Phase-space formulation of quantum mechanics and quantum state reconstruction for physical systems with Lie-group symmetries

Abstract

We present a detailed discussion of a general theory of phase-space distributions, introduced recently by the authors [J. Phys. A {\bf 31}, L9 (1998)]. This theory provides a unified phase-space formulation of quantum mechanics for physical systems possessing Lie-group symmetries. The concept of generalized coherent states and the method of harmonic analysis are used to construct explicitly a family of phase-space functions which are postulated to satisfy the Stratonovich-Weyl correspondence with a generalized traciality condition. The symbol calculus for the phase-space functions is given by means of the generalized twisted product. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction method based on measurements of displaced projectors, which comprises a number of recently proposed quantum-optical schemes and is also related to the standard methods of signal processing. A general group-theoretic description of this method is developed using the technique of harmonic expansions on the phase space.

REVTeX, 18 pages, no figures

Cited by in corpus (101)

• Control of quantum phenomena: Past, present, and future
• Alternative commutation relations, star products and tomography
• Quasi-probability representations of quantum theory with applications to quantum information science
• Wigner Functions for Arbitrary Quantum Systems
• Generalized spin mapping for quantum-classical dynamics
• Spin-mapping approach for nonadiabatic molecular dynamics
• Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase space approach
• Variational-State Quantum Metrology
• Wigner functions, squeezing properties and slow decoherence of atomic Schrodinger cats
• Spin tomography
• Exact number conserving phase-space dynamics of the M-site Bose-Hubbard model
• Quantum phase space measurement and entanglement validation made easy
• Visualizing operators of coupled spin systems
• Discrete Moyal-type Representations for a Spin
• Overview of the phase space formulation of quantum mechanics with application to quantum technologies
• Continuous phase-space representations for finite-dimensional quantum states and their tomography
• Generation of time-frequency grid state with integrated biphoton frequency combs
• New Phase Space Formulations and Quantum Dynamics Approaches
• Quantum reconstruction of an intense polarization squeezed optical state
• A general approach to quantum mechanics as a statistical theory
• The Pauli Equation for Probability Distributions
• Wigner tomography of multispin quantum states
• Wigner-Weyl isomorphism for quantum mechanics on Lie groups
• Quantum polarization tomography of bright squeezed light
• Wigner functions for curved spaces I: On hyperboloids
• SU(N)-symmetric quasi-probability distribution functions
• Dynamical phase transitions in dissipative quantum dynamics with quantum optical realization
• Hybrid discrete-continuous truncated Wigner approximation for driven, dissipative spin systems
• Collective Radiative Interactions in the Discrete Truncated Wigner Approximation
• Rapid and slow decoherence in conjunction with dissipation in a system of two-level atoms
• Quantum phase-space representation for curved configuration spaces
• Bopp operators and phase-space spin dynamics: Application to rotational quantum brownian motion
• Wigner distributions and quantum mechanics on Lie groups: the case of the regular representation
• Simple Minimal Informationally Complete Measurements for Qudits
• Wigner negativity in spin-$j$ systems
• Inverted spectroscopy and interferometry for quantum-state reconstruction of systems with SU(2) symmetry
• Wigner function for the orientation state
• Spin coherent states in NMR quadrupolar system: experimental and theoretical applications
• Quantum entanglement from classical trajectories
• On detailed balance in nonadiabatic dynamics: From spin spheres to equilibrium ellipsoids
• Measurement of complete and continuous Wigner functions for discrete atomic systems
• Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics
• The quantum state vector in phase space and Gabor's windowed Fourier transform
• Predicting correlations in superradiant emission from a cascaded quantum system
• Time evolution of coupled spin systems in a generalized Wigner representation
• Wigner distributions for non Abelian finite groups of odd order
• Fast computation of spherical phase-space functions of quantum many-body states
• Orientational order parameters for arbitrary quantum systems
• Visualizing entanglement in atoms and molecules
• Quantum symmetries and the Weyl-Wigner product of group representations
• JIMWLK Evolution, Lindblad Equation and Quantum-Classical Correspondence
• Quantum phase-space description of light polarization
• Non-Markovian Quantum State Diffusion for Spin Environments
• Quantum speed limits in arbitrary phase spaces
• On the family of Wigner functions for N-level quantum system
• Quantum decoherence in the rotation of small molecules
• The Weyl-Wigner-Moyal formalism on a discrete phase space. I. A Wigner function for a nonrelativistic particle with spin
• Exact variational dynamics of the multimode Bose-Hubbard model based on SU(M) coherent states
• Phase-space path-integral calculation of the Wigner function
• Quantum metrology and coherence
• $SU(1,1)$ covariant $s$-parametrized maps
• Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems
• Phase Spaces, Parity Operators, and the Born-Jordan Distribution
• Qutrit squeezing via semiclassical evolution
• Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems
• How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of $\mathrm{SL}(2,\mathbb{C})$
• Testing Bell's inequality with two-level atoms via population spectroscopy
• Generalised coherent states and the diagonal representation for operators
• Non-classicality of positive relativistic Wigner function
• Discrete phase-space mappings, tomographic condition and permutation invariance
• Simultaneous Measurements of Noncommuting Observables. Positive Transformations and Instrumental Lie Groups
• Continuous phase-space methods on discrete phase spaces
• Non-adiabatic Ring Polymer Molecular Dynamics in the Phase Space of the SU(N) Lie Group
• Transition probabilities and transition rates in discrete phase space
• Spin relaxation in phase space
• Permutation symmetry for the tomographic probability distribution of a system of identical particles
• Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group
• Quantum and statistical mechanics in open systems: theory and examples
• Polytopes of Absolutely Wigner Bounded Spin States
• Quantum canonical transformations in Weyl-Wigner-Groenewold-Moyal formalism
• Star product and ordered moments of photon creation and annihilation operators
• Coherent-state path integral calculation of the Wigner function
• Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems
• Schrödinger model and Stratonovich-Weyl correspondence for Heisenberg motion groups
• Stellar representation of extremal Wigner-negative spin states
• Phase-Space Methods for Many-Body Quantum Optics
• Frame representations of qudit quantum mechanics
• Visualization of correlations in hybrid quantum systems
• Lueders Theorem for Coherent-State POVMs
• Correspondence rules for Wigner functions over SU(3)/U(2)
• SU(3) truncated Wigner approximation for strongly interacting Bose gases
• Macroscopic Schrödinger-cat states of nonequilibrium electrons induced by cat-state optical driving and projective measurements on the light field
• Particles as superoscillations of spacetime with a nonlocal metric?
• Fermion quadrature bases for Wigner functionals
• Classical simulation of circuits with realistic odd-dimensional Gottesman-Kitaev-Preskill states
• Quantum information in Riemannian spaces
• Weyl calculus on the Fock space and Stratonovich-Weyl correspondence for Heisenberg motion groups
• Operational and harmonic-analytic aspects of quasi-probability distributions
• Noise-tailored Constructions for Spin Wigner Function Kernels
• On deformed quantum mechanical schemes and $\star$-value equations based on the space-space noncommutative Heisenberg-Weyl group
• A new information metric and a possible higher bound for a class of measurements in the Quantum Estimation Problem