arXiv:2212.03807

Generalizing Choi map in $M_3$ beyond circulant scenario

Anindita Bera, Scala, Giovanni, Gniewomir Sarbicki, Dariusz Chruściński

Published Dec 7, 2022• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2212.03807

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arXiv:2212.03807

Generalizing Choi map in $M_3$ beyond circulant scenarioGeneralizing Choi map in $M_3$ beyond circulant scenario

Anindita Bera, Scala, Giovanni, Gniewomir Sarbicki, Dariusz Chruściński

Published Dec 7, 2022• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2212.03807

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Generalizing Choi map in M 3 beyond circulant scenario

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Linear and Multilinear AlgebraMar 18, 20243 citations

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A problem of further generalization of generalized Choi maps Φ[a,b,c] acting on $\mathbb M_{3}$ introduced by Cho, Kye, and Lee is discussed. Some necessary conditions for positivity of the generalized maps are provided as well as some sufficient conditions. Also, some sufficient conditions for decomposability of these maps are shown.A problem of further generalization of generalized Choi maps Φ[a,b,c] acting on $\mathbb M_{3}$ introduced by Cho, Kye, and Lee is discussed. Some necessary conditions for positivity of the generalized maps are provided as well as some sufficient conditions. Also, some sufficient conditions for decomposability of these maps are shown.

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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.

A problem of further generalization of generalized Choi maps Φ[a,b,c] acting on

\mathbb M_{3}

introduced by Cho, Kye, and Lee is discussed. Some necessary conditions for positivity of the generalized maps are provided as well as some sufficient conditions. Also, some sufficient conditions for decomposability of these maps are shown.

Optimality of generalized Choi maps in

M_3

A family of linear positive maps in the algebra of

3 \times 3

complex matrices proposed recently in Bera et al. arXiv:2212.03807 is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in

M_3

. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.