Comparison of estimation limits for quantum two-parameter estimation

TL;DR

This paper introduces a framework to compare different quantum multiparameter estimation bounds, revealing their differences in attainable precision and applicability, with implications for quantum measurement theory.

Contribution

It provides a general framework for directly comparing quantum estimation bounds, analyzing their attainability, and clarifying their physical significance.

Findings

Nagaoka Cramér-Rao bound and Lu--Wang uncertainty relation can give different precision limits.

Examples show cases where both bounds are attainable and where Lu--Wang is not.

Unattainability relates to the figure of merit used in the bounds.

Abstract

Measurement estimation bounds for local quantum multiparameter estimation, which provide lower bounds on possible measurement uncertainties, have so far been formulated in two ways: by extending the classical Cram\'er--Rao bound (e.g., the quantum Cram\'er--Rao bound and the Nagaoka Cram'er--Rao bound) and by incorporating the parameter estimation framework with the uncertainty principle, as in the Lu--Wang uncertainty relation. In this work, we present a general framework that allows a direct comparison between these different types of estimation limits. Specifically, we compare the attainability of the Nagaoka Cram\'er--Rao bound and the Lu--Wang uncertainty relation, using analytical and numerical techniques. We show that these two limits can provide different information about the physically attainable precision. We present an example where both limits provide the same attainable precision and an example where the Lu--Wang uncertainty relation is not attainable even for pure states. We further demonstrate that the unattainability in the latter case arises because the figure of merit underpinning the Lu--Wang uncertainty relation (the difference between the quantum and classical Fisher information matrices) does not necessarily agree with the conventionally used figure of merit (mean squared error). The results offer insights into the general attainability and applicability of the Lu--Wang uncertainty relation. Furthermore, our proposed framework for comparing bounds of different types may prove useful in other settings.