measuring inconsistency in probabilistic logic consistency independence and continuity are incompatible postulates minimal inconsistent sets are not suitable to analyze probabilistic incon sistencies independence can be weakened considering the underlying consolidation process inconsistency and incoherence measures based on distances and dutch books coincide inconsistency measures have been proposed as a way to manage inconsistent knowledge bases in the ai community to deal with inconsistencies in the context of conditional probabilistic logics rationality postulates and computational efficiency have driven the formulation of inconsistency measures independently investigations in formal epistemol ogy have used the betting concept of dutch book to measure an agent s degree of incoherence in this paper we show the impossibility of joint satisfiability of the proposed postulates proposing to replace them by more suitable ones thus we reconcile the rationality postulates for inconsistency measures in probabilistic bases and show that several inconsistency measures suggested in the literature and computable with linear programs satisfy the reconciled postulates additionally we give an interpretation for these feasible measures based on the formal epistemology concept of dutch book bridging the views of two so far separate communities in ai and philosophy in particular we show that incoherence degrees in formal epistemology may lead to novel approaches to inconsistency measures in the ai view

limits for entanglement measures the basic principle of entanglement processing says that entanglement cannot increase under local operations and classical communication based on this principle we show that any entanglement measure e suitable for the regime of a high number of identically prepared entangled pairs satisfies ed or e or ef where ed and ef are the entanglement of distillation and formation respectively moreover we exhibit a theorem establishing a very general form of bounds for distillable entanglement

a note on the realignment criterion for a quantum state in a bipartite system represented as a density matrix researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state we obtain bounds for elementary symmetric functions of singular values of realignment matrices this answers some open problems proposed by lupo aniello and scardicchio as a consequence we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension

tensor norms and the classical communication complexity of nonlocal quantum measurement we initiate the study of quantifying nonlocality of a bipartite measurement by the minimum amount of classical communication required to simulate the measurement we derive general upper bounds in terms of some tensor norms of the measurement operator as applications we show that a if the amount of communication is a constant then quantum and classical communication protocols with an unlimited amount of shared entanglement or shared randomness compute the same class of functions and b it requires only a constant amount of communication to classically generate an approximation of the output distribution resulting from local measurements on an entangled quantum state as long as the number of measurement outcomes is a constant

evenly distributed unitaries on the structure of unitary designs we clarify the mathematical structure underlying unitary t designs these are sets of unitary matrices evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group we present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design lower bounds for the number of elements of 2 designs are derived we show how to turn mutually unbiased bases into approximate 2 designs whose cardinality is optimal in leading order designs of higher order are discussed and an example of a unitary 5 design is presented we comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups further we sketch connections to problems in linear optics and questions regarding typical entanglement

anytime deduction for probabilistic logic abstract this paper proposes and investigates an approach to deduction in probabilistic logic using as its medium a language that generalizes the propositional version of nilsson s probabilistic logic by incorporating conditional probabilities unlike many other approaches to deduction in probabilistic logic this approach is based on inference rules and therefore can produce proofs to explain how conclusions are drawn we show how these rules can be incorporated into an anytime deduction procedure that proceeds by computing increasingly narrow probability intervals that contain the tightest entailed probability interval since the procedure can be stopped at any time to yield partial information concerning the probability range of any entailed sentence one can make a tradeoff between precision and computation time the deduction method presented here contrasts with other methods whose ability to perform logical reasoning is either limited or requires finding all truth assignments consistent with the given sentences

on the expressivity of inconsistency measures we survey recent approaches to inconsistency measurement in propositional logic and provide a comparative analysis in terms of their expressivity for that we introduce four different expressivity characteristics that quantitatively assess the number of different knowledge bases that a measure can distinguish our approach aims at complementing ongoing discussions on rationality postulates for inconsistency measures by considering expressivity as a desirable property we evaluate 16 different measures on the proposed characteristics and conclude that the distance based measure i dalal from grant and hunter 2013 8 and the proof based measure i p m from jabbour and raddaoui 2013 16 have maximal expressivity along all considered characteristics in our study we discovered several interesting relationships of inconsistency measurement to e g set theory and boolean functions and we also report these findings

multipartite generalization of the schmidt decomposition we find a canonical form for pure states of a general multipartite system in which the constraints on the coordinates with respect to a factorizable orthonormal basis are simply that certain ones vanish and certain others are real for identical particles they are invariant under permutations of the particles as an application we find the dimension of the generic local equivalence class

