9,346 research outputs found
Entanglement and nonclassical properties of hypergraph states
Hypergraph states are multi-qubit states that form a subset of the locally
maximally entangleable states and a generalization of the well--established
notion of graph states. Mathematically, they can conveniently be described by a
hypergraph that indicates a possible generation procedure of these states;
alternatively, they can also be phrased in terms of a non-local stabilizer
formalism. In this paper, we explore the entanglement properties and
nonclassical features of hypergraph states. First, we identify the equivalence
classes under local unitary transformations for up to four qubits, as well as
important classes of five- and six-qubit states, and determine various
entanglement properties of these classes. Second, we present general conditions
under which the local unitary equivalence of hypergraph states can simply be
decided by considering a finite set of transformations with a clear
graph-theoretical interpretation. Finally, we consider the question whether
hypergraph states and their correlations can be used to reveal contradictions
with classical hidden variable theories. We demonstrate that various
noncontextuality inequalities and Bell inequalities can be derived for
hypergraph states.Comment: 29 pages, 5 figures, final versio
Multipartite purification protocols: upper and optimal bounds
A method for producing an upper bound for all multipartite purification
protocols is devised, based on knowing the optimal protocol for purifying
bipartite states. When applied to a range of noise models, both local and
correlated, the optimality of certain protocols can be demonstrated for a
variety of graph and valence bond states.Comment: 15 pages, 16 figures. v3: published versio
Graph Theory with Applications to Statistical Mechanics
This work will have two parts. The first will be related to various types of graph connectivity, and will consist of some exposition on the work of Andreas Holtkamp on local variants of vertex connectivity and edge connectivity in graphs. The second part will consist of an introduction to the field of physics known as percolation theory, which has to do with infinite connected components in certain types of graphs, which has numerous physical applications, especially in the field of statistical mechanics
Combinatorial laplacians and positivity under partial transpose
Density matrices of graphs are combinatorial laplacians normalized to have
trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320
(2006)). If the vertices of a graph are arranged as an array, then its density
matrix carries a block structure with respect to which properties such as
separability can be considered. We prove that the so-called degree-criterion,
which was conjectured to be necessary and sufficient for separability of
density matrices of graphs, is equivalent to the PPT-criterion. As such it is
not sufficient for testing the separability of density matrices of graphs (we
provide an explicit example). Nonetheless, we prove the sufficiency when one of
the array dimensions has length two (for an alternative proof see Wu,
\emph{Phys. Lett. A}\textbf{351} (2006), no. 1-2, 18--22).
Finally we derive a rational upper bound on the concurrence of density
matrices of graphs and show that this bound is exact for graphs on four
vertices.Comment: 19 pages, 7 eps figures, final version accepted for publication in
Math. Struct. in Comp. Sc
Combinatorial and Geometric Properties of Planar Laman Graphs
Laman graphs naturally arise in structural mechanics and rigidity theory.
Specifically, they characterize minimally rigid planar bar-and-joint systems
which are frequently needed in robotics, as well as in molecular chemistry and
polymer physics. We introduce three new combinatorial structures for planar
Laman graphs: angular structures, angle labelings, and edge labelings. The
latter two structures are related to Schnyder realizers for maximally planar
graphs. We prove that planar Laman graphs are exactly the class of graphs that
have an angular structure that is a tree, called angular tree, and that every
angular tree has a corresponding angle labeling and edge labeling.
Using a combination of these powerful combinatorial structures, we show that
every planar Laman graph has an L-contact representation, that is, planar Laman
graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that
planar Laman graphs and their subgraphs are the only graphs that can be
represented this way.
We present efficient algorithms that compute, for every planar Laman graph G,
an angular tree, angle labeling, edge labeling, and finally an L-contact
representation of G. The overall running time is O(n^2), where n is the number
of vertices of G, and the L-contact representation is realized on the n x n
grid.Comment: 17 pages, 11 figures, SODA 201
- …