proceedings. 9. H. Grothe, Statistiche Hefte 28, 233 (1987) 10. Giuseppe F. Italiano, Vittorio Ottaviani ,Antonio Grillo, Alessandro Lentini: BENCHMARKING FOR THE QP CRYPTOGRAPHIC SUITE August (2009) 11. P. L'Ecuyer and P. Hellekalek, in Random and Quasi-Random Point Sets, No. 138 in Lectures Notes In Statistics Springer, New York (1998) 12. H. Niederreiter, Math. Japonica 31, 759 (1986) 13. H. Niederreiter, J. Comput. Appl. Math. 31, 139 (1990) 14. Mattew Robshaw, Olivier Billet (Eds.): New Stream Cipher Designs, The eSTREAM Finalists State-of-the-Art Survey, LNCS 4986 Springer (2008) 15. Regoli, M., pre-mRNA Introns as a Model for Cryptographic Algorithm: Theory and Experiments, proceedings: QUANTUM BIO-INFORMATICS

15

III From Quantum Information to Bio-Informatics Tokyo University of Science, Japan, 11-14 March 2009 16. Regoli, M., A redundant cryptographic symmetric algorithm that confounds statistical tests, Open Systems and Information Dynamics (2011) to appear

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Quantum BiD-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 17-28)

STUDY OF TRANSCRIPTIONAL REGULATORY NETWORK BASED ON CIS MODULE DATABASE SHIZU AKASAKAt, TOMOKO URUSHIBARA, TOMONORl SUZUKI AND SATORU MIYAZAKI Graduate School a/Pharmaceutical Sciences, Tokyo University a/Science 2641 Yamazaki, Noda-city, Chiba, 278-8510, Japan Microarray analysis is a high-throughput method for analyzing expression levels of multiple genes, therefore the microarray have been regarded by many investigators as a powerful method. Treating a huge amount of data and judgment of differentially expressed genes require appropriate statistical analysis. When the microarray analysis suggests there are co-expressed genes under a specific condition, there is high possibility that the common transcriptional factors (TFs) control them. It is also difficult to identify the TFs involved in co-expression through only biochemical experiments. In view of cis-element pattern related to co expressed genes might be one of the solutions to infer the gene expression mechanism clearly. So far, we have constructed Cis-Module database in order to specify cis-element location and distribution on genome. Using this database and rat microarray data, we have investigated the TFs network related to co-expression of genes. If we could also extract the human genes that are orthologous to co-expressed gene in rat, it will allow us to compare their cis-elements and TFs and to consider difference of gene expression profiles between rat and human. It will be very useful to find out attention to drug discovery targeting gene expression mechanism.

1. Introduction

In 2003, Human Genome Project was finished [1]. And all human genome sequence data has been determined and mapped genes on it. After that, many researchers have been studying gene expression in detail. That's because they want to find differentially expressed genes from these data for clarifYing the function of genes. However, it is not efficient to test huge number of genes individually so that we analyze gene expression. Recently, micro array analysis is a good method for analyzing expression levels of multiple genes. Treating a huge amount of data and judgment of differentially expressed genes require appropriate statistical analysis. When the micro array analysis suggests a set of gene expresses under some biological

t

Work partially supported by grant 2-4570.5 of the Swiss National Science Foundation. 17

18

condition, one has a valuable clue as to the detection of the function of the genes. If there are co-expressed genes under a specific condition, it is high possibility that these genes are controlled by the common transcriptional factors (TFs). As Fig.l, if we confirm co-expression of gene A and gene D, TF3 and TF5 may be common for each other. However, the number of co-expression gene are too large in micro array analysis, so it is difficult to identify TFs involved in co-expression through only biochemical experiments. Here we tried to look to cis-element pattern related to co expressed genes by bioinformatics and predict genes Figure I Co-expressed genes and controlled by same TFs. And we aim at common transcriptional factors making gene expression mechanism clear.

2. Transcriptional control and cis-modules Like Fig.l, some gene transcription is controlled by multiple transcriptional factors (TFs). Each transcriptional factor (TF) recognizes specific sequence in up or down stream of gene and the sequence is called cis-element (CE). When some TF recognize specific CE, gene transcription become activated or suppressed depending on the situation. Sets of cis-elements are involved in control of gene expression and they are called cis-modules especially. So it is important to study about cis-modules for clarifying gene expression mechanism. 3. Available Cis-Module Data in public database Currently, there are several available databases for transcriptional factors and their DNA binding site. So, a lot of cis-element sequences have been researched and stored in databases. However, there are few databases collect them as cis-modules. For example, JASPAR and TRANSFAC, which are both transcriptional factor databases, have cis-element patterns each TF recognizes. However, on those database, there are no information associated gene to cis-elements.

19

Transcriptional factor

ｦ Ｈ ｒＦ

', ".

!

ｔ ｇａｔｎｍ

ｾｔｆＹ＠

...,

ｦｾ＠

ＧＺｩｾ Ｑ＠

r

TF5

Ci s element pattern

Figure 2 Cis-element database and cis-element information

In addition to that, we can see another problems for some information of cis-modules based on bio-chemical experiment in International Nucleotide Sequence Database Collaboration (INSDC). Sometimes, cis-module for a gene is registered by different researchers as Entry_ A and Entry_ B (Fig.3). In this case, cis-elements SPI and AP2 (SPI and AP2 are name of transcriptional factor) are defined upstream region of Entry_A but not Entry_B. In Entry_B, 3 cis-elements for NFKB are described but 2 of them in Entry_A. That is, we can see each entry has different cis-element information To resolve this problem, we need to integrate a number of entries, described about same gene, in one entry. Therefore, grouping together cis-elements per a gene as cis-module and re-construction of cis-module database is required.

Figure 3 Current situation of cis-module entry and reconstruction cis-module database

20

4.

Construction Cis-module Datbase

4.1. Data sources and Data Collection (Fig4-1) In this work, the database responsible for five organic species (Homo sapiens, Mus musculus, Rattus norvegicus, Dorosophila melanogaster and Saccharomyces serevisiae) was constructed. To collect cis-element information, we used DDBJ (http://www.ddbj.nig.ac.jpl) which is one of three members of International Nucleotide Sequence Database Collaboration (INSDC), and this database stores information assured by biochemical experiments. At the same time, we extracted genome data from Ensembl database (http://www.ensembl.org/index.html).whichcontainedlocationofgeneloci. By using Ensembl data, we could bring together data.

4.2. Identification oj CDS location on Genome (Fig4-2) We extracted coding sequences on genome (ensembVCDSs) from genome sequence data. And then, CDSs of DDBJ entries (DDBJ/CDSs) were extracted from DDBJ data, and compared with ensmebl/CDSs with SSEARCH program. This process allowed us to identify locations of records on genome.

4.3. Extraction and Comparison upstream region (Fig4-3, 4-4) Upstream region of CDSs of DDBJ records (USRlDDBJ) and that of ensembl genome data (USRlensembl) were extracted. And, USRlDDBJ were compared with corresponding USRlensembl, leading to identification of cis-element location on genome. Through this process, we get cis-module entry which is cis-module information per gene.

21

(1) Obtaining data

Get records including cis elements information fromDDBJ

Get genome data of each species (Human, Mouse, Rat, Fly, Yeast) from Ensembl genome browser

(2)Identification of DDBJ records on genome

ｾ

(CDS) from DDBJ records using keywords search

Ｈａｾ

ｩＭ＼ｬ Ｚｨ ＢＧ ａｾ Ｇ ｾ ｾ＠

ｾＺ＠ ,"-"" "'ＲＺ _>"ｾ Ｉ＠ - - " .,Ｋ Ｏ ＺＧ ＠ｾ Ｚ Ｇ［ ｾ＠

(CDS) from EnsembI genome data (2b)

ｾ＠

oai,...;'H ,·- - ',-. ,-

Ｇｾ Ｌｾ

•.

.

ｾ

Mappmg (2a) based on (2b)

Mapping (3a) based on (3b) and identification cis element location on genome

Get cis-module information per gene

Figure 4 Flow chart about construction of Cis-module database

ｾｃＱﾣ＠

22

genome data D(from each species)

•

•

Extraction CDS from INSDB and Ensembl(2a,2b)

Extraction upstream region from DDBJ records (3a)

5'

--

BLAST ( (Mapping)

d

c:::{I ｾ＠

USRlDDBJ

c

-

II 3'

Extraction upstream region from genome data (3b) (USRlEnsembl)

5'

3'

Identification of cis-element location on genome Figure 5 Reconstruction of Cis-module database. Each number of this figure correspondent to the number offig.4 (flow chart)

23

5. Explanation of User Interface and Advantage of Cis-Module Database This database is freely available on the web (http://www.pharmacoinformatics.jp/cis/). Fig.6 illustrates how the cis-element information is displayed. By using this database, we can refer to cis-elements of each gene upstream region so that researchers are able to utilize the information. At the same time, we can check the distribution of cis-elements in upstream region of gene visually. In addition to that, we also get the information of cis-element location on genome. So, this database reflects total image of gene expression mechanism and databases like this are not before now.

Fieference

ｃ ｩ ｳｾｲ

･ｧｵｬ

-

Ilk

｡ｴｯｲｹ＠

• • \:

Module Database Fielease 1 .0

.SI:

-Figure 6 User interface ofeis-Module Database.

24

6. Application of Cis-module Database 6.1. Summary of Co-occurrence ofCEs and TFsfrom Cis-module Database A gene expression is regulated by sets of cis-elements and transcriptional factors. Therefore, we extracted entries that have more than two TF information from Cis-Module Database (CMDB) to know TFs working together when some gene express. In table 1, when the gene (module_7) changes the expression, TF1, TF3 and TF6 work for the change together. So we could get co-occurrence of CEs and TFs from CMDB. By this summary, we could know which TFs and CEs involved in a gene expression. Table I : The example of co-occurrence information about transcriptional factors (TFs).

TFI module 7

ｾﾥＬｓ＠ 0

...

TF3 ｲＢ［ｾａｆＬｙＰ

... module 431

TF2

Ｚ＠

...

TF6

ｾ｜＿ｩ

［Ｇｾ＠

0

0

0

6.2. Prediction ofgene network for caloric restriction Rat Recently, relationship between anti-aging and calorie control are suggesting. As the reason for that caloric restriction rodents have longer life. So we performed below process (Fig.7) to identify common transcriptional factors, which regulate differentially-expressed genes under calorie restricted condition. It is expected that they are involved in a key role for anti-aging. (1) Data analysis (about Microarray data) We used rat micro array data, which has 31099 genes proved and was kindly provided by Dr. Higami. In the present study, 3 groups were prepared: nontransgenic male Wi star rats ad libitum intake (AL), calorie restricted (CR) rats and heterotransgenic rats (tgl-: TO). Each group had 4 rats. We targeted at AL and CR rat to analysis to specify which genes change the expression under the caloric restricted condition. AL group rats continued to receive food at libtium, and CR group rats were provided with 70% of the mean daily intake of AL average. After the comparison of AL data and CR data, differentially expressed 54 genes were identified.

25

(2) Bio-informational approach for the analysis of 54 genes with cis-module database In this section, we described how our bio-informational approach were applied to identify the common transcriptional factors (TFs) for 54 differentially expressed-genes in caloric restricted rats and clustering 54 genes based on co-regulation of these TFs. First, we performed statistical analysis to identify co-expressed genes. Second, we extracted rat genome sequence and upstream regions (USRs) from about each gene gained by analysis result. In particular, we aligned cis-elements (CEs) for TFs in Cis-DB to know what types of TF involve in those genes expression. Then, we tried to evaluate co-occurrence pattern of CEs in co-expressed genes.

Ｍｾ＠

ｾＮ＠

Statistical

..ｾ＠ : Analysis·

31099 genes

--..

Co-expressed genes (54 genes)

ｧ･ｮｯｾ＠

ＮＺ［Ｂｾ＠

-:::>-+

ｾｃｙｍａｔｧ･Ｎｮｬ＠

USR

t:;;

Figure 7 Statistical analyses of microarray data and search cis-elements pattern

6.3. Evaluation CE pattern predicted in USR ofgenes We could find cis-element patterns automatically by use of the procedure mentioned above(Fig. 7). However, many false positive data might be included in the prediction. Therefore we need to propose the method to improve the accuracy of cis-elements patterns by use of Cis-module database (CMDB). If Co-expressed genes are detected under some condition, we can predict common cis-elements which involve in co-expression regulation of genes. For example, if we get co-expressed genes under certain condition and multiple CEs are found in the USRs of each gene, CE 1 and CE3 are predicted as common cis-elements involved in co-expression regulation of genes. And then, we can predict that TFs recognize those CEs are thought to relate to co-expression under the condition. CE I

｣ｾＺｉＡｄ］＠

.. :::1 .. ｉＡｩｬｾ］Ｚ｟＠

CE3

ｾ｣ﾷ］＠

CEI CE2 CEI ｟ｬｉ］ＺｩＡＱ｛ｾＧﾷ＠

CE3

CE4

•.］ｉｾ＠

CES

gene A

CE3

c::::J gene C CE5

.:= ..:= ...1'l"m. • .,.""••••I'!I!J., gene F

Figure 8 Co-occurrence of cis-elements in co-expressed genes

26

6.4. Identification ofgenes regulated by SREBPI There are reports by other research group that SREBP 1 is a key TF for calorie control. But no bio-chemically supported information of genes regulated by SREBP 1. So it is very useful to find genes regulated by SREBP 1. By using the cis element pattern recognized by SREBP 1, we aligned the cis element pattern for upstream regions of 54 genes, co-expressed under caloric restrict condition. As a result, we could identify some genes successfully.

7. Conclusion and Future works In this review, we introduced you the method of construction and application of Cis-Module Database (CMDB). Our database includes 5 species and grouped together cis-elements per a gene as cis-module. Again, uniqueness of CMDB is that we can check sets of cis-element, involving in each gene expression actually. So we can know the relationship between gene and cis-elements. In general, available databases on transcriptional factors and their DNA binding sites having cis-element patterns recognized by each transcriptional factor do not have any information between gene and cis-elements. That is, our database is only the database including cis-elements and their associated genes. Our database also will contribute to predict common cis-elements and transcriptional factors which related to co-expressed genes. As a next step, we are planning to predict of transcriptional factors network based on co-occurrence of cis-elements of differentially expressed genes. If we recognize some genes are controlled by same transcriptional factors, we get together these genes as one group. For example, if we recognize gene A and gene C are controlled by TFI and TF3, gene A and gene C get together as one group. In the same manner, we can construct a cluster for the other TFs. Then it will be possible to perform cluster analysis of TFs and construct gene network based on transcriptional factors.

08

Figure 9 Gene network based on transcriptional factors

Currently, there is some report describing "gene group are same, but TF group controlling these gene group are different". To evaluate this idea, we are also planning to extract homologous human genes to co-expressed rat genes and

27

construct gene network of Human (For example, gene D and gene D' have homology). And then, we intend to compare gene network for other species. It may indicate that transcriptional regulatory mechanisms are variable among species. We believe that our result will admit to find out attention to drug discovery targeting gene expression regulation mechanism. T F2&9

TF I&4

TF7&8 TFI &3

O,n, N 0, where [A, B] := AB - BA. Lemma 3.4. Let G be a strongly closed subgroup of U(9J1). Then 9 is a real Lie algebra with the Lie bracket [X, Y] := XY - YX. Based on the above preliminaries, we can prove the following main result. Theorem 3.1. Let G be a strongly closed subgroup of the unitary group U(9J1) of a finite von Neumann algebra 9J1. Then 9 is a complete topological real Lie algebra with respect to the strong resolvent topology. Moreover, 9c is a complete topological Lie * -algebra. Remark 3.3. It is easy to see that for G = U(9J1), its Lie algebra u(9J1) is equal to {A E 9J1; A * = -A} and the exponential map exp : u(9J1)

->

U(9J1)

is continuous and surjective. Proposition 3.1. Let 9J1 1 , 9J1 2 be finite von Neumann algebras on Hilbert spaces H 1 , H2 respectively. Let G i be a strongly closed subgroup of U(9J1 i ) (i = 1,2). For any strongly continuous group homomorphism

36

rp : G 1 ----t G 2 , there exists a unique SRT-continuous Lie algebra homomorphism M, n ;:::: k + 1. Then, for all 'Ij;

E

M n , n = 1,··· ,k, and

D(T), 2

(2.19) co

Mn

L L

1(ena,T'Ij;) 12



102

follows: There is a set of complex signals stored in the memory. Choosing one of these signals may be interpreted as generating a hypothesis concerning an "expected view of the world". Then the brain compares a signal arising from our senses with the signal chosen from the memory leading to a change of the state of both signals. Furthermore, measurements of that procedure like EEG or MEG are based on the fact that recognition of signals causes a certain loss of excited neurons, i.e. the neurons change their state from excited to non-excited. As it was pointed out by R. Penrose [25] this change that comes along with recognition of a signal can be identified with a process of self-collapses. A quantum statistical model of the recognition process should reflect the change of the signals and the loss of excited neurons. In a (still incomplete) series of papers the procedures of creation of signals from the memory, amplification, accumulation and transformation of input signals, and measurements like EEG and MEG are treated in detail (cf. [17, 8, 9, 10, 11, 12, 14, 13, 22]). In the present note we will not present this approach in detail and in full mathematical strength. A few of the basic ideas and structures of the proposed model of the recognition process will be sketched and we will explain the relation between this model and a certain process of self-collapses. In Section 2 we will collect some biological facts and experiments described in [26]. Moreover, we will cite some opinions given by R. Penrose, W. Singer and others allowing to formulate some basic postulates and requirements each brain model has to fulfill. Our model seems to be appropriate and in accordance with the experimental results. The subsequent section contains the basic mathematical notions. We introduce the underlying Hilbert space representing the space of signals and the basic operators acting on the signal space. In the sequel we explain the basic ingredients of our model of the recognition process. For simplicity we will restrict our considerations to the case of interaction of two signals one coming from our senses and the other one created from our memory. In Section 4 the single steps of the process of recognition will be modelled by a Markov chain with discrete time.

2. Some biological facts and experiments In this section we want to present some biological facts and experiments. Each mathematical model describing certain aspects of brain activities should be in accordance with these experimentally proven facts. Firstly, using EEG measurements one gets information on the densities of excited neurons located in the regions of the brain. These densities depend on time.

103

The curves are of the following type:

no activity

no activity

activity ... time

The behaviour of that curve shows two types of periods: periods of oscillations interpreted by specialists as phases of no activity, and periods interpreted as phase of special activity in the considered region of the brain. As an example we mention some experiments described in [26]. All regions of the cortex seem to have their own electromagnetic rhythms. For example, the so-called p.,- rhythm which is produced in certain parts of the cortex that process tactile stimuli and control movement, consists of two main frequency components, one around 10 Hz and the other around 20 Hz. The 20 Hz component of the p.,-rhythm seems to be closely related to motor functions, while the other one is linked more to the sense of touch. "What exactly causes the oscillation is still a big puzzle" [26]. The experiments show that the functional state of the cortex can be monitored by measuring changes in the 20 Hz rhythm in that area of the brain [26]. Using MEG one could show that this rhythm is significantly suppressed if the subject moves his fingers, thereby activating the motor cortex. Interestingly, a similar suppression occurs when the subject merely imagines making such movements or just views someone else moving their finger. Similar effects were observed by G. Rizzolatti (University of Parma/Italy), who used needle electrodes to measure the response of neurons in the brain of a monkey. It was observed that the neurons discharge both when the monkey picks a

104

raisin and when it sees the experimenter make the same movement. The conclusion is that one has to unravel the significance (or unimportance) of the cortical rhythms during the periods of recognition of signals or other actions of the brain. Summarizing one can state that there are periods of no activity in certain regions indicated by the rhythms mentioned above, and there are periods of action where the rhythms are suppressed. Considering a quantum model of the brain the periods of no activity in certain regions of the brain should be represented by a unitary evolution (the quantum system resp. a part of the system remains isolated from the enviroment), i. e. it is a process of type 2 in the sense of J. v. Neumann [40]. Now, if there will be a signal arising from the senses the process of recognition starts. That process represents a rapid sequence of "trials" checking whether the signal from the senses and the signal created by the brain (at least partially) coincide. Let us mention still Stapp (cf. [39]) who uses the second quantization of harmonic oscillators to explain the observed oscillations. Stapp identifies these trials with measurements performed by a mysterious "observer" in the brain what seems to be non-realistic. In our model of recognition these trials are not represented by a process of measurements. Nevertheless, the results of the trials are represented by projections like in the case of measurements. Now, the experimentally verified quantum Zeno effect tells us that a rapid repetition of a certain measurement will suppress the unitary evolution of the system, i. e. the effects of the measurements dominate the evolution of the system. We will see that the quantum model proposed by us enables one to explain the different phases of the curves obtained as the outcomes of EEG measurements using the quantum Zeno effect. Hameroff and Penrose discuss in [25] the concept of quantum theory related to some biological aspects of brain activities. Especially, they deal with the problem how recognition of signals is connected with a process of selfcollapses. We would like to mention some basic statements taken from their paper [25]: - As long as a quantum system remains isolated from its enviroment, it can be satisfactorily described in terms of a deterministic, unitarily evolving process. That process is computable, non-random, and reversible. - The conventional quantum theory view is that the quantum state reduces by enviroment entanglement, measurement or observation (subjective reduction). The mesurement process is non-computable, random, and irreversible, and it is known in various contexs as collapse of the wave function.

105

- A number of physicists have argued in support of special models in which the rules of standard quantum mechanics are modified by the inclusion of some additional procedure according to which the reduction of the state becomes an objectively real process (objective reduction) the system abruptly self-collapses. That would give a new non-computable theory, i. e. they are convinced that processes of self-collapses cannot be described in terms of the conventional quantum mechanics it requires additional postulates (relations to Gadel's incompleteness theorem ?) - Consciousness, it is argued, requires non-computability. The only readily available apparent source of non-computability are self-collapses. The self-collapse, irreversible in time, creates an instantanoues "now" event. Sequences of such events create a flow of time, and consciousness. As it was mentioned in Section 1 in a series of papers we developed a quantum statistical model describing certain aspects of the recognition process. In the sequel we will present now certain aspects of this model. We will argue that the model is in accordance with the experimental results and the opinions cited above. The paper is focussed on the relation between the process of recognition and a certain process of self-collapses. 3. The Space of Signals In the present section we introduce briefly notions and notations needed in the sequel. For interpretation and motivation of the introduced notions we refer to the above mentioned papers. Starting point will be a set G representing the space where the process of recognition and processing of the signals takes place. For the mathematical model it is irrelevant what is the concrete structure of G. So let G be an arbitrary complete separable metric space equipped with a fixed finite diffuse measure on the a-algebra of Borel sets of G. The elements of the Hilbert space L2 (G, fJ) can be interpreted as functions of the excited neurons. We assume that G decomposes into disjoint regions G I , ... , Gn being responsible for different tasks. So L2(G k ) represents the space of the excited neurons in the region G k . Now, let r(L2(G)) denote the symmetric Fock space over L2(G):

Incoming signals are identified with states on the symmetric Fock space 1{ = r(L2(G)). Why do we choose this Fock space as signal space? The main reason is the possibility to identify the Fock space over L2 (G) with

106

the tensor product of the Fock spaces over L 2 (Gd, ... , L2(G n ):

r(L2(G l U ... U G n )) = r(L2(Gd)0 ... 0r(L2(G n )). The signal decomposes according to the decomposition of the space. Now, we assume that the decomposition is fixed and maximal (very fine) in that sense that each region G r is responsible only for one simple task represented by an E L2(G r ), Ilrll = 1, r::; n. For each r E {I, ... ,n}, k ｾ＠ 1 define functions fl E r(L2(G r )) by

r

{VkT.fIr(Xj),X=(Xl, ... ,Xk)EGk ,

r-

fk (x)

:=

]=1

r-

,

o

fo (x)

_

:=

1I0(X),

elsewhere

Observe that for each r E {I, ... , n} UIh'2o gives an orthonormal system in r(L2(G r )). We denote by ｈｾｩｧ＠ c r(L2(G r )) the Hilbert space with basis UIh'2o and interpret this space as space of signals of type r. An especially important class of functions in the Fock space are the so-called exponential vectors. Exponential vectors in ｈｾｩｧ＠ are given by 00

exp{a· r} =

L k=O

ｾｦｉ＠

k

(a E C).

v k!

Observe that

m d exp{t· fr} I dtm t=O

and

= ｾ＠

. frm' m > O.

One easily concludes that the linear span of the exponential vectors is dense i. e. in ｈｾｩｧＬ＠ ｈｾｩｧ＠

= Lin{exp{a· tr}:

a E IC},

r E {I, ... , n}.

The space '1.Isig . IL

.

=

'1.I si g,o, ILl

,o,'1.Isig

we will call the space of regular signals. States on signals. The states Wg

:=

(1)

'U • • . 'U1Ln

Hsig

are called regular

e-llgl12 . (exp{g},· exp{g})

are called coherent states on ｈｾｩｧ＠ if g E L2(G r ) resp. on Hsig if g E L2(G). Hereby, (,.) denotes the scalar product in the corresponding Hilbert space. Roughly speaking, coherent states describe states of systems of quantum particles where each particle is in the same one-particle state. Furthermore, Wo = (exp{O}, . exp{O}) is called the vacuum state.

J07

4. The Process of Recognition The recognition process is based on a comparison of signals: one signal will be the input signal coming from our senses, the other one is taken from the memory. Both signals are modeled by the Hilbert space Hsig introduced above. The memory space Hmem is a further Hilbert space the structure of which we will not discuss in the present paper. Also we will not describe here the mechanism how the signal is taken out from memory. A detailed discussion and interpretation one can find in [7] and [8, 14, 15]. The whole processing procedure will take place step by step on the space

The comparison procedure between the two mentioned signals is done with the aid of operators (proj ections) on H := Hsig®Hsig , and we concentrate our considerations to these first two spaces of the tensor product space. Basic for our considerations will be the symmetric beam splitter being a well-known operator in quantum optics describing the splitting of coherent light into two beams. Definition 4.1. Let r E {I, ... ,n} be fixed. The linear operator

Vr(exp{f}®exp{g}) := exp

ｻｾＮ＠

(j

+ g)} ®exp ｻｾＮ＠

(j - g)}

(2)

we call symmetric beam splitter in the region r. It is well-known that tensor products of exponential vectors are total in r(L2(G r )) ®r(L2(G r )). So the symmetric beam splitter Vr is fully characterized by formula (2). Proposition 4.1. (Properties of the symmetric beam splitter) For r E

{I, ... ,n} the symmetric beam splitter Vr is unitary and self-adjoint: V; = Vr and V; =

][r(£2(G r ))0r(L2(G r

))

(][

denoting the identical opera-

108

tor). Moreover, jor all j, g E L2(G r

and a, (3 E C

)

Vr (exp{aj}0exp{(3j}) = exp

{a+(3} v'2 j 0exp {a-(3} v'2 j

(4)

Vr (exp{j}0exp{j}) = exp{ v2j}0exp{O} 1

1

Vr(exp{ O}0exp{j}) = exp{ v'2 j}0exp{ 1

(3)

v'2 j}

(5)

1

Vr (exp{j}0exp{ O}) = exp {v'2 j}0exp {v'2 j}

(6)

V; (exp{J}0exp{g} ) = exp{J}0exp{g}.

(7)

A good survey on properties of general (usually non-selfadjoint) beam splitting operators with an arbitrary number of in- and outputs is given in [24]. In the sequel Prw := (\{f, .)\{f denotes the projection onto (the subspace generated by) \{f. For r E {I, ... , n} define projections T[, T[, To, To on ｈｾｩｧＰ＠ by

T[ := ｖｲＨＱｉｴｾｩｧＰｐ･ｸｰｻｏｽＩＬ＠ = ＱｉｴｾｩｧＰ＠

To

:=

V;

To

:= Vr(Ir ＭＱｉｴｾｩｧＰｐｲ･ｸｰｻｏｽＩｖ＠

=:

IT) T[

= Ir -

= Ir -

T[

Observe that T[ has the property

T[ (exp{ar}0exP{br}) = exp ｻｾＨ｡＠

+ b)r} 0exp ｻｾＨ｡＠

+ b)r}.

So, T[ corresponds to a projection onto the subspace of ｈｾｩｧ Ｐ ｈｾｩｧ＠ with equal first and second factor of the tensor product. T[ is interpreted as recognition in the r-th region whereas To = Ir - T[ indicates that there was no recognition in the area r. Now we join together these operators T[ and fg of partial recognition resp. "no" recognition in the single areas to operators on the whole tensor product signal space H := Hsig0Hsig. We put

n := {O, I}n =

{E

(EI,'"

=

, En):

E {O, I}, k::; n}

n

n

'i

to.Tr

'= '6'

(£1 , . .. ,E n ) ·

r=l

Ek

r=l

Er '

109

Because of notational convenience we have put in the above definition T[ := T[ (however To =I- ｾＩ＠ This assembly of the operat ors in the single areas was possible because of

Now we want to sketch the process of recognition. Starting point will be a state {}o on H representing a pair of signals the first arisen from the senses the second one created from the brain. Now, there is chosen a certain element ;Sl = (cL ... ,ｅｾＩ＠ E n indicating what happens in the first step of recognition: - EX:

= 1 indicates recognition in the k-th region,

- EX:

= 0 means there

is no recognition in the k-th region.

The corresponding event will be identified with the projection TCcL ... ,e;). Like in the case of measurements the probability of the event (EL ... , ｅｾＩ＠ is given by

tr ( (}o . TCci ,... ,e;)) where tr( ·) denote the usual trace of an operator. Together with the probability tr({}o . TE" ) representing the subjective reduction of the state {} caused by the measurement according to TE" we can consider the following transformation of the state {}o representing objective reduction caused by the self-collapse indicated by ;Sl:

K E'l ((}o ) .._- TE'l {}o TE" tr({}o . TE") provided tr({}o . TE") > O. So for each given sequence (EL ... , ｅｾＩ＠ we get a channel mapping the initial state {}o into the new state KE" ({}o) where partial recognition appears in all regions k with EX: = 1. Starting point for the second step of recognition will be {}l := KE" ({}o) and a second sequence ;S2 E n. Applying the same procedure replacing {}o by {}l the probability of the event indicated by ;S2 will be

110

and caused by the self-collapse indicated by E'2 the state 01 be transformed to

= KE'l (00) will

provided tr(Ol . TE'2) > O. This procedure can be repeated arbitrarily often - given an initial state 00 and a sequence (E'k)k=l one obtains this way a sequence (Ok)k=O of states on 1i and a sequence (tr(Ok _ l . TE'k Ｉｾｬ＠ of probabilities of the events (E'k)k=l provided these expressions exist (i. e. if tr(Ok_l . TE'k) > 0 for k ｾ＠ 1). A necessary condition for this to hold is that the sequence ＨｅＧｫＩｾｬ＠ has to be in some sense increasing. More precisely, we require

(rE{l, ... ,n})

(8)

for all kEN. The relation (8) defines a semi-ordering of the elements in

n.

We write E'k ｾ＠ E'k+l to indicate relation (8). This property is in accordance with the requirements our model should fulfill. If once partial recognition in the area r occurs there will be no change for this region in the future, only further zeros may be transformed into 1. The full recognition of the signal would be obtained if for some kEN we have E'k = (1, .. . ,1). The sequence ＨｏｫＩｾ＠ is in accordance with this relation. Especially, we have if E'k - l ｾ＠

E'k

does not hold,

(9)

and 1'f

-k-l oo p>lul>l/p

(F'(u) -

ｾ＠

1

+u

)dn(u),

where dn(u) is the Levy measure. See [1] Section 3.2. In the present setup, the approximation of single Poisson component does correspond to the approximation of the delta measure on u-space. It is in line with the passage from digital to analogue.

