fast algorithms for close to toeplitz plus hankel systems and two sided linear prediction the low displacement rank definition of close to toeplitz ct matrices is extended to close to toeplitz plus hankel ctph matrices fast algorithms for solving ctph systems of equations are presented a matrix is defined as ctph if the sum of a ct matrix and a second ct matrix postmultiplied by an exchange matrix an equivalent definition in terms of uv rank is also given this definition is motivated by the application of the algorithms to two sided linear prediction tsp autocorrelation and covariance forms of tsp analogous to those for one sided linear prediction osp are defined the covariance form of tsp is solved using the ctph fast algorithms just as the covariance form of osp is solved using ct fast algorithms numerical examples show that tsp produces smaller residuals than osp and resolves sharp spectral peaks better than osp and that covariance tsp produces smaller residuals than autocorrelation tsp

new analogs of split algorithms for arbitrary toeplitz plus hankel matrices fast algorithms for solving arbitrary toeplitz plus hankel systems of equations are presented the algorithms are analogs of the split levinson and schur algorithms although the more general toeplitz plus hankel structure requires that the algorithms be based on a four term recurrence relations with the previous split algorithms are considered the algorithms require roughly half as many multiplications as previous fast algorithms for toeplitz plus hankel systems

the geometric approach to quantum correlations computability versus reliability we propose a modified metric based on the hilbert schmidt norm and adopt it to define a rescaled version of the geometric measure of quantum discord such a measure is found not to suffer from pathological dependence on state purity although the employed metric is still non contractive under quantum operations we show that the resulting indicator of quantum correlations is in agreement with other bona fide discord measures in a number of physical examples we present a critical assessment of the requirements of reliability versus computability when approaching the task of quantifying or measuring general quantum correlations in a bipartite state

characterizing nonclassical correlations via local quantum uncertainty quantum mechanics predicts that measurements of incompatible observables carry a minimum uncertainty which is independent of technical deficiencies of the measurement apparatus or incomplete knowledge of the state of the system nothing yet seems to prevent a single physical quantity such as one spin component from being measured with arbitrary precision here we show that an intrinsic quantum uncertainty on a single observable is ineludible in a number of physical situations when revealed on local observables of a bipartite system such uncertainty defines an entire class of bona fide measures of nonclassical correlations for the case of 2 ifmmode times else texttimes fi d systems we find that a unique measure is defined which we evaluate in closed form we then discuss the role that these correlations which are of the discord type can play in the context of quantum metrology we show in particular that the amount of discord present in a bipartite mixed probe state guarantees a minimum precision as quantified by the quantum fisher information in the optimal phase estimation protocol

a class of iteration methods based on the hss for toeplitz systems of weakly nonlinear equations for toeplitz systems of weakly nonlinear equations combining the separability and strong dominance between the linear and the nonlinear terms with the hermitian and skew hermitian splitting hss iteration technique we establish two nonlinear composite iteration schemes called picard csss and nonlinear csss like iteration methods which are based on a special case of the hss where the symmetric part h 1 2 a a t is a centrosymmetric matrix and the skew symmetric part h 1 2 a a t is a skew centrosymmetric matrix the advantages of these methods are that they can transfer the linear sub systems involved in inner iteration to two linear systems of half an order besides fast methods are available for computing the two half steps involved in the inner iteration numerical results are provided to further show that both picard csss and nonlinear csss like iteration methods are feasible and effective

two methods for toeplitz plus hankel approximation to a data covariance matrix recently fast algorithms have been developed for computing the optimal linear least squares prediction filters for nonstationary random processes fields whose covariances have block toeplitz hankel form if the covariance of the random process field must be estimated from the data the following problem is presented given a data covariance matrix computer from the available data find the toeplitz plus hankel matrix closest to this matrix in some sense the authors give two procedures for computing the toeplitz plus hankel matrix that minimizes the hilbert schmidt norm of the difference between the two matrices the first approach projects the data covariance matrix onto the subspace of toeplitz plus hankel matrices for which basis functions can be computed using a gram schmidt orthonormalization the second approach projects onto the subspace of symmetric toeplitz plus skew persymmetric hankel matrices resulting in a much simpler algorithm the extension to block toeplitz plus hankel data covariance matrix approximation is also addressed

