Modern Quantum Theory: From Quantum Mechanics to Entanglement and Quantum Information

✍ Scribed by Reinhold Bertlmann, Nicolai Friis

• Publisher: Oxford University Press
• Year: 2023
• Tongue: English
• Leaves: 1028
• Category: Library

✦ Table of Contents

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