Quasi-probability representations of quantum theory with applications to quantum information science

Abstract

This article comprises a review of both the quasi-probability representations of infinite-dimensional quantum theory (including the Wigner function) and the more recently defined quasi-probability representations of finite-dimensional quantum theory. We focus on both the characteristics and applications of these representations with an emphasis toward quantum information theory. We discuss the recently proposed unification of the set of possible quasi-probability representations via frame theory and then discuss the practical relevance of negativity in such representations as a criteria for quantumness.

v3: typos fixed, references added

References in corpus (17)

• Negativity and contextuality are equivalent notions of nonclassicality
• Tight informationally complete quantum measurements
• Preparation contextuality powers parity-oblivious multiplexing
• Classicality in discrete Wigner functions
• Discrete Wigner functions and quantum computational speedup
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Experimental determination of a nonclassical Glauber-Sudarshan P function
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Chaos, entanglement and decoherence in the quantum kicked top
• Non-negative Wigner functions in prime dimensions
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Optimal quantum tomography for states, measurements, and transformations
• Quantumness Witnesses
• Bopp operators and phase-space spin dynamics: Application to rotational quantum brownian motion
• Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model
• Inadequacy of a classical interpretation of quantum projective measurements via Wigner functions
• Is there contextuality for a single qubit?

Cited by in corpus (112)

• Negative Quasi-Probability as a Resource for Quantum Computation
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Estimating outcome probabilities of quantum circuits using quasiprobabilities
• Jarzynski-like equality for the out-of-time-ordered correlator
• The quasiprobability behind the out-of-time-ordered correlator
• Quantum Advantage in Postselected Metrology
• Weak Values are Interference Phenomena
• Quantifying the magic of quantum channels
• Quantum fluctuation theorems, contextuality and work quasi-probabilities
• Efficiently computable bounds for magic state distillation
• Pair-cat codes: autonomous error-correction with low-order nonlinearity
• Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation
• Equivalence between contextuality and negativity of the Wigner function for qudits
• Visualizing operators of coupled spin systems
• Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables
• Kirkwood-Dirac nonclassicality, support uncertainty and complete incompatibility
• All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed sets of operational equivalences
• Wigner phase space distribution as a wave function
• Axiomatic and operational connections between the $l_1$-norm of coherence and negativity
• Probabilistic Simulation of Quantum Circuits with the Transformer
• Quasiprobability representations of quantum mechanics with minimal negativity
• Properties and Applications of the Kirkwood-Dirac Distribution
• Negativity of quasiprobability distributions as a measure of nonclassicality
• A hidden variable model for universal quantum computation with magic states on qubits
• Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers
• Quasiprobabilities in quantum thermodynamics and many-body systems
• Quasi-probability distributions for observables in dynamic systems
• Nonnegative subtheories and quasiprobability representations of qubits
• Quasiprobability distributions for quantum-optical coherence and beyond
• Simulating all non-signalling correlations via classical or quantum theory with negative probabilities
• Discrete Wigner Formalism for Qubits and Non-Contextuality of Clifford Gates on Qubit Stabilizer States
• General phase spaces: from discrete variables to rotor and continuum limits
• Superintegrability of Geodesic Motion on the Sausage Model
• Wigner function for a particle in an infinite lattice
• Probability representation of quantum dynamics using pseudostochastic maps
• Discrete Wigner Functions from Informationally Complete Quantum Measurements
• Generalized Delta Functions and Their Use in Quantum Optics
• Which multiphoton states are related via linear optics?
• Classical Simulation of Noncontextual Pauli Hamiltonians
• Entanglement quasiprobabilities of squeezed light
• Spin squeezing and entanglement via finite-dimensional discrete phase-space description
• Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations
• Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)
• The classical-quantum limit
• The role of cohomology in quantum computation with magic states
• Time evolution of coupled spin systems in a generalized Wigner representation
• Towards a complete, continuous, Wigner function for an ensemble of spins or qubits
• Quasistates and quasiprobabilities
• Testing quantum mechanics: a statistical approach
• Fast computation of spherical phase-space functions of quantum many-body states
• Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators
• Quantum space-time marginal problem: global causal structure from local causal information
• Experimental Reconstruction of Entanglement Quasiprobabilities
• Quantum state tomography from sequential measurement of two variables in a single setup
• The magic of universal quantum computing with permutations
• Detecting quantumness in uniform precessions
• Quantum Theory is a Quasi-stochastic Process Theory
• Epistemically restricted phase space representation, weak momentum value, and reconstruction of quantum wave function
• Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable
• States that "look the same" with respect to every basis in a mutually unbiased set
• Minimal informationally complete measurements for probability representation of quantum dynamics
• $\psi$-epistemic interpretations of quantum theory have a measurement problem
• Quantum Bayesian Inference in Quasiprobability Representations
• Observers of quantum systems cannot agree to disagree
• Optical experiment to test negative probability in context of quantum-measurement selection
• Semiclassical Formulation of Gottesman-Knill and Universal Quantum Computation
• On the discrete Wigner function for SU(N)
• Multi-time correlations in the positive-P, Q, and doubled phase-space representations
• How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of $\mathrm{SL}(2,\mathbb{C})$
• Imprecise probability for non-commuting observables
• Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems
• Excluding joint probabilities from quantum theory
• Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well
• Measurement Contextuality and Planck's Constant
• Efficient classical computation of expectation values in a class of quantum circuits with an epistemically restricted phase space representation
• Moments of nonclassicality quasiprobabilities
• Verification of joint measurability using phase-space quasiprobability distributions
• Quantum Bochner's theorem for phase spaces built on projective representations
• Simulations of quantum nonlocality with local negative bits
• Quantum Path Computing: Computing Architecture with Propagation Paths in Multiple Plane Diffraction of Classical Sources of Fermion and Boson Particles
• Wigner representation of the rotational dynamics of rigid tops
• Kenfack Zyczkowski indicator of nonclassicality for two non-equivalent representations of Wigner function of qutrit
• On Quantum Steering and Wigner Negativity
• Ontological models for quantum theory as functors
• Continuous phase-space methods on discrete phase spaces
• Algebraic Probability-Theoretic Characterization of Quantum Correlations
• Geometric view on noneikonal waves
• From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations
• Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations
• Witnessing Bell violations through probabilistic negativity
• Quantum entanglement as an extremal Kirkwood-Dirac nonreality
• Leggett-Garg test of macrorealism using indefinite causal order of measurements
• Measurement of a spin-1 system
• Why quantum correlations are shocking
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space
• Wigner formalism for a particle on an infinite lattice: dynamics and spin
• Quasiprobability Thermodynamic Uncertainty Relation
• Quantum fluctuating theory for one-dimensional shock waves
• Frame representations of qudit quantum mechanics
• Reexamination of the Kochen-Specker theorem: Relaxation of the completeness assumption
• On measures of classicality/quantumness in quasiprobability representations of finite-dimensional quantum systems
• Symmetries and Wigner representations of operational theories
• The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity
• Probing Kirkwood-Dirac nonpositivity and its operational implications via moments
• Circuit connectivity boosts by quantum-classical-quantum interfaces
• Limitations imposed by complementarity
• Generalized conditional expectations for quantum retrodiction and smoothing
• Operational quasiprobabilities for continuous variables
• Exploring entanglement resource in Si quantum dot systems with operational quasiprobability approach
• Efficient measure of information backflow with a quasistochastic process
• Maximally symmetric stabilizer MUBs in even prime-power dimensions
• Ideal stochastic process modeling with post-quantum quasiprobabilistic theories