  In bipartite quantum systems of dimension 3x3 entangled states that are
positive under partial transposition (PPT) can be constructed with the use of
unextendible product bases (UPB). As discussed in a previous publication all
the lowest rank entangled PPT states of this system seem to be equivalent,
under special linear product transformations, to states that are constructed in
this way. Here we consider a possible generalization of the UPB constuction to
low-rank entangled PPT states in higher dimensions. The idea is to give up the
condition of orthogonality of the product vectors, while keeping the relation
between the density matrix and the projection on the subspace defined by the
UPB. We examine first this generalization for the 3x3 system where numerical
studies indicate that one-parameter families of such generalized states can be
found. Similar numerical searches in higher dimensional systems show the
presence of extremal PPT states of similar form. Based on these results we
suggest that the UPB construction of the lowest rank entangled states in the
3x3 system can be generalized to higher dimensions, with the use of
non-orthogonal UPBs.
