14,146 research outputs found
Definitions of entanglement entropy of spin systems in the valence-bond basis
The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely
related to quantum entanglement. We investigate measures of entanglement
entropy based on transition graphs, which characterize state overlaps in the
overcomplete valence-bond basis. The transition graphs can be generated using
projector Monte Carlo simulations of ground states of specific hamiltonians or
using importance-sampling of valence-bond configurations of amplitude-product
states. We consider definitions of entanglement entropy based on the bonds or
loops shared by two subsystems (bipartite entanglement). Results for the
bond-based definition agrees with a previously studied definition using
valence-bond wave functions (instead of the transition graphs, which involve
two states). For the one dimensional Heisenberg chain, with uniform or random
coupling constants, the prefactor of the logarithmic divergence with the size
of the smaller subsystem agrees with exact results. For the ground state of the
two-dimensional Heisenberg model (and also Neel-ordered amplitude-product
states), there is a similar multiplicative violation of the area law. In
contrast, the loop-based entropy obeys the area law in two dimensions, while
still violating it in one dimension - both behaviors in accord with
expectations for proper measures of entanglement entropy.Comment: 9 pages, 8 figures. v2: significantly expande
Local Boxicity, Local Dimension, and Maximum Degree
In this paper, we focus on two recently introduced parameters in the
literature, namely `local boxicity' (a parameter on graphs) and `local
dimension' (a parameter on partially ordered sets). We give an `almost linear'
upper bound for both the parameters in terms of the maximum degree of a graph
(for local dimension we consider the comparability graph of a poset). Further,
we give an time deterministic algorithm to compute a local box
representation of dimension at most for a claw-free graph, where
and denote the number of vertices and the maximum degree,
respectively, of the graph under consideration. We also prove two other upper
bounds for the local boxicity of a graph, one in terms of the number of
vertices and the other in terms of the number of edges. Finally, we show that
the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page
Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI
With the objective of employing graphs toward a more generalized theory of
signal processing, we present a novel sampling framework for (wavelet-)sparse
signals defined on circulant graphs which extends basic properties of Finite
Rate of Innovation (FRI) theory to the graph domain, and can be applied to
arbitrary graphs via suitable approximation schemes. At its core, the
introduced Graph-FRI-framework states that any K-sparse signal on the vertices
of a circulant graph can be perfectly reconstructed from its
dimensionality-reduced representation in the graph spectral domain, the Graph
Fourier Transform (GFT), of minimum size 2K. By leveraging the recently
developed theory of e-splines and e-spline wavelets on graphs, one can
decompose this graph spectral transformation into the multiresolution low-pass
filtering operation with a graph e-spline filter, and subsequent transformation
to the spectral graph domain; this allows to infer a distinct sampling pattern,
and, ultimately, the structure of an associated coarsened graph, which
preserves essential properties of the original, including circularity and,
where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017
- …