Dynamics, Information and Complexity in Quantum Systems (eBook)

Preface 7
Contents 8
1 Introduction 11
Classical Dynamical Systems 15
2 Classical Dynamics and Ergodic Theory 17
2.1 Classical Dynamical Systems 18
2.2 Symbolic Dynamics 33
2.3 Ergodicity and Mixing 48
2.4 Information and Entropy 64
3 Dynamical Entropy and Information 78
3.1 Dynamical Entropy 78
3.2 Codes and Shannon Theorems 93
4 Algorithmic Complexity 112
4.1 Effective Descriptions 113
4.2 Algorithmic Complexity and Entropy Rate 129
4.3 Prefix Algorithmic Complexity 134
Quantum Dynamical Systems 142
5 Quantum Mechanics of Finite Degrees of Freedom 144
5.1 Hilbert Space and Operator Algebras 144
5.2 C** Algebras 148
5.3 von Neumann Algebras 171
5.4 Quantum Systems with Finite Degrees of Freedom 183
5.5 Quantum States 195
5.6 Dynamics and State-Transformations 232
6 Quantum Information Theory 260
6.1 Quantum Information Theory 260
6.2 Bipartite Entanglement 266
6.3 Relative Entropy 292
7 Quantum Mechanics of Infinite Degrees of Freedom 321
7.1 Observables, States and Dynamics 327
7.2 von Neumann Entropy Rate 380
7.3 Quantum Spin Chains as Quantum Sources 385
Quantum Dynamical Entropies and Complexities 411
8 Quantum Dynamical Entropies 413
8.1 CNT Entropy: Decompositions of States 415
8.2 AFL Entropy: OPUs 453
9 Quantum Algorithmic Complexities 484
9.1 Effective Quantum Descriptions 485
9.2 qubit Quantum Complexity 495
9.3 cbit Quantum Complexity 507
References 517
Index 527