arXiv:math/0510138

On decomposability of positive maps between $M_2$ and $M_n$

W. A. Majewski, Marcin Marciniak

Published Oct 7, 2005• 2 citations

Abstract

A map $ϕ:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form $ϕ=ϕ_1+ϕ_2$ where $ϕ_1$ is a CP map while $ϕ_2$ is a co-CP map. A partial characterization of decomposability for maps $ϕ: M_2(\bC) \to M_3(\bC)$ is given.

DOI: 10.48550/arxiv.math/0510138

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arXiv:math/0510138

On decomposability of positive maps between $M_2$ and $M_n$ On decomposability of positive maps between $M_2$ and $M_n$

W. A. Majewski, Marcin Marciniak

Published Oct 7, 2005• 2 citations

arXiv HTML PDF

Abstract

A map

ϕ:M_m(\bC)\to M_n(\bC)

is decomposable if it is of the form

ϕ=ϕ_1+ϕ_2

where

ϕ_1

is a CP map while

ϕ_2

is a co-CP map. A partial characterization of decomposability for maps

ϕ: M_2(\bC) \to M_3(\bC)

is given.

DOI: 10.48550/arxiv.math/0510138

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On decomposability of positive maps between

M_2

and

M_n

A map

ϕ:M_m(\bC)\to M_n(\bC)

is decomposable if it is of the form

ϕ=ϕ_1+ϕ_2

where

ϕ_1

is a CP map while

ϕ_2

is a co-CP map. A partial characterization of decomposability for maps

ϕ: M_2(\bC) \to M_3(\bC)

is given.

Decomposability of extremal positive maps on

M_2

A map $\phi:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form

\phi=\phi_1+\phi_2

where

\phi_1

is a CP map while

\phi_2

is a co-CP map. It is known that if

m=n=2

then every positive map is decomposable. Given an extremal unital positive map $\phi:M_2(\bC)\to M_2(\bC)$ we construct concrete maps (not necessarily unital)

\phi_1

and

\phi_2

which give a decomposition of

\phi

. We also show that in most cases this decomposition is unique.

Decomposability of extremal positive unital maps on

M_2

A map $ϕ:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form

ϕ=ϕ_1+ϕ_2

where

ϕ_1

is a CP map while

ϕ_2

is a co-CP map. It is known that if

m=n=2

then every positive map is decomposable. Given an extremal unital positive map $ϕ:M_2(\bC)\to M_2(\bC)$ we construct concrete maps (not necessarily unital)

ϕ_1

and

ϕ_2

which give a decomposition of

ϕ

. We also show that in most cases this decomposition is unique.

A map

\varphi:M_m(\mathbb C)\to M_n(\mathbb C)

is decomposable if it is of the form

\varphi=\varphi_1+\varphi_2

where

\varphi_1

is a CP map while

\varphi_2

is a co-CP map. It is known that if

m=n=2

then every positive map is decomposable. Give

Generalizing Choi map in

M_3

beyond circulant scenario

We present a generalization of the family of linear positive maps in

M_3

proposed thirty years ago by Cho et al. (Linear Algebra Appl.

{\bf 171}

, 213 (1992)) as a generalization of the seminal Choi non-decomposable map. The necessary and sufficient conditions for decomposability are provided.