There Are Many More Positive Maps Than Completely Positive Maps. (23rd September 2017) - Klep, Igor; McCullough, Scott; Šivic, Klemen; Zalar, Aljaž
Abstract: A * -linear map Φ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations I_nΦ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the 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 that are not completely positive is given.