Properties and construction of extreme bipartite states having positive partial transpose

Details Properties and construction of extreme bipartite states having positive partial transpose Properties and construction of extreme bipartite states having positive partial transpose

2012-03-07

arxiv.org/abs/1203.1364v3

Physics

Mathematical Physics

Definition 13 corrected

Scientific paper

We investigate the set E of extreme points of the compact convex set of PPT states (i.e., the states having positive semidefinite partial transpose) of a bipartite M x N quantum system. Let E(M,N,r) denote the subset of E consisting of states of rank r which are supported on M x N. We show that for M,N>2 the sets E(M,N,M+N-2) are nonempty. On the other hand we show that for M,N>3 the sets E(M,N,N+1) are empty. It is known that the set E(M,N,MN) is empty, and we show that also the set E(M,N,MN-1) is empty. We divide the set of all states into the good and the bad states (the definition is too technical to be given here). We show that the good states have many good properties. In particular, we solve the separability problem for good M x N PPT states of rank M+N-2 when M=3 or 4. All pure bipartite states are good. We obtain a simple characterization of good PPT states.

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