Nonadditivity of bipartite distillable entanglement follows from a conjecture on bound entangled Werner states

Assuming the validity of a conjecture given by DiVincenzo et al. [Phys. Rev. A 61, 062312 (2000)] and by Dür et al. [Phys. Rev. A 61, 062313 (2000)], we show that the distillable entanglement for two bipartite states, each of which individually has zero distillable entanglement, can be nonzero. We show that this also implies that the distillable entanglement is not a convex function. Our example consists of the tensor product of a bound entangled state based on an unextendible product basis with an entangled Werner state which lies in the class of conjectured undistillable states.     

The entanglement of several graph states

We exactly evaluate the entanglement of a six vertex and a nine vertex graph states which correspond to non ??two-colorable?? graphs. to non ??two-colorable?? graphs. The upper bound of entanglement for five vertex ring graph state is improved to 2. is improved to 2.9275, less than the upper bound determined by local operations and classical communication. communication. An upper bound of entanglement is proposed based on the definition of graph state. state.     

No signaling, entanglement breaking, and localizability in bipartite channels

A bipartite quantum channel represents the interaction between systems, generally allowing for the exchange of information. A special class of bipartite channels is the no-signaling ones, which do not allow for communication. Piani et al. [Phys. Rev. A 74, 012305 (2006)] conjectured that all no-signaling channels are mixtures of entanglement breaking and localizable channels, which require only local operations and entanglement. Here we provide the general realization scheme, and give a counterexample to the conjecture, achieving no-signaling superquantum correlations while preserving entanglement.     

A Lower Bound for Nodal Count on Discrete and Metric Graphs

We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under a certain genericity condition, we show that the number of nodal domains of the n ^th eigenfunction is bounded below by n  ?  ?, where ? is the number of links that distinguish the graph from a tree.Our results apply to operators on both discrete (combinatorial) and metric (quantum) graphs. They complement already known analogues of a result by Courant who proved the upper bound n for the number of nodal domains.To illustrate that the genericity condition is essential we show that if it is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.In the Appendix we review the proof of the case ?  =  0 on metric trees which has been obtained by other authors.     

Virtual Crystals and Kleber's Algorithm

 Kirillov and Reshetikhin conjectured what is now known as the fermionic formula for the decomposition of tensor products of certain finite dimensional modules over quantum affine algebras. This formula can also be extended to the case of q-deformations of tensor product multiplicities as recently conjectured by Hatayama et al. In its original formulation it is difficult to compute the fermionic formula efficiently. Kleber found an algorithm for the simply-laced algebras which overcomes this problem. We present a method which reduces all other cases to the simply-laced case using embeddings of affine algebras. This is the fermionic analogue of the virtual crystal construction by the authors, which is the realization of crystal graphs for arbitrary quantum affine algebras in terms of those of simply-laced type. Received: 10 September 2002 / Accepted: 22 January 2003 Published online: 7 May 2003 Communicated by L. Takhtajan     

Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems

In two papers Franz et al. proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution (Franz and Leone in J Stat Phys 111(3–4):535–564, 2003) and a general distribution (Franz et al. in J Phys A 36(43), 10967, 2003). Panchenko and Talagrand (Probab Theo Relat Fields 130(3):319–336, 2004) simplified the proof and generalized the result of Franz and Leone (J Stat Phys 111(3–4):535–564, 2003) for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks (Salez in Combin Probab Comput 25(03):436–447, 2016) and allows to bypass the arguments in Franz et al. (J Phys A 36(43):10967, 2003) based on the study of the Sherrington–Kirkpatrick (SK) model.     

Kolmogorov Basic Graphs and Their Application in Network Complexity Analysis

Throughout the years, measuring the complexity of networks and graphs has been of great interest to scientists. The Kolmogorov complexity is known as one of the most important tools to measure the complexity of an object. We formalized a method to calculate an upper bound for the Kolmogorov complexity of graphs and networks. Firstly, the most simple graphs possible, those with O(1) Kolmogorov complexity, were identified. These graphs were then used to develop a method to estimate the complexity of a given graph. The proposed method utilizes the simple structures within a graph to capture its non-randomness. This method is able to capture features that make a network closer to the more non-random end of the spectrum. The resulting algorithm takes a graph as an input and outputs an upper bound to its Kolmogorov complexity. This could be applicable in, for example evaluating the performances of graph compression methods.     

Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases

We prove upper bounds on the ground state energies of the one- and two-component charged Bose gases. The upper bound for the one-component gas agrees with the high density asymptotic formula proposed by L. Foldy in 1961. The upper bound for the two-component gas agrees in the large particle number limit with the asymptotic formula conjectured by F. Dyson in 1967. Matching asymptotic lower bounds for these systems were proved in references [10] and [11]. The formulas of Foldy and Dyson which are based on Bogolubov’s pairing theory have thus been validated.     

Network synchronizability analysis: The theory of subgraphs and complementary graphs

In this paper, subgraphs and complementary graphs are used to analyze network synchronizability. Some sharp and attainable bounds are derived for the eigenratio of the network structural matrix, which characterizes the network synchronizability, especially when the network’s corresponding graph has cycles, chains, bipartite graphs or product graphs as its subgraphs.     

Critical Behavior of the Annealed Ising Model on Random Regular Graphs

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by \(n^{3/4}\) converges to a specific random variable, with n the number of vertices of random regular graphs.     

Couplings and Asymptotic Exponentiality of Exit Times

The goal of this note is simply to call attention to the resulting simplification in the proof of asymptotic exponentiality of exit times in the Freidlin–Wentzell regime (as proved by F. Martinelli et al.) by using the coupling proposed by T. Lindvall and C. Rogers.     

General Properties of Thermal Entanglement in an Arbitrary-Length Heisenberg Spin Chain

We investigate general properties of thermal entanglement in arbitrary-length 1D Helsenberg spin-1/2 chain based on classifications of its eigenstates. The influences of magnetic field and temperature on entanglement are qualitatively discussed and three features are presented. The conclusions hold for both bipartite and multipartite entanglement, and are in agreement with the results numerically proven by Arnesen et al. [Phys. Rev. Lett. 50 （2001） 017901].     

Three-dimensional elastic wave scattering by a layer containing vertical periodic fractures

Elastic wave scattering off a layer containing a single set of vertical periodic fractures is examined using a numerical technique based on the work of Hennion et al. [J. Acoust. Soc. Am. 87, 1861-1870 (1990)]. This technique combines the finite element method and plane wave method to simulate three-dimensional scattering off a two-dimensional fractured layer structure. Each fracture is modeled explicitly, so that the model can simulate both discrete arrivals of scattered waves from individual fractures and multiply scattered waves between the fractures. Using this technique, we examine changes in scattering characteristics of plane elastic waves as a function of wave frequency, angle of incidence, and fracture properties such as fracture stiffness, height, and regular and irregular spacing.     

On a Graph Theoretic Formula of Gammelgaard for Berezin–Toeplitz Quantization

We give a proof of (a slightly refined version of) a graph theoretic formula due to Gammelgaard, Karabegov and Schlichenmaier for Berezin–Toeplitz quantization on Kähler manifolds. We obtain the formula by inverting the Berezin transform using a composition formula for the ring of differential operators encoded by linear combinations of strongly connected graphs. The same method is also used to identify the dual Karabegov–Bordemann–Waldmann star product. Our proof has the merit of giving more insight into Karabegov–Schlichenmaier’s identification theorem (Karabegov in J Reine Angew Math 540:49–76, 2001) that the Karabegov classifying form of the Berezin and Berezin–Toeplitz star products are, respectively, obtained by deforming the Kähler metric along the Ricci curvature and the logarithm of the Bergman kernel.     

Directed Dominating Set Problem Studied by Cavity Method: Warning Propagation and Population Dynamics

The minimal dominating set for a digraph (directed graph) is a prototypical hard combinatorial optimization problem. In a previous paper, we studied this problem using the cavity method. Although we found a solution for a given graph that gives very good estimate of the minimal dominating size, we further developed the one step replica symmetry breaking theory to determine the ground state energy of the undirected minimal dominating set problem. The solution space for the undirected minimal dominating set problem exhibits both condensation transition and cluster transition on regular random graphs. We also developed the zero temperature survey propagation algorithm on undirected Erdös-Rényi graphs to find the ground state energy. In this paper we continue to develope the one step replica symmetry breaking theory to find the ground state energy for the directed minimal dominating set problem. We find the following. (i) The warning propagation equation can not converge when the connectivity is greater than the core percolation threshold value of 3.704. Positive edges have two types warning, but the negative edges have one. (ii) We determine the ground state energy and the transition point of the Erdös-Rényi random graph. (iii) The survey propagation decimation algorithm has good results comparable with the belief propagation decimation algorithm.     

Oscillation of the NCO model

We study opinion oscillation of the nonconsensus opinion model (NCO) on graphs, and in particular on bipartite graphs. Using intensive numerical simulations, we investigate the relationship between amplitude A$A$ (the percentage of nodes whose opinions oscillate) and (p,q)$(p,q)$, which are the initial configuration fractions with opinion 1 on two sets of two bipartite graphs. Finally, for the general graph, we present several definitions and develop three propositions as regards whether an oscillation can occur or not on a certain graph.     

Bipartite graphs as models of complex networks

It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model which achieves the following challenges: it produces graphs which have the three main wanted properties (clustering, degree distribution, average distance), it is based on some real-world observations, and it is sufficiently simple to make it possible to prove its main properties. This model consists in sampling a random bipartite graph with prescribed degree distribution. Indeed, we show that any complex network may be viewed as a bipartite graph with some specific characteristics, and that its main properties may be viewed as consequences of this underlying structure. We also propose a growing model based on this observation.     

The operator product expansion for minimally subtracted operators

The operator product expansion is proved for minimally subtracted operators using an extension of a simplified version of Zimmermann's momentum subtraction proof. An explicit algorithm for calculating the coefficient functions is obtained, and it is shown that they are analytic in masses and can therefore be calculated perturbatively in asymptotically free theories. This result, which is implicitly assumed in many papers, has only recently been proved by Tkachov et al. who used a very different method.     

Evolutionary games defined at the network mesoscale: the Public Goods game

The evolutionary dynamics of the Public Goods game addresses the emergence of cooperation within groups of individuals. However, the Public Goods game on large populations of interconnected individuals has been usually modeled without any knowledge about their group structure. In this paper, by focusing on collaboration networks, we show that it is possible to include the mesoscopic information about the structure of the real groups by means of a bipartite graph. We compare the results with the projected (coauthor) and the original bipartite graphs and show that cooperation is enhanced by the mesoscopic structure contained. We conclude by analyzing the influence of the size of the groups in the evolutionary success of cooperation.     

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.