Advantages of the Kirkwood–Dirac distribution among general quasi-probabilities on finite-state quantum systems

Abstract

Abstract We investigate the properties of quasi-joint-probability (QJP) distributions on finite-state quantum systems, especially two- and three-state systems, based on the general framework of quantum/quasi-classical representations. We show that the Kirkwood–Dirac distribution is a prime candidate among the QJP distributions that behave well in view of the following two perspectives: the information contained in the QJP distribution and its affinity to genuine joint-probability distributions. Regarding the first criterion, we show that the Kirkwood–Dirac distributions on two- and three-state quantum systems yield faithful quasi-classical representations of quantum states with a minimal set of observables, namely a pair of two different directions of spin, and thereby point out that in general the imaginary parts of the QJP distributions play essential roles in this respect. As for the second criterion, we prove that the Kirkwood–Dirac distributions on finite-state quantum systems are supported on the product set of the spectra of the quantum observables involved.