903 research outputs found
The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states
We study entanglement properties of mixed density matrices obtained from
combinatorial Laplacians. This is done by introducing the notion of the density
matrix of a graph. We characterize the graphs with pure density matrices and
show that the density matrix of a graph can be always written as a uniform
mixture of pure density matrices of graphs. We consider the von Neumann entropy
of these matrices and we characterize the graphs for which the minimum and
maximum values are attained. We then discuss the problem of separability by
pointing out that separability of density matrices of graphs does not always
depend on the labelling of the vertices. We consider graphs with a tensor
product structure and simple cases for which combinatorial properties are
linked to the entanglement of the state. We calculate the concurrence of all
graph on four vertices representing entangled states. It turns out that for
some of these graphs the value of the concurrence is exactly fractional.Comment: 20 pages, 11 figure
On the homology of the space of knots
Consider the space of `long knots' in R^n, K_{n,1}. This is the space of
knots as studied by V. Vassiliev. Based on previous work of the authors, it
follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson
algebra. A partial description of a basis is given here. In addition, the mod-p
homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'.
Recursive application of this theorem allows us to deduce that there is
p-torsion of all orders in the integral homology of K_{3,1}.
This leads to some natural questions about the homotopy type of the space of
long knots in R^n for n>3, as well as consequences for the space of smooth
embeddings of S^1 in S^3.Comment: 36 pages, 6 figures. v3: small revisions before publicatio
Ising Model Observables and Non-Backtracking Walks
This paper presents an alternative proof of the connection between the
partition function of the Ising model on a finite graph and the set of
non-backtracking walks on . The techniques used also give formulas for
spin-spin correlation functions in terms of non-backtracking walks. The main
tools used are Viennot's theory of heaps of pieces and turning numbers on
surfaces.Comment: 33 pages, 11 figures. Typos and errors corrected, exposition
improved, results unchange
Combinatorial nullstellensatz and its applications
In 1999, Noga Alon proved a theorem, which he called the Combinatorial Nullstellensatz, that gives an upper bound to the number of zeros of a multivariate polynomial. The theorem has since seen heavy use in combinatorics, and more specifically in graph theory. In this thesis we will give an overview of the theorem, and of how it has since been applied by various researchers. Finally, we will provide an attempt at a proof utilizing a generalized version of the Combinatorial Nullstellensatz of the GM-MDS Conjecture
The algebraic structure behind the derivative nonlinear Schroedinger equation
The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schr\"
odinger equation (DNLSE) amongst others interesting and important nonlinear
integrable equations. In this paper, a general higher grading affine algebraic
construction of integrable hierarchies is proposed and the KN hierarchy is
established in terms of a Kac-Moody algebra and principal
gradation. In this form, our spectral problem is linear in the spectral
parameter. The positive and negative flows are derived, showing that some
interesting physical models arise from the same algebraic structure. For
instance, the DNLSE is obtained as the second positive, while the Mikhailov
model as the first negative flows, respectively. The equivalence between the
latter and the massive Thirring model is explicitly demonstrated also. The
algebraic dressing method is employed to construct soliton solutions in a
systematic manner for all members of the hierarchy. Finally, the equivalence of
the spectral problem introduced in this paper with the usual one, which is
quadratic in the spectral parameter, is achieved by setting a particular
automorphism of the affine algebra, which maps the homogeneous into principal
gradation.Comment: references adde
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