What is special about the Kirkwood-Dirac distributions?
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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 $\hat X$.