Proposition: A product-free support gives a witness

Let $0\ne Z⪰0$ on ${\mathbb{C}}^a\otimes {\mathbb{C}}^b$, and let $P$ be its range projection and $Q=I-P$. If $\mathrm{ran}Z$ is product-free, then

For every $0<ϵ\le {\delta }_{\ast }$,

is nonnegative on every positive separable matrix and satisfies

Proof. The product of the two unit spheres is compact. A zero minimum would give a unit product vector in $\ker Q=\mathrm{ran}Z$. Thus the minimum is positive. The defining lower bound proves block positivity of $W$, which extends to the separable cone. Since $QZ=0$, the negative expectation follows.