Dynamics, Information and Complexity in Quantum Systems

This book offers a self-contained overview of the entropic approach to quantum dynamical systems. In it, complexity in quantum dynamics is addressed by comparison with the classical ergodic, information, and algorithmic complexity theories.

The leading theme of the book is complexity in quantum dynamics. This issue is addressed by comparison with the classical ergodic, information and algorithmic complexity theories. Of particular importance is the notion of Kolmogorov-Sinai dynamical entropy and of its inequivalent quantum extensions formulated by Connes, Narnhofer and Thirring on one hand and Alicki and Fannes on the other. Their connections with extensions to quantum systems of Kolmogorov-Chaitin-Solomonoff algorithmic complexity theory is also presented. The technical tools employed are those of the algebraic approach to quantum statistical mechanics which offers a unifying view of classical and quantum dynamical systems. Proofs and examples are provided in order to make the presentation self consistent.

Contenu
1 Introduction Part I Classical Dynamical Systems 2 Classical Dynamics and Ergodic Theory 2.1 Classical Dynamical Systems 2.2 Symbolic Dynamics 2.3 Ergodicity and Mixing 2.4 Information and Entropy 3 Dynamical Entropy and Information 3.1 Dynamical Entropy 3.2 Codes and Shannon Theorems 4 Algorithmic Complexity 4.1 Kolmogorov Complexity 4.2 Algorithmic Complexity and Entropy Rate 4.3 Pre_x Algorithmic Complexity Part II Quantum Dynamical Systems 5 Quantum Mechanics of Finite Degrees of Freedom 5.1 Hilbert Space and Operator Algebras 5.2 Algebras of Bounded Operators on H 5.3 Quantum Systems with Finite Degrees of Freedom 5.4 Quantum States 5.5 Dynamics and State-Transformations 6 Quantum Information Theory 6.1 Quantum Information Theory 6.2 Bipartite Entanglement 6.3 Relative Entropy 7 Quantum Mechanics of Infinite Degrees of Freedom 7.1 Observables, States and Dynamics 7.2 Entropy Density 7.3 Quantum Spin Chains as Quantum Sources Part III Quantum Dynamical Entropies and Complexities 8 Quantum Dynamical Entropies 8.1 CNT Entropy: Decompositions of States 8.2 AFL Entropy: OPUs 9 Quantum Algorithmic Complexity 9.1 Quantum Algorithmic Complexities References