Quantum Information Theory: Mathematical Foundation

✍ Scribed by Masahito Hayashi

• Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg
• Year: 2016
• Tongue: English
• Leaves: 672
• Series: Graduate Texts in Physics
• Edition: 2
• Category: Library

✦ Synopsis

This graduate textbook provides a unified view of quantum information theory. Clearly explaining the necessary mathematical basis, it merges key topics from both information-theoretic and quantum- mechanical viewpoints and provides lucid explanations of the basic results. Thanks to this unified approach, it makes accessible such advanced topics in quantum communication as quantum teleportation, superdense coding, quantum state transmission (quantum error-correction) and quantum encryption. Since the publication of the preceding book Quantum Information: An Introduction, there have been tremendous strides in the field of quantum information. In particular, the following topics – all of which are addressed here – made seen major advances: quantum state discrimination, quantum channel capacity, bipartite and multipartite entanglement, security analysis on quantum communication, reverse Shannon theorem and uncertainty relation. With regard to the analysis of quantum security, the present book employs an improved method for the evaluation of leaked information and identifies a remarkable relation between quantum security and quantum coherence. Taken together, these two improvements allow a better analysis of quantum state transmission. In addition, various types of the newly discovered uncertainty relation are explained. Presenting a wealth of new developments, the book introduces readers to the latest advances and challenges in quantum information. To aid in understanding, each chapter is accompanied by a set of exercises and solutions.