References 1. T. Hida, Stationary stochastic processes. Princeton University Press. 1970. 2. T. Hida, Analysis of Brownian functionals. Carleton Math. Lecture Notes no. 13, Carleton University, 1975. 3. T. Hida, Brownian motion. Springer-Verlag. 1980. 4. T. Hida and Si Si, An innovation approach to random fields. Application of white noise theory. World Scientific Pub. Co. 2004. 5. T. Hida and Si Si, Lectures on white noise functionals. World. Sci. Pub. Co. 2008. 6. T. Hida, Si Si and Win Win Htay, A noise of Poisson type and its gdneralized functionals, preprint (submitted). 7. J.L. Lions, The earth, planet, the role of mathematics and supercomputers. (original in Spanish). Spanish Inst. 1990. 8. Si Si, Effective determination of Poisson noise. IDAQP 6 (2003), 609-617. 9. P. Levy, Theorie de l'addition des variables aleatoires. Gauthier-Villars, 1937, 10. P. Levy, Processus stochastiques et mouvement brownien. Gauthier-Villars. 1948. 2eme ed. with supplement 1965. 11. P. Levy, Problemes concrets d'analyse fonctionnelle. Gauthier-Villars. 1951. 12. W. Feller, An introduction to probability theory and its applications. vol.1. Wiley, 1950. Chapt. in particular Chapt. III. 13. I. Ojima, Levy process and innovation theory in the context of Micro-Macro duality, Proc. The 5th Nagoya Levy Seminar. 2006. 65-69.

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 145-156)

REMARKS ON THE DEGREE OF ENTANGLEMENT

DARIUSZ CHRUSCINSKIl, YUJI HIROTA 2 , TAKASHI MATSUOKA 3 AND MASANORI OHYA 4 1 Institute of Physics, Nicolaus Copernicus University 2 Quantum Bio-Informatics Center, Tokyo University of Science, 3 Department of Business Administration and Information, Tokyo University of Science, Suwa, 4 Department of Information Science, Tokyo University of Science We analyze a measure of quantum entanglement called degree of entanglement (DEN). It is shown how DEN behaves for well known classes of bipartite states. Moreover, we compare DEN for quantum states having the same marginals. Contrary to naive expectation it is shown that separable state might possesses stronger correlation (measured by DEN) than an entangled state. Keywords: Quantum entanglement, Quantum entropy

1. Introduction

In recent years, due to the rapid development of quantum information theory 1 the necessity of classifying entangled states as a physical resource is of primary importance. It is well known that it is extremely hard to check whether a given density matrix describing a quantum state of the composite system is separable or entangled. There are several operational criteria which enable one to detect quantum entanglement (see e.g. 2 for the recent review). The most famous Peres-Horodecki criterion is based on the partial transposition: if a state p is separable then its partial transposition pr = (ll ® T)p is positive. States which are positive under partial transposition are called PPT states. Clearly each separable state is necessarily PPT but the converse is not true. We stress that it is easy to test wether a given state is PPT, however, there is no general methods to construct PPT states. There are several measure of entanglement 2. However, there is no universal measure which shows that the problem of quantifying quantum entanglement can not be reduced to computing a single quantity. Moreover, 145

146

various measures are not compatible: if EI and E2 are two measures, then one can find two states p and pi such that EI (p) < EI (pi) but E2 (p) > E2(p'). It shows that various measures shows different aspects of quantum correlations. In the present paper we analyze a particular measure called degree of entanglement (DEN) and based on the mutual entropy.1O,1l,18 As other measures DEN uniquely characterized the entanglement of pure states. However, it gives only a partial answer for mixed states. This paper is organized as follows: in Section 2 we introduce basic properties of DEN. Sections 3 and 4 provide several examples of quantum states for which one easily compute DEN. Moreover, since they possess the same marginal states (maximally mixed) one can compare the corresponding degree of entanglement. Surprisingly, it turned out that separable state can have stronger correlation (with respect to DEN) than an entangled state. Final conclusions are collected in the last section.

2. The Degree of Entanglement We begin our discussion by recalling the definition of quantum entanglement. Throughout this paper, Hilbert spaces are assumed to be finite dimensional. If is the state on the Hilbert space HI ® H 2 , then Tr1i2e denotes the partial trace of with regard to H 2 .

e

e

Definition 2.1. Let H be a tensor product Hilbert space of two Hilbert spaces HI and H2 and B(H) the set of bounded operators on H.

(1) A state

e

on B(H) is said to be separable if there exist finite sequences of density operators {pJt'-,1 C B(Hd and {O'dt'-,l C B(H 2) such that

(1) with

"\,,N ｾＲ＠

(2) A state

A2 = 1 and A > 0 (Vi = 1 ... N) 1,

-

" .

e on B(H) is said to be entangled if it is not separable.

The classical example of an entangled pure state is given by w el - el ® eo) - Bell state of two qubits.

=

Jz (eo ®

Definition 2.2. Let HI, H2 be Hilbert spaces and e a density operator on HI ® H2 with marginal states p = Tr1i2e and 0' = Tr1il e. The DEN for e

147

with regards to p, a is defined by the following formula 1

D(() : p, a) = 2{S(P)

+ S(a)} - Ie(p, a),

where Ie (p, a) is the mutual entropy for () : Ie(p, a)

(2)

= tr()(log () -log p (>9 a).

In terms of von Neumann entropy, Ie (p, a) can be rewritten in the form + S(a) - S(()). Therefore, one obtains finally

Ie(p,a) = S(p)

1

D(() : p, a) = S(()) - 2{S(P)

+ S(a)}.

(3)

We will often use this form to calculate DEN in this and all subsequent sections. As an example let us calculate DEN for the singlet state w. The marginal states WI = W2 = ｾｨ＠ and hence one finds D(w : Wl,W2) = -log2 < O. Actually, one has the following Theorem 2.1. M. Ohya and T. Matsuoka 18 Let () be a pure state with marginal states p, a. Then, we have the following classification:

(1) () is separable if and only if D(() : p, a) = 0; (2) () is entangled if and only if D(() : p, a) < O. According to the above theorem, DEN gives us the necessary and sufficient condition for the separability of pure states. For mixed states one has 3 ,10,1l Theorem 2.2. Assume that the compound state () is a mixed state. If () is separable, then D(() : p, a) > O. Now, we compare quantum bipartite states with respect to DEN. Definition 2.3. Let ()1, ()2 be states which have common marginal states p, a. The state ()1 possesses stronger correlations than ()2 if the following inequality holds:

D(()l : p,a) < D(()2 : p,a).

(4)

The less value of DEN one gets, the stronger correlations one has. Here a natural question arises: Problem Let ()1 and ()2 be compound states having the common marginal states p, a. Assume that D(()l : p, a) < D(()2 : p, a). If ()2 is entangled, is ()1 entangled as well? In what follows we show that it is not the case.

148

3. DEN for 2-qudit states 3.1. Circulant states We start this section by recalling the definition of circulant state. 12 (see also Refs. [13, 14]). Consider the finite-dimensional Hilbert space Cd (d E N) with the standard basis {eo, el, ... ,ed-d. Let ｾｯ＠ be the subspace of Cd 0 Cd generated by ei 0 ei (i = 0, 1, ... , d - 1) :

= span{ eo 0 eo, el 0 el, ... ,ed-l 0 ed-I}. ｾｯ＠

(5)

For any non-negative integer 0:, we define the operator sa on Cd by

ek

f---+

(k = 0,1, ... , d -1),

ek+a(mod d),

and denote by ｾ｡＠ the image of ｾｯ＠ by Id 0 sa : ｾ｡＠ = (Id 0 ｳ｡ＩｾｯＮ＠ It turns out that ｾ｡＠ and ［Ｓ＠ｾ (0: =I- {3) are orthogonal to each other and

Cd 0 Cd ｾ＠

ｾｯ＠

EB ｾｬ＠

EB ... EB ｾ､ＭｬＮ＠

(6)

This decomposition is called the circulant decomposition. Let Po, PI' ... , Pd-l be positive d x d matrices with entries in C which satisfy

tr(po For each matrix Pa (0: ｰｾ＠ on (C d )®2 as

+ ... + Pd-l) = 1.

(7)

= 0,1,··· ,d - 1), we define the new linear operator d-l

ｰｾ＠

=

L

(8)

(ei' Paej leij 0 sa eij (S")*,

i,j=O where eij means leil(ejl. Since Skeij(Sk)* be also written as

= ei+k,j+k, we note that

ｰｾ＠

can

d-l

ｰｾ＠

=

L

(ei' p"ej leij 0 eHa,j+a·

(9)

i,j=O

One can easily check that the sum of these operators d-l

p" = ｌｐｾ＠

(10)

,,=0

defines a density matrix on (C d )®2. For further details of circulant states we refer to Ref. 12.

149

Let us consider a particular example of a circulant state for d

= 3:

1 (111) Po = A lII , 111 where

(12) and

E

> o. It can easily be checked that tr(po + PI + P2)

ti Po

1

= A

100010001 000000000 000000000 000000000 100010001 000000000 000000000 000000000 100010001

,

ti _ E PI - A

= 1. One finds

000000000 010000000 000000000 000000000 000000000 000001000 000000100 000000000 000000000

(13)

and

ti _ liE P2-A

000 000 000 000 000 000 001 000 000 000 100 000 000 000 000 000 000 000 000 000 000 000 000 010 000 000 000

(14)

Finally, one obtains the following family of circulant states

10 0 0 00 liE 00 0 10 0 00 0 00 0 00 0 10 0

oE

0 100 0 1 0 000 0 0 o 000 0 0 liE 0 0 0 0 0 o 100 0 1 0 OEO 0 0 0 00 E 0 0 0 00 o liE 0 0 10 001

(15)

150

0.8

0.7

0.6 0.5

0.4

0.3

0.2

0.1

x Figure 1.

The graph of D EN for

(/1 (x)

The marginal states p, (J of 8 1 (c:) can be calculated as

p = (J =

1

3 (eoo + ell + e22)

(16)

Therefore, we have

D (81 (c:): p,(J )

=

-A3

[ 3 log A + clog 3c: j[ + ｾｬｯｧａ＠ 1

3/c: +log3 J ,

(17)

with A defined in (12). Actually, one can classify the state of 8 1 (c:) by the value of c: (see Ref. 17). Theorem 3.1.

(1) 81 (c:) is separable iff c: = 1; (2) 81 (c:) is both PPT and entangled for c:

cJ l.

The corresponding graph of DEN for 81 (c:) is shown in Fig. 1. The maximal value corresponds to c: = 1 and is given by D (8 1 (1) : p, (J) = ｾ＠ log 3 ｾ＠ 0.7324.

151

3.2. Horodecki state

Let us consider another example of circulant states introduced in Refs. 15, 16: (0 ｾ＠

0: ｾ＠

5),

(18)

where

1/J= 8+

1 J3(eo0eo

+

e10 e 1

+ e20 e2),

(19)

1

= "3 (eoo 0 ell + ell 0 e22 + e22 0 eoo),

8- =

1

"3 (ell 0

eoo

+

e22 0 ell

+

eoo 0 e22) ,

(20) (21)

and hence 0 0

0 0 0

2 21

0

0 21 0 0 5- 0 0 ----:2l 0 0 5- 0 0 0 ----:2l 0 0 2 2 0 0 21 0 21 0

0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 21 21 0 221

0

0 0 0 0 0 0

0 0 0 0 0 0 0

2 21

0 0 0 2 21

(22)

0

0 21 5-a 0 ----:2l 0 2 0 0 21

The marginal states p and (J are the same as (16). The von Neumann entropy of 82 (0:) is calculated to be 2 2 0: 0: 5-0: 5-0: 8(8 2 (0:)) = -7 log 7 -7 log 7 - -7- log -7-

5

+ 7 log 3.

(23)

Therefore, we have

D(8 2 (0:): p,(J)

=

2 2 0: 0: 5-0: 5-0: 2 -7 log 7 -7log7 - -7-log-7- -7log3.

(24)

As in the case of the circulant state 8 1 (E), it is known that the state 82 (0:) can be classified by the values of 0: (see Refs. [15, 16]). Theorem 3.2.

(1) 82 (0:) is separable if and only if 0: E [2,3]; (2) 82 (0:) is both entangled and PPT if and only if 0: E [1,2) U (3,4]; (3) 82 (0:) is not PPT if and only if 0: E [0,1) U (4,5].

152

0.8

0.75 0.7

1:::

1;' 0.65

'" g;

0.6

ｾ＠

i

0.55

C')

OJ o

ｾ＠

0.5

c:: 0.45 X .;,

f

0.4

X

0.

0.35

ｾ＠

OJ o 0.3

ｾ＠ ｾ＠

C>

o

0.25

2

3

2.5

4

3.5

4.5

5

x

Figure 2.

The graph of DEN for (l2(X)

Now, since 81(10) and 82 (a) have common marginal states, we can compare DEN for them. One has for example

D(8 1 (1) : p,O") ｾ＠ 0.7324 < D(8 2 (3.1) : p,O") ｾ＠ 0.7587.

(25)

Note, however, that 81 (1) is separable while 82 (3.1) is entangled. This example gives negative answer to our original question. The corresponding graph of DEN for 82 (a) is shown in Fig. 2. 4. Bell diagonal states

In this section, we analyze Bell diagonal states for d = 3 (see Refs. [4, 5, 6, 7, 8, 9]). Consider the Hilbert space 1t =

re 3

with the standard basis

{eo, e1, e2 }. We set

(26) and define flk,e for any k, C (0 :S k, C :S 2) as

flk ,e = (Wk ,e ® 13)flo,o ,

(27)

153

where Wk,e means the circle action given by

(i=0,1,2).

(28)

Finally, we define

e( a, p) =

P( 0

l-a-p 9 I3 0 I3 + aIOo,o) (0 0 ,0 1 + 2"

1 1,0) (0 1 ,01

+ 10 2,0) (0 2 ,0 1) (29)

Note that e( a, p) can be written in the form 1+2(0:+(3) 9

0 O 0 20:-(3 -6-

0

0 0

1-0:-(3 -g-

1-0:-(3

0 0

0

0 0

0 0

0 0

20:-(3 -6-

0

0 0

0 0 0 0

Note that tr e(a, p)

=

20:-(3 -6-

20:-(3 -6-

0 0 0

0 0 0 0

0 0 0

0 0 0

0

0

1+2(0:+(3) 9

0

0

0 0

200-(3 -6-

0 1-00-(3 -g-

0 0

0

1+2(00+(3) 9

0

1-00-(3

0 0

0 0

0

0

200-(3 -6-

0

0

0 0 0 0

1 and the eigenvalues of e(a, p) are given by

l-a-p 9

-2a + 7p + 2 18

8a -

p+ 1

(30)

9

Therefore, e(a, p) defines a state if and only if the parameters a, p satisfy

a+p:::; 1,

2a -7p:::; 2,

-8a + p:::; 1.

(31)

On the other hand, let us ask for the condition when e(a, p) can be positive under partial transpose. The partial transpose Te(a, p) of e(a, p) is given by 1+2(00+(3) 9

0

0

0

1-00-(3 -g-

0

20:-(3 -6-

0

1-0:-(3 -g-

0

006

0

1-00-g-

0 0 0

0 0 0 0

0 0

O

0

0

0 0

O O

0 0 0 0

0 0 0 0

0 0 200-(3

0 0

1+2(0:+(3) 9

0

0

0 0 0

0

0 0 0 0

0 0 0 0

0

1-00-(3

20:-(3

0 0

200-(3 -6-

1-00-(3 -g-

0 0 0 0

0

0

0

1+ 2(00+(3) 9

0

154

From this one can check that the trace is equal to 1 and the eigenvalues are

1+2(a+,8)

-8a +,8 + 2

9

18

4a - 5,8 + 2 18

(32)

Hence, the PPT condition holds for a, ,8 such that

a

1

+ ,8 ?: -"2'

8a - ,8 ::; 2,

-4a + 5,8 ::; 2.

(33)

On the set which consists of a, ,8 satisfying both (31) and (33), the state is either separable or bound entangled. However, it was shown 7 that for this class all PPT states are separable. Let us consider for example the line ,8 = 3/ 5. One has: B(a , 3/ 5) is entangled if and only if

a E [0,1/4) U (13/40,2/5] ,

(34)

and it is separable iff

a

E

[1 / 4,13/ 40] .

(35)

The marginal states p and a are calculated to be the same as (16). Hence we obtain DEN for e(a, 3/5)

(2 - 5a)_ 31 - 10a l (31-lOa) / 5 ) .. p, a ) -_ lOa-4 ( ( DBa,3 15 l og 45 45 og 90 - 40a + 2 1og (40a + 2) - 1og 3 . 45 45

(36)

By a simple calculation, it easily be verified that D(e(a, 3/5) p, a) is monotonically decreasing if a E [1/20, 2/5). Hence, the values of DEN for a E [1 / 20,1/4) are greater than the ones for a E (1 / 4,13/40) (see Fig. 3). This gives also the negative answer to our original question. 5. Conclusions We provided several examples of bipartite quantum states for which one can easily compute DEN. Moreover, since they possess the same marginal states (maximally mixed) one can compare the corresponding degree of entanglement. It turned out that separable state can have stronger correlation (with respect to DEN) than an entangled state. This observation is inconsistent with the conventional understanding of quantum entanglement. Consequently, we propose that the meaning of DEN should be changed to express the intensity not of entanglement but of correlation. The details will be discussed in the forthcoming paper 19.

155

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 ·0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x Figure 3.

The graph of DEN for B(x, 3/5)

Acknowledgments

This work was partially supported by the Polish Ministry of Science and Higher Education Grant No 3004/B/H03/2007/33.

References 1. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum infor· mation, Cambridge University Press, Cambridge, 2000. 2. R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki: Quantum en· tanglement, Rev. Mod. Phys. 81 (2009). pp 865-942. 3. L. Accardi, T. Matsuoka, M. Ohya:Entangled Markov chains are indeed en· tangled, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 9 (2006), pp 379-390. 4. R. A. Bertlmann, and Ph. Krammer: Simplex of bound entangled multipartite qubit states, Phys. Rev. A 78(2008), 10pp. 5. R. A. Bertlmann, and Ph. Krammer: Geometric entanglement witnesses and bound entanglement, Phys. Rev. A 77, 024303 (2008), 4pp. 6. B. Baumgartner, B. Hiesmayr, and H. Narnhofer: State space for two qutrits has a phase space structure in its cone, Phys. Rev. A 74 (2006), l4pp.

156

7. B. Baumgartner, B. Hiesmayr, and H. Narnhofer: A special simplex in the state space for entangled qudits, J. Phys. A: Math. Theor. 40, 7919 (2007). 8. B. Baumgartner, B.C. Hiesmayr and H. Narnhofer: The geometry of biparticle qutrits including bound entanglement, Physics Letters A, 372 (2008), pp 2190-2195. 9. D. Chruscinski, A. Kossakowski, T. Matsuoka, K. Mlodawski, A class of Bell diagonal states and entanglement witnesses, to apper in Open Systems and Inf. Dynamics 10. V. P. Belavkin and M. Ohya: Quantum entropy and information in discrete entangled state, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 4 (2001), pp 137-160. 11. V. P. Belavkin and M. Ohya: Entanglement, quantum entropy and mutual information, Proc. R. Soc. London A 458 (2002), pp 209-231. 12. D. Chruscinski and A. Kossakowski: Circulant states with positive partial transpose, Physical Review A 76 (2007), 14 pp. 13. D. Chrusciilski and A. Pittenger: Generalized Circulant Densities and a Sufficient Condition for Separabil ity, J. Phys. A: Math. Theor. 41 (2008) 385301. 14. D. Chruscinski and A. Kossakowski, Multipartite Circulant States with Positive Partial Transpose, Open Sys. Information Dyn. 15 (2008) pp 189-212. 15. P. Horodecki, M. Horodecki, and R. Horodecki: Bound entanglement can be activated, Phys. Rev. Lett. 82, (1999), pp 1056-1059. 16. M. Horodecki, P. Horodecki and R. Horodecki: Mixed state entanglement and quantum condition, In Quantum Information, Springer Tracts in Modern Physics 173 (2001), pp 151-195. 17. J. Jurkowski, D. Chrusciilski and A. Rutkowski: A class of bound entangled states of two qutrits, Open Syst. Inf. Dyn., 16 (2009), no 2-3, pp 235-242. 18. M. Ohya and T. Matsuoka: Quantum entangled state and its characterization, Found. Probab. Phys. no 3, 750 (2005), pp 298-306. 19. D. Chrusciilski, Y. Hirota, T. Matsuoka and M. Ohya: Quantum correlation and essential q-entanglement, in preparation.

Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 157-171)

A COMPLETELY DISCRETE PARTICLE MODEL DERIVED FROM A STOCHASTIC PARTIAL DIFFERENTIAL EQUATION BY POINT SYSTEMS

KARL-HEINZ FICHTNER\ KEI INOUE 2 AND MASANORI OHYA 3 1 Friedrich-Schiller- Universitiit Jena, Fakultiit fur Mathematik und Informatik, Institut fur Angewandte Mathematik, 07737 Jena, Germany. E-mail: [email protected] 2 Department of Electrical Engineering, Tokyo University of Science, Yamaguchi, Sanyo-Onoda, Yamaguchi 756-0884, Japan. E-mail: [email protected] 3 Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan. [email protected]

Several scientific and technical problems can be described by a stochastic partial differential equation. The solution of the equation could be considered as the limit of a suitable discrete particle model. The existence of such a kind of approximation was discussed in 5. A completely discrete particle model, which is constructed to simulate by computer, is considered in 3. In this paper we give proofs of some lemmas which are used to prove the main theorem in 3.

1. Introduction

Several scientific and technical problems can be described by partial differential equations of the type Ｍ］ｾｶＫｦＮ＠

av

1

at

2

(1)

A more precise model is to take into account stochastic disturbances, i.e., to the right hand of (1) there have to be added corresponding source terms. Often this disturbance occurs as the so-called white noise. So we regard the stochastic partial differential equation

av at (x, t) = ＲｾｶＨｸＬ＠

1

t)

+ f(x, t) + O"(x, t) ｾＨｸＬ＠ 157

t).

(2)

158

Note that (2) is connected with the idea of diffusion and generation of particles in random spatial-temporal points. Therefore the solution of (2) could be considered as the limit of a suitable discrete particle model. The existence of such a kind of approximation was discussed in 5, and the related problems were considered in 4,6. In 3 we construct a completely discrete particle model approximating a continuous system being different from the limit considered in 5. In this paper we give proofs of some lemmas which are used to prove the main theorem in 3.

2. Basic notations

The Poisson process Following 7, we introduce the notions and notations in this section. Let G be a Polish space, i.e., a separable topological space for which there is a complete metric. We denote the (I-algebra of the Borel sets in G by ® and the ring of the bounded Borel sets of ® by .e. In the case G = R d we use notations ® = ®d, .e = .ed . Further, M denotes the set of all integervalued measures on ® being finite on.e. Let 9Jt be the smallest (I-algebra of M-subsets which makes the function

f---+

(X),

E M

measurable for each X from.e. A measurable mapping from a probability space [n, F, P] into [M,9Jt] is called a random point system in G, the distribution on [M,9Jt] generated by such a random point system is said to be a point process with the phase space G. To each measure H on [M,9Jt], a measure PH on ®, called the intensity measure of H, is assigned by PH(X):=

J

(X)H(d 0 denotes the Poisson distribution with expectation A. For all 'c-finite measure f.L on ® it holds (p(p)t *

where

(p(p)t*

= p(np)

is the n-th convolution-power of

(3) p(J-t)

(cf. 7).

l7-additive processes

Definition 2.1. A random process 7] = (7](B); BE ,c) on [n, F, Pl is called a-additive if for all sequences ＨｂｩＩｾｬ＠ of pairwise disjoint elements from ,c such that ｕｾｬ＠ Bi E ,c, we have

Definition 2.2. A a-additive process 7] = (7](B); B E ,c) is called deof pairwise disjoint bounded composable if for all finite sequences ＨｂｩＩｾｬ＠ measurable sets the random variables 7](Bd, ... , 7](Bm) are independent

(cf. 2). If \fF is a random point system in G then the family \II := (\fF(B); B E ,c) represents a a-additive process. Moreover, if the distribution of \fF is a Poisson process then \II is a decomposable process. The white noise based on the Lebesgue measure on [Rd, ®dl as well as the so-called generalized Brownian motion represents a decomposable process in this sense (cf. 8). Here U is called to be a generalized Brownian motion on G with noise intensity measure f.L if

(i) each random variable U(B) has law N(O, f.L(B)) , (ii) the variables U(B 1 ), ... , U(Bm) are independent B 1 , ... ,Bm are pairwise disjoint.

whenever

Definition 2.3. For each n E N let 7](n) and 7] be a-additive processes on [n, F, Pl. The sequence 7](n) is said to converge toward 7] if for every finite sequence ＨｂｩＩｾｬ＠ of elements from ,C the random vector

160

[7)(n) (Bd, ... ,7)(n) (Bm)] converges in distribution ( i.e. , weakly convergence of the probability distribution) to the random vector [7)(B 1 ), . . . , 7)(Bm)]. 3. A Special Type of a Stochastic Partial Differential Equation We consider the stochastic partial differential equation

av at (x, t) = ＲｾｶＨｸＬ＠

1

t)

+ f(x, t) + (J"(x, t) ｾＨｸＬ＠

t)

(4)

with initial value v(x,O) = 0) or negative(f(x, t) < 0) charged particles are added. Finally, a stochastic source term describing the stochastic disturbance is added. (J"2(x, t) is the intensity of the noise. It must be noted that in the higher dimensional case (x E R d, d > 1) the mathematical concept symbolized by (4) is not quite clear.

A Partially Discrete Particle Model A treatment of this case was discussed in 5. Firstly let us explain the main idea of the model. Let

0, be the stochastic kernel on [Rd, Qjd] given by

K)..(q,·) = N(q,)..1,·) Here I denotes the identity matrix and N(q, ),,1,') denotes the normal distribution with the expectation vector q and the diagonal matrix I as covariance matrix. Then

(5) describes how the charge present at initial time zero diffuses until the time

t. Here Zd denotes the Lebesgue measure on [Rd, Qjd]. The discretization of the process corresponding to the source term f occurs as follows: Let f be a bounded measurable function defined on R d X R +. We define functions f+, f - putting

f+

:= max{O,

f}, f-:= max{O, - f} .

161

f-L1 denotes the measure on R d+ 1 X f-L1(B x {I}) f-L1(B x {-I})

{

-1,1} characterized by

J =J =

j+(XhB(X) Zd+1(dx), BE £d+1,

j-(X)XB(x) Zd+l(dx), BE £d+1.

(j?n denotes a random point system in Rd+1 x {-I, I} with distribution P(nJ.Ll)' Thus (j?n describes the_ configuration of charge points which are added spatially-temporally. If {(In{[x, S, I]) = 1 then a charge unit of the amount l/n is added. If (j?n{[x, s, -I]) = 1 then a charge unit of the amount -1/ n is added. After occurring a particle, it diffuses. Thus the contribution of this source process to the charge density until t > 0 is given by

ｄｾｮＩｏ＠

= ｾ＠

J

C

(Kt-s(x, ')X(O,t)(s)

+ 8xOX{t}(s)) (j?n(d[x, s, c]).

(6)

Here XB denotes the indicator function of a set B. The disturbance term is transformed similarly. Let IJ be a bounded measurable function defined on R d X R +. 1J2 is considered as the density of a measure f-L2 on [Rd+l, Qjd+1]. We define a measure f-L on Rd+1 x {-I, I} by f-L := f-L2 x

1

2" (8 1 + L d·

Let {(In be a random point system in R d+1 X { -1, I} with distribution p(nJ.L)' Analogously to (j?n the random point system {(In describes the configuration of charge points which are added spatially-temporally with mean intensity nIJ 2 including the "sign" of the unit charges. Differently to the effect of the first source process individual charges of the value 1/ Vn are generated. Thus the contribution of this second source process to the charge density until t > 0 is given by c;n) 0

=

In J

c (Kt-s(x,.) X(O,t)(s)

+ 8xOX{t}(s)) {(In (d[x, s, c]).

(7)

Each particle should give a contribution to the entire charge and evaluate independently on the other ones. These requirements do not follow from the properties of the Poisson process. Therefore, we assume that (j?n and {(In are independent. Consequently the entire process can be defined as the sum of independent random variables

(8)

162

By the normalization lin, respectively lifo, the "potential" of one generated particle decreases when n increases, whereas the average number of the particle increases, i.e., the produced charge is "smeared over" in a certain manner. From theorem 1.3.6 in 5 we can conclude: Theorem 3.1. Let be t > O. Then it holds

(a) The sequence of the (J"-additive processes ｵｾｮＩ＠ = ＨｵｾｮＩ＠ (B); B E £d) converges (in the sense of definition 2.3) to a (J"-additive process Ut = (ut(B); BE £d). (b) For each finite sequence ＨｂｩＩＧｾｬ＠ from £d the random vector [ut(BI), ... , ut(Bm)] is normally distributed. The distribution is characterized by the expectation values

and by the covariances

Let us note that the processes (Ut)t>o can be interpreted as the solution of the integral equation corresponding to (4) (cf. 5).

A Completely Discrete Particle Model In 3 a completely discrete particle model is considered, i.e., in addition to the source term j and the diffusion term (J"2 the initial condition cp has to be discretized. Furthermore the particles of the considered system are moving according to a Brownian motion. In the following we assume that cp(x) is an integrable function on Rd. Furthermore we assume that for all t > 0 the functions j(x,s)x(O,t)(s) , (J"2(x,s)X(0,t)(s) are integrable. Those conditions insure that the considered random point systems are always finite. In principle one can consider the case that cp, j, (J"2 are bounded measurable functions like in 5. Now the initial condition cp will be discretized as follows: /-to denotes the measure on R d+ 1 X {-1, I} characterized by

J =J

/-to(B x {I}) = /-to(B x {-I})

cp+ (X)xB (x) ld(dx), BE £d, CP- (X)XB(X) ld(dx), BE £d.

163

Let ｾｮ＠

be a Poisson random point system on R d X { -1, I} with distribution p(nl-'o)' ｾｮ＠ describes the configuration of charge points. Further ((wx(t))t>O)xERd denotes a family of independent standard Wiener processes on Rd. A charge point starts from x at initial time 0 and moves to x + wx(t) at time t. The contribution of this diffusion process to the charge density until t > 0 is given by

(9) We have already discretized the source term f and the diffusion term J2, i.e., we already have the point configurations of ｾｮ＠ and q,n. A charge point occurring at time s on position x moves to x + Wx (t - s) at time t. The contributions of these source processes to the charge density until t > 0 are given by

D-en) t (-)

11 In 1

=;

c;n)(-) =

-

c X(O,t)(s) Dx+wx(t-s)(-) q,n(d[x, s, c]), C

X(O,t)(S) Dx+wx(t-s)(-) q,n(d[x, s, c]).

(10) (11)

Furthermore we assume that ((w x (t))t2:0)XERd, ｾｮＬ＠ ｾｮＬ＠ q,n are independent. The entire process can be defined as the sum of independent random variables

(12) Similarly as the partially discrete particle model, the discrete system Ut(n) should approximate a continuous system. The following theorem makes that more precise 3. Theorem 3.2. Let be t given by

Vt(B) = At(B) at(B)

=

1

> 0, B

+

E £d. Further let Vb at measures on Rd

1

Kt-s(x, B)f(x, S)x(O,t)(S) ld+l(d[x, s])

Kt-s(x, B)J 2(x, S)x(O,t)(S) 1d+1(d[x, s]).

We assume that "\it is a generalized Brownian motion with noise intensity measure at. Then the sequence of the J-additive processes (Ut(n))nEN converges to a decomposable process Ut given by the following equation

164

Remark 3.1. This theorem means that the completely discrete particle model approximates a continuous system being different from the limit considered in 5, i.e., the limit of the completely discrete model has the same expectation values but different covariances, compared with the limit considered in 5. 4. Some Lemmas

For the proof of the main theorem in necessary to use the following lemmas.

