constructing exact symmetric informationally complete measurements from numerical solutions recently several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements sic povms or sics and algebraic number theory these conjectures relate the sics to their minimal defining algebraic number field testing or sharpening these conjectures requires that the sics are expressed exactly rather than as numerical approximations while many exact solutions of sics have been constructed previously using grobner bases this method has probably been taken as far as is possible with current computer technology except in special cases where there are additional symmetries here we describe a method for converting high precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the galois symmetries of an sic using this method we have calculated 69 new exact solutions including nine new dimensions where previously only numerical solutions were known which more than triples the number of known exact solutions in some cases the solutions require number fields with degrees as high as 12 288 we use these solutions to confirm that they obey the number theoretic conjectures and address two questions suggested by the previous work

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

the hanna neumann conjecture for demushkin groups abstract we confirm the hanna neumann conjecture for topologically finitely generated closed subgroups u and w of a nonsolvable demushkin group g namely we show that g u g w d u g w g 1 d u d w where d k max d k 1 0 and d k is the least cardinality of a topological generating set for the group k

weighted complex projective 2 designs from bases optimal state determination by orthogonal measurements we introduce the problem of constructing weighted complex projective 2 designs from the union of a family of orthonormal bases if the weight remains constant across elements of the same basis then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases being equivalent whenever the design is composed of d 1 bases in dimension d we show that for the purpose of quantum state determination these designs specify an optimal collection of orthogonal measurements using highly nonlinear functions on abelian groups we construct explicit examples from d 2 orthonormal bases whenever d 1 is a prime power covering dimensions d 6 10 and 12 for example where no complete sets of mutually unbiased bases have thus far been found

formal state determination following the general results of park and band it is shown that any state of a quantal ensemble allowed by the hilbert space formulation of nonrelativistic quantum mechanics may be determined from the results of measurements under the assumption that all measurements those of energy position and spin projection are in accordance with the projection postulate

wigner weyl correspondence in quantum mechanics for continuous and discrete systems a dirac inspired view drawing inspiration from dirac s work on functions of non commuting observables we develop an approach to phase space descriptions of operators and the wigner weyl correspondence in quantum mechanics complementary to standard formulations this involves a two step process introducing phase space descriptions based on placing position dependences to the left of momentum dependences or the other way around then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators the method works uniformly for both continuous cartesian degrees of freedom and for systems with finite dimensional state spaces it is interesting that the kernel encountered is naturally expressible in terms of geometric phases and its removal involves extracting its square root in a suitable manner

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

symmetric informationally complete positive operator valued measures a new computer study we report on a new computer study of the existence of d2 equiangular lines in d complex dimensions such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory we provide numerical solutions in all dimensions d 67 and moreover a putatively complete list of weyl heisenberg covariant solutions for d 50 a symmetry analysis of this list leads to new algebraic solutions in dimensions d 24 35 and 48 which are given together with algebraic solutions for d 4 15 and 19

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

galois quantum systems irreducible polynomials and riemann surfaces finite quantum systems in which the position and momentum take values in the galois field gf pl are studied ideas from the subject of field extension are transferred in the context of quantum mechanics the frobenius automorphisms in galois fields lead naturally to the frobenius formalism in a quantum context the hilbert space splits into frobenius subspaces which are labeled with the irreducible polynomials associated with the ypl y the frobenius maps transform unitarily the states of a galois quantum system and leave fixed all states in some of its galois subsystems where the position and momentum take values in subfields of gf pl an analytic representation of these systems in the l sheeted complex plane shows deeper links between galois theory and riemann surfaces

the curious nonexistence of gaussian 2 designs ensembles of pure quantum states whose 2nd moments equal those of the unitarily uniform haar ensemble 2 designs are optimal solutions for several tasks in quantum information science especially state and process tomography we show that gaussian states cannot form a 2 design for the continuous variable quantum optical hilbert space l 2 mathbb r this is surprising because the affine symplectic group hwsp the natural symmetry group of gaussian states is irreducible on the symmetric subspace of two copies in finite dimensional hilbert spaces irreducibility guarantees that hwsp covariant ensembles such as mutually unbiased bases in prime dimensions are always 2 designs this property is violated by continuous variables for a subtle reason the well defined hwsp invariant ensemble of gaussian states does not have a density matrix because its defining integral does not converge in fact no gaussian ensemble is even close in a precise sense to being a 2 design this surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography

affine constellations without mutually unbiased counterparts it has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d we introduce affine constellations and compare their existence properties with those of mutually unbiased constellations the observed discrepancies make a deeper relation between the two existence problems unlikely

a multiple criteria decision support system for testing integrated environmental models abstract spatial models of ecological and hydrological processes are widely used tools for studying natural systems over large areas however these models lack specific mechanisms for reporting output uncertainty contributed by model structure and so testing their suitability for studying a large range of problems is difficult this paper describes a method of evaluating the uncertainty contributed by underlying assumptions used in constructing integrated environmental models from two or more sub models that were developed for different purposes integrated environmental models are typically constructed from many individual process based models conflicting assumptions between these sub models e g spatial scale differences are easily overlooked during model development and application this semantic error cannot be predicted prior to simulation as it may only emerge through the interaction of sub models applied to a particular set of data used to drive a simulation model agreement is proposed and demonstrated as a way to detect problems of model integration at the state variable level within an integrated ecosystem model this model agreement is then propagated to model response variables using multiple criteria to examine their sensitivity predictability and synchronicity to the measured uncertainty in state variables these three properties are combined under fuzzy logic in order to provide decision support on where for a given time during simulation the sub models agree on a particular response variable this paper describes the details of the approach and its application using an existing integrated environmental model the results show that for a given set of model inputs and application integrated environmental models may have spatially variable levels of agreement at the sub model level the results using rhessysd a spatially integrated ecosystem hydrology model indicate that semantic error in estimates of plant available soil moisture are consistent with observations of the need for resetting events such as flooding to initialize the model to a point where further simulation results can be trusted these results suggest that a dynamic selection of sub models may be warranted given a reasonable method of determining sub model disagreement during simulation fuzzy set theory may be a useful tool in arriving at such a model selection process as it allows for a relatively straightforward synthesis of numerous model evaluation criteria with a large quantity of output from the model

mubs inequivalence and affine planes there are fairly large families of unitarily inequivalent complete sets of n 1 mutually unbiased bases mubs known in cn for various prime powers n the number of such sets is not bounded above by any polynomial as a function of n while it is standard that there is a superficial similarity between complete sets of mubs and finite affine planes there is an intimate relationship between these large families and affine planes this note briefly summarizes old results that do not appear to be well known concerning known families of complete sets of mubs and their associated planes

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

the weight distribution of a class of linear codes from perfect nonlinear functions in this correspondence the weight distribution of a class of linear codes based on perfect nonlinear functions also called planar functions is determined the class of linear codes under study are either optimal or among the best codes known and have nice applications in cryptography

galois quantum systems a finite quantum system in which the position and momentum take values in the galois field gf p is constructed from a smaller quantum system in which the position and momentum take values in using field extension the galois trace is used in the definition of the fourier transform the heisenberg weyl group of displacements and the sp 2 gf p group of symplectic transformations are studied a class of transformations inspired by the frobenius maps in galois fields is introduced the relationship of this galois quantum system with its subsystems in which the position and momentum take values in subfields of gf p is discussed

averaging sets a generalization of mean values and spherical designs 0 1 r one cannot of course expect there to be a single point x at which every one assumes its mean value but there is an easy generalization in terms of weighted averages there exist at most m points xi and positive weights oi summing to 1 so that aifj xi i fi dt for j 1 2 m