✦ Table of Contents

Preface
Preface to the First English Edition
Preface to the Japanese Edition
Contents
Notations
About the Author
Prologue
1 Mathematical Formulation of Quantum Systems
1.1 Quantum Systems and Linear Algebra
1.2 State and Measurement in Quantum Systems
1.3 Quantum Two-Level Systems
1.4 Composite Systems and Tensor Products
1.5 Matrix Inequalities and Matrix Monotone Functions
1.6 Solutions of Exercises
References
2 Information Quantities and Parameter Estimation in Classical Systems
2.1 Information Quantities in Classical Systems
2.1.1 Entropy
2.1.2 Relative Entropy
2.1.3 Mutual Information
2.1.4 The Independent and Identical Condition and Rényi Entropy
2.1.5 Conditional Rényi Entropy
2.2 Geometry of Probability Distribution Family
2.2.1 Inner Product for Random Variables and Fisher Information
2.2.2 Bregman Divergence
2.2.3 Exponential Family and Divergence
2.3 Estimation in Classical Systems
2.4 Type Method and Large Deviation Evaluation
2.4.1 Type Method and Sanov's Theorem
2.4.2 Cramér Theorem and Its Application to Estimation
2.5 Continuity and Axiomatic Approach
2.6 Large Deviation on Sphere
2.7 Related Books
2.8 Solutions of Exercises
References
3 Quantum Hypothesis Testing and Discrimination of Quantum States
3.1 Information Quantities in Quantum Systems
3.1.1 Quantum Entropic Information Quantities
3.1.2 Other Quantum Information Quantities
3.2 Two-State Discrimination in Quantum Systems
3.3 Discrimination of Plural Quantum States
3.4 Asymptotic Analysis of State Discrimination
3.5 Hypothesis Testing and Stein's Lemma
3.6 Hypothesis Testing by Separable Measurements
3.7 Proof of Direct Part of Stein's Lemma and Hoeffding Bound
3.8 Information Inequalities and Proof of Converse Part of Stein's Lemma ƒ
3.9 Proof of Theorem 3.1
3.10 Historical Note
3.11 Solutions of Exercises
References
4 Classical-Quantum Channel Coding (Message Transmission)
4.1 Formulation of the Channel Coding Process in Quantum Systems
4.1.1 Transmission Information in C-Q Channels and Its Properties
4.1.2 C-Q Channel Coding Theorem
4.2 Coding Protocols with Adaptive Decoding and Feedback
4.3 Channel Capacities Under Cost Constraint
4.4 A Fundamental Lemma
4.5 Proof of Direct Part of C-Q Channel Coding Theorem
4.6 Proof of Converse Part of C-Q Channel Coding Theorem
4.7 Pseudoclassical Channels
4.8 Historical Note
4.8.1 C-Q Channel Capacity
4.8.2 Hypothesis Testing Approach
4.8.3 Other Topics
4.9 Solutions of Exercises
References
5 State Evolution and Trace-Preserving Completely Positive Maps
5.1 Description of State Evolution in Quantum Systems
5.2 Examples of Trace-Preserving Completely Positive Maps
5.3 State Evolutions in Quantum Two-Level Systems
5.4 Information-Processing Inequalities in Quantum Systems
5.5 Entropy Inequalities in Quantum Systems
5.6 Conditional Rényi Entropy and Duality
5.7 Proof and Construction of Stinespring and Choi--Kraus Representations
5.8 Historical Note
5.8.1 Completely Positive Map and Quantum Relative Entropy
5.8.2 Quantum Relative Rényi Entropy
5.9 Solutions of Exercises
References
6 Quantum Information Geometry and Quantum Estimation
6.1 Inner Products in Quantum Systems
6.2 Metric-Induced Inner Products
6.3 Geodesics and Divergences
6.4 Quantum State Estimation
6.5 Large Deviation Evaluation
6.6 Multiparameter Estimation
6.7 Relative Modular Operator and Quantum f-Relative Entropy
6.7.1 Monotonicity Under Completely Positivity
6.7.2 Monotonicity Under 2-Positivity
6.8 Historical Note
6.8.1 Quantum State Estimation
6.8.2 Quantum Channel Estimation
6.8.3 Geometry of Quantum States
6.8.4 Equality Condition for Monotonicity of Relative Entropy
6.9 Solutions of Exercises
References
7 Quantum Measurements and State Reduction
7.1 State Reduction Due to Quantum Measurement
7.2 Uncertainty and Measurement
7.2.1 Uncertainties for Observable and Measurement
7.2.2 Disturbance
7.2.3 Uncertainty Relations
7.3 Entropic Uncertainty Relation
7.4 Measurements with Negligible State Reduction
7.5 Historical Note
7.6 Solutions of Exercises
References
8 Entanglement and Locality Restrictions
8.1 Entanglement and Local Quantum Operations
8.2 Fidelity and Entanglement
8.3 Entanglement and Information Quantities
8.4 Entanglement and Majorization
8.5 Distillation of Maximally Entangled States
8.6 Dilution of Maximally Entangled States
8.7 Unified Approach to Distillation and Dilution
8.8 Maximally Correlated State
8.9 Dilution with Zero-Rate Communication
8.10 Discord
8.11 State Generation from Shared Randomness
8.12 Positive Partial Transpose (PPT) Operations
8.13 Violation of Superadditivity of Entanglement Formation
8.13.1 Counter Example for Superadditivity of Entanglement Formation
8.13.2 Proof of Theorem 8.14
8.14 Secure Random Number Generation
8.14.1 Security Criteria and Their Evaluation
8.14.2 Proof of Theorem 8.15
8.15 Duality Between Two Conditional Entropies
8.15.1 Recovery of Maximally Entangled State from Evaluation of Classical Information
8.15.2 Duality Between Two Conditional Entropies of Mutually Unbiased Basis
8.16 Examples
8.16.1 2 times2 System
8.16.2 Werner State
8.16.3 Isotropic State
8.17 Proof of Theorem 8.2
8.18 Proof of Theorem 8.3
8.19 Proof of Theorem 8.8 for Mixed States
8.20 Proof of Theorem 8.9 for Mixed States
8.20.1 Proof of Direct Part
8.20.2 Proof of Converse Part
8.21 Historical Note
8.21.1 Entanglement Distillation
8.21.2 Entanglement Dilution and Related Topics
8.21.3 Additivity
8.21.4 Security and Related Topics
8.22 Solutions of Exercises
References
9 Analysis of Quantum Communication Protocols
9.1 Quantum Teleportation
9.2 C-Q Channel Coding with Entangled Inputs
9.3 C-Q Channel Coding with Shared Entanglement
9.4 Quantum Channel Resolvability
9.5 Quantum-Channel Communications with an Eavesdropper
9.5.1 C-Q Wiretap Channel
9.5.2 Relation to BB84 Protocol
9.5.3 Secret Sharing
9.5.4 Distillation of Classical Secret Key
9.5.5 Proof of Direct Part of C-Q Wiretap Channel Coding Theorem
9.5.6 Proof of Converse Part of C-Q Wiretap Channel Coding Theorem
9.6 Channel Capacity for Quantum-State Transmission
9.6.1 Conventional Formulation
9.6.2 Proof of Hashing Inequality (8.121)
9.6.3 Decoder with Assistance by Local Operations
9.7 Examples
9.7.1 Group Covariance Formulas
9.7.2 d-Dimensional Depolarizing Channel
9.7.3 Transpose Depolarizing Channel
9.7.4 Generalized Pauli Channel
9.7.5 PNS Channel
9.7.6 Erasure Channel
9.7.7 Phase-Damping Channel
9.8 Proof of Theorem 9.3
9.9 Historical Note
9.9.1 Additivity Conjecture
9.9.2 Channel Coding with Shared Entanglement
9.9.3 Quantum-State Transmission
9.10 Solutions of Exercises
References
10 Source Coding in Quantum Systems
10.1 Four Kinds of Source Coding Schemes in Quantum Systems
10.2 Quantum Fixed-Length Source Coding
10.3 Construction of a Quantum Fixed-Length Source Code
10.4 Universal Quantum Fixed-Length Source Codes
10.5 Universal Quantum Variable-Length Source Codes
10.6 Mixed-State Case and Bipartite State Generation
10.7 Compression with Classical Memory
10.8 Compression with Shared Randomness
10.9 Relation to Channel Capacities
10.10 Proof of Lemma 10.3
10.11 Historical Note
10.12 Solutions of Exercises
References
11 Erratum to: Quantum Information Theory
Erratum to:M. Hayashi, Quantum Information Theory, Graduate Texts in Physics, DOI 10.1007/978-3-662-49725-8
Appendix Limits and Linear Algebra
Postface to Japanese version
Index