2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94674A theorem of Davies states that for symmetric quantum states there exists a symmetric POVM maximizing the mutual information. To apply this theorem the representation of the symmetry group has to be irreducible. We obtain a similar yet weaker result for reducible representations. We apply our results to the double trines ensemble and show numerically that for this ensemble the pretty good measurement is optimal.21 pages, 5 figuresQuantum Physicstext