arXiv:2312.02814

Optimality of generalized Choi maps in $M_3$

Giovanni Scala, Anindita Bera, Gniewomir Sarbicki, Dariusz Chruściński

Published Dec 5, 2023• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2312.02814

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arXiv:2312.02814

Optimality of generalized Choi maps in $M_3$ Optimality of generalized Choi maps in $M_3$

Giovanni Scala, Anindita Bera, Gniewomir Sarbicki, Dariusz Chruściński

Published Dec 5, 2023• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2312.02814

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Optimality of generalized Choi maps in M 3

Giovanni Scala, Anindita Bera, Gniewomir Sarbicki +1 more

Abstract A family of linear positive maps in the algebra of 3 × 3 complex matrices proposed recently by Bera et al 2024 Linear and Multilinear Algebra 1–16) 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.Abstract A family of linear positive maps in the algebra of 3 × 3 complex matrices proposed recently by Bera et al 2024 Linear and Multilinear Algebra 1–16) 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.

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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

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.

Bound entanglement detection in

4 \otimes 4

systems via generalized Choi maps

We give an affirmative answer to the question posed by Yang et al. on the existence of indecomposable 2-positive linear maps from

\mathcal{M}_3(\mathbb{C})

\mathcal{M}_4(\mathbb{C})

. In this study, we construct an explicit class of linear maps in

\mathcal{B}(\mathcal{M}_3(\mathbb{C}), \mathcal{M}_4(\mathbb{C}))

that are positive, yet fail to be completely positive or completely copositive. Furthermore, we demonstrate that the associated Choi matrices of these maps serve as entanglement witnesses.

M_m(\mathbb{C}) \otimes M_n(\mathbb{C})

. Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.