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Rank properties of exposed positive maps

By Marcin Marciniak

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 $\phi(X)=AXA^*$ or $\phi(X)=AX^TA^*$ is an exposed point of $\fP$. We also show that if a map $\phi$ is an exposed point of $\fP$ then either $\phi$ is rank 1 non-increasing or $\rank\phi(P)>1$ for any one-dimensional projection $P\in\fB(\cK)$.Comment: 6 pages, last section removed - it will be a part of another pape

Topics: Mathematics - Functional Analysis, Quantum Physics, 47H07, 46L05
Year: 2012
DOI identifier: 10.1080/03081087.2012.721360
OAI identifier: oai:arXiv.org:1103.3497
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