Exposedness of elementary positive maps between matrix algebras

TL;DR

This paper investigates the exposedness of elementary positive maps between matrix algebras, providing alternative proofs using Choi matrices and Woronowicz's method to deepen understanding in quantum information and matrix theory.

Contribution

It offers two new proofs of the exposedness of positive maps, enhancing theoretical understanding with methods from Choi matrices and Woronowicz's approach.

Findings

Confirmed the exposedness of the maps using Choi matrices.

Provided an alternative proof via Woronowicz's method.

Strengthened the theoretical foundation of positive maps in quantum theory.

Abstract

The positive linear maps $\ad_s$ which send matrices $x$ to $s^*xs$ play important roles in quantum information theory as well as matrix theory. It was proved by Marciniak [Linear Multilinear Alg. 61 (2013), 970--975] that the map $\ad_s$ generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.