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

the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

bounds on universal sequences universal sequences for graphs a concept introduced by aleliunas m sc thesis university of toronto toronto ontario canada january 1978 and aleliunas et al proc 20th annual symposium on foundation of computer science 1979 pp 218 223 are studied by letting u d n denote the minimum length of a universal sequence for d regular undirected graphs with n nodes the latter paper has proved the upper bound u d n o d 2 n 3 log n using a probabilistic argument here a lower bound of u 2 n omega n log n is proved from which u d n omega n log n for all d is deduced also for complete graphs u n 1 n omega n log 2 n log log n an explicit construction of universal sequences for cycles d 2 of length n o log n is given

determining the first order perturbation of a polyharmonic operator on admissible manifolds abstract we consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator l g x q with x being a w 1 vector field and q being an l function on compact riemannian manifolds with boundary which are conformally embedded in a product of the euclidean line and a simple manifold we show that the knowledge of the dirichlet to neumann map determines x and q uniquely the method is based on the construction of complex geometrical optics solutions using the carleman estimate for the laplace beltrami operator due to dos santos ferreira kenig salo and uhlmann notice that the corresponding uniqueness result does not hold for the first order perturbation of the laplace beltrami operator

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

an inverse boundary value problem for schrodinger operators with vector potentials we consider the schrodinger operator for a magnetic potential a and an electric potential q which are supported in a bounded domain in r n with n 3 we prove that knowledge of the dirichlet to neumann map associated to the schrodinger operator determines the magnetic field rot a and the electric potential 4 simultaneously provided rot a is small in the l topology

determining the first order perturbation of a bi harmonic operator on bounded and unbounded domains from partial data abstract in this paper we study inverse boundary value problems with partial data for the bi harmonic operator with first order perturbation we consider two types of subsets of r n n 3 one is an infinite slab the other is a bounded domain in the case of a slab we show that from dirichlet and neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane one can uniquely determine the magnetic potential and the electric potential in the case of a bounded domain we show the unique determination of the magnetic potential and the electric potential from partial dirichlet and neumann data under two different assumptions the first assumption is that the magnetic and electric potentials are known in a neighborhood of the boundary in this situation we obtain the uniqueness result when the dirichlet and neumann data are only given on two arbitrary open subsets of the boundary the second assumption is that the dirichlet and neumann data are known on the same part of the boundary whose complement is a part of a hyperplane we also establish the unique determination result in this local data case

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

complex hadamard matrices and equiangular tight frames abstract in this paper we give a new construction of parametric families of complex hadamard matrices of square orders and connect them to equiangular tight frames the results presented here generalize some of the recent ideas of bodmann et al 3 and extend the list of known equiangular tight frames in particular a 36 21 frame coming from a nontrivial cube root signature matrix is obtained for the first time

states that are far from being stabilizer states stabilizer states are eigenvectors of maximal commuting sets of operators in a finite heisenberg group states that are far from being stabilizer states include magic states in quantum computation mub balanced states and sic vectors in prime dimensions the latter two fall under the umbrella of minimum uncertainty states muss in the sense of wootters and sussman we study the correlation between two ways in which the notion of far from being a stabilizer state can be quantified two theorems valid for all prime dimensions are given as well as detailed results for low dimensions in dimension 7 we identify the mub balanced states as being antipodal to the sic vectors within the set of mus in a sense that we make definite in dimension 4 we show that the states that come closest to being mus with respect to all of the six stabilizer mubs are the fiducial vectors for alltop mubs

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

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

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

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

equiangular tight frames and unistochastic matrices we demonstrate that a complex equiangular tight frame composed of n vectors in dimension d denoted etf d n exists if and only if a certain bistochastic matrix univocally determined by n and d belongs to a special class of unistochastic matrices this connection allows us to find new complex etfs in infinitely many dimensions and to derive a method to introduce non trivial free parameters in etfs we present an explicit six parametric family of complex etf 6 16 which defines a family of symmetric povms minimal and maximal possible average entanglement of the vectors within this qubit qutrit family are described furthermore we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix which we apply to find all existing classes of complex etfs containing up to 20 vectors

an anisotropic inverse boundary value problem we consider the impedance tomography problem for anisotropic conductivities given a bounded region in space a diffeomorphism from to itself which restricts to the identity on and a conductivity on it is easy to construct a new conductivity which will produce the same voltage and current measurements on we prove the converse in two dimensions i e if 1 and 2 produce the same boundary measurements then 1 2 for an appropriate for conductivities which are near a constant

determining anisotropic real analytic conductivities by boundary measurements if an electrical potential is applied to the surface of a solid body the current flux across the surface depends on the conductivity in the interior of the body we want to consider the inverse problem of determining the conductivity by these boundary measurements

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

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

calderon inverse problem with partial data on riemann surfaces on a fixed smooth compact riemann surface with boundary m0 g we show that for the schrodinger operator v with potential v c m0 for some 0 the dirichlet to neumann map n measured on an open set m0 determines uniquely the potential v we also discuss briefly the corresponding consequences for potential scattering at 0 frequency on riemann surfaces with asymptotically euclidean or asymptotically hyperbolic ends

