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

lifschitz tail for alloy type models driven by the fractional laplacian abstract we establish precise asymptotics near zero of the integrated density of states for the random schrodinger operators 2 v in l 2 r d for the full range of 0 2 and a fairly large class of random nonnegative alloy type potentials v the ids exhibits the lifschitz tail singularity we prove the existence of the limit lim 0 d ln n c d d d with c 0 the constant c is finite if and only if the common distribution of the lattice random variables charges 0 in this case the constant c is expressed explicitly in terms of this distribution in the limit formula d denotes the dirichlet ground state eigenvalue of the operator 2 in the unit ball in r d and d is the volume of this ball

existence of the density of states for multi dimensional continuum schrodinger operators with gaussian random potentials a wegner estimate is proved for quantum systems in multi dimensional euclidean space which are characterized by one particle schrodinger operators with random potentials that admit a certain one parameter decomposition in particular the wegner estimate applies to systems with rather general gaussian random potentials as a consequence these systems possess an absolutely continuous integrated density of states whose derivative the density of states is locally bounded an explicit upper bound is derived

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

low lying eigenvalues of randomly curved quantum waveguides abstract we consider the negative dirichlet laplacian on an infinite waveguide embedded in r 2 and finite segments thereof the waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature we derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling i e weak disorder regime this allows us to estimate the probability of low lying eigenvalues a tool which is relevant in the context of anderson localization for random schrodinger operators

large deviations and lifshitz singularity of the integrated density of states of random hamiltonians we consider the integrated density of states ids of random hamiltonian h v v being a random field on d which satisfies a mixing condition we prove that the probability of large fluctuations of the finite volume ids 1 h d around the thermodynamic limit is bounded from above by exp k k 0 in this case can be recovered from a variational principle furthermore we show the existence of a lifshitztype of singularity of as 0 in the case where v is non negative more precisely we prove the following bound exp k d 2 as 0 k 0 this last result is then discussed in some examples

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

graph states in phase space the phase space for a system of n qubits is a discrete grid of 2n 2n points whose axes are labeled in terms of the elements of the finite field to endow it with proper geometrical properties we analyze the representation of graph states in that phase space showing that these states can be identified with a class of nonsingular curves we provide an algebraic representation of the most relevant quantum operations acting on these states and discuss the advantages of this approach

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

the quantum hall effect for electrons in a random potential we assume the existence of sufficiently localised states near the edges of each landau band we then prove that the hall conductivity is quantised and the parallel conductivity vanishes when the filling factor stays close to an integer the hall integer is a topological invariant given by the landau band index

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

quantum particle in a hierarchical potential with tunnelling over arbitrarily large scales the authors introduce a notion of hierarchical potential by iterating a basic elementary geometric construction over an increasing sequence of length scales they then show that a quantum particle in such a potential exhibits a subdiffusive behaviour characterised by r2 or c ln t beta if the initial state is a normalised wavepacket superposition of states of sufficiently low energy

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

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

localization estimates for a random discrete wave equation at high frequency it is shown that at high frequencies matrix elements of the green s function of a random discrete wave equation decay exponentially at long distances this is the input to the proof of dense point spectrum with localized eigenfunctions in this frequency range the proof uses techniques of frohlich and spencer a sequence of renormalization transformations shows that large regions where wave propagation is easily maintained become increasingly sparse as resonance is approached

an eigensystem approach to anderson localization abstract we introduce a new approach for proving localization pure point spectrum with exponentially decaying eigenfunctions dynamical localization for the anderson model at high disorder in contrast to the usual strategy we do not study finite volume green s functions instead we perform a multiscale analysis based on finite volume eigensystems eigenvalues and eigenfunctions information about eigensystems at a given scale is used to derive information about eigensystems at larger scales this eigensystem multiscale analysis treats all energies of the finite volume operator at the same time establishing level spacing and localization of eigenfunctions in a fixed box with high probability a new feature is the labeling of the eigenvalues and eigenfunctions by the sites of the box

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

nonparametric estimation of mixing densities for discrete distributions by a mixture density is meant a density of the form d where is a family of probability densities and is a probability measure on we consider the problem of identifyin

on the spectrum of schrodinger operators with a random potential we investigate the spectrum of schrodinger operatorsh of the type h qi f x xi i qi and i independent identically distributed random variables i d we establish a strong connection between the spectrum ofh and the spectra of deterministic periodic schrodinger operators from this we derive a condition for the existence of forbidden zones in the spectrum ofh for random one and three dimensional kronig penney potentials the spectrum is given explicitly

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

local wegner and lifshitz tails estimates for the density of states for continuous random schrodinger operators we introduce and prove local wegner estimates for continuous generalized anderson hamiltonians where the single site random variables are independent but not necessarily identically distributed in particular we get wegner estimates with a constant that goes to zero as we approach the bottom of the spectrum as an application we show that the differentiated density of states exhibits the same lifshitz tails upper bound as the integrated density of states

generation of bases with definite factorization for an n qubit system and mutually unbiased sets construction we propose a systematic procedure to construct all the possible bases with a definite factorization structure eigenstates of n commuting monomials constructed as products of pauli operators for an n qubit system as well as the possibility of collecting them into mutually unbiased sets we also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases the results are applied to generate non isomorphic complete sets of mutually unbiased bases

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions