This wide-ranging book introduces information as a key concept not only in physics, from quantum mechanics to thermodynamics, but also in the neighboring sciences and in the humanities. The central part analyzes dynamical processes as manifestations of information flows between microscopic and macroscopic scales and between systems and their environment. Quantum mechanics is interpreted as a reconstruction of mechanics based on fundamental limitations of information processing on the smallest scales. These become particularly manifest in quantum chaos and in quantum computing. Covering subjects such as causality, prediction, undecidability, chaos, and quantum randomness, the book also provides an information-theoretical view of predictability.
More than 180 illustrations visualize the concepts and arguments. The book takes inspiration from the author's graduate-level topical lecture but is also well suited for undergraduate studies and is a valuable resource for researchers and professionals.
Author(s): Thomas Dittrich
Series: The Frontiers Collection
Publisher: Springer
Year: 2022
Language: English
Pages: 566
City: Cham
Preface
Contents
Part I Natural Systems as Information Processors
1 The Concept of Information
1.1 Some History
1.2 The “Three Dimensions” of Information
1.3 From Boltzmann's Entropy to Shannon's Information
1.4 Sign: Entropy and Negentropy: Actual Versus Potential Information
1.5 Hierarchical Structures
1.6 Properties of Shannon’s Definition of Information
1.6.1 An Extremum Property
1.6.2 Equal Probabilities Imply Maximum Entropy
1.6.3 Information Content of Tree Structures
1.7 Joint, Conditional, Mutual Information, Bayes’ Law, Correlations and Redundancy
1.8 Information in Continuous Physical Quantities
2 Simple Applications
2.1 Logics
2.1.1 Propositional Logics
2.1.2 Boolean Algebra and Electronic Implementations
2.1.3 Set Theory
2.1.4 Inference Chains
2.2 The Genetic Code
2.2.1 Syntax
2.2.2 Semantics, The Central Dogma
2.2.3 Pragmatics and Discussion
2.2.4 Genetic Information in Phylogenesis
2.3 Fourier Transform
2.3.1 Discrete Symmetries in Fourier Transformation
2.3.2 Sampling
2.3.3 Uncertainty Relations
2.3.4 Fast Fourier Transformation
3 Epistemological Aspects
3.1 Causality
3.1.1 Causality from Topology: Markov Chains and Bayesian Networks
3.1.2 Causality from Information Flow: Transfer Information
3.1.3 Causality in Continuous Time: Kolmogorov-Sinai Entropy
3.1.4 Records and Memory
3.1.5 Causality and Special Relativity Theory
3.1.6 Finality
3.2 Prediction
3.2.1 Prediction, Anticipation, Simulation
3.2.2 Prediction from Within: Self-Fulfilling and Self-Destroying Prophecy
3.2.3 Self-Reference and Information-Theoretical Limits of Self-Prediction
3.3 Learning and Adaption
3.3.1 Detectors of Correlation and Causality
3.3.2 Predictors in Society
3.3.3 Darwin's Demons: Anticipatory Systems and Entropy Flow in Ontogeny and Phylogeny
4 Information and Randomness
4.1 Quantifying Randomness
4.2 Randomness According to Structure: Redundancy, Data Compression, and Scientific Induction
4.2.1 Induction
4.2.2 Pattern Recognition and Algorithmic Complexity
4.3 Gödel’s Theorem and Incompleteness
4.3.1 Formal Systems
4.3.2 Gödel’s Incompleteness Theorem and Provability of Randomness
4.3.3 Interpretations and Consequences of Gödel’s Incompleteness Theorem
5 Information in Classical Hamiltonian Dynamics
5.1 Review of Hamiltonian Dynamics and Symplectic Geometry
5.2 Hamiltonian Dynamics of Continuous Density Distributions
5.3 Information Density, Information Flow, and Conservation of Information in Hamiltonian Systems
5.4 Conservation of Information Without Energy Conservation: Harmonic Oscillator Driven at Resonance
5.5 Information Processing in Chaotic Hamiltonian Systems: Bernoulli Shift and Baker Map
5.6 Information Exchange Between Degrees of Freedom: Normal Modes in Pairs and Chains of Harmonic Oscillators