stationary measures and invariant subsets of homogeneous spaces iii let g be a real lie group be a lattice in g and g be a compactly generated closed subgroup of g if the zariski closure of the group ad g is semisimple with no compact factor we prove that every g orbit closure in g is a finite volume homogeneous space we also establish related equidistribution properties

a fast algorithm for optimal linear interpolation a fast algorithm for computing the optimal linear interpolation filter is developed the algorithm is based on the sherman morrison inversion formula for symmetric matrices the relationship between the derived algorithm and the levinson algorithm is illustrated it is shown that the new algorithm in comparison with the well known algorithms requires fewer multiplications and hence is of lower complexity

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

a systolic architecture for new split algorithms for arbitrary toeplitz plus hankel matrices recently new fast algorithms have been developed for computing the optimal linear least squares prediction filters for arbitrary toeplitz plus hankel covariances the authors propose a systolic architecture that can fully express the inherent concurrency of this highly parallelizable algorithm the simplification of this array structure for centrosymmetric covariances is also addressed

analogues of split levinson schur and lattice algorithms for three dimensional random field estimation problems fast algorithms for computing the linear least squares estimate of a three dimensional random field from noisy observations inside a sphere are proposed the algorithms can be viewed as three dimensional analogues of the split levinson schur and lattice algorithms of linear prediction since they exploit an assumed toeplitz plus hankel structure of the double radon transform of the random field covariance therefore these algorithms require fewer computations than would the solution of the three dimensional wiener hopf integral equation unlike previous generalized levinson algorithms no quarter plane or asymmetric half plane support assumptions for the filter are necessary nor is the three dimensional filtering problem treated as a multichannel vector filtering problem the algorithms work in three stages first the three dimensional split schur algorithm computes a potential from the covariance of the random field this potential is a three dimensional analogue of the parameter appearing in the

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

efficient solution of a toeplitz plus hankel coefficient matrix system of equations there are well known fast algorithms such as the levinson recursion for solving linear equations with a toeplitz or hankel coefficient matrix this paper extends the saving obtained by the levinson recursion for inversion of a toeplitz matrix to problems involving the inversion of matrices which are the sum of toeplitz and hankel matrices

discrete fast algorithms for two dimensional linear prediction on a polar raster new generalized split levinson and schur algorithms for the two dimensional linear least squares prediction problem on a polar raster are derived the algorithms compute the prediction filter for estimating a random field at the edge of a disk from noisy observations inside the disk the covariance function of the random field is assumed to have a toeplitz plus hankel structure for both its radial part and its transverse angular part this assumption is valid for some types of random fields such as isotropic random fields the algorithms generalize the split levinson and schur algorithms in two ways 1 to two dimensions and 2 to toeplitz plus hankel covariances

optimality of gaussian discord in this letter we exploit the recently solved conjecture on the bosonic minimum output entropy to show the optimality of gaussian discord so that the computation of quantum discord for bipartite gaussian states can be restricted to local gaussian measurements we prove such optimality for a large family of gaussian states including all two mode squeezed thermal states which are the most typical gaussian states realized in experiments our family also includes other types of gaussian states and spans their entire set in a suitable limit where they become choi matrices of gaussian channels as a result we completely characterize the quantum correlations possessed by some of the most important bosonic states in quantum optics and quantum information

peres horodecki separability criterion for continuous variable systems the peres horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states the partial transpose operation admits in the continuous case a geometric interpretation as mirror reflection in phase space this recognition leads to uncertainty principles stronger than the traditional ones to be obeyed by all separable states for all bipartite gaussian states the peres horodecki criterion turns out to be a necessary and sufficient condition for separability

