32,253 research outputs found
Local unitary versus local Clifford equivalence of stabilizer and graph states
The equivalence of stabilizer states under local transformations is of
fundamental interest in understanding properties and uses of entanglement. Two
stabilizer states are equivalent under the usual stochastic local operations
and classical communication criterion if and only if they are equivalent under
local unitary (LU) operations. More surprisingly, under certain conditions, two
LU equivalent stabilizer states are also equivalent under local Clifford (LC)
operations, as was shown by Van den Nest et al. [Phys. Rev. \textbf{A71},
062323]. Here, we broaden the class of stabilizer states for which LU
equivalence implies LC equivalence () to include all
stabilizer states represented by graphs with neither cycles of length 3 nor 4.
To compare our result with Van den Nest et al.'s, we show that any stabilizer
state of distance is beyond their criterion. We then further prove
that holds for a more general class of stabilizer states
of . We also explicitly construct graphs representing
stabilizer states which are beyond their criterion: we identify all 58 graphs
with up to 11 vertices and construct graphs with () vertices
using quantum error correcting codes which have non-Clifford transversal gates.Comment: Revised version according to referee's comments. To appear in
Physical Review
Nested cycles in large triangulations and crossing-critical graphs
We show that every sufficiently large plane triangulation has a large
collection of nested cycles that either are pairwise disjoint, or pairwise
intersect in exactly one vertex, or pairwise intersect in exactly two vertices.
We apply this result to show that for each fixed positive integer , there
are only finitely many -crossing-critical simple graphs of average degree at
least six. Combined with the recent constructions of crossing-critical graphs
given by Bokal, this settles the question of for which numbers there is
an infinite family of -crossing-critical simple graphs of average degree
Manifolds associated with -colored regular graphs
In this article we describe a canonical way to expand a certain kind of
-colored regular graphs into closed -manifolds by
adding cells determined by the edge-colorings inductively. We show that every
closed combinatorial -manifold can be obtained in this way. When ,
we give simple equivalent conditions for a colored graph to admit an expansion.
In addition, we show that if a -colored regular graph
admits an -skeletal expansion, then it is realizable as the moment graph of
an -dimensional closed -manifold.Comment: 20 pages with 9 figures, in AMS-LaTex, v4 added a new section on
reconstructing a space with a -action for which its moment graph is
a given colored grap
The First Order Definability of Graphs with Separators via the Ehrenfeucht Game
We say that a first order formula defines a graph if is
true on and false on every graph non-isomorphic with . Let
be the minimal quantifier rank of a such formula. We prove that, if is a
tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar
graph, then , where denotes the order of . This bound is
optimal up to a constant factor. If is a constant, for connected graphs
with no minor and degree , we prove the bound
. This result applies to planar graphs and, more generally, to
graphs of bounded genus.Comment: 17 page
- …