Rank properties of exposed positive maps

Abstract

Let $\cK$ and $\cH$ be finite dimensional Hilbert spaces and let $\fP$ denote the cone of all positive linear maps acting from $\fB(\cK)$ into $\fB(\cH)$. We show that each map of the form $\varphi (X)=AX{A}^{\ast }$ or $\varphi (X)=A{X}^T{A}^{\ast }$ is an exposed point of $\fP$. We also show that if a map $\varphi$ is an exposed point of $\fP$ then either $\varphi$ is rank 1 non-increasing or $\rankϕ(P)>1$ for any one-dimensional projection $P\in\fB(\cK)$.

6 pages, last section removed - it will be a part of another paper

References in corpus (4)

• Optimal entanglement criterion for mixed quantum states
• Entanglement witnesses arising from exposed positive linear maps
• Exposed positive maps: a sufficient condition
• On extremal positive maps acting between type I factors

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• Exposedness of Choi type entanglement witnesses and applications to lengths of separable states
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• Merging of positive maps: a construction of various classes of positive maps on matrix algebras
• On the structure of positive maps II: low dimensional matrix algebras
• Dimension formula for induced maximal faces of separable states and genuine entanglement