What Is Special About the Kirkwood–Dirac Distributions? Only They Produce Natural Conditional Expectations

Among the many quasiprobability representations of quantum mechanics, the family of Kirkwood–Dirac (KD) representations has come to the foreground in recent years. Each such KD representation is determined by the choice of two complementary complete sets of commuting observables $\widehat{A}$ and $\widehat{B}$ with respect to which it is Born-compatible, meaning that it correctly reproduces their Born probabilities for every state. In this paper, we identify what property uniquely characterizes the KD representations among all such $\widehat{A}$ and $\widehat{B}$ Born-compatible quasiprobability representations. For that purpose, we first define a natural notion of a quantum conditional expectation of an observable $\widehat{X}$, given an observable $\widehat{Y}$, in a state $\widehat{\rho }$, as a best estimator, and we show that it has the basic properties generally expected of a conditional expectation. We then show that only the KD representations provide a notion of conditional, expectation given $\widehat{B}$ (or given $\widehat{A}$) that coincides with the above quantum conditional expectation. As a byproduct of our analysis, we show a state-dependent no-go theorem. We prove that, if the quantum conditional expectation of an observable $\widehat{X}$, given an observable $\widehat{Y}$ in a state $\widehat{\rho }$ admits an anomalous value (meaning a value lying outside the interval $[x_{min},x_{max}]$), then there cannot exist a Born-compatible joint probability distribution $\mu (x,y)$ for $\widehat{X}$ and $\widehat{Y}$ in the state $\widehat{\rho }$ for which the associated conditional probability $\mu \left(x\right|y)$ yields a conditional expectation that coincides with the quantum conditional expectation. We further apply our findings to revisit a standard model for phase estimation in quantum metrology. We show in particular that, within the real sector of a given KD representation, the classical Fisher information of this phase estimation problem vanishes identically.