  We investigate minimax estimators for quantum state tomography under general
Bregman divergences. First, generalizing the work of Komaki et al.
$\href{http://dx.doi.org/10.3390/e19110618}{\textrm{[Entropy 19, 618 (2017)]}}$
for relative entropy, we find that given any estimator for a quantum state,
there always exists a sequence of Bayes estimators that asymptotically perform
at least as well as the given estimator, on any state. Second, we show that
there always exists a sequence of priors for which the corresponding sequence
of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case
risk). Third, by re-formulating Holevo's theorem for the covariant state
estimation problem in terms of estimators, we find that any covariant
measurement is, in fact, minimax (i.e. it minimizes the worst-case risk).
Moreover, we find that a measurement is minimax if it is only covariant under a
unitary 2-design. Lastly, in an attempt to understand the problem of finding
minimax measurements for general state estimation, we study the qubit case in
detail and find that every spherical 2-design is a minimax measurement.
