Encoding parameters by measurement: Forgetting can be better in quantum metrology

TL;DR

This paper explores quantum parameter estimation where encoding is through measurements, revealing that in many cases, discarding measurement outcomes can improve estimation precision, and it establishes criteria for optimal measurement strategies.

Contribution

It introduces a framework for quantum parameter estimation via measurements, deriving conditions when ignoring outcomes enhances precision and when the quantum Cramér-Rao bound is attainable.

Findings

Forgetting measurement outcomes often yields higher precision.

Derived necessary and sufficient conditions for optimal measurement strategies.

Identified when the quantum Cramér-Rao bound is valid and achievable.

Abstract

We introduce quantum parameter estimation with the encoding being via a quantum measurement. We quantify the precision for estimating parameters characterizing a general two-outcome qubit measurement, considering two cases: when the outcomes of the encoding measurement are recorded and when the same are ignored. We find that in a large variety of such estimation scenarios, forgetting the outcomes yields higher precision. We derive a necessary criterion under which remembering the measurement outcomes provides better precision in comparison to the outcome-forgotten strategy. Furthermore, we establish a necessary and sufficient criterion for the simultaneous estimation of two parameters encoded by an arbitrary quantum process, including those involving measurements, using qubit probes, and find when the quantum Cram\'er$-$Rao bound is valid and achievable. For simultaneous estimation of two parameters characterizing the measurement, we find that the achievable quantum Cram\'er$-$Rao bound can be a valid precision bound only when the measurement direction depends on the parameters of interest.