injectivity of compressing maps on primitive sequences over bbz p e let zopf pe be the integer residue ring with odd prime p and integer eges2 for a sequence a over zopf pe one has a unique p adic expansion a a 0 a 1 p a e 1 pe 1 where a i can be regarded as a sequence over zopf p for 0lesilese 1 let f x be a strongly primitive polynomial over zopf pe and g f x pe be the set of all primitive sequences generated f x by over zopf pe recently the authors xuan yong zhu and wen feng qi have proved that for a function phi x0 xe 1 g xe 1 eta x0 xe 2 over zopf p and a b ising f x pe where 2lesdeg glesp 1 phi a 0 a 1 a e 1 phi b 0 b 1 b e 1 if and only if a b to further complete their work we show that such injectivity also holds for deg g 1 that is for a function phi x0 xe 1 xe 1 eta x0 xe 2 over zopf p and a b ising f x pe phi a 0 a 1 a e 1 phi b 0 b 1 b e 1 if and only if a b

interaction of light with the polarization devices a vectorial pauli algebraic approach a theoretical approach to the interaction between polarized light and polarization devices based on the vectorial and pure operatorial form of the pauli algebra is presented unlike the standard jones and mueller approaches this formalism is coordinate free i e it does not appeal to any matrix representation of the involved operators this vectorial approach establishes a mathematical bridge between the hilbert space of the density operators of the polarization states and the poincar space of their geometric representations and gives a rigorous justification of the handling of the interactions between the polarization states and polarization systems on the poincar sphere in the poincar ball in such an approach unlike the standard ones the three relevant quantities that characterize the interaction the gain the poincar vector of the outgoing light and its degree of polarization result straightforwardly in block in the pauli vectorial expressions of the density operator of the output state the final equations are symmetric compact and physically expressive a generalized form of malus law for any dichroic device and partially polarized light is obtained this way

bell type inequalities for nonlocal resources we present bipartite bell type inequalities which allow the two partners to use some nonlocal resource such inequalities can only be violated if the parties use a resource which is more nonlocal than the one permitted by the inequality we introduce a family of n input nonlocal machines which are generalizations of the well known pr popescu rohrlich box then we construct bell type inequalities that cannot be violated by strategies that use one of these new machines finally we discuss implications for the simulation of quantum states

bifurcations of relative periodic orbits in nls gp with a triple well potential abstract the nonlinear schrodinger gross pitaevskii nls gp equation is considered in the presence of three equally spaced potentials the problem is reduced to a finite dimensional hamiltonian system by a galerkin truncation families of oscillatory orbits are sought in the neighborhoods of the system s nine branches of standing wave solutions normal forms are computed in the neighborhood of these branches various hamiltonian hopf and saddle node bifurcations showing how the oscillatory orbits change as a parameter is increased numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and nls gp near the hamiltonian hopf bifurcations and some subtle disagreements near the saddle node bifurcations due to exponentially small terms in the asymptotics

distribution properties of compressing sequences derived from primitive sequences over bbz p e let z pe be the integer residue ring with odd prime p and integer e 2 any sequence a over z pe has a unique p adic expansion a a0 a1 p ae 1 pe 1 where ai can be regarded as a sequence over z p for 0 i e 1 let f x be a strongly primitive polynomial over z pe and a b be two primitive sequences generated by f x over z pe assume x0 xe 1 xe 1 x0 xe 2 is an e variable function over z p with the monomial p 1 2 xe 2 p 1 x1 p 1 not pearing in the expression of x0 x1 xe 2 it is shown that if there exists an s z p such that a0 t ae 1 t s if and only if b0 t be 1 t s for all nonnegative t with i 0 where is an m sequence determined by f x and a0 then a b this implies that for compressing sequences derived from primitive sequences generated by f x over z pe single element distribution is unique on all positions t with t 0 in particular when x0 x1 xe 2 0 it is a completion of the former result on the uniqueness of distribution of element 0 in highest level sequences

computationally efficient nonlinear bell inequalities for quantum networks the correlations in quantum networks have attracted strong interest with new types of violations of the locality the standard bell inequalities cannot characterize the multipartite correlations that are generated by multiple sources the main problem is that no computationally efficient method is available for constructing useful bell inequalities for general quantum networks in this work we show a significant improvement by presenting new explicit bell type inequalities for general networks including cyclic networks these nonlinear inequalities are related to the matching problem of an equivalent unweighted bipartite graph that allows constructing a polynomial time algorithm for the quantum resources consisting of bipartite entangled pure states and generalized greenberger horne zeilinger ghz states we prove the generic nonmultilocality of quantum networks with multiple independent observers using new bell inequalities the violations are maximal with respect to the presented tsirelson s bound for einstein podolsky rosen states and ghz states moreover these violations hold for werner states or some general noisy states our results suggest that the presented bell inequalities can be used to characterize experimental quantum networks

