New examples of extremal positive linear maps

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.