multiple summing operators on banach spaces abstract in this paper we improve some previous results about multiple p summing multilinear operators by showing that every multilinear form from l 1 spaces is multiple p summing for 1 p 2 the proof is based on the existence of a predual for the banach space of multiple p summing multilinear forms we also show the failure of the inclusion theorem in this class of operators and improve some results of y melendez and a tonge about dominated multilinear operators

on the distinguishability of random quantum states we develop two analytic lower bounds on the probability of success p of identifying a state picked from a known ensemble of pure states a bound based on the pairwise inner products of the states and a bound based on the eigenvalues of their gram matrix we use the latter and results from random matrix theory to lower bound the asymptotic distinguishability of ensembles of n random quantum states in d dimensions where n d approaches a constant in particular for almost all ensembles of n states in n dimensions p 0 72 an application to distinguishing boolean functions the oracle identification problem in quantum computation is given

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

the notion of weak contradiction definition and measures in this paper we present a way to represent contradiction between fuzzy sets this representation is given in terms of the notion of f weak contradiction unlike other approaches we do not define contradiction just by using one of the relations of f weak contradiction but by considering the whole set of relations this consideration avoids the need to fix an operator beforehand in order to take into account all the information between two fuzzy sets as a result we characterize the contradiction between fuzzy sets and define a family of measures of contradiction satisfying four interesting properties symmetry antitonicity if the intersection is empty then the measure is one and if there is an element in the intersection with degree of membership hbox 1 then the measure is zero

separability criterion induced from cross norm is not equivalent to positive partial transpose recently a computational criterion of separability induced by the greatest cross norm was proposed by rudolph 2002 preprint quant ph 0202121 there rudolph conjectured that the new criterion is neither weaker nor stronger than the positive partial transpose criterion for separability we show that there exists a counterexample to the this claim that is the proposed criterion is not equivalent to the positive partial transpose criterion

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

automorphic forms with degenerate fourier coefficients the main theme of this paper is that singular automorphic forms on classical groups are given by theta series liftings we establish several inequalities relating the automorphic multiplicities of a given representation and that of its abstract theta lift our methods allow one to lift noncuspidal square integrable automorphic forms when the dual pair involved is in the stable range in this way we construct new families of singular automorphic forms many of which are clearly unipotent in fact starting from one dimensional representations and repeating the procedure of lifting in the stable range one may obtain all automorphic forms which are quadratic unipotent in the sense of moeglin

on the largest bell violation attainable by a quantum state abstract we study the projective tensor norm as a measure of the largest bell violation of a quantum state in order to do this we consider a truncated version of a well known sdp relaxation for the quantum value of a two prover one round game one which has extra restrictions on the dimension of the sdp solutions our main result provides a quite accurate upper bound for the distance between the classical value of a bell inequality and the corresponding value of the relaxation along the way we give a simple proof that the best complementation constant of l 2 n in l 1 l is of order ln n as a direct consequence we show that we cannot remove a logarithmic factor when we are computing the largest bell violation attainable by the maximally entangled state

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

inconsistency measures for probabilistic logics inconsistencies in knowledge bases are of major concern in knowledge representation and reasoning in formalisms that employ model based reasoning mechanisms inconsistencies render a knowledge base useless due to the non existence of a model in order to restore consistency an analysis and understanding of inconsistencies are mandatory recently the field of inconsistency measurement has gained some attention for knowledge representation formalisms based on classical logic an inconsistency measure is a tool that helps the knowledge engineer in obtaining insights into inconsistencies by assessing their severity in this paper we investigate inconsistency measurement in probabilistic conditional logic a logic that incorporates uncertainty and focuses on the role of conditionals i e if then rules we do so by extending inconsistency measures for classical logic to the probabilistic setting further we propose novel inconsistency measures that are specifically tailored for the probabilistic case these novel measures use distance measures to assess the distance of a knowledge base to a consistent one and therefore takes the crucial role of probabilities into account we analyze the properties of the discussed measures and compare them using a series of rationality postulates

singular unitary representations of classical groups on presente une description de la partie relativement singuliere du dual unitaire d un groupe classique de type i sur un corps local f de caracteristique autre que 2