instabilities of breathers in a finite nls lattice abstract we study some aspects of the dynamics of unstable breathers in a three site discrete cubic nls chain with dirichlet boundary conditions we view breathers as fixed points of the energy in the reduced phase space obtained by eliminating directions related to the global phase symmetry of the system we use a combination of numerical calculations and morse theoretical arguments to see that there are two breathers that correspond to critical energies where the energy hypersurface changes its connectivity these breathers are elliptic hyperbolic fixed points of the reduced four dimensional system we compute the periodic orbits in their center manifolds lyapunov orbits and see evidence for homoclinic intersections of their stable and unstable manifolds we also examine the possibility of heteroclinic connections between lyapunov orbits these however appear not to exist for the energies near the energy where the energy hypersurface becomes connected

small violations of bell inequalities for multipartite pure random states for any finite number of parts measurements and outcomes in a bell scenario we estimate the probability of random n qudit pure states to substantially violate any bell inequality with uniformly bounded coefficients we prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity in addition we also prove that if the number of parts is at least 3 this probability also goes to zero as the local hilbert space dimension goes to infinity

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

the lindstedt poincare technique as an algorithm for computing periodic orbits the lindstedt poincare technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations it uses a nearby periodic orbit of the unperturbed differential equation as the first approximation we derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems the algorithm unlike the lindstedt poincare technique does not require the dynamical system to be a small perturbation of a solvable differential equation this makes it more broadly applicable the algorithm is quadratically convergent it works with equal facility as examples show irrespective of whether the periodic orbit is attracting or repelling or a saddle one of the examples presents what is possibly the most accurate computation of hill s orbit of lunation since its justly celebrated discovery in 1878

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

grothendieck s theorem past and present probably the most famous of grothendieck s contributions to banach space theory is the result that he himself described as the fundamental theorem in the metric theory of tensor products that is now commonly referred to as grothendieck s theorem gt in short or sometimes as grothendieck s inequality this had a major impact first in banach space theory roughly after 1968 then later on in c algebra theory roughly after 1978 more recently in this millennium a new version of gt has been successfully developed in the framework of operator spaces or non commutative banach spaces in addition gt independently surfaced in several quite unrelated fields in connection with bell s inequality in quantum mechanics in graph theory where the grothendieck constant of a graph has been introduced and in computer science where the grothendieck inequality is invoked to replace certain np hard problems by others that can be treated by semidefinite programming and hence solved in polynomial time in this expository paper we present a review of all these topics starting from the original gt we concentrate on the more recent developments and merely outline those of the first banach space period since detailed accounts of that are already available for instance the author s 1986 cbms notes

a relevant two qubit bell inequality inequivalent to the chsh inequality we computationally investigate the complete polytope of bell inequalities for two particles with small numbers of possible measurements and outcomes our approach is limited by pitowsky s connection of this problem to the computationally hard np problem despite this we find that there are very few relevant inequivalent inequalities for small numbers for example in the case with three possible 2 outcome measurements on each particle there is just one new inequality we describe mixed 2 qubit states which violate this inequality but not the chsh the new inequality also illustrates a sharing of bi partite non locality between three qubits something not seen using the chsh inequality it also inspires us to discover a class of bell inequalities with m possible n outcome measurements on each particle

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

correlation polytopes their geometry and complexity a family of polytopes correlation polytopes which arise naturally in the theory of probability and propositional logic is defined these polytopes are tightly connected to combinatorial problems in the foundations of quantum mechanics and to the ising spin model correlation polytopes exhibit a great deal of symmetry exponential size symmetry groups which leave the polytope invariant and act transitively on its vertices are defined using the symmetries a large family of facets is determined a conjecture concerning the full facet structure of correlation polytopes is formulated the conjecture however implies that np co np various complexity results are proved it is shown that deciding membership in a correlation polytope is an np complete problem and deciding facets is probably not even in np the relations between the polytope symmetries and its complexity are indicated

context invariant quasi hidden variable qhv modelling of all joint von neumann measurements for an arbitrary hilbert space we prove the existence for each hilbert space of the two new quasi hidden variable qhv models statistically noncontextual and context invariant reproducing all the von neumann joint probabilities via non negative values of real valued measures and all the quantum product expectations via the qhv classical like average of the product of the corresponding random variables in a context invariant model a quantum observable x can be represented by a variety of random variables satisfying the functional condition required in quantum foundations but each of these random variables equivalently models x under all joint von neumann measurements regardless of their contexts the proved existence of this model negates the general opinion that in terms of random variables the hilbert space description of all the joint von neumann measurements for dimh 3 can be reproduced only contextually the existence of a statistically noncontextual qhv model in particular implies that every n partite quantum state admits a local quasi hidden variable model introduced in loubenets j math phys 53 022201 2012 the new results of the present paper point also to the generality of the quasi classical probability model proposed in loubenets j phys a math theor 45 185306 2012

