Matéo Spriet | Searcharxiv

arXiv2026

Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

We define the Kirkwood-Dirac quasiprobability representation of quantum mechanics associated with the Fourier transform over second countable locally compact abelian groups. We discuss its link with the Kohn-Nirenberg quantization of the phase space $G\times \widehat{G}$. We use it to argue that in this abstract setting the Wigner-Weyl quantization, when it exists, can still be interpreted as a symmetric ordering. Then, we identify all generalized (non-normalizable) pure states having a positive Kirkwood-Dirac distribution. They are, up to the natural action of the Weyl-Heisenberg group, Haar measures on closed subgroups. This generalizes a result known for finite abelian groups. We then show that the classical fragment of quantum mechanics associated with the Kirkwood-Dirac distribution is non-trivial if and only if the group has a compact identity component. Finally, we provide for connected compact abelian groups a complete geometric description of this classical fragment.

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arXiv2025

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 $\hat A$ and $\hat B$ with respect to which it is Born-compatible, meaning that it correctly reproduces their Born probabilities for every state. We identify in this paper what property uniquely characterizes the KD representations among all such $\hat A$ and $\hat B$ Born-compatible quasiprobability representations. For that purpose, we first define a natural notion of \emph{quantum conditional expectation} of an observable $\hat X$, given an observable $\hat Y$, in a state $\hat \rho$, 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 $\hat B$ (or given $\hat A$) 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 $\hat X$, given an observable $\hat Y$ in a state $\hat \rho$ admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution $\mu(x,y)$ for $\hat X$ and $\hat Y$ in the state $\hat \rho$ for which the associated conditional probability $\mu(x|y)$ 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.

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