conditional logic and the principle of entropy abstract the conditional three valued logic of calabrese is applied to the language l of conditionals on propositional variables with finite domain the conditionals in l serve as a means for the construction and manipulation of probability distributions respecting the principle of maximum entropy and of minimum relative entropy this principle allows a sound inference even in the presence of uncertain evidence the inference is directed it respects a probabilistic version of modus ponens not of modus tollens it permits transitive chaining and supports a cautious monotony conjunctive conditional and material deduction are manageable in this probabilistic logic too the concept is not merely theoretical but enables large scale applications in the expert system shell spirit

extendible polynomials on banach spaces abstract we are concerned with the following question when can a polynomial p e x e and x are banach spaces be extended to a banach space containing e we prove that the polynomials that are extendible to any larger space are precisely those which can be extended to c b e if x is complemented in its bidual and l b e in general we also show that the extendibility is a property that is preserved by aron berner extensions and composition with linear operators we construct a predual of the space of extendible polynomials for the case that x is a dual space

generalized schmidt decomposition and classification of three quantum bit states we prove for any pure three quantum bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form this leads to a canonical form which generalizes the two quantum bit schmidt decomposition it is uniquely characterized by the five entanglement parameters it leads to a complete classification of the three quantum bit states it shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties

a separability criterion for density operators we give a necessary and sufficient condition for a mixed quantum mechanical state to be separable the criterion is formulated as a boundedness condition in terms of the greatest cross norm on the tensor product of trace class operators

pauli diagonal channels constant on axes we define and study the properties of channels which are analogous to unital qubit channels in several ways a full treatment can be given only when the dimension d pm a prime power in which case each of the d 1 mutually unbiased bases mub defines an axis along each axis the channel looks like a depolarizing channel but the degree of depolarization depends on the axis when d is not a prime power some of our results still hold particularly in the case of channels with one symmetry axis we describe the convex structure of this class of channels and the subclass of entanglement breaking channels we find new bound entangled states for d 3 for these channels we show that the multiplicativity conjecture for maximal output p norm holds for p 2 we also find channels with behaviour not exhibited by unital qubit channels including two pairs of orthogonal bases with equal output entropy in the absence of symmetry this provides new numerical evidence for the additivity of minimal output entropy

equivalence of norms on operator space tensor products of algebras the haagerup norm 11 ilh on the tensor product a x b of two c algebras a and b is shown to be banach space equivalent to eith er the banach space projective norm 11 11 or the operator space projective norm 11 iia if and only if either a or b is finite dimensional or a and b are infinite dimensional and subhomogeneous the banach space projective norm and the operator space projective nortn are equivalent on a x b if and only if a or b is subhomogeneous

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

unconditional basis and gordon lewis constants for spaces of polynomials abstract no infinite dimensional banach space x is known which has the property that for m 2 the banach space of all continuous m homogeneous polynomials on x has an unconditional basis following a program originally initiated by gordon and lewis we study unconditionality in spaces of m homogeneous polynomials and symmetric tensor products of order m in banach spaces we show that for each banach space x which has a dual with an unconditional basis x i the approximable nuclear m homogeneous polynomials on x have an unconditional basis if and only if the monomial basis with respect to x i is unconditional moreover we determine an asymptotically correct estimate for the unconditional basis constant of all m homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates x with the above property

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

multiplicativity of completely bounded p norms implies a new additivity result we prove additivity of the minimal conditional entropy associated with a quantum channel represented by a completely positive cp trace preserving map when the infimum of s 12 s 1 is restricted to states of the form mathcal i otimes phi left psi rangle langle psi right we show that this follows from multiplicativity of the completely bounded norm of considered as a map from l1 lp for lp spaces defined by the schatten p norm on matrices and give another proof based on entropy inequalities several related multiplicativity results are discussed and proved in particular we show that both the usual l1 lp norm of a cp map and the corresponding completely bounded norm are achieved for positive semi definite matrices physical interpretations are considered and a new proof of strong subadditivity is presented

maximal vectors in hilbert space and quantum entanglement let v be a norm closed subset of the unit sphere of a hilbert space h that is stable under multiplication by scalars of absolute value 1 a maximal vector for v is a unit vector h whose distance to v is maximum d v sup 1d v d v denoting the distance from to the set v maximal vectors generalize the maximally entangled unit vectors of quantum theory in general under a mild regularity hypothesis on v there is a norm on h whose restriction to the unit sphere achieves its minimum precisely on v and its maximum precisely on the set of maximal vectors this entanglement measuring norm is unique there is a corresponding entanglement measuring norm on the predual of b h that faithfully detects entanglement of normal states we apply these abstract results to the analysis of entanglement in multipartite tensor products h h1 hn and we calculate both entanglement measuring norms in cases for which dimhn is relatively large with respect to the others we describe the set of maximal vectors in explicit terms and show that it does not depend on the number of factors of the hilbert space h1 hn 1

