What is special about the Kirkwood-Dirac distributions?
Matéo Spriet, Christopher Langrenez, Raymond Brummelhuis, Stephan De Bièvre
TL;DR
This work characterizes what makes Kirkwood-Dirac (KD) quasiprobabilities special among Born-compatible representations for two noncommuting observables. It juxtaposes classical and quantum notions of conditional expectation, introducing two quantum constructions via minimization (left and right) and via quasiprobability representations, and shows KD representations uniquely align the representation-based conditional expectations with the corresponding quantum best-predictor. The main result demonstrates that among all Born-compatible frames on $\sigma(\hat{A})\times\sigma(\hat{B})$, only the left and right KD representations yield conditional expectations that match the optimal estimator for all observables, with pull-through identities characterizing this compatibility. This clarifies the foundational role of KD distributions in quantum conditioning and has implications for their use in applications relying on conditional structure in quantum systems.
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 $\hat X$.