In this paper we introduce a lemma. This lemma states that for a unitary covariant channel Ψ and a channel M such that M(ρ0) is of rank one for some state ρ0 ∈ S(H) the additivity of the minimal output entropy of Ψ implies that of Ψ ◦M , and the multiplicativity of the maximal output p-norm of Ψ, that of Ψ ◦M . For the product channel of the depolarising channel and an arbitrary channel the add...
