the gauss bonnet chern theorem a probabilistic perspective we prove that the euler form of a metric connection on a real oriented vector bundle e over a compact oriented manifold m can be identified as a current with the expectation of the random current defined by the zero locus of a certain random section of the bundle we also explain how to reconstruct probabilistically the metric and the connection on e from the statistics of random sections of e contents

quantifying entanglement resources we present an overview of the quantitative theory of single copy entanglement in finite dimensional quantum systems in particular we emphasize the point of view that different entanglement measures quantify different types of resources which leads to a natural interdependence of entanglement classification and quantification apart from the theoretical basis we outline various methods for obtaining quantitative results on arbitrary mixed states this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

spectral decomposition of bell operators for multiqubit systems the spectral decomposition of multipartite bell operators for two dichotomic observables per site as introduced by werner and wolf phys rev a 64 032112 2001 is done implications on the characterization of bell inequalities as criteria of entanglement are discussed

a complete set of multidimensional bell inequalities we give a multidimensional generalization of the complete set of bell correlation inequalities given by werner and wolf 2001 phys rev a 64 032112 and by z ukowski and brukner 2002 phys rev lett 88 210401 for the two dimensional case our construction applies to the n party two observable case where each observable is d valued the inequalities obtained involve homogeneous polynomials they define the facets of a polytope in a complex vector space of dimension dn we detail the inequalities obtained in the case d 3 and from them we recover known inequalities we finally explain how the violations of our inequalities by quantum mechanics can be computed and could be observed when using unitary observables

looking for symmetric bell inequalities finding all bell inequalities for a given number of parties measurement settings and measurement outcomes is in general a computationally hard task we show that all bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full bell polytope as an illustration of our method we generate 238 885 new bell inequalities and 1085 new svetlichny inequalities we find in particular facet inequalities for bell experiments involving two parties and two measurement settings that are not of the collins gisin linden massar popescu type

on the number of nodal domains of random spherical harmonics let n f be a number of nodal domains of a random gaussian spherical harmonic f of degree n we prove that as n grows to infinity the mean of n f n 2 tends to a positive constant a and that n f n 2 exponentially concentrates around a this result is consistent with predictions made by bogomolny and schmit using a percolation like model for nodal domains of random gaussian plane waves

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

multipartite nonlocality as a resource and quantum correlations having indefinite causal order the characterization of quantum correlations in terms of information theoretic resource has been a fruitful approach to understand the power of quantum correlations as a resource while bipartite entanglement and bell inequality violation in this setting have been extensively studied relatively little is known about their multipartite counterpart in this paper we apply and adapt the recently proposed definitions of multipartite nonlocality bancal et al 2013 phys rev a 88 014102 to the three and four partite scenarios to gain new insight on the resource aspect of multipartite nonlocal quantum correlations specifically we show that reproducing certain tripartite quantum correlations requires mixtures of classical resources be it the ability to change the groupings or the time orderings of measurements thus when seen from the perspective of biseparable one way classical signaling resources certain tripartite quantum correlations do not admit a definite causal order in the four partite scenario we obtain a superset description of the set of biseparable correlations which can be produced by allowing two groups of bipartite non signaling resources quantum violation of the resulting bell like inequalities is investigated as a byproduct we obtain some new examples of device independent witnesses for genuine four partite entanglement and also device independent witnesses that allow one to infer the structure of the underlying multipartite entanglement this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

a framework for the study of symmetric full correlation bell like inequalities full correlation bell like inequalities represent an important subclass of bell like inequalities that have found applications in both a better understanding of fundamental physics and in quantum information science loosely speaking these are inequalities where only measurement statistics involving all parties play a role in this paper we provide a framework for the study of a large family of such inequalities that are symmetrical with respect to arbitrary permutations of the parties as an illustration of the power of our framework we derive i a new family of svetlichny inequalities for arbitrary numbers of parties settings and outcomes ii a new family of two outcome device independent entanglement witnesses for genuine n partite entanglement and iii a new family of two outcome tsirelson inequalities for arbitrary numbers of parties and settings we also discuss briefly the application of these new inequalities in the characterization of quantum correlations

