Abstract Methods In Information Theory (Second Edition)

by Yuichiro Kakihara

Overview

Information Theory is studied from the following points of view: (1) the theory of entropy as amount of information; (2) the mathematical structure of information sources (probability measures); and (3) the theory of information channels. Shannon entropy and Kolmogorov-Sinai entropy are defined and their basic properties are examined, where the latter entropy is extended to be a linear functional on a certain set of measures. Ergodic and mixing properties of stationary sources are studied as well as AMS (asymptotically mean stationary) sources.The main purpose of this book is to present information channels in the environment of functional analysis and operator theory as well as probability theory. Ergodic, mixing, and AMS channels are also considered in detail with some illustrations. In this second edition, channel operators are studied in many aspects, which generalize ordinary channels. Also Gaussian channels are considered in detail together with Gaussian measures on a Hilbert space. The Special Topics chapter deals with features such as generalized capacity, channels with an intermediate noncommutative system, and von Neumann algebra method for channels. Finally, quantum (noncommutative) information channels are examined in an independent chapter, which may be regarded as an introduction to quantum information theory. Von Neumann entropy is introduced and its generalization to a C*-algebra setting is given. Basic results on quantum channels and entropy transmission are also considered.

Product Details

ISBN-13:	9789814759236
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/15/2016
Series:	Series On Multivariate Analysis , #10
Pages:	412
Product dimensions:	5.90(w) x 9.10(h) x 1.20(d)

Table of Contents

Preface to the First Edition vii

Preface to the Second Edition xi

Chapter I Entropy 1

1.1 The Shannon entropy 1

1.2 Conditional expectations 12

1.3 The Kolmogorov-Sinai entropy 18

1.4 Algebraic models 31

1.5 Entropy functionals 43

1.6 Relative entropy and Kullback-Leibler information 53

Bibliographical notes 67

Chapter II Information Sources 69

2.1 Alphabet message spaces and information sources 69

2.2 Ergodic theorems 74

2.3 Ergodic and mixing properties 78

2.4 A MS sources 94

2.5 Shannon-McMillan-Breiman theorem 102

2.6 Ergodic decompositions 110

2.7 Entropy functionals, revisited 114

Bibliographical notes 123

Chapter III Information Channels 125

3.1 Information channels 125

3.2 Mixing channels 134

3.3 Semiergodic channels 141

3.4 Ergodic channels 149

3.5 AMS channels 155

3.6 Capacity and transmission rate 166

3.7 Coding theorems 178

Bibliographical notes 188

Chapter IV Channel Operators 190

4.1 Channel operators 190

4.2 Generalized channels and topological properties 206

4.3 Pseudo channel operators: General theory 220

4.4 Pseudo channel operators: Topological structure 229

Bibliographical notes 234

Chapter V Gaussian Channels 236

5.1 Probability measures on a Hilbert space 236

5.2 Gaussian measures: Equivalence and singularity 250

5.3 Gaussian channels 265

5.4 Additive Gaussian channels 272

Bibliographical notes 284

Chapter VI Special Topics 285

6.1 Channels with a noise source 286

6.2 Channel capacity, revisited 293

6.3 Measurability of channels 299

6.4 Approximation of channels 306

6.5 Harmonic analysis for channels 309

6.6 Channels with a noncommutative intermediate system 317

6.7 Von Neumann algebras generated by stochastic processes 323

Bibliographical notes 331

Chapter VII Quantum Channels 333

7.1 Quantum entropy 333

7.2 Quantum channels 344

7.3 Entropy transmission 351

Bibliographical notes 356

References 359

Glossaries of Axioms 377

Indices 383

Notation index 383

Author index 390

Subject index 393