systematic encoding via grobner bases for a class of algebraic geometric goppa codes any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring the theory of grobner bases for modules gives a compact description and implementation of a systematic encoder we present examples of algebraic geometric goppa codes that can be encoded by these methods including the one point hermitian codes

non kolmogorov probability models and modified bell s inequality we analyze the proof of bell s inequality and demonstrate that this inequality is related to one particular model of probability theory namely kolmogorov measure theoretical axiomatics from 1933 we found a numerical statistical correction to bell s inequality such an additional term e on the right hand side of bell s inequality can be considered as a probability invariant of a quantum state this is a measure of nonreproducibility of hidden variables in different runs of experiments experiments to verify bell s inequality can be considered as just experiments to estimate the constant e it seems that bell s inequality could not be used as a crucial reason to deny local realism we consider deterministic as well as stochastic hidden variables models

lemma for linear feedback shift registers and dfts applied to affine variety codes in this paper we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of e g reed solomon codes algebraic geometry codes and affine variety codes our lemma corresponds to the nonsystematic encoding of affine variety codes and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a grobner basis and a generalized inverse discrete fourier transform we clarify that our lemma yields the error value estimation in the fast erasure and error decoding of a class of dual affine variety codes moreover we show that systematic encoding corresponds to a special case of erasure only decoding the lemma enables us to reduce the computational complexity of error evaluation from o n3 using gaussian elimination to o qn2 with some mild conditions on n and q where n is the code length and q is the finite field size

the extension problem for partial boolean structures in quantum mechanics alternative partial boolean structures implicit in the discussion of classical representability of sets of quantum mechanical predictions are characterized with definite general conclusions on the equivalence of the approaches going back to bell and kochen specker an algebraic approach is presented allowing for a discussion of partial classical extension amounting to reduction of the number of contexts classical representability arising as a special case as a result known techniques are generalized and some of the associated computational difficulties overcome the implications on the discussion of boole bell inequalities are indicated

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

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

entropic tests of multipartite nonlocality and state independent contextuality we introduce a multipartite extension of an information theoretic distance introduced by zurek nature london 341 119 1989 we use it to develop a new tool for studying quantum correlations from an information theoretic perspective in particular we derive entropic tests of multipartite nonlocality for three qubits and for an arbitrary even number of qubits as well as a test of state independent contextuality in addition we rederive the tripartite mermin inequality and a state independent noncontextuality inequality by cabello phys rev lett 101 210401 2008 this suggests that the information theoretic distance approach to multipartite nonlocality and state independent contextuality can provide a more general treatment of nonclassical correlations than the orthodox approach based on correlation functions

on pseudo hyperkahler prepotentials an explicit surjection from a set of locally defined unconstrained holomorphic functions on a certain submanifold of sp1 4n onto the set hkp q of local isometry classes of real analytic pseudo hyperkahler metrics of signature 4p 4q in dimension 4n is constructed the holomorphic functions called prepotentials are analogues of kahler potentials for kahler metrics and provide a complete parameterisation of hkp q in particular there exists a bijection between hkp q and the set of equivalence classes of prepotentials this affords the explicit construction of pseudo hyperkahler metrics from specified prepotentials the construction generalises one due to galperin ivanov ogievetsky and sokatchev their work is given a coordinate free formulation and complete self contained proofs are provided the appendix provides a vital tool for this construction a reformulation of real analytic g structures in terms of holomorphic frame fields on complex manifolds

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

quantum field theory and local contextual extensions it is shown that quantum field theory admits at least one local and causal extension which gives the same statistical predictions a general algebraic formalism for the study of such extensions is presented quantum field theory and the corresponding extensions are considered in the framework of the haag kastler approach 1964 obstruction due to the bell inequalities is overcome by the use of a generalized probability theory

computing the feng rao distances for codes from order domains abstract we compute the feng rao distance of a code coming from an order domain with a simplicial value semigroup the main tool is the apery set of a semigroup that can be computed using a grobner basis

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

joint measurements and svetlichny s inequality we prove that svetlichny s inequality can be derived from the existence of joint measurements and the principle of no signaling then we show that on the basis of the assumption of quantum measurement it would imply the breach of causality if the magnitude of violation of svetlichny s inequality exceeds the quantum bound

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

footprints or generalized bezout s theorem in two previous papers the first by feng rao berg and zhu see ibid vol 43 p 1799 810 1997 and the second by feng zhu shi and rao see proc 35th afferton conf communication control and computing p 205 14 1997 the authors use a generalization of bezout s theorem to estimate the minimum distance and generalized hamming weights for a class of error correcting codes obtained by evaluation of polynomials in points of an algebraic curve the main aim of this article is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so called footprint from grobner basis theory we also develop the theory further such that the minimum distance and the generalized hamming weights not only can be estimated but also can actually be determined

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

all bipartite entangled states display some hidden nonlocality one of the most significant and well known properties of entangled states is that they may lead to violations of bell inequalities and are thus inconsistent with any local realistic theory however there are entangled states that cannot violate any bell inequality and in general the precise relationship between entanglement and observable nonlocality is not well understood we demonstrate that a violation of the clauser horne shimony holt chsh inequality can be demonstrated in a certain kind of bell experiment for all entangled states our proof of the result consists of two main steps we first provide a simple characterization of the set of states that do not violate the chsh inequality even after general local operations and classical communication second we prove that for each entangled state there exists another state not violating the chsh inequality such that violates the chsh inequality

joint distributions quantum correlations and commuting observables we provide necessary and sufficient conditions for several observables to have a joint distribution when applied to the bivalent observables of a quantum correlation experiment we show that these conditions are equivalent to the bell inequalities and also to the existence of deterministic hidden variables we connect the no hidden variables theorem of kochen and specker to these conditions for joint distributions we conclude with a new theorem linking joint distributions and commuting observables and show how violations of the bell inequalities correspond to violations of commutativity as in the theorem

a serial in serial out hardware architecture for systematic encoding of hermitian codes via gro spl uml bner bases when a nontrivial permutation of a hermitian code is given the code will have a module structure over a polynomial ring of one variable by exploiting the theory of gro spl uml bner bases for modules a novel and elegant systematic encoding scheme for hermitian codes is proposed by heegard et al 1995 the goal of this paper is to develop a serial in serial out hardware architecture similar to a classical cyclic encoder for such a systematic encoding scheme moreover we demonstrate that under a specific permutation the upper bounds of the numbers of memory elements and constant multipliers in the proposed architecture are both proportional to o n where n is the length of the hermitian code to encode a codeword of length n this architecture takes n clock cycles without any latency therefore the hardware complexity of the proposed architecture is much less than that of the brute force systematic encoding by matrix multiplication

the two bell s theorems of john bell many of the heated arguments about the meaning of bell s theorem arise because this phrase can refer to two different theorems that john bell proved the first in 1964 and the second in 1976 his 1964 theorem is the incompatibility of quantum phenomena with the dual assumptions of locality and determinism his 1976 theorem is the incompatibility of quantum phenomena with the unitary property of local causality this is contrary to bell s own later assertions that his 1964 theorem began with that single and indivisible assumption of local causality even if not by that name while there are other forms of bell s theorems which i present to explain the relation between jarrett completeness fragile locality and epr completeness i maintain that bell s two versions are the essential ones although the two bell s theorems are logically equivalent their assumptions are not and the different versions of the theorem suggest quite different conclusions which are embraced by different communities for realists the notion of local causality ruled out by bell s 1976 theorem is motivated implicitly by reichenbach s principle of common cause and explicitly by the principle of relativistic causality and it is the latter which must be forgone operationalists pay no heed to reichenbach s principle but wish to keep the principle of relativistic causality which bolstered by an implicit principle of agent causation implies their notion of locality thus for operationalists bell s theorem is the 1964 one and implies that it is determinism that must be forgone i discuss why the two camps are drawn to these different conclusions and what can be done to increase mutual understanding

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

joint measurability of quantum effects and the matrix diamond in this work we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra free spectrahedra typically arise as matricial relaxations of linear matrix inequalities an example of a free spectrahedron is the matrix diamond which is a matricial relaxation of the l1 ball we find that joint measurability of binary positive operator valued measures is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study this connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements in particular we completely characterize the case in which the dimension is exponential in the number of measurements conversely we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond