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.
Author(s): Fabio Benatti
Series: Theoretical and Mathematical Physics
Edition: 2
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 618
City: Cham
Tags: Classical and Quantum Dynamical Systems, Ergodic Theory, Entropy, Information Theory, Algorithmic Complexity
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
Part I Classical 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
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
4.1.2 Kolmogorov Complexity
4.2 Algorithmic Complexity and Entropy Rate
4.3 Prefix Algorithmic Complexity
4.3.1 Bibliographical Notes
Part II Quantum 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
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
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
7.2 Inequivalent Representations
7.3 Factor Types
7.4 Observables, States and Dynamics
7.4.1 Bosons and Fermions
7.4.2 GNS Representation and Dynamics
7.4.3 Quantum Ergodicity and Mixing
7.4.4 Algebraic Quantum K-Systems
7.4.5 Quantum Spin-Chains
7.5 Von Neumann Entropy Rate
7.6 Quantum Spin-Chains as Quantum Sources
7.6.1 Quantum Compression Theorems
7.6.2 Quantum Capacities
7.7 Quantum Fluctuations and Mesoscopic Physics
7.7.1 Mean-Field Observables
7.7.2 Quantum Fluctuations: Mesoscopic Limit
7.7.3 Mesoscopic Dissipative Dynamics
7.7.4 Mesoscopic Entanglement Through Dissipation
7.8 Quantum Perceptrons
7.8.1 A Model of Continuous Quantum Perceptron
7.8.2 Gaussian Input States
7.8.3 Quantum Storage Capacity: Statistical Approach
7.8.4 Quantum Storage Capacity: Explicit Computation
Part III Quantum Dynamical Entropies and Complexities
8 Quantum Dynamical Entropies
8.1 CNT Entropy: Decompositions of States
8.1.1 CNT Entropy Rate and CNT Entropy
8.1.2 CNT Entropy: Quasi-local Algebras
8.1.3 CNT Entropy: Stationary Couplings
8.1.4 CNT Entropy: Applications
8.1.5 Entropic Quantum K-Systems
8.2 AFL Entropy: OPUs
8.2.1 Quantum Symbolic Models and AFL Entropy
8.2.2 AFL Entropy: Interpretation
8.2.3 AFL -Entropy: Applications
8.2.4 AFL Entropy and Quantum Channel Capacities
9 Quantum Algorithmic Complexities
9.1 Effective Quantum Descriptions
9.1.1 Effective Descriptions by qubit Strings
9.1.2 Quantum Turing Machines
9.2 qubit Quantum Complexity
9.2.1 Quantum Brudno's Theorem
9.3 bit Quantum Complexity
9.3.1 Circuit Algorithmic Complexity
9.3.2 Quantum Universal Semi-density Matrix
References
Index