the lie algebraic significance of symmetric informationally complete measurements examples of symmetric informationally complete positive operator valued measures sic povms have been constructed in every dimension 67 however it remains an open question whether they exist in all finite dimensions a sic povm is usually thought of as a highly symmetric structure in quantum state space however its elements can equally well be regarded as a basis for the lie algebra gl d c in this paper we examine the resulting structure constants which are calculated from the traces of the triple products of the sic povm elements and which it turns out characterize the sic povm up to unitary equivalence we show that the structure constants have numerous remarkable properties in particular we show that the existence of a sic povm in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl d c we hope that transforming the problem in this way from a question about quantum state space to a question about lie algebras may help to make the existence problem tractable

universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement we present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system the algorithm specifies a physically realizable i e finite positive operator valued measurement on a finite number of identically prepared systems we illustrate the general formalism by applying it to different scenarios of the state estimation of n independent and identically prepared two level systems qubits

nice error frames canonical abstract error groups and the construction of sics abstract nice error bases are generalisations of the pauli matrices which have applications in quantum information theory these orthonormal bases for the d d matrices m d c also generalise the projective action of the heisenberg group on c d here we extend nice error bases to nice error frames these are equal norm tight frames for m d c consisting of d d unitary matrices with a group indexing structure we show that each nice error frame irreducible faithful projective representation is associated with a canonical abstract error group this is calculated in number of examples e g for all nice error bases for d 14 which then allows us to investigate which nice error bases might give rise to sics symmetric informationally complete positive operator valued measures these results show that the current catalogue of nice error bases over counts in particular we give an explicit example of a sic for d 6 with a nonabelian index group and show that the hoggar lines appear for various nice error bases some of which are subgroups of the clifford group thus all known sics appear as orbits of subgroups of the clifford group

maximally symmetric stabilizer mubs in even prime power dimensions one way to construct a maximal set of mutually unbiased bases mubs in a prime power dimensional hilbert space is by means of finite phase space methods mubs obtained in this way are covariant with respect to some subgroup of the group of all affine symplectic phase space transformations however this construction is not canonical as a consequence many different choices of covariance subgroups are possible in particular when the hilbert space is 2n dimensional it is known that covariance with respect to the full group of affine symplectic phase space transformations can never be achieved here we show that in this case there exist two essentially different choices of maximal subgroups admitting covariant mubs for both of them we explicitly construct a family of 2n covariant mubs we thus prove that contrary to the odd dimensional case maximally covariant mubs are very far from being unique in even prime power dimensions

maximally informative ensembles for sic povms in dimension 3 in order to find out for which initial states of the system the uncertainty of the measurement outcomes will be minimal one can look for the minimizers of the shannon entropy of the measurement in case of group covariant measurements this question becomes closely related to the problem of how informative the measurement is in the sense of its informational power namely the orbit under group action of the entropy minimizer corresponds to a maximally informative ensemble of equiprobable elements we give a characterization of such ensembles for three dimensional group covariant weyl heisenberg symmetric informationally complete positive operator valued measures sic povms in both geometric and algebraic terms it turns out that a maximally informative ensemble arises from the input state orthogonal to a subspace spanned by three linearly dependent vectors defining a sic povm geometrically or from an eigenstate of a certain weyl matrix algebraically

linguistic modelling of imperfect spatial information as a basis for simplifying spatial analysis spatial analysis is at the basis of several decision processes in fields such as urban planning and environmental management in order to carry it out effectively more flexible geographic information systems are needed to be able to represent and manage the imperfection that invariably affects geographic information starting from the consideration that experts usually deal with imperfect spatial information through linguistic terms e g they identify the approximate position of a phenomenon on a map and classify spatial properties through linguistic labels this contribution proposes the use of linguistic granule of information to represent and manage imperfect spatial information a fuzzy object based data model is proposed as a tool for supporting spatial analysis based on the management of linguistic granule in particular the problem of defining methods to manage imperfect information depending on the type of imperfection is discussed finally an example of spatial analysis applied to support a decision problem in environmental impact assessment is described

