Covariant quantum measurements that maximize the likelihood

We derive the class of covariant measurements that are optimal according to the maximum likelihood criterion. The optimization problem is 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. The general result is applied to the case of covariant state estimation for finite dimension, and to the Weyl- Heisenberg displacement estimation in infinite dimension. We also consider estimation with multiple copies, and analyze the behavior of the likelihood versus the number of copies. A "continuous-variable" analog of the measurement of direction of the angular momentum with two antiparallel spins by Gisin and Popescu is given.