  We investigate conditions on a finite set of multi-partite product vectors
for which separable states with corresponding product states have unique
decomposition, and show that this is true in most cases if the number of
product vectors is sufficiently small. In the three qubit case, generic five
dimensional spaces give rise to faces of the convex set consisting of all
separable states, which are affinely isomorphic to the five dimensional simplex
with six vertices. As a byproduct, we construct three qubit entangled PPT edge
states of rank four with explicit formulae. This covers those entanglement
which cannot be constructed from unextendible product basis.