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

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

fibonacci lucas sic povms we present a conjectured family of symmetric informationally complete positive operator valued measures which have an additional symmetry group whose size is growing with the dimension the symmetry group is related to fibonacci numbers while the dimension is related to lucas numbers the conjecture is supported by exact solutions for dimensions d 4 8 19 48 124 and 323 as well as a numerical solution for dimension d 844

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

the calderon problem with partial data in two dimensions we prove for a two dimensional bounded domain that the cauchy data for the schrodinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential this implies for the conductivity equation that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset we can determine uniquely the conductivity we use carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results

bccb complex hadamard matrices of order 9 and mubs abstract a new type of complex hadamard matrices of order 9 are constructed the studied matrices are symmetric block circulant with circulant blocks b c c b and form an until now unknown non reducible and non affine two parameter orbit several suborbits are identified including a one parameter intersection with the fourier orbit f 9 4 the defect of this new type of hadamard matrices is observed to vary from a generic value 2 to the anomalous values 4 and 10 for some sub orbits and to 12 and 16 for some single matrices the latter matrices are shown to be related to complete sets of mubs in dimension 9

a time space tradeoff for undirected graph traversal by walking automata we prove a time space tradeoff for traversing undirected graphs using a structured model that is a nonjumping variant of cook and rackoff s jumping automata for graphs

equiangular tight frames from paley tournaments abstract we prove the existence of equiangular tight frames having n 2 d 1 elements drawn from either c d or c d 1 whenever n is either 2 k 1 for k n or a power of a prime such that n 3 mod 4 we also find a simple explicit expression for the prime power case by establishing a connection to a 2 d element equiangular tight frame based on quadratic residues

two constructions of approximately symmetric informationally complete positive operator valued measures symmetric informationally complete positive operator valued measures sic povms have many applications in quantum information however it is not easy to construct sic povms and there are only a few known classes of them and we do not even know whether there exists an infinite class of them thus constructing approximately symmetric informationally complete positive operator valued measures asic povms has its own meaning in this paper we use character sums over finite fields to present two constructions of asic povms we show that there are some classes of infinite families of asic povms by using some special functions over finite fields

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

stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials abstract in this article stability estimates are given for the determination of the zeroth order bounded perturbations of the biharmonic operator when the boundary neumann measurements are made on the whole boundary and on slightly more than half the boundary respectively for the case of measurements on the whole boundary the stability estimates are of ln type and for the case of measurements on slightly more than half of the boundary we derive estimates that are of ln ln type

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

a note on equiangular tight frames we settle a conjecture of joseph renes about the existence and construction of certain equiangular tight frames

frames for vector spaces and affine spaces a finite frame for a finite dimensional hilbert space is simply a spanning sequence we show that the linear functionals given by the dual frame vectors do not depend on the inner product and thus it is possible to extend the frame expansion and other elements of frame theory to any finite spanning sequence for a vector space the corresponding coordinate functionals generalise the dual basis the case when the vectors are linearly independent and are characterised by the fact that the associated gramian matrix is an orthogonal projection existing generalisations of the frame expansion to banach spaces involve an analogue of the frame bounds and frame operator the potential applications of our results are considerable whenever there is a natural spanning set for a vector space computations can be done directly with it in an efficient and stable way we illustrate this with a diverse range of examples including multivariate spline spaces generalised barycentric coordinates and vector spaces over the rationals such as the cyclotomic fields

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

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

inverse boundary value problems for the perturbed polyharmonic operator author s krupchyk k lassas m uhlmann g abstract we show that a first order perturbation a x d q x of the polyharmonic operator m m 2 can be determined uniquely from the set of the cauchy data for the perturbed polyharmonic operator on a bounded domain in n n 3 notice that the corresponding result does not hold in general when m 1 2013 american mathematical society

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

symmetric space bounded computation abstract a symmetric turing machine is one such that the yields relation between configurations is symmetric the space complexity classes for such machines are found to be intermediate between the corresponding deterministic and nondeterministic space complexity classes certain natural problems are shown to be complete for symmetric space complexity classes and the relationship of symmetry to determinism and nondeterminism is investigated

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

determining a first order perturbation of the biharmonic operator by partial boundary measurements abstract we consider an operator 2 a x d q x with the navier boundary conditions on a bounded domain in r n n 3 we show that a first order perturbation a x d q can be determined uniquely by measuring the dirichlet to neumann map on possibly very small subsets of the boundary of the domain notice that the corresponding result does not hold in general for a first order perturbation of the laplacian

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

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

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

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

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

determining a magnetic schrodinger operator from partial cauchy data in this paper we show in dimension n 3 that knowledge of the cauchy data for the schrodinger equation in the presence of a magnetic potential measured on possibly very small subsets of the boundary determines uniquely the magnetic field and the electric potential we follow the general strategy of 7 using a richer set of solutions to the dirichlet problem that has been used in previous works on this problem

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

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

surveying points in the complex projective plane fields institute ontario the perimeter insti tute ontario and tulane university new orleans s abramsky d appleby r blume kohout d brody h brown s flammia c fuchs l hardy and h zhu for stimulating discussions sms acknowledges support arising from visits to the university of nijmegen the university of sofia and the university of turin and thanks n lora lamia for helpful comments j armstrong

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