Some new additivity results on quantum channels

In this paper we introduce a lemma. This lemma states that for a unitary covariant channel $\Psi$ and a channel $M$ such that $M(\rho_0)$ is of rank one for some state $\rho_0$ the additivity of the minimal output entropy of $\Psi$ implies that of $\Psi \circ M$, and the multiplicativity of the maximal output $p$-norm of $\Psi$, that of $\Psi \circ M$. Applying the lemma to the depolarising channel we produce a new class of memoryless quantum channels for which the additivity of the minimal output entropy and the multiplicativity of the maximal output $p$-norm hold. This class includes the shifted depolarising channel.