Extremal states of positive partial transpose in a system of three qubits

We have studied mixed states in the system of three qubits with the property that all their partial transposes are positive, these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. In random numerical searches we find entangled PPT states with a large variety of rank combinations. For ranks equal to five or higher we find both extremal and nonextremal PPT states of nearly every rank combination, with the restriction that the square sum of the four ranks of an extremal PPT state can be at most 193. We have studied especially the rank four entangled PPT states, which are found to have rank four for every partial transpose. These states are all extremal, because of the previously known result that every PPT state of rank three or less is separable. We find two distinct classes of rank 4444 entangled PPT states, identified by a real valued quadratic expression invariant under local SL(2,C) transformations, mathematically equivalent to Lorentz transformations. This quadratic Lorentz invariant is nonzero for one class of states (type I in our terminology) and zero for the other class (type II). The previously known states based on unextendible product bases is a nongeneric subclass of the type I states. We present analytical constructions of states of both types, general enough to reproduce all the rank 4444 PPT states we have found numerically. We can not exclude the possibility that there exist nongeneric rank four PPT states that we do not find in our random numerical searches.