New examples of extremal positive linear maps

TL;DR

This paper constructs explicit examples of extremal positive linear maps on symmetric matrices with a maximal number of real zeros in biquadratic forms, demonstrating the upper bound and extending to complex matrices.

Contribution

It provides the first explicit examples of nonnegative biquadratic forms with 8, 9, or 10 real zeros, showing the upper bound is attainable and identifying extremal maps.

Findings

Constructed new families of biquadratic forms with 8, 9, or 10 zeros.

Demonstrated the extremality of certain positive linear maps.

Extended examples to positive maps on complex matrices.

Abstract

New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in $\mathbb{P}^2\times \mathbb{P}^2$ are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros; our examples show that the upper bound is obtained. Such biquadratic forms define positive linear maps on real symmetric $3\times 3$ matrices that are not completely positive. Our constructions are explicit, and moreover we are able to determine which of the examples are extremal. We extend the examples to positive maps on complex matrices and find families of extreme rays in the cone of positive maps.