arXiv:quant-ph/0605177 · Journal of Mathematical Physics

On Weyl channels being covariant with respect to the maximum commutative group of unitaries

Grigori Gennadievich Amosov

Published Jan 1, 2007• 29 citations

Abstract

We investigate the Weyl channels being covariant with respect to the maximum commutative group of unitary operators. This class includes the quantum depolarizing channel and the “two-Pauli” channel as well. Then, we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the two-Pauli channel in the qubit case.

DOI: 10.1063/1.2406054

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arXiv:quant-ph/0605177 · Journal of Mathematical Physics

On Weyl channels being covariant with respect to the maximum commutative group of unitaries

Grigori Gennadievich Amosov

Published Jan 1, 2007• 29 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.2406054

Related papers

On estimation of the entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group of unitaries

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d=2

our approach allows to prove the additivity conjecture for the Holevo-Schumacher-Westmoreland bound. Our method is based upon using the decreasing property of the relative entropy. Earlier we had shown that the method gives the proof of the additivity conjecture for the quantum depolarizing channel in any dimension.

We consider bistochastic quantum channels generated by unitary representations of the discret group. The proof of the additivity conjecture for the quantum depolarizing channel

Φ

based on the decreasing property of the relative entropy is given. We show that the additivity conjecture is true for the channel

Ξ=Ψ\circ Φ

, where

Ψ

is the phase damping.

\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})

. Here

x

runs the set of states

\\sigma (H)

in the Hilbert space

H, dimH=3,

W_{m,n}

are the Weyl operators satisfying the relation

W_{m,n}W_{m',n'}=e^{\\frac {2\\pi i(m'n-mn')}{3}}W_{m+m',n+n'}, 0\\le m,n\\le 2,

and

0\\le r_{m},p_{1},p_{2}\\le 1, \\sum \\limits _{m=0}^{2}r_{m}+3(p_{1}+p_{2})=1

. This class includes the quantum depolarizing channel as well as a number of the other channels being covariant with respect to the maximum commutative group of unitaries.