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

lifting bell inequalities a bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers measurement settings or measurement outcomes in this article such liftings of bell inequalities are studied it is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope

bounding the set of quantum correlations we introduce a hierarchy of conditions necessarily satisfied by any distribution p ensuremath alpha ensuremath beta representing the probabilities for two separate observers to obtain outcomes ensuremath alpha and ensuremath beta when making local measurements on a shared quantum state each condition in this hierarchy is formulated as a semidefinite program among other applications our approach can be used to obtain upper bounds on the quantum violation of an arbitrary bell inequality it yields for instance tight bounds for the violations of the collins et al inequalities

bell type inequalities for partial separability in n particle systems and quantum mechanical violations we derive n particle bell type inequalities under the assumption of partial separability i e that the n particle system is composed of subsystems which may be correlated in any way e g entangled but which are uncorrelated with respect to each other these inequalities provide upon violation experi mentally accessible sufficient conditions for full n particle entanglement i e for n particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled the inequalities are shown to be maximally violated by the n particle greenberger horne zeilinger states

two scale kinematics and linearization for simultaneous two scale analysis of periodic heterogeneous solids at finite strain abstract we introduce the notion of two scale kinematics and the procedure of two scale linearization which are indispensable to the simultaneous two scale analysis method for the mechanical behavior of periodic heterogeneous solids at finite strain these are accomplished by formulating the two scale boundary value problem in both material and spatial descriptions with reference to the two scale modeling strategy developed in comput methods appl mech engrg 190 40 41 2001 5427 that utilized the convergence results of mathematical homogenization the formulation brings the intimate relationship between micro and macro scale kinematics in describing the micro macro coupling behavior inherent in heterogeneous media it is also shown that the two scale linearization necessitates the strict consistency with the micro scale equilibrated state and naturally invites the tangential homogenization process for both material and spatial descriptions several numerical examples of simultaneous two scale computations are presented to illustrate the two scale nature of the deformation of a heterogeneous solid at finite strain

inequalities for experimental tests of the kochen specker theorem we derive inequalities for n partite states under the assumption that the hidden variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one fine showed that this assumption is connected to the no hidden variables theorem of kochen and specker ks theorem these inequalities give a way to experimentally test the ks theorem the fidelity to the bell states which is larger than 1 2 is sufficient for the experimental confirmation of the ks theorem hence the werner state is enough to test experimentally the ks theorem furthermore it is possible to test the ks theorem experimentally using uncorrelated states an n partite uncorrelated state violates the n partite inequality derived here by an amount that grows exponentially with n

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

a stochastic gauss bonnet chern formula we prove that a gaussian ensemble of smooth random sections of a real vector bundle e over compact manifold m canonically defines a metric on e together with a connection compatible with it additionally we prove a refined gauss bonnet theorem stating that if the bundle e and the manifold m are oriented then the euler form of the above connection can be identified as a current with the expectation of the random current defined by the zero locus of a random section in the above gaussian ensemble

concurrent multi scale analysis of elastic composites by a multi level computational model abstract in this paper an adaptive multi level computational model is developed to comprehensively address issues of discretization and physically based modeling error in structures consisting of heterogeneous micro structures the model consists of three essential levels i a purely macroscopic domain level 0 with homogenized material parameters ii a macro micro domain level 1 with the micro domain represented by the periodic repetition of a representative volume element rve and iii purely microscopic domain level 2 where the rve ceases to exist and extended micro structural regions need to be modeled all micro structural computations of arbitrary heterogeneous domains are conducted using the adaptive voronoi cell finite element model an intermediate layer is used between the macroscopic and microscopic regions for smooth transition the level 1 domain is utilized to estimate criteria for switching from macroscopic to microscopic calculations physically based error indicators are developed for transition from macroscopic to microscopic domain statistical functions are used to assess the extent of the rve domains for non uniform micro structures for the macroscopic domains h and hp adaptations are executed using conventional error indicators based on the energy norm the numerical simulations are compared with results from pure micro mechanics and other multi level models in the literature the model is also tested for a realistic problem with macroscopic stress risers

convergent relaxations of polynomial optimization problems with noncommuting variables we consider optimization problems with polynomial inequality constraints in noncommuting variables these noncommuting variables are viewed as bounded operators on a hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints such problems arise naturally in quantum theory and quantum information science to solve them we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution we also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem

