MATHEMATICA SCANDINAVICA

A Decomposition Theorem for Positive Maps, and the Projection Onto a Spin Factor

Erling Størmer

Published Mar 7, 2016• 2 citations

Abstract

It is shown that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal. The result is studied in detail for the projection onto a spin factor.

DOI: 10.7146/math.scand.a-23300

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MATHEMATICA SCANDINAVICA

A Decomposition Theorem for Positive Maps, and the Projection Onto a Spin Factor

Erling Størmer

Published Mar 7, 2016• 2 citations

Publisher

Abstract

DOI: 10.7146/math.scand.a-23300

Related papers

A decomposition theorem for positive maps, and the projection onto a spin factor

Erling Størmer

arXiv:1308.3332Aug 15, 20131 citation

Decomposable and atomic projection maps

Erling Størmer

arXiv:0904.0103Apr 1, 20090 citations

A factorization property of positive maps on C*-algebras

B. V. Rajarama Bhat, Hiroyuki Osaka

The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.

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A factorization property of positive maps on $C^*$ -algebrasA factorization property of positive maps on $C^*$-algebras

B. V. Rajarama Bhat, Hiroyuki Osaka

The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$ , $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$ . We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$ , 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 $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $τ$ is completely positive.The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$, $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, 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 $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $τ$ is completely positive.

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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.

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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.

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.