Mathematical Proceedings of the Cambridge Philosophical Society

On the convex set of completely positive linear maps in matrix algebras

Seung-Hyeok Kye

Published Jul 1, 1997• 15 citations

Abstract

Let PI (respectively CPI) be the convex compact set of all unital positive (respectively completely positive) linear maps from the matrix algebra Mm([Copf ]) into Mn([Copf ]). We show that maximal faces of CPI correspond to one dimensional subspaces of the vector space Mm, n([Copf ]). Furthermore, a maximal face of CPI lies on the boundary of PI if and only if the corresponding subspace is generated by a rank one matrix.

DOI: 10.1017/s0305004196001508

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Mathematical Proceedings of the Cambridge Philosophical Society

On the convex set of completely positive linear maps in matrix algebras

Seung-Hyeok Kye

Published Jul 1, 1997• 15 citations

Publisher

Abstract

DOI: 10.1017/s0305004196001508

Related papers

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Let

D

be a space of

2\times n

matrices. Then the face of the cone of all completely positive maps from

M_2

into

M_n

given by

D

is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in

D

forms a subspace of

D

together with zero and

D^\perp

is spanned by rank one matrices.

A linear map

L

from

{\mathbb C}^{n \times n}

into

{\mathbb C}^{n \times n}

is called a quantum channel if it is completely positive and trace preserving. The set

{\cal L}_n

of all such quantum channels is known to be a compact convex set. While the extreme points of

{\cal L}_n

can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix

Z(L)

associated with the quantum channel

L

, we compute the maximum dimension of a proper face of

{\cal L}_n

, and in addition the possible dimensions of faces generated by

L

when $rank \ Z(L)=2 $.

A factorization property of positive maps on

C^*

-algebras

A_i

C_i

be unital C*-algebras and let

α_i

be positive linear maps from

A_i

C_i,

i=1,2

. We obtain conditions under which any positive map

β

from the minimal C*-tensor product

A_1 \otimes_{min} A_2

C_1 \otimes_{min} C_2

, such that $ α_1 \otimes α_2 \geq β$, factorizes as

β= γ\otimes α_2

for some positive map

γ

. In particular we show that when

α_i \colon A_i \rightarrow B(\mathcal H_i)

are completely positive (CP) maps for some Hilbert spaces

\mathcal H_i

(i=1,2)

, and

α_2

is a pure CP map and

β

is a CP map so that

α_1 \otimes α_2 - β

is also CP, then

β= γ\otimes α_2

for some CP map

γ

. We show that a similar result holds in the context of positive linear maps when

A_2 = C_2 = B(\mathcal H)

and

α_2 = id

. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map

τ

from a unital C*-algebra

A

to a C*-algebra

C

, if

τ\otimes id_k

is decomposable for some

k \geq 2

, where

id_k

is the identity map on the algebra

M_k(\mathbb {C} )

k\times k

matrices, then

τ

is completely positive.