Quantum Information: An Introduction

by Masahito Hayashi

Springer, 2006, ISBN 3-540-30265-4

Chapter 0: Prologue
0.1 Invitation to Quantum Information Theory
0.2 History of Quantum Information Theory
0.3 Structure of This Book

Chapter 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

Chapter 2: Information Quantities and Parameter Estimation in Classical Systems
2.1 Information Quantities in Classical Systems
2.2 Extensions to Quantum Systems
2.3 Geometry of Probability Distribution Family
2.4 Estimation in Classical Systems
2.5 Type Method and Large Deviation Evaluation
2.6 Related Books

Chapter 3: Quantum Hypothesis Testing and Discrimination of Quantum States
3.1 Two-State Discrimination in Quantum Systems
3.2 Discrimination of Plural Quantum States
3.3 Asymptotic Analysis of State Discrimination
3.4 Hypothesis Testing and Stein's Lemma
3.5 Hypothesis Testing by Separable Measurements
3.6 Proof of Direct Part of Stein's Lemma
3.7 Information Inequalities and Proof of Converse Part of Stein's Lemma
3.8 Historical Note

Chapter 4: Classical-Quantum Channel Coding (Message Transmission)
4.1 Formulation of Channel Coding Process in Quantum System
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 Classical-Quantum Channel Coding Theorem
4.6 Proof of Converse Part of Classical-Quantum Channel Coding Theorem
4.7 Pseudo Classical Channels
4.8 Historical Note

Chapter 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 Historical Note

Chapter 6: Quantum Information Geometry and Quantum Estimation
6.1 Inner Products in Quantum Systems
6.2 Metrics induced Inner Products
6.3 Geodesics and Divergences
6.4 Quantum State Estimation
6.5 Large Deviation Evaluation
6.6 Multi-Parameter Estimation
6.7 Historical Note

Chapter 7: Quantum Measurements and State Reduction
7.1 State Reduction due to Quantum Measurement
7.2 Uncertainty and Measurement
7.3 Measurements with Negligible State Demolition
7.4 Historical Note

Chapter 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 State
8.7 Unified Approach to Distillation and Dilution
8.8 Dilution with zero-rate communication
8.9 State Generation From Shared Randomness
8.10 Positive Partial Transpose (PPT) Operation
8.11 Examples
8.12 Historical Note

Chapter 9: Analysis of Quantum Communication Protocols
9.1 Quantum Teleportation
9.2 Classical-Quantum Channel Coding with Entangled Inputs
9.3 Classical-Quantum Channel Coding with Shared Entanglement
9.4 Quantum Channel Resolvability
9.5 Quantum Channel Communications with an Eavesdropper
9.6 The Channel Capacity for Quantum State Transmission
9.7 Examples
9.8 Historical Note

Chapter 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 States Case
10.7 Compression by Classical Memory
10.8 Compression with Shared Randomness
10.9 Relation to Channel Capacities
10.10 Historical Note

Chapter A: Limits and Linear Algebra

Chapter B: Proofs of Theorems and Lemmas

Chapter C: Hints and Brief Solutions to Exercises