On exposed positive maps: Robertson and Breuer-Hall maps

TL;DR

This paper demonstrates that Breuer-Hall positive maps are not only optimal but also exposed, and that the Robertson map in 4x4 matrices is both extreme and exposed, revealing deeper structural properties in entanglement theory.

Contribution

It proves that Breuer-Hall maps are exposed and that the Robertson map is both extreme and exposed, advancing understanding of positive maps in quantum entanglement.

Findings

Breuer-Hall maps are exposed.

Robertson map is both extreme and exposed.

Provides new insights into the structure of positive maps.

Abstract

It is well known that so called Breuer-Hall positive maps used in entanglement theory are optimal. We show that these maps possess much more subtle property --- they are exposed. As a byproduct it proves that a Robertson map in the algebra of 4 x 4 complex matrices is not only extreme, which was already shown by Robertson, but also exposed.