Properties and Construction of Extreme Bipartite States Having Positive Partial Transpose

We consider a bipartite quantum system with and . We study the set of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets . Our main results pertain to the subsets of consisting of states whose reduced density operators have ranks M and N, respectively. The set is just the set of pure product states. It is known that for and for r = MN. We prove that also . Leinaas, Myrheim and Sollid have conjectured that for all M, N > 2 and that for 1 < r < M + N - 2. We prove the first part of their conjecture. The second part is known to hold when min(M, N) = 3 and we prove that it holds also when min(M, N) = 4. This is a consequence of our result that if M, N > 3. We introduce the notion of "good" states, show that all pure states are good and give a simple description of the good separable states. For a good state , we prove that the range of contains no product vectors and that the partial transpose of has rank M + N-2 as well. In the special case M = 3, we construct good extreme states of rank N + 1 for all N >= 4.