3

(theorem 3.2 in this paper) it is

Lemma 4.1. Let \[Tn be a Poisson random point system with intensity measure nf.1 where f.1 is a locally finite measure on G and n E N. u(n) denotes the CT-additive process defined by u(n) := ｾ＠ \[Tn. Then the sequence (u(n) )nEN converges to the (trivial) CT-additive process f.1, i. e., AUCn) (g) -+ exp{ifg(x)f.1(dx)} (n-++oo).

Lemma 4.2. Let ＼ｰｲＬｾ＠ be independent Poisson random point systems with intensity measure nf.1 where f.1 is a locally finite measure on G and n E N. u(n) denotes the CT-additive process defined by u(n) := vk(. Then for each t > 0 1>t := J 1>(d[x, s])Ox+wx(t-s)X(O,t)(s) becomes a Poisson random point system in R d with finite intensity measure f-Lt being absolutely continuous w. r. t. the Lebesgue measure with density

5. Proof

In this section we give the proofs of the lemmas in section 4. Using the lemmas the proof of theorem 3.2 is given in 3. Proof of Lemma 4.1 Let us consider the characteristic functional Cu(n) (g) of u(n). Firstly m

= ｌ｣ｾｭＩｘｂＨＢＧ＠

we consider the special case gem)

with pairwise disjoint k

k=l

subsets Bi m), ... , ｂｾｭＩＮ＠

Then we have

J

gCm)(x)du(n)(x) =

ｦ｣ｾｭＩｵＨｮ＠

(Bkm)) =

ｾ＠ ｦ｣ｾｭＩｷｮ＠

k=l

(Bkm)). (13)

k=l

From (13) we obtain Cu(n) (g(m))

=

Eexp

=

Eexp

+00

=L

h=O

(i Jg(m) (x)du(n) (x)) Ｈｩｾ＠

ｾ｣ｭＩｷｮ＠ +00

(Bkm))) m

... L II exp

((m)) ｩｃｾ＠

lk

lk=O k=l

xPr{wn(Bim)) =h,'" ＬｗｮＨｂｾｭＩ＠

= ｦｌｾ＠

exp

ＨｩｃｾＩ＠

lk) Pr {w

=lm} n

(Bkm)) =

lk}'

(14)

166

Since t(g(m)) = Eexp

(i / g(m) (X)diJ>t(X)) L

= Eexp {

g(m)(x

+ wx(t - S))}

[x,sjE,OO)xERd and

(31)

iJ>.

is

obtained

from

the

independence

of

Further it holds

Eexp{ig(m)(x+wx(t-s))} = / exp {ig(m)(y)} kt-s(x,y)dy.

(32)

From (31),(32) we obtain

Ct (g(m)) =

exp ( / f.l(d[x, s]h(o,t) (s) [/ exp {ii m)(y) } kt-s(x , y)dy - 1])

= exp ( / { / / h(x, s )X(O,t) (s )kt-s(x, y)dxds } [ex p {ig(m) (y) } - 1] dY) . (33)

171

Setting h t as

ht(y)

:= /

/

h(x, s)X(O,t)(s)kt-s(x, y)dxds,

we obtain from (33)

CiPt(g(m)) = exp ( / ht(y) [exp {ig(m)(y)} -1] dY)

= exp ( / ILt(dy)

J) .

[exp {ig(m)(y)} - 1

(34)

Similarly as (18) let us approximate a general function g by the sequence (g(m))mEN. Then it holds from (34)

CiPt(g)

=

exp ( / ILt(dy) [exp{ig(y)} - l l)

.

That proves lemma 4.4 . •

References 1. L. Breiman, Probability, Reading, Mass., 1968. 2. J. Feldman, Decomposable processes and continuous products of probability spaces, J. Function. Analysis, 8, I-51, 1971. 3. K.-H. Fichtner, K. Inoue, M. Ohya, Approximative approaches to a stochastic partial differential equation by point systems, Preprint. 4. K.-H. Fichtner, R. Manthey, Weak approximation of stochastic equations, Stochastics and Stochastics Reports, 43, 139-160, 1993. 5. K.-H. Fichtner, M. Schmidt, Approximation of a continuous system by point systems, SERDIeA, 13, 396-402, 1987. 6. K.-H. Fichtner, G. Winkler, Generalized Brownian motion, point processes and stochastic calculus for random fields, Math. Nachr. 161, 291-307, 1993. 7. K. Matthes, J. Kerstan, J. Mecke, Infinitely divisible point processes, J.Wiley, New York, 1978. 8. J.B. Walsh, A stochastic model of neural response, Adv.Appl.Probability, 13, 231-281, 1981. 9. H. Zessin, The method of moments for random measure, Z. Wahrsch. Verw. Gebiete 62, 395-409, 1983.

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 173- 183)

ON QUANTUM ALGORITHM FOR EXPTIME PROBLEM S. IRIYAMA AND M. OHYA

Department of Information Sciences , Tokyo University of Science 2641, Yamazaki, Noda City, Chiba, Japan There exists a quantum algorithm with chaos dynamics solving an NP-complete problem in polynomial time, called OMV SAT algorithm . The language class EXPTIME is larger class than NP, there is no classical algorithm to solve it in polynomial time. In this paper we propose a quantum algorithm for one of the problems in EXPTIME, Pebble Game, and compare the computational complexity of it with the classical one. We show that a quantum algorithm with Oracle solves it in polynomial time while a classical algorithm with same Oracle does in exponential time.

1. Introduction

We have studied on quantum algorithm for several years. Ohya, and Volovich discovered the quantum algorithm with chaos dynamics called the OMV quantum algorithm which can solve NP complete problem in polynomial time. We applied this quantum algorithm to the other problems, multiple alignment of amino acid sequence, Hamilton closed path problem, protein folding problem and EXPTIME problem. Therefore we found that OMV quantum algorithm is useful for searching problems, i.e., to search the objects which satisfy the given conditions. In the field of Bio-Information there exist many searching problems, and we can apply our quantum algorithm to them. In this paper, we show a quantum algorithm for EXPTIME problem, Pebble Game. Then we discuss the computational complexity of it.

2. Quantum Algorithm A quantum algorithm is constructed by the following steps: (1) Prepare a Hilbert space (2) Construct an initial state (3) Construct unitary operators to solve the problem 173

174

(4) Apply them for the initial state and obtain a result state (5) If necessary, amplify the probability of correct result (6) Measure an observable with the result state In the first step, we define the Hilbert space depending on the problem. Let

((:2

be a Hilbert space spanned by 10)

=

ＨｾＩ＠

and 11)

=

ＨｾＩＬ＠

a

normalized vector 1'If!) = a 10) + (311) on this space is called a qubit. Since we can use a superposition of 10) and 11) as an initial state vector, the quantum algorithm is more effective than classical one. One can apply Hadamard transformation H __ 1 (1 1 )

- y'2

1-1

to create a superposition. For 10) and 11), it works as H 10)

1

1

= y'210) + y'211)

1 1 H 11) = -10) - -11) .

y'2

y'2

Hadamard transformation has a very important role in a quantum algorithm. Here we introduce logical gates, which are NOT gate, C-NOT gate and CC-NOT gate. We call these gates fundamental gates. We can also construct AND and OR gate by considering the product of fundamental gates and some imprementations. The NOT gate UNOT is defined on a Hilbert space ((:2 as UNO T =

11) (01

+ 10) (11·

It works for an arbitrary qubit as

C-NOT UCN gate and CC-NOT Hilbert space as UCN

UCCN

are given on two and three qubit

= 10) (01 ® I + 11) (11 ® UNO T

UCCN =

10) (01 ® I ® I

+ 11) (11 ® 10) (01

®I

+ 11) (11 ® 11) (11 ® UNO T ,

respectively. The unitary operator to solve the problem is constructed by these fundamental gates.

175

3. OMV SAT Algorithm In this section, we explain OMV(Ohya-Masuda-Volovich) quantum algorithm which contains two part, that are unitary computation and chaos amplification process. It is discussed precisely in the papers l ,2,4,9. Let X == {Xl, ... ,xn},n E N be a set. Xk and its negation Xk (k = 1, ... , n) are called literals. Let X == {Xl, .. " Xn} be a set, then the set of all literals is denoted by X' == X UX = {Xl, ... , X n , Xl,"" X n }. The set of all subsets of X' is denoted by :F (X') and an element G E :F (X') is called a clause. We take a truth assignment t to all variables Xk. If we can assign the truth value to at least one element of G, then G is called satisfiable. Let L = {O, I} be a Boolean lattice with usual join V and meet 1\, and t (x) be the truth value of a literal X in X. Then the truth value of a clause G is written as t(G) == VxEct(x). Moreover the set C of all clauses Gj (j = 1,2,'" ,m) is called satisfiable iff t (C) == I\'j=l t (Gj ) = 1. Thus the SAT problem is written as follows: [SAT problem] Given a Boolean set X == {Xl,'" ,xn}and a set C = {Gl ,'" ,Gm } of clauses, determine whether C is satisfiable or not. That is, this problem is to ask whether there exists a truth assignment to make C satisfiable. It is known that we can check the satisfiability in polynomial time when a specific truth assignment is given, however we do not determine it in polynomial time when an assignment is not specified. We first calculate the total number of qubits, and show that this number depends on the input data. This calculation is done in polynomial time of input size. Since the total number of qubit required the quantum algorithm, we decide the Hilbert space and the initial state vector on it. Let C {Gl , ... ,Gm } be a set of clauses on X' {Xl, ... ,X n , Xl, ... ,x n }. The computational basis of this algorithm is on the Hilbert space H = (C 2)'l9 n +I-'+l where f.L is a number of dust qubits, it is shown that f.L is less than 2mn g. Let

be an initial state vector. For X we put

where

Cl, C2, ... ,Cn

E

=

{Xl, ... ,x n }

and a truth assignment t,

{O, I} , and we write t as a sequence of binary sym-

176

boIs:

A unitary operator Uc : 1i follows

-t

1i computes t (C) for all truth assignment as

where

Id P )

lei)

leI, e2, ... ,en) is a binary representation of t. Accardi and Sabbadini

=

is dust qubits denoted by p, strings of binary symbols, and

pointed out that OMV SAT algorithm is combinatorial 6 . Theorem 3.1. (l) For a set of clauses C = {Gl , ... , Gm } on X' == {Xl, ... ,X n , Xl,···, xn}, the number p, of dust qubits for algorithm of SAT

problem is p,::::: 2nm

For a set of clauses C = {Gl , ... , Gm }, we can construct the unitary operator Uc to calculate the truth value of C as

Uc ==

m- l

m

i=l

j=l

II UAND (i) IIUoR (j) H (n)

where, H (k) is a unitary operator to apply Hadamard transformation to first k qubits, that is

The computational complexity of quantum computation depends on the number of unitary operator in the quantum circuit. Let U be the unitary operator, it is written as

where Un,· .. ,Ul are fundamental gates. The computational complexity T (U) is considered as n.

177

We need to combine some fundamental gates such as UNOT, UCN and UCCN to construct the quantum circuit in fact. UAN D and UOR can be written as a combination of fundamental gates. Here we obtain the computational complexity T (Uc) of SAT algorithm by the number of UNOT, UAN D and UOR. Theorem 3.2. f) For a set of clauses C = {G 1 , ... , Gm {X1, ... ,X n ,X1, ... ,X n }, T(Uc) is

}

and literal X'

=

m

T (Uc) = m - 1 +

L

(IGkl

+ Ｒｩｾ＠

- 1)

k-1

:::; 4mn-l

3.1. Chaos Amplifier

Here we will briefly review how chaos can playa constructive role in computation (see 1,2 for the details). Consider the so called logistic map which is given by the equation

The properties of the map depend on the parameter a. If we take, for example, a = 3.71, then the Lyapunov exponent is positive, the trajectory is very sensitive to the initial value and one has the chaotic behavior 2. It is important to notice that if the initial value Xo = 0, then Xn = for all n. The state 1'IjJ) of the previous subsection is transformed into the density matrix of the form

°

where PI and Po are projectors to the state vectors 11) and 10) . One has to notice that PI and Po generate an Abelian algebra which can be considered as a classical system. The following theorems is proven in 1,2,3. Theorem 3.3. For the logistic map Xn+1 = aXn (1 - Xn) with a E [0,4] and Xo E [0, 1], let Xo be Ｒｾ＠ and a set J be {O, 1,2, ... , n , ... , 2n}. If a is 3.71, then there exists an integer k in J satisfying Xk > ｾＮ＠ Theorem 3.4. Let a and n be the same in above theorem. If there exists

k in J such that Xk > ｾＬ＠

then k > ｬｯｧＲｾａ

Ｍ ＱＮ＠

178

Theorem 3.5. Let It (C)I be the cardinality of these assignments, if Xo ==

.{n with r = It (C)I and there exists k in J such that exists k satisfying the following inequality if C is SAT.

- 13.71 -log2 r] [nlog2 - 1

< k < [-45

-

-

Xk

>

ｾＬ＠

then there

(n _1)]

From these theorems, for all k, it holds k (2) g3.71 q

{=>

0 0

iff C is not SAT iff C is SAT

Ohya and Volovich proposed quantum algorithm to calculate truth function by unitary operators and mentioned that it is necessary to use some amplification processes to detect the result in case of very small probability. Then they discovered that one can construct this process by using chaos dynamics. The computational complexity of OMV SAT algorithm is discussed precisely in the papers 10 ,9,1l

4. Language Classes

There exist several classical language classes defined by a deterministic Turing machine. Definition 4.1. Let M be a deterministic Turing machine such that halts for all input. For a length n of input, let f (n) be a maximum length of tape cell of M. We define a space complexity of M as

f.

Definition 4.2. PSPACE is the language class which is recognized by a deterministic Turing machine in a polynomial space. Definition 4.3. NPSPACE is the language class which is recognized by a non-deterministic Turing machine in a polynomial space. Definition 4.4. EXPTIME is the language class which is recognized by a deterministic Turing machine in an exponential time . The following relation is known:

P

ｾ＠

NP

ｾ＠

PSPACE

= NPSPACE ｾ＠

EXPTIME

179

5. Pebble Game Pebble Game is the two players game using a play board and stones(pebbles). Players move one stone at a time along a given rule alternatively. A player wins when he moves a stone to the winning position on the board given by the rule, or he loses when he cannot move any stones. We want to know whether there exist strategies such that a first mover can win every time. The computational complexity of this problem is obviously very large, and in some cases depending on a size of board and rule it belongs to EXPTIME. Then we propose a quantum algorithm for Pebble Game. First, we explain a representation of game and a definition of Pebble Game.

5.1. Representation of a Game Let I be a set of players, M a set of moves (or strategies) available to those players, and P a set of payments for each combination of moves. A game G is given a triplet (I, M, P). The set of moves is given by a rule of the game. Players choice their move in their turn alternatively, so then these choices are described by a sequence of moves. We denote this by a position Pi where a player i E I has the move. A set of ordered pair (pi, Pj) where Pi and Pj are positions such that Pi -7 Pj is a feasible move implies the rule of the game. The game is finished when there does not exist any moves in someone's turn. When the game is finished, the payments are given to players by a function of the position. In this study we assume that the game must be finished so that the length of position is finite. We say a player A wins if the payments of A is greater than a player B in two players game. Then we consider the following problem: [Game Probleml] Does there exist a way such that a player A wins independently of how a player B plays? To answer this, a winning position of A was introduced by Konig.as a set of positions where A wins in finite moves in spite of moves of B. A set of winning positions of A is constructed inductively from a set of positions where A wins by only one move. Let P is a winning position, q is also a winning position if there exists a move r A of A such that for all moves rB of B, it holds

180

Therefore, the Problem 1 is equivalent as the following: [Game Problem2] Determine whether an initial position of the game is in a winning position of A?

5.2. Pebble Game Let n be a positive finite integer denoted by a size of Pebble Game. n- Pebble Game is given by a triplet G Pebble = (I, M, P) where I =

{A,B}, M = {r=(r l ,r 2 ,r3)E{1, ... ,n}3;risanavailablemove} and P = {f: Mk -> {O, I}}. M is a set of available moves depending on a situation of the board and positions of stones. The rule of Pebble Game is the following: • Prepare n lattices denoted by a board and put m ( < n) stones on nodes. Each nodes has a unique number i = 1, ... n. • Players move one of stones alternatively if the stone so chosen can jump the neighbor stone. • If a player puts a stone on the node n, he wins. If a player can not move any stones, he loses. Let x = (Xl, X2, ... , xn) be a vector of Boolean variables denoted by a situation of board, where Xi has one to one correspondence with a node i: Xi

=

{O no stone on a node i 1 node i has a stone

We prepare m stones on the board Xs arbitrary and call it an initial situation. And we call a finished situation X f if there are no available moves. A move ri (i E {A, B}) for a player i is represented by (rl, r2, r3) E {I , ... , n} 3 where rl indicates a position of stone, r2 the neighbor and r3 a position of destination. If there is a stone on rl and r2 and there is not a stone on r3 , the move ri = (rl,r2,r3) is available. After one move, the situation of board is changed by

_ {Xi i = rl or i = r3

Xi -

Xi

otherwise

we denote ri (x) by a situation of board after a move rio If there no moves available or he can move a stone to the node n, the game is finished. Then we have a sequence of moves. For a sequence of moves Pi (i E {A, B}) = (r1, r§, ... , rf) a payment for a player A is given

181

by a function fA E P:

and for a player B is by fB E P: fB (Pi)

=

{ o1 ii == BA

Let us define the following problem: [Pebble Probleml] Does there exist a way such that a player A wins independently of how a player B plays in n-Pebble Game? This problem belongs to EXPTIME if m is not fixed 8 . We check whether an initial situation is a winning position of A for all number m of stones. Then the Pebble Problem1 is translated to the following problem: [Pebble Problem2] For all finished situations xf determine whether there exists k such that Ｓｲｾ｜ｬＭＱ＠

...

3d ｜ｬｲＱＳｾ＠

(xs)

= xf

where Xs is an initial situation. We propose a quantum algorithm with Oracle to solve the Pebble Problem2, and show that even if we assume an Oracle, a classical algorithm is still an exponential time.

5.3. Computational Complexity of a Classical Algorithm for Pebble Game In order to discuss a relative computational complexity between classcal and quantum algorithm, we assume the following Oracle Mo:

Mo : {x; x is a situation} ---; N U {O} For a finished situation x f' Mo cutputs the time of it immediately, just one step. If a situation x is not a final situation, Mo outputs o. A classical algorithm to solve Pebble Problem2 is the following. Step1 For all situations, we do: Step2 Calculate time k (x) for a situation x Step3 If x is a final situation, construct a set Wi (i = 1,2, ... , k (x))of winning situations: Wi

where Wo Step4 Check Xs

= {y; ＳｲｾＬ＠

= {x}

= Wk(x).

\lrB, 3r A, ｲｾｂ＠

AY E

Wi-I}

182

The number of all situations 2n , and the upper bound of number of available moves is 4n since a player can move one pebble into four directions at most. Therefore the computational complexity of a classical algorithm Tc (n) is

Tc (n)

rv

(4n)3 x 2n

exp (n)

rv

Even if we assume the Oracle, this problem belongs to EXPTIME.

6. Quantum Algorithm for Pebble Game As we assume the Oracle Mo, we use a quantum Oracle UMo which works as same as Mo. Here, we construct the following quantum algorithm: Step 1 Step2 Step3 Step4

Create a superposition of all situations. For the superposition, apply Oracle UMo. We construct Wi (i = 1,2, ... k (xf)) for the superposition. If Xs E Wk(x), make the final qubit 11).

All situations are represented by a binary form, so then we can create a superposition of them using Hadamard transformation. Step3 is achieved by a product of unitary gates. In Step4, AND operation is constructed by unitary gates 9 . If final qubit of superposition is 11), there exists a way of winning. Using the chaos amplifier, we obtain the result.

7. Computational Complexity Here, we calculate the computational complexity of the quantum algorithm as the total number of fundamental gates. The Step 1 is constructed by n Hadamard gates. The Step2 is done by only one Oracle UMo. The computational complexity of Step3 has the same order as the classical algorithm. In the Step4 AND operation requires n gates for n qubit. The upper bound of IWk(x) I is ＨｾＩ＠ where m is the number of pebbles. Therefore, computational complexity TQ (n) of quantum algorithm of Pebble Problem2 is

TQ (n) where ｛ｾＨ＠

rv

{n

+ 1 + (4n)3 + (:) }

x

｛ｾＨｮ＠

-1)]

rv

poly (n)

n - 1)] is for chaos amplification to obtain the correct result.

183

8. Conclusion

The computational complexity Tc (n) of a classical algorithm for n-Pebble Game with Oracle is

Tc (n) ｾ＠

(4n)3 x 2n ｾ＠ exp (n)

This is a exponential time of input size n. We constructed a quantum algorithm for this, the computational complexity TQ (n) is

TQ (n)

cv {

n+

1+ (4n)

3

+ (:) }

x

｛ｾＨ＠

n-

1)]

cv

poly (n)

In this study, we assume the Oracle Mo and the unitary operator UMo which works as Mo. Even if we use this Oracle, the computational complexity of the classical algorithm for Pebble game is an exponential time while it of the quantum algorithm is a polynomial of n.

References 1. M.Ohya and I.V.Volovich, Quantum computing and chaotic amplification, J. opt. B, 5 ,No.6 639-642, 2003. 2. M .Ohya and I.V.Volovich , New quantum algorithm for studying NP-complete problems, Rep.Math.Phys. , 52 , No.l,25-33 2003. 3. M.Ohya and I.V.Volovich, Mathematical Foundation of Quantum Computers, Teleportations and Cryptography, to be published. 4. M.Ohya and N.Masuda, NP problem in Quantum Algorithm, Open Systems and Information Dynamics, 7 No.1 33-39 , 2000 . 5. L. Accardi and M.Ohya, A Stochastic limit approach to the SAT problem, Open systems and Information Dynamics, 11-3, 219-233 , 2004 6. L.Accardi and R.Sabbadini, On the Ohya- Masuda quantum SAT Algorithm, Preprint Volterra, N. 432, 2000. 7. E.Bernstein and U.Vazirani, Quantum Complexity Theory, In Proc. 25th ACM Symp. on Theory of Computation, 11-20, 1993. 8. T.Kasai, A. Adachi , S. Iwata, Classes of Pebble Games and Complete Problems,SIAM J. Comput. Volume 8, Issue 4, pp. 574-586 (1979) 9. S.Iriyama and M.Ohya, Rigorous Estimate for OMV SAT Algorithm, Open Systems & Information Dynamics, 15, 2, 173-187, 2008 10. S.Iriyama, M.Ohya and I.V.Volovich (2006) Generalized Quantum Turing Machine and its Application to the SAT Chaos Algorithm, QP-PQ:Quantum Prob. White Noise Anal., Quantum Information and Computing, 19, World Sci. Publishing, 204-225 11. S.Iriyama and M.Ohya (2008) Language Classes Defined by Generalized Quantum Turing Machine, Open System and Information Dynamics 15:4, 383-396. 12. S.Iriyama and M.Ohya (2009) The problem to construct Unitary Quantum Turing Machine for compute partial recursive function , TUS preprint. 13. S.Iriyama and M.Ohya (2010) Quantum Algorithm for Pebble Game and Its Computational Complexity, TUS preprint.

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 185-197)

ON SUFFICIENT ALGEBRAIC CONDITIONS FOR IDENTIFICATION OF QUANTUM STATES

ANDRZEJ JAMIOLKOWSKI Institute of Physics, Nicholas Copernicus University, 87- 100 Torun, Poland E-mail: [email protected] The aim of this paper is to discuss the relationship between some problems of the identification of quantum states by geometric methods used in stroboscopic tomography and, on the other hand , by a lgebraic approach typical for the quantum generalization of classical sufficient statistics. Some examp les of necessary and sufficient conditions which must be fulfilled by generators of algebras in order to estimate states of quantum systems are discussed.

Keywords: algebras of observables; stroboscopic tomography; generators of subalgebras.

1. Introduction

One of the basic purpose of physical theories, in both classical and quantum sectors, is to describe events that are observed in Nature or in experiments conducted in laboratories. Usually, we have a physical system under investigation, and we try to obtain information about it by making some experiments. As results, measurement outcomes are registered. In fact, in many situations in classical systems, and in all cases in the quantum regime, we can not predict the individual measurement outcomes and we obtain only some probabilities of results. In other words, we obtain, as the outputs of experiments, only probability distributions on a set of possible measurement outcomes. The statistical theory of classical systems is based on classical probability theory. However, atoms and molecules obey the statistical laws of quantum mechanics and one has to develop a parallel theory based on quantum probability (noncommutative probability), which is an essential generalization of classical probability theory. In classical statistical physics we consider probability distributions as natural representation of states and random variables as representation of observables (physical quantities). In the description of microsystems it 185

186

is natural to start with the idea of observables as a quantum analogue of random variables and to define states as a derived concept. This means that we introduce quantum states through the concept of algebra of observables, taking the states to belong to the dual space. The algebra, furnished with the operation of conjugation (*-operation) is postulated to be a C*-algebra with identity. It can be considered as a natural generalization of the algebra of classical functions with *-operation given by complex conjugation, and whose real random variables are self-adjoint elements. A nice discussion of these issues is given by W. Thirring in his lecture notes 1. From the physical point of view, by a state we understand the description of statistical properties of a system when prepared repeatedly in the same way. In other words, one of the basic assumptions of quantum mechanics is based on the observation that determination of a completely unknown state can be achieved by appropriate measurements only if we have at our disposal a set of identically prepared copies of the system in question. From the mathematical point of view, we say that a state on a C*- algebra of observables is an assignment of a number (the expectation value) for every element of the algebra. This assignment should obey the natural laws of linearity and positivity. Thus, if we denote by A a C*algebra with identity (a set of observables) then a state on A is a linear map (1)

such that P(C"1Q1 + a2Q2) = a1P(Qd + a2P(Q2) for all a1, a2 in C and Q1,Q2 in A , and p(Q) ｾ＠ 0 for all positive Q E A. Usually, we also assume that p(lI) = 1. Moreover, to set up an effective approach to the above problem of state determination, one has to identify a collection of observables, a quorum 2 , such that their expectation values contain complete information about the state of the system under consideration. In the standard formulation of quantum mechanics, usually we introduce a Hilbert space H associated with a given microsystem and we identify an algebra of observables A with the set of hermitian elements of the Hilbert-Schmidt space B(H). For any normalized vector Iw) E H the formula

p,p(Q) =

(wIQlw)

(2)

defines a state on A. Such a state is called vector state. In general, the expectation values are given by convex combinations of some vector states

187

and we have the following equality

f5(Q)

=

Tr(pQ) ,

(3)

where p denotes a state of the system in question. The problems of state determination have gained new relevance in recent years , following the realization that quantum systems and their evolutions can perform tasks such as teleportation , secure communication or dense coding (c.f. e.g. 3,4). It is important to realize that if we identify the quorum of observables, then we also have a possibility to determine the expectation values of physical quantities (observables) for which no measuring apparatuses are available 5. The idea of stroboscopic tomography for open quantum systems appeared for the first time in the beginning of 1980 's (although it was expressed in different terms 6 ,7 from the ones used presently). In particular, the question of the minimal number of observables Ql, . .. , Q", for which quantum states can be (Ql , . . . , Q",)-reconstructible was discussed. Simultaneously, it was shown that reconstructibility of states in a finite dimensional systems can be achieved if a sequence of the so-called K rylov subspaces which are defined by

(4) where Q is a fixed observable and lL is a generator of time evolution of the system in question, span the Hilbert-Schmidt space B(H) (cf. below Sect. 3). That is, more precisely, if the following equality is satisfied

(5) In the above equality I-" denotes the degree of the minimal polynomial of the superoperator lL and Ql, ... , Qr represent fixed observables. The symbol ffi denotes the Minkowski sum of subspaces (5) (cf. e.g. 8). We recall that for two subspaces Kl and K2 of the vector space H, by Kl ffi K2 one understands the smallest subspace of H which contains both Kl and.K2 . It is well known that the Krylov subspaces Kk(lL, Q) for k = 1,2, ... form a nested sequence of subspaces of increasing dimensions that eventually become invariant under lL. Hence for a given Q, there exists an index I-" = 1-"( Q) , often called the grade of Q with respect to lL, for which Kl(lL,Q) 0, that is, p represents a faithful state. It follows from (16) and (17) that the operator 7r1! is the quantum analogue of a classical conditional probability. Indeed, in the case A is an Abelian algebra, 7r1! coincides with a classical conditional probability. Definition 3.1. An operator 7r E A is called a quantum conditional probability operator (QCPO) if 7r satisfies condition (16) and (17).

One easily finds the following formula n

7r (log e4> -log p 0 0') = Tr Ｈｰｾ＠ 01) 7f

(p; ａｾＩ＠ where

I4> (p; ａｾＩ＠

ａｾ＠

(p) =Tr 1 (p ｾ＠ 0 1)

7f oo 'P wQ ,h . If hx is invertible for all x E L , then 'P(b) h is a backward QMC on B L . WQ,

On the other hand, it is known 8 that ｻＧｐｾ Ｌ ｾｽ＠ satisfy the compatibility condition if a sequence {Wnj} is projective with respect to Trnj, i.e. Trn - 1j (Wn j)

=

W n - 1j,

Vn E N.

(15)

One has the following Theorem 3.2. 11 Let the boundary conditions Wo E B(o) ,+ and h = {h x E Bx,+ }xEL satisfy (13) and (14). Then {Wnj} is a projective sequences of density operators, i. e. there is a unique forward QM C 'P WQ, (f) h on B L such U ) = W - lim that (n (n (n,J). rWQ ,h n---+oo Ywo ,h

273

Definition 3.1. We say that there exists a phase transition for a family of operators {K} if (13) and (14) have at least two (wo, {hx}xEL) and Cwo, {hx}xEL) solutions such that the corresponding QMC 'Pw 0, hand 'Pw 0, fi are disjoint. Otherwise, we say there is no phase transition. 4. Quantum d-Markov chains associated with XY-model

In this section, we prove the existence of a phase transition of the quantum d-Markov chain associated with XY-model on a Cayley tree of order three. Let us consider a semi-infinite Cayley tree ｲｾ＠ = (L, E) of order 3. Our starting C* -algebra is the same EL but with Ex = M2 (q for x E L. By ｏＢｾｵＩＬ＠ O"£u) , ｏＢｾｵＩ＠ we denote the Pauli spin operators for at site u E L. Here (u)=(OI) O"x 10'

=

(0 -i) °'

O"z

=

exp{,6H}, ,6

(u)

O"y

i

(u)

(1 °)

0-1'

(16)

For every edge < u, v >E E put K

=

> 0,

(17)

H

= -21 (O"(u)O"(v) x x + O"(u)O"(v)) y y .

(18)

where

Now taking into account the following equalities 2m _ H2 _ 1 (n H -:2

-

(u) (v)) o"z O"z ,

2m-l H

== H ,

mEN,

one finds K =

n+ sinh,6H + (cosh,6 - ｉＩｈｾｵＬｶ＾Ｇ＠

We are going to describe all solutions h = {h x } and Wo of the equations (13),(14). Furthermore, we shall assume that hx = hy for every x, y E W n , n E N. Hence, we denote ｨｾｮＩ＠ := hx, if x E W n . Now from (17),(18) therefore, the equation (14) can be one can see that K = ｋｾｵＬｶ＾Ｇ＠ rewritten as follows K K ) - h(n-l) Trx ( K K K h y(n)h(n)h(n)K z v - x ,

(19) for every x E L. After a little algebra the equation (19) reduces to the following system {

ｂｬ｡ｩｾＩｉＨＲ＠

(n))3 B 2 (a 11

2+ A 2 a(n)la(n)1 11 12 + ａｬ｡ｩｾＩＱＳ＠ = ｬ｡ｩｾＭｉＩＱ＠

a(n-l) 11

(20)

274

where Al = sinh 3 ;3 cosh;3, BI = sinh;3 cosh 2 ;3(1 + cosh;3 + cosh 2 ;3), (21) A2 = sinh2 ;3 cosh 2 ;3(1 + 2 cosh ;3), B2 = cosh 6 ;3, (22)

hen) y

= hen) = hen) = z v

Remark 4.1. Note that according to positivity of ｨｾｮＩ＠ > lai;) I for all n E N. conclude that ｡ｩｾＩ＠

(n) ( all (n)

(n))

l2 a (n) a 21 a 22

and ai';) = ｡ｾＩ＠

.

we

Now we are going to investigate the derived system (20). To do this, let us define a mapping f : (x, y) E ｬｒｾ＠ --+ (Xl, yl) E ｬｒｾ＠ given by

{

B2(XI)3 + A 2x ' (y')2 = X BI (Xl?yl + Al (yl)3 = Y

(23)

Furthermore, due Remark 4.1, we restrict the dynamical system (23) to the following domain ,6.

