arXiv:quant-ph/0202124

On quantum channels

Frank Verstraete, Henri Verschelde

Published Feb 21, 2002• 26 citations

Abstract

One of the most challenging open problems in quantum information theory is to clarify and quantify how entanglement behaves when part of an entangled state is sent through a quantum channel. Of central importance in the description of a quantum channel or completely positive map (CP-map) is the dual state associated to it. The present paper is a collection of well-known, less known and new results on quantum channels, presented in a unified way. We will show how this dual state induces nice characterizations of the extremal maps of the convex set of CP-maps, and how normal forms for states defined on a Hilbert space with a tensor product structure lead to interesting parameterizations of quantum channels.

DOI: 10.48550/arxiv.quant-ph/0202124

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arXiv:quant-ph/0202124

On quantum channels

Frank Verstraete, Henri Verschelde

Published Feb 21, 2002• 26 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0202124

Related papers

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X

to another one

Y

is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.

H_1

and

H_2

are finite-dimensional Hilbert spaces, a channel from

H_1

H_2

is a completely positive, linear map

\mathcal{I}

that takes the set of states

\mathcal{S}(H_1)

for

H_1

to the set of states

\mathcal{S}(H_2)

for

H_2

. Corresponding to

\mathcal{I}

there is a unique dual map

\mathcal{I}^*

from the set of effects

\mathcal{E}(H_2)

for

H_2

to the set of effects

\mathcal{E}(H_1)

for

H_1

. We call

\mathcal{I}^*(b)

the effect

b

conditioned by

\mathcal{I}

and the set

\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))

the conditioned set of

\mathcal{I}

. We point out that

\mathcal{I}^c

is a convex subeffect algebra of the effect algebra

\mathcal{E}(H_1)

. We extend this definition to the conditioning

\mathcal{I}^*(B)

for an observable

B

H_2

and say that an observable

A

is in

\mathcal{I}^c

A=\mathcal{I}^*(B)

for some observable

B

. We show that

\mathcal{I}^c

is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.

We investigate the quantum interaction between two quantum sys- tems in terms of

C^{*}

-algebras.We introduce the

co

rrelation CP-map between the bipartite systems and study the operational and statistical structure of the quantum interaction.We then generalize the informational quantities into this quantum setting, and introduce a new entropy of CP-maps, which vanishes at extreme CP-maps in CP-convexity.Introduction.We prepare two quantum states described by density operators

\rho

and

\sigma

on Hilbert spaces

H_{1}

and

K_{1}

respectively, and so initially we have the compound state

\omega_{0}=\rho\otimes\sigma

on the tensor product Hilbert space

H_{1}\otimes K_{1}

.Suppose that there exists an interaction between the two subsystems for some time interval, and that the systems are in equilibrium with the exterior for this time period, then the compound state is changed to a state

\omega

described by

\omega=U(\rho\otimes\sigma)U^{*}

where

U

is a unitary operator representing the quantum evolution of the systems.After the interaction, the states of the subsystems on

H_{2}

and

K_{2}

are described by

Tr_{K_{2}}w

and

R_{H_{2}}\omega

respectively.Then the channels

\varphi^{*}

from

T(H_{1})

T(H_{2})

, and

\phi^{*}

from

T(K_{1})

T(K_{2})

, are defined by

\varphi^{*}(\rho)=h_{K_{2}}\omega=r_{b_{K_{2}}U(\rho}\otimes\sigma)U^{*}

and

\phi^{*}(\sigma)=b_{H_{2}}\omega=h_{H_{2}}U(\rho\otimes\sigma)U^{*}

,The detailed version of this paper will be submitted for publication elsewhere.Typaeet by

A_{A}\#?ffl