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.

Résumé

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.

Contenu

Classical Dynamical Systems.- Classical Dynamics and Ergodic Theory.- Dynamical Entropy and Information.- Algorithmic Complexity.- Quantum Dynamical Systems.- Quantum Mechanics of Finite Degrees of Freedom.- Quantum Information Theory.- Quantum Mechanics of Infinite Degrees of Freedom.- Quantum Dynamical Entropies and Complexities.- Quantum Dynamical Entropies.- Quantum Algorithmic Complexities.