| Description |
1 online resource (xiv, 618 pages) : illustrations |
| Series |
Theoretical and mathematical physics |
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Theoretical and mathematical physics (Springer (Firm))
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| Contents |
Intro -- Preface to the Second Edition -- Preface to the First Edition -- Contents -- 1 Introduction -- Part IClassical Dynamical Systems -- 2 Classical Dynamics and Ergodic Theory -- 2.1 Classical Dynamical Systems -- 2.1.1 Hamiltonian Mechanics -- 2.1.2 Shift Dynamical Systems -- 2.2 Symbolic Dynamics -- 2.2.1 Algebraic Formulations -- 2.2.2 Conditional Probabilities and Expectations -- 2.2.3 Dynamical Shifts and Classical Spin Chains -- 2.3 Ergodicity and Mixing -- 2.3.1 K-systems -- 2.3.2 Ergodicity and Convexity -- 2.4 Information and Entropy -- 2.4.1 Transmission Channels |
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2.4.2 Stationary Information Sources -- 2.4.3 Shannon Entropy -- 2.4.4 Conditional Entropy -- 2.4.5 Mutual Information -- 3 Dynamical Entropy and Information -- 3.1 Dynamical Entropy -- 3.1.1 KS Entropy and Lyapounov Exponents -- 3.1.2 Entropic K-Systems -- 3.2 Codes and Shannon Theorems -- 3.2.1 Source Compression -- 3.2.2 Channel Capacity -- 3.3 Classical Machine Learning -- 3.3.1 Classification Tasks with Classical Perceptrons -- 3.3.2 Storage Capacity -- 3.3.3 Storage Capacity: Statistical Approach -- 4 Algorithmic Complexity -- 4.1 Effective Descriptions -- 4.1.1 Classical Turing Machines |
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4.1.2 Kolmogorov Complexity -- 4.2 Algorithmic Complexity and Entropy Rate -- 4.3 Prefix Algorithmic Complexity -- 4.3.1 Bibliographical Notes -- Part IIQuantum Dynamical Systems -- 5 Quantum Mechanics of Finite Degrees of Freedom -- 5.1 Hilbert Space and Operator Algebras -- 5.2 C* Algebras -- 5.2.1 Positive Operators -- 5.2.2 Positive and Completely Positive Maps -- 5.3 Von Neumann Algebras -- 5.3.1 States and GNS Representation -- 5.3.2 C* and von Neumann Abelian Algebras -- 5.4 Quantum Systems with Finite Degrees of Freedom -- 5.5 Quantum States -- 5.5.1 Beam splitters |
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5.5.2 Uncertainty Relations -- 5.5.3 Gaussian States -- 5.5.4 States in the Algebraic Approach -- 5.5.5 Density Matrices and von Neumann Entropy -- 5.5.6 Composite Systems -- 5.5.7 Entangled States -- 5.6 Dynamics and State-Transformations -- 5.6.1 Thermal States -- 5.6.2 Quantum Operations -- 5.6.3 Open Quantum Dynamics -- 5.6.4 Quantum Dynamical Semigroups -- 5.6.5 Physical Operations and Positive Maps -- 5.6.6 Non-Markovianity -- 5.6.7 Back-Flow of Information -- 6 Quantum Information Theory -- 6.1 Quantum Information Theory -- 6.2 Bipartite Entanglement |
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6.2.1 Distillability and Bound Entanglement -- 6.2.2 Entanglement Cost -- 6.2.3 Concurrence -- 6.2.4 Two-Mode Gaussian States -- 6.2.5 Positive Maps and Semigroups -- 6.2.6 Dissipative Entanglement Generation -- 6.3 Relative Entropy -- 6.3.1 Holevo's Bound and the Entropy of a Subalgebra -- 6.3.2 Entropy of a Subalgebra and Entanglement of Formation -- 6.4 Entanglement, Non-locality and Quantum Metrology -- 6.4.1 Identical Particles -- 6.4.2 Quantum Metrology -- 6.4.3 Identical Particles -- 7 Quantum Mechanics of Infinite Degrees of Freedom -- 7.1 Relaxation to Equilibrium |
| Summary |
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 |
| Notes |
Description based upon print version of record |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based on online resource; title from digital title page (viewed on October 16, 2023) |
| Subject |
Quantum theory.
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Kolmogorov complexity.
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Quantum entropy.
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Differentiable dynamical systems.
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Computational complexity.
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Computational complexity
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Differentiable dynamical systems
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Kolmogorov complexity
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Quantum entropy
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Quantum theory
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| Form |
Electronic book
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| ISBN |
9783031342486 |
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3031342488 |
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