= {(x,y)

E ｬｒｾ＠

:x

> y}.

Denote Pg(t) = t g

+ 2t4 + 2t 3 -

t - 1,

(24)

+ 2 cosh;3) + D cosh6 ;3'

(25)

D:= A2 - AI. B I -B2 Further, we will need the following auxiliary fact:

(26)

-

tS

-

t7

-

t6 1

E := sinh2;3 cosh 2 ;3(1

Lemma 4.1. Let AI, B I , A 2 , B 2 , D be numbers defined by (24), (22), (26) and Pg(t) be polynomial given by (24), where ;3 > O. Then the following statements holds true:

(i) The polynomial P g (t) has only tree positive roots 1, t*, and t* such that 1.05 < t* < 1.1 and 1.5 < t* < 1.6. Moreover, if t E (1, t*) u (t*,oo) then Pg(t) > 0 and t E (t*,t*) then Pg(t) < O. Denote by ;3* = cosh -1 t* and;3* = cosh -1 t*; (ii) For any ;3 E (0, (0) we have Al < A 2 ; (iii) If;3 E (0,;3*] U [;3*,(0) then BI :s; B2 and If;3 E (;3*,;3*) then BI > B 2 ;

275

(iv) (v) (vi) (vii)

For any (3 E (0,00) we have Al + BI < A2 + B 2; If (3 E ((3*, (3*) then D > 1 and E > 0; For any (3 E (0,00) we have AIA2 < BIB2 and AIB2 < A 2B I ; If (3 E ((3*, (3*) then A2BI < AIA2 + 3A I B 2 + BIB2 and 2AIA2 3A I B 2 < A 2B I ·

+

Let us first find all of the fixed and periodic points of (23). Theorem 4.1. Let f be a dynamical system given by (23). Then the following assertions hold true:

(i) If (3 E (0, (3*] U [(3*,00) then there is a unique fixed point Ｈ｣ｯｳｾＳ＠

f3' 0)

in the domain 6.; (ii) If (3 E ((3*, (3*) then there are two fixed points in the domain 6., which are ｃｯｳｾＳｦＧＰＩ＠ and (VDE,VE). (iii) For any (3 E (0,00) the dynamical system f does not have any k periodic points, where k ;::: 2.

Now let us formulate results concerning limiting behavior of f. Theorem 4.2. Let f : 6. ---+ ｬｒｾ＠ be the dynamical system given by (23) and (3 E (0, (3*] U [(3*,00). Then the following assertions hold true:

(i) ify(O) > 0 then the trajectory ｻＨｸｮＩＬｹｽｾ］ｯ＠

of f starting from

the point (x(O), y(O)) is finite. (ii) if yeO) = 0 then the trajectory ｻＨｸｮＩＬｹｽｾ］ｯ＠ point (x(O), yeO)) has the following form

starting from the

3\1x(O) cosh 3 (3 -n--{ x(n) = - - cosh 3 (3 yen)

= 0,

and it converges to the fixed point Ｈ｣ｯｳｾＳ＠

f3 ' 0).

Theorem 4.3. Let f : 6. ---+ ｬｒｾ＠ be the dynamical system given by (23) and (3 E ((3*, (3*). The following assertions hold true: (i) There are two invariant lines w.r.t. f defined by y = O} and 12 = {(x,y) E 6.: y = ]v};

h = {(x, y)

E

6. :

(ii) if an initial point (x(O), yeO)) belongs to the invariant lines lk' of dynamical system (23), then the trajectory {(x(n) , ｹＨｮＩｽｾ］ｯＬ＠ starting from the point (x(O), yeO)), converges to the fixed point which belongs an invariant line lk' where k = l,2;

276

(iii) if an initial point (x(O), yeO)) satisfies the following condition

yeO) x(O)

( E

1)

0, y75

,

then the trajectory {(x(n), ｹＨｮＩｽｾ］ｯＬ＠ starting from the point (x(O) , y(O)), converges to the fixed point Ｈ｣ｯｳｾＳ＠ f3 ,0) which belongs an invariant line h; (iv) if an initial point (x(O), y(O)) satisfies the following condition yeO) x(O) E

(1 ) y75,1

then the trajectory ｻＨｸｮＩＬｹｽｾ］ｯ＠ (x(O) , y(O)), is finite.

,

starting from the point

Let 13 E (0,13*] u [13*,00). From Theorem 4.2, we infer that equation (19) has a lot of parametrical solutions (wo (a), {h x (a)}) given by

3\1a cosh3 13 wo(a) =

ｨｾｮＩ＠

(a) =

(

cosh 3 f3

(27)

° for every x E V., here a is any positive real number. The boundary conditions corresponding to the fixed point of (23) is give by (27) at value of ao =

1

3 in. Therefore, further, we denote such cosh 13 operators by Wo (ao) and ｨｾｮＩ＠ (ao). Let us consider the states ＼ｰｾＧｬＩＬｨＨ｡＠ corresponding to the solutions

Ｈｷｯ｡ＩＬｻｨｾｮ｜ｽＮ＠ One can show that n(X'; - 1) n

where an = 1 or -1. Of course cp is a generalized functional if test functionals are of the form n

bn being real and rapidly decreasing. The bilinear form is given by

(18) the right side is absolutely convergent, i.e.

318

And _ 6.L

mesK1 mesK2

,

t

--->

±oo,

where mes denotes the Lebesgue measure on Qn (so one could say that the probability distribution on the configuration space tends to the uniform distribution) . The proof uses Theorem 3.2 and is similar to the proof of Theorem 4.1 from [10]. This proposition somewhat strengthens a result of [8] which goes back to Poincare [1]. In [8] it was actually proved in the special case when the initial probability measure v(O) is the product of copies of a one-dimensional probability measure. A completely similar proposition is valid for quantum systems because for systems with quadratic Hamiltonian functions the equations for the Wigner measure coincide with the Liouville equations (cf. [10]).

References 1. H.Poincare. J.de Physique theorique et appliquee, 4 serie, 5, (1906),369-403. 2. N.N.Bogolyubov. Problems of dynamical theory in statistical physics. Moscow, 1946 (in Russian; there exists an English translation). 3. E.Wigner. Phys.Rev., 40, (1932), 749. 4. J. E. Moya!. Quantum mechanics as a statistical theory. Proc. Cambridge Philos. Soc. , v. 45 , (1949), 99-24. 5. G. B. Folland. Harmonic Analysis in Phase Space. (Princeton Univ. Press, 1989) . 6. Kim Y. S., Noz M. E. Phase-Space Picture of Quantum Mechanics. Group Theoretical Approach. (World Scientific, 1991). 7. Radu Balescu, Equilibrium and nonequilibrium statistical mechanics, vo!'l (John Wiley and Sons, 1975). 8. V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincare (Moscow-Izhevsk, 2002) [in Russian]. 9. V.V.Kozlov. Reg. Chaotic Dyn. 1. (2004), 23-34. 10. V.V.Kozlov, O.G.Smolyanov. Theory Probability and Applications. Vo!' 51, 1, (2006) , pp. 1-13. 11. O.G.Smolyanov, S.V.Fomin. Soviet Mathematical Surveys, V. 31, 4. (1976), 3-56. 12. O.G.Smolyanov, H.von Weizsaecker.Comptes Rend. Acad. Sci. Paris. T. 321, ser. 1. (1995), 103-108. 13. N.Bourbaki , Integration, Chapitre 6 (Springer, 2007).

337

14. O.G.Smolyanov, H.v.Weizsacker. Smooth probability measures and associated differential operators. Inf. Dimens. Anal., Quantum Probab. and Relat. Top. V.2, 1, (1999),51-78. 15. L. Accardi and O. G. Smolyanov. Generalized Levy Laplacians and Cesaro Means. Doklady Mathematics, Vol. 79, 1, (2009), 1-4. 16. T.Hida, H.H.Kuo, J.Pothoff, L.Streit. White noise. An infinite dimensional calculus. Kluwer Academic, 1993.

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 339-354)

ANALYSIS OF SEVERAL CATEGORICAL DATA USING MEASURE OF PROPORTIONAL REDUCTION IN VARIATION

KOUJI YAMAMOTO, KOUJI TAHATA, NOBUKO MIYAMOTO AND SADAO TOMIZAWA * Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan * E-mail: [email protected] For a two-way contingency table with nominal row and column variables, the measures which describe the proportional reduction in variation (PRV) from the marginal distribution of one variable to the conditional distribution given the other variable are proposed by Goodman and Kruskal (1954), Theil (1970), and Freeman (1987, p. 101). Tomizawa, Seo and Ebi (1997), and Miyamoto, Usui and Tomizawa (2005) proposed the generalization of those measures. Tomizawa, Miyamoto and Yajima (2002), and Yamamoto and Tomizawa (2009) proposed the PRV measures for a nominal-ordinal contingency table and for an ordinal-ordinal contingency table, respectively. The present paper (1) reviews these PRY measures and (2) analyzes and compares between several categorical data using these PRY measures.

Keywords: Concentration coefficient; Measure, Proportional reduction in variation; Square contingency table; Total uncertainty coefficient.

1. Introduction

The data in Table 1 taken from the Meteorological Agency in Japan are obtained from the daily temperatures at Nagasaki City, Japan, in two years, 2001 and 2002, using three levels, (1) below normals, (2) normals and (3) above normals (see Tahata, Takazawa and Tomizawa, 2008). The observations, say, Iij, in the (i, j)-th cell indicate that for each of Iij days in 365 days (i.e., from 1 January to 31 December), the temperatures in two years are i in 2001 and j in 2002. Table 2 is taken from Tallis (1962) and constructed from the crossclassified data of Merino ewes according to the numbers of lambs born in consecutive years, 1952 and 1953 (also see Bishop, Fienberg and Holland, 1975, p. 288; Miyamoto, Niibe and Tomizawa, 2005). 339

340

Table 3 is the data on unaided distance vision of 7477 women aged 30-39 employed in Royal Ordnance factories in Britain from 1943 to 1946. The row variable is the right eye grade and the column variable is the left eye grade with the categories ordered from the Best (1) to the Worst (4). The vision data in Table 3 have been analyzed by many statisticians, including Stuart (1955), Bishop et al. (1975, p. 284), McCullagh (1978), Goodman (1979) , Agresti (1983) , Tomizawa (1985 , 1993, 2009), Miyamoto, Ohtsuka and Tomizawa (2004), Tomizawa, Miyamoto and Yamamoto (2006), Tomizawa and Tahata (2007), and Tahata, Yamamoto, Nagatani and Tomizawa (2009). Table 4 is the data on unaided distance vision of 3168 pupils comprising nearly equal number of boys and girls aged 6-12 at elementary schools in Tokyo, Japan, examined in June 1984. The data in Table 4 have also been analyzed by Tomizawa (1985), Miyamoto et al. (2004), and Tahata and Tomizawa (2006). Table 5 is the data on unaided distance vision of 4746 students aged 18 to about 25 including about 10 percent women in Faculty of Science and Technology, Science University of Tokyo in Japan examined in April 1982. The data in Table 5 have been analyzed by Tomizawa (1984, 1985) and Tahata et al. (2009). The data in Table 6 represent the cross-classification of a sample of individuals according to their socioprofessional category in 1954 and in 1962 (see Caussinus, 1965; Bishop et al., 1975, p. 298). Tables 1 through 6 are the data of square contingency tables having the same row and column classifications. In addition, the categories in each of Tables 1 through 6 are ordered. Many observations concentrate on (or near) the main diagonal cells in the table. Therefore the row classification tends to be strongly associated with the column classification, namely, the model of independence (i.e., null association) between the row and column classifications does not hold. For those data we are interested in whether or not the row value of an individual is symmetric to the column value. Many models of symmetry and asymmetry have been proposed by many statisticians; for instance, Bowker (1948), Caussinus (1965), Bishop et al. (1975, Chap. 8), McCullagh (1978), Goodman (1979), Agresti (1983, 2002), Tomizawa (1993, 2009) , and Tomizawa and Tahata (2007). We omit here the details of models of symmetry or asymmetry. For the data in Tables 1 through 6 we are also interested in measuring the relative improvement in variation in predicting the value of the other variable when the value of one variable is known, opposed to when it is not

341

known. Consider an r x c contingency table with both nominal categories of the explanatory variable X and the response variable Y. Let Pij denote the probability that an observation will fall in the (i, j)-th cell (i = 1, ... ,rj j = 1, ... ,c). A measure which describes the proportional reduction in variation (PRV) from the marginal distribution of Y to the conditional distribution of Y given the value of X has form

V(Y) - E[V(YIX)] V(Y)

(1.1 )

where V(Y) is an index of variation for the marginal distribution of Y, and E[V(YIX)] is the expectation of the conditional variation taken with respect to the distribution of X (Agresti, 2002. p. 56). Tomizawa, Seo and Ebi (1997) proposed the generalized PRY measure defined by

(A> -1),

where c

Pi·

=

r

LPit, p.j

=

t=l

LPSj, s=l

and the value at A = 0 is taken to be the continuous limit as A - 7 0, and where A is a real value that is chosen by the user. Note that Tomizawa and Ebi (1998) and Tomizawa and Machida (1999) extended the measure T(A) into the multi-way contingency tables. The variation index used in T(A) is

V(Y) =

ｾ＠

(1 -ｴｰｾＫｬＩ＠

,

J=l

which includes the Shannon entropy (when A = 0) and Gini concentration (when A = 1). In special cases, when A = 1, T(l) is identical to Goodman and Kruskal's (1954) measure (called the concentration coefficient) defined

342

by

T=

and when).. = 0, T(O) is identical to Theil's (1970) measure (called the uncertainty coefficient) defined by

t U

=

i=l

tpij log ( Pi j ) j=l p"'P'J

---'----c-----

- LP.j logp.j j=l

In a nominal-nominal contingency table, for a situation in which the explanatory and response variables are not defined clearly, a measure which describes the PRY from the marginal distribution of one variable (of X and Y) to the conditional distribution of the variable given the value of the other variable has a general form V(Y)

+ V(X)

- E[V(YIX)]- E[V(X/y)] V(Y) + V(X)

(1.2)

Miyamoto, Usui and Tomizawa (2005) proposed a generalized PRY measure, i.e., a generalized total uncertainty measure ｔｴｾｬ＠ with)" > -1 (in a similar idea to TP.·»). In a special case, when).. = 0, ｔｴｾｌｬ＠ is identical to Freeman's (1987, p. 101) total uncertainty measure defined by 2

Utotal = ｾＭｲｃ｣

t

i=l

t Pij log ( Pi j j=l p",p']

)

- LPi.logpi. - LP.j logp.j i=l j=l

For a nominal-ordinal table with a nominal variable X and an ordinal variable Y, Tomizawa, Miyamoto and Yajima (2002) proposed a PRY measure. For ordinal-ordinal tables, Tomizawa and Yukawa (2003, 2004) proposed some PRY measures. Also, for an ordinal-ordinal table in which the explanatory and response variables are not defined clearly, Yamamoto and Tomizawa (2009) considered a PRY measure ｾｬ｡＠ with ).. > -1 (see Section 2).

343

For the data in Tables 1 through 6, we cannot define clearly which of row and column variables is the explanatory variable and the response variable. So, for these data we are interested in applying Yamamoto-Tomizawa PRY (A) measure I]> total' The purpose of the present paper is (1) to review the PRY measure ｉ｝＾ｾｌ＠ and (2) to analyze and compare between the data in Tables 1 through 6 . h ,T..(A) usmg t e measure 'l'total' 2. Review of generalized total uncertainty measure

Consider an r x c contingency table with ordered categories in which the explanatory and response variables are not defined clearly. This section reviews briefly the generalized total uncertainty measure I]> ｾｌＮ＠ The measure is defined as follows: for A > -1,

(A + 1)

｛ｾＨｆｫＩａＫＱ＠ ｾ＠

I]>(A) _ total -

J(A) l(Jk) ｾＧｊ＠

ｾ＠ ｾＨ､ｫＩａＫＱ＠

+

ｾ＠

j=l k = l

ｾ＠ i= l k= l

1

J(A) 2(2k)

2'

r- 1

c-1

L ｈｾ｝Ｉ＠

'

+ L ｈｾＩ＠

j=l

i=l

where c

j

ｆＮｾＱＩ＠

=

LP.t,

L

F.j2) =

t= l

r

i

(l) C i·

=

'" ｾｐｳＮＬ＠

C(2) 2'

= '" ｾ＠

ｾ＠ A

Ps· ,

s=i+1

s=l

H(A) = l(J)

p·t ,

t= j + 1

(1 _ｾＨｆｫＩａ

Ｋ ＱＩ＠

ｾＧｊ＠

'

k=l

H(A) 2(2)

= ｾ＠ A

(1 _ｾＨ､ｫＩａＫＱ＠ ｾ＠

2'

,

k=l

(A)

J'Uk)

=

1

'\(H 1)

ｾ＠

r

F(k) ｾ＠

Flk)

{(

p,) A} ,

F(k)jF(k) 2J

'J

_

1

344

with j

FS) =

aU) =

F(2) =

LPit, t=1

2J

c

Pit, L t=j+l

i

r

0 IS gIven '¥ total WIt Pij rep ace by {Pij}, where Pij = Iij / nand n = L L Iij. Using the delta method (Bishop et al., 1975, Sec. 14.6), ｖｮＨ＼ｩ＾ｾｌｉ＠ - ＼ｉ＾ｾｌＩ＠ has asymptotically (n ----> (0) a normal distribution with mean zero and variance (72 ｾｩ｡ｴｬＮ＠ For

[

345

the detail of ＨＷＲ｛＼ｉ＾ｾｌ｝Ｌ＠ see Yamamoto and Tomizawa (2009). Therefore we can obtain an approximate confidence interval for ＼ｉ＾ｾｩ｡ｬ＠ using the estimated approximate standard error ｯＭ｛＼ｉ＾ｾｌｬＯｖｮ＠ with {Pij} replaced by {Pij}.

(72 [ ｾ｡ｬ＠

for ｾｩ｡ｬＧ＠

where ｯＭＲ｛＼ｉ＾ｾｩ｡ｬ＠

denote

4. Analysis of data

We shall analyze the data in Tables 1 through 6 using the total uncertainty measure ＼ｉ＾ｾｩ｡ｬＧ＠ Table 7 gives the value of estimated measure ｾｌ＠ and the approximate 95% confidence interval for the measure ＼ｉ＾ｾｌ＠ applied to the data in Tables 1 through 6. We see from Table 7 that the confidence interval for ＼ｉ＾ｾｌ＠ applied to the data in Table 1 includes zero for all A. Therefore this would indicate that there is a structure of independence, i.e., null association, between the daily temperature at Nagasaki City in 2001 and in 2002. Namely, when we know the temperature of a day in 2001 (in 2002), the knowledge would not be useful for predicting the temperature of the day in 2002 (in 2001). We also see from Table 7 that for the data of Merino ewes in Table 2 the values of estimated measure ｾｌｬ＠ are close to zero, however, the confidence intervals for ＼ｉ＾ｾｩ｡ｬ＠ do not include zero. Therefore these would indicate that the number of lambs born in 1953 may be somewhat associated with the number of lambs born in 1952. Namely, when we know the number of lambs born in 1952 (in 1953), the knowledge may be somewhat useful for predicting the number of lambs born in 1953 (in 1952). Moreover we see from Table 7 that for three kinds of vision data in are greater than Tables 3, 4 and 5, the values of estimated measure ｾｌ＠ zero and the confidence intervals for ＼ｉ＾ｾｩ｡ｬ＠ do not include zero. Therefore for each of vision data in Tables 3, 4 and 5, the right eye grade for an individual is strongly associated with the left eye grade for the individual. Namely, when we know the grade of one eye (of right eye and left eye) for an individual, the knowledge would be useful for predicting the grade of the other eye for the individual. In addition, we see from Table 7 that (1) the value of estimated measure ｾｌ＠ and the values in confidence interval for ＼ｉ＾ｾｩ｡ｬ＠ applied to the vision data of pupils in Table 4 are greater than the corresponding values of them applied to the vision data of women in Table 3, and (2) the values of them applied to the vision data of students in Table 5 are greater than the corresponding values of them applied to the vision data of pupils in Table

346

4. Thus, when we want to predict the grade of one eye for an individual by obtaining the knowledge of the grade of the other eye for the individual, (1) the knowledge would be useful for the data of pupils in Table 4 rather than for the data of women in Table 3, and also (2) the knowledge would be useful for the data of students in Table 5 rather than for the data of pupils in Table 4. We see from Table 7 that for example, when A = 1, the value of estimated measure ＼ｬ＾ｾＡｌ＠ is 0.650 for the data of students in Table 5. Thus, when we predict the grade of one eye for a student by obtaining the knowledge of the grade of the other eye for the student, the prediction becomes 65% better when we know the information than when we do not know the information. Similarly, for the vision data of pupils in Table 4, the prediction becomes 53% better when we know the information than when we do not know the information. Also, for the vision data of women in Table 3, the prediction becomes 44% better. We see further from Table 7 that for the data of socioprofessional status in Table 6, the value of estimated measure ＼ｬ＾ｾ｡＠ are greater than zero and the confidence intervals for ＼ｉ＾ｾｌ＠ do not include zero. In addition, the value of ＼ｬ＾ｾｌ＠ applied to the data in Table 6 is greater than any other value of ＼ｬ＾ｾｩ｡＠ applied to the data in Tables 1 through 5. Therefore, for the data of socioprofessional status in Table 6, the socioprofessional status in 1962 for an individual is strongly associated with the status in 1954 for the individual. Namely, when we know the socioprofessional status in 1954 (in 1962) for an individual, the knowledge would be useful for predicting the status in 1962 (in 1954) for the individual. We see from Table 7 that for example, when A= 1, the value of estimated measure ＼ｬ＾ｾＡ｡＠ is 0.699 for the data of socioprofessional status in Table 6. Thus, we want to predict the socioprofessional status in one year of 1954 and 1962 for an individual by obtaining the knowledge of the status in the other year for the individual, the prediction becomes about 70% better when we know the information than when we do not know the information. 5. Remarks

For a two-way contingency table with the explanatory variable X and the response variable Y, the PRY measure of form (1.1) including, e.g., TCA), T and U, would be useful for seeing what degree the relative improvement in variation for predicting the value of response variable Y when we know the value of explanatory variable X is toward the perfect prediction (i.e.,

347

when the measure equals 1). For a two-way contingency table in which the explanatory and response variables are not defined clearly, the PRY measure of form (1.2) including, e.g., ｔｴｾｬＧ＠ Utotal and ＼ｉ＾ｾｩ｡ｬＧ＠ would be useful for seeing what degree the relative improvement in variation for predicting the value of one variable when we know the value of the other variable is toward the perfect prediction. Yamamoto and Tomizawa (2009) applied the total uncertainty measure ＼ｉ＾ｾｩ｡ｬ＠ to the data of cross-classification of father's and his son's occupational status in Denmark, in British and in Japan (though the details are omitted here). When we want to predict the son's occupational status for a pair of father and his son by obtaining the knowledge of his father's occupational status for the pair, and conversely we want to predict the father 's occupational status for the pair by obtaining the knowledge of his son's occupational status for the pair, we are interested in what degree the prediction becomes better when we know the information for one of the pair than when we do not know the information. In such a case, the PRY measure as ｾｌ＠ would be useful for measuring the degree of the proportional reduction in variation. Note that (1) the measures T()') , T, U, ｔｌｾＱ＠ and Utotal are usually used when the row and column classifications have both nominal categories, (2) the measure ＼ｉ＾ｾｌ＠ is used when those have both ordered categories, and (3) the PRY measure proposed by Tomizawa et al. (2002) is used when one of row and column classifications has the nominal category and the other has the ordered category.

6. Conclusions The present paper has analyzed several categorical data and compared the degree of PRY between them using the total uncertainty measure ＼ｉ＾ｾｌﾷ＠ In the present paper we have seen that when we want to predict the value of one variable by knowing the value of the other variable, the prediction based on the information would be useful for the unaided distance vision data and for the data of socioprofessional status rather than for the data of temperatures in two years and for the data of numbers of lambs born in consecutive years.

7. Discussion The data in Table 8, taken from Everitt (1992, p. 56) show the frequencies obtained when 284 consecutive admissions to a psychiatric hospital are

348

classified with respect to social class and diagnosis. Everitt examined what degree the knowledge of a patient's social class is useful for predicting his diagnostic category using Goodman and Kruskal's (1954) lambda measure. We are now interested in applying the PRY measures, TeA), ｔｴｾｬ＠ and

Ｑ＾ｾｌ＠ to the data in Table 8. However, since the categories in Table 8 seem to be nominal (i.e., not ordinal), it would not be suitable to apply the measure Ｑ＾ｾｬ｡Ｎ＠ Therefore we shall apply TeA) and ｔｴｾｬ＠ to these data. We now see that the estimated values of TeA) ＨｔｌｾｉＩ＠ are, for instance, 0.041 (0.046) when A = 0, 0.043 (0.050) when A = 0.4, and 0.039 (0.049) when A = 1, and the confidence intervals for TeA) ＨｔｴｾｬＩ＠ do not include zero though the details are omitted. Therefore the knowledge of a patient's social class would be useful for predicting his diagnostic category, and also conversely the knowledge of a patient's diagnostic category would be useful for predicting his social class. So, using the PRY measure, it would be important to examine what degree the prediction of his diagnostic category becomes better when one knows the patient's social class than when one does not know it.

References 1. Agresti, A. (1983). A simple diagonals-parameter symmetry and quasi-

symmetry model. Statistics and Probability Letters, 1, 313-316. 2. Agresti, A. (2002). Categorical Data Analysis, second edition. Wiley, New York. 3. Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. The MIT Press, Cambridge, Massachusetts. 4. Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association, 43, 572-574. 5. Caussinus, H. (1965). Contribution a l'analyse statistique des tableaux de correlation. Annales de la Faculte des Sciences de l'Universite de Toulouse, 29, 77-182. 6. Cressie, N. and Read, T. R. C. (1984). Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Ser. B, 46, 440-464. 7. Everitt, B. S. (1992). The Analysis of Contingency Tables, second edition. Chapman and Hall, London. 8. Freeman, D. H. (1987). Applied Categorical Data Analysis. Marcel Dekker, New York. 9. Goodman, L. A. (1979). Multiplicative models for square contingency tables with ordered categories. Biometrika, 66, 413-418. 10. Goodman, L. A. and Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732-764.

349 11. McCullagh, P. (1978). A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65, 413-418. 12. Miyamoto, N., Niibe, K. and Tomizawa, S. (2005). Decompositions of marginal homogeneity model using cumulative logistic models for square contingency tables with ordered categories. Austrian Journal of Statistics, 34, 361-373. 13. Miyamoto, N., Ohtsuka, W. and Tomizawa, S. (2004). Linear diagonalsparameter symmetry and quasi-symmetry models for cumulative probabilities in square contingency tables with ordered categories. Biometrical Journal, 46, 664-674. 14. Miyamoto, N., Usui, E. and Tomizawa, S. (2005). Generalized total uncertainty measure for two-way contingency table with nominal categories. The Pacific and Asian Journal of Mathematical Sciences, 1, 23-39. 15. Patil, G. P. and Taillie, C. (1982). Diversity as a concept and its measurement. Journal of the American Statistical Association, 77, 548-561. 16. Stuart, A. (1955). A test for homogeneity of the marginal distributions in a two-way classification. Biometrika, 42, 412-416. 17. Tahata, K. and Tomizawa, S. (2006). Decompositions for extended double symmetry models in square contingency tables with ordered categories. Journal of the Japan Statistical Society, 36, 91-106. 18. Tahata, K. and Tomizawa, S. (2008). Orthogonal decomposition of pointsymmetry for multi-way tables. Advances in Statistical Analysis, 92, 255-269. 19. Tahata, K., Takazawa, A. and Tomizawa, S. (2008). Collapsed symmetry model and its decomposition for multi-way tables with ordered categories. Journal of the Japan Statistical Society, 38, 325-334. 20. Tahata, K., Yamamoto, K., Nagatani, N. and Tomizawa, S. (2009). A measure of departure from average symmetry for square contingency tables with ordered categories. Austrian Journal of Statistics, 38, 101-108. 21. Tallis, G. M. (1962). The maximum likelihood estimation of correlation from contingency tables. Biometrics, 18, 342-353. 22. Theil, H. (1970). On the estimation of relationships involving qualitative variables. American Journal of Sociology, 76, 103-154. 23. Tomizawa, S. (1984). Three kinds of decompositions for the conditional symmetry model in a square contingency table. Journal of the Japan Statistical Society, 14, 35-42. 24. Tomizawa, S. (1985). Analysis of data in square contingency tables with ordered categories using the conditional symmetry model and its decomposed models. Environmental Health Perspectives, 63, 235-239. 25. Tomizawa, S. (1993). Diagonals-parameter symmetry model for cumulative probabilities in square contingency tables with ordered categories. Biometrics, 49, 883-887. 26. Tomizawa, S. (2009). Analysis of square contingency tables in statistics. American Mathematical Society Translations, 227, 147-174. 27. Tomizawa, S. and Ebi, M. (1998). Generalized proportional reduction in variation measure for multi-way contingency tables. Journal of Statistical Research, 32, 75-84.

350

28. Tomizawa, S. and Machida, M. (1999). Measure of proportional reduction in variation for multi-way contingency tables with multiple response variables. The Egyptian Statistical Journal, 43, 167-182. 29. Tomizawa, S. and Tahata, K. (2007). The analysis of symmetry and asymmetry: orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables. Journal de la Societe Francaise de Statistique, 148, 3-36. 30. Tomizawa, S. and Yukawa, T. (2003). Proportional reduction in variation measures of departure from cumulative dichotomous independence for square contingency tables with same ordinal classifications. Far East Journal of Theoretical Statistics, 11, 133-165. 31. Tomizawa, S. and Yukawa, T. (2004). Proportional reduction in variation measure for two-way contingency tables with ordered categories. Journal of Statistical Research, 38, 45-59. 32. Tomizawa, S., Miyamoto, N. and Yajima, R. (2002). Proportional reduction in variation measure for nominal-ordinal contingency tables. Calcutta Statistical Association Bulletin, 53, 167-183. 33. Tomizawa, S., Miyamoto, N. and Yamamoto, K. (2006). Decomposition for polynomial cumulative symmetry model in square contingency tables with ordered categories. Metron, 64, 303-314. 34. Tomizawa, S., Seo, T. and Ebi, M. (1997). Generalized proportional reduction in variation measure for two-way contingency tables. Behaviormetrika, 24, 193-201. 35. Yamamoto, K. and Tomizawa, S. (2009). Measure of proportional reduction in variation and measure of agreement for contingency tables with ordered categories. International Journal of Applied Mathematics and Statistics, 14, 3-23.