correlation inequalities and hidden variables a counterexample is given to a recent suggestion that if the inequalities of clauser and horne hold then there exists a hidden variable model for a spin textonehalf correlation experiment of the einstein podolsky rosen type

branches of solutions to semilinear elliptic equations on mathbb r n for a large class of functions f we consider the nonlinear elliptic eigenvalue problem we describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem a sharper result is obtained in the case of the lowest eigenvalue the discussion is based on the degree theory for c 2 proper fredholm maps developed by p m fitzpatrick j pejsachowicz and p j rabier

on the inequivalence of the ch and chsh inequalities due to finite statistics different variants of a bell inequality such as chsh and ch are known to be equivalent when evaluated on nonsignaling outcome probability distributions however in experimental setups the outcome probability distributions are estimated using a finite number of samples therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant we explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario and propose a method to optimize the statistical robustness of a bell inequality in the process we describe the finite group composed of relabeling of parties measurement settings and outcomes and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization signaling or having uniform marginals

bell inequalities from variable elimination methods tight bell inequalities are facets of pitowsky s correlation polytope and are usually obtained from its extreme points by solving the hull problem here we present an alternative method based on a combination of algebraic results on extensions of measures and variable elimination methods e g the fourier motzkin method our method is shown to overcome some of the computational difficulties associated with the hull problem in some non trivial cases moreover it provides an explanation for the arising of only a finite number of families of bell inequalities in measurement scenarios where one experimenter can choose between an arbitrary number of different measurements

multigraph approach to quantum non locality non contextuality nc and bell inequalities can be expressed as bounds for positive linear combinations s of probabilities of events exclusive events in s can be represented as adjacent vertices of a graph called the exclusivity graph of s in the case that events correspond to the outcomes of quantum projective measurements quantum probabilities are intimately related to the grotschel lovasz schrijver theta body of the exclusivity graph then one can easily compute an upper bound to the maximum quantum violation of any nc or bell inequality by optimizing s over the theta body and calculating the lovasz number of the corresponding exclusivity graph in some cases this upper bound is tight and gives the exact maximum quantum violation however in general this is not the case the reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in bell inequality and similar scenarios an interesting question is whether there is a graph theoretical concept which accounts for this problem here we show that for any given n partite bell inequality an edge coloured multigraph composed of n single colour graphs can be used to encode the relationships of exclusivity between each party s parts of the events then the maximum quantum violation of the bell inequality is exactly given by a refinement of the lovasz number that applies to these edge coloured multigraphs we show how to calculate upper bounds for this number using a hierarchy of semi definite programs and calculate upper bounds for i3 i3322 and the three bipartite bell inequalities whose exclusivity graph is a pentagon the multigraph theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

compression mappings on primitive sequences over z p sup e let z p sup e be the integer residue ring with odd prime p spl ges 5 and integer e spl ges 2 for a sequence a over z p sup e there is a unique p adic expansion a a sub 0 a spl middot p a sub e 1 spl middot p sup e 1 where each a sub i is a sequence over 0 1 p 1 and can be regarded as a sequence over the finite field gf p naturally let f x be a primitive polynomial over z p sup e and g f x p sup e the set of all primitive sequences generated by f x over z p sup e set spl phi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 1 x sub 0 x sub 1 x sub e 2 spl psi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 2 x sub 0 x sub 1 x sub e 2 where spl eta sub e 2 1 and spl eta sub e 2 2 are arbitrary functions of e 1 variables over gf p and 2 spl les k spl les p 1 then the compression mapping spl phi sub e 1 g f x p sup e spl rarr gf p sup spl infin a spl rarr spl phi sub e 1 a sub 0 a sub e 1 is injective that is a b if and only if spl phi sub e 1 a sub 0 a sub e 1 spl phi sub e 1 b sub 0 b sub e 1 for a b spl isin g f x p sup e furthermore if f x is a strongly primitive polynomial over z p sup e then spl phi sub e 1 a sub 0 a sub e 1 spl psi sub e 1 b sub 0 b sub e 1 if and only if a b and spl phi sub e 1 x sub 0 x sub e 1 spl psi sub e 1 x sub 0 x sub e 1 for a b spl isin g f x p sup e

local quasi hidden variable modelling and violations of bell type inequalities by a multipartite quantum state we introduce for a general correlation scenario a new simulation model a local quasi hidden variable lqhv model where locality and the measure theoretic construction inherent to a local hidden variable lhv model are preserved but positivity of a simulation measure is dropped we specify a necessary and sufficient condition for lqhv modelling and based on this prove that every quantum correlation scenario admits a lqhv simulation via the lqhv approach we construct analogs of bell type inequalities for an n partite quantum state and find a new analytical upper bound on the maximal violation by an n partite quantum state of s1 sn setting bell type inequalities either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type discrete or continuous this general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an s1 sn setting lhv de

