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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.
List of contents
1 IntroductionPart I Classical Dynamical Systems2 Classical Dynamics and Ergodic Theory2.1 Classical Dynamical Systems2.2 Symbolic Dynamics2.3 Ergodicity and Mixing2.4 Information and Entropy3 Dynamical Entropy and Information3.1 Dynamical Entropy3.2 Codes and Shannon Theorems4 Algorithmic Complexity4.1 Kolmogorov Complexity4.2 Algorithmic Complexity and Entropy Rate4.3 Pre_x Algorithmic ComplexityPart II Quantum Dynamical Systems5 Quantum Mechanics of Finite Degrees of Freedom5.1 Hilbert Space and Operator Algebras5.2 Algebras of Bounded Operators on H5.3 Quantum Systems with Finite Degrees of Freedom5.4 Quantum States5.5 Dynamics and State-Transformations6 Quantum Information Theory6.1 Quantum Information Theory6.2 Bipartite Entanglement6.3 Relative Entropy7 Quantum Mechanics of Infinite Degrees of Freedom7.1 Observables, States and Dynamics7.2 Entropy Density7.3 Quantum Spin Chains as Quantum SourcesPart III Quantum Dynamical Entropies and Complexities8 Quantum Dynamical Entropies8.1 CNT Entropy: Decompositions of States8.2 AFL Entropy: OPUs9 Quantum Algorithmic Complexity9.1 Quantum Algorithmic ComplexitiesReferences
Summary
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.
Additional text
From the reviews:
“This book fits very well with the recent trends in theoretical and mathematical physics exploiting the interrelations between quantum theory, statistical mechanics and information theory. … The book is clearly written and contains a reasonable amount of elementary background material illustrated by numerous examples, and therefore can serve very well students at the graduate level and researchers entering the field of quantum dynamics and quantum information.” (Robert Alicki, Mathematical Reviews, Issue 2010 c)
Report
From the reviews: "This book fits very well with the recent trends in theoretical and mathematical physics exploiting the interrelations between quantum theory, statistical mechanics and information theory. ... The book is clearly written and contains a reasonable amount of elementary background material illustrated by numerous examples, and therefore can serve very well students at the graduate level and researchers entering the field of quantum dynamics and quantum information." (Robert Alicki, Mathematical Reviews, Issue 2010 c)