351 Table 1. The daily temperatures at Nagasaki City, Japan, in 2001 and 2002; from Tahata et al. (2008).

2001

Below normals (1)

2002 Normals (2)

Above normals (3)

Total

Below normals (1) Normals (2) Above normals (3)

11 38 19

18 79 64

30 64 42

59 181 125

Total

68

161

136

365

Table 2. Merino ewes according to number of lambs born in consecutive years; from Tallis (1962). Number of Lambs in 1953

Number of Lambs in 1952 0 1 2

Total

2

58 26 8

52 58 12

1 3 9

111 87 29

Total

92

122

13

227

0

Table 3. D naided distance vision of 7477 women aged 30-39 employed in Royal Ordnance factories in Britain from 1943 to 1946; from Stuart (1955). Left eye grade Third Second (2) (3)

Right eye grade

Best (1)

Best (1) Second (2) Third (3) Worst (4)

1520 234 117 36

266 1512 362 82

Total

1907

2222

Worst (4)

Total

124 432 1772 179

66 78 205 492

1976 2256 2456 789

2507

841

7477

352 Table 4. Unaided distance vision of 3168 pupils comprising nearly equal number of boys and girls aged 6-12 at elementary schools in Tokyo, Japan, examined in June 1984; from Tomizawa (1985). Left eye grade Second Third (2) (3)

Right eye grade

Best (1)

Best (1) Second (2) Third (3) Worst (4)

2470 96 10 12

126 138 42 7

Total

2588

313

Worst (4)

Total

21 33 75 16

10 5 15 92

2627 272 142 127

145

122

3168

Table 5. Unaided distance vision of 4746 students aged 18 to about 25 including about 10% women in Faculty of Science and Technology, Science University of Tokyo in Japan examined in April 1982; from Tomizawa (1984). Left eye grade Second Third (2) (3)

Right eye grade

Best (1)

Best (1) Second (2) Third (3) Worst (4)

1291 149 64 20

130 221 124 25

Total

1524

500

Worst (4)

Total

40 114 660 249

22 23 185 1429

1483 507 1033 1723

1063

1659

4746

Table 6. Cross-classification of individuals according to Socioprofessional status; from Caussinus (1965). Status in 1954

(1)

(2)

Status in 1962 (3) (4) (5)

(1) (2) (3) (4) (5) (6)

187 4 22 6 1 0

13 191 8 6 3 2

17 4 182 10 4 2

11 9 20 323 2 5

Total

220

223

219

370

(6)

Total

3 22 14 7 126 1

1 3 4 17 153

232 231 249 356 153 163

173

179

1384

353

Table 7.

Estimate of measure ＼ｰｾｬ｡Ｇ＠

estimated approximate standard

error for ＴＺ＾ｾｬ｡Ｇ＠ and approximate 95% confidence interval for ＼ｰｾｬ｡Ｇ＠ applied to Tables 1 through 6. .A.

-0.4 0.0 0.4 1.0 -0.4 0.0 0.4 1.0 -0.4 0.0 0.4 1.0 -0.4 0.0 0.4 1.0 -0.4 0.0 0.4 1.0 -0.4 0.0 0.4 1.0

Estimated measure Standard error (a) For Table 1 0.009 0.006 0.011 0.007 0.012 0.008 0.012 0.008 (b) For Table 2 0.085 0.030 0.093 0.033 0.094 0.034 0.095 0.035 (c) For Table 3 0.278 0.007 0.363 0.008 0.408 0.009 0.435 0.009 (d) For Table 4 0.432 0.019 0.503 0.020 0.524 0.021 0.530 0.021 (e) For Table 5 0.470 0.010 0.574 0.010 0.622 0.010 0.650 0.009 (f) For Table 6 0.518 0.019 0.624 0.019 0.673 0.019 0.699 0.019

Confidence interval (-0.003, (-0.004, (-0.004, (-0.004,

0.020) 0.025) 0.028) 0.029)

(0.027, (0.029, (0.028, (0.026,

0.143) 0.157) 0.161) 0.163)

(0.265, (0.347, (0.390, (0.417,

0.292) 0.379) 0.425) 0.453)

(0.395, (0.463, (0.484, (0.488,

0.470) 0.543) 0.565) 0.572)

(0.450, (0.554, (0.603, (0.631,

0.490) 0.593) 0.641) 0.668)

(0.481, 0.554) (0.587, 0.662) (0.636,0.710) (0.663, 0.736)

354

Table 8. Social class and diagnostic category for a sample of psychiatric patients; from Everitt (1992, p. 56). Diagnosis Depressed Personality disorder

Social class

Neurotic

(1) (2) (3)

45 10 17

25 45 21

Total

72

91

Schizo phrenic

Total

21 24 18

18 22 18

109 101 74

63

58

284

Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 355- 361)

THE ELECTRON RESERVOIR HYPOTHESIS FOR TWO-DIMENSIONAL ELECTRON SYSTEMS

K. YAMADA1, T. UCHIDA1 , M. FUJITAl, H. KOIZUMI 2

AND T. TOYODA h 1 Department of Physics, Tokai University, Kitakaname 1117, Hiratsuka, Kanagawa 259-1292, Japan 2Nippon Gear Co., Ltd., Kirihara-cho 7, Fujisawa, Kanagawa 252-0811 , Japan * Corresponding author. E-mail: [email protected]

The electron reservoir model for the integer quantum Hall effects, the magnetoplasmon dispersion plateaus, and the radiation-induced magnetoresistance oscillations, are briefly reviewed.

Keywords: Two-dimensional electron systems; Quantum Hall effects; Magnetoplasmon dispersion ; Magnetoresistance oscillations.

1. Introduction

The quantum statistical theory of many-electron systems is based on the grand canonical ensemble, which requires the existence of an electron reservoir. If the number of electrons of the system under consideration is fixed, then it is necessary to calculate the chemical potential as a function of the electron number N exp and other thermodynamic variables by solving the equation (1)

where ｾ＠ oR is the second quantized field operator in the Heisenberg picture describing the electrons, and a is the spin variable. The notation < ... >G stands for the grand canonical ensemble expectation value defined as 1

(.. ')G == -tr{e- i3 Zo

355

(

ｾ＠

ｾ＠

H-j1.N

)

... },

(2)

356

where Ze is the grand partition function, f3 = l/kBT, J-L is the chemical potential, and iI and N are the Hamiltonian and number operators for the electrons, respectively. Except for the zero-temperature limit of an ideal electron gas, the above equation (1) cannot be solved to find the chemical potential

(3) This difficulty is inherent in the grand canonical ensemble formulation of quantum many-body theories. However, if the system is an open system with respect to the electron number, the situation is totally different. In such a system, the chemical potential is one of the independent thermodynamic variables that are directly controlled in the experiment. Then one need not solve the above equation. The aim of this paper is to show that there exist such cases in the two-dimensional electron systems (2DES) in various semiconductors such as MOSFET and GaAs heterostructure FET 1 ,2,3. Three prominent cases, i.e., the integer quantum Hall effects, the magnetoplasmon dispersion plateaus, and the radiation-induced magnetoresistance oscillations, are briefly reviewed in the following sections. 2. Quantum Hall Effects In 1980 von Klitzing4 discovered that the Hall conductivity of a twodimensional electron system in the inversion layer of MOSFET at a very low temperature (T = 1.5K) is quantized

(j=1,2 , ... )

(4)

when the system is subjected to a strong perpendicular magnetic field (B = 18.9T). In 1985, Toyoda, Gudmundsson, and Takahashi3 showed that this phenomenon can be fully explained by introducing the second quantized Schrodinger field operators to describe the electrons and by assuming the Hamiltonian

(5) On the right-hand side, the first term HA is the kinetic energy with minimal coupling to the magnetic field; the second term HE, the energy due to the electric field; the third term H spin , the coupling between the spin and the magnetic field; the fourth term H e - e , the electron-electron interaction; and the fifth term H imp , the effects of impurities or lattice defects. Using this

357

Hamiltonian, the canonical equations of motion for the currents

Iv =

J

d2r [- ien ｬｪｉｾＨｸ 2m

Ｉ＠ '8 vljla(X) - ｾ＠me ａｶＨ

ｘＩｬｪｉｾＨ

ｘＩｬｪｉ｡Ｈ｝＠

(6)

are calculated. Then taking the grand canonical ensemble average of the equations and considering the boundary conditions, the Hall conductivity formula

Nee

(7)

O"H=B

is obtained. If the 2DES is an open system with respect to electrons, then

eB N = -

1

LL = N((3 fl) he a n= al+exp(3(Ena-fl) " 00

(8)

where Ena is the energy spectrum of the electron asymptotic field, i.e., the Landau levels with renormalized parameters. Equations (7) and (8) yield the quantum Hall conductivity formula

e2 O"H

=h

ｾ＠ ｾ＠

1

00

1 + exp (3 (Ena - fl) .

(9)

The significance of this result is that the electron-electron interaction and the impurity term in the Hamiltonian do not appear explicitly. Their effects should be in the energy spectrum Ena. Adopting the linear model for the relation between the chemical potential and the gate voltage fl

= a(Vg + Va)

(10)

and assuming the Landau level energy spectrum Ena

=

eB n-m*e

(1) +- + n

2

g* flB sgn(a)-2-B ,

(11)

where flB is the Bohr magneton, m* and g* are the effective mass and effective g-factor, respectively, the Hall conductivity can be written as a function of B, Vg , and temperature. The obtained theoretical results show excellent quantitative agreement with the experimental data3 . 3. Magnetoplasmon Dispersion Plateaus

In 2004, Holland et al. 6 measured the explicit filling factor dependence of the dispersion of the long-wavelength magnetoplasmon in a high-mobility 2DES realized in a GaAs quantum well, using the coupling between the

358

plasmon with THz radiation. The observed dispersion seemed to deviate violently from the well-established semi-classical dispersion 271"e 2 N 2DES (12) q, em where e is the dielectric constant of GaAs semiconductor into which the 2DES is embedded, m is the electron effective mass, -e is the electron charge, N 2DE S is the electron number density, q is the wave-number vector, and We = eBjmc is the cyclotron frequency. U sing the measured magnetoplasmon frequency they defined the renormalized magnetoplasmon frequency, 2

2

W mp

= We +

w!;P,

nEXP =

{

2 - We

We

mp

They plotted this

EXp }2

W mp

.

(13)

n!;P versus the filling factor defined as hcNsample

v = ---"--

(14) eB ' where Nsample is the electron number density of the samples, whose explicit values were given for three samples in Ref. 6. By substituting Eqs. (12) and (14) into Eq. (13), and assuming N 2DE S = Nsample, one straightforwardly finds

nEXP

_

271"ecNsample

mp

-

eB

_

q-

271"e 2 eh vq.

(15)

For a fixed value of the wave-number vector q, this Eq. (15) shows that n!;p is simply proportional to the filling factor . After the measurement, however, they found an astonishing deviation from such a simple linear relation. They found a quantized dispersion with plateaus forming around even filling factors. If we follow carefully the quantum statistical mechanical derivation of the dispersion relation (12)10, then we find that the quantity N 2DES in the formula is actually the grand canonical ensemble expectation value for the electron number density in the system and that it should be given by (8). Then, the renormalized magnetoplasmon frequency defined by Eq. (13) should be replaced by 271"ecN2DEs eB q,

(16)

where w mp is given by Eq. (12). The only difference between Eqs. (15) and (16) is the electron number density. In Eq. (15) , Nsample is a given

359

quantity. On the other hand, in Eq. (16), N 2DE S is a function of the temperature T, the magnetic field strength B, and the chemical potential /-i, as given by Eq. (8). Substituting Eq. (8) into Eq. (16), we find 7

Apart from the factor 27rq/c , the resemblance to the quantum Hall conductivity formula is derived in Refs. 2 and 8 is unmistakable. This result (17) shows excellent agreement with the experimental data.

4. Radiation-induced Magnetoresistance Oscillations

In 2004, Zudov et alY and Mani et aP2 found a new class of magnetoresistance oscillations in a high-mobility two-dimensional electron system (2DES) in a GaAs/ AlxGal-xAs heterostructure subjected to weak magnetic fields and millimeterwave radiation. The period of these new oscillations is governed by the ratio of the millimeterwave to cyclotron frequencies, and the minima of the oscillations are characterized by an exponentially vanishing diagonal resistance. When the millimeterwave radiation is turned off, the magnetoresistance shows the well-established SdHvA oscillation whose period is governed by the ratio of the chemical potential to the cyclotron frequency. Hence this magnetoresistance oscillation is apparently induced by the illumination with millimeterwaves. Although various different theoretical models have been proposed, one crucial question remains to be solved. The experiment by Smet et al. 13 shows that the resistance oscillations are notably immune to the polarization of the radiation field. This observation is discrepant with these theories and seems to cast doubt on the validity of the theoretical models so far proposed. The Fermi liquid theory of electrical conductivity was originally formulated by Eliashberg. The core of the theory is the analytic continuation of the finite temperature current correlation function with respect to the Matsubara frequency to obtain the retarded real-time current response function, which directly yields the conductivity. Here the formulation given in Ref. 5 is applied to an electron gas that is confined in the xy-plane and subjected to a perpendicular magnetic field B = (0,0, B). The general expression of

360

conductivity can be found as -e 2 !'i

O'xx = 4m 2

LL 00

J

dw

M=O

0:

x

1 27f

jPrna x

2

dp P ｾｇｍ･ｸＨｰＬｗＩ｡ｐ＠

of(w)

R

A

-Prnax

{I + ｾｒ･ａｉＨｰＬｷＩｽ＠ (18)

where f(w) = (l+exp(,8hw))-I, GR (G A ) is the retarded (advanced) Green function, and An is the vertex function. The theories so far proposed consider the effects ofthe millimeterwave radiation on the 2DES only. However, if there is an electron reservoir, the effects of the radiation on the electrons in the reservoir should also be taken into account. Since the amount of energy that an excited electron can receive from millimeterwave radiation is !'iv, the condition under which the electron can join the 2DES should be hw e/ 2 + E res < !'iv. This condition can also be written as B < (2mc/!'ie)(!'iv - Eres) == Be. As the chemical potential is the minimum free energy to add an electron to the system, the emergence of such a process may be described by introducing another singularity in the retarded Green function at !'iv. This singularity of the Green function may be expressed as an effective chemical potential. Then the conductivity formula (18) yields I4

O'xx

e2 A

= -m

{WIBI

e2 A m

(19)

== -WOFF

(20)

+ W2B2 + ｾｂｽ＠

== -WON

when the radiation is on, and

e2 A O'xx = - {W2 m

+ ｾｂｽ＠

e2 A m

when the radiation is off. Here ｾ＠ == (-e/47f 2 mcA) LM AO is assumed to be a constant, and Wi'S are given as Wi

=

L ex

f

M =O

,8 hw e

{1 + e(3(c M ,,-TJ;l} {I + e-(3(C M ,,-TJil} .

(21)

In the measurement by Zudov et al. l l the current Ix is measured by controlling the electric field Ex, while the current Iy as well as the external electric field Ey are kept zero. Therefore, the resistivity Rxx observed in their measurement should correspond to 1/0' xx in this theory. The resistivity corresponding to Rxx in RefY is given as (22)

361

The theoretical pattern shows excellent agreement with the experimental curve. The B-dependence of the oscillatory patterns of the millimeterwave induced magnetoresistance oscillations observed by Zudov et al. l l is almost perfectly reproduced from our theoretical model based on the FLH and the ERH, including its immunity to the polarization of the radiation field in perfect accordance with the experimental observation by Smet et al. 13. 5. Concluding Remarks We have shown that the electron reservoir model can perfectly explain the three prominent phenomena in semiconductor 2DES. Although experimental identification of the microscopic mechanism of the electron reservoir still needs to be carried out, there seems to be no doubt that the electron reservoir should exist in those systems. References 1. G. A. Baraff and D. C. Tsui, Phys. Rev. B 24, 2274 (1981). 2. T. Toyoda, V. Gudmundsson, and Y. Takahashi, Phys. Lett. 102A, 130 (1984) 3. T. Toyoda, V. Gudmundsson, and Y. Takahashi, Physica 132A, 164 (1985). 4. K. von Klitzin, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). 5. T. Toyoda, Phys . Rev. A 39, 2659(1989) 6. S. Holland , Ch. Heyn, D. Heitmann, E. Batke, R. Hey, K. J. Friedland, and C.-M. Hu, Phys. Rev. Lett. 93, 186804 (2004). 7. T. Toyoda, N. Hiraiwa, T. Fukuda, and H. Koizumi, Phys. Rev. Lett. 100, 036802 (2008). 8. T. Toyoda, V. Gudmundsson , and Y . Takahashi, Physica 132A, 164 (1985). 9. M. P. Greene, H. J . Lee, J. J. Quinn, and S. Rodriguez, Phys. Rev. 177, 1019 (1969). 10. N. Hiraiwa and T. Toyoda, in preparation. 11. M.A. Zudov, D.R. Du, J.A. Simmons, and J.L. Reno, Phys. Rev. B 64, 201311 (2001). 12 . R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, and V. Umansky, Nature 420,646 (2002). 13. J. H. Smet , B. Gorshunov, C. Jiang, L. Pfeiffer, K. West, V. Umansky, M. Dressel, R. Meisels, F. Kuchar, and K. von Klitzing, Phys. Rev. Lett. 95, 116804 (2005). 14. T. Toyoda, Modern Physics Letters B, 24, 1923 (2010).

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 363-372)

ON THE CORRESPONDENCE BETWEEN NEWTONIAN AND FUNCTIONAL MECHANICS E.V. PISKOVSKIY, LV. VOLOVICH

Moscow Institute of Physics and Technology Institustkiy lane 9, 141700 Dolgoprudny, Moscow Region, Russia email: [email protected] Steklov Mathemati cal Institute Gubkin St.8, 119991 Moscow, Russia email: volovich@mi. ms.ru

The world view underlying traditional science is based on reductionism and determinism when there is an empty space (vacuum) and material points which move along the Newtonian trajectories. This approach may be called "mechanistic" or " Newtonian" . Quantum mechanics, in its Copenhagen interpretation, also adopts this world view. However this world view is not satisfactory by at least two reasons. First, there is uncertainty in the derivation of the position and velocity of the material point and second, it can not solve the time irreversibility problem. Moreover, the Newtonian approach is not well suited for applications of mathematics and physics to life science. Recently a new approach to classical mechanics was proposed in which the basic notion is not the trajectory but a probability distribution. In this functional mechanics approach one deals with the mean trajectories and one has corrections to the Newtonian equation of motion. In this note we consider correspondence between the Newtonian trajectories for an anharmonic oscillator and the averaged trajectories in the functional mechanics and compute the dependence of the characteristic time from the dispersion.

1. Introduction

Classical mechanics, as first formulated by Newton and further developed by others, was seen, until the early 20th century, as the foundation for science as a whole. Other disciplines, such as physics, biology, or economics, did accept a general mechanistic or Newtonian approach and world view. Even quantum mechanics is based on the classical deterministic Newtonian mechanics , according to the Copenhagen interpretation. The basic principle behind Newtonian science is reductionism: to understand any complex phenomenon, you need to reduce it to its individual components. The smallest possible parts are called atoms or "elementary 363

364

particles". The only property that fundamentally distinguishes particles is their position in space and velocities. If you know the initial positions and velocities of the particles constituting a system together with the forces acting on those particles, then you can in principle predict the further evolution of the system with complete certainty and accuracy. The evolution will be regular, reversible and predictable. Such categories as life, mind, or organization are to be seen as particular arrangements of particles in space and time. However the Newtonian world view is not satisfactory by at least two reasons. First, there is uncertainty in the derivation of the position and velocity of the material point. Second, it can not solve the time irreversibility problem, see 1; for a discussion of the irreversibility problem see 2. Moreover, the Newtonian approach is not well suited for applications of mathematics and physics to life science. Recently a new approach to classical mechanics was proposed in which the basic notion is not the trajectory but a probability distribution. In this functional mechanics approach 1, see also 3,4,5,6 ,7, one deals with the mean trajectories and one has corrections to the Newtonian equation of motion. Emphasize that the exact derivation of the coordinate and momentum can not be done, not only in quantum mechanics, where there is the Heisenberg uncertainty relation, but also in classical mechanics. Always there are some errors in setting the coordinates and momenta. There are classical uncertainty relations 1: !:,.q > 0, !:,.p> 0, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). The concept of arbitrary real numbers, given by the infinite decimal series, is a mathematical idealization, such numbers can not be measured in the experiment. In this note we consider correspondence between the Newtonian trajectories for an anharmonic oscillator and the averaged trajectories in the functional mechanics and compute the dependence of the characteristic time from the dispersion. We are motivated by the consideration of the quantum classical correspondence for the baker's map performed by Inoue, Ohya and one of the present authors 8. Note that the conventional widely used concept of the microscopic state of the system at some moment in time as the point in phase space, as well as the notion of trajectory and the microscopic equations of motion have no direct physical meaning, since arbitrary real numbers not observable

365

(observable physical quantities are only presented by rational numbers). The fundamental equation of the microscopic dynamics of the proposed functional probabilistic approach is not Newton's equation, but a Liouville equation for distribution function. It is well known that the Liouville equation is used in statistical mechanics for the description of the motions of gas. Let us stress that we shall use the Liouville equation for the description of a single particle in the empty space. A Liouville equation on the manifold f with coordinates x = (xl, ... , xk) has the form [)p [)t

k

+

,,[) i L [)x Jpv )

= 0.

(1)

i=1

Here p

= p(x, t) is the probability density function and v = v(x) =

(vI, ... , v k )

vector field on f. The solution of the Cauchy problem for the equation (1) with initial data -

(2) might be written in the form

(3) Here CPt (x) is a phase flow along the solutions of the characteristic equation

X=v(x).

(4)

Let (q,p) be co-ordinates on the plane ffi.2 (phase space), t E ffi. is time. The state of a classical particle at time t will be described by the function p = p( q, p, t), it is the density of the probability that the particle at time t has the coordinate q and momentum p. Therefore, in the functional approach to classical mechanics the concept of precise trajectory of a particle is absent, the fundamental concept is a distribution function p = p(q,p, t) and D-function as a distribution function is not allowed. We assume that the continuously differentiable and integrable function p = p( q, p, t) satisfies the conditions:

p ｾ＠ 0,

r

ill!.2

p(q,p, t)dqdp = 1, t E ffi..

(5)

If f = f(q,p) is a function on phase space, the average value of f at time t is given by the integral

(f)(t) =

J

f(q,p)p(q,p, t)dqdp.

(6)

366

In a sense we are dealing with a random process ｾＨｴＩ＠ with values in the phase space. Motion of a point body along a straight line in the potential field will be described by the equation

8p p 8p 8t = - m 8q

8V(q) 8p

+ --;)g 8p .

(7)

Here V(q) is the potential field and mass m > O. If the distribution Po(q,p) for t = 0 is known, we can consider the Cauchy problem for the equation (7):

plt=o

=

(8)

Po(q,p)·

The mean trajectories defined as follows

(q)(t) =

J

qp(q,p, t)dqdp =

J

q(t)Po(q,p)dqdp,

(9)

where q(t) - is a classical trajectory of a point mass, the function q(t) is governed by Newton equation

.. mq

8V(q)

=--;)g'

Therefore the mean trajectory can be obtained by averaging classical trajectory with reference to probability distribution function for initial conditions. This fact will be widely used in the present work. 2. Anharmonic oscillator In the present work an anharmonic oscillator is considered. Namely a point mass is moving within a field that is described by the potential V(q): 1

1

V(q) = 2w6q2 + = t:: q4

(10)

where Wo > 0 and E > 0 is a small coupling constant. The coordinate q(t) changes in accordance with the Newton equation:

q + W6q = _ Eq 3.

(11)

We set the following initial conditions:

(12) The exact solution to (11) is well-known but for simplicity we shall use the approximate Krylov-Bogolyubov method described for example in 9. The solution reads: a3

qKB(t) = acos(tw)

+ E32w5

cos(3tw)

+ O(E2),

367

where a is an arbitrary constant and is the frequency w depends on parameter a as follows:

3

w = w(a) = wo(1 + Sa 2 c + O(c 2 )). From the initial conditions (12) we get (13) We write

and we have qKB(O, b) = b. The mean value of coordinate in the functional mechanics we define by the integral

(q)(t, CJ)

=

11qKB(t, b)e

r;;:;; V

CJ1r

(b - bO)2

CJ

db

(14)

IR

°

with the distribution function Po(b) = exp{ -(b- bO)2 /CJ} / ViW. Here CJ > is dispersion. Note that (q)(O, CJ) = boo We want to compare two time dependent functions: (q)(t, CJ) and qKB(t, bo). For small t the difference between them is small since (q)(O, CJ) = qKB(O, bo) = boo The question is what is the characteristic threshold value tc such that for t > tc the difference between the two functions become rather big and what is the dependence of tc = tc(CJ) from the dispersion CJ? Note that the analogous problem of the dependence of the characteristic time from the Planck constant is considered in 8.

3. Newtonian and Averaged Trajectories Comparison It was proved in

1

that for any time t one has lim (q)(t, CJ) = qKB(t, bo).

a-+O

That is why one should expect tc(CJ) to increase with CJ --> 0. One can assume without loss of generality that Wo = 1. The real roots of the cubic equation (13) are given by the following function 10:

a = a(b) =

S{£ ｳｩｮｨＨｾ｡ｲｃｓｦﾥ｢ｯＩ＠

368

Let us introduce the following change of variables in the integral (14): + bo = b. It yields

zva

(q)(t, O") =

In 1

(a(zVa + bo) cos(tw(zVa + bo))+

+

E(a(zva+bo))3 32

2

cos(3tw(zva + bo)))e- Z dz

(15)

One can make a rough estimate of the threshold time tc by using the method of steepest descent. In this way we get tc = 0(1/ as 0" --7 O.

va)

4. Numerical Approach One of the ways to compare functional mechanical mean value of coordinate and trajectory obtained by means of perturbations theory is estimate dependence time tthr of convergence on dispersion 0" of initial condition b. To define the threshold time we consider the following function

L\(t) = I(q)(t, 0")

-

qKBM(t, bo)1

(16)

so that the moment of time tc = tthr(O", C) is the minimal value of time t when the absolute value of the function L\(t) equals some positive value C

(17) For instance, constant C can be put equal to bo. In order to carry out numerical estimations one has to define constants as follows E = 0.1, bo = 0.5. The classical and averaged trajectories for different time intervals are plotted on the figures below. The classical and averaged trajectories are represented by the dashed and solid lines respectively. One can see from Fig. 1 that the divergence between classical and averaged trajectories is less than E for t E [0; 15]. On the Fig. 2 it is shown that the amplitude of the averaged trajectory is decreasing with time, consequently the divergence between the trajectories is increasing. Also one can see that the phase shift between oscillations becomes easily observed (Fig. 2). The averaged trajectory amplitude tends to zero with time (Fig 3) while the phase difference of the oscillations does not seem to have any limit value. In order to move further and make estimations of tc for different values of 0" it is necessary to define constant C = O.lqKB(O, bo). So that the

369

q f

0.4 i

!

I

I

f I

/'. '

\

\

\

,

\

\\

/

0.2

I

\

I

\\

I

\

,

-0.2

-0.4

Figure 1.

Numerically estimated classical and averaged trajectories in t E [0; 15]

q

Figure 2.

Numerically estimated classical and averaged trajectories in t E [60; 75]

following equation for threshold time is considered:

= 0.1, bo = 0.5, qKB(O, bo) = bo, ＱｾＨｴ｣Ｉ＠

E

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The numerical estimations and nonlinear regression mentioned in the present work are performed with "Wolfram Mathematica 6.0" program suite licensed to Steklov Mathematical Institute.

372

Acknowledgements

This work was partially supported by grants RFBR-OS-OI-00727-a and RFBR-09-0l-12161-ofi-m, and also by grants NSh-3224.200S.1 and by Division of Mathematics of RAS. References 1. LV. Volovich, Randomness in Classical Mechanics and Quantum Mechanics. Foundations of Physics, DOl 1O.1007/s10701-01O-9450-2; Time Irreversibility Problem and Functional Formulation of Classical Mechanics, arXiv:0907.2445. 2. V.V. Kozlov, Gibbs ensembles and nonequilibrium statistical mechanics, Moscow-Ijevsk (in Russian), 2008. 3. LV. Volovich. Functional mechanics and time irreversibility problem. In "Quantum Bio-Informatics III", ed. L. Accardi, W. Freudenberg, M. Ohya. World Scientific, Singapore, 2010, pp. 393-404. 4. A. S. Trushechkin, 1. V . Volovich, Functional Classical Mechanics and Rational Numbers, P-Adic Numbers, Ultra metric Analysis and Applications, 1:4 (2009), 365-371; arXiv: 0910.1502. 5. A. S. Trushechkin, Irreversibility and the measurement procedure in the functional mechanics, Theor. Mathern. Phys. 164:3 (2010), 435-440. 6. LV. Volovich, Bogolyubov equations and functional mechanics, Theor. Mathern. Phys. 164:3 (2010) 354-362. 7. E.V . Piskovskiy, A study of some model systems of functional mechanics in the functional mechanics framework, Abst., The Second International Conference on Mathematical Physics and Its Applications, Samara, 2010. 8. K. Inoue, M. Ohya, LV. Volovich, Semiclassical properties and chaos degree for the quantum baker's map, Journal of Mathematical Physics, 43, 734-755 (2002) . 9. N.N. Bogolyubov, Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York, 1961. 10. G. Bikhoff, S. Mac Lane, A Survey of Modern Algebra, 4th ed., Macmillan Publishing Co Inc., New York, 1977.

Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 373-386)

QUANTILE-QUANTILE PLOTS: AN APPROACH FOR THE INTER-SPECIES COMPARISON OF PROMOTER ARCHITECTURE IN EUKARYOTES

KASPAR FELDMEIER, JOACHIM KILIAN, KLAUS HARTER, DIERK WANKE* AND KENNETH W. BERENDZEN* 2MBP Pfianzenphysiologie, Universitiit Tiibingen Auf der Morgenstelle 1, D-72076 Tiibingen, Germany * e-mail: [email protected] * e-mail: [email protected] Regulatory non-coding DNA is important to drive gene transcription and thereby influence mRNA and consequently protein abundance. Therefore, biologists and bioinformation scientists aim to extract meaningful information from these sequence regions, in particular upstream regulation regions called promoters, and conclude on regulatory sequence function. While some approaches have been successful for single genes or a single genome, it is an open question whether information on promoter function can readily be transferred between different species. Thus, it is useful for biologists to know more about the general structure and composition of promoters including the occurrence of cis regulatory DNA-elements (CREs) to be able to compare promoter architecture between organisms. To approach this task, we utilized the fully sequenced genomes of the plant model organisms: mouse-ear cress (Arabidopsis thaliana), western balsam poplar (Populus trichocarpa) , Sorghum bicolor and rice (Oryza sativa). For the interspecies comparison we made use of quantile-quantile (QQ)-plots of the variances of hexanucleotides or known functional CREs of core-promoter regions. Here, we suggest that the differences in promoter architecture correlate with the sizes of the intergenic space, i.e. regions, in which the promoters are located. In contrast, analysis of CREs is hampered by the general lack of well characterized transcription factorCRE-relationships.

Keywords: Promoter code; cis-regulatory elements CRE); cross-species promoter analysis; promoter evolution; QQ-plot of variances.

