Low rank extremal PPT states and unextendible product bases

Abstract: It is known how to construct, in a bipartite quantum system, a unique low rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (a UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by non-singular product transformations, unitary or non-unitary. The characteristic property of a state $\rho$ in such a family is that its kernel $\Ker\rho$ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in $\Im\rho$, the orthogonal complement of $\Ker\rho$. The generalized UPB in $\Ker\rho$ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension $3\times 3$, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, non-orthogonal UPBs in similar ways.