grothendieck constant is norm of strassen matrix multiplication tensor we show that two important quantities from two disparate areas of complexity theory strassen s exponent of matrix multiplication omega and grothendieck s constant k g are different measures of size for the same underlying object the matrix multiplication tensor i e the 3 tensor or bilinear operator mu l m n mathbb f l times m times mathbb f m times n rightarrow mathbb f l times n a b mapsto ab defined by matrix matrix product over mathbb f mathbb r or mathbb c it is well known that strassen s exponent of matrix multiplication is the greatest lower bound on the log of the tensor rank of mu l m n we will show that grothendieck s constant is the least upper bound on a tensor norm of mu l m n taken over all l m n in mathbb n aside from relating the two celebrated quantities this insight allows us to rewrite grothendieck s inequality as a norm inequality begin aligned mu l m n 1 2 infty max x y m ne 0 frac mathrm tr xmy x 1 2 y 2 infty m infty 1 leqslant k g end aligned we prove that grothendieck s inequality is unique in the sense that if we generalize the 1 2 infty norm to arbitrary p q r in 1 infty begin aligned mu l m n p q r max x y m ne 0 frac mathrm tr xmy x p q y q r m r p end aligned then p q r 1 2 infty is up to cyclic permutations the only choice for which mu l m n p q r is uniformly bounded by a constant independent of l m n

on the no signaling approach to quantum nonlocality the no signaling approach to nonlocality deals with separable and inseparable multiparty correlations in the same set of probability states without conflicting causality the set of half spaces describing the polytope of no signaling probability states that are admitted by the most general class of bell scenarios is formulated in full detail an algorithm for determining the skeleton that solves the no signaling description is developed upon a new strategy that is partially pivoting and partially incremental the algorithm is formulated rigorously and its implementation is shown to be effective to deal with the highly degenerate no signaling descriptions several applications of the algorithm as a tool for the study of quantum nonlocality are mentioned applied to a large set of bipartite bell scenarios we found that the corresponding no signaling polytopes have a striking high degeneracy that grows up exponentially with the size of the bell scenario

grothendieck type inequalities in combinatorial optimization we survey connections of the grothendieck inequality and its variants to combinatorial optimization and computational complexity 2011 wiley periodicals inc

symmetry breaking bifurcation in the nonlinear schrodinger equation with symmetric potentials we consider the focusing attractive nonlinear schrodinger nls equation with an external symmetric potential which vanishes at infinity and supports a linear bound state we prove that the symmetric nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non degenerate local maxima at zero under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork in the particular case of double well potentials with large separation the power of nonlinearity determines the subcritical or supercritical character of the bifurcation the results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter

on the distinctness of modular reductions of maximal length sequences modulo odd prime powers we discuss the distinctness problem of the reductions modulo m of maximal length sequences modulo powers of an odd prime p where the integer m has a prime factor different from p for any two different maximal length sequences generated by the same polynomial we prove that their reductions modulo m are distinct in other words the reduction modulo m of a maximal length sequence is proved to contain all the information of the original sequence

invariant manifolds for a class of dispersive hamiltonian partial differential equations we construct an invariant manifold of periodic orbits for a class of non linear schrodinger equations using standard ideas of the theory of center manifolds we rederive the results of soffer and weinstein comm math phys 133 119 146 1997 j differential equations98 376 390 1992 on the large time asymptotics of small solutions scattering theory

perfectly matched layers in photonics computations 1d and 2d nonlinear coupled mode equations extending the general approach for first order hyperbolic systems developed in d appelo t hagstrom g kreiss perfectly matched layers for hyperbolic systems general formulation well posedness and stability siam j appl math 2006 to appear we construct pml equations for the mixed type system governing propagation of optical wave packets in both 1d and 2d bragg resonant photonic waveguides with a cubic nonlinearity i e the coupled mode equations we prove that in the linear case the layer equations are absorbing and perfectly matched we also prove they are stable for constant parameters a number of numerical experiments are performed to assess the layer s performance in both the linear and nonlinear regimes

stronger violation of local theories with equalities bell type inequalities are used to test local realism against quantum theory in this paper we consider a two party system with two settings and two possible outcomes on each side and derive equalities in local theories which are violated by quantum theory by a factor of 1 522 tolerating 0 586 fraction of white noise admixture which is twice that of the previous results

local hidden variable modelling classicality quantum separability and the original bell inequality we introduce a general condition sufficient for the validity of the original bell inequality 1964 in a local hidden variable lhv frame this condition can be checked experimentally and incorporates only as a particular case the assumption on perfect correlations or anticorrelations usually argued for this inequality in the literature specifying this general condition for a quantum bipartite case we introduce the whole class of bipartite quantum states separable and nonseparable that i admit an lhv description under any bipartite measurements with two settings per site ii do not necessarily exhibit perfect correlations and may even have a negative correlation function if the same quantum observable is measured at both sites but iii satisfy the perfect correlation version of the original bell inequality for any three bounded quantum observables a1 a2 b1 b2 at sites a and b respectively analysing the validity of this general lhv condition under classical and quantum correlation scenarios with the same physical context we stress that unlike the clauser horne shimony holt inequality the original bell inequality distinguishes between classicality and quantum separability

maximal violation of the collins gisin linden massar popescu inequality for infinite dimensional states we present a much simplified version of the collins gisin linden massar popescu inequality for the 2 ifmmode times else texttimes fi 2 ifmmode times else texttimes fi d bell scenario numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled while the best measurements are the same as conjectured best measurements for the maximally entangled state for very large values of d the inequality seems to reach its minimal value given by the probability constraints this gives numerical evidence for a tight quantum bell inequality or generalized csirelson inequality for the 2 ifmmode times else texttimes fi 2 ifmmode times else texttimes fi ensuremath infty scenario

hidden variables joint probability and the bell inequalities it is shown that the following statements about a quantum correlation experiment are mutually equivalent 1 there is a deterministic hidden variables model for the experiment 2 there is a factorizable stochastic model 3 there is one joint distribution for all observables of the experiment returning the experimental probabilities 4 there are well defined compatible joint distributions for all pairs and triples of commuting and noncommuting observables 5 the bell inequalities hold

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

loopholes in bell inequality tests of local realism bell inequalities are intended to show that local realist theories cannot describe the world a local realist theory is one where physical properties are defined prior to and independent of measurement and no physical influence can propagate faster than the speed of light quantum mechanical predictions for certain experiments violate the bell inequality while a local realist theory cannot and this shows that a local realist theory cannot give those quantum mechanical predictions however because of unexpected circumstances or loopholes in available experiment tests local realist theories can reproduce the data from these experiments this paper reviews such loopholes what effect they have on bell inequality tests and how to avoid them in experiment avoiding all these simultaneously in one experiment usually called a loophole free or definitive bell test remains an open task but is very important for technological tasks such as device independent security of quantum cryptography and ultimately for our understanding of the world this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

new concise upper bounds on quantum violation of general multipartite bell inequalities last years bounds on the maximal quantum violation of general bell inequalities were intensively discussed in the literature via different mathematical tools in the present paper we analyze quantum violation of general bell inequalities via the lqhv local quasi hidden variable modelling framework correctly reproducing the probabilistic description of every quantum correlation scenario the lqhv mathematical framework allows us to derive for all d and n a new upper bound 2d 1 n 1 on the maximal violation by an n qudit state of all general bell inequalities also new upper bounds on the maximal violation by an n qudit state of general bell inequalities for s settings per site these new upper bounds essentially improve all the known precise upper bounds on quantum violation of general multipartite bell inequalities for some s d and n the new upper bounds are attainable

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

branches of solutions to semilinear elliptic equations on r n these ecole polytechnique federale de lausanne epfl n 1700 1997 institut d analyse et calcul scientifiquechaire d analyse reference doi 10 5075 epfl thesis 1700print copy in library catalog record created on 2005 03 16 modified on 2016 08 08

long periodic orbits and invariant tori in a singularly perturbed hamiltonian system in this paper we study a singularly perturbed two degree of freedom hamiltonian system with a normally elliptic slow manifold we prove that the slow manifold persists but can have a large number e 1 of exponentially small e c e gaps we demonstrate the existence of kam tori in a neighborhood of the slow manifold in addition we investigate a bifurcation which describes the creation of a gap in the slow manifold and derive its normal form 2002 elsevier science b v all rights reserved

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 et al 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 suppl 8 329 45 to represent hidden deterministic behaviours quantum behaviours and no signalling behaviours 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 behaviours 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

geometry of the set of synchronous quantum correlations we provide a complete geometric description of the set of synchronous quantum correlations for the three experiment two outcome scenario we show that these correlations form a closed set moreover every correlation in this set can be realized using projection valued measures on a hilbert space of dimension no more than 16

normal form for the symmetry breaking bifurcation in the nonlinear schrodinger equation abstract we derive and justify a normal form reduction of the nonlinear schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double well symmetric potential we prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time dependent solutions of the nonlinear schrodinger equation over long but finite time intervals