1. Introduction

Gene expression in eukaryotes is controlled by several intermingling levels of regulation that involves both pre- and post-transcriptional processes. One level comprises the intermingling of proteins, termed transcription factors 373

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(TFs), which make direct physical contact with specific short stretches of DNA called cis-regulatory DNA-motif elements (CREs) and subsequently act on the transcription machinery. These CREs are typically positioned upstream of the DNA sequence encoding the RNA transcript of a gene and follow a cryptic regulatory code 1. The interface between the transcription factors and their regulatory DNA sequence companions harbor essential information to control specific gene expression and integrate information derived from upstream signaling cascades 2. From an evolutionary point of view, it has been noted that the same or highly similar TF-CRE-relations exist in closely related species and control the expression of the same or highly similar genes. However, it still remains to be elucidated, how TF-CRE-relations change or are retained over more distant evolutionary time. Previous studies have shown that several DNA-motifs are conserved in the regulatory promoter sequences between many eukaryote genomes, e.g. Arabidopsis, Saccharomyces, Drosophila and Caenorhabditis, and this is especially true for the core-promoter region proximal to the genes 1. To draw conclusions on promoter- and CRE-evolution from comparative cross-species approaches, it is of importance to integrate the genomic sequences of closely and more distantly related species in the same analysis. Plant genomes are especially suited for studies on promoter sequences, as most CREs are located upstream of the transcription start site (TSS) close to the gene sequences 1,3. Fortunately, genomic sequences from several plant species are publicly available which one can use for interspecies comparisons. One of the best studied eukaryote model organisms to date is the flowering plant Arabidopsis thaliana. Its physiological responses and its lifecycle have been studied for more than 100 years. With only five chromosomes, the overall complexity of its genome is low and the intergenic regions that harbor the promoters are small. Thus far, Arabidopsis has risen to one of the best understood eukaryote model organisms with the best annotated genome sequence of all eukaryotes available 4. While Arabidopsis thaliana is a modern dicot plant of little agricultural importance, the evolutionary more distantly related grasses or trees are of a high economical value world wide 5. A major drawback for genetic approaches in grasses lies in the complex genomes of cereal crops that have multimerized and duplicated during the breeding and inbreeding processes of domestication. Rice has a small size and relatively simple genomic organization and is favored in laboratory research as it is the second fully sequenced eukaryote plant species 6. The mono cot Sorghum bicolor, whose

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genome sequence will be about to be finished in the near future, is closer related to rice than it is to the dicot Arabidopsis 7. Despite their economical importance, trees have a great disadvantage for genetic studies due to their naturally a long life cycle compared to cryptogams, making laboratory experiments tedious or not feasible 8. Nevertheless, the genomic sequence of the western balsam poplar (Populus trichocarpa) has been submitted to public databases 8 and can readily been analyzed by bioinformatics routines. Here, we conduct an interspecies promoter motif analysis to assess whether the architecture of the core-promoters and CREs distribution therein is evolutionary conserved. Therefore, we extracted upstream sequences from Arabidopsis, rice, poplar and Sorghum annotated genes, which contain the proximal promoters and most essential CREs. The variances of all hexanucleotide motifs and known functional CREs were computed for these four species. For the pair-wise visualization of the interspecies differences in the promoters, we employed quantile-quantile (QQ)-plots of these variances. This approach disclosed that a higher information density is contained in the promoters of those species with more compact genomes.

2. Material and Methods 2.1. Plant genome information

The genome sequence of the chromosome pseudomolecules for the plant model organisms Arabidopsis thaliana, western balsam poplar (Populus trichocarpa) , Sorghum bicolor and rice (Oryza sativa) were retrieved from GenBanks Plant Genomes Central (http://www . ncbi . nlm. nih. gOY / genomes/PLANTS/PlantList .html): Arabidopsis thaliana GenBank accessions: NC_003070.9, NC_003071.7, NC_003074.8, NC_003075.7 and NC_003076.8; Populus trichocarpa GenBank accessions: NC_008467.1, NC_008468.1, NC_008469.1, NC_008470.1, NC_008471.1 , NC_008472.1, NC_008473.1, NC_008474.1, NC_008475.1, NC_008476.1, NC_008477.1, NC_008478.1, NC_008479.1, NC_008480.1, NC_008481.1, NC_008482.1, NC_008483.1, NC_008484.1 and NC_008485.1; Oryza sativa GenBank accessions: NC_008394.1, NC_008395.1, NC_008396.1, NC_008397.1, NC_008398.1, NC_008399.1, NC_008400.1, NC_008401.1, NC_008402.1, NC_008403.1, NC_008404.1 and NC_008405.1. The genomic sequences of Sorghum bicolor 1v4 were retrieved from the Sorghum bicolor Genome at Plant genome data base, PlantGDB (http://www . plantgdb . org/ /SbGDB/). Promot-

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ers were extracted as 1000 bp 5 of the annotated start for each genes (without redundancy) using Motif Mapper .NET (5.1.1.39) and python scripts (1.2) (http://www.zmbp.uni-tuebingen.de/PlantPhysiologyI ResearehGroups/harter/berendzen/programs.html).

2.2. Hexanucleotide Motifs and cisRegulatory Elements Known CREs were retrieved from the PLACE databases (http://www . dna. afire. go. jp/PLACE/) as has been published previously in 1. Hexamers were generated with Motif Mapper All Oligos function.

2.3. Variance based Promoter Motif Analysis To compute the variances for all DNA-motifs, frequency distribution curves were compiled with the Motif Mapper python scripts. All frequency distribution curves were rooted to the annotated start of the transcriptional unit of the genes pointing to the right. Subsequently, the variance for each motif was computed from its distribution curve.

2.4. Phylogenetic Analysis Comparison of information from the Arabidopsis, Populus, Sorghum and rice (Oryza) genomes as well as correlation data has been compared by using ClustalW (http://www.ebi.ae . ukl elustalw/). Default settings were used to calculate the dendrogram files. Tree graphs were visualized using TreeView software version 1.6.6 (http://taxonomy . zoology. gla. ae. ukl rod/treeview . html). The evolutionary distance was computed on RbcL and Cytc sequences. Distance trees for the variances of DNA-motifs in the promoter were computed from the Euclidean distance of the variance vectors by using statistiXL 1.8 for MS Excel. The variances for each of the motifs and all four organisms have been used as attribute data to compute a derived distance matrix according to the statistiXL descriptions. Next, a correlation coefficient was calculated, to indicate how similar the final hierarchical pattern and initial distance matrix were. A dendrogram file was provided to graphically summarize the similarity patterns.

2.5. Interspecies Quantile-Quantile Plots The a-quantiles of the datasets were computed using Math Works MATLAB routines 9. For each of the DNA-motifs in each of the organisms promoter

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datasets the a-quantiles have been assessed on the basis of the variances of motif occurrence. To conclude on the conservation of motifs in the distantly related plant species, the quantiles of the datasets were plotted against each other. In a QQ-plot two variance datasets were compared by their quantiles. The same quantiles are calculated for both datasets and used as x- /y-coordinates for the QQ-plot. The a-quantiles used for QQ-plot comparisons were as follows: 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98 and 0.99. 3. Results and Discussion

3.1. DNA-motif variance as a measure of information content Biologists and bioinformation scientists are working handin-hand to gain better insight into how regulatory non-coding DNA the promoter sequences is capable of mediating changes in gene expression. The approaches taken so far to compare gene regulation on a genomic scale are hampered by the varying occurrence of these small DNA-motifs between species due to different genome sizes or DNA-base composition (GC/AT-content) 1. On the other hand, the positional frequency of a motif in promoters of one species can be graphed as frequency distribution curves for each motif, whereby each motif has its own trajectory (Fig. 1A). Strong deviations from the mean background frequency are characteristic for disequilibria, i.e. regions within a promoter that do not follow the average distribution, and presumptively are from information states of DNA-protein evolution. It has been demonstrated that higher order information is indeed responsible for these disequilibria since many of these motifs are often DNA binding sites for proteins 1. Based on these observations, one can conclude that a positional bias of a motif in promoters from a single genome reflects functional information. A comparative approach of DNA-motif distribution curves is complicated, as necessary normalization procedures will have a transforming effect on the curves (Fig. 1B) and distort the biological data. Thus, we used the variance of a motifs frequency-distribution within the promoter as a measure of its information content. DNA-motifs with high variance will have distributioncurve signatures that are distinct from the background mean frequency and, hence, harbor more information than others. By using the variance of a motif as a measure for information content, no additional normalization or correction is needed, thus preserving as much biological information as

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possible. This approach provides us with several observations: First, two different frequency-distribution curves for two different DNA motifs with different means can still have the same variance (Fig. lA); the variance is invariant under translation and will not be altered, when the plot is shifted along the y-axis. Second, normalized distribution curves display different variances with different means (Fig. 1B). As a consequence, we can also evaluate rare motives and omit compensating for DNA-base composition between the different genomes of the organisms. We extracted the -1000 bp upstream of the annotated start sites of genes. This resulted in in 33238 promoter sequences for the Arabidopsis genome, 10483 for Populus, 36250 for Sorghum and 18328 for Oryza. Next, the frequency distribution curves for all 4096 hexanucleotides and 426 known CREs were compiled and the variances were calculated.

3.2. Quantile-quantile (QQ)-plots for interspecies promoter comparison To compare the variances of the motifs of different species, we made use of QQ-plots 9. In a QQ-plot, two datasets are compared by their quantiles. The same quantiles were calculated for each dataset and used as coordinates for the QQ-plot. If, for example, two normal distributions with different variances are plotted, the resulting QQ-plot will exhibit a linear function with a steepness dependent on the variances; if they have different means the resulting QQ-plot will be affine. If the variances of the hexanucleotide or CRE maps show a similar distribution, the QQ-plots could have an affine shape. The steepness would correlate with the variance of the variances. A higher consensus (positional bias) in one species could hint to higher average information content in the underlying promoter sequences. First we examine the QQ-plots for hexanucleotides for all species. Arabidopsis and Populus are both dicot species and are considered to be more related to each other than to the two mono cots Oryza and Sorghum. When the quantiles of the variances of Arabidopsis and Populus were QQ-plotted against each other (Fig. 2A) , a graph with only a small steepness was the result. This can be explained by higher information content in the promoters of Arabidopsis compared to Populus. However, there was a strong increase in the last quantile [0.99] in Populus, which can be accounted to 41 hexanucleotides with the highest variance in both organisms. Among these motifs, we identified several motifs of putative functionality, e.g. the TATA-box-like motifs 1.

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Figure 1. Assessing the variances of DNA motifs from frequency distribution curves (A) Two different frequency distribution curves for two different DNA motifs with different means [dashed lines] can have the same variance. The variance is invariant under translation and will not be altered, when the plot is shifted along the y-axis. (B) Interspecies comparison is hampered by different motif frequencies. Thus , normalized distribution curves, which were gained by division through their means for comparative reasons, display a different variance with different means of two otherwise identical frequency distribution curves.

Figure 2B shows the quantiles of the variances of Arabidopsis and Oryza. Although the QQ-plots were located under the expected linear function with a steepness of 1, the quantiles were highly similar, which is indicative for high similarities in the promoter architecture of these two species. Simi-

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larly, the QQ-plots of the Arabidopsis and Sorghum hexanucleotide variances (Fig. 2C) were also proximal to the bisecting line, but with slightly

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higher variances in Sorghum. Here again, the majority of difference was found in the quantiles between 0.90 and 0.96. This can not readily be explained by e.g. different sizes of the promoter dataset - both species had a similar number of annotated promoters. However, it might be possible that the Sorghum genome data or the dataset contains a certain degree of redundancy and, thus, there were more highly conserved motifs with a higher variance found that lead to higher quantiles. Therefore, further research on the nature of these elements and on the Sorghum dataset needs to be done to clarify this matter. The comparison ofthe hexanucleotides for Populus and Oryza (Fig. 2D) displays overall similarities and trends to the Populus and Arabidopsis comparison (Fig. 2A). Hence, the Populus promoter dataset contains unique features that account for these differences. This observation can not be explained by differences in the sizes of the dataset as the number of promoters contains in the Oryza and Populus dataset are of a similar magnitude, while Arabidopsis is larger. Nevertheless, the same trends were observed for all comparisons with the Populus promoters. Figure 2E displays the comparison of hexanucleotide variances of Sorghum and Oryza. Interestingly, the quantiles of Sorghum were much higher than those of Oryza. As has been noted previously, an explanation would be that the underlying promoter regions in Sorghum have a higher amount of high-consensus hexanucleotides - probably hinting to a higher number of regulatory active sequences or to redundance within the dataset. In Figure 2F, Sorghum is compared to the Populus dataset. While Populus displayed the lowest values in its quantiles, the Sorghum dataset contains the highest values in its quantiles of our organisms and, hence, the resulting plot has a very high steepness due to the high amount of high-consensus hexanucleotides (those with specific positional bias) in the Sorghum promoters. While we consider the variance in hexanucleotides a measure for overall promoter architecture and motif composition, we expected that the variances of CREs would provide us with information on conserved corepromoters and retained CRE function in gene regulation. Figure 3 gives QQ-plots of the CRE-motives for the four plant species under investigation. For comparative reasons, we also included the data of the hexanucleotide variances [in grey] from the previous study in the CRE-plots. This will provide us with the information, whether the variances of the CREs differ from the general promoter architecture. Since the CRE-motives harbor known regulatory information, we could expect from them to have high variances in the promoter regions and, thus, harbor a higher order of information.

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In Figure 3A the quantiles of Populus were plotted against Ambidopsis. Similar to the hexanucleotide comparison, the quantiles of Populus were much lower than those of A mbidopsis. On the one hand this finding could be explained by the on average smaller sizes of promoters with higher information content in Ambidopsis. On the other hand, the Populus dataset is of a poor quality due to mis-annotation or problems to curate the genome information. Next, the quantiles of the CRE variances of Oryza were compared to Ambidopsis (Fig. 3B). It was obvious that the variances were of approximately of the same values for the promoters of both species and, thus, close to the bisecting line. Here, also the steepness of the QQ-plot for the CRE variances was similar to that of the QQ-plot of the hexanucleotides, which points towards the fact, that the amount of relative information between two species is similar for hexanucleotides and CRE-motifs. Figure 3C shows the QQ-plot of the CRE variances of Sorghum and Ambidopsis. The quantiles of the variances were very similar for both species, which is indicative of similar frequency distribution curves for the CREs in the promoters of the two species. Especially for the lower quantiles the graph follows the dissecting line. Hence, one can conclude that the motif composition in the promoters of the two species and their architectures are highly similar in general. However, this similarity was highest for the known functional CREs. Only for the quantiles 0.97 - 0.99, the variances diverged. The simplest explanation for this observation was the presence of longer CRE-consensi or naturally rare motifs, which occur slightly more in one genome than the other and, thus, have exorbitantly high variance over a very low mean frequency. The QQ-plots of the CRE variances of Populus against the Oryza (Fig. 3D) or Sorghum (Fig. 3F) datasets revealed a very low information content for Populus. Again, the values in the quantiles of Populus were the smallest in the whole dataset and, therefore, Populus also displayed a low CRE consensus. The same trend was seen for the Sorghum promoters. In Figure 3E, the quantiles of CRE variance in Sorghum were plotted against the Oryza dataset; these quantiles displayed similarities for both species and, thus, the distribution of the known functional elements in the promoters likely follows a similar trend in frequency. Since the quantiles of Sorghum have also been similar to that of Ambidopsis (Fig. 3C), we could conclude that CRE distribution in the promoters might indeed be conserved between the three species. Moreover, these similarities could be detected with QQ-plots of the variance as a measure irrespective of the

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Figure 4. Distance matrix tree graphs for evolutionary distance of the species and Euclidean distance of the DNA-motifs. Tree graphs display the relative relatedness of the organisms on the basis of their protein sequence similarities (A) or the similarities between the DNA-motif variances of all hexanucleotides (B) and known CREs (C). Similarities and dissimilarities between the tree topologies in A - C are highlighted by lines that interconnect the four organisms between the graphs.

different dataset sizes or DNA-base composition (GC/AT-content). Our presumption was that information content within promoters is retained as positional disequilibria in promoter sequences when observed at a genomic scale. As a simple, but effective measurement, we took the variance of DNA-motifs frequency-distribution to indicate the degree of information a motif carries, and compared these variances between species to capture global similarities in promoter architecture. To make conclusions on these interspecies comparisons, we established distance matrices for protein sequence similarity and for the variance of hexanucleotides or CREs; these results were displayed as tree graphs (Fig. 4). The last common ancestor of dicot (Arabidopsis and Populus) and monocot (Sorghum and Oryza) species lived at approx. 140 Mio years before present 10. The phylogenetic tree in Figure 4A captures this evolutionary time and serves as our reference. The Oryza dataset appears to be closer related to Arabidopsis than to its evolutionarly close relative Sorghum when looking at the hexanucletide variances. One possible explanation for this similarity in hexanucleotide composition is that both species have relatively small, compressed genome sizes 11. As a consequence, the intergenic space, i.e. the DNA-region between the genes, is relatively small, and as such, more information must be packed into shorter promoter regions 3. The Arabidopsis thaliana genome is one of the smallest eukaryote genome 4 and, thus, regulatory promoter sequence must be short and enriched for motifs with a high order of information. This notion might be true for Oryza as well, as its genome size is one of the smallest amongst other Poaceae crop plants. The CRE variance is shown in Figure 4C. Here, the overall tree topology follows our reference tree (Fig. 4A), with the exception of the Populus

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dataset, which could likely be considered as an outlier. The reasons for the aberrant Populus set might be manifold, but most likely originate from a low quality in the annotation of the gene starts. 4. Conclusions We have shown that promoter architecture is strongly dependent on the sizes of intergenic regions. The observation that the CRE variances for Arabidopsis, Oryza and Sorghum were highly similar supports the idea that the changes in protein sequence or in cis-regulatory DNA over time might be proportional. Thus, known functional elements display the same evolutionary concepts as are considered for genome or protein evolution. To further support and refine our findings, well annotated genome information of more species is needed and has to be integrated in our analysis. The sequence information of species with different genome sizes and larger evolutionary distances would be of special interest. Moreover, the analysis of the CRE variances is hampered by too little information on transcription factor (TF)-CRE relationships. Hence, our approach is well worth to be repeated with updated information on both, TF-CREs and a larger number of promoter datasets. Acknowledgements We thank Jochen Supper for his constant input and critical discussions. This work was supported by the DFG (HA2146/11-1). References 1. K. W. Berendzen, K. Stueber, K. Harter and D. Wanke, Bmc Bioinformatics

7(NOV 30 2006). 2. J. L. Riechmann, J. Heard, G. Martin, L. Reuber, C. Jiang, J. Keddie, L. Adam, O. Pineda, O. J. Ratcliffe, R. R. Samaha, R. Creelman, M. Pilgrim, P. Broun, J. Z. Zhang, D. Ghandehari, B. K. Sherman and G. Yu, Science 290, 2105(Dec 2000). 3. D. Walther, R. Brunnemann and J. Selbig, PLoS Genet 3, p. ell(Feb 2007). 4. Arabidopsis Genome Initiative, Nature 408, 796(Dec 2000). 5. Y. Yamazaki and P. Jaiswal, Plant Cell Physiol46, 63(Jan 2005). 6. International Rice Genome SequencingProject, Nature 436, 793(Aug 2005). 7. X. Wang, G. Haberer and K. F. X. Mayer, BMC Genomics 10, p. 284 (2009). 8. G. A. Tuskan, S. Difazio, S. Jansson, J. Bohlmann, I. Grigoriev, U. Hellsten, N. Putnam, S. Ralph, S. Rombauts, A. Salamov, J. Schein, L. Sterck, A. Aerts, R. R. Bhalerao, R. P. Bhalerao, D. Blaudez, W. Boerjan, A. Brun,

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A. Brunner, V. Busov, M. Campbell, J. Carlson, M. Chalot, J. Chapman, G.-L. Chen, D. Cooper, P. M. Coutinho, J. Couturier, S. Covert, Q. Cronk, R. Cunningham, J. Davis, S. Degroeve, A. Dejardin, C. Depamphilis, J. Detter, B. Dirks, I. Dubchak, S. Duplessis, J. Ehlting, B. Ellis, K. Gendler, D. Goodstein, M. Gribskov, J. Grimwood, A. Groover, L. Gunter, B. Hamberger, B. Heinze, Y. Helariutta, B. Henrissat, D. Holligan, R. Holt, W. Huang, N. Islam-Faridi, S. Jones, M. Jones-Rhoades, R. Jorgensen, C. Joshi, J. Kangasjarvi, J. Karlsson, C. Kelleher, R. Kirkpatrick, M. Kirst, A. Kohler, U. Kalluri, F. Larimer, J. Leebens-Mack, J.-C. Leple, P. Locascio, Y. Lou, S. Lucas, F. Martin, B. Montanini, C. Napoli, D. R. Nelson, C. Nelson, K. Nieminen, O. Nilsson, V. Pereda, G. Peter, R. Philippe, G. Pilate, A. Poliakov, J. Razumovskaya, P. Richardson, C. Rinaldi, K. Ritland, P. Rouze, D. Ryaboy, J. Schmutz, J. Schrader, B. Segerman, H. Shin, A. Siddiqui, F. Sterky, A. Terry, C.-J. Tsai, E. Uberbacher, P. Unneberg, J. Vahala, K. Wall, S. Wessler, G. Yang, T. Yin, C. Douglas, M. Marra, G. Sandberg, Y. Van de Peer and D. Rokhsar, Science 313, 1596(Sep 2006). 9. Y.-Y. Ho, L. Cope, M. Dettling and G. Parmigiani, Methods Mol Biol408, 171 (2007). 10. S.-M. Chaw, C.-C. Chang, H.-L. Chen and W.-H. Li, J Mol Evol58, 424(Apr 2004). 11. M. Spannagl, O. Noubibou, D. Haase, L. Yang, H. Gundlach, T. Hindemitt, K. Klee, G. Haberer, H. Schoof and K. F. X. Mayer, Nucleic Acids Res 35, DS34(Jan 2007).

Quantum Bio-Informatics IV eds. L. Accardi, W . Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 387-401)

ENTROPY TYPE COMPLEXITIES IN QUANTUM DYN AMICAL PROCESSES

NOBORU WATANABE Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan E-mail: [email protected]

In a study of several systems, we are interested to examine (1) the dynamics of state change and (2) the complexity of states . Ohya introduced in 1991 (ref.[21]) a general idea so-called a Information Dynamics (ID), which constitutes a theory under a frame of the ID by synthesizing the formalities of the investigations of (1) and (2). There are two kind of complexities in ID. One is a complexity of state describing system itself and another is a transmitted complexity between two systems. Entropies of classical and quantum systems are the example of these complexities. In order to treat a flow of dynamics process, dynamical entropies were introduced in not only classical but also quantum systems . The main purpose of this paper is to compare with these mean entropies and the complexities in rD, we calculate these mean entropies for some simple models to discuss the complexity of information transmission for OOK and PSK modulations.

1. Introduction

In several complicated systems, it is important to study (1) the dynamics of state change and (2) the complexity of states of systems. Information Dynamics (ID) introduced by Ohya is a general idea constructing a theory by synthesizing the research schemes of (1) and (2) under a frame of the ID. In ID, there are two type of complexities, that is, (a) a complexity of state describing system itself and (b) a transmitted complexity between two systems. Entropies of classical and quantum information theory are the example of the complexities of (a) and (b). In quantum information theory, Ohya introduced a compound state and defined Ohya mutual entropy 18 based on quantum relative entropy of Umegaki 31 in 1983, he extended it 19 to general quantum systems by using the relative entropy of Araki 5 and Uhlmann 32. Based on the quantum mutual entropy, he quantum capacity is discussed in 24,25,28. One can discuss the coding theorems by means of 387

388

the mean entropy and the mean mutual entropy defined by the dynamical entropy. The KS entropy 14 was introduced in classical systems. Several quantum dynamical entropy were studied by Emch 10 , Connes-Stormer 9, Connes, Narnhoffer and Thirring 8, Park 29, .Alicki and Fannes 4, Hudetz 12. Ohya 21, Voiculescu 33, Accardi, Ohya and Watanabe 2 ,3, Kossakowski, Ohya and Watanabe 15,34, Ohya and Petz 23 and Choda 7, Bennati 6, Muraki and Ohya 16 and so on. In this paper, we briefly explain the mean entropy and the mean mutual entropy defined by Ohya 21. We calculate these mean entropies for some simple models to discuss the complexity of information transmission for subsets of the initial state space.

2. Quantum Channels The concept of channel has been played an important role in the progress of the quantum communication theory. Here we briefly review the notion of the quantum channels. Let Hk (k = 1,2) be complex separable Hilbert spaces. We denote the set of all bounded linear operators on Hk by B(Hd (k = 1,2) and we express the set of all density operators on Hk by 6(Hk) (k = 1,2). Let (B(H k ),6(H k )) (k = 1,2) be input (k = 1) and output (k = 2) quantum systems, respectively. (1) A mapping from 6(Hd to 6(H2) is called a quantum channel A*. (2) A* is called a linear channel if A* satisfies the affine property such as A*(2:k AkPk) = 2:k Ak A* (Pk) for any Pk E 6(Hl) and any nonnegative number Ak E [0,1] with 2:k Ak = 1. For the quantum channel A *, the dual map A of A * is defined by

trA*(p)B = trpA(B),

Vp

E

6(H 1 ), VB

E

B(H2)'

(3) A * is called a completely positive (CP) channel if A * is linear channel and its dual map A : B(H 2) --+ B(H 1 ) of A* holds

( x , t AiA(A;Aj)AjX);::::: 0

(Vx

E

HI)

c B(H 2) and any {Ai}

C

B(Hd·

t,J=1

for any n E N, any {Ad

Almost all physical transform of states can be denoted by the CP channels 18,19,13,23,27.

389

2.1. Noisy optical channel

Now we explain the noisy optical channel such as an example of the quantum communication channels 19,26. Let Kl and K2 be complex separable Hilbert spaces of noise and loss systems, respectively. For an input state p in 6(Hd and a noise state ｾ＠ E 6 (Kd, we defined in 26 a mapping IT* by

== V (p @ ｾＩ＠ V*,

IT* (p ｀ｾＩ＠

which is called a generalized beam splitting, where V is a linear mapping from HI @ Kl to H 2 @K 2 given by nl+ml

V(lnl; @ Iml;) =

L

Cj",m"lj; @ Inl + ml - j;

j=O

for the nl,ml,j,(nl+ml-j) photon number state vectors Inl; HI, Iml; E Kl , Ij; E H2, Inl + ml - j; E K 2, and Cn1,m1 J

K j - r = ''"'(-It"+

.I

, "(

_ ')'

ynl·ml·]· nl +ml ]. r!(nl - j)!(j - r)!(ml - j + r)!

ｾ＠

r=L xaml-j+2r

,

E

(-73) nl +j-2r .

(1)

a and {3 are complex numbers satisfying lal 2 + 1{31 2 = l. K and L are constants given by K = min{nl,j}, L = max{ml - j, O}. For the coherent input state p = Ie; (el @ Ih;; (h;I E 6 (Hl@Kd, the output state of IT* is obtained by IT* (Ie; (el @ Ih;; (h;I)

= lae + @

IT* with the vacuum noise state given by

{3h;;

(ae +

{3h;1

1-73e + (ih;) \ -73e + (ih;I·

= 10; (01 is called the beam splitter ITo ｾｯ＠

= lae; (ael @ 1-73e) \-73 e l for the coherent input state p @ ｾｯ＠ = Ie; (el @ 10; (01 E 6 (Hl@Kd· ITo was ITo (Ie; (el ｀ｾｏＩ＠

described by means of the lifting Eo from 6 (H) to 6 (H@K) in the sense of Accardi and Ohya 1 as follows

Eo (Ie; (el) =

lae; (ael @ l{3e) ({3el·

Based on the liftings, the beam splitting was studied by Accardi - Ohya and Fichtner - Freudenberg - Libsher 11. Noisy quantum channel A* introduced in 26 with a fixed noise state ｾ＠ E 6 (Kd was defined by A*(p)

==

trJC 2IT*(p ｀ｾＩ＠

= trJC2 V (p ｀ｾＩ＠

V*.

(2)

390

Moreover, the noisy quantum channel with the vacuum noise state 10) (01 is called the attenuation channel given by Ohya 19 as

Ao

ｾｯ＠

= (3)

which are important to discuss the quantum communication processes.

3. Complexities In ID 22, two kind of complexities CS (p), TS (p; A*) are used for studying the complex systems. CS (p) is a complexity of a state p measured from a subset Sand T S (p; A*) is a transmitted complexity associated with the state change from p to A* p. These complexities should satisfy the following conditions: Let S ,S, St be subsets of 6 (Hd , 6 (H 2) , 6 (H1 ® H2), respectively. (1) For any PES, CS (p) and TS (p; A*) are nonnegative. (2) For a bijection j from ex6(H 1) to ex6(H 1), CS(p) is equal to CS(j (p)), where ex6 (Hd is the set of extremal point of 6 (Hd. (3) For p ® a E 6 (H1 ® H 2), p E 6 (Hd, a E 6 (H 2), the complexity CSt (p ® a) of the state p ® a of totally independent systems is equal to the sum C S (p) + C S (a) of the complexities of the states p and a. (4) The transmitted complexity TS (p; A*) is greater than 0 and it is less than the complexity C S (p) of the state p. (5) If the channel A* is the identity map id, then T S (p ;id) is equal to

CS(p) . One of the example of the above complexities are the Shannon entropy

S (p) for CS (p) and classical mutual entropy I (p; A*) for TS (p; A *). Let us consider these complexities for quantum systems.

3.1. Example of Complexity C S (p) 3.1.1. (1) von Neumann entropy One of the example of the complexity C S (p) of ID in quantum system is the von Neumann entropy 17 S (p) described by

C S (p) {:} S(p) = -trplogp for any density operators p E 6 (Hd, which satisfies the above conditions (1), (2), (3).

391

3.1.2. (2) S-mixing entropy Let (A, 6(A), a(G)) be a C*-dynamical system and S be a weak* compact and convex subset of 6(A). For example, S is given by 6(A) (the set of all states on A), I(a) (the set of all invariant states for a ), K(a) (the set of all KMS states), and so on. Every state rp E S has a maximal measure fL pseudosupported on exS such that rp

=

is

(4)

wdfL,

where exS is the set of all extreme points of S. The measure fL giving the above decomposition is not unique unless S is a Choquet simplex. We denote the set of all such measures by M",(S), and define

D",(S) = {M",(S); s.t.

c

jR+

= 1,

fL

3fLk

ｾｦｌｫ＠

and {rpd

=

c

ｾｦｌｫＸＨｲｰＩｽＧ＠

exS

(5)

where 8(rp) is the Dirac measure concentrated on an initial state rp. For a measure fL E D",(S), we put H(fL)

=-

LfLklogfLk·

(6)

k

The C*-entropy of a state rp E S with respect to S (S-mixing entropy) is defined in 21 by SS(rp)

={

inf {H (fL);

+00

fL E

D",(S)}

if D",(S) = 0.

(7)

It describes the amount of information of the state rp measured from the subsystem S. We denote S6(AJ(rp) by S(rp) if S = 6(A). It is an extension of von Neumann's entropy. This entropy (mixing S-entropy) of a general state rp satisfies the following properties 21. Theorem 3.1. When A = B(H) and at = Ad(Ut ) (i.e., at(A) = Ut AUt for any A E A) with a unitary operator Ut, for any state rp given by rp(.) = trp· with a density operator p, the following facts hold:

(1) S(rp)

= -trplogp.

(2) If rp is an a-invariant faithful state and every eigenvalue of p is non-degenerate, then SI(a.J(rp) = S(rp), where I (a) is the set of all a-invariant faithful states.

392

(3) If ep E K(n:), then sK(al(ep) states.

= 0,

where K (n:)is the set of all KMS

Theorem 3.2. For any ep E K(n:) , one can obtain

(1) sK(al(ep) :::; sI(al(ep). (2) sK(al(ep) :::; Seep).

rs (Pi A *)

3.2. Example of Transmitted Complexity

3.2.1. (1) Ohya mutual entropy for density operator An example of the transmitted complexity T S (p; A*) of ID in quantum system is the Ohya mutual entropy with respect to the initial state p and the quantum channel A* defined in 19 by T S (p; A*)

ｾ＠

J (p; A*)

== sup

ｻｾｓＨｏＧｅｐ｀ａＪＩ＠

=

ｾａｮｅｽＧ＠

(8)

where 0' E is the compound state given by 0' E = 2::n AnEn@A *En associated with the Schatten-von Neumann (one dimensional spectral) decomposition 30 p = 2::n AnEn of the input state p, and S (', .) is the Umegaki's relative entropy denoted by

S(

) = {trp(lOgp-IogO') (whenranpcranO') p,O' (h .) 00 ot erWlse

(9)

which was extended to more general quantum systems by Araki and Uhlmann 5,20,23,32. The Ohya mutual entropy holds the above conditions (4) such as

0:::; J(p,A*):::; S(p). 3.2.2. (2) Ohya mutual entropy for general C*-system Let (A,6(A),n:(G)) be a unital C*-system and S be a weak* compact convex subset of 6(A). For an initial state ep E S and a channel A* : 6 (A) -+ 6 (B), two compound states are

＼ｐｾ＠ o)

and the mutual entropy with respect to S is defined by Ohya

IS (c.p ; A*) = sup ｻｉｾ＠

21

as

(c.p ; A*) ; f.L E Mtp (S)} .

