Seung-Hyeok Kye: Indecomposable exposed multi-linear maps and separable states with unique decomposition

Abstract: We discuss duality between positivity of multi-linear maps and multi-partite entanglement. We will focus on several notions of extremeness such as exposedness and full spanning properties which detect PPT entanglement with nonzero volume. We will exhibit examples in the 3 × 3 and 2 × 2 × 2 cases, and look inside the dual faces of them. They have subfaces which are affinely isomorphic to the 9-dimensional simplex. In this way, we find separable states with unique decomposition whose lengths are strictly greater than the dimensions of the systems.