further result of compressing maps on primitive sequences modulo odd prime powers let be the integer residue ring with odd prime and integer for a sequence over there is an unique adic expansion where each is a sequence over and can be regarded as a sequence over the prime field gf naturally let be a strongly primitive polynomial over and the set of all primitive sequences generated by over suppose that gf gf it is shown that any function in is an injective map from to gf and the derived sequences of different functions are also different that is if and only if and for and these injective functions in can be considered as good candidates for the keys of a stream cipher

classifying 50 years of bell inequalities since john s bell demonstrated the interest of studying linear combinations of probabilities in relation with the epr paradox in 1964 bell inequalities have lead to numerous developments unfortunately the description of bell inequalities is subject to several degeneracies which make any exchange of information about them unnecessarily hard here we analyze these degeneracies and propose a decomposition for bell like inequalities based on a set of reference expressions which is not affected by them these reference expressions set a common ground for comparing bell inequalities we provide algorithms based on finite group theory to compute this decomposition implementing these algorithms allows us to set up a compendium of reference bell like inequalities available online at www faacets com this website constitutes a platform where registered bell like inequalities can be explored new inequalities can be compared to previously known ones and relevant information on bell inequalities can be added in a collaborative manner this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

on the probabilistic description of a multipartite correlation scenario with arbitrary numbers of settings and outcomes per site we consistently formalize the probabilistic description of multipartite joint measurements performed on systems of any nature this allows us 1 to specify in probabilistic terms the difference between nonsignaling the einstein podolsky rosen epr locality and bell s locality 2 to introduce the notion of a local hidden variable lhv model for an s1 sn setting n partite correlation experiment with outcomes of any spectral type discrete or continuous and to prove both general and specifically quantum statements on an lhv simulation in an arbitrary multipartite case 3 to classify lhv models for a multipartite quantum state in particular to show that any n partite quantum state pure or mixed admits an arbitrary s1 1 1 setting lhv description 4 to evaluate a threshold visibility for an arbitrary bipartite noisy quantum state to admit an s1 s2 setting lhv description under any generalized quantum measurements of two parties

further result on distribution properties of compressing sequences derived from primitive sequences over bf z p e let p be an odd prime number e an integer greater than 1 and z pe the integer residue ring modulo pe in this paper we obtain an improved result of the previous paper ieee trans inf theory 56 1 2010 555 563 on distribution properties of compressing sequences derived from primitive sequences over z pe it is shown that two primitive sequences and b generated by a strongly primitive polynomial f x over z pe are the same if there exist s z p and k z p such that the distribution of in their compressing sequences ae 1 a0 ae 2 and be 1 b0 be 2 is coincident at the positions t with t k where x0 xe 2 is an e variable polynomial over z p with the coefficient of xe 2p 1 x1p 1x0p 1 not equal to 1 e p 1 2 and is an m sequence over z p determined by f x and a compared with the previous result this gives a more precise characterization on the positions of a compressing sequence i e of the form ae 1 a0 ae 2 derived from a primitive sequence a over z pe that completely determines a in particular the result is also true for the highest level sequence ae 1 by taking x0 xe 2 0

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

necessary conditions for extended noncontextuality in general sets of random variables in this paper we explore the graph approach to contextuality to restate the extended definition of noncontextuality as given by kujala et al phys rev lett 115 150401 2015 using graph theoretical terms this extended definition avoids the assumption of the pre sheaf or non disturbance condition which states that if two contexts overlap then the marginal distribution obtained for the intersection must be the same a restriction that will never be perfectly satisfied in real experiments with this we are able to derive necessary conditions for extended noncontextuality for any set of random variables based on the geometrical aspects of the graph approach which can be tested directly with experimental data in any contextuality experiment and which reduce to traditional necessary conditions for noncontextuality if the non disturbance condition is satisfied

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

generalizing optimal bell inequalities bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing identifying inequalities for many particles or measurements is however difficult due to the computational complexity of characterizing the set of local correlations we develop a method to characterize bell inequalities under constraints which may be given by symmetry or other linear conditions this allows one to search systematically for generalizations of given bell inequalities to more parties as an example we find all possible generalizations of the two particle inequality by froissart nuovo cimento soc ital fis b 64 241 1981 also known as i3322 inequality to three particles for the simplest of these inequalities we study their quantum mechanical properties and demonstrate that they are relevant in the sense that they detect nonlocality of quantum states for which all two setting inequalities fail to do so