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

mutually orthogonal latin squares from the inner products of vectors in mutually unbiased bases mutually unbiased bases mubs are important in quantum information theory while constructions of complete sets of d 1 mubs in c d are known when d is a prime power it is unknown if such complete sets exist in non prime power dimensions it has been conjectured that complete sets of mubs only exist in if a maximal set of mutually orthogonal latin squares mols of side length d also exists there are several constructions roy and scott 2007 j math phys 48 072110 paterek daki and brukner 2009 phys rev a 79 012109 of complete sets of mubs from specific types of mols which use galois fields to construct the vectors of the mubs in this paper two known constructions of mubs alltop 1980 ieee trans inf theory 26 350 354 wootters and fields 1989 ann phys 191 363 381 both of which use polynomials over a galois field are used to construct complete sets of mols in the odd prime case the mols come from the inner products of pairs of vectors in the mubs

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

wigner distributions for finite state systems without redundant phase point operators we set up wigner distributions for n state quantum systems following a dirac inspired approach in contrast to much of the work in this study requiring a 2n 2n phase space particularly when n is even our approach is uniformly based on an n n phase space grid and thereby avoids the necessity of having to invoke a quadrupled phase space and hence the attendant redundance both n odd and even cases are analysed in detail and it is found that there are striking differences between the two while the n odd case permits full implementation of the marginal property the even case does so only in a restricted sense this has the consequence that in the even case one is led to several equally good definitions of the wigner distributions as opposed to the odd case where the choice turns out to be unique

a family of alltop functions that are ea inequivalent to the cubic function sequences with optimal correlation properties are much sought after for applications in communication systems in 1980 alltop ieee trans inf theory 26 3 350 354 1980 described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation subsequently these sequences were used to construct mutually unbiased bases mubs a structure of importance in quantum information theory the key feature of this cubic function is that its difference function is a planar function functions with planar difference functions have been called alltop functions this paper provides a new family of alltop functions and establishes the use of alltop functions for construction of sequence sets and mubs

on approximately symmetric informationally complete positive operator valued measures and related systems of quantum states we address the problem of constructing positive operator valued measures povms in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform this is motivated by the related question of constructing symmetric informationally complete povms sic povms for which the inner products are perfectly uniform however sic povms are notoriously hard to construct and despite some success of constructing them numerically there is no analytic construction known we present two constructions of approximate versions of sic povms where a small deviation from uniformity of the inner products is allowed the first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power the second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases moreover we construct vector systems in cn which are almost orthogonal and which might turn out

mutually unbiased bases and hadamard matrices of order six we report on a search for mutually unbiased bases mubs in six dimensions we find only triplets of mubs and thus do not come close to the theoretical upper bound 7 however we point out that the natural habitat for sets of mubs is the set of all complex hadamard matrices of the given order and we introduce a natural notion of distance between bases in hilbert space this allows us to draw a detailed map of where in the landscape the mub triplets are situated we use available tools such as the theory of the discrete fourier transform to organize our results finally we present some evidence for the conjecture that there exists a four dimensional family of complex hadamard matrices of order 6 if this conjecture is true the landscape in which one may search for mubs is much larger than previously thought

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

dimensions lengths and separability in finite dimensional quantum systems many important sets of normalized states in a multipartite quantum system of finite dimension d such as the set s of all separable states are real semialgebraic sets we compute dimensions of many such sets in several low dimensional systems by using dimension arguments we show that there exist separable states which are not convex combinations of d or less pure product states for instance such states exist in bipartite m n systems when m 2 n 2 1 this solves an open problem proposed by divincenzo terhal and thapliyal about 12 years ago we prove that there exist a separable state and a pure product state whose mixture has smaller length than that of we show that any real s which is invariant under all partial transpose operations is a convex sum of real pure product states in the case of the 2 n system the number r of product states can be taken to be r rank we also show that the general multipartite separability problem can be reduced to the case of real states regarding

mining spatial association rules in image databases in this paper we propose a novel spatial mining algorithm called 9dlt miner to mine the spatial association rules from an image database where every image is represented by the 9dlt representation the proposed method consists of two phases first we find all frequent patterns of length one next we use frequent k patterns k 1 to generate all candidate k 1 patterns for each candidate pattern generated we scan the database to count the pattern s support and check if it is frequent the steps in the second phase are repeated until no more frequent patterns can be found since our proposed algorithm prunes most of impossible candidates it is more efficient than the apriori algorithm the experiment results show that 9dlt miner runs 2 5 times faster than the apriori algorithm

