Asymptotic Efficiency and Finite Sample Performance of Frequentist Quantum State Estimation

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.