Precision Limits and Uncertainty Relations for Quantum Multiparameter Estimation

Abstract

Determining optimal measurements for simultaneously estimating multiple parameters of a physical system is an important task for many applications. For estimating parameters in quantum systems, the uncertainty principle fundamentally limits the estimation of incompatible parameters—there is a trade-off in terms of the accuracy with which each parameter can be measured. Limits on the precision of estimation have been formulated, in analogy to classical theory, as lower bounds on a weighted sum of estimation variances, such as the Nagaoka–Hayashi Cramér–Rao bound (NHCRB). Recently, an alternative limit called the Lu–Wang uncertainty relation (LWUR) was proposed based on an existing uncertainty relation. It is a trade-off relation between estimation variances, quantifying the effect of Heisenberg’s uncertainty principle on the joint estimation of two parameters. Of interest for any theoretical estimation limit is its attainability: whether or not there exists a measurement saturating the limit. In this work, I investigate the differences between the limits that the NHCRB and LWUR place on estimation variances. I compare the NHCRB and the minimum sum of variances allowed by the LWUR, using a model problem of estimating orthogonal rotations of a quantum state. I find estimation problems where both approaches place an identical limit on the estimation variances. However, there are also estimation problems where the difference between the limits is infinitely large and the LWUR cannot be attained. In particular, I highlight a flaw in the figure-of-merit of estimation performance used to derive the LWUR—there are estimation problems in which measurements optimising this figure-of-merit do not yield any information about the system’s parameters. To circumvent this, I present an uncertainty relation for parameter estimation based on the NHCRB, which can define an attainable uncertainty relation. Additionally, I show that this approach can provide non-trivial trade-off relations for estimating more than two parameters, something that the LWUR cannot do.