Stable subspaces of positive maps of matrix algebras

TL;DR

This paper analyzes the structure of extremal bistochastic positive maps on 3x3 matrices, establishing a decomposition, classifying maps based on stable subspaces, and linking them to entanglement detection.

Contribution

It introduces an isometric-sweeping decomposition for such maps, classifies all extremal 3x3 bistochastic maps into three categories, and provides the first example of an extremal atomic positive map with a non-trivial stable subspace.

Findings

Classified extremal bistochastic maps on 3x3 matrices into three categories.

Established the existence of an isometric-sweeping decomposition for these maps.

Computed entanglement witnesses and identified entangled states detected by them.

Abstract

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3x3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.