Quantum Continuous Variables. A Primer of Theoretical Methods

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Acknowledgments
Preface to the Second Edition
List of Problems
SECTION I: Preliminaries
	Chapter 1: Introduction
		1.1. What is this book about?
			1.1.1. Synopsis
		1.2. How to use this book
		1.3. Mathematical notation, conventions and basic formulae
			1.3.1. Hilbert spaces and operators
			1.3.2. Linear algebra and direct sums
				1.3.2.1. Singular Value Decomposition
				1.3.2.2. Schur Complements
			1.3.3. Compact outer product notation
			1.3.4. Delta functions
			1.3.5. Gaussian integrals
			1.3.6. Miscellanea
	Chapter 2: Quantum Mechanics: Instructions for Use
		2.1. Quantum states and quantum measurements
		2.2. CP-maps and unitary transformations
		2.3. Dynamics: Hamiltonians and master equations
		2.4. Continuous variables
		2.5. Entropies
		2.6. Quantum entanglement
			2.6.1. Entanglement of pure quantum states
			2.6.2. Partial transposition and logarithmic negativity
		2.7. Further reading
SECTION II: Foundations
	Chapter 3: Gaussian States of Continuous Variable Systems
		3.1. Canonical commutation relations
		3.2. Quadratic Hamiltonians and Gaussian states
			3.2.1. Displacement operators
			3.2.2. The symplectic group of linear canonical transformations
			3.2.3. Normal modes
			3.2.4. Normal mode decomposition of a Gaussian state
			3.2.5. The Fock basis
		3.3. Statistical moments of a Gaussian state and the covariance matrix
		3.4. The uncertainty principle
		3.5. Purity and entropies of Gaussian states
		3.6. Further reading
	Chapter 4: Phase Space Methods
		4.1. Coherent states
		4.2. The Fourier–Weyl relation
		4.3. Characteristic functions and quasi-probability distributions
			4.3.1. General properties of the characteristic function
			4.3.2. Quasi-probability distributions
		4.4. Characteristic function of a Gaussian state
		4.5. Further reading
SECTION III: Dynamics
	Chapter 5: Gaussian Operations
		5.1. Gaussian unitary transformations
			5.1.1. Linear displacements
			5.1.2. Symplectic transformations
				5.1.2.1. Passive Transformations: Phase Shifters and Beam Splitters
				5.1.2.2. Squeezing Transformations
		5.2. Tensor products and partial traces of Gaussian states
		5.3. Deterministic Gaussian CP-maps
			5.3.1. Dual Gaussian CP-maps and action onWeyl operators
			5.3.2. Classical mixing
			5.3.3. Losses, attenuators and amplifiers
		5.4. Gaussian measurements
			5.4.1. Homodyne detection
				5.4.1.1. Homodyne Generating Function
			5.4.2. Bell measurements and heterodyne detection
			5.4.3. Ideal general-dyne detections
			5.4.4. Noisy measurements
			5.4.5. Conditional Gaussian dynamics
		5.5. Choi–Jamiolkowski description of the most general Gaussian CP-map
			5.5.1. Choi isomorphism and gate teleportation
			5.5.2. Choi isomorphism in infinite dimension
			5.5.3. The most general Gaussian CP-map
		5.6. Further reading
	Chapter 6: Diffusive Dynamics and Continuous Monitoring
		6.1. Linear and quadratic Hamiltonian dynamics
		6.2. Open diffusive dynamics
			6.2.1. Master equations
			6.2.2. Quantum Langevin equations
		6.3. General-dyne filtering of diffusive dynamics
			6.3.1. Stochastic master equations
		6.4. Linear feedback control
		6.5. Optimal filtering of quantum squeezing
		6.6. Diffusive coherent feedback
			6.6.1. Interferometric feedback
			6.6.2. Optimal squeezing through coherent feedback
		6.7. Further reading
SECTION IV: Correlations
	Chapter 7: Entanglement of Continuous Variable Systems
		7.1. Separability criteria for Gaussian states
			7.1.1. Separability of two-mode Gaussian states
		7.2. A general criterion for Gaussian separability
		7.3. Separability of multi-mode Gaussian states
			7.3.1. 1 vs. n mode Gaussian states
			7.3.2. Locally symmetric states
			7.3.3. Pure and isotropic Gaussian states
		7.4. Logarithmic negativity of Gaussian states
		7.5. Entanglement distillation
			7.5.1. Distilling Gaussian entanglement from non-Gaussian states
		7.6. Higher-order separability criteria
		7.7. Probabilistic entanglement enhancement: Photon subtracted states
		7.8. Quantum nonlocality with continuous variables
		7.9. Further reading
SECTION V: Technologies
	Chapter 8: Quantum Information Protocols with Continuous Variables
		8.1. Quantum teleportation
			8.1.1. Quantum teleportation of Gaussian states
			8.1.2. Classical threshold for coherent states
		8.2. Classical communication over bosonic channels
			8.2.1. Maximum output entropy of phase-insensitive channels: Gaussian extremality
			8.2.2. Minimum output entropy of phase-insensitive channels
			8.2.3. Classical capacity of phase-insensitive channels
		8.3. Quantum metrology
			8.3.1. Gaussian quantum Fisher information
			8.3.2. Quantum estimation with single-mode Gaussian states
		8.4. Quantum key distribution
			8.4.1. Quantum key distribution with coherent states
		8.5. Boson sampling
			8.5.1. Gaussian boson sampling
		8.6. Further reading
	Chapter 9: A Grand Tour of Continuous Variable Platforms
		9.1. Quantum light
			9.1.1. Classical light
			9.1.2. Canonical quantisation
			9.1.3. Quantum electromagnetic fields in free space
			9.1.4. Input-output interfaces and quantum Langevin equations
			9.1.5. Driven cavities
			9.1.6. Linear optical quantum computing
			9.1.7. Nonlinearities and universal computation with continuous variables
		9.2. Atom-light interactions
			9.2.1. The rotating wave approximation
			9.2.2. Dispersive interactions
		9.3. Quantum optomechanics
			9.3.1. Linearised dynamics
			9.3.2. Sideband driving
		9.4. Trapped ions
			9.4.1. The Cirac–Zoller quantum computer
		9.5. Collective excitations
			9.5.1. Atomic ensembles
			9.5.2. Spin waves and magnons
		9.6. Superconducting degrees of freedom and circuit QED
			9.6.1. Circuit QED
		9.7. Cold bosonic atoms in optical lattices
		9.8. Further reading
Appendix A: A note on fermions
Appendix B: Some Notable Facts About the Symplectic Group
	B.1. The orthogonal compact subgroup
	B.2. The singular value decomposition
Appendix C: The Wiener process
Appendix D: Selected Mathematical Lore on Quantum Channels
	D.1. The Holevo bound
	D.2. Entanglement breaking channels
	D.3. Phase-contravariant Gaussian channels
		D.3.1. Conjugate channels
Appendix E: Classical and Quantum Estimation Bounds
	E.1. Classical Fisher information
	E.2. Quantum Fisher information
References
Index