What is special about the Kirkwood-Dirac distributions?

TL;DR

This paper characterizes the unique properties of Kirkwood-Dirac quasiprobability distributions in quantum mechanics, showing they are distinguished by their optimal conditional expectation properties among Born-compatible representations.

Contribution

It proves that Kirkwood-Dirac distributions uniquely have conditional expectations that match the best predictor based on a given observable, among all Born-compatible quasiprobability representations.

Findings

Kirkwood-Dirac distributions have a unique optimal conditional expectation property.

All Born-compatible quasiprobability representations admit a natural conditional expectation.

This property characterizes Kirkwood-Dirac distributions among quasiprobability representations.

Abstract

Among all possible quasiprobability representations of quantum mechanics, the family of Kirkwood-Dirac representations has come to the foreground in recent years because of the flexibility they offer in numerous applications. This raises the question of their characterisation: what makes Kirkwood-Dirac representations special among all possible choices? We show the following. For two observables $\hat A$ and $\hat B$, consider all quasiprobability representations of quantum mechanics defined on the joint spectrum of $\hat A$ and $\hat B$, and that have the correct marginal Born probabilities for $\hat A$ and $\hat B$. For any such Born-compatible quasiprobability representation, we show that there exists, for each observable $\hat{X}$, a naturally associated conditional expectation, given $\hat B$. In addition, among the aforementioned representations, only the Kirkwood-Dirac representation has the following property: its associated conditional expectation of $\hat{X}$ given $\hat{B}$ coincides with the best predictor of $\hat{X}$ by a function of $\hat B$, for all $\hat X$.