



02530nas a2200205   4500008004100000245006700041210006600108260001400174520187400188100003502062700002502097700002302122700002002145700002302165700002702188700002202215700002202237700002802259856003702287       2024                        eng d00aProperties and Applications of the Kirkwood-Dirac Distribution0 aProperties and Applications of the KirkwoodDirac Distribution  c3/27/20243 a<p>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&#39;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&#39;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&#39; incompatibility.</p>1 aArvidsson-Shukur, David, R. M.1 aBraasch, William, F.1 aDe Bievre, Stephan1 aDressel, Justin1 aJordan, Andrew, N.1 aLangrenez, Christopher1 aLostaglio, Matteo1 aLundeen, Jeff, S.1 aHalpern, Nicole, Yunger  uhttps://arxiv.org/abs/2403.1889902365nas a2200145   4500008004100000245006700041210006000108260001200168490000600180520192900186100002802115700001902143700002002162856003702182       2018                        eng d00aThe quasiprobability behind the out-of-time-ordered correlator0 aquasiprobability behind the outoftimeordered correlator  c04/20180 vA3 a<p>Two topics, evolving rapidly in separate fields, were combined recently: The out-of-time-ordered correlator (OTOC) signals quantum-information scrambling in many-body systems. The Kirkwood-Dirac (KD) quasiprobability represents operators in quantum optics. The OTOC has been shown to equal a moment of a summed quasiprobability. That quasiprobability, we argue, is an extension of the KD distribution. We explore the quasiprobability&#39;s structure from experimental, numerical, and theoretical perspectives. First, we simplify and analyze the weak-measurement and interference protocols for measuring the OTOC and its quasiprobability. We decrease, exponentially in system size, the number of trials required to infer the OTOC from weak measurements. We also construct a circuit for implementing the weak-measurement scheme. Next, we calculate the quasiprobability (after coarse-graining) numerically and analytically: We simulate a transverse-field Ising model first. Then, we calculate the quasiprobability averaged over random circuits, which model chaotic dynamics. The quasiprobability, we find, distinguishes chaotic from integrable regimes. We observe nonclassical behaviors: The quasiprobability typically has negative components. It becomes nonreal in some regimes. The onset of scrambling breaks a symmetry that bifurcates the quasiprobability, as in classical-chaos pitchforks. Finally, we present mathematical properties. The quasiprobability obeys a Bayes-type theorem, for example, that exponentially decreases the memory required to calculate weak values, in certain cases. A time-ordered correlator analogous to the OTOC, insensitive to quantum-information scrambling, depends on a quasiprobability closer to a classical probability. This work not only illuminates the OTOC&#39;s underpinnings, but also generalizes quasiprobability theory and motivates immediate-future weak-measurement challenges.</p>1 aHalpern, Nicole, Yunger1 aSwingle, Brian1 aDressel, Justin  uhttps://arxiv.org/abs/1704.01971