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

geometry of quantum systems density states and entanglement various problems concerning the geometry of the space of hermitian operators on a hilbert space are addressed in particular we study the canonical poisson and riemann jordan tensors and the corresponding foliations into k hler submanifolds it is also shown that the space of density states on an n dimensional hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state thus the space of rank k states k 1 n is a smooth manifold of real dimension 2nk k2 1 and this stratification is maximal in the sense that every smooth curve in viewed as a subset of the dual to the lie algebra of the unitary group at every point must be tangent to the strata it crosses for a quantum composite system i e for a hilbert space decomposition an abstract criterion of entanglement is proved

an entanglement measure based on two order minors in this paper a new entanglement measure called emm the entanglement measure based on minors has been constructed by the convex roof method and proved to be a good entanglement measure according to the axiomatic point of view computation of emm can be finished directly by the two order minors of a coefficient matrix instead of the eigenvalues of the density operator required by most of the other entanglement measures so it is very fast and very easy on the other hand emm is related to the modified bures distance between and the closest separable state so it has geometry meaning we also investigate the relations between emm and the entanglement of formation negativity and logarithmic negativity and discover that emm is always smaller than or equal to them emm is equivalent to concurrence however their definitions and methods of proof are completely different emm will improve the efficiency of searching for maximally entangled multipartite states

combining probabilistic logic programming with the power of maximum entropy this paper is on the combination of two powerful approaches to uncertain reasoning logic programming in a probabilistic setting on the one hand and the information theoretical principle of maximum entropy on the other hand more precisely we present two approaches to probabilistic logic programming under maximum entropy the first one is based on the usual notion of entailment under maximum entropy and is defined for the very general case of probabilistic logic programs over boolean events the second one is based on a new notion of entailment under maximum entropy where the principle of maximum entropy is coupled with the closed world assumption cwa from classical logic programming it is only defined for the more restricted case of probabilistic logic programs over conjunctive events we then analyze the nonmonotonic behavior of both approaches along benchmark examples and along general properties for default reasoning from conditional knowledge bases it turns out that both approaches have very nice nonmonotonic features in particular they realize some inheritance of probabilistic knowledge along subclass relationships without suffering from the problem of inheritance blocking and from the drowning problem they both also satisfy the property of rational monotonicity and several irrelevance properties we finally present algorithms for both approaches which are based on generalizations of recent techniques for probabilistic logic programming under logical entailment the algorithm for the first approach still produces quite large weighted entrophy maximization problems while the one for the second approach generates optimization problems of the same size as the ones produced in probabilistic logic programming under logical entailment

first eigenvalue of laplace operator on locally symmetric space this paper is concerned with the lower bound for the first positive eigenvalue lambda 1 gamma of the laplace operator bigtriangleup on the space of square integrable functions on some locally symmetric space gamma backslash g k where g so 2n 1 mathbb c k is a maximal compact subgroup of g and gamma is a lattice in g which arises from some imaginary quadratic field it is well known that the casimir operator fancyscript c acts by a scalar lambda pi on every irreducible admissible representation pi of g by diximir s schur lemma in this paper we prove that when n ge 5 there exists a spectral gap between lambda 1 gamma and the infimum lambda 1 g of lambda pi when pi passes through all non trivial irreducible spherical unitary representations of g

on the congruence subgroup problem determination of the metaplectic kernel let f be a global field i e either a number field or a function field in one variable over a finite field and let oo be the set of its archimedean places let g be a finite non empty set of places of f containing oo let o o denote the ring of g integers of f let a denote the ring of g adeles i e the restricted direct product of the completions f v for r eg let f be an absolutely simple simply connected subgroup of sl defined over f recall that a subgroup f of c f is an g arithmetic subgroup if fnsl n o has finite index in f as well as in o fy f c sl n o an g arithmetic subgroup f is a congruence subgroup if there exists a non zero ideal a in o o g such that

summing multilinear operators by blocks the isotropic and anisotropic cases abstract unifying several directions of the development of the study of summing multilinear operators between banach spaces we construct a general framework that studies under one single definition multilinear operators that are summing with respect to sums taken over any number of indices iterated and non iterated sums the isotropic and the anisotropic cases sums over arbitrary blocks and over several different sequence norms a large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances ideal properties and coincidence theorems for the general classes are proved

a new class of entanglement measures we introduce new entanglement measures on the set of density operators on tensor product hilbert spaces these measures are based on the greatest cross norm on the tensor product of the sets of trace class operators on hilbert space we show that they satisfy the basic requirements on entanglement measures discussed in the literature including convexity invariance under local unitary operations and nonincrease under local quantum operations and classical communication

