Maximal output purity and capacity for asymmetric unital qudit channels

We consider generalizations of depolarizing channels to maps of the form Φ(ρ) = Σk ak Vk ρ Vk† + (1-a)(Tr ρ 1/d I with Vk being unitary and Σkak = a < 1. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions on Vk under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy can be proved for Φ x Ω with Ω arbitrary. We also show that the Holevo capacity need not equal log d - Smin(Φ) as one might expect for a convex combination of unitary conjugations