Properties and construction of extreme bipartite states having positive partial transpose

Abstract

We consider a bipartite quantum system H_A x H_B with M=dim H_A and N=dim H_B. We study the set E of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets E_r={rho in E: rank rho=r}. Our main results pertain to the subsets E_r^{M,N} of E_r consisting of states whose reduced density operators have ranks M and N, respectively. The set E_1 is just the set of pure product states. It is known that E_r^{M,N} is empty for 1< r <= min(M,N) and for r=MN. We prove that also E_{MN-1}^{M,N} is empty. Leinaas, Myrheim and Sollid have conjectured that E_{M+N-2}^{M,N} is not empty for all M,N>2 and that E_r^{M,N} is empty 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 E_{N+1}^{M,N} is empty 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 rho in E_{M+N-2}^{M,N}, we prove that the range of rho contains no product vectors and that the partial transpose of rho has rank M+N-2 as well. In the special case M=3, we construct good 3 x N extreme states of rank N+1 for all N>3.