  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.
