Intro
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
Part IClassical Dynamical Systems
2 Classical Dynamics and Ergodic Theory
2.1 Classical Dynamical Systems
2.1.1 Hamiltonian Mechanics
2.1.2 Shift Dynamical Systems
2.2 Symbolic Dynamics
2.2.1 Algebraic Formulations
2.2.2 Conditional Probabilities and Expectations
2.2.3 Dynamical Shifts and Classical Spin Chains
2.3 Ergodicity and Mixing
2.3.1 K-systems
2.3.2 Ergodicity and Convexity
2.4 Information and Entropy
2.4.1 Transmission Channels

2.4.2 Stationary Information Sources
2.4.3 Shannon Entropy
2.4.4 Conditional Entropy
2.4.5 Mutual Information
3 Dynamical Entropy and Information
3.1 Dynamical Entropy
3.1.1 KS Entropy and Lyapounov Exponents
3.1.2 Entropic K-Systems
3.2 Codes and Shannon Theorems
3.2.1 Source Compression
3.2.2 Channel Capacity
3.3 Classical Machine Learning
3.3.1 Classification Tasks with Classical Perceptrons
3.3.2 Storage Capacity
3.3.3 Storage Capacity: Statistical Approach
4 Algorithmic Complexity
4.1 Effective Descriptions
4.1.1 Classical Turing Machines

4.1.2 Kolmogorov Complexity
4.2 Algorithmic Complexity and Entropy Rate
4.3 Prefix Algorithmic Complexity
4.3.1 Bibliographical Notes
Part IIQuantum Dynamical Systems
5 Quantum Mechanics of Finite Degrees of Freedom
5.1 Hilbert Space and Operator Algebras
5.2 C* Algebras
5.2.1 Positive Operators
5.2.2 Positive and Completely Positive Maps
5.3 Von Neumann Algebras
5.3.1 States and GNS Representation
5.3.2 C* and von Neumann Abelian Algebras
5.4 Quantum Systems with Finite Degrees of Freedom
5.5 Quantum States
5.5.1 Beam splitters

5.5.2 Uncertainty Relations
5.5.3 Gaussian States
5.5.4 States in the Algebraic Approach
5.5.5 Density Matrices and von Neumann Entropy
5.5.6 Composite Systems
5.5.7 Entangled States
5.6 Dynamics and State-Transformations
5.6.1 Thermal States
5.6.2 Quantum Operations
5.6.3 Open Quantum Dynamics
5.6.4 Quantum Dynamical Semigroups
5.6.5 Physical Operations and Positive Maps
5.6.6 Non-Markovianity
5.6.7 Back-Flow of Information
6 Quantum Information Theory
6.1 Quantum Information Theory
6.2 Bipartite Entanglement

6.2.1 Distillability and Bound Entanglement
6.2.2 Entanglement Cost
6.2.3 Concurrence
6.2.4 Two-Mode Gaussian States
6.2.5 Positive Maps and Semigroups
6.2.6 Dissipative Entanglement Generation
6.3 Relative Entropy
6.3.1 Holevo's Bound and the Entropy of a Subalgebra
6.3.2 Entropy of a Subalgebra and Entanglement of Formation
6.4 Entanglement, Non-locality and Quantum Metrology
6.4.1 Identical Particles
6.4.2 Quantum Metrology
6.4.3 Identical Particles
7 Quantum Mechanics of Infinite Degrees of Freedom
7.1 Relaxation to Equilibrium