Athanasius Kircher A Renaissance Man and the Quest for Lost Knowledge

In the last few decades quantum theory has experienced an extensive revival owing to the rapid development of quantum information and quantum technologies. Based on a series of courses taught by the authors, the book takes the reader on a journey from the beginnings of quantum theory in the early twentieth century to the realm of quantum-information processing in the twenty-first. The central aim of this textbook, therefore, is to offer a detailed introduction to quantum theory that covers both physical and information-theoretic aspects, with a particular focus on the concept of entanglement and its characteristics, variants, and applications. Suitable for undergraduate students in physics and related subjects who encounter quantum mechanics for the first time, this book also serves as a resource for graduate students who want to engage with more advanced topics, offering a collection of derivations, proofs, technical methods, and references for graduate students and more experienced readers engaged with teaching and active research. The book is divided into three parts: Part I - Quantum Mechanics, Part II - Entanglement and Non-Locality, and Part III - Advanced Topics in Modern Quantum Physics. Part I provides a modern view on quantum mechanics, a central topic of theoretical physics. Part II is dedicated to the foundations of quantum mechanics and entanglement: starting with density operators, hidden-variable theories, the Einstein-Podolsky-Rosen Paradox, and Bell Inequalities, but also touching upon philosophical questions, followed by a deeper study of entanglement-based quantum communication protocols like teleportation, before giving a detailed exposition of entanglement theory, including tools for the detection and quantification of entanglement. Part III is intended as a collection of standalone chapters to supplement the contents of Parts I and II, covering more advanced topics such as classical and quantum entropies, quantum operations and measurements, decoherence, quantum metrology and quantum optics, and entanglement in particle physics. Title Page Copyright Page Dedication Acknowledgements Preface Content Part I Quantum Mechanics 1 Wave-Particle Duality 1.1 Planck's Law of Black-Body Radiation 1.1.1 Quantization of Energy 1.1.2 Black-Body Radiation 1.1.3 Derivation of Planck's Law 1.2 The Photoelectric Effect 1.2.1 Observation of the Photoelectric Effect 1.2.2 Einstein's Explanation for the Photoelectric Effect 1.2.3 The Millikan Experiment 1.3 The Compton Effect 1.3.1 The Compton Shift Formula 1.3.2 The Experiment of Compton 1.4 Bohr's Theses 1.5 Wave Properties of Matter 1.5.1 Louis de Broglie's Hypothesis 1.5.2 Electron Diffraction from a Crystal 1.6 Heisenberg's Uncertainty Principle 1.6.1 Heisenberg's Microscope 1.6.2 Energy-Time Uncertainty Principle 1.7 The Double-Slit Experiment 1.7.1 Comparison of Classical and Quantum-Mechanical Results 1.7.2 Interpretation of Quantum-Mechanical Results 1.7.3 Interferometry with Large Molecules 1.8 Schrödinger's Cat 2 The Time-Dependent Schrödinger Equation 2.1 Wave Function and Time-Dependent Schrödinger Equation 2.1.1 Discovering the Schrödinger Equation 2.1.2 First Quantization 2.1.3 The Interpretation of the Wave Function 2.1.4 The Normalization of the Wave Function 2.2 The Continuity Equation 2.3 States and Observables 2.3.1 The