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arXiv:quant-ph/0412034 · International Journal of Quantum Information

ADDITIVITY FOR TRANSPOSE DEPOLARIZING CHANNELS

Nilanjana Datta, Alexander Semenovich Holevo, Yuri Suhov

Published Feb 1, 2006• 33 citations
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Abstract

Additivity of the minimal output entropy for the family of transpose depolarizing channels introduced by Fannes et al.5 is considered. It is shown that using the method of Ref. 3 allows us to prove the additivity for the range of the parameter values for which the problem was left open in Ref. 5. Together with the result of Ref. 5, this covers the whole family of transpose depolarizing channels. In addition, the multiplicativity of the maximal p-norm for this family of channels is proved for all 1 ≤ p ≤ 2.

DOI
10.1142/s0219749906001633

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Additivity of the minimal output entropy for the family of transpose depolarizing channels introduced by Fannes et al.5 is considered. It is shown that using the method of Ref. 3 allows us to prove the additivity for the range of the parameter values for which the problem was left open in Ref. 5. Together with the result of Ref. 5, this covers the whole family of transpose depolarizing channels. In addition, the multiplicativity of the maximal p-norm for this family of channels is proved for all 1 ≤ p ≤ 2.
We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for p>1p>1p>1, this formulation can be expressed in terms of a constrained maximal output Schatten ppp-norm. More precisely, for a completely positive map Ω:L(B′)→L(A)Ω:L(B')\to L(A)Ω:L(B′)→L(A), we consider the quantity υp(Ω)\upsilon_p(Ω)υp​(Ω) defined by optimizing ∥(Ω⊗idE)(σB′E)∥p\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p∥(Ω⊗idE​)(σB′E)∥p​ over all bipartite states σB′Eσ^{B'E}σB′E whose B′B'B′-marginal is maximally mixed. We focus on the case p=2p=2p=2. First, we compute υ2\upsilon_2υ2​ for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map N:L(B′)→L(A)N:L(B')\to L(A)N:L(B′)→L(A) satisfies N†∘N=a idA+b Tr[⋅] IdN^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_dN†∘N=aidA​+bTr[⋅]Id​ for some constants a,b≥0a,b\ge 0a,b≥0, where N†N^{\dagger}N† denotes the Hilbert-Schmidt adjoint of NNN, the quantity υ2(N)\upsilon_2(N)υ2​(N) is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map ΩΩΩ, multiplicativity of υp(Ω)\upsilon_p(Ω)υp​(Ω) implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.
We prove that the minimal Renyi entropy of order 2 (RE2) output of a positive-partial-transpose (PPT)-inducing channel joint to an arbitrary other channel is equal to the sum of the minimal RE2 output of the individual channels. PPT-inducing channels are channels with a Choi matrix which is bound entangled or separable. The techniques used can be easily recycled to prove additivity for some non-PPT-inducing channels such as the depolarizing and transpose depolarizing channels, though not all known additive channels. We explicitly make the calculations for generalized Werner-Holevo channels as an example of both the scope and limitations of our techniques.