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

on the residue fields of henselian valued stable fields abstract let k v be a henselian valued field satisfying the following conditions for a given prime number p i central division k algebras of finite p primary dimensions have schur indices equal to their exponents ii the value group v k properly includes its subgroup p v k the paper shows that if k is the residue field of k v and r is an intermediate field of the maximal p extension k p k then the natural homomorphism br k br r of brauer groups maps surjectively the p component br k p on br r p it proves that br k p is divisible if p 2 or k is a nonreal field and that br k 2 is of order 2 when k is formally real we also obtain that r embeds as a k subalgebra in a central division k algebra if and only if the degree r k divides the index of

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

feynman s path integral and mutually unbiased bases our previous work on quantum mechanics in hilbert spaces of finite dimension n is applied to elucidate the deep meaning of feynman s path integral pointed out by g svetlichny he speculated that the secret of the feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases we confirm the corresponding property of the short time propagator by using a specially devised n n approximation of quantum mechanics in applied to our finite dimensional analogue of a free quantum particle

achieving the orthoplex bound and constructing weighted complex projective 2 designs with singer sets abstract equiangular tight frames are examples of grassmannian line packings for a hilbert space more specifically according to a bound by welch they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit norm frame this paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames the orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs we construct frames of unit norm vectors in k dimensional complex hilbert spaces that achieve the orthoplex bound when k 1 is a prime power we obtain a tight frame with k 2 1 vectors and when k is a prime power with k 2 k 1 vectors in addition we show that these frames form weighted complex projective 2 designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography our construction is based on singer s family of difference sets and the related concept of relative difference sets

partial order and a t0 topology in a set of finite quantum systems a whole part theory is developed for a set of finite quantum systems n with variables in z n the partial order subsystem is defined by embedding various attributes of the system m quantum states density matrices etc into their counterparts in the supersystem n for m n the compatibility of these embeddings is studied the concept of ubiquity is introduced for quantities which fit with this structure it is shown that various entropic quantities are ubiquitous the sets of various quantities become t0 topological spaces with the divisor topology which encapsulates fundamental physical properties these sets can be converted into directed complete partial orders by adding top elements the continuity of various maps among these sets is studied

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

evaluating structural and topological consistency of complex regions with broad boundaries in multi resolution spatial databases multi resolution or multi scale spatial databases store and manage multiple representations of spatial objects in the same area so consistency among multiple representations of the same objects should be evaluated and maintained although many approaches have been proposed to check inconsistencies in multi resolution databases there is still a lack of effective approaches working for complex objects especially for regions with broad boundaries which is a general model for representing various types of uncertainties this paper presents approaches for evaluating structural and topological consistency among multiple representations of complex regions with broad boundaries cbbrs based on map generalization operators merging dropping and hybrid of these two for evaluation of structural consistency all possible multiple representations of a cbbr are generated automatically and organized into a structured neighborhood graph and then correspondences and equivalences among the multiple representations are defined to determine whether two representations at different levels of detail are structurally consistent for evaluation of topological consistency the topological relations between all pairs of regions in two cbbrs are considered and their variation with change of spatial scale is analyzed since the approaches in this paper are built on a hiearchical representation of cbbrs with arbitrarily complex structure they will also work well for evaluating consistency among multiple representations of complex objects

tight informationally complete quantum measurements we introduce a class of informationally complete positive operator valued measures which are in analogy with a tight frame as close as possible to orthonormal bases for the space of quantum states these measures are distinguished by an exceptionally simple state reconstruction formula which allows painless quantum state tomography complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures are both members of this class the latter being the unique minimal rank one members recast as ensembles of pure quantum states the rank one members are in fact equivalent to weighted 2 designs in complex projective space these measures are shown to be optimal for quantum cloning and linear quantum state tomography

minimal optimal generalized quantum measurements optimal and finite positive operator valued measurements on a finite number n of identically prepared systems have been presented recently with physical realization in mind we propose here optimal and minimal generalized quantum measurements for two level systems we explicitly construct them up to n 7 and verify that they are minimal up to n 5 we finally propose an expression which gives the size of the minimal optimal measurements for arbitrary n

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

optimizing quantum process tomography with unitary 2 designs we show that weighted unitary 2 designs define optimal measurements on the system ancilla output state for ancilla assisted process tomography of unital quantum channels examples include complete sets of mutually unbiased unitary operator bases each of these specifies a minimal series of optimal orthogonal measurements general quantum channels are also considered

discrete phase space approach to mutually orthogonal latin squares we show there is a natural connection between latin squares and commutative sets of monomials defining geometric structures in finite phase space of prime power dimensions a complete set of such monomials defines a mutually unbiased basis mub and may be associated with a complete set of mutually orthogonal latin squares mols we translate some possible operations on the monomial sets into isomorphisms of latin squares and find a general form of permutations that map between latin squares corresponding to unitarily equivalent mutually unbiased sets

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

factorization in finite quantum systems unitary transformations in an angular momentum hilbert space h 2j 1 are considered they are expressed as a finite sum of the displacement operators which play the role of su 2j 1 generators with the weyl function as coefficients the chinese remainder theorem is used to factorize large qudits in the hilbert space h 2j 1 in terms of smaller qudits in hilbert spaces h 2ji 1 all unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits

orbits of mutually unbiased bases we express alltop s construction of mutually unbiased bases as orbits under the weyl heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4 we reproduce alltop s mutually unbiased bases using abelian subgroups of the clifford group in prime dimensions in direct analogy to the well known construction of mutually unbiased bases using abelian subgroups of the weyl heisenberg group finally we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases

dimension towers of sics i aligned sics and embedded tight frames algebraic number theory relates sic povms in dimension d 3 to those in dimension d d 2 we define a sic in dimension d d 2 to be aligned to a sic in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d d 2 in a specified way we give 19 mostly numerical examples of aligned sics we conjecture that given any sic in dimension d there exists an aligned sic in dimension d d 2 in all our examples the aligned sic has lower dimensional equiangular tight frames embedded in it if d is odd so that a natural tensor product structure exists we prove that the individual vectors in the aligned sic have a very special entanglement structure and the existence of the embedded tight frames follows as a theorem if d 2 is an odd prime number we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned sic to this dimension

symplectic transformations and quantum tomography in finite quantum systems quantum systems where the position and momentum are in the ring zd d is an odd integer are considered symplectic transformations are studied and the order of sp 2 zd is calculated quantum tomography is also discussed it is shown that measurements used in the inverse radon transform need to be made on j2 d lines where j2 d is the jordan totient function

hjelmslev geometry of mutually unbiased bases the basic combinatorial properties of a complete set of mutually unbiased bases mubs of a q dimensional hilbert space with p being a prime and r a positive integer are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective hjelmslev plane defined over a galois ring of characteristic p2 and rank r the q vectors of a basis of correspond to the q points of a so called neighbour class and the q 1 mubs answer to the total number of pairwise disjoint neighbour classes on the conic

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

beyond stabilizer codes i nice error bases nice error bases have been introduced by knill 1996 as a generalization of the pauli basis these bases are shown to be projective representations of finite groups we classify all nice error bases of small degree and all nice error bases with abelian index groups we show that in general an index group of a nice error basis is necessarily solvable

