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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.

Computational complexity. --- Differentiable dynamical systems. --- Kolmogorov complexity. --- Quantum entropy. --- Quantum theory. --- Quantum theory --- Kolmogorov complexity --- Quantum entropy --- Differentiable dynamical systems --- Computational complexity --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Complexity, Computational --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics. --- Dynamics. --- Ergodic theory. --- Quantum physics. --- Quantum computers. --- Spintronics. --- Statistical physics. --- Dynamical systems. --- Quantum Information Technology, Spintronics. --- Dynamical Systems and Ergodic Theory. --- Quantum Physics. --- Mathematical Methods in Physics. --- Statistical Physics, Dynamical Systems and Complexity. --- Mathematical statistics --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Magnetoelectronics --- Spin electronics --- Microelectronics --- Nanotechnology --- Computers --- Thermodynamics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Statistical methods --- Electronic data processing --- Machine theory

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This second edition of Dynamics, Information and Complexity in Quantum Systems widens its scope by focussing more on the dynamics of quantum correlations and information in microscopic and mesoscopic systems, and their use for metrological and machine learning purposes. The book is divided into three parts: Part One: Classical Dynamical Systems Addresses classical dynamical systems, classical dynamical entropy, and classical algorithmic complexity. Includes a survey of the theory of simple perceptrons and their storage capacity. Part Two: Quantum Dynamical Systems Focuses on the dynamics of entanglement under dissipative dynamics and its metrological use in finite level quantum systems. Discusses the quantum fluctuation approach to large-scale mesoscopic systems and their emergent dynamics in quantum systems with infinitely many degrees of freedom. Introduces a model of quantum perceptron whose storage capacity is computed and compared with the classical one. Part Three: Quantum Dynamical Entropies and Complexities Devoted to quantum dynamical entropies and algorithmic complexities. This book is meant for advanced students, young and senior researchers working in the fields of quantum statistical mechanics, quantum information, and quantum dynamical systems. It is self-contained, and the only prerequisites needed are a standard knowledge of statistical mechanics, quantum mechanics, and linear operators on Hilbert spaces.

Quantum physics. --- Dynamical systems. --- System theory. --- Quantum computing. --- Statistical Physics. --- Quantum Physics. --- Dynamical Systems. --- Complex Systems. --- Quantum Information. --- Physics --- Mathematical statistics --- Computation, Quantum --- Computing, Quantum --- Information processing, Quantum --- Quantum computation --- Quantum information processing --- Electronic data processing --- Systems, Theory of --- Systems science --- Science --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Thermodynamics --- Statistical methods --- Philosophy --- Quantum theory. --- Dynamics. --- Statistical physics.

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Irreversible processes --- Space and time --- Symmetry (Physics) --- Processus irréversibles. --- Espace et temps --- Symétrie (physique)

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Discrete mathematics --- Mathematics --- Quantum mechanics. Quantumfield theory --- Classical mechanics. Field theory --- Statistical physics --- Computer. Automation --- quantumfysica --- grafentheorie --- quantumcomputers --- statistiek --- systeemtheorie --- fysica --- dynamica

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Ergodic theory. Information theory --- Mathematical physics --- Quantum mechanics. Quantumfield theory --- Statistical physics --- Computer. Automation --- computergestuurd meten --- quantumfysica --- statistiek --- wiskunde --- fysica --- informatietheorie