Qubitqudit states with positive partial transpose
Abstract
We show that the length of a qubitqutrit separable state is equal to max(r,s), where r is the rank of the state and s the rank of its partial transpose. We refer to the ordered pair (r,s) as the birank of this state. We also construct examples of qubitqutrit separable states of any feasible birank (r,s). We determine the closure of the set of normalized twoqutrit entangled states having positive partial transpose (PPT) of rank 4. The boundary of this set consists of all separable states of length at most 4. We prove that the length of any qubitqudit separable state of birank (d+1,d+1) is equal to d+1. We also show that all qubitqudit PPT entangled states of birank (d+1,d+1) can be built in a simple way from edge states. If V is a subspace of dimension k<d in a 2⊗d space such that V contains no product vectors, we show that the set of all product vectors in V^{⊥} is a vector bundle of rank dk over the projective line. Finally, we explicitly construct examples of qubitqudit PPT states (both separable and entangled) of any feasible birank.
 Publication:

Physical Review A
 Pub Date:
 December 2012
 DOI:
 10.1103/PhysRevA.86.062332
 arXiv:
 arXiv:1210.0111
 Bibcode:
 2012PhRvA..86f2332C
 Keywords:

 03.67.Mn;
 03.65.Ud;
 Entanglement production characterization and manipulation;
 Entanglement and quantum nonlocality;
 Quantum Physics;
 Mathematical Physics
 EPrint:
 13 pages, 2 tables