Topology of the cone of positive maps on qubit systems

TL;DR

This paper offers a geometric perspective on the structure of positive maps on qubit systems, providing new decomposition methods, conditions for complete positivity, and topological characterizations of special map classes.

Contribution

It introduces a geometric proof for positive map decomposition, new conditions for complete positivity, and a topological description of maps that are both completely positive and copositive.

Findings

Decomposition of positive maps via boundary-preserving operators

Conditions for complete positivity of maps on 2x2 matrices

Topological characterization of maps that are both completely positive and copositive

Abstract

An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra $M_{2}(\mathbb{C})$ has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of $M_{2}(\mathbb{C})$ have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of $M_{n}(\mathbb{C})$. Lastly, following the same geometrical approach, a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented.