Construction of exposed indecomposable positive linear maps between matrix algebras

TL;DR

This paper constructs a broad class of indecomposable positive linear maps between specific matrix algebras, revealing their geometric properties and connections to separable states and combinatorial topology.

Contribution

It introduces new indecomposable positive maps that generate exposed extreme rays and explores their geometric and topological properties.

Findings

Generated a large class of indecomposable positive maps.

Connected extreme points to the Riemann sphere and trigonometric moment curves.

Identified boundary separable states with full ranks.

Abstract

We construct a large class of indecomposable positive linear maps from the $2\times 2$ matrix algebra into the $4\times 4$ 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 the exactly same properties with the convex hull of the trigonometric moment curve studied from combinatorial topology. Any interior points of the dual faces are $2\otimes 4$ boundary separable states with full ranks. We exhibit concrete examples of such states.