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Linear and Multilinear Algebra

Generalizing Choi map in M 3 beyond circulant scenario

Anindita Bera, Giovanni Scala, Gniewomir Sarbicki, Dariusz Chruściński

Published Mar 18, 2024• 3 citations
Publisher

Abstract

We introduce a family of positive linear maps in the algebra of 3×3 complex matrices, which generalizes the seminal positive non-decomposable map originally proposed by Choi. Necessary and sufficient conditions for decomposability are derived and demonstrated. Specifically, the proposed maps evolve the circulant structure with three parameters into a doubly stochastic structure with five parameters.

DOI
10.1080/03081087.2024.2326249

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We introduce a family of positive linear maps in the algebra of 3×3 complex matrices, which generalizes the seminal positive non-decomposable map originally proposed by Choi. Necessary and sufficient conditions for decomposability are derived and demonstrated. Specifically, the proposed maps evolve the circulant structure with three parameters into a doubly stochastic structure with five parameters.
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We present a generalization of the family of linear positive maps in M3M_3M3​ proposed thirty years ago by Cho et al. (Linear Algebra Appl. 171{\bf 171}171, 213 (1992)) as a generalization of the seminal Choi non-decomposable map. The necessary and sufficient conditions for decomposability are provided.
Optimality of generalized Choi maps in M3M_3M3​
A family of linear positive maps in the algebra of 3×33 \times 33×3 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 M3M_3M3​. 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.
A problem of further generalization of generalized Choi maps Φ[a,b,c] acting on M3\mathbb M_{3}M3​ 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.
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.
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