Lemma about Additivity and Some Variation of the Depolarising Channel

In this paper, we introduce two results. First we prove Lemma about the additivity and the multiplicativity. 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 . Next we consider a new class of memoryless quantum channels. In a recent paper [1], C. King analysed the quantum memoryless depolarising channel and established the multiplicativity of the p-norm for p ∈ [0,∞], the additivity of the minimal output entropy and the additivity of the Holevo capacity for the product channel of this channel and an arbitrary channel. We apply the Lemma to his result and also extend his method to produce the multiplicativity and the additivities for the new class of channels. The channels Φ are defined as a convex combination of a channel M satisfying the above property and the completely noisy channel: Φ(ρ) = λM(ρ)+(1−λ) d I, where λ ∈ [0, 1], d is the dimension of the signal Hilbert space H, I is the identity operator. A similar property was used in a recent paper by Wolf and Eisert [2]. In our paper the multiplicativity of p-norm and the additivity of the minimal output entropy for a channel from the above class are established without additional assumptions, while the additivity of the Holevo capacity is proved under an additional condition that the channel M is irreducibly covariant.