(13)

4. Quantum Mean Mutual Entropy of K-S type In this section, we briefly review quantum mean entropy and quantum mean mutual entropy introduced by Ohya 21. A stationary information source in quantum information theory is described by a C*triple (A, 6(A), () A) with a stationary state c.p with respect to () A; that is, A is a unital C* -algebra, 6(A) is the set of all states on A, ()A is an automorphism of A, and c.p E 6(A) is a state over A with

c.p 0

()A

= c.p.

Let an output C*-dynamical system be the triple (8,6(8), ()s), and A* : 6(A) ---> 6(8) be a covariant c.p. channel: A : 8 ---> A such that A 0 ()s = () A 0 A. In this section we explain new functionals ｓｾＨ｣Ｎｰ［｡ｍＩＬ＠ SS(c.p;a M ), ｉｾ＠ (c.p; aM, f3N) and IS (c.p; aM, f3N) introduced in 21,16 for a pair of finite sequences of aM = (aI, a2, . .. , aM), f3N = (13 1,13 2, . .. ,13 N) of completely positive unital maps am : Am ---> A , f3 n : 8 n ---> 8 where Am and 8 n (m = 1, · · · ,M, n = 1, ··· ,N) are finite dimensional unital C*-algebras. For a given finite sequences of completely positive unital maps am : Am ---> A from finite dimensional unital C* -algebras Am (m = 1,·· . ,M) and a given measure f.L of c.p E Mtp(S), the compound state of M

aic.p, a2c.p, ... ,aMc.p on the tensor product algebra

®

Am is given by

21 ,16

m=l

ｩｦ＾ｾＨ｡ｍＩ＠

=

M

r

Q9 a:nw df.L(w).

(14)

JS(A) m=l

Furthermore ｩｦ＾ｾＨ｡ｍ＠ with aM U f3N

ｩｦ＾ｾ＠

U f3N) is a compound state of ｩｦ＾ｾＨ｡ｍＩ＠

and ｩｦ＾ｾＨＳｎＩ＠

== (aI, a2,··· ,aM, 131, 132,··· ,f3N ) constructed as (aM

U

f3N) =

1

SeA )

Ｈｾ｡ＺｮｷＩ＠

m=l

(db ｦＳｾｗＩ＠ n=l

df.L.

(15)

394

For any pair (aM, (3N) of finite sequences aM = (aI, ... , aM) and (3N = ((31' ... , (3N) of completely positive unital maps (c.p.u. maps for short) am : Am ----; A, (3n : 8 n ----; A from finite dimensional unital C* -algebras and any extremal decomposition measure J.L of rp, the entropy functional SIL and the mutual entropy functional III are defined by 21,16

ｓｾＨｲｰ［｡ｍＩ］＠

r

JSeA)

ｉｾ＠ (rp; aM, (3N) = S ( ＼ｐｾ＠

ｓＨＶ＾Ｚ［］ｬ｡ｮｷＬ＼ｐｾｍＩ､ｊＮｌ＠

(16)

(aM U (3N), ＼ｐｾ＠

(aM)

(6)

＼ｉ＾ｾ＠

((3N)) ,

(17)

where S(·,·) is the relative entropy for a finite algebra. The relative entropy of two states was introduced in 31 for O'-finite and semi finite von Neumann algebras. Araki 5 and Uhlmann 32 extended this relative entropy for more general quantum systems 23. For a given pair of finite sequences of completely positive unital maps aM = (a1,'" ,aM), (3N = ((31'''' , (3N), the functional SS(rp;a M ) (resp. IS(rp;a M,(3N)) is given by taking the supremum of ｓｾＨｲｰ［｡ｍＩ＠ (resp. ｉｾＨｲｰ［＠ aM, (3N)) for all possible extremal decompositions J.L'S of rp:

SS(rp;a M) = ｳｵｰｻｓｾＨｲ［｡ｍＩ＠ IS(rp;a M ,(3N) = ｵｰｻｉｾＨｲ［｡ｍＬＳｎＩ＠ｳ

J.L

E

M

i= - CXJ

00

00

00

j =-oo

j' = -oo

j = -oo

Let a (resp. (3) be the embedding map from Ao to A , (resp. Bo to B) given by a( A) = ... I ® I ® A ® I ® . .. E A,

for any A E Ao,

(3(B) = ... I ® I ® B ® I ® ...

for any B E Bo.

E

B,

We denote the set of all density operators on rto (resp. rto) by 60 (resp. ( 0 ), and let 6 (resp. 6) be the set of all states p on A (resp. p on B) . The maps afM)' (3r(M) are given by N aiM)

(3r(M)

== ==

(

BAoao'(M),"', BA aO'(M)'

(1(M)

N

0

1

) oao'(M)'

J... 0 (3, 1(M) 0 J... 0 Bs 0 (3,'" , 1(M) 0 J... 0 B;-l 0 (3),

396 00

where we took a special channel and modulators M such that

A == ®

A

i = - oo 00

and

i(M)

== ®

I(M)'

i=-(X)

4.1.1. Mean mutual entropy for modulated state of OOK

and PSK 00

For an initial state

00

to\

p(M)

(M)

Pi

I.(Y

E

i=-oo

ｰｾｏｋＩ＠

and ｰｾｐｓｋＩ＠

®

6

i

(M = OaK, PSK),

i=-CXJ

are given by

ｰｾｏｋＩ＠

+ (1 - v) I,,;) (,,;1, = v 1-,,;) (-,,;1 + (1- v) I,,;) (,,;1

=

ｰｾｐｓｋＩ＠

v 10) (01

The Schatten decomposition of ｰｾｍＩ＠

(0::; v ::; 1).

is obtained as 2

pi M ) = L ａｾＩ＠

ｅｾＩ＠

,

ni=l

where the eigenvalues ａｾＩ＠

ａｾ ｾｏ ｋ Ｉ＠

ｾ＠

=

Ar:,SK) =

ｾ＠

{l+(_l)k-l

=

pi M ) are

V1 - 4v(1- v) (1- exp (-1,,;1 2)) },

{l+(_l)k-l V1- 4v(1- v) (1- exp (-2 1,,;1 2 ) ) }

Two projections ｅｾ

(ni

of

Ｉ＠ (ni

1,2) are given by

1, 2) and the eigenvectors ｬ･ｾ

Ｉ＠

(k = 1, 2) .

of

ａｾＩ＠

397

where

For the above initial state ｅｾａＨＩＬ＠ one can obtain the output state for the attenuation channel as follows:

Ao

ａＪｅｾＨＩ＠

2

L

=

Ｚ｜ｾ［ｅＷ＠

(ni

= 1, 2),

ｮｾ］ｬ＠

.

ｾＨｍＩ＠

(M)

where the eIgenvalues Ani, n; of A* Eni

.

are gIven by (ni = 1,2), (k = 1,2)

398

When

AD

is given by the attenuation channel, we get

The compound states through the attenuation channel

Ao becomes

iI> E (a[11)) ® iI> E ({3;:;'o (M))

=

i;l··· nNf:=l (IT ａｾＩ＠ xJo

mtｾｯ＠

(IT ｾ［Ｚ＠

mt=l Crr: ａｾＺＩ＠ ｭｾｬ＠ ｭｾＩ＠

ＨｾｅＺＧｬＩ＠

ＨｾｅＺ＠

m:, )

399 00

Lemma 4.1. For an initial state

00

(8)

p(M) =

(8)

pi M ) E

i=-oc>

6

i,

we have

i=-OCJ

IE(p(M); arM)' fJraOK))

ｾ＠

nNt m: ａｾＧｬｘ［Ｉｮ＠

nt,·· ﾷｮＧｎｴＮ､ｾ＠

By using the above lemma, we have the following theorems. 00

Theorem 4.2. For an initial state

p(M)

=

(8)

00

pi M )

E

i=-oo

(8)

6i

i=-oo

(M = OOK, PSK), we have S(p(M); ()A, arM)) =

ｊｾｯｓＨｰｍＩ［＠

arM)) 2

=-

L ａｾｍＩ＠

log ａｾｍＩ＠

n=l

and I-(p( M)., A* , 2

2

_ ""

- 0

0

n'=l n=l

ＨｍＩｾ＠

An

()A ()B n,N a. N ) ''-'(M)' fJ(M) , ｾＨｍＩ＠

An, n' An, n' log -----=-2----'--" A(M)>..(M) ｾ＠

m

m,n'

m=l 00

00

Theorem 4.3. For an initial state

p(M)

=

(8)

pi M )

E

i=-OCJ

(M

=

OOK, PSK), one has the following inequalities:

(1) () N ) S-( P(PSK).,A, a(PSK)

2':

S-( (OOK). () N ) p ,A, a(OOK) ,

(8) i=-OCJ

6

i

400

(2) I-( p(PSK) ;A* , _( (OOK) A* ?: 1 p ;,

eA, eB, a(PSK)' N (3N ) (PSK) eA, eB, a(OOK)' N (3N ) (OOK)'

References 1. Accardi, L., and Ohya, M., Compound channels, transition expectation and liftings , Appl. Math, Optim., 39 , 33-59 (1999). 2. Accardi, L. , Ohya, M. and Watanabe, N., Dynamical entropy through quantum Markov chain, Open System and Information Dynamics, 4, 71-87, (1997) . 3. Accardi, L., Ohya, M. and Watanabe, W., Note on quantum dynamical entropies , Rep. Math. Phys. , 38, 457-469, (1996). 4. Alicki, R . and Fannes, M. , Defining quantum dynamical entropy, Lett . Math. Physics, 32 , 75-82 , (1994). 5. Araki, H., Relative entropy for states of von Neumann algebras, Publ. RIMS Kyoto Univ. 11, 809-833, 1976. 6. Benatti, F., Deterministic Chaos in Infinite Quantum Systems, Springer, Berlin, (1993). 7. Choda, M. , Entropy for extensions of Bernoulli shifts , Ergodic Theory Dynam. Systems, 16, No.6 , 1197-1206 (1996). 8. Connes, A., Narnhoffer, H. and Thirring, W., Dynamical entropy of C*algebras and von Neumann algebras, Commun. Math. Phys., 112, 691719, (1987) . 9. Connes, A. and Stormer , E. , Entropy for automorphisms of von Neumann algebras, Acta Math. , 134, 289-306, (1975). 10. Emch, G.G., Positivity of the K- entropy on non-abelian K-fiows, Z. Wahrscheinlichkeitstheory verw. Gebiete, 29, 241 (1974). 11. Fichtner, K.H., Freudenberg, W., and Liebscher, V., Beam splittings and time evolutions of Boson systems, Fakultat fur Mathematik und Informatik, Math/ Inf/96/ 39, J ena, 105 (1996). 12. Hudetz, T., Topological entropy for appropriately approximated C*-algebras, J. Math. Phys. 35, No .8, 4303-4333 (1994). 13. Ingarden , R.S., Kossakowski, A., and Ohya, M., Information Dynamics and Open Systems, Kluwer, (1997). 14. Kolmogorov, A.N., Theory of transmission of information, Amer. Math. Soc. Transla tion, Ser. 2, 33, 291 (1963). 15 . Kossakowski, A., Ohya, M. and Watanabe, N., Quantum dynamical entropy for completely positive map, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2, No.2, 267-282, (1999) 16. Muraki, N. and Ohya, M., Entropy functionals of Kolmogorov Sinai type and their limit theorems , Letter in Mathematical Physics., 36, 327-335, (1996) .

401

17. von Neumann, J., Die Mathematischen Grundlagen der Quantenmechanik, Springer-Berlin, (1932). 18. Ohya, M., Quantum ergodic channels in operator algebras, J. Math. Anal. Appl., 84, 318-328, (1981). 19. Ohya, M., On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, 770-774 (1983). 20. Ohya, M., Note on quantum probability, L. Nuovo Cimento, 38, 402-404, (1983) . 21. Ohya, M., Some aspects of quantum information theory and their applications to irreversible processes, Rep. Math. Phys. 27, 19-47, (1989). 22. Ohya, M., Information dynamics and its applications to optical communication processes, Springer Lecture Note in Physics, 378, 81-92, (1991). 23. Ohya, M., and Petz, D., Quantum Entropy and its Use, Springer, Berlin, (1993) . 24. Ohya, M., Petz, D., and Watanabe, N., On capacity of quantum channels, Probability and Mathematical Statistics, 17, 179-196 (1997). 25. Ohya, M., Petz, D., and Watanabe, N., Numerical computation of quantum capacity, International Journal of Theoretical Physics, 37, No.1, 507-510 (1998). 26. Ohya, M., and Watanabe, N., Construction and analysis of a mathematical model in quantum communication processes, Electronics and Communications in Japan, Part 1, 68, No.2, 29-34 (1985). 27. Ohya, M., and Watanabe, N., Foundations of Quantum Communication Theory (in Japanese), Makino Pub. Co., (1998). 28. Ohya, M., and Watanabe, N., Quantum capacity of noisy quantum channel, Quantum Communication and Measurement, 3, 213-220 (1997). 29. Park, Y.M., Dynamical entropy of generalized quantum Markov chains, Lett. Math. Phys. 32, 63-74, (1994) 30. Schatten, R., Norm Ideals of Completely Continuous Operators, SpringerVerlag, (1970). 31. Umegaki, H., Conditional expectations in an operator algebra IV (entropy and information), Kodai Math. Sem. Rep., 14, 59-85 (1962). 32. Uhlmann, A., Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory, Commun. Math. Phys., 54, 21-32, 1977. 33. Voiculescu, D., Dynamical approximation entropies and topological entropy in operator algebras, Comm. Math. Phys., 170, 249, (1995) 34. Watanabe, N., Some Aspects of Complexities for Quantum Processes, Open Systems and Information Dynamics, 16, No.2&3, 293-304, (2009).

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Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 403-412)

A FAIR SAMPLING TEST FOR EKERT PROTOCOL GUILLAUME ADENIER* and NOBORU WATANABE

Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan * E-mail: [email protected] Andrei Yu. Khrennikov

Linnaeus University, Vejdes plats 7, SE-351 95 Viixjo, Sweden

We propose a local scheme to enhance the security of quantum key distribution in Ekert protocol (E91). Our proposal is a fair sampling test meant to detect an eavesdropping attempt that would use a biased sample to mimic an apparent violation of Bell inequalities. The test is local and non disruptive: it can be unilaterally performed at any time by either Alice or Bob during the production of the key, and together with the Bell inequality test.

Keywords: Ekert protocol, Entangled states, Fair sampling.

1. Introduction

Ekert protocoP-3 uses entangled states to guarantee the secrecy of a key distributed to two parties (Alice and Bob) that wished to communicate secretly through a public channel. Identical measurements performed on a maximally entangled state yield perfect correlation, which can be used to produce a shared key. The secrecy of the key can be guaranteed by the violation of Bell inequalities measured for non-identical measurements. An unconditional violation of Bell inequalities would guarantee that no local (hidden) variables exist that an eavesdropper (Eve) could exploit. It would mean unconditional privacy: Eve could have full control of the detectors and the source, more advanced theory and technology, it would still be secure. 2 However, actual implementations of Ekert protocol presently require the use of photons, because a key distribution protocol is useful only if Alice 403

404

and Bob can be separated by macroscopic distances 4, which makes photons the only practical solution. The downside is that the type of photons is limited in practice by the pair creation process (parametric down conversion) and by the optical components that are used (fiber optics, polarizing beamsplitters) to wavelength at which standard photon counters have a poor detection efficiency 5 ,6. It means that optical implementations of Ekert protocols cannot avoid a rather heavy postselection: Alice and Bob must discard all measurements for which either of them failed to register a click. The trouble is that the validation of a violation of Bell inequalities observed in such a post selection protocol requires an extra assumption: the sample of detected pairs must represent fairly the population of emitted pairs (fair sampling assumption) . It has long been known that a breakdown of this assumption allows local hidden-variable models to reproduce exactly the predictions of Quantum Mechanics on the subset of detected pairs 7, unless the detection efficiency is higher than 83% 8 ,9,10 ,11. In the context of experiments on the foundations of Quantum Mechanics, the fair sampling assumption is usually considered reasonable, with the idea that Nature is not conspiratory. In Quantum Key Distribution however, Eve is expected to conspire 12, so that Alice and Bob must assume that Eve is actually attempting to bias their sample. a Naturally, this issue becomes critical if Eve manufactured the detectors, which means that Alice and Bob should thoroughly check that their detectors are functioning according to specifications 2. We will argue here that photomultipliers or avalanche photo diodes can in principle be subjected to such a bias sampling attack, by exploiting the thresholds of these detectors, in particular if they exhibit nonlinearities.

2. A biased-sampling attack on Ekert protocol The motivation for the possibility of a biased-sampling attack is that avalanche photo diodes and photomultipliers are fundamentally threshold detectors. They are sometimes referred to as such because they cannot distinguish between the absorption of one or of several photons, but it should also be pointed out that the principle of detection itself relies on thresholds, and that these thresholds are essential to the proper functioning of these aThis possibility should not be underestimated. A successful quantum hacking has already been successfully implemented experimentally with a time-shifting attack 13. The attack essentially introduced a hidden-variable in the protocol (the time-shift) to influence the probability for Alice and Bob to get either a 0 or a 1.

405

ipB

Figure 1. Standard Ekert protocol. Alice and Bob randomly switch their measurement settings. Pairs associated with identical measurement settings (if A = if B) are used to produce a correlated key, while those associated to non-identical measurement settings (if A i= if B) are used to check the violation of Bell inequality (and the security of the key).

detectors. At the input, the energy must be higher than the band gap to trigger an avalanche or a photoelectron; while at the output, the current must be higher than a discriminator value to be counted as a click 14. This combined threshold could be exploited by Eve to obtain an apparent violation of Bell inequalities on the detected sample using only local states entangled state 15 . The idea of the attack would be to replace the genuine source of entangled photons with a source of near threshold pulses correlated in polarization. These pulses would lead to a probability of generating a click in either output channel that would depend on the characteristics of the pulse and the measurement settings. Consider a near-threshold pulse linearly polarized along >- impinging on a polarizer set at a measurement angle cp. It would have a significantly greater chance to produce a click when the angle difference 0: between these two variables is 0: = 1>- - cpl ｾ＠ k1r /2 (parallelor orthogonal) than when 0: ｾ＠ k1r/2 + 7f/4 (diagonal). Indeed, in the first case most of the energy of the pulse would go in one specific channel, retaining enough energy in this channel to remain above the threshold, while in the other case the energy of the pulse would be split between both channels, in principle below the threshold. In the context of low efficiency of detection, it would bias the sampling of the pulses with the effect of artificially increasing the visibility of the coincidence counts, thus leading to an apparent violation of Bell inequalities 15.

406

In principle, Eve could even obtain an apparent violation of Bell inequalities reproducing exactly the predictions of Quantum Mechanics on the detected sample, by reproducing the asymmetrical detection pattern of a Larsson-Gisin model 10,11. For this purpose, Eve would have to send pairs of correlated pulses with energy Eo = 2 and polarization A, with A being a random variable uniformly distributed on the interval [-Jr /2, Jr /2]. On one side, Eve would have to send pulses to which the detectors react with a very steep rising detection probability at the threshold (ideal threshold), while on the other side, Eve would have to send pulses to which the detectors react with a detection probability rising linearly at the threshold (linear threshold). With the condition Eo = 2, what happens is that Malus' Law is cut by the bottom precisely at the intersection of the two channels (at I'P - AI = Jr /4). Consequently, Alice would always records a click in exactly one channel: channell if I'PA - AI < Jr/4, channel 0 otherwise; whereas on Bob's side the probability to get a click in the channel 1 would vary with cos2('PB - A) when I'PB - AI < Jr/4, and with sin2('PB - A) when I'P B - AI > Jr / 4 in channel O. The crucial feature of the resulting detection pattern is that the probability to obtain a click in either channel on Bob's side depends explicitly on A: it is maximum for I'PB - AI = 0, and decreases down to zero for I'PB - AI = Jr /4. The sampling is thus unfair, or biased, and leads to an apparent violation of Bell inequalities on the detected sample 10,11,15

Note that if instead of scanning the full correlation, Alice and Bob are only checking a few points of the correlation predicted by Quantum Mechanics, as is the case in Ekert protocol, Eve would not need to aim at reproducing the full correlation and could therefore implement her attack with a symmetrical design, inducing the same detection pattern for Alice and Bob. However, if each pulse contains at most one photon the biased sampling described here would be ineffective, because the energy seen at a detector would always be the same regardless of the measurement settings (when this photon does reach a detector). Eve would therefore have to produce nearthreshold pulses consisting of several photons of lower frequencies. In order for her attack to work however, she needs that the probability that a pulse produces a click in either channel be close to zero for angle differences close to diagonals, that is for angle differences close to a = I'P - AI = Jr / 4 + kJr /2, while it should be non zero, and significantly greater than the probability of a dark count, for other angles differences, in particular for a = I'P - AI = k7r /2.

407

3. Countermeasures In order to prevent Eve from using this bias sampling attack, Alice and Bob can in principle use several countermeasures. The first countermeasure consists in increasing the efficiency to reach 83%. However, this proves difficult with threshold detectors. Decreasing the band gap threshold does increase the efficiency of the detectors, but only at the cost of higher dark count rates. Unless special detectors operating near absolute zero temperature are used, such as Transition-Edge Sensors (which are too cumbersome and slow to be practical solution to QKD), this can be considered a general rule that applies to any detectors, and fundamentally limits their efficiencies, so that this desirable solution can be considered unrealistic in a quantum key distribution framework. The second countermeasure would be to guarantee that Alice's and Bob's detectors are not susceptible to nonlinearities at any frequency. The probability that a photon produces a click in a detector should remain the same regardless of the circumstances, in particular it should not depend on the number of other photons reaching the detector simultaneously. If this can be guaranteed, then Eve cannot exploit the threshold because each photon sees the threshold independently of the presence of other photons, and is therefore either above the threshold or below, regardless of the instrument settings. In practice, it might be difficult to guarantee that detectors do not exhibit nonlinearities at any frequency, and as much as this question has been studied experimentally, the result is that detectors do exhibit nonlinearities 16. A third countermeasure would consist in using filters to prevent Eve from using lower frequency photons. This would however come at the expanse of a lower overall quantum efficiency. The narrower the bandwidth of the filter, the lower the quantum efficiency. With photon detectors that have significant amount of dark count, this would mean an increased of the quantum bit error rate. The imperfections of the filters could also become the target of Eve's attack. If for instance the filter does not provide a complete extinction at a frequency usable for an attack, Eve would only have to send brighter pulses to allow enough of these photons to go through and perform the bias sampling attack. In principle, any of the three above countermeasure would be enough to prevent Eve's attack, but they can be very difficult or impractical to implement. This leads us to suggest a fourth possibility, which is to test the fairness of the sampling.

408

Figure 2. Fair Sampling test on Alice's side. The detector Al is replaced by a polarimeter with two detectors At and A 1 , whereas the detector Ao is replaced by a polarimeter with two detectors At and Ail. Ekert protocol is unaltered by our test if all detectors have the same efficiency "I): the light green area is equivalent to detector Al with efficiency "I), whereas the light red area is equivalent to detector Ao with efficiency "I).

For this purpose, we propose to analyzing the output channels of the polarizing beamsplitters, instead of simply feeding detectors with them. We keep the standard design of Ekert protocol, with two polarizing beamsplitters on each side (Alice and Bob) projecting the incoming pulses on random bases rp A and rp B, as depicted on Fig. 1, but we replace each detectors by a polarimeter: a polarizing beamsplitter followed by a detector at each output. Consider Alice's side (see Fig. 2). We label the additional PBS in channell by its orientation eA, , and the one in channel 0 by its orientation eAo. A click in any of the two detectors following eA, is counted as aI, whereas a click in any of the two detectors following eAo is counted as a o. Bob proceeds similarly with two polarimeters labeled eB, and eBo. From the point of view of Ekert protocol and of a genuine source of entangled photons, nothing is changed. Consider Alice's channell. The setup constituted by the PBS eA, and its corresponding detectors can be seen as one big single detector in channell, in which the orientation eA, has no influence on the result, that is, if we assume a balanced quantum efficiencies

409

TJ of the detectors at the two outputs. A photon exiting the PBS oriented

along r.p A through channell will be detected in either output channel after () Al with a probability TJ. Similarly, the setup constituted by the () Ao and its corresponding detectors can be seen as one single detector in channell, where the orientation () Ao plays no role whatsoever, and the same goes for Bob's setup. As long as all the detectors have the same efficiency TJ, each polarimeter can then be considered as one detector with quantum efficiency TJ· The polarimeter () Al can be seen as one single detector in channell, in which the orientation () Al has no influence on the result: a photon exiting the PBS r.p A through channel 1 will be detected in either output channel of polarimeter () Al with a probability TJ. Similarly, polarimeter () Ao can be seen as one single detector in channel 0, where the orientation () Ao plays no role whatsoever, and the same goes for Bob's setup. The production of the key and the verification of the violation of Bell inequalities is thus unaltered by our fair sampling test setup in case of a genuine source of entangled photons, because the additional measurement settings () AI' () A o , () BI and () BI controlled by Alice and Bob have no influence on the measured results. However, they have a strong influence on the result in the case of a biased sample attack by Eve. Let us consider the simpler case of an ideal threshold detector. By ideal threshold detector, we mean a detector that produces a click with certainty if an only if the pulse impinging on the detector carries an energy Eo greater than the detector threshold . Eve sends pairs of correlated pulses with energy Eo and polarization .x, where .x is a random variable uniformly distributed on the interval [0,271"[. By Malus law, the energy of the pulse reaching Alice's detector is

At

(1) Starting from a uniform distribution of the polarization .x of pulses on the circle, we write that !PAdAI = IFA +I (E A+ )dEA+ I, so that the probability I I to get an energy between E A +1 and E A +I + dEA +I in the A + channel is given by

(2)

where Emax = Eo cos 2 (r.p A detector (by Malus' law).

-

() AI)

is the maximum energy reaching the

410

The probability to obtain a click in an ideal threshold detector placed at the transmitted output (+) of polarimeter Al is then simply the integral of this density distribution over the energy reaching the detector, from the threshold to Emax:

(3)

(4) Similarly the probability to obtain a click in an ideal threshold detector positioned at the reflected output (-) of polarimeter Al is

2

Eo sin ('P A

-

BAI)

.

(5)

We can also consider a less ideal threshold detector, that would be more likely to resemble the characteristics of a real detector. In order to keep the calculations of the integrals simple enough, we considered the case of threshold detector with linear rise, that is, a threshold detector for which the probability to generate a click increases linearly starting from the threshold , possibly with a saturation value after which increasing the impinging energy no longer increases the probability to generate a click. In these cases, the analytical results are more complicated since the probability to get a click for an energy E + dE is not always equal to 1, but the principle of calculation remains the same: integrate the product of the probability density distribution by the probability of obtaining a click for a given energy. The analytical results are qualitatively similar to that of ideal threshold detectors. The results in the case Eo = 2- which is leading to a violation of Bell inequalities exactly reproducing the predictions of Quantum Mechanics- are displayed in Fig. 3: the probability to get a click in the polarimeter oriented along B Al depends on I'P A - BAli . It is maximum for I'P A - B Al I = 0 + k7r /2, and reaches zero for I'P A - BAli = 7r /4+ k7r /2. Similar results would be obtained for Alice's BAo polarimeter, and for Bob's polarimeters. This fair sampling test can be implemented very simply on Alice's side by fixing B Al = B Ao = O. The random switching in Ekert protocol (Fig. 1 and Fig. 2) ensures that the points at 0 and 7r/4 are both scanned automatically. Any significant difference in the number of single counts recorded

411

OM predictions

0.4

0.3 0.2 0.1

o

7r

7r

37r

4

2

4

Figure 3. Analytical Results in case of biased-sampling attack on threshold detectors in Alice's Al channel, with Eo = 2. The blue plots represent the probability to get a click in detector whereas the purple plots represent the probability to get a click in detector Ai. The probability to get a click in polarimeter () A, thus depends on I'P A - () All· By contrast, in case of a genuine source of entangled photons, Quantum Mechanics predicts independence.

At,

when 'P A = 0 and 'P A = 7f / 4 would betray Eve's attempt to bias the sample through a biased-sampling attack on the threshold detectors. Similarly, Bob would chose B, = Bo = 7f /8, and compare the number of singles when 'PB = -7f/8 and 'PB = 7f/8.

e

e

4. Conclusion This fair sampling test can be implemented during the production of the key and together with the violation of Bell inequality check, so that it seems hard to fool it without reducing the visibility of the violation. For instance, increasing the energy of the pulses with respect to the threshold would tend to reduce the dip in the fair sampling test, but it would reduce the visibility of the correlation (weaker violation of Bell inequalities) at the same time, and it would also give rise to double counts. The combination of a Bell inequality test with a monitoring of the double counts and our local fair sampling test therefore constitutes a solid scheme

412

against eavesdropping a E91 protocol using a biased-sampling attack. In principle, the same idea could be implemented in other QKD protocol, by replacing passive detectors in each channel by a device with the same efficiency, that would analyze further whichever degree of freedom is used to encode the key, instead of simply feeding detectors with it b.

Acknowledgments We are grateful to Hoi-Kwong Lo, Jan-Ake Larsson, Takashi Matsuoka and Masanori Ohya for useful discussions on Quantum Key Distribution.

References 1. A. K. Ekert, Phys. Rev. Lett. 67, 661(Aug 1991). 2. N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Rev. Mod. Phys. 74, 145(Mar 2002). 3. G. Jaeger, Quantum Information (Springer New-York, 2007). 4. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Ltitkenhaus and M. Peev, Rev. Mod. Phys. 81, 1301(Sep 2009). 5. H.-K. LO and Y. Zhao, Quantum cryptography (2008). 6. A. S. H. Z. J. G. R. T. W. D. Stucki, G. Ribordy, Journal of Modern Optics 48, p. 1967 (2001). 7. P. M. Pearle, Phys. Rev. D 2, 1418(Oct 1970). 8. A. Garg and N. D. Mermin, Phys. Rev. D 35, 3831(Jun 1987). 9. P. H. Eberhard, Phys. Rev. A 47, R747(Feb 1993). 10. J. Ake Larsson, Physics Letters A 256, 245 (1999). 11. N. Gisin and B. Gisin, Physics Letters A 260, 323 (1999). 12. A. Ekert, Less reality, more security(September, 2009). 13. Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen and H.-K. Lo, Phys. Rev. A 78, p. 042333(Oct 2008). 14. G. F. Knoll, Radiation Detection and Measurement (Wiley &Sons, 1999). 15. G. Adenier, Violation of bell inequalities as a violation of fair sampling in threshold detectors FOUNDATIONS OF PROBABILITY AND PHYSICS5 1101 (AlP, 2009). 16. K. J. Resch, J. S. Lundeen and A. M. Steinberg, Phys. Rev. A 63, p. 020102(Jan 2001). 17. H.-K. L. Bing Qi, Li Qian, A brief introduction of quantum cryptography for engineers (2010).

bThe use of four detectors on each side can also serve other purpose, like shielding Alice and Bob from a time-shift attack 17

Quantum Bio-Informatics IV eds. L. Accardi, W. Freudenberg and M. Ohya © 2011 World Scientific Publishing Co. (pp. 413-426)

BROWNIAN DYNAMICS SIMULATION OF MACROMOLECULE DIFFUSION IN A PROTOCELL TADASHI ANDO

Center for the Study of Systems Biology, School of Biology, Georgia Institute of Technology250 14th Street NW, Atlanta, GA 30318-5304, USA

JEFFREY SKOLNICK

Center for the Study ofSystems Biology, School ofBiology, Georgia Institute of Technology250 14th Street NW, Atlanta, GA 30318-5304, USA The interiors of all living cells are highly crowded with macromolecules, which differs considerably the thermodynamics and kinetics of biological reactions between in vivo and in vitro. For example, the diffusion of green fluorescent protein (GFP) in E. coli is -IO-fold slower than in dilute conditions. In this study, we performed Brownian dynamics (BD) simulations of rigid macromolecules in a crowded environment mimicking the cytosol of E. coli to study the motions of macromolecules. The simulation systems contained 35 70S ribosomes, 750 glycolytic enzymes, 75 GFPs, and 392 tRNAs in a 100 nm x 100 nm x 100 nm simulation box, where the macromolecules were represented by rigid-objects of one bead per amino acid or four beads per nucleotide models. Diffusion tensors of these molecules in dilute solutions were estimated by using a hydrodynamic theory to take into account the diffusion anisotropy of arbitrary shaped objects in the BD simulations. BD simulations of the system where each macromolecule is represented by its Stokes radius were also performed for comparison. Excluded volume effects greatly reduce the mobility of molecules in crowded environments for both molecular-shaped and equivalent sphere systems. Additionally, there were no significant differences in the reduction of diffusivity over the entire range of molecular size between two systems. However, the reduction in diffusion of GFP in these systems was still 4-5 times larger than for the in vivo experiment. We will discuss other plausible factors that might cause the large reduction in diffusion in vivo.

