Journal of Mathematical Analysis and Modeling

Decomposable of positive map from M3(C) to M2(M2(C))

Winda C. Akatch, Okelo B. Nyaare, Priscah Mogotu Omoke

Published Dec 29, 2023• 0 citations

Abstract

In most literature, the decomposition of positive maps from M3 to M2 are discussed where the matrix elements are complex numbers. In this paper we construct a positive maps φ(µ,c1,c2) from M3(C) to M2(M2(C)). The Choi matrices for complete positivity and complete copositivity ares visualized as tensor matrix M3 ⊗M2 with M2(C) as the entry elements. The construction allow us describe decomposability on positive semidefinite matrices.

DOI: 10.48185/jmam.v4i2.826

Related papers

Characterization of Maps on Positive Semidefinite Choi Matrices

C. A. Winda, Benard Okelo, Omolo Ongati

MathLAB JournalAug 31, 20200 citations

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

W. A. Majewski, Marcin Marciniak

arXiv:math/0510138Oct 7, 20052 citations

Decomposability of extremal positive maps on $M_2$

W. A. Majewski, Marcin Marciniak

Sep 30, 20050 citations

All 2-positive linear maps from M3 to M3 are decomposable

Yang Yu, Denny H. Leung, Wai-Shing Tang

Following an idea of Choi, we obtain a decomposition theorem for k-positive linear maps from Mm to Mn, where 2<=k

arXiv:1603.03534Mar 11, 20162 citations

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

B. V. Rajarama Bhat, Hiroyuki Osaka

arXiv:1912.02381Dec 5, 20192 citations

Decomposability of extremal positive unital maps on $M_2$

W. A. Majewski, Marcin Marciniak

arXiv:math/0510005Sep 30, 20050 citations

Journal of Mathematical Analysis and Modeling

Decomposable of positive map from M3(C) to M2(M2(C))

Winda C. Akatch, Okelo B. Nyaare, Priscah Mogotu Omoke

Published Dec 29, 2023• 0 citations

Publisher

Abstract

DOI: 10.48185/jmam.v4i2.826

Related papers

Characterization of Maps on Positive Semidefinite Choi Matrices

C. A. Winda, Benard Okelo, Omolo Ongati

Several investigations have been done on positive maps on their algebraic structures with more emphasis on completely positive maps. In this study, we have described the structure of the Choi matrices for 2-positive maps on positive semidefinite matrices and the conditions for complete positivity of positive linear maps from n to n+1. The motivation behind these objectives is work done by Majewski and Marciniak on the structure of positive maps ф from Mn to Mn + 1(2 ≥ 2) between matrix algebras.Several investigations have been done on positive maps on their algebraic structures with more emphasis on completely positive maps. In this study, we have described the structure of the Choi matrices for 2-positive maps on positive semidefinite matrices and the conditions for complete positivity of positive linear maps from n to n+1. The motivation behind these objectives is work done by Majewski and Marciniak on the structure of positive maps ф from Mn to Mn + 1(2 ≥ 2) between matrix algebras.

MathLAB JournalAug 31, 20200 citations

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

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.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.

arXiv:math/0510138Oct 7, 20052 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

All 2-positive linear maps from M3 to M3 are decomposable

Yang Yu, Denny H. Leung, Wai-Shing Tang

Following an idea of Choi, we obtain a decomposition theorem for k-positive linear maps from Mm to Mn, where 2<=k

arXiv:1603.03534Mar 11, 20162 citations

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.

arXiv:1912.02381Dec 5, 20192 citations

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

W. A. Majewski, Marcin Marciniak

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 $ϕ:M_m(\bC)\to M_n(\bC)$ 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.

arXiv:math/0510005Sep 30, 20050 citations

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.

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 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.