Properties and Applications of the Kirkwood-Dirac Distribution

Abstract

Recent years have seen the Kirkwood-Dirac (KD) distribution come to the forefront as a powerful quasi-probability distribution for analysing quantum mechanics. The KD distribution allows tools from statistics and probability theory to be applied to problems in quantum-information processing. A notable difference to the Wigner function is that the KD distribution can represent a quantum state in terms of arbitrary observables. This paper reviews the KD distribution, in three parts. First, we present definitions and basic properties of the KD distribution and its generalisations. Second, we summarise the KD distribution's extensive usage in the study or development of measurement disturbance; quantum metrology; weak values; direct measurements of quantum states; quantum thermodynamics; quantum scrambling and out-of-time-ordered correlators; and the foundations of quantum mechanics, including Leggett-Garg inequalities, the consistent-histories interpretation and contextuality. We emphasise connections between operational quantum advantages and negative or non-real KD quasi-probabilities. Third, we delve into the KD distribution's mathematical structure. We summarise the current knowledge regarding the geometry of KD-positive states (the states for which the KD distribution is a classical probability distribution), describe how to witness and quantify KD non-positivity, and outline relationships between KD non-positivity, coherence and observables' incompatibility.

42 pages, 14 figures; as published in NJP

References in corpus (75)

• Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification
• Fluctuation theorems: Work is not an observable
• A simple test for hidden variables in spin-1 system
• Experimentally testable state-independent quantum contextuality
• Using entanglement against noise in quantum metrology
• Experimental joint weak measurement on a photon pair as a probe of Hardy's Paradox
• Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System
• Is the quantum state real? An extended review of $\psi$-ontology theorems
• Negativity and contextuality are equivalent notions of nonclassicality
• Information Scrambling in Computationally Complex Quantum Circuits
• Weak values and the Leggett-Garg inequality in solid-state qubits
• Anomalous Weak Values Are Proofs of Contextuality
• Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair
• Optimizing the Signal to Noise Ratio of a Beam Deflection Measurement with Interferometric Weak Values
• Kochen-Specker Contextuality
• Precision frequency measurements with interferometric weak values
• Non-equilibrium quantum fluctuations of work
• Measurement of many-body chaos using a quantum clock
• Jarzynski-like equality for the out-of-time-ordered correlator
• Probing quantum information propagation with out-of-time-ordered correlators
• Weak Values are Interference Phenomena
• Weak Value Amplification Can Outperform Conventional Measurement in the Presence of Detector Saturation
• Quantum theory of successive projective measurements
• Approaching quantum-limited metrology with imperfect detectors by using weak-value amplification
• Signatures of quantum behavior in single-qubit weak measurements
• Information Scrambling and Chaos in Open Quantum Systems
• Negative quasiprobabilities enhance phase estimation in quantum-optics experiment
• Quantifying quantum coherence via nonreal Kirkwood-Dirac quasiprobability
• Power-recycled weak-value-based metrology
• Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables
• Quantum circuits for measuring weak values, Kirkwood--Dirac quasiprobability distributions, and state spectra
• Probing Quantum Interference Effects in the Work Distribution
• Direct measurement of a nonlocal entangled quantum state
• Kirkwood-Dirac nonclassicality, support uncertainty and complete incompatibility
• Quantum scrambling with classical shadows
• Uncertainty limits for quantum metrology obtained from the statistics of weak measurements
• Uniqueness of noncontextual models for stabilizer subtheories
• Contextual advantages and certification for maximum confidence discrimination
• Noncommuting conserved charges in quantum thermodynamics and beyond
• Experimental observation of thermalization with noncommuting charges
• Quasiprobabilities in quantum thermodynamics and many-body systems
• Enhanced weak-value amplification via photon recycling
• What is nonclassical about uncertainty relations?
• Contextuality and Wigner negativity are equivalent for continuous-variable quantum measurements
• Compressive Wavefront Sensing with Weak Values
• Projective measurements can probe non-classical work extraction and time-correlations
• Class of quasiprobability distributions of work and initial quantum coherence
• Benchmarking Information Scrambling
• Operational interpretation and estimation of quantum trace-norm asymmetry based on weak value measurement and some bounds
• How quantum paradoxes originate from the non-classical statistics of physical properties related to each other by half-periodic transformations
• What is the most general class of quasiprobabilities of work?
• Work statistics, quantum signatures and enhanced work extraction in quadratic fermionic models
• Quantum coherence as asymmetry from complex weak values
• Quantum Physics and Human Language
• How to build Hamiltonians that transport noncommuting charges in quantum thermodynamics
• Virtual quantum broadcasting
• Exploring quasiprobability approach to quantum work in the presence of initial coherence: Advantages of the Margenau-Hill distribution
• Universal Optimal Quantum Correlator
• Non-Abelian transport distinguishes three usually equivalent notions of entropy production
• Quantum Mechanics of Successive Measurements with Arbitrary Meter Coupling
• Contextuality, Coherences, and Quantum Cheshire Cats
• Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)
• Characterizing the geometry of the Kirkwood-Dirac positive states
• Classification of incompatibility for two orthonormal bases
• Quantum simulation of operator spreading in the chaotic Ising model
• General quantum correlation from nonreal values of Kirkwood-Dirac quasiprobability over orthonormal product bases
• Theory and experiment for resource-efficient joint weak-measurement
• Demonstration of the quantum principle of least action with single photons
• Nonclassical advantage in metrology established via quantum simulations of hypothetical closed timelike curves
• Characterization of the non-classical relation between measurement outcomes represented by non-orthogonal quantum states
• Direct measurement of ultrafast temporal wavefunctions
• Counterfactuality, back-action, and information gain in multi-path interferometers
• Quantitative relations between different measurement contexts
• Direct tomography of quantum states and processes via weak measurements of Pauli spin operators on an NMR quantum processor
• Increase the Efficiency of Post-selection in Direct Measurement of Quantum Wave Function

