Minimax quantum state estimation under Bregman divergence

Abstract

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.