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

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

visualizing two qubits the notions of entanglement witnesses and separable and entangled states for a two qubit system can be visualized in three dimensions using the slocc equivalence classes this visualization preserves the duality relations between the various sets and allows us to give proof by inspection of a nonelementary result of horodecki et al phys lett a 223 1 8 1996 that for two qubits peres ppt positive partial transpose test for separability is iff we then show that the chsh bell inequalities can be visualized as circles and cylinders in the same diagram this allows us to give a geometric proof of yet another result of horodecki et al phys lett a 200 340 344 1995 which optimizes the violation of the chsh bell inequality finally we give numerical evidence that remarkably allowing alice and bob to use three rather than two measurements each does not help them to distinguish any new entangled slocc equivalence class beyond the chsh class

local hidden variable models for entangled quantum states while entanglement and violation of bell inequalities were initially thought to be equivalent quantum phenomena we now have different examples of entangled states whose correlations can be described by local hidden variable models and therefore do not violate any of the bell inequalities we provide an up to date overview of the existing literature regarding local hidden variable models for entangled quantum states in both the bipartite and multipartite cases and discuss some of the most relevant open questions in this context our review covers twenty five years of this line of research beginning with the seminal work by werner 1989 phys rev a 40 8 which provided the first example of an entangled state with a local model werner s work in turn appeared twenty five years after the seminal work by bell 1964 physics 1 195 about the impossibility of recovering the predictions of quantum mechanics using a local hidden variable theory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

hamiltonian hopf bifurcations in the discrete nonlinear schrodinger trimer oscillatory instabilities quasi periodic solutions and a new type of self trapping transition oscillatory instabilities in hamiltonian anharmonic lattices are known to appear through hamiltonian hopf bifurcations of certain time periodic solutions of multibreather type here we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear the three site discrete nonlinear schrodinger model with periodic boundary conditions the stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes we numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types close to the lower amplitude threshold stable two frequency quasi periodic solutions exist surrounding the unstable stationary solution and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small by contrast close to the higher amplitude threshold all two frequency quasi periodic solutions are detached from the unstable stationary solution and the resulting dynamics is of population inversion type involving also the originally unexcited site

maximal non classicality in multi setting bell inequalities the discrepancy between maximally entangled states and maximally non classical quantum correlations is well known but still not well understood we aim to investigate the relation between quantum correlations and entanglement in a family of bell inequalities with n settings and d outcomes using analytical as well as numerical techniques we derive both maximal quantum violations and violations obtained from maximally entangled states furthermore we study the most non classical quantum states in terms of their entanglement entropy for large values of d and many measurement settings interestingly we find that the entanglement entropy behaves very differently depending on whether n 2 or n 2 when n 2 the entanglement entropy is a monotone function of d and the most non classical state is far from maximally entangled whereas when n 2 the entanglement entropy is a non monotone function of d and converges to that of the maximally entangled state in the limit of large d

two party bell inequalities derived from combinatorics via triangular elimination we establish a relation between the two party bell inequalities for two valued measurements and a high dimensional convex polytope called the cut polytope in polyhedral combinatorics using this relation we propose a method triangular elimination to derive tight bell inequalities from facets of the cut polytope this method gives two hundred million inequivalent tight bell inequalities from currently known results on the cut polytope in addition this method gives general formulae which represent families of infinitely many bell inequalities these results can be used to examine general properties of bell inequalities

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

bifurcation and stability for nonlinear schrodinger equations with double well potential in the semiclassical limit we consider the stationary solutions for a class of schrodinger equations with a symmetric double well potential and a nonlinear perturbation here in the semiclassical limit we prove that the reduction to a finite mode approximation give the stationary solutions up to an exponentially small term and that symmetry breaking bifurcation occurs at a given value for the strength of the nonlinear term the kind of bifurcation picture only depends on the nonlinearity power we then discuss the stability instability properties of each branch of the stationary solutions finally we consider an explicit one dimensional toy model where the double well potential is given by means of a couple of attractive dirac s delta pointwise interactions

a normal form for hamiltonian hopf bifurcations in nonlinear schroźdinger equations with general external potentials a normal form is derived for hamiltonian hopf bifurcations of solitary waves in nonlinear schro dinger equations with general external potentials this normal form is a simple second order nonlinear ordinary differential equation ode that is asymptotically accurate in describing solution dynamics near hamiltonian hopf bifurcations when the nonlinear coefficient in this normal form is complex which occurs if the second harmonic of the hopf bifurcation frequency falls inside the continuous spectrum of the system the solution of this normal form will blow up to infinity in finite time meaning that solution oscillations near hamiltonian hopf bifurcations will strongly amplify and eventually be destroyed when the nonlinear coefficient of the normal form is real the normal form can admit periodic solutions which correspond to long lasting solution oscillations in the original partial differential equation pde system quantitative comparison between the normal form s predictions and true pde solutions is also made in several numerical examples and good agreement is obtained

on the bound states of the nonlinear schrodinger equation with a linear potential abstract we study the nonlinear schrodinger equation with an attractive linear potential i t v x 2 0 n 2 nls which arises in the mathematical description of phenomena in nonlinear optics and plasma physics nonlinear bound states are finite energy localized solutions which if dynamically stable play an important role in the structure of general solutions of nls we discuss the existence and nonlinear orbital stability of nonlinear ground states of nls in particular if v supports a linear bound state then nls has stable nonlinear ground states in the supercritical and critical cases 2 n where blow up self focusing or collapse can occur this is a phenomenon not present in the case where v 0 in addition islands of stability in the regime of large h1 norm exist

quantum correlation and grothendieck s constant the best upper bound for the violation of the clauser home shimony holt chsh inequality was first derived by tsirelson for increasing number of 1 valued observables on both sites of the correlation experiment tsirelson obtained the grothendieck s constant k g 1 73 0 06 as a limit for the maximal violation in this paper we construct a generalization of the chsh inequality with four 1 valued observables on both sites of a correlation experiment and show that the quantum violation approaches 1 58 moreover we estimate the maximal quantum violation of a correlation experiment for large and equal number of 1 valued observables on both sites in this case the maximal quantum violation converges to 3 1 73 for very large n which coincides with the approximate value of grothendieck s constant

distribution properties of compressing sequences derived from primitive sequences modulo odd prime powers let a and b be primitive sequences over z pe with odd prime p and e 2 for certain compressing maps we consider the distribution properties of compressing sequences of a and b and prove that a b if the compressing sequences are equal at the times t such that t k where is a sequence related to a we also discuss the s uniform distribution property of compressing sequences for some compressing maps we have that there exist different primitive sequences such that the compressing sequences are s uniform we also discuss that compressing sequences can be s uniform for how many elements s

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

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

tight multipartite bell s inequalities involving many measurement settings which quantum states do not allow a local realistic lr description this question still remains open mainly because our present tools to test local realism are not optimal most of bell s inequalities are for the case in which only two measurement settings can be chosen by each observer e g the clauser horneshimony holt chsh inequality 1 inequalities for bipartite higher dimensional systems 2 multipartite bell s inequalities 3 6 one can call such inequalities standard ones one can expect that allowing the observers to choose between more than two observables should give more stringent constraints on lr models thus new inequalities may extend the class of nonseparable states which cannot be described by lr variables violation of local realism is an important ingredient for building quantum information protocols that decrease the communication complexity 7 and is a criterion for the efficient quantum key distribution 8 multisetting bell s inequalities may lead to such novel schemes although various nonstandard bell s inequalities are known 9 13 we still lack an efficient method for their derivation a general way is to define the facets of the correlation polytope 9 13 yet this is a computationally hard np problem 14 recently wu and zong 15 derived an inequality for three parties which involves four local settings for two observers and two settings for the third one and showed that it is stronger than the standard three particle bell inequalities here by generalizing the method of ref 15 we derive a wide class of tight bell s inequalities for n 2 parties and many measurement settings we find a single general bell s inequality that generates all inequalities from this class we also give a necessary and sufficient condition for the violation of the new inequalities by quantum predictions most importantly the inequalities reveal violation of local realism for the classes of states for which all standard bell s inequalities 5 6 are satisfied we start with the case ofn 3 observers suppose that the first two observers can choose between four settings and the third one between two settings we denote such problem as 4 4 2 l etai withi 2f 1 2 3 4g stand for the predetermined local realistic values for the first observer under the local settingi bj withj 2f 1 2 3 4g for similar values for the second observer and ck withk 2 f1 2g for the values for the third observer for the given run of the experiment we assume thata i b j a ndc k can take values 1 or 1 then the lr values satisfy the following algebraic identity a12 s 0 x k l 1 2 s 0 k l a1 1 k a 2 b1 1 l b 2 4

all clauser horne shimony holt polytopes the correlations that admit a local hidden variable model are described by a family of polytopes whose facets are the bell inequalities the clauser horne shimony holt chsh inequality is the simplest such bell inequality and is a facet of every bell polytope we investigate for which bell polytopes the chsh inequality is also the unique non trivial facet we prove that the chsh inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements irrespective of the number of measurement settings and outcomes for the other party based on numerical results we conjecture that it is also the unique facet for all bipartite polytopes involving two measurements per party where at least one measurement is dichotomic finally we remark that these two situations can be the only ones for which the chsh inequality is the unique facet i e any polytope that does not correspond to one of these two cases necessarily has facets that are not of the chsh form as a byproduct of our approach we derive a new family of facet inequalities this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

a new device independent dimension witness and its experimental implementation a dimension witness is a criterion that sets a lower bound on the dimension needed to reproduce the observed data three types of dimension witnesses can be found in the literature device dependent ones in which the bound is obtained assuming some knowledge on the state and the measurements device independent prepare and measure ones that can be applied to any system including classical ones and device independent bell based ones that certify the minimal dimension of some entangled systems here we consider the collins gisin linden massar popescu bell type inequality for four outcomes we show that a sufficiently high violation of this inequality witnesses d 4 and present a proof of principle experimental observation of such a violation this presents a first experimental violation of the third type of dimension witness beyond qutrits

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples