Qubitqudit states with positive partial transpose
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by
Lin Chen, Dragomir Z. Djokovic
2012
Abstract
We show that the length of a qubitqutrit separable state is equal to the
max(r,s), where r is the rank of the state and s is 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 of rank four having positive partial transpose (PPT). The boundary of
this set consists of all separable states of length at most four. We prove that
the length of any qubitqudit separable state of birank (d+1,d+1) is 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
the tensor product of C^2 and C^d such that V contains no product vectors, we
show that the set of all product vectors in the orthogonal complement of 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.
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article
Stage
submitted
Date 20120929
Version
v1
Language
en
^{?}
1210.0111v1
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