Scalar Product 2.3.2 Operators 2.3.3 The Commutator 2.4 Expectation Values and Variances 2.4.1 Expectation Values of Operators 2.4.2 Uncertainty of Observables 2.5 The Uncertainty Principle 2.5.1 Uncertainty Relation for Observables 2.5.2 Position-Momentum Uncertainty 2.5.3 Uncertainty of Gaussian Wave Packets 2.5.4 Wave Packets in Momentum Space 2.6 Time Evolution in Quantum Mechanics 2.6.1 The Propagator 2.6.2 Schrödinger Picture and Heisenberg Picture 2.6.3 Time Evolution of Expectation Values 2.6.4 Time Evolution of Free Wave Packets 2.6.5 Energy-Time Uncertainty 2.7 Recovering Classical Physics 2.7.1 The Ehrenfest Theorem 2.7.2 Dirac's Rule 3 Mathematical Formalism of Quantum Mechanics 3.1 Hilbert Space 3.1.1 Norm and Completeness 3.1.2 Dimensionality of Hilbert Spaces 3.1.3 The Dual Hilbert Space and Dirac Notation 3.1.4 The Hilbert Space for Photon Polarization 3.2 Operators on Finite-Dimensional Hilbert Spaces 3.2.1 Projectors 3.2.2 The Spectral Theorem 3.2.3 Unitary Operators 3.3 Infinite-Dimensional Hilbert Spaces 3.3.1 Self-adjoint Operators on Infinite-Dimensional Hilbert Spaces 3.3.2 Continuous Spectra 3.3.3 Distributional Aspects of Quantum Mechanics 4 The Time-Independent Schrödinger Equation 4.1 Solving the Schrödinger Equation 4.1.1 Stationary States 4.1.2 The Schrödinger Equation as an Eigenvalue Problem 4.1.3 Expansion into Stationary States 4.1.4 Physical Interpretation of the Expansion Coefficients 4.2 Bound States 4.2.1 The Finite Potential Well 4.2.2 The Infinite Potential Well 4.2.3 The Dirac-Delta Potential Well 4.2.4 The Double Well and the Ammonia Molecule 4.3 Scattering and the Tunnel Effect 4.3.1 The Finite Potential Barrier 4.3.2 Reflection and Transmission 4.3.3 Tunnelling and the Gamow Factor 4.3.4 Transmission Resonances 5 The Quantum Harmonic Oscillator 5.1 Algebraic Method 5.1.1 Annihilation and Creation Operators 5.1.2 The Occupation-Number Operator 5.1.3 The Ground State of the Harmonic Oscillator 5.1.4 Eigenstates of the Harmonic Oscillator 5.2 Analytic Method 5.2.1 The Differential Equation of the Harmonic Oscillator 5.2.2 The Hermite Polynomials 5.3 Zero-Point Energy 5.3.1 Uncertainty Relation for the Harmonic Oscillator 5.3.2 The Zero-Point Energy of the Harmonic Oscillator 5.4 Comparison with the Classical Oscillator 5.5 The Three-Dimensional Harmonic Oscillator 5.5.1 Eigenstates of the Three-Dimensional Harmonic Oscillator 5.5.2 Systems of Multiple Degrees of Freedom 6 Orbital Angular Momentum 6.1 Angular Momentum and the Rotation Group 6.1.1 The Orbital Angular Momentum Operator 6.1.2 The Rotation Group in Three Dimensions 6.1.3 Lie Groups and Lie Algebras 6.2 Rotations in the Hilbert Space 6.2.1 Unitary Representations of the Rotation Group 6.2.2 The Lie Algebra of the Rotation Group 6.2.3 Rotation of the Wave Function 6.2.4 Rotation of Operators 6.2.5 Rotation of Vector Operators 6.2.6 Rotation of Scalar Operators 6.3 Angular Momentum Eigenstates and Eigenvalues 6.3.1 Angular Momentum Ladder Operators 6.3.2 Angular Momentum Eigenvalues 6.3.3 Angular Momentum Eigenstates 6.4 Angular Momentum Eigenfunctions 6.4.1 Spherical Polar Coordinates 6.4.2 Angular Momentum Operators in Spherical Coordinates 6.4.3 The Spherical Harmonics 6.4.4 Uncertainty of Angular-Momentum Operators 7 The Three-Dimensional Schrödinger Equation 7.1 The Radial Schrödinger Equation 7.1.1 Angular Momentum in the Schrödinger Equation 7.1.2 Reduced