aqora

Semidefinite descriptions of low-dimensional separable matrix cones

Roland Hildebrand

Linear Algebra and its ApplicationsJun 3, 20088 citations

Geometry of Entanglement Witnesses Parametrized by SO (3) Group

Dariusz Chruściński, Filip A. Wudarski

We characterize a set of positive maps in matrix algebra of 4 × 4 complex matrices. Equivalently, we provide a subset of entanglement witnesses in ℂ4⊗ℂ4 parametrized by the rotation group SO (3). Interestingly, these maps/witnesses define two intersecting convex cones in the 3-dimensional parameter space. The existence of two cones is related to the topological structure of the underlying orthogonal group. We perform detailed analysis of the corresponding geometric structure.

arXiv:1204.4830Open Systems & Information DynamicsSep 1, 20125 citations

Positive maps of low dimensional matrix algebras

Stanisław Lech Woronowicz

Reports on Mathematical PhysicsOct 1, 1976446 citations

Construction of exposed indecomposable positive linear maps between matrix algebras

Kil-Chan Ha, Seung-Hyeok Kye

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.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.

arXiv:1410.5545Linear and Multilinear AlgebraFeb 17, 20168 citations

Exposedness of elementary positive maps between matrix algebras

Seung-Hyeok Kye

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.

arXiv:2301.05788Linear and Multilinear AlgebraJan 18, 20242 citations

Cones with a mapping cone symmetry in the finite-dimensional case

Łukasz Skowronek

Linear Algebra and its ApplicationsFeb 16, 201115 citations

Convex cones in mapping spaces between matrix algebras

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