A unifying formalism of generalized conditional expectations (GCEs) for quantum mechanics has recently emerged, but its physical implications regarding the retrodiction of a quantum observable remain controversial. To address the controversy, here I offer operational meanings for a version of the GCEs in the context of quantum parameter estimation. When a quantum sensor is corrupted by decoherence, the GCE is found to relate the operator-valued optimal estimators before and after the decoherence. Furthermore, the error increase, or regret, caused by the decoherence is shown to be equal to a divergence between the two estimators. The real weak value as a special case of the GCE plays the same role in suboptimal estimation – its divergence from the optimal estimator is precisely the regret for not using the optimal measurement. For an application of the GCE, I show that it enables the use of dynamic programming for designing a controller that minimizes the estimation error. For the frequentist setting, I show that the GCE leads to a quantum Rao-Blackwell theorem, which offers significant implications for quantum metrology and thermal-light sensing in particular. These results give the GCE and the associated divergence a natural, useful, and incontrovertible role in quantum decision and control theory.
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
A^ and
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
A^ and
B^ Born-compatible quasiprobability representations. For that purpose, we first define a natural notion of \emph{quantum conditional expectation} of an observable
X^, given an observable
Y^, in a state
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
B^ (or given
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
X^, given an observable
Y^ in a state
ρ^ admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution
μ(x,y) for
X^ and
Y^ in the state
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.
In classical probability theory, the best predictor of a future observation
of a random variable
X, is its expected value
EP[X] when no other
information is available, whereas if information consisting in the observation
of another random variable is available, then the best predictor is is another
random variable
EP[X∣Y]. It is the purpose of this note to explore the
analogue of this in the case of quantum mechanics. we shall see that when the
result of an observation is taken into account by means of a non-commutative
conditional expectation, exactly as in classical prediction theory, some of the
usual paradoxes cease to be so. The moral of this note is that the mystery of
the probabilistic interpretation of quantum mechanics lies in the superposition
principle