Wave Function and Effective Potential 7.2 Bound States in Three Dimensions 7.2.1 Normalization of the Radial Wave Function 7.2.2 Rayleigh–Ritz Variational Principle 7.3 The Spherical Potential Well 7.3.1 General Solutions for the Spherical Potential Well 7.3.2 The Ground State of the Spherical Potential Well 7.4 The Coulomb Potential and the Stability of Matter 7.4.1 The Coulomb Potential 7.4.2 An Upper Bound on the ground-state energy of the H-Atom 7.4.3 A Lower Bound from Heisenberg's Uncertainty 7.4.4 A Lower Bound: Sobolev Inequalities 7.5 The Hydrogen Atom 7.5.1 The Radial Schrödinger Equation for the Hydrogen Atom 7.5.2 The Energy Levels of the Hydrogen Atom 7.5.3 The Laguerre Polynomials 7.5.4 Properties of the Hydrogen Atom 8 Spin and Atomic Structure 8.1 The Magnetic Dipole Moment 8.1.1 Classical Magnetic Dipoles 8.1.2 The Magnetic Dipole Moment of the Hydrogen Atom 8.1.3 Magnetic Dipoles in External Magnetic Fields 8.2 Spin 8.2.1 The Stern–Gerlach Experiment 8.2.2 Spin 1/2 8.2.3 Mathematical Formulation of Spin 8.2.4 Spin Measurements 8.2.5 Spinors and the Relation of SO(3) and SU(2) 8.3 The Atomic Structure—Revisited 8.3.1 Total Angular Momentum 8.3.2 Addition of Angular Momenta 8.3.3 Indistinguishable Particles and Pauli Principle 8.3.4 Electronic Orbitals 8.3.5 Term Symbols and Hund's Rules 9 Electromagnetism in Quantum Mechanics 9.1 The Pauli Equation 9.1.1 Hamiltonian for the Interaction with the Electromagnetic Field 9.1.2 Paramagnetic and Diamagnetic Contributions 9.1.3 The Stern–Gerlach Term 9.2 Gauge Symmetries in Quantum Mechanics 9.3 The Aharonov–Bohm effect 9.4 Geometric Phases 9.4.1 Holonomy 9.4.2 The Berry Phase 9.4.3 The Aharonov–Anandan Phase 9.4.4 Spin 1/2 in Adiabatically Rotating Magnetic Field 9.5 A Rush through Differential Geometry and Topology 9.5.1 Differential Geometry 9.5.2 Fibre Bundles 9.5.3 Connection and Curvature 9.6 Topological Interpretation of Physical Effects 9.6.1 Aharonov–Bohm Effect and Topology 9.6.2 Berry Phase and Topology 9.6.3 Dirac Monopole and Topology 10 Perturbative Methods in Quantum Mechanics 10.1 Time-Independent Perturbation Theory 10.1.1 Rayleigh–Schrödinger Perturbation Theory 10.1.2 Non-Degenerate Perturbation Theory 10.1.3 Degenerate Perturbation Theory 10.1.4 Avoided Crossings 10.2 The Fine Structure of the Hydrogen Atom 10.2.1 Relativistic Correction to the Kinetic Energy 10.2.2 Spin-Orbit Correction 10.2.3 The Darwin Correction 10.2.4 Combined Fine-Structure Correction 10.3 The Zeeman Effect 10.3.1 Weak Field—Anomalous Zeeman Effect 10.3.2 Strong Field—Paschen–Back Effect 10.4 The Stark Effect for the Hydrogen Atom 10.4.1 The Wigner–Eckart Theorem 10.4.2 First-Order Stark Effect 10.4.3 Second-Order Stark Effect 10.5 Time-Dependent Perturbation Theory 10.5.1 Time-Dependent Hamiltonians 10.5.2 The Interaction Picture 10.5.3 Fermi's Golden Rule Part II Entanglement and Non-Locality 11 Density Matrices 11.1 Pure States 11.2 Mixed States 11.3 Time Evolution of Density Matrices 11.4 Density Matrices for Quantum Systems in Thermal Equilibrium 11.5 Density Matrices for Two-Level Quantum Systems 11.5.1 Pure and Mixed States of a Single Qubit 11.5.2 The Bloch Decomposition 11.5.3 Spin 1/2 in an External Magnetic Field 11.6 Geometry of the State Space 11.7 Density Matrices for Bipartite Quantum Systems 12 Hidden-Variable Theories 12.1 Historical Overview and Hidden-Variable Basics 12.2 Von Neumann and