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

analysing inconsistent first order knowledgebases it is well known that knowledgebases may contain inconsistencies we provide a framework of measures based on a first order four valued logic to quantify the inconsistency of a knowledgebase this allows for the comparison of the inconsistency of diverse knowledgebases that have been represented as sets of first order logic formulae we motivate the approach by considering some examples of knowledgebases for representing and reasoning with ontological knowledge and with temporal knowledge analysing ontological knowledge including the statements about which concepts are subconcepts of other concepts and which concepts are disjoint can be problematical when there is a lack of knowledge about the instances that may populate the concepts and analysing temporal knowledge such as temporal integrity constraints can be problematical when considering infinite linear time lines isomorphic to the natural numbers or the real numbers or more complex structures such as branching time lines we address these difficulties by providing algebraic measures of inconsistency in first order knowledgebases

quantum entanglement and communication complexity we consider a variation of the communication complexity scenario where the parties are supplied with an extra resource particles in an entangled quantum state we note that quantum nonlocality can be naturally expressed in the language of communication complexity these are communication complexity problems where the output is embodied in the correlations between the outputs of the individual parties without entanglement the parties must communicate to produce the required correlations whereas with entanglement no communication is necessary to produce the correlations in this sense nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication we show how to transform examples of nonlocality into more traditional communication complexity problems where the output is explicitly determined by each individual party the resulting problems require communication with or without entanglement but the required communication is less when entanglement is available all these results are a noteworthy contrast to the well known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties

linear rank preservers of tensor products of rank one matrices abstract let n 1 n k be integers larger than or equal to 2 we characterize linear maps m n 1 n k m n 1 n k such that r a n k a 1 a k 1 whenever r a n k a 1 a k 1 for all a i m n i i 1 k applying this result we extend two recent results on linear maps that preserve the rank of special classes of matrices

generalized schmidt decomposability and its relation to projective norms in multipartite entanglement projective norms are capable of measuring entanglement of multipartite quantum states however typically the explicit computation of these distance based geometric entanglement monotones is very difficult even for finite dimensional systems motivated by the significance of schmidt decompositions for our quantitative understanding of bipartite quantum entanglement a generalization of this concept to multipartite scenarios is proposed in the sense that generalized schmidt decomposability of a multipartite pure state implies that its projective norm can be calculated in a simple way analogous to the bipartite case thus this concept of generalized schmidt decomposability of multipartite quantum states is linked in a natural way to projective norms as entanglement monotones therefore it may not only be a convenient tool for calculations but may also shed new light onto the intricate features of multipartite entanglement in an analogous way as the classical schmidt decomposition does for bipartite quantum systems

operator monotones the reduction criterion and the relative entropy we introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy we present applications of this new inequality and in particular we prove a new lower bound on the relative entropy of entanglement and other properties of entanglement measures

bipartite quantum systems on the realignment criterion and beyond inspired by the computable cross norm or realignment criterion we propose a new point of view about the characterization of the states of bipartite quantum systems we consider a schmidt decomposition of a bipartite density operator the corresponding schmidt coefficients or the associated symmetric polynomials are regarded as quantities that can be used to characterize bipartite quantum states in particular starting from the realignment criterion a family of necessary conditions for the separability of bipartite quantum states are derived we conjecture that these conditions which are weaker than the parent criterion can be strengthened in such a way to obtain a new family of criteria that are independent of the original one this conjecture is supported by numerical examples for the low dimensional cases these ideas can be applied to the study of quantum channels leading to a relation between the rate of contraction of a map and its ability to preserve entanglement

hahn banach extension of multilinear forms and summability the aim of this paper is to investigate close relations between the validity of hahn banach extension theorems for multilinear forms on banach spaces and summability properties of sequences from these spaces a case of particular importance occurs when we consider banach spaces which have the property that every bilinear form extends to any superspace

