Two-stage quantum estimation and asymptotics of quantum-enhanced transmittance sensing
Phys. Rev. A 113, 052456 – Published 28 May, 2026
DOI: https://doi.org/10.1103/kmnp-11ch
Abstract
We consider the estimation of a single unknown parameter embedded in a quantum state. Quantum Cramér-Rao bound (QCRB) is the ultimate limit of the mean squared error for any unbiased estimator. While it can be achieved asymptotically for a large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. Prior work addresses this paradox using a two-stage approach: in the first stage, a preliminary estimate is obtained by applying, on a vanishing fraction of quantum state copies, a suboptimal measurement that does not depend on the parameter of interest. In the second stage, the preliminary estimate is used to construct the QCRB-achieving measurement, which is then applied to the remaining quantum state copies. This is akin to two-step estimators for classical problems with nuisance parameters. Unfortunately, the original analysis imposes conditions that severely restrict the class of classical estimators applicable to quantum-measurement outcomes, hindering the application of this method. We relax these conditions to substantially broaden the class of usable estimators for single-parameter problems at the cost of slightly weakening the asymptotic properties of the two-stage method. We also account for nuisance parameters. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.
