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arXiv:2301.05788 · Linear and Multilinear Algebra

Exposedness of elementary positive maps between matrix algebras

Seung-Hyeok Kye

Published Jan 18, 2024• 2 citations
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Abstract

The positive linear maps Ads which send matrices x to s∗xs play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ads generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

DOI
10.1080/03081087.2024.2304694

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The positive linear maps Ads which send matrices x to s∗xs play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ads generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.
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.
We construct a large class of indecomposable positive linear maps from the matrix algebra into the matrix algebra, which generate exposed extreme rays of the convex cone of all positive maps. We show that extreme points of the dual faces for separable states arising from these maps are parametrized by the Riemann sphere, and the convex hulls of the extreme points arising from a circle parallel to the equator have exactly the same properties with the convex hull of the trigonometric moment curve studied from combinatorial topology. Any interior points of the dual faces are boundary separable states with full ranks. We exhibit concrete examples of such states.
Let DDD be a space of 2×n2\times n2×n matrices. Then the face of the cone of all completely positive maps from M2M_2M2​ into MnM_nMn​ given by DDD is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in DDD forms a subspace of DDD together with zero and D⊥D^\perpD⊥ is spanned by rank one matrices.