Some new additivity results on quantum channels

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 additivity of the minimal output entropy, the multiplicativity of the maximal output p-norm for p ∈ [0,∞] and the additivity of the Holevo capacity are established [1],[2]. Applying the lemma to these results 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 of channels are defined as a linear combination of a channel M satisfying the above property and the completely noisy channel: Φ(ρ) = λM(ρ) + (1− λ) d I, where λ ∈ [− 1 d−1 , 1], d is the dimension of the signal Hilbert space H and I is the identity operator. A similar property was used in a recent paper [3]. An interesting example of this class is the shifted depolarising channel, for which the multiplicativity is shown for p = 2 in [4],[5]. The additivity of the Holevo capacity for the above class of channels is also proved under an additional condition that M is irreducibly covariant. ∗Email: [email protected] 1