Separable states with unique decompositions

We search for faces of the convex set consisting of all separable states, which are affinely isomorphic to simplices. In the two-qutrit case, we found that six product vectors spanning a five dimensional space give rise to a face isomorphic to the 5-dimensional simplex with six vertices, under suitable linear independence assumption. This face is inscribed in the face for PPT states whose boundary shares the fifteen 3-simplices on the boundary of the 5-simplex. The remaining boundary points consist of PPT entangled edge states of rank four. We also show that every edge state of rank four arises in this way. We also construct a face isomorphic to the 9-simplex. As applications, we give answers to questions in the literature \citechen_dj_ext_PPT,chen_dj_semialg. For the qubit-qudit cases with $d≥ 3$, we also show that $(d+1)$-dimensional subspaces give rise to faces isomorphic to the $d$-simplices, in most cases.