164 research outputs found
Bounds on Quantum Correlations in Bell Inequality Experiments
Bell inequality violation is one of the most widely known manifestations of
entanglement in quantum mechanics; indicating that experiments on physically
separated quantum mechanical systems cannot be given a local realistic
description. However, despite the importance of Bell inequalities, it is not
known in general how to determine whether a given entangled state will violate
a Bell inequality. This is because one can choose to make many different
measurements on a quantum system to test any given Bell inequality and the
optimization over measurements is a high-dimensional variational problem. In
order to better understand this problem we present algorithms that provide, for
a given quantum state, both a lower bound and an upper bound on the maximal
expectation value of a Bell operator. Both bounds apply techniques from convex
optimization and the methodology for creating upper bounds allows them to be
systematically improved. In many cases these bounds determine measurements that
would demonstrate violation of the Bell inequality or provide a bound that
rules out the possibility of a violation. Examples are given to illustrate how
these algorithms can be used to conclude definitively if some quantum states
violate a given Bell inequality.Comment: 13 pages, 1 table, 2 figures. Updated version as published in PR
Metric trees of generalized roundness one
Every finite metric tree has generalized roundness strictly greater than one.
On the other hand, some countable metric trees have generalized roundness
precisely one. The purpose of this paper is to identify some large classes of
countable metric trees that have generalized roundness precisely one.
At the outset we consider spherically symmetric trees endowed with the usual
combinatorial metric (SSTs). Using a simple geometric argument we show how to
determine decent upper bounds on the generalized roundness of finite SSTs that
depend only on the downward degree sequence of the tree in question. By
considering limits it follows that if the downward degree sequence of a SST satisfies , then has generalized roundness one. Included among the
trees that satisfy this condition are all complete -ary trees of depth
(), all -regular trees () and inductive limits
of Cantor trees.
The remainder of the paper deals with two classes of countable metric trees
of generalized roundness one whose members are not, in general, spherically
symmetric. The first such class of trees are merely required to spread out at a
sufficient rate (with a restriction on the number of leaves) and the second
such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table
Euclidean Distances, soft and spectral Clustering on Weighted Graphs
We define a class of Euclidean distances on weighted graphs, enabling to
perform thermodynamic soft graph clustering. The class can be constructed form
the "raw coordinates" encountered in spectral clustering, and can be extended
by means of higher-dimensional embeddings (Schoenberg transformations).
Geographical flow data, properly conditioned, illustrate the procedure as well
as visualization aspects.Comment: accepted for presentation (and further publication) at the ECML PKDD
2010 conferenc
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
Universality-class dependence of energy distributions in spin glasses
We study the probability distribution function of the ground-state energies
of the disordered one-dimensional Ising spin chain with power-law interactions
using a combination of parallel tempering Monte Carlo and branch, cut, and
price algorithms. By tuning the exponent of the power-law interactions we are
able to scan several universality classes. Our results suggest that mean-field
models have a non-Gaussian limiting distribution of the ground-state energies,
whereas non-mean-field models have a Gaussian limiting distribution. We compare
the results of the disordered one-dimensional Ising chain to results for a
disordered two-leg ladder, for which large system sizes can be studied, and
find a qualitative agreement between the disordered one-dimensional Ising chain
in the short-range universality class and the disordered two-leg ladder. We
show that the mean and the standard deviation of the ground-state energy
distributions scale with a power of the system size. In the mean-field
universality class the skewness does not follow a power-law behavior and
converges to a nonzero constant value. The data for the Sherrington-Kirkpatrick
model seem to be acceptably well fitted by a modified Gumbel distribution.
Finally, we discuss the distribution of the internal energy of the
Sherrington-Kirkpatrick model at finite temperatures and show that it behaves
similar to the ground-state energy of the system if the temperature is smaller
than the critical temperature.Comment: 15 pages, 20 figures, 1 tabl
Enhancement of Stochastic Resonance in distributed systems due to a selective coupling
Recent massive numerical simulations have shown that the response of a
"stochastic resonator" is enhanced as a consequence of spatial coupling.