quantifying multipartite nonlocality the nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs the maximal number m of groups and the minimal number k of broadcasting parties that allow for the reproduction of a given set of correlations quantify their multipartite nonlocal content we show how upper bounds on m and lower bounds on k can be computed from the violation of the mermin svetlichny inequalities while n partite greenberger horne zeilinger states violate these inequalities maximally we find that w states violate them only by a very small amount

a two scale deformation model for polycrystalline solids using a strongly coupled finite element methodology a two scale finite element framework for analysis of polycrystalline solids at the continuum and crystal scales is demonstrated the framework is strongly coupled in the sense that the response at the continuum scale directly bears on the response at the crystal scale and vice versa data needed at one scale from computations at the other scale are generated on the fly two important issues are addressed the projection of continuum scale motion onto crystal scale aggregates and the averaging crystal scale stresses for use at the continuum scale the framework is implemented in a scalable parallel computing environment and applied to two examples with different geometries and loading modes it is shown that combining formulations for the crystal and continuum scales can provide more detailed and accurate results in comparison to a single scale finite element approach that invokes a simpler scale linking methodology

asymptotic violation of bell inequalities and distillability a multipartite quantum state violates a bell inequality asymptotically if after jointly processing by general local operations an arbitrarily large number of copies of it the result violates the inequality in the bipartite case we show that asymptotic violation of the clauser horne shimony holt inequality is equivalent to distillability hence bound entangled states do not violate it in the multipartite case we consider the complete set of full correlation bell inequalities with two dichotomic observables per site we show that asymptotic violation of any of these inequalities by a multipartite state implies that pure state entanglement can be distilled from it although the corresponding distillation protocol may require that some of the parties join into several groups we also obtain the extreme points of the set of distributions generated by measuring n quantum systems with two dichotomic observables per site

nonclassical correlations from randomly chosen local measurements we show that correlations inconsistent with any locally causal description can be a generic feature of measurements on entangled quantum states specifically spatially separated parties who perform local measurements on a maximally entangled state using randomly chosen measurement bases can with significant probability generate nonclassical correlations that violate a bell inequality for n parties using a greenberger horne zeilinger state this probability of violation rapidly tends to unity as the number of parties increases we also show that even with both a randomly chosen two qubit pure state and randomly chosen measurement bases a violation can be found about 10 of the time among other applications our work provides a feasible alternative for the demonstration of bell inequality violation without a shared reference frame

do all pure entangled states violate bell s inequalities for correlation functions any pure entangled state of two particles violates a bell inequality for two particle correlation functions gisin s theorem we show that there exist pure entangled ng2 qubit states that do not violate any bell inequality for n particle correlation functions for experiments involving two dichotomic observables per local measuring station we also find that mermin ardehali belinskii klyshko inequalities may not always be optimal for refutation of local realistic description

unbounded violation of tripartite bell inequalities we prove that there are tripartite quantum states constructed from random unitaries that can lead to arbitrarily large violations of bell inequalities for dichotomic observables as a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model this is in sharp contrast with the bipartite case where all violations are bounded by grothendieck s constant we will discuss the possibility of determining the hilbert space dimension from the obtained violation and comment on implications for communication complexity theory moreover we show that the violation obtained from generalized greenberger horne zeilinger ghz states is always bounded so that in contrast to many other contexts ghz states do not lead to extremal quantum correlations in this case in order to derive all these physical consequences we will have to obtain new mathematical results in the theories of operator spaces and tensor norms in particular we will prove the existence of bounded but not completely bounded trilinear forms from commutative c algebras finally we will relate the existence of diagonal states leading to unbounded violations with a long standing open problem in the context of banach algebras

a geometric comparison of entanglement and quantum nonlocality in discrete systems we compare entanglement with quantum nonlocality employing a geometric structure of the state space of bipartite qudits the central object is a regular simplex spanned by generalized bell states the collins gisin linden massar popescu bell inequality is used to reveal states of this set that cannot be described by local realistic theories optimal measurement settings necessary to ascertain nonlocality are determined by means of a recently proposed parameterization of the unitary group combined with robust numerical methods the main results of this paper are descriptive geometric illustrations of the state space that emphasize the difference between entanglement and quantum nonlocality namely it is found that the shape of the boundaries of separability and bell inequality violation are essentially different moreover it is also shown that for mixtures of states sharing the same amount of entanglement bell inequality violations and entanglement measures are non monotonically related

