There are many more positive maps than completely positive maps

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.

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

References in corpus (6)

• Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• How often is a random quantum state k-entangled?
• The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
• The probability of entanglement
• Multihomogenous Nonnegative Polynomials and Sums of Squares

Cited by in corpus (5)

• Positive maps and entanglement in real Hilbert spaces
• Multihomogenous Nonnegative Polynomials and Sums of Squares
• A random copositive matrix is completely positive with positive probability
• New examples of extremal positive linear maps
• Cross-positive linear maps, positive polynomials and sums of squares