Positive maps, positive polynomials and entanglement witnesses

TL;DR

This paper explores the connections between positive quantum maps, entanglement witnesses, and multivariate polynomials, providing explicit conditions for positivity in specific cases and linking algebraic properties to quantum entanglement detection.

Contribution

It establishes a novel link between positive maps and polynomial inequalities, offering explicit positivity conditions and insights into entanglement witnesses.

Findings

Indecomposable block positive operators relate to non-sums-of-squares biquadratic forms

Explicit positivity conditions are derived for certain polynomial inequalities

The general problem of characterizing positive maps remains open

Abstract

We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.