arXiv:1803.10446 · Advances in Operator Theory

On tensors of factorizable quantum channels with the completely depolarizing channel

Yuki Ueda

Published Jun 8, 2018• 0 citations

arXiv HTML PDF

DOI: 10.15352/aot.1803-1340

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arXiv:1803.10446 · Advances in Operator Theory

On tensors of factorizable quantum channels with the completely depolarizing channel

Yuki Ueda

Published Jun 8, 2018• 0 citations

arXiv HTML PDF

DOI: 10.15352/aot.1803-1340

Related papers

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We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for

p>1

, this formulation can be expressed in terms of a constrained maximal output Schatten

p

-norm. More precisely, for a completely positive map

Ω:L(B')\to L(A)

, we consider the quantity

\upsilon_p(Ω)

defined by optimizing

\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p

over all bipartite states

σ^{B'E}

whose

B'

-marginal is maximally mixed. We focus on the case

p=2

. First, we compute

\upsilon_2

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')\to L(A)

satisfies

N^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_d

for some constants

a,b\ge 0

, where

N^{\dagger}

denotes the Hilbert-Schmidt adjoint of

N

, the quantity

\upsilon_2(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

Ω

, multiplicativity of

\upsilon_p(Ω)

implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.