5.6.1 Two Coupled Harmonic Oscillators in Resonance
5.6.2 Chains of N Coupled Harmonic Oscillators
6 Information in Classical Dissipative Dynamics
6.1 Lyapunov Exponents Measure Vertical Information Flows
6.2 Entropy Loss into Microscales: The Dissipative Harmonic Oscillator
6.3 The Generic Case: Coexistence of Chaos and Dissipation
6.4 Fractals, Dimension, and Information
7 Fluctuations, Noise, and Microscopic Degrees of Freedom
7.1 Noise, Diffusion, and Information Loss
7.2 Fluctuation–Dissipation Theorems: Einstein’s Relation and Nyquist’s Theorem
7.3 The Second Law of Thermodynamics in the Light of Information Flows
7.3.1 Mixing and Thermalization
7.3.2 Diffusion and Coarse-Graining
7.3.3 Grand Total: The Second Law of Thermodynamics
8 Information and Quantum Mechanics
8.1 Information Theory Behind the Principles of Quantum Mechanics
8.1.1 Postulates of Quantum Mechanics Related to Information
8.1.2 Hilbert Space Vectors as Basic Information Carriers
8.1.3 Heisenberg’s Uncertainty Principle and Information in Phase Space
8.1.4 Entanglement and Non-Locality
8.2 Quantum Information
8.2.1 The Density Operator and Von-Neumann Entropy
8.2.2 Entanglement and Quantum Information
8.2.3 Decoherence and Quantum Information
8.3 Dynamics of Quantum Information
8.3.1 Unitary Time Evolution
8.3.2 Unitary Transformations Conserve Information
8.3.3 Incoherent Processes and Classicality
8.4 Quantum Measurement
8.4.1 Overview
8.4.2 Von-Neumann Theory of Quantum Measurement
8.4.3 Entanglement and Non-Locality in Quantum Measurement
8.4.4 The Quantum Zeno Effect
8.4.5 Quantum Randomness
8.4.6 Quantum Causality
8.5 Quantum Death and Resurrection of Chaos
8.5.1 Quantum Chaos: A Deep Probe into Quantum Information Processing
8.5.2 Discretizing Classical Chaos
8.5.3 Quantum Death of Classical Chaos
8.5.4 Resurrection of Chaos by Decoherence and Dissipation
8.6 Mixing, Irreversibility, and Information Production in Quantum Systems
8.6.1 The Role of Chaos: Berry’s Conjecture
8.6.2 Typicality and the Eigenstate Thermalization Hypothesis
8.6.3 Many-Body Localization: Threatening Thermalization?
8.6.4 Perspectives: Equilibration and Entanglement
Part II Computers as Natural Systems
9 Physical Aspects of Computing
9.1 What’s so Special About Computing?
9.1.1 Computers as Man-Made Tools
9.1.2 Computers and Computing in Natural Dynamical Systems
9.2 Physical Conditions of Computation
9.2.1 Implementing and Controlling a Single Bit: Macroscopic Discretization
9.2.2 Implementing Gates: Reversible and Irreversible Operations
9.3 Global Structure of Classical Computing: Moving on Granular Spaces
9.3.1 Granular State Spaces
9.3.2 Navigation on Granular Spaces
9.3.3 Models of Classical Computing: The Turing Machine
9.3.4 Cellular Automata: Parallel Computing on Granular Spaces
9.3.5 Conway’s Game of Life
9.4 The Hierarchical Structure of Computation
9.4.1 Structured Organization of Computers: An Overview
9.4.2 Emergence in the Hierarchy of Computing
9.4.3 Emergent Dynamics: Vertical Information Flow and Downward Causation
10 Quantum Computation
10.1 What’s so Special About Quantum Computing?
10.2 Tools for Quantum Computation: Qubits and Quantum Gates
10.2.1 The Qubit
10.2.2 Unitary Operators, Reversible Computation, and Quantum Gates
10.3 Strategies for Quantum Computation: Quantum Algorithms
10.3.1 Quantum Dense Coding
10.3.2 Quantum Parallelism
10.3.3 The Deutsch and Deutsch-Jozsa Algorithms
10.3.4 Quantum Fourier Transform
10.3.5 Quantum Search Algorithms
10.4 Decoherence and Error Correction
10.4.1 Sources, Types, and Effects of Noise
10.4.2 Error Protection and Correction
10.4.3 Error Prevention: Computing In Decoherence-Free Subspaces
10.5 Physical Implementations
10.5.1 Peepholes: Communicating with a Quantum Computer
10.5.2 Prototypical Platforms for Quantum Computing
11 Epilogue
Bibliography
Index
