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

TL;DR

This paper demonstrates that the Kirkwood-Dirac distribution uniquely behaves like a genuine probability and provides more accessible information in finite-state quantum systems, outperforming other quasi-probabilities like the Wigner function.

Contribution

It establishes the Kirkwood-Dirac distribution as the most suitable quasi-probability for finite-state quantum systems, highlighting its similarity to true probabilities and its ability to distinguish quantum states.

Findings

Kirkwood-Dirac distribution behaves more like a genuine probability than other quasi-probabilities.

States of two- and three-state systems can be fully distinguished using the Kirkwood-Dirac distribution.

The imaginary part of the distribution is crucial for state distinguishability in two-state systems.

Abstract

We investigate features of the quasi-joint-probability distribution for finite-state quantum systems, especially the two-state and three-state quantum systems, comparing different types of quasi-joint-probability distributions based on the general framework of quasi-classicalization. We show from two perspectives that the Kirkwood-Dirac distribution is the quasi-joint-probability distribution that behaves nicely for the finite-state quantum systems. One is the similarity to the genuine probability and the other is the information that we can obtain from the quasi-probability. By introducing the concept of the possible values of observables, we show for the finite-state quantum systems that the Kirkwood-Dirac distribution behaves more similarly to the genuine probability distribution in contrast to most of the other quasi-probabilities including the Wigner function. We also prove that the states of the two-state and three-state quantum systems can be completely distinguished by the Kirkwood-Dirac distribution of only two directions of the spin and point out for the two-state system that the imaginary part of the quasi-probability is essential for the distinguishability of the state.