fast algorithms for linear least squares smoothing problems in one and two dimensions using generalized discrete bellman siegert krein resolvent identities new fast algorithms for linear least square smoothing problems in one and two dimensions are derived these are discrete and multidimensional generalizations of the bellman siegert krein resolvent identity which has been applied to the continuous one dimensional stationary smoothing problem by t kailath 1969 the new equations relate the linear least squares prediction filters associated with discrete random fields to the smoothing filters for those fields this results in new fast algorithms for deriving the latter from the former in particular used in conjunction with recently developed generalized one two dimensional split levinson and schur algorithms for covariances with block toeplitz plus hankel structure these algorithms can be used to compute smoothing filters for random fields defined on a polar raster using fewer computations than those required by previous algorithms

hellinger distance as a measure of gaussian discord especially investigated in recent years the gaussian discord can tentatively be quantified by a distance between a given two mode gaussian state gs and the set of all the zero discord two mode gss however as this set consists only of product states such a distance captures all the correlations quantum and classical between modes therefore it is merely an upper bound for the geometric discord no matter which is the employed distance in this work we choose for this purpose the hellinger metric that is known to have many beneficial properties recommending it as a good measure of quantum behaviour in general this metric is determined by affinity a relative of the uhlmann fidelity with which it shares many important features as a first step of our work the affinity of a pair of n mode gss is written then in the two mode case we succeeded in determining exactly the closest gaussian product state and computed the gaussian discord accordingly the obtained general formula is remarkably simple and becomes still friendlier in the significant case of symmetric two mode gss we then analyse in detail two special classes of two mode gss of theoretical and experimental interest as well the squeezed thermal states and the mode mixed thermal ones the former are separable under a well known threshold of squeezing while the latter are always separable it is worth stressing that for symmetric states belonging to either of these classes we find consistency between their geometric hellinger discord and the originally defined discord in the gaussian approach at the same time the gaussian hellinger discord of such a state turns out to be a reliable measure of the total amount of its cross correlations

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

unified view of quantum and classical correlations we discuss the problem of the separation of total correlations in a given quantum state into entanglement dissonance and classical correlations using the concept of relative entropy as a distance measure of correlations this allows us to put all correlations on an equal footing entanglement and dissonance whose definition is introduced here jointly belong to what is known as quantum discord our methods are completely applicable for multipartite systems of arbitrary dimensions we investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states

correction to comments on a curiosum concerning discrete time convolution plus a remark in the author s comments to an article by hall and wise see ibid vol 38 no 6 p 1059 1990 included a typographic error which is corrected he also points out that the sequences used in the example become l sub 1 norm sequences

no local broadcasting theorem for multipartite quantum correlations we prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled as long as it does not simply encode a multipartite classical probability distribution said quantumness is revealed by the new task of local broadcasting i e of locally sharing preestablished correlations which is feasible if and only if correlations are stricly classical our operational approach leads to natural definitions of measures for quantumness of correlations it also reproduces the standard no broadcasting theorem as a special case

einstein podolsky rosen like separability indicators for two mode gaussian states we investigate the separability of the two mode gaussian states tmgss by using the variances of a pair of einstein podolsky rosen epr like observables our starting point is inspired by the general necessary condition of separability introduced by duan et al 2000 phys rev lett 84 2722 we evaluate the minima of the normalized forms of both the product and sum of such variances as well as that of a regularized sum making use of simon s separability criterion which is based on the condition of positivity of the partial transpose ppt of the density matrix simon 2000 phys rev lett 84 2726 we prove that these minima are separability indicators in their own right they appear to quantify the greatest amount of epr like correlations that can be created in a tmgs by means of local operations furthermore we reconsider the epr like approach to the separability of tmgss which was developed by duan et al with no reference to the ppt condition by optimizing the regularized form of their epr like uncertainty sum we derive a separability indicator for any tmgs we prove that the corresponding epr like condition of separability is manifestly equivalent to simon s ppt one the consistency of these two distinct approaches epr like and ppt affords a better understanding of the examined separability problem whose explicit solution found long ago by simon covers all situations of interest

