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Note on multiple additivity of minimal entropy output of extreme SU(d)-covariant channels

Robert Alicki, M. Fannes

Published Jul 5, 2004• 12 citations
OpenAlex

Abstract

We give an elementary self-contained proof that the minimal entropy output of arbitrary products of channels $\\rho \\mapsto \\frac{1}{d-1}(1-\\rho^T)$ is additive.

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arXiv:quant-ph/0403072•Mar 9, 2004•19 citations
We give an elementary self-contained proof that the minimal entropy output of arbitrary products of channels rhomapstofrac1d−1(1−rhoT)\\rho \\mapsto \\frac{1}{d-1}(1-\\rho^T)rhomapstofrac1d−1(1−rhoT) is additive.
Additivity of minimal entropy output is proven for the class of quantum channels Λt(A):=tAT+(1−t)τ(A)Λ_t (A):=t A^{T}+(1-t)τ(A)Λt​(A):=tAT+(1−t)τ(A) in the parameter range −2/(d2−2)≤t≤1/(d+1)-2/(d^2-2)\le t \le 1/(d+1)−2/(d2−2)≤t≤1/(d+1).
In this paper we introduce a lemma. This lemma states that for a unitary covariant channel Psi\\PsiPsi and a channel MMM such that M(rho0)M(\\rho_0)M(rho0​) is of rank one for some state rho0\\rho_0rho0​ the additivity of the minimal output entropy of Psi\\PsiPsi implies that of PsicircM\\Psi \\circ MPsicircM, and the multiplicativity of the maximal output ppp-norm of Psi\\PsiPsi, that of PsicircM\\Psi \\circ MPsicircM. 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 ppp-norm hold. This class includes the shifted depolarising channel.