Properties and Applications of the Kirkwood-Dirac Distribution
arxiv(2024)
摘要
The most famous quasi-probability distribution, the Wigner function, has
played a pivotal role in the development of a continuous-variable quantum
theory that has clear analogues of position and momentum. However, the Wigner
function is ill-suited for much modern quantum-information research, which is
focused on finite-dimensional systems and general observables. Instead, 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 and
observables' incompatibility.
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