302 research outputs found
The Structure of Qubit Unextendible Product Bases
Unextendible product bases have been shown to have many important uses in
quantum information theory, particularly in the qubit case. However, very
little is known about their mathematical structure beyond three qubits. We
present several new results about qubit unextendible product bases, including a
complete characterization of all four-qubit unextendible product bases, which
we show there are exactly 1446 of. We also show that there exist p-qubit UPBs
of almost all sizes less than .Comment: 20 pages, 3 tables, 7 figure
Three-by-three bound entanglement with general unextendible product bases
We discuss the subject of Unextendible Product Bases with the orthogonality
condition dropped and we prove that the lowest rank non-separable
positive-partial-transpose states, i.e. states of rank 4 in 3 x 3 systems are
always locally equivalent to a projection onto the orthogonal complement of a
linear subspace spanned by an orthogonal Unextendible Product Basis. The
product vectors in the kernels of the states belong to a non-zero measure
subset of all general Unextendible Product Bases, nevertheless they can always
be locally transformed to the orthogonal form. This fully confirms the
surprising numerical results recently reported by Leinaas et al. Parts of the
paper rely heavily on the use of Bezout's Theorem from algebraic geometry.Comment: 36 page
Distinguishability of complete and unextendible product bases
It is not always possible to distinguish multipartite orthogonal states if
only local operation and classical communication (LOCC) are allowed. We prove
that we cannot distinguish the states of an unextendible product basis (UPB) by
LOCC even when infinite resources (infinite-dimensional ancillas, infinite
number of operations). Moreover we give a necessary and sufficient condition
for the LOCC distinguishability of complete product bases.Comment: added necessary and sufficient condition for complete product bases,
example Lagarias-Shor ten-parties complete basi
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
A Family of Indecomposable Positive Linear Maps based on Entangled Quantum States
We introduce a new family of indecomposable positive linear maps based on
entangled quantum states. Central to our construction is the notion of an
unextendible product basis. The construction lets us create indecomposable
positive linear maps in matrix algebras of arbitrary high dimension.Comment: 16 pages LaTex: updated and a derivation of a lower bound on epsilon
is added and calculated for one of the examples. Submitted to Lin. Alg. and
Its App
Understanding entanglement as resource: locally distinguishing unextendible product bases
It is known that the states in an unextendible product basis (UPB) cannot be
distinguished perfectly when the parties are restricted to local operations and
classical communication (LOCC). Previous discussions of such bases have left
open the following question: What entanglement resources are necessary and/or
sufficient for this task to be possible with LOCC? In this paper, I present
protocols which use entanglement more efficiently than teleportation to
distinguish certain classes of UPB's. The ideas underlying my approach to this
problem offer rather general insight into why entanglement is useful for such
tasks.Comment: Final, published version. Many revisions following very useful
suggestions of the referee have been added. In particular, Appendix A has
been completely rewritte
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