1.

Introduction

One of the most characteristic features of the interiors of all living cells is the extremely high total concentration of biological macromolecules. Typically, 20-30% of the total volume of cytoplasm is occupied by a variety of proteins, nucleic acids and other macromolecules. Under these conditions, the distance between neighboring proteins is comparable to the protein size itself, though the molar concentration of each protein ranges from nM to 11M. In this crowded, heterogeneous environment, biomolecules work to maintain living systems and 413

414

they have evolved over several billion years. Therefore, modeling the crowded cellular environment is not only an important first step toward whole cell simulation but also a crucial factor in understanding the nature of living systems. In this study, we performed Brownian dynamics (BD) simulations of rigid macromolecules in a crowded environment mimicking the cytosol of E. coli to study the motions of macromolecules. BD simulations using an equivalent sphere system, where macromolecules were represented by their Stokes radius were also performed. It has been reported that the diffusion of green fluorescent protein (GFP) in E. coli is ｾＱＰＭｦｯｬ､＠ slower than in dilute conditions (1, 2). Our aim is to investigate the mechanism(s) that causes this large reduction in diffusion in vivo. 2.

Methods

2.1. Estimation of diffusion tensor of a macromolecule from atomic structure To account for the diffusion anisotropy of macromolecules in our simulation, the diffusion tensors of macromolecules were calculated by using the rigid-particle formalism method (3-5). Here, we will describe this approach briefly. The diffusion of an arbitrarily shaped object undergoing Brownian motion is expressed by a 6 x 6 diffusion tensor, D, which is related to a frictional or resistance tensor, S, through the generalized Einstein relationship, D = kBT S·l. Both D and S can be partitioned into 3 x 3 sub-matrices, which correspond to translation (tt), rotation (rr), and translation-rotation coupling (tr and rt) tensors: D= (

_ )-1 D,,) = k T(SII D - '

DII

.:I

B

D"

rr

.:I

tr

ｾｲ＠

"

(1)

where (2) Here, the superscript T indicates transposition. Translational and rotational diffusion coefficients in a dilute solution are given by Do" = 1/3 Tr{D,,),

(3)

1/3 Tr{n,,),

(4)

Do" where Tr is the trace of the tensor.

=

415

The components of E can be obtained by the following procedure. From the Cartesian coordinates of the object consisting of N beads with the same radius, a, the 3 x 3 hydrodynamic interaction tensors between beads i and}, Tij (i,) = 1, ... , N) are calculated using the expression formulated by Rotne and Prager (6) and Yamakawa (7), the so-called RPY tensor, l";j ｾ＠

T.= 'J

2a, (5)

Here, 1] is the viscosity of the solvent and rij is the distance vector between beads i and j. It is important to note that the radius of bead is the only parameter to be optimized to reproduce hydrodynamic properties in dilute conditions. In what follows, we ignore intermolecular hydrodynamic interactions. Now, consider a 3N x 3N supermatrix, B, consisting of N x N Bij blocks at an arbitrary origin 0

B= [

Bij

1

B." ;

:.. ..

B:N ; ,

BNl

...

BNN

(6)

=6ij 6:17a + (1-6ij)rij.

Here, ｾｩｪ＠ is the Kronecker delta function. This supermatrix is then inverted to obtain a 3N x 3N supermatrix, C, ...

1

C IN . , ... C NN

.

in which each of the written as

Cij

block is a 3 x 3 matrix. Now, the elements ofE can be

Ell

= LLCij'

E" = LLUi ·Cij' E"

where

(7)

=-LLUi·Cij.U

(8) j ,

416

U,

ｾ｛＠ ｾ＠

-Zi

0

-Y;

x;

-;;

y, ]

(9)

.

Here, Xi, Yi, and ziare the components of the position vector of bead i at origin 0. So far the choice of the origin of the coordinates has been left arbitrary. However, the diffusion tensor, D, especially translational and translation-rotation coupling tensors, depends on the origin. At a certain origin, the so-called center of diffusion Q, the translational diffusion coefficient reaches a minimum. The position of Q with respect to the arbitrary origin 0, is calculated with the diffusion tensor obtained at 0, Do, as

[ 1[ XOQ

ROQ = YOQ = Z Oil

DYY + D" Tr,D Tr,O -

XY _DTr,O

-D" Tr,O

]

- ｄＺＧｾｯ＠

D Tr,O + D" Tr,O

D;;,o xz D Tr,O

XX

DTr,O + DTr,O XX

YY

_1[ D

Y'

fr,O

-D'Y fr,O

ｄｴｾｏ＠

]

- ｴ［ｾｏ＠ｄ . (10) DXY D tr,O tr,O YX

D at the center of diffusion Q are calculated by D !t,Q =DIt,D -U OQ ·Drr,O ·U OQ +Drt,O ·U OQ -U OQ ·Dtr,O' D rl,!) =D rl,O -U OQ ·DTr,O'

(11)

D rr,Q =DTr,O'

where

U OQ

=( ｚｾｑ＠

l-YoQ

o Ｍｾｏｑ＠

YOQ]

.

(12)

A volume correction term for rotational and intrinsic viscosity estimation is applied in Eq. 8 in some studies (3, 4). However, significant deviations of calculated diffusion properties from experimental values were not observed even without the correction.

2.2. Brownian dynamics for arbitrarily shaped objects BD is one of the most important simulation approaches to investigate the Brownian motion of arbitrary shaped objects, in which solvent molecules are treated implicitly and the influence of solvent on solute particles is incorporated through frictional and stochastic forces (8). In the high-friction limit, where it is assumed that momentum relaxation is much faster than position relaxation, and when we treat the diffusion tensor as a constant, a BD propagation scheme for an arbitrarily-shaped object can be written as (9)

417

Xi

=

ｘｾ＠

+

ｫｬＺＬＮｾ＠

Di ·FiP +gi(M),

(13)

B

where I:,.t is the time step and Xi is the vector describing the position of the center of diffusion and orientation of the i-th object, (14) Here, rJ, r2, and r3 are the position of the center of diffusion, and qJ I, qJ2, qJ3 describe its orientation. FP is a generalized system force having two components, the force acting on the center of diffusion (f) and the torque Cr):

(15) g(l:,.t) is a 6 x 1 random displacement vector during time step I:,.t due to the Brownian noise, which satisfies the following relations: (16) Here Di is the 6 x 6 diffusion tensor of object i at the center of diffusion. Once this diffusion tensor is calculated as described above, we can compute the random displacement vector using a Cholesky decomposition technique (9). The Cholesky decomposition of the diffusion tensor D is determined as

D=S·ST,

(17)

where S is a lower triangle matrix. The desired vector geM) is then obtained by the following: (18) where Z is a 6 x 1 vector, which has elements chosen from a Gaussian distribution so that (19) In BD simulations, quatemions, q = (qQ, ql, q2, q3), were used for handling rotations of rigid objects (8). Diffusion tensors of objects were evaluated in body-fixed frames only once at the start of the simulation. The force and torque on each object calculated in the laboratory or space-fixed frame (f' or T S) were converted to their body-fixed frame (f' or T b) using the rotation matrix Q obtained with quatemions, (20)

418

For each step, quatemions were scaled usmg Lagrange's method of undetermined multipliers to satisfY q2 = I (10).

2.3. Potential function In this study, we considered only repulsive interactions between intermolecular particles in BD simulations using a soft-sphere potential described by (21) is the distance between particles i andj, and kss is a force constant. rm is ai + aj + fl, in which ai and aj are radii of particles i and j and fl is an arbitrary parameter representing buffer distance between particles. In this study, fl of 2 A and kss of 5kB TI fl2 was used, which means Vss = 5kB T at the distance rij = ai + aj. where

rij

2.4.

Simulation conditions and analysis

All simulations were performed at 298 K with periodic boundary conditions. For all simulation systems, ten independent simulations were run with different initial configurations. 35 I-ls simulations were performed with time step of 0.5 ps. Configurations of the systems were sampled every 1 ns. Trajectories for the first 5 I-lS were discarded for analysis. The translational diffusion coefficient of a particle in three dimensions is estimated by

([r(t + T)- r(t )]') = 6D T ,

(22)

where ret) is position of the particle at time t and T is time interval. () indicates the ensemble average over the same particle type and time t.

419

3. Results

3.1. Estimation of diffusion tensor of a macromolecule from atomic structure 0.18

Translational --B-Rotational -e-

0.16 ,;

la\g> antibonding orbital

bonding orbitals

(b)

(a) Figure 1.

Spatial extension of lai g

)

antibonding orbital and

I bl g

)

bonding orbital

Undoped copper oxide La2Cu04 is an anti ferromagnetic Mott insulator, in which an electron correlation plays an important role. Thus we may say that undoped cuprates are governed by Mott physics. Most of models of high temperature superconductivity, assumed that the doped holes itinerate through an orbital of b 1g symmetry extended over a CU02 plane in the systems consisting of the CU06 octahedrons elongated by the JT effect. Those models are called "single-component theory", because they consider only the orbitals of hole carriers extend only over a CU02 plane. This singlecomponent theory, however, met a serious difficulty that, in the presence of anti-ferromagnetic (AF) order constructed from the localized spins, the hole carriers cannot move smoothly. In order to remove this difficulty, RYB model 2, t-J model 3, etc. were proposed. When Sr2+ ions are substituted for La3+ ions in LSCO, first-principles calculations showed that the apical oxygen in the CU06 octahedrons tend to approach towards Cu2+ ions 4,5. As a result the CU06 octahedron elongated by the JT effect shrink by doping holes. This deformation against the JT distortion is called "anti-J ahn-Teller effect" 6.

471

By this effect, the energy separation between the two kinds of orbital states, the ai g anti-bonding orbital state and the bIg bonding orbital state, becomes smaller 7,8. The spatial extension of the ai g anti-bonding orbital and the bIg bonding orbital is shown in figure l. By taking account of the anti-Jahn-Teller effect Kamimura and Suwa proposed that one must consider the ai g and bIg states equally in forming the metallic state of cuprates, and they constructed a metallic state coexisting with the local AF order. This model is now called "Kamimura-Suwa model" (K-S model) following the two authors' names of the paper by H. Kamimura and Y. Suwa 9. In this paper we will discuss two important results obtained from the interplay of Jahn-Teller Physics and Mott Physics. One is concerned with the Fermi surface. On the basis of the K-S model we will show that the "Fermi arcs" observed in ARPES should be one of the edges of real small Fermi pockets in the nodal region. This prediction is shown to be consistent with recent ARPES experimental results reported by Tanaka et al 11 and by Meng et al 22 . As the second important result obtained from the interplay of JahnTeller Physics and Mott Physics, we will show that the mechanism of superconductivity is caused by the interplay of strong electron-phonon interactions and an AF order. It is shown that (1) the characteristic phase difference of wave functions between up- and down-spin carriers in the presence of the AF order leads to the superconducting gap of d x 2_y2 symmetry even in the phonon-involved mechanism, (2) the out-of plane phonon modes in LSCO with tetragonal symmetry contribute to the formation of Cooper pairs while the in-plane modes do not, (3) Tc is higher in a cuprate with CU05 pyramid than that with CU06 octahedron because the number of out-of plane modes which contribute to d-wave superconductivity is larger in the former than in the latter, and (4) the calculated hole-concentration dependences of Tc and of the isotope effects for LSCO are consistent with recent experimental results. The organization of the present paper is the following: It consists of three parts. In the first part (Section 2) we will review briefly the K-S model. In the second part (Section 3) we will give the key features of the many-body-effects included energy bands and Fermi surfaces. And in the final part (Section 4) we will show how the d-symmetry superconducting gap appears from the K-S model. Section 5 is devoted to conclusion and concluding remarks.

472

2. Brief review of the Kamimura-Suwa (K-S) model

2.1. Anti-Jahn-Teller Effect When Sr2+ ions are substituted for La3+ions in LSCO, one may think intuitively that apical oxygen in the CU06 octahedrons tend to approach toward central Cu 2+ ions in order to gain the attractive electrostatic energy. Theoretically it was shown by the first-principles variational calculations of the spin-density-functional approach 4,5 that the optimized distance between apical 0 and Cu in LSCO which minimizes the total energy of LSCO decreases with increasing Sr concentration. As a result the elongated CU06 octahedrons by the Jahn-Teller (JT) interactions shrink by doping holes. We have called this shrinking effect against the Jahn-Teller distortion "antiJ ahn-Teller effect" 6.

2.2. The energy landscape of CU06 octahedron In order to clealify the the character of carriers hole,the first discribe many electron state CU06 octahedrons. Figure 2 shows the energy-level landscape starting from the e g and t2g orbitals of a Cu2+ ion in a regular CU06 octahedron with octahedral symmetry shown at the left column in the case of LSCO. By the JT effect the Cu eg orbital state splits into aIg and bIg orbital states, which form antibonding and bonding molecular orbitals with oxygen p orbitals. The molecular orbitals are denoted by ai g , aIg, big and bIg, as shown at the middle column. In an undoped case, 7 electrons occupy these cluster orbitals, so that the highest occupied big state is half filled, resulting in an S = 1/2 state, where a big orbital has Cu d x 2_y2 character. Following Mott physics, we introduce the Hubbard U interaction (U = 10eV) as a strong electron-correlation effect. Then the big state splits into the lower and upper Hubbard bands denoted by L.H. and V.H. in the figure, and this gives rise to a localized spin around a Cu site. These localized spins form the antiferromagnetic (AF) order by the superexchange interaction via an intervening 0 2- ion in undoped La2Cu04. Now let us remove one electron from the present system. In other words, let us consider the case that one hole is introduced to this CU06 cluster which has an up-spin electron as the localized spin. Due to the strong electron-electron interaction, it is not the big electron which is removed from the original electron configuration as expected from one-electron energy diagram. Instead, there are two possibilities of removing an electron;

473

U.H. fl······ /

Electron energy

/ /

/

/"

/

IA-site IIB-site I

/ /

/ /

3d

CU 2 1

Exchange interaction

,

3eV

\

\

... +

\ \

-- Ｍ

\ ｾ＠

\ \ \

\

L.H. ｜Ｎ

ｾ＠

••.

•• '1-"

Localized spin

Figure 2. Energy diagram showing the interplay of the Mott physics (electron correlation and exchange interaction between the spins of a doped hole and a localized hole) and of the J ahn- Teller physics (anti-JT effects) (Results of the first-principles calculations by Kamimura and Eto 7,8).

the bIg and ai g states. Since the multiplet energy of the six electron state with an absent big electron becomes very large due to the Hubbard repulsion of bIg, other multiplet states become more favorable. That is, the many-electron states with one electron occupying the big state and the other one in the bIg or ai g orbital with other doubly occupied orbitals. The former state must have total spin S = 0 due to the exchange interaction between bIg and big states while the latter has spin triplet structure because of Hund's interaction between big and ai g states. In the right column of figure 2, the present situation is described by assigning effective "one-electron energy" to each eg-electrons in a CU06 octahedron. For example, removing an up-spin electron form an ai g orbital costs O.5eV more than removing a down-spin electron from the same orbital as shown in the figure. Next, we show the energy landscape in "hole picture" in figure 3. As is

474

ｾ

Localized spin

/

Ｏ＠

Localized spin

\

\ \ \

-- - ----\ -.... + \

\

\

Hole energy

\

L.H. \. .••. . ..... Localized spin Figure 3_ Energy diagram in hole picture, when one down-spin hole is doped into CU06 cluster embedded in LSCO material.

shown in figure 3, when one down-spin hole is doped, it occupies bIg orbital in A-site. Then this hole can transfer from bIg orbital at A-site to ai g orbital at B-site. While the localized up-spin at A- and the localized down-spin at B-sites occupy U.H. band, in the hole picture. So that, the dopant downspin hole in ai g orbital at B-site becomes parallel to the localized spin in the localized big orbital by Hund's coupling. Thus this spin-triplet multiplet is called Hund's coupling 3B Ig . On the other hand, the dopant down-spin hole in bIg orbital at A-site becomes anti-parallel to the localized spin in the localized big orbital. This spin-singlet multiplet is called Zhang-Rice 1 A lg . The spatial distribution of these two kinds of multiplets are shown in the upper part of figure 3. By the first-principles cluster calculations which takes into account the Madelung potential due to all the ions surrounding a CU06 cluster in LSCO, Kamimura and Eto showed that the lowest-state energies of these two multiplets are nearly equal when both the Hund's coupling and the spin-singlet

475

exchange interaction are taken into account 7,8. Consequently the energy difference between the highest occupied orbital states ai g in 3B 1g multiplet and b 1g in 1 A 1g multiplet becomes only O.leV for the optimum doping (x = 0.15) in La2-xSrxCu04' Thus, when the two CU06 clusters with localized up and down spins are placed at the nearest neighbors, these states are easily mixed by the transfer interaction between ai g and b 1g orbitals, which is about 0.3eV for LSCO.

2.3. Kamimura-Suwa model (K-S model)

By using the calculated results of Kamimura and Eto 7,8 and assuming that the localized spins form an AF order in a spin-correlated region, Kamimura and Suwa 9 constructed a metallic state of LSCO for its underdoped regime. The feature of the K-S model is that the hole-carriers in the under doped regime of LSCO form a metallic state, by taking the Hund's coupling triplet and the Zhang-Rice singlet alternately in the presence of the local AF order without destroying the AF order. In order to understand the coexistence of AF order and metallic state, we will show the cartoon (figure 4), where the localized hole state and carrier hole state has been compared with 1st and 2nd story. In figure 4 the first story of a Cu house with (yellow) roof is occupied by the Cu localized spins, which form the AF order in the spin-correlated region by the superexchange interaction. The second story in a Cu house consists of two floors, lower ai g floor and upper b 1g floor. These second stories between neighboring eu houses are connected by Oxygen rooms with (blue-color) roof. In the second story a hole-carrier with up spin enters into the ai g floor at the lefthand eu house due to Hund's coupling with eu localized up-spin in the first story (Hund's coupling triplet), as shown in the extreme left column of the figure. By the transfer interaction, the hole is transferred into the b 1g floor at the neighboring eu house (the second from the left) through the Oxygen room, where the hole with up spin forms a spin-singlet state with a localized down spin at the second Cu house from the left (Zhang-Rice singlet). Thus the AF order is preserved when a hole-carrier itinerates. The important feature of K-S model is the coexistence of the AF order and a metallic (superconducting) state in the underdoped regime. This feature is different from that of the single-component theory.

476

dopant hole with up spin

5= 1 5= 0 5= 1 5= 05= 1 5= 0 big

floor 2 nd story

1st story Up-spin Down

rFigure 40

Up-

Down-

Up-

Down-

o 0 o 0 spin-correlated region (antiferromagnetic order) o

--I

An extended two-story house model (K-S model)

2.4. Effective Hamiltonian for the Kamimura-Suwa model (K-S) model The following effective Hamiltonian is introduced in order to describe the extended two-story house (K-S) model by Kamimura and Suwa 9 It consists of four parts: The effective one-electron Hamiltonian Heff for ｡ｾｧＨｩＩ＠ and b1g(b 1) orbital states, the transfer interaction between neighboring CU06 octahedrons (CU05 pyramids) H tn the superexchange interaction between the Cu d x 2_y2 localized spins H AF , and the exchange interactions between the spins of dopant holes and of d x 2_y2 localized holes within the same CU06 octahedron (CU05 pyramid) Hex Thus we have 0

0

H

= Heff + H tr + HAF + Hex = L i,m,a

EmCJmuCimu

+

L

tmn (CJmuCjnu

+ hoc o)

(i,j),m,n,a

+JLSiOSj+LKmSi,moSi, (i,j) i,m

(1)

477

where

Em

(m

=

aig(ai) or b1g (bd) represents the effective one-electron

energy of the aig(ai) and b 1g (bd orbital states, clma- and Cima- are the creation and annihilation operators of a dopant hole with spin (J in the i-th CU06 octahedron (i-th CU05 pyramid), respectively, tmn the effective transfer integrals of a dopant hole between m-type and n-type orbitals of neighboring CU06 octahedrons (CU05 pyramids), J the superexchange interaction between the spins S i and S j of dX 2 _y2 localized holes in the big (bi) orbital at the nearest neighbor Cu sites i and j (J > 0 for AF interaction), and K m the exchange integral for the exchange interaction between the spin of a dopant hole Sim and the dX 2 _y2 localized spin S i in the i-th CU06 octahedron (i-th CU05 pyramid). The values of the parameters in Eq. (1) for the case of LSCO are: J ｾ｟Ｍ［Ｌｉｧ＠ = 0.1, Ka* = -2.0, Kb l g = 4.0, ta*19 a*19 = 0.2, tb l g b , g = 0.4, ta*19 b , g = ｊｴ｡ｾｧ｢ｉｬ＠ '" 0.28, ｅ｡ｾｧ＠ = 0, Eb 'g = 2.6 in units of eV, where the values of Hund's coupling exchange constant K a *Ig and Zhang-Rice exchange constant K b'g are taken from the first principles cluster calculations for a CU06 octahedron in LSCO 7,8, and the energy difference of the effective - Eb ,g = -2.6, one-electron energies between ai g and bIg orbital states, ｅ｡ｾｧ＠ is determined so as to reproduce the energy difference between the 3B 1g and 1A 1g multiplets in LSCO in the MCSCF cluster calculations 8, while those of tmn's are due to band structure calculations 4,5.

3. Effective energy band and the shape of Fermi surface Ushio and Kamimura 13,12 have started from tight binding Hamiltonian. Then they have obtained the effective many-body-effects included Hamiltonian (1) by taking into account the exchange term as a molecular field, which is determined so that the energies of 1A 1g and 3B 1g coincide with that calculated by Eto and Kamimura 8.

3.1. Effective energy band In figure 5 (b), the calculated many-body effect included energy band structure for up-spin (or down-spin) dopant holes for LSCO is shown for various values of wave-vector k and symmetry points in the AF Brillouin zone. Here one should note that the energy in this figure is taken for electronenergy but not hole-energy, and the Hubbard bands for localized big holes do not appear in this figure. In the undoped La2Cu04, all the energy bands in figure 5 (2) are fully occupied by electrons so that La2Cu04 is an insulator, consistent with ex-

478

In doping, a hole carrier begins to occupy from 11 point.

AF Brillouin zone ｾ＠

＾ｾＮＭ ｾ＠

0

ｾ＠

00

ｾ＠ ｾ＠

ｾ

ｾＭＫ

ｾ＠

o

....

ｾＲｪＺｳ］＠ ＶｌＭｾ＠

(0,0,0)

(a)

('1C, O,O) .t::.. (0,0,0) (0,0, '1C) G 1 ('1C / 2, '1C /2,0)

(b)

Figure 5. (a) The Fermi surface of hole-carriers for x = 0.15 calculated for the #1 band. Here the kx axis is taken along rG1, corresponding to the x-axis (the Gu-O-Gu direction) in a real space. (b) The many-body-effect included band-structure for upspin dopant holes, obtained by Ushio and Kamimura 13,12. In this figure the Gu-O-Gu distance a is taken to be unity.

perimental results. In this respect the present effective energy band structure is completely different from the ordinary LDA energy bands 16,17. When Sr are doped, holes begin to occupy the top of the highest band in figure 5 (b) marked by #1 at ｾ＠ point which corresponds to Clf j2a, 7r j2a, 0) in the AF Brillouin zone. At the onset concentration of superconductivity, the Fermi level is located at the energy of E = 9.04 eV just below the top of the #1 band at ｾＬ＠ which is a little higher than that of the G 1 point. Here the G 1 point in the AF Brillouin zone lies at (7rja, 0, 0), and corresponds to a saddle point of the van Hove singularity. 3.2. The shape of Fermi surface

Based on the calculated band structure shown in figure 5 (b), U shio and Kamimura 13,6 calculated the Fermi surfaces for the underdoped regime of LSCO. This Fermi surface structure is also completely different from that

479

of an ordinary Fermi liquid picture, in which a Fermi surface is large. In figure 5 (a) the Fermi surface structure of hole-carriers calculated for x = 0.15 is shown as an example, where the Fermi surface (FS) consists of two pairs of extremely flat tubes. Thus the feature of FS in the underdoped regime is a small FS, and its volume is proportional to the doping concentration of dopant holes.

1.0

0.0 0.0

, 03 :, 05 . 0.5

0 .0

0.5

' 0.0

0.5

0.5

1.0

Figure 6. Observed doping dependence of Fermi arc (the inner section of FS) of La2-xSrxCu04 from x = 0.03 to 0.15, observed in ARPES by T. Yoshida et al 20,21 The calculated results based on the K-S model for x = 0.05 and x = 0.15 shown by thin curves are superimposed on the experimental results obtained by Yoshida et al.

Since the Bloch function of top band has Fourier component mainly in AF BZ, only the inner section of FS might be observed. ARPES researchers have called it "Fermi arc". Recently the Fermi arcs of hole carriers have been observed for the underdoped regime of LSCO by Yoshida et al 20,21. In figure 6 the calculated FS for x = 0.05 and x = 0.15 are compared with experimental results of La2-xSrxCu04 with x = 0.03, x = 0.05, x = 0.07 and x = 0.15. As seen in the figure, the agreement on the doping dependence of Fermi arcs between theory and experiment is fairly good. Recently, Meng et.al. 22 report ARPES measurements of Bi2Sr2-xLaxCu06+8 (La-Bi2201) that reveal Fermi pockets, supporting our calculated Fermi surface. We extend our calculations to the overdoped regime of LSCO. Since

480

the superexchange interaction becomes destroyed with increasing the hole concentration in the overdoped regime, the K-S model is considered not to hold beyond a certain hole concentration XO' As a result FS will change from small ones to a large one beyond XO' In the case of LSCO we think that Xo is 0.2 from experimental results of Loram et al 23 and of Nakano et al 24. 4. High temperature superconductivity

4.1. Spin-dependent electron-phonon interaction In this subsection we describe that the electron-phonon interaction in the K-S model depends on a spin direction of a hole carrier, in doing so, we describe only a stream of theoretical derivation. Therefore, for those who have interests in the derivation of equations, please read chapters 13 and 14 in our Springer text book entitled "Theory of Copper Oxide Superconductors" 6. On the basis of the K-S model we have shown that the interplay between electron-phonon interaction and the AF order leads to the d-wave phonon-involved mechanism in LSCO 26,27,6. As seen in figure 7, the atomic wave function for up-spin coincides with that for down-spin, if it is displaced by vector a (a is Cu-O-Cu distance). Thus, the wavefunctions of a hole-carrier with up and down spin in K-S model have the following phase relation:

(2) From this relation (2), the electron-phonon interaction matrix elements from states k to k' with up spin scattered by phonon with wave vector q, V l' (k, k', q), has the following spin-dependent property:

V l' (k, k', q)

= exp(iK . a)V 1 (k, k', q),

(3)

where K = k - k' - q is a reciprocal lattice vector in the AF Brillouin zone and a is a Cu-O-Cu distance. Since a reciprocal lattice vector in the AF Brillouin zone is expressed as K = (mr/a,m1f/a,O) with n+m = even, the electron-phonon interactions for up spin and down spin may have a different sign depending on a value of K.

4.2. Effective inter-hole interaction via phonon From the relation (3) and K = (n1f/a,m1f/a,O) with n+m = even, the effective interactions of a pair of holes from (k 1', - k 1) to (k' 1', - k' 1),

481

(a)

-

a

(b)

Figure 7. Schematic picture showing the phase relation between the wavefunctions of a hole-carrier with up and down spin on the extended two-story house mod el for La2-xSrxCu04

scattered by phonon with wave vector q, is expressed as

v l' (k , k' , q)V 1 (-k, -k', q) = exp(iK . a)1V l' (k , k' , q)12.

(4)

Since exp(iK . a) = +1 for n = even and exp(iK . a) = -1 for n = odd, the effective interaction for forming a Cooper pair becomes attractive for n = even while repulsive for n = odd. This remarkable result in the K-S model leads to the superconducting gap of d x 2_y2 symmetry. In figure 8 we show how attractive and repulsive subprocesses compete each other in a scattering process of a conduction hole by phonon. Suppose a hole occupies the state A in figure 8. A hole is scattered by phonon with wavevector q from A to shaded region because q is in normal BZ. So that , scattering process from the state A to the state B consists of two kinds of subprocesses. One subprocess is that a hole is scattered by phonon with wavevector q from A to state C' on FS. Since state C' is transferred to an equivalent state B by the translation of a reciprocal lattice vector (-Ql - Q2), the effective interaction for this subprocess is attractive

482

q: phonon wave vector

QIt Q2: AF reciproca l lattice vectors

..........•

umklapp sub-process (repulsive) normal sub-process (attractive)

Figure 8. Competition of attractive and repulsive subprocesses in the same scattering process of a carrier hole by phonon from state A to B in the AF Brillouin zone. A hole is first scattered by phonon to the point in the shadowed area , and then to the point equivalent to it.

according to equation (4) with n = 2. The other subprocess is that the hole at the state A on FS is also scattered from A to state C by phonon with wavevector q. Since the state C is equivalent to state B by the translation of a reciprocal lattice vector ( - Ql), the effective interaction is repulsive by equation (4) with n = m = 1. State C I is inside an AF Brillouin zone while state C is outside the AF Brillouin zone, one may say that a scattering process from A to B via C I is a Normal scattering while a scattering process from A to B via Cis Umklapp scattering. Normal and Umklapp scattering have different sign, so that the effective interaction between holes at state A and B becomes attractive or repulsive depending on the strengths of two scatterings. These strengths vary by depending a wave vector q. As an example of calculated results, we present the calculated results of the k and kl dependence of the electron-phonon spectral function 0;2 F r 1 ([1, k, k/) with spin-singlet for one of out-of-plane modes, A 1g mode (see the lower left corner ofthe Fig. 9(a)) in LSCO with tetragonal symmetry in figure 9, where Fr 1 ([1, k, k/) is the momentum-dependent density of phonon states and 0;2 is the square of the electron-phonon coupling constant

483

La

(b)

attractive

(a) repulsive

(C)

repulsive

Figure 9. Calculated result of a 2 P, 1 (0, k, k') and a gap function for one of out-of-plane phonon modes, Al g mode in LSCO as a function of k'

26,6. From the Fig. 9( c) we can see that the momentum-dependent spectral function varies by taking values with + and - sign, when the wave vector k' changes from the section CD of small FS to the section @)while k is fixed. This is clearly a d-wave behavior. By using this spectral function, we can obtain the Fig. 9(c), the obtained d-component gap function 6.(k) vary as a function of (cos(kxa) - cos(kya)) and the wavefunction of a Cooper pair has a spatial extension of d x 2_y2 symmetry. Kamimura et al calculated the electron-phonon spectral function a 2F(n) for all phonon modes of LSCO with tetragonal symmetry, and their calculated result of d-wave component of the spectral function for LSCO with tetragonal symmetry, a 2F i 1(2) (0,), is shown in figure 10 as a function of phonon frequency n. The phonon modes of LSCO with tetragonal symmetry are classified into in-plane modes and out-of-plane modes with regard to a CU02 plane. Figure 10 shows that the out-of-plane modes

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