Operational meanings of a generalized conditional expectation in quantum metrology

TL;DR

This paper provides operational interpretations of generalized conditional expectations in quantum metrology, linking them to estimation errors, decoherence effects, and optimal control, thereby clarifying their physical significance.

Contribution

It introduces operational meanings for GCEs in quantum parameter estimation, connecting them to decoherence, regret, and control strategies, and establishes a quantum Rao-Blackwell theorem.

Findings

GCE relates pre- and post-decoherence estimators

Decoherence-induced error equals divergence between estimators

GCE enables dynamic programming for optimal control

Abstract

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.