Properties and Construction of Extreme Bipartite States Having Positive Partial Transpose

Abstract

We consider a bipartite quantum system HA ⊗ HB with M = Dim HA and N = Dim HB. We study the set ε of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets εr = {ρ ∈ ε: rank ρ = 1}. Our main results pertain to the subsets εr M, N of εr 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 εr M, N = ∅ for 1 < r ≤ min(M, N) and for r = MN. We prove that also εMN-1 M, N = ∅. Leinaas, Myrheim and Sollid have conjectured that εM+N-2 M, N ≠ ∅ for all M, N > 2 and that εr 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 εN+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 ∈ εM+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. © 2013 Springer-Verlag Berlin Heidelberg.