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arXiv:quant-ph/0410195

Additivity of minimal entropy output for a class of covariant channels

M. Fannes, Bart Haegeman, Milán Mosonyi, Dimitri Vanpeteghem

Published Oct 25, 2004• 24 citations
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Abstract

Additivity of minimal entropy output is proven for the class of quantum channels $Λ_t (A):=t A^{T}+(1-t)τ(A)$ in the parameter range $-2/(d^2-2)\le t \le 1/(d+1)$.

DOI
10.48550/arxiv.quant-ph/0410195

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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).
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.
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.
We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. This class is the one for which the norm of the output is maximized for an output state being a normalized projection. For such channels - which we characterize in their general form - we prove the additivity of the minimal output Renyi entropies with entropic parameters contained in [0, 2], generalizing an argument by Alicki and Fannes, and present several examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.
We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels, the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters α ∊ [0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify various instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.