Covariant quantum measurements that maximize the likelihood

Abstract

The class of measurements of covariant parameters were derived that are optimal according to the maximum likelihood criterion. The optimization problem was fully resolved in the case of pure input states, under the physically meaningful hypotheses of unimodularity of the covariance group and measurability of the stability subgroup. Some useful mathematical identities for group integrals were derived that are then used to algebraically characterize covariant measurements. It is concluded that the optimal covariant estimation for mixed input states is still an open problem, and an explicit analytical optimization seems a very difficult task.