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 $\hat ρ$ , 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 $\hat ρ$ admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution $μ(x,y)$ for $\hat X$ and $\hat Y$ in the state $\hat ρ$ for which the associated conditional probability $μ(x|y)$ 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.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 $\hat ρ$, 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 $\hat ρ$ admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution $μ(x,y)$ for $\hat X$ and $\hat Y$ in the state $\hat ρ$ for which the associated conditional probability $μ(x|y)$ 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.

quant-pharXiv:2511.01996SymmetryNov 5, 20250 citations

Conditional expectation and repeated measurements of continuous quantum observables

Masanao Ozawa

Lecture notes in mathematicsJan 1, 198323 citations

Conditional expectations in generalized probability theory

H. Cycon, K. -E. Hellwig

Expectations have been considered as dual objects of instruments in several papers on generalized probability theory and quantum theory. Here, we relate a generalized conditional (GC) expectation to a given instrument and a given state by two requirements, which are analogous to the axioms by which the classical conditional expectation is related to a given sub-σ-algebra and a given probability measure. In examples we illustrate the close similarity of GC expectations with classical conditional expectations. Eventually, we introduce a rich class of quantum stochastic processes, which are Markovian.Expectations have been considered as dual objects of instruments in several papers on generalized probability theory and quantum theory. Here, we relate a generalized conditional (GC) expectation to a given instrument and a given state by two requirements, which are analogous to the axioms by which the classical conditional expectation is related to a given sub-σ-algebra and a given probability measure. In examples we illustrate the close similarity of GC expectations with classical conditional expectations. Eventually, we introduce a rich class of quantum stochastic processes, which are Markovian.

Journal of Mathematical PhysicsJun 1, 197717 citations

\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 ρ

, 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 ρ

admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution

μ(x,y)

for

\hat X

and

\hat Y

in the state

\hat ρ

for which the associated conditional probability

μ(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.