  We consider four norms on tensor product spaces that have appeared in quantum
information theory and demonstrate duality relationships between them. We show
that the product numerical radius is dual to the robustness of entanglement,
and we similarly show that the S(k)-norm is dual to the projective tensor norm.
We show that, analogous to how the product numerical radius and the S(k)-norm
characterize k-block positivity of operators, there is a natural version of the
projective tensor norm that characterizes Schmidt number. In this way we obtain
an elementary new proof of the cross norm criterion for separability, and we
also generalize both the cross norm and realignment criteria to the case of
arbitrary Schmidt number.
