Note on Multiple Additivity of Minimal Renyi Entropy Output of the Werner-Holevo Channels

We give an elementary self-contained proof that the minimal entropy output of arbitrary products of channels ρ → 1 d−1 ½−ρ T is additive. This paper is concerned with efficiency of transmission of classical information through a particular quantum channel. Generally, the minimal entropy output of a general quantum channel Γ, given in terms of a completely positive trace-preserving map on the d-dimensional density matrices , quantifies the minimal noise produced by the channel in terms of the von Neumann entropy of transformed pure states MEO Γ := inf ϕ, ϕ=1 S Γ |ϕϕ| where S(ρ) is the von Neumann entropy of a density matrix ρ. Recalling the definition of the p-Renyi entropy of a density matrix ρ, p > 1 S p (ρ) := − 1 p − 1 log Tr ρ p , the minimal p-Renyi entropy output is MEO p Γ := inf ϕ, ϕ=1 S p Γ(|ϕϕ|). (1) Obviously, lim p↓1 MEO p Γ = MEO Γ and we can therefore extend the notation MEO p to p = 1. The p-Renyi entropy of a density matrix is directly related to its p-norm. Let X be a general, not necessarily square, complex matrix. The p-norm of X is given by X p := Tr|X| p 1 p = j χ r j 1 p 1