There are many more positive maps than completely positive maps
Functional Analysis
2019-07-10 v2 Optimization and Control
Quantum Physics
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.
Cite
@article{arxiv.1611.02838,
title = {There are many more positive maps than completely positive maps},
author = {Igor Klep and Scott McCullough and Klemen Šivic and Aljaž Zalar},
journal= {arXiv preprint arXiv:1611.02838},
year = {2019}
}
Comments
v2: 47 pages; includes a more thorough discussion of (completely) positive maps on complex matrices; v1: 37 pages; supplementary material (a Mathematica notebook) is available from Other formats