Indecomposable exposed positive bi-linear maps between two by two matrices

Abstract: Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between $2\times 2$ matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose whose volume is nonzero.