  We consider bi-linear analogues of $s$-positivity for linear maps. The dual
objects of these notions can be described in terms of Schimdt ranks for
tri-tensor products and Schmidt numbers for tri-partite quantum states. These
tri-partite versions of Schmidt numbers cover various kinds of bi-separability,
and so we may interpret witnesses for those in terms of bi-linear maps. We give
concrete examples of witnesses for various kinds of three qubit entanglement.
