Properties and Construction of Extreme Bipartite States Having Positive Partial Transpose

We consider a bipartite quantum system H_A ⊗ H_B with M = Dim H_A and N = Dim H_B. We study the set ε of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets ε_r = {ρ ∈ ε: rank ρ = 1}. Our main results pertain to the subsets ε_r^{M, N} of ε_r consisting of states whose reduced density operators have ranks M and N, respectively. The set ε₁ is just the set of pure product states. It is known that ε_r^{M, N} = ∅ for 1 < r ≤ min(M, N) and for r = MN. We prove that also ε_MN-1^{M, N} = ∅. Leinaas, Myrheim and Sollid have conjectured that ε_M+N-2^{M, N} ≠ ∅ for all M, N > 2 and that ε_r^{M, N} = ∅ 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 ε_N+1^{M, N} = ∅ 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 ρ ∈ ε_M+N-2^{M, N}, we prove that the range of ρ contains no product vectors and that the partial transpose of ρ has rank M + N - 2 as well. In the special case M = 3, we construct good 3 × N extreme states of rank N + 1 for all N ≥ 4.