13,458 research outputs found
Classification of GHZ-type, W-type and GHZ-W-type multiqubit entanglements
We propose the concept of SLOCC-equivalent basis (SEB) in the multiqubit
space. In particular, two special SEBs, the GHZ-type and the W-type basis are
introduced. They can make up a more general family of multiqubit states, the
GHZ-W-type states, which is a useful kind of entanglement for quantum
teleporatation and error correction. We completely characterize the property of
this type of states, and mainly classify the GHZ-type states and the W-type
states in a regular way, which is related to the enumerative combinatorics.
Many concrete examples are given to exhibit how our method is used for the
classification of these entangled states.Comment: 16 pages, Revte
On the number of fully packed loop configurations with a fixed associated matching
We show that the number of fully packed loop configurations corresponding to
a matching with nested arches is polynomial in if is large enough,
thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11
(2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
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