polynomials in operator space theory the aim of this article is to start a metric theory of homogeneous polynomials in the category of operator spaces for this purpose we take advantage of the basic fact that the space pm e of all m homogeneous polynomials on a vector space e can be identified with the algebraic dual of the m th symmetric tensor product m se given an operator space v we study several different types of completely bounded polynomials on v which form the operator space duals of m sv endowed with related operator structures of special interest are what we call haagerup kronecker and schur polynomials polynomials associated with different types of matrix products

a notion of rank for unitary representations of general linear groups a notion of rank for unitary representations of general linear groups over a locally compact nondiscrete field is defined rank measures how singular a representation is when restricted to the unipotent radical of a maximal parabolic subgroup irreducible representations of small rank are classified it is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations i definition and basic properties of rank 0 introduction the unitary dual of gln f f a locally compact nondiscrete field has been classified by d vogan v in the archimedean case and up to the construction of all cuspidal representations by m tadic in the non archimedean case much remains to be done however in terms of the study of specific properties of the unitary representations involved the inspiration for this paper comes form the following well known theorem of howe and moore theorem 0 1 hm let g be a zariski connected reductive algebraic group over a nondiscrete locally compact field f and let p be a unitary representation of dimension greater than 1 such that the center z of g acts by a character then the absolute values of the matrix coefficients of p approach zero at infinity on g z an easy consequence of this is corollary 0 2 in the hypotheses for theorem 0 1 if n is a subgroup of g which is not compact modulo z any vector that is invariant under n transforms according to a character of g o in particular if p is irreducible and has a nontrivial n invariant vector p must be a character received by the editors august 15 1988 1980 mathematics subject classification 1985 revisiont primary 22e46 22e50 this paper was submitted in a different form in partial fulfillment of the requirements for doctor of philosophy in the department of mathematics of yale university the author acknowledges support from the consiglio nazionale delle ricerche italy 1990 american mathematical society 0002 9947 90 1 00 25 per page

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

the uniqueness theorem for entanglement measures we explore and develop the mathematics of the theory of entanglement measures after a careful review and analysis of definitions of preliminary results and of connections between conditions on entanglement measures we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von neumann entropy on pure states we also prove several versions of a theorem on extreme entanglement measures in the case of mixed states we analyze properties of the asymptotic regularization of entanglement measures proving for example convexity for the entanglement cost and for the regularized relative entropy of entangle ment

unextendible product bases and bound entanglement an unextendible product basis upb for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state we give examples of upbs and show that the uniform mixed state over the subspace complementary to any upb is a bound entangled state we exhibit a tripartite 2 3 2 3 2 upb whose complementary mixed state has tripartite entanglement but no bipartite entanglement i e all three corresponding 2 3 4 bipartite mixed states are unentangled we show that members of a upb are not perfectly distinguishable by local positive operator valued measurements and classical communication s0031 9007 99 09360 6

on the measure of conflicts shapley inconsistency values there are relatively few proposals for inconsistency measures for propositional belief bases however inconsistency measures are potentially as important as information measures for artificial intelligence and more generally for computer science in particular they can be useful to define various operators for belief revision belief merging and negotiation the measures that have been proposed so far can be split into two classes the first class of measures takes into account the number of formulae required to produce an inconsistency the more formulae required to produce an inconsistency the less inconsistent the base the second class takes into account the proportion of the language that is affected by the inconsistency the more propositional variables affected the more inconsistent the base both approaches are sensible but there is no proposal for combining them we address this need in this paper our proposal takes into account both the number of variables affected by the inconsistency and the distribution of the inconsistency among the formulae of the base our idea is to use existing inconsistency measures in order to define a game in coalitional form and then to use the shapley value to obtain an inconsistency measure that indicates the responsibility contribution of each formula to the overall inconsistency in the base this allows us to provide a more reliable image of the belief base and of the inconsistency in it

counterexamples to the maximal p norm multiplicativity conjecture for all p 1 for all p 1 we demonstrate the existence of quantum channels with non multiplicative maximal output p norms equivalently for all p 1 the minimum output renyi entropy of order p of a quantum channel is not additive the violations found are large in all cases the minimum output renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors since p 1 corresponds to the von neumann entropy these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true it cannot be proved as a consequence of any channel independent guarantee of maximal p norm multiplicativity we also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p 2

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces