Symmetric Measurements Attaining the Accessible Information

A theorem of Davies shows that for symmetric quantum states there exists a symmetric positive operator-valued measure (POVM) that has a simple structure and that maximizes the mutual information. To apply this theorem, the representation of the symmetry group has to be irreducible. In the following discussion, similar yet weaker results are obtained for reducible representations. The results are applied to the double trines and it is shown with numerical methods that for these states the pretty good measurement is optimal.