Similar results have been analytically obtained in a reaction-diffusion model,
using "nonequilibrium potential" techniques. We now consider a field-dependent
diffusivity and show that the "selectivity" of the coupling is more efficient
for achieving stochastic-resonance enhancement than its overall value in the
constant-diffusivity case.Comment: 10 pgs (RevTex), 4 figures, submitted to Phys.Rev.Let
Lassoing and corraling rooted phylogenetic trees
The construction of a dendogram on a set of individuals is a key component of
a genomewide association study. However even with modern sequencing
technologies the distances on the individuals required for the construction of
such a structure may not always be reliable making it tempting to exclude them
from an analysis. This, in turn, results in an input set for dendogram
construction that consists of only partial distance information which raises
the following fundamental question. For what subset of its leaf set can we
reconstruct uniquely the dendogram from the distances that it induces on that
subset. By formalizing a dendogram in terms of an edge-weighted, rooted
phylogenetic tree on a pre-given finite set X with |X|>2 whose edge-weighting
is equidistant and a set of partial distances on X in terms of a set L of
2-subsets of X, we investigate this problem in terms of when such a tree is
lassoed, that is, uniquely determined by the elements in L. For this we
consider four different formalizations of the idea of "uniquely determining"
giving rise to four distinct types of lassos. We present characterizations for
all of them in terms of the child-edge graphs of the interior vertices of such
a tree. Our characterizations imply in particular that in case the tree in
question is binary then all four types of lasso must coincide
A generalization of the concept of distance based on the simplex inequality
We introduce and discuss the concept of n-distance, a generalization to n elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality
d(x1,…,xn)≤K∑i=1nd(x1,…,xn)zi,x1,…,xn,z∈X,
where K=1. Here d(x1,…,xn)zi is obtained from the function d(x1,…,xn) by setting its ith variable to z. We provide several examples of n-distances, and for each of them we investigate the infimum of the set of real numbers K∈]0,1] for which the inequality above holds. We also introduce a generalization of the concept of n-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function
On the Relationship between Convex Bodies Related to Correlation Experiments with Dichotomic Observables
In this paper we explore further the connections between convex bodies
related to quantum correlation experiments with dichotomic variables and
related bodies studied in combinatorial optimization, especially cut polyhedra.
Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J.
Phys. A: Math. Gen. 38 10971-87) with respect to Bell inequalities. We show
that several well known bodies related to cut polyhedra are equivalent to
bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329-45) to
represent hidden deterministic behaviors, quantum behaviors, and no-signalling
behaviors. Among other things, our results allow a unique representation of
these bodies, give a necessary condition for vertices of the no-signalling
polytope, and give a method for bounding the quantum violation of Bell
inequalities by means of a body that contains the set of quantum behaviors.
Optimization over this latter body may be performed efficiently by semidefinite
programming. In the second part of the paper we apply these results to the
study of classical correlation functions. We provide a complete list of tight
inequalities for the two party case with (m,n) dichotomic observables when
m=4,n=4 and when min{m,n}<=3, and give a new general family of correlation
inequalities.Comment: 17 pages, 2 figure
Nonequilibrium phase transitions induced by multiplicative noise: effects of self-correlation
A recently introduced lattice model, describing an extended system which
exhibits a reentrant (symmetry-breaking, second-order) noise-induced
nonequilibrium phase transition, is studied under the assumption that the
multiplicative noise leading to the transition is colored. Within an effective
Markovian approximation and a mean-field scheme it is found that when the
self-correlation time of the noise is different from zero, the transition is
also reentrant with respect to the spatial coupling D. In other words, at
variance with what one expects for equilibrium phase transitions, a large
enough value of D favors disorder. Moreover, except for a small region in the
parameter subspace determined by the noise intensity and D, an increase in the
self-correlation time usually preventsthe formation of an ordered state. These
effects are supported by numerical simulations.Comment: 15 pages. 9 figures. To appear in Phys.Rev.
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