gisin s theorem for arbitrary dimensional multipartite states we present a set of bell inequalities which are sufficient and necessary for separability of general pure multipartite quantum states in arbitrary dimensions the relations between bell inequalities and distillability are also studied we show that any quantum states that violate one of these bell inequalities are distillable

generating facets for the cut polytope of a graph by triangular elimination the cut polytope of a graph arises in many fields although much is known about facets of the cut polytope of the complete graph very little is known for general graphs the study of bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or more generally the complete k partite graph lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope in this paper we introduce a lifting operation named triangular elimination applicable to the cut polytope of a wide range of graphs triangular elimination is a specific combination of zero lifting and fourier motzkin elimination using the triangle inequality we prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing the proof is based on a variation of the lifting lemma adapted to general graphs the result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph we also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

device independent witnesses of genuine multipartite entanglement we consider the problem of determining whether genuine multipartite entanglement was produced in an experiment without relying on a characterization of the systems observed or of the measurements performed we present an n partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states but which can be violated by n partite entangled states such as greenberger horne zeilinger states in contrast to traditional entanglement witnesses the violation of this inequality implies that the state is not biseparable independently of the hilbert space dimension and of the measured operators violation of this inequality does not imply however genuine multipartite nonlocality we show more generically how the problem of identifying genuine tripartite entanglement in a deviceindependent way can be addressed through semidefinite programming

universality results for zeros of random holomorphic sections in this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular hermitian holomorphic line bundles on a compact kahler complex space x namely under mild moment assumptions we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections in the case when x is a compact kahler manifold we also prove an off diagonal exponential decay estimate for the bergman kernels of a sequence of positive line bundles on x

spectral decomposition of bell s operators for qubits the spectral decomposition is given for the n qubit bell operators with two observables per qubit it is found that the eigenstates when non degenerate are n qubit ghz states even for those operators that do not allow the maximal violation of the corresponding inequality we present two applications of this analysis in particular we discuss the existence of pure entangled states that do not violate the mermin klyshko inequality for n 3

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

bell inequalities for arbitrarily high dimensional systems we develop a novel approach to bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy this is used to construct a family of bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise in particular our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

multipartite bell type inequalities for arbitrary numbers of settings and outcomes per site we introduce a single general representation incorporating in a unique manner all bell type inequalities for a multipartite correlation scenario with an arbitrary number of settings and any spectral type of outcomes at each site specifying this general representation for correlation functions we prove that the form of any correlation bell type inequality does not depend on spectral types of outcomes in particular on their numbers at different sites and is determined only by extremal values of outcomes at each site we also specify the general form of bounds in bell type inequalities on joint probabilities our approach to the derivation of bell type inequalities is universal concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type discrete or continuous we in particular prove that for an n partite quantum state possibly infinite dimensional admitting the setting lhv description the mermin klyshko inequality holds for any two bounded quantum observables per site not necessarily dichotomic

on symmetric nonlocal games abstract nonlocal games are used to display differences between the classical and quantum world in this paper we study symmetric xor games which form an important subset of nonlocal games we give simple methods for calculating the classical and the quantum values for symmetric xor games with one bit input per player we illustrate those methods with two examples one example is an n player game due to ardehali 1992 3 that provides the maximum quantum over classical advantage the second example comes from generalization of chsh game by letting the referee to choose arbitrary symmetric distribution of players inputs

family of bell like inequalities as device independent witnesses for entanglement depth we present a simple family of bell inequalities applicable to a scenario involving arbitrarily many parties each of which performs two binary outcome measurements we show that these inequalities are members of the complete set of full correlation bell inequalities discovered by werner wolf ifmmode dot z else z fi ukowski brukner for scenarios involving a small number of parties we further verify that these inequalities are facet defining for the convex set of bell local correlations moreover we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many body entangled in other words our bell inequalities when supplemented with the appropriate quantum bounds naturally serve as device independent witnesses for entanglement depth allowing one to certify genuine k partite entanglement in an arbitrary n ensuremath ge k partite scenario without relying on any assumption about the measurements being performed or the dimension of the underlying physical system a brief comparison is made between our witnesses and those based on some other bell inequalities as well as quantum fisher information a family of witnesses for genuine k partite nonlocality applicable to an arbitrary n ensuremath ge k partite scenario based on our bell inequalities is also presented

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions