  We study informationally overcomplete measurements for quantum state
estimation so as to clarify their tomographic significance as compared with
minimal informationally complete measurements. We show that informationally
overcomplete measurements can improve the tomographic efficiency significantly
over minimal measurements when the states of interest have high purities.
Nevertheless, the efficiency is still too limited to be satisfactory with
respect to figures of merit based on monotone Riemannian metrics, such as the
Bures metric and quantum Chernoff metric. In this way, we also pinpoint the
limitation of nonadaptive measurements and motivate the study of more
sophisticated measurement schemes. In the course of our study, we introduce the
best linear unbiased estimator and show that it is equally efficient as the
maximum likelihood estimator in the large-sample limit. This estimator may
significantly outperform the canonical linear estimator for states with high
purities. It is expected to play an important role in experimental designs and
adaptive quantum state tomography besides its significance to the current
study.
