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. GiveA 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

Banach Center PublicationsJan 1, 20065 citations

Decomposability of extremal positive maps on $M_2$ Decomposability of extremal positive maps on $M_2$

W. A. Majewski, Marcin Marciniak

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.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.

Sep 30, 20050 citations

Completely Positive Maps

Princeton University Press eBooksJan 10, 20090 citations

Completely Positive Maps

WORLD SCIENTIFIC eBooksFeb 12, 20170 citations

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

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.