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

geometry of the welch bounds abstract a geometric perspective involving grammian and frame operators is used to derive the entire family of welch bounds this perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k tensors tight frames homogeneous polynomials and t designs in particular a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k tensors it is also shown that tightness of the bounds requires tight frames the lack of tight frames of symmetric k tensors in many cases however leads to consideration of sets that come as close as possible to attaining the bounds the geometric derivation is then extended in the setting of generalized or continuous frames the welch bounds for finite sets and countably infinite sets become special cases of this general setting

projective ring line of an arbitrary single qudit as a continuation of our previous work havlicek and saniga 2007 j phys a math gen 40 f943 52 preprint 0708 4333 an algebraic geometrical study of a single d dimensional qudit is made with d being any positive integer the study is based on an intricate relation between the symplectic module of the generalized pauli group of the qudit and the fine structure of the projective line over the modular ring explicit formulae are given for both the number of generalized pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of we find remarkably that a perp set is not a set theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes the operators are also seen to be structured into disjoint layers according to the degree of their representing vectors a brief comparison with some multiple qudit cases is made

a sharpening of the welch bounds and the existence of real and complex spherical t designs the welch bounds for a finite set of unit vectors are a family of inequalities indexed by t 1 2 ldots which describe how evenly spread the vectors are they have important applications in signal analysis where sequences giving equality in the first welch bound are known as welch bound equality sequences or as unit norm tight frames here we consider sequences of vectors giving equality in the higher order welch bounds these are seen to correspond to tight frames for the complex symmetric t tensors which we prove always exist we show that for t 1 the welch bounds can be sharpened for real vectors and again vectors giving equality always exist we give a unified treatment of various conditions for equality in both the real and complex cases in particular we give an explicit description of the corresponding cubature rules t designs our results set up a framework for the construction and classification several configurations of vectors of recent interest these include mutually unbiased bases complex equiangular lines spherical half designs projective t designs and minimisers of the higher order frame potential one interesting consequence is a construction of sets of complex equiangular lines which were previously unknown

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

the isotropic lines of bb z _ d 2 we show that the isotropic lines in the lattice are the lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice the set of isotropic lines decomposes into orbits under the action of we give an explicit description of those orbits as well as their number and respective cardinalities we also develop two group actions on the group related to the topic

geometrical approach to the discrete wigner function in prime power dimensions we analyse the wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field we separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non uniqueness of the representation of the wigner function explicit expressions for the wigner kernel are given in both cases

partial ordering of weak mutually unbiased bases a quantum system with variables in where with prime numbers is considered the non near linear geometry of the phase space is studied the lines through the origin are factorized in terms of prime factor lines in weak mutually unbiased bases wmub which are products of the mutually unbiased bases in the prime factor hilbert spaces are also considered the factorization of both lines and wmub is analogous to the factorization of integers in terms of prime numbers the duality between lines and wmub is discussed it is shown that there is a partial order in the set of subgeometries of isomorphic to the partial order in the set of subsystems of

spectra of phase point operators in odd prime dimensions and the extended clifford group we analyze the role of the extended clifford group in classifying the spectra of phase point operators within the framework laid out by gibbons et al phys rev a 70 062101 2004 for setting up wigner distributions on discrete phase spaces based on finite fields to do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field auxiliary results include a derivation of the conjugacy classes of esl 2 fn

hamiltonians of quantum systems with positions and momenta in gf pl a quantum system with positions and momenta in gf pl is considered such a system can be constructed from l smaller systems in which the positions and momenta take values in zp if the hamiltonian of this l partite system is compatible with gf pl the concept of compatibility of a hamiltonian with gf pl allows the quantum formalism in the l partite system to be expressed in terms of galois arithmetic transformations of the basis in gf pl produce unitary transformations of the quantum states which form a representation of gl l zp they are used to define which subset of the general set of hamiltonians in the l partite system is compatible with gf pl

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

the sylow subgroups of the absolute galois group gal q abstract we describe the l sylow subgroups of gal q for an odd prime l by observing and studying their decomposition as f z l where f is a free pro l group and z l are the l adic integers we determine the finite z l quotients of f and more generally show that every split embedding problem of z l groups for f is solvable moreover we analyze the z l action on generators of f

discrete wigner function by symmetric informationally complete positive operator valued measure we construct a version of the discrete wigner function making use of symmetric informationally complete positive operator valued measure we will show that this version is the natural discrete analog of continuous wigner function in addition to this discrete wigner function has many properties of the continuous wigner function