blind deconvolution of symmetric noncausal impulse responses using two sided linear prediction considers the problem of estimating the time symmetric noncausal impulse response of a linear time invariant system from measurements of the response of the system to an unknown input signal which is assumed to be a realization of a white random process the symmetric impulse response is modeled by a two sided ar or arma system model the two sided ar coefficients are estimated using a two step procedure first an estimate of an unconstrained parameter vector is computed by solving a close to toeplitz plus hankel system of equations using previously developed fast algorithms then the polynomial square root of the result is obtained by solving a constrained least squares problem which has a simple solution unlike previous methods this approach requires no iterative procedure however it may lead to an unstable model in some extreme cases simulation results illustrate the performance of the proposed methods

cryptographic distinguishability measures for quantum mechanical states this paper mostly expository in nature surveys four measures of distinguishability for quantum mechanical states this is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions namely the probability of an identification error the kolmogorov distance the bhattacharyya coefficient and the shannon 1948 distinguishability as defined through mutual information these measures have a long history of use in statistical pattern recognition and classical cryptography we obtain several inequalities that relate the quantum distinguishability measures to each other one of which may be crucial for proving the security of quantum cryptographic key distribution in another vein these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states this is a tool that may prove useful in the analysis of various quantum cryptographic protocols

fast triangular factorization of the sum of quasi toeplitz and quasi hankel matrices abstract the literature contains several recent fast algorithms for the triangular factorization of strongly regular toeplitz plus hankel matrices in this paper we study the rather more general sum of quasi toeplitz and quasi hankel matrices both hermitian and non hermitian quasi toeplitz and quasi hankel matrices are those that are congruent to toeplitz and hankel matrices in a special sense the derivation is based on the concept of displacement structure and its intimate relation to the schur reduction procedure for triangular factorization various special cases covering displacement ranks from two to eight are considered several other problems e g factorization of the inverse matrix solution of exact or overdetermined linear systems can be reduced to the direct factorization problem

inseparability of mixed two mode gaussian states generated with a su 1 1 interferometer we study the entanglement properties of two mode gaussian light emerging from a generic su 1 1 interferometer our tool is the two mode characteristic function which is determined by the 4 4 covariance matrix for an initial product of two mixed single mode gaussian states we investigate the output two mode covariance matrix its structure displays the noise properties of the reduced states as well as the correlations between modes classicality of the output two mode state is characterized by the existence of the glauber sudarshan p representation for testing separability we apply the peres simon criterion requiring preservation of the positivity of the density matrix under partial transposition since inseparability entails nonclassicality the threshold gain above which nonclassicality of the output state becomes manifest is lower than that allowing for its inseparability we find that only for a thermal input do nonclassicality and inseparability of the output have the same threshold gain

on the hss iteration methods for positive definite toeplitz linear systems we study the hss iteration method for large sparse non hermitian positive definite toeplitz linear systems which first appears in bai golub and ng s paper published in 2003 z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 and hss stands for the hermitian and skew hermitian splitting of the coefficient matrix a in this note we use the hss iteration method based on a special case of the hss splitting where the symmetric part h 12 a a t is a centrosymmetric matrix and the skew symmetric part s 12 a a t is a skew centrosymmetric matrix for a given toeplitz matrix hence fast methods are available for computing the two half steps involved in the hss and ihss iteration methods some numerical results illustrate their effectiveness

random walks on finite volume homogeneous spaces extending previous results by a eskin and g margulis and answering their conjectures we prove that a random walk on a finite volume homogeneous space is always recurrent as soon as the transition probability has finite exponential moments and its support generates a subgroup whose zariski closure is semisimple

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

a new autoregressive method for high performance spectrum analysis in this paper a new method of autoregressive ar spectrum estimation is presented it shall be called two sided autoregressive spectrum estimation because an interpolative or smoothing model is postulated as opposed to the predictive one sided model used in ar modeling the matrix equations arising in the estimation procedures proposed in this paper exhibit a special structure the exploitation of these structures leads to fast solutions that reduce the total number of computations by an order of magnitude compared with straightforward approaches also special attention is directed to the constrained two sided ar model simulation examples show higher resolution capability of the proposed method relative to the least squares ar method

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16