3,221 research outputs found
Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
We report new results and generalizations of our work on unextendible product
bases (UPB), uncompletable product bases and bound entanglement. We present a
new construction for bound entangled states based on product bases which are
only completable in a locally extended Hilbert space. We introduce a very
useful representation of a product basis, an orthogonality graph. Using this
representation we give a complete characterization of unextendible product
bases for two qutrits. We present several generalizations of UPBs to arbitrary
high dimensions and multipartite systems. We present a sufficient condition for
sets of orthogonal product states to be distinguishable by separable
superoperators. We prove that bound entangled states cannot help increase the
distillable entanglement of a state beyond its regularized entanglement of
formation assisted by bound entanglement.Comment: 24 pages RevTex, 15 figures; appendix removed, several small
corrections, to appear in Comm. Math. Phy
Qubit-qudit states with positive partial transpose
We show that the length of a qubit-qutrit 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 qubit-qutrit separable states of any feasible birank
(r,s). We determine the closure of the set of normalized two-qutrit 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 qubit-qudit separable state of birank (d+1,d+1) is d+1. We
also show that all qubit-qudit 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 d-k over the projective line. Finally, we explicitly
construct examples of qubit-qudit PPT states (both separable and entangled) of
any feasible birank.Comment: 13 pages, 2 table
Maximally-Disordered Distillable Quantum States
We explore classical to quantum transition of correlations by studying the
quantum states located just outside of the classically-correlated-states-only
neighborhood of the maximally mixed state (the largest separable ball (LSB)).
We show that a natural candidate for such states raises the possibility of a
layered transition, i.e., an annular region comprising only classical and the
classical-like bound entangled states, followed by free or distillable
entanglement. Surprisingly, we find the transition to be abrupt for bipartite
systems: distillable states emerge arbitrarily close to the LSB. For
multipartite systems, while the radius of the LSB remains unknown, we determine
the radius of the largest undistillable ball. Our results also provide an upper
bound on how noisy shared entangled states can be for executing quantum
information processing protocols.Comment: Published Version, 7 pages, Late
Class of PPT bound entangled states associated to almost any set of pure entangled states
We analyze a class of entangled states for bipartite systems,
with non-prime. The entanglement of such states is revealed by the
construction of canonically associated entanglement witnesses. The structure of
the states is very simple and similar to the one of isotropic states: they are
a mixture of a separable and a pure entangled state whose supports are
orthogonal. Despite such simple structure, in an opportune interval of the
mixing parameter their entanglement is not revealed by partial transposition
nor by the realignment criterion, i.e. by any permutational criterion in the
bipartite setting. In the range in which the states are Positive under Partial
Transposition (PPT), they are not distillable; on the other hand, the states in
the considered class are provably distillable as soon as they are Nonpositive
under Partial Transposition (NPT). The states are associated to any set of more
than two pure states. The analysis is extended to the multipartite setting. By
an opportune selection of the set of multipartite pure states, it is possible
to construct mixed states which are PPT with respect to any choice of bipartite
cuts and nevertheless exhibit genuine multipartite entanglement. Finally, we
show that every -positive but not completely positive map is associated to a
family of nondecomposable maps.Comment: 12 pages, 3 figures. To appear in Phys. Rev.
New classes of n-copy undistillable quantum states with negative partial transposition
The discovery of entangled quantum states from which one cannot distill pure
entanglement constitutes a fundamental recent advance in the field of quantum
information. Such bipartite bound-entangled (BE) quantum states \emph{could}
fall into two distinct categories: (1) Inseparable states with positive partial
transposition (PPT), and (2) States with negative partial transposition (NPT).
While the existence of PPT BE states has been confirmed, \emph{only one} class
of \emph{conjectured} NPT BE states has been discovered so far. We provide
explicit constructions of a variety of multi-copy undistillable NPT states, and
conjecture that they constitute families of NPT BE states. For example, we show
that for every pure state of Schmidt rank greater than or equal to three, one
can construct n-copy undistillable NPT states, for any . The abundance
of such conjectured NPT BE states, we believe, considerably strengthens the
notion that being NPT is only a necessary condition for a state to be
distillable.Comment: Latex, 10 page
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