Additivity of Measurement Values 12.2.1 Von Neumann's Assumption 12.2.2 Bell's Two-Dimensional Hidden-Variable Model 12.3 Contextuality 12.4 Statements Incompatible with Quantum Mechanics 12.5 The Kochen–Specker Theorem 12.5.1 Kochen–Specker Theorem for Spin-1 System 12.5.2 Peres' Nonet for Two Qubits 12.5.3 Mermin's Pentagram for Three Qubits 13 Bell Inequalities 13.1 The EPR Paradox 13.1.1 The EPR Criteria 13.1.2 The EPR Paradox—Aharonov–Bohm Scenario 13.1.3 Bohr's Reply to EPR 13.1.4 Schrödinger's Reply to EPR 13.2 Bell Inequalities—Theory 13.2.1 The Setup 13.2.2 The CHSH Inequality 13.2.3 Bell's Inequality 13.2.4 Wigner's Inequality 13.2.5 Clauser–Horne Inequality 13.3 Bell Inequalities—Experiments 13.3.1 First-Generation Experiments in the 1970s 13.3.2 Second-Generation Experiments in the 1980s 13.3.3 Third-Generation Experiments in the 1990s 13.3.4 Fourth-Generation Experiments After 2000 13.4 Interpretations of Quantum Mechanics 13.4.1 Realism 13.4.2 Information 14 Quantum Teleportation 14.1 Quantum Teleportation 14.1.1 The Teleportation Protocol 14.1.2 Remarks on the Teleportation Protocol 14.2 Experiments on Quantum Teleportation 14.2.1 Milestones of Experimental Teleportation 14.2.2 Experimental Bell-state Measurements via a Beam Splitter 14.3 Primitives of Quantum Communication 14.3.1 Entanglement Swapping 14.3.2 The Formalism of Isometries 14.3.3 Delayed-Choice Entanglement Swapping 14.3.4 Quantum Teleportation versus Classical Information Transfer 14.3.5 The Dense-Coding Protocol 14.4 Quantum Key Distribution 14.4.1 The BB 84 Protocol 14.4.2 The Ekert-91 Protocol 15 Entanglement and Separability 15.1 Composite Quantum Systems 15.1.1 Entanglement and Separability for Pure States 15.1.2 The Schmidt Decomposition 15.1.3 Subsystems and Reduced States 15.1.4 Purification of Quantum States 15.2 Entanglement and Separability 15.2.1 Entanglement and Separability for Mixed States 15.2.2 Quantum Correlations versus Classical Correlations 15.2.3 Bloch Decomposition for Two Qubits 15.2.4 The Peres–Horodecki Criterion 15.3 Entanglement and Non-Locality 15.3.1 Separable States Cannot Violate a Bell Inequality 15.3.2 The CHSH-Operator Criterion 15.3.3 Werner States 15.3.4 Tsirelson's Bound 15.3.5 Hidden Non-Locality 15.4 Separability Criteria from Positive Maps 15.4.1 The Positive-Map Theorem 15.4.2 Proof of the PPT Criterion in Dimension 6 15.4.3 The Reduction Criterion 15.4.4 Isotropic States 15.5 Geometry of Two-Qubit Quantum States 15.5.1 Weyl States and Their Geometric Representation 15.5.2 Entanglement and Separability of Weyl States 16 Quantification and Conversion of Entanglement 16.1 Quantifying Entanglement 16.1.1 Entropy of Entanglement—Quantifying Pure-State Entanglement 16.1.2 LOCC and Majorization 16.1.3 The Asymptotic Setting—Cost and Distillation of Entanglement 16.2 Entanglement Measures for Mixed States 16.2.1 Requirements for Entanglement Measures and Monotones 16.2.2 Entanglement of Formation and Concurrence 16.2.3 Entanglement Measures Based on Distance 16.2.4 Prominent Entanglement Monotones—Negativities 16.3 Entanglement Witnesses 16.3.1 Entanglement-Witness Theorem 16.3.2 Construction of an Entanglement Witness 16.3.3 Bertlmann–Narnhofer–Thirring Theorem 16.3.4 Entanglement Witness for Werner States 16.3.5 Entanglement Witness for Isotropic States 16.4 Entanglement and Separability—A Choice of Factorization 16.4.1 Factorization Algebra 16.4.2 Alice and Bob Factorizations 17 High-Dimensional Quantum Systems 17.1 Bases for Density Matrices 17.1.1 Generalized Gell–Mann Matrix Basis 17.1.2 Polarization-Operator Basis 17.1.3 Weyl-Operator Basis 17.2 Applications of Operator Bases 17.2.1 Generalized Bloch Decomposition for Two Qudits 17.2.2 Isotropic Two-Qudit States 17.2.3 Entanglement Witness in Terms of Spin-1 Operators 17.3 Entanglement of Qutrits, Qudits ... 