Facial structures for various notions of positivity and applications to the theory of entanglement

TL;DR

This paper explores the facial structures of convex cones related to positive linear maps and applies these insights to the study of entangled states and entanglement witnesses in quantum information theory.

Contribution

It provides an expository analysis of facial structures in convex cones of positive maps and applies this to entanglement theory, offering new perspectives on entangled states and witnesses.

Findings

Facial structures of convex cones are characterized.

Applications to entangled edge states with positive partial transposes.

Insights into the optimality of entanglement witnesses.

Abstract

In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.