Minimax quantum state estimation under Bregman divergence

TL;DR

This paper explores minimax estimators for quantum state tomography under Bregman divergences, establishing their asymptotic optimality, the role of covariant measurements, and characterizing minimax measurements for qubits.

Contribution

It generalizes minimax estimation results to Bregman divergences, links covariant measurements to minimaxity, and characterizes minimax measurements for qubits.

Findings

Existence of asymptotically minimax Bayes estimators.

Covariant measurements under unitary 2-designs are minimax.

All spherical 2-designs are minimax measurements for qubits.

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.