Compatibility in Multiparameter Quantum Metrology

TL;DR

This paper introduces the concept of compatibility in multiparameter quantum metrology, establishing a framework to analyze when multiple parameters can be estimated simultaneously with optimal precision, and explores theoretical bounds and examples.

Contribution

It defines and rigorously analyzes the concept of compatibility in quantum metrology, clarifying ambiguities and deriving new bounds for simultaneous parameter estimation.

Findings

Compatibility allows for optimal simultaneous estimation of multiple parameters.

Theoretical bounds are established for phase and dephasing estimation.

Examples illustrate differences between unitary and non-unitary estimation.

Abstract

Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous theoretical framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and non-unitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.