17.3.1 Two-Parameter Entangled States—Qubits 17.3.2 Two-Parameter Entangled States—Qutrits 17.3.3 Three-Parameter States and Bound Entanglement—Qutrits 17.4 Detecting and Quantifying High-Dimensional Entanglement 17.4.1 Detecting Entanglement in High Dimensions 17.4.2 Bounds on Entanglement Measures 17.4.3 The Schmidt Number 18 Multipartite Entanglement 18.1 Tripartite Systems 18.1.1 Tripartite Pure States 18.1.2 Tripartite Mixed States 18.2 GHZ Theorem à la Mermin 18.3 Multipartite Systems 18.3.1 k-Separability and Genuine Multipartite Entanglement 18.3.2 Specific Multipartite Entangled States 18.3.3 Detecting Genuine Multipartite Entanglement 18.3.4 Characterizing Genuine Multipartite Entanglement 18.4 Entangled Entanglement 18.4.1 Physical Aspects and Mathematical Structure 18.4.2 Construction of Entangled Entanglement Part III Advanced Topics in Modern Quantum Physics 19 Entropy of Classical Systems 19.1 Entropy in Thermodynamics and Statistical Physics 19.1.1 Thermodynamics 19.1.2 Statistical Mechanics 19.2 Shannon Entropy in Classical Information Theory 19.2.1 Shannon Entropy 19.2.2 Shannon Entropy—Message Compression 19.2.3 Shannon Entropy—Measure of Uncertainty 19.2.4 Binary Entropy Function 19.3 Relative Entropy, Joint and Conditional Entropy 19.3.1 Classical Relative Entropy 19.3.2 Joint and Conditional Entropy 19.4 Mutual Information 19.5 Rényi Entropy 20 Quantum Entropy and Correlations in Quantum Information 20.1 Von Neumann Entropy 20.2 Quantum Rényi Entropy 20.3 Quantum Relative Entropy and Quantum Joint Entropy 20.3.1 Quantum Relative Entropy 20.3.2 Quantum Joint Entropy 20.4 Quantum Conditional Entropy and Mutual Information 20.4.1 Quantum Conditional Entropy 20.4.2 Quantum Mutual Information 20.4.3 Entropies of Multipartite Systems 20.5 Conditional and Mutual Amplitude Operators 20.5.1 Conditional Amplitude Operator 20.5.2 Mutual Amplitude Operator 20.5.3 Properties of the Conditional Amplitude Operator 20.6 Negative Conditional Entropy and Geometry of Quantum States 20.6.1 Geometry of the Conditional Entropy and Cerf–Adami Operator 20.6.2 Inequivalence of the CAO and PPT Criteria 20.7 Conditional Rényi Entropy and Non-Locality 21 Quantum Channels and Quantum Operations 21.1 Purification of Quantum States Revisited 21.2 Quantum Operations and Quantum Channels 21.2.1 CPTP Maps 21.2.2 Dephasing and Depolarizing Channel 21.3 Kraus Decomposition 21.3.1 Unitary Dynamics on Larger Hilbert Spaces 21.3.2 Kraus Decomposition for the Amplitude-Damping Channel 21.3.3 Kraus Decomposition and CPTP Maps 21.4 The Church of the Larger Hilbert Space 21.4.1 The Choi–Jamiołkowski Isomorphism 21.4.2 Kraus-Decomposition Theorem 21.4.3 Kraus Decomposition for the Depolarizing Channel 21.4.4 Stinespring Dilation 21.5 Impossible Operations—No Cloning 21.5.1 The No-Cloning Theorem 21.5.2 Cloning with More General Operations 21.5.3 Cloning Classical Information 21.5.4 Imperfect Cloning 22 Open Quantum Systems, Decoherence, Atom-Field Coupling 22.1 Interaction