There Are Many More Positive Maps Than Completely Positive Maps. (23rd September 2017)

• Record Type: Journal Article
• Authors: Klep, Igor; McCullough, Scott; Šivic, Klemen; Zalar, Aljaž
• Is Part Of: International mathematics research notices. Volume 2019:Number 11(2019)
• Page Start: 3313
• Page End: 3375
• Publication Date: 2017-09-23
• Subjects: (Primary) 13J30, 46L07, 52A40 -- (Secondary) 47L25, 81P45, 90C22; Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx203
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: A $\ast$ -linear map $\Phi$ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations $I_n\otimes \Phi$ 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.