Asymptotic Efficiency and Finite Sample Performance of Frequentist Quantum State Estimation

TL;DR

This paper evaluates the finite sample and asymptotic performance of maximum likelihood estimators in quantum state estimation, highlighting limitations of frequentist methods and advocating for Bayesian approaches for improved accuracy.

Contribution

It provides a detailed comparison of finite sample and asymptotic properties of ML estimators in quantum state estimation, revealing limitations and proposing Bayesian methods as a better alternative.

Findings

ML estimators sometimes exclude the true parameter value in finite samples

Asymptotic properties may not hold for realistic sample sizes

Bayesian methods achieve lower estimation errors by incorporating prior knowledge

Abstract

We undertake a detailed study of the performance of maximum likelihood (ML) estimators of the density matrix of finite-dimensional quantum systems, in order to interrogate generic properties of frequentist quantum state estimation. Existing literature on frequentist quantum estimation has not rigorously examined the finite sample performance of the estimators and associated methods of hypothesis testing. While ML is usually preferred on the basis of its asymptotic properties - it achieves the Cramer-Rao (CR) lower bound - the finite sample properties are often less than optimal. We compare the asymptotic and finite-sample properties of the ML estimators and test statistics for two different choices of measurement bases: the average case optimal or mutually unbiased bases (MUB) and a representative set of suboptimal bases, for spin-1/2 and spin-1 systems. We show that, in both cases, the standard errors of the ML estimators sometimes do not contain the true value of the parameter, which can render inference based on the asymptotic properties of the ML unreliable for experimentally realistic sample sizes. The results indicate that in order to fully exploit the information geometry of quantum states and achieve smaller reconstruction errors, the use of Bayesian state reconstruction methods - which, unlike frequentist methods, do not rely on asymptotic properties - is desirable, since the estimation error is typically lower due to the incorporation of prior knowledge.