arXiv:quant-ph/0510148

On Weyl-covariant channels

Motohisa Fukuda, Alexander Semenovich Holevo

Published Oct 19, 2005• 25 citations

Abstract

Formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weyl-covariant channels. We then extend a result in quant-ph/0509126 concerning a bound of the maximal output 2-norm of a Weyl-covariant channel. A class of channels which attain the bound is introduced, for which the multiplicativity of the maximal output 2-norm is proven. Complementary channels are described which share the multiplicativity properties with the Weyl-covariant channels.

DOI: 10.48550/arxiv.quant-ph/0510148

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

Grigori Gennadievich Amosov

May 19, 20061 citation

The additivity conjecture for the Weyl channels in the dimension d=3

Grigori Gennadievich Amosov

Jun 6, 20060 citations

ADDITIVITY CONJECTURE AND COVARIANT CHANNELS

Alexander Semenovich Holevo

In this survey paper we discuss the relation between the minimal output entropy and the χ-capacity for irreducibly covariant quantum channels implying equivalence of the additivity property for both quantities for such channels. The structure of the Weyl-covariant channels is described in detail.

International Journal of Quantum InformationDec 16, 200441 citations

Multiplicativity of the maximal output 2-norm for depolarized Werner-Holevo channels

Spyridon Michalakis

We study the multiplicativity of the output 2-norm for depolarized Werner-Holevo channels and show that multiplicativity holds for a product of two identical channels in this class. Moreover, it is shown that the depolarized Werner-Holevo channels do not satisfy the entrywise positivity condition introduced by King and Ruskai (e-print quant-ph∕0909181v2; e-print quant-ph∕0981026v1), which suggests that the main result is nontrivial.

arXiv:0707.1722Journal of Mathematical PhysicsDec 1, 20079 citations

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

Grigori Gennadievich Amosov

arXiv:quant-ph/0605177Journal of Mathematical PhysicsJan 1, 200729 citations

On complementary channels and the additivity problem

Alexander Semenovich Holevo

arXiv:quant-ph/0509101Sep 14, 200539 citations

arXiv:quant-ph/0510148

On Weyl-covariant channels

Motohisa Fukuda, Alexander Semenovich Holevo

Published Oct 19, 2005• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0510148

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

Grigori Gennadievich Amosov

We estimate the von Neumann entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group. In qubit case, such the class includes the quantum depolarizing channel and the "two-Pauli" channel as well. For the dimesion $d=2$ 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 estimate the von Neumann entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group. In qubit case, such the class includes the quantum depolarizing channel and the "two-Pauli" channel as well. For the dimesion $d=2$ 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.

May 19, 20061 citation

The additivity conjecture for the Weyl channels in the dimension d=3

Grigori Gennadievich Amosov

Following the method introduced in quant-ph/0408004 and quant-ph/0605177 we prove that the additivity conjecture for the Holevo-Schumacher-Westmoreland bound of the output of a quantum channel holds for the channels of the form $\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})$ . 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.Following the method introduced in quant-ph/0408004 and quant-ph/0605177 we prove that the additivity conjecture for the Holevo-Schumacher-Westmoreland bound of the output of a quantum channel holds for the channels of the form $\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})$. 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.

Jun 6, 20060 citations

ADDITIVITY CONJECTURE AND COVARIANT CHANNELS

Alexander Semenovich Holevo

International Journal of Quantum InformationDec 16, 200441 citations

Multiplicativity of the maximal output 2-norm for depolarized Werner-Holevo channels

Spyridon Michalakis

arXiv:0707.1722Journal of Mathematical PhysicsDec 1, 20079 citations

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

Grigori Gennadievich Amosov

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.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.

arXiv:quant-ph/0605177Journal of Mathematical PhysicsJan 1, 200729 citations

On complementary channels and the additivity problem

Alexander Semenovich Holevo

We explore complementarity between output and environment of a quantum channel (or, more generally, CP map), making an observation that the output purity characteristics for complementary CP maps coincide. Hence, validity of the mutiplicativity/additivity conjecture for a class of CP maps implies its validity for complementary maps. The class of CP maps complementary to entanglement-breaking ones is described and is shown to contain diagonal CP maps as a proper subclass, resulting in new class of CP maps (channels) for which the multiplicativity/additivity holds. Covariant and Gaussian channels are discussed briefly in this context.We explore complementarity between output and environment of a quantum channel (or, more generally, CP map), making an observation that the output purity characteristics for complementary CP maps coincide. Hence, validity of the mutiplicativity/additivity conjecture for a class of CP maps implies its validity for complementary maps. The class of CP maps complementary to entanglement-breaking ones is described and is shown to contain diagonal CP maps as a proper subclass, resulting in new class of CP maps (channels) for which the multiplicativity/additivity holds. Covariant and Gaussian channels are discussed briefly in this context.

arXiv:quant-ph/0509101Sep 14, 200539 citations

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.

\\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.