arXiv:1301.2886

On estimating the output entropy of a tensor product of the quantum phase-damping channel with an arbitrary channel

Grigori Gennadievich Amosov

Published Jan 14, 2013• 0 citations

Abstract

We obtained the estimation from below for the output entropy of a tensor product of the quantum phase-damping channel with an arbitrary channel. It is shown that from this estimation immediately follows that the strong superadditivity of the output entropy holds for this channel as well as for the quantum depolarizing channel.

DOI: 10.48550/arxiv.1301.2886

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arXiv:1301.2886

On estimating the output entropy of a tensor product of the quantum phase-damping channel with an arbitrary channel

Grigori Gennadievich Amosov

Published Jan 14, 2013• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1301.2886

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We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.

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Given a quantum channel

Φ

in a Hilbert space

H

put

\hat H_Φ(ρ)=\min \limits_{ρ_{av}=ρ}Σ_{j=1}^{k}π_{j}S(Φ(ρ_{j}))

, where

ρ_{av}=Σ_{j=1}^{k}π_{j}ρ_{j}

, the minimum is taken over all probability distributions

π=\{π_{j}\}

and states

ρ_{j}

H

S(ρ)=-Trρ\logρ

is the von Neumann entropy of a state

ρ

. The strong superadditivity conjecture states that

\hat H_{Φ\otimes Ψ}(ρ)\ge \hat H_Φ(Tr_{K}(ρ))+\hat H_Ψ(Tr_{H}(ρ))

for two channels

Φ

and

Ψ

in Hilbert spaces

H

and

K

, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in prime dimensions. The estimation of the quantity

\hat H_{Φ\otimes Ψ}(ρ)

for the special class of Weyl channels

Φ

of the form

Φ=Ξ\circ Φ_{dep}

, where

Φ_{dep}

is the quantum depolarizing channel and

Ξ

is the phase damping is given.