of System and Environment 22.1.1 Open Quantum Systems 22.1.2 Dynamics of Open Quantum Systems—Dynamical Maps 22.2 Markovian Dynamics and Master Equations 22.2.1 The Born–Markov Approximation 22.2.2 Markovian Master Equations—GKLS Equation 22.2.3 Derivation of the GKLS Equation 22.3 Emission and Absorption of Photons 22.3.1 Spontaneous Emission 22.3.2 Emission Process—Amplitude-Damping Channel 22.3.3 Emission-Absorption Process 22.4 The Jaynes–Cummings Model 22.4.1 Two-Level Atom 22.4.2 Quantization of Electric and Magnetic Field 22.4.3 Uncoupled Atom and Field 22.4.4 Interacting Atom and Field 22.4.5 Eigenvalues and Eigenstates of the Jaynes–Cummings Hamiltonian 22.4.6 Atom-Level Probability 23 Quantum Measurements 23.1 Von Neumann Measurements 23.2 Positive Operator-Valued Measures (POVMs) 23.2.1 Mathematical Description of POVMs 23.2.2 Symmetric Informationally Complete POVMs 23.2.3 Naimark Dilation 23.3 Non-Ideal Projective Measurements 23.3.1 A Model for Ideal Projective Measurements 23.3.2 Unbiased Non-Ideal Measurements 23.4 Distinguishing Quantum States 23.4.1 Distinguishing Orthogonal States 23.4.2 Distinguishing Non-Orthogonal States 24 Quantum Metrology 24.1 Quantum Parameter Estimation 24.1.1 Measurement Statistics 24.1.2 Local Parameter Estimation 24.1.3 The Cramér–Rao Bound 24.1.4 Phase Estimation with Individual Qubits 24.2 The Quantum Cramér–Rao Bound and Heisenberg Scaling 24.2.1 The Quantum Cramér–Rao Bound 24.2.2 The Uhlmann Fidelity 24.2.3 The Quantum Fisher Information 24.2.4 Phase Estimation with N-Qubit Probe States 24.3 Bayesian Parameter Estimation 24.3.1 The Bayesian-Estimation Paradigm 24.3.2 Bayesian Phase Estimation with Single Qubits 24.3.3 A Bayesian Cramér–Rao Bound 24.3.4 Bayesian Phase Estimation with N Qubits 25 Quantum States of Light 25.1 Quantization of the Electromagnetic Field 25.1.1 Modes of the Classical Electromagnetic Field 25.1.2 The Quantized Electromagnetic Field 25.1.3 The Fock Space 25.2 Coherent States 25.2.1 Definition and Properties of Coherent States 25.2.2 Coordinate Representation of Coherent States 25.3 Phase Space in Quantum Mechanics—The Wigner Function 25.4 Gaussian States and Operations 26 Particle Physics—Bell Inequalities 26.1 K Mesons 26.1.1 Strangeness and CP 26.2 Analogies and Quasi-Spin 26.3 Entanglement of Strangeness 26.3.1 Time Evolution and Unitarity 26.4 Analogies and Differences for K Mesons 26.5 Bell Inequalities for K Mesons 26.5.1 Bell Inequality for Time Variation 26.5.2 Bell Inequality for Quasi-Spin States—CP Violation 27 Particle Physics—Entanglement and Decoherence 27.1 Decoherence Model for Entangled Particle Systems 27.2 Measurement of Entangled Kaons 27.2.1 Entangled Kaons 27.2.2 Measurement 27.2.3 CPLEAR Experiment 27.3 Connection to Phenomenological Model 27.4 Decoherence and Entanglement Loss for Kaonic Systems 27.4.1 Von Neumann Entropy for K0 0 Systems 27.4.2 Separability and Entanglement of Kaonic Systems 27.4.3 Entanglement of Formation and Concurrence of Kaonic Systems 27.5 Entanglement of Beauty 27.5.1 B mesons 27.5.2 Production of B Mesons 27.5.3 Entanglement of B Mesons 27.6 Decoherence of Entangled Beauty 27.6.1 Decoherence Model 27.6.2 Experiment at KEK-B 27.7 Open Quantum System and Particle Decay 27.7.1 Physical Setup—Open Quantum System 27.7.2 Extended Formalism 27.7.3 Extended Master Equation 27.7.4 Case of Non-Singular Decay References Copyright Notices Index