Unextendible product bases and the construction of inseparable states

Let H[N] denote the tensor product of n finite dimensional Hilbert spaces H(r). A state |phi> of H[N] is separable if |phi> is the tensor product of states in the respective product spaces. An orthogonal unextendible product basis is a finite set B of separable orthonormal states |phi(k)> such that the non-empty space B9perp), the set of vectors orthogonal to B, contains no separable projection. Examples of orthogonal UPB sets were first constructed by Bennett et al [1] and other examples appear, for example, in [2] and [3]. If F denotes the set of convex combinations of the projections |phi(k)><phi(k)|, then F is a face in the set S of separable densities. In this note we show how to use F to construct families of positive partial transform states (PPT) which are not separable. We also show how to make an analogous construction when the condition of orthogonality is dropped. The analysis is motivated by the geometry of the faces of the separable states and leads to a natural construction of entanglement witnesses separating the inseparable PPT states from S.