Cited by in corpus (32)

• Characterizing the geometry of the Kirkwood-Dirac positive states
• Quantum speed limit for Kirkwood-Dirac quasiprobabilities
• Snapshotting Quantum Dynamics at Multiple Time Points
• Quantum unital Otto heat engines: using Kirkwood-Dirac quasi-probability for the engine's coherence to stay alive
• Convex roofs witnessing Kirkwood-Dirac nonpositivity
• Multipartite quantum states over time from two fundamental assumptions
• Non-positive energy quasidistributions in coherent collision models
• Compatibility of Quantum Measurements and the Emergence of Classical Objectivity
• Quasiprobability distributions with weak measurements
• Experimental demonstration of generalized quantum fluctuation theorems in the presence of coherence
• Impact of quantum coherence on the dynamics and thermodynamics of quenched free fermions coupled to a localized defect
• Quasiprobability Thermodynamic Uncertainty Relation
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space
• Contextuality Can be Verified with Noncontextual Experiments
• Exploring weak value arguments and Bargmann invariants in $N$-level quantum systems through the Majorana symmetric representation
• Quantum Coherence and Anomalous Work Extraction in Qubit Gate Dynamics
• The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity
• An Elementary Characterization of Bargmann Invariants
• Probing Kirkwood-Dirac nonpositivity and its operational implications via moments
• Numerical ranges of Bargmann invariants
• Quasi-probability distribution of work in a measurement-based quantum Otto engine
• Experimental evidence for the physical delocalization of individual photons in an interferometer
• Quantum Non-Demolition Measurements and Leggett-Garg inequality
• Quantum advantage from negativity of a work quasiprobability distribution
• Estimation of multivariate traces of states given partial classical information
• Quantum coherence and negative quasi probabilities in a contextual three-path interferometer
• Orthogonalization speed-up from quantum coherence after a sudden quench
• Efficient measure of information backflow with a quasistochastic process
• Almost no experiments have classical Kirkwood-Dirac representations
• Hermitian Kirkwood-Dirac real operators for discrete Fourier transformations
• Measure of set imaginarity
• Spin coherence scale: operator-ordering sensitivity beyond the Heisenberg-Weyl group