Properties and construction of extreme bipartite states having positive partial transpose

TL;DR

This paper investigates the structure and existence of extreme states with positive partial transpose in bipartite quantum systems, proving new emptiness results for certain rank subsets and constructing specific examples.

Contribution

It proves the conjecture that E_{MN-1}^{M,N} is empty and extends emptiness results for E_r^{M,N} when min(M,N)=4, also introducing the concept of good states.

Findings

E_{MN-1}^{M,N} is empty for all M,N.

E_{N+1}^{M,N} is empty if M,N>3.

Constructed good 3 x N extreme states of rank N+1 for N>3.

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.