There are many more positive maps than completely positive maps

TL;DR

This paper investigates the prevalence of positive maps that are not completely positive, providing quantitative bounds, theoretical insights using algebraic geometry, and an algorithm for constructing such maps.

Contribution

It offers the first quantitative bounds on the ratio of positive to completely positive maps and introduces an algorithm to generate positive maps that are not completely positive.

Findings

Quantitative bounds on the fraction of positive maps that are not completely positive.

Application of real algebraic geometry techniques to analyze positive maps.

An algorithm to produce positive maps which are not completely positive.

Abstract

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.