triangularly implicit iteration methods for ode ivp solvers it often happens that iteration processes used for solving the implicit relations arising in ode ivp methods only start to converge rapidly after a certain number of iterations fast convergence right from the beginning is particularly important if we want to use so called step parallel iteration in which the iteration method is concurrently applied at a number of step points in this paper we construct highly parallel iteration methods that do converge fast from the first iteration on our starting point is the pdirk method parallel diagonally implicit iterated runge kutta method designed for solving implicit runge kutta equations on parallel computers the pdirk method may be considered as a newton type iteration in which the newton jacobian is simplified to block diagonal form however when applied in a step parallel mode it turns out that its relatively slow convergence or even divergent behavior reduces the effectiveness of the step parallel scheme by replacing the block diagonal newton jacobian approximation in pdirk by a block triangular approximation we do achieve convergence right from the beginning at a modest increase of the computational costs our convergence analysis of the block triangular approach will be given for the wide class of general linear methods but the derivation of iteration schemes is limited to runge kutta based methods a number of experiments show that the new parallel triangularly implicit iterated runge kutta method ptirk method is a considerable improvement over the pdirk method

convexity of quasi entropy type functions lieb s and ando s convexity theorems revisited given a positive function f on 0 and a non zero real parameter we consider a function if a b x tr x f larb 1 rb x in three matrices a b 0 and x this generalizes the notion of monotone metrics on positive definite matrices and in the literature 1 has been typical we investigate how operator monotony of f is sufficient and or necessary for joint convexity concavity of if a b x similar discussions are given for quasi entropies and quantum skew informations

alpha divergence is unique belonging to both f divergence and bregman divergence classes a divergence measure between two probability distributions or positive arrays positive measures is a useful tool for solving optimization problems in optimization signal processing machine learning and statistical inference the csiszar f divergence is a unique class of divergences having information monotonicity from which the dual alpha geometrical structure with the fisher metric is derived the bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the alpha structure in general csiszar gave an axiomatic characterization of divergences related to inference problems the kullback leibler divergence is proved to belong to both classes and this is the only such one in the space of probability distributions this paper proves that the alpha divergences constitute a unique class belonging to both classes when the space of positive measures or positive arrays is considered they are the canonical divergences derived from the dually flat geometrical structure of the space of positive measures

fisher information in quantum statistics braunstein and caves braunstein s l and caves c m 1994 phys rev lett 72 3439 43 proposed to use helstrom s quantum information number to define meaningfully a metric on the set of all possible states of a given quantum system they showed that the quantum information is nothing other than the maximal fisher information in a measurement of the quantum system maximized over all possible measurements combining this fact with classical statistical results they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished based on arbitrary measurements on n identical copies of the given quantum system we show that the measurement which maximizes the fisher information typically depends on the true unknown state of the quantum system we close the resulting loophole in the argument by showing that one can still achieve the same optimal rate of distinguishability by a two stage adaptive measurement procedure when we consider states lying not on a smooth curve but on a manifold of higher dimension the situation becomes much more complex we show that the notion of distinguishability of close by states depends strongly on the measurement resources one allows oneself and on a further specification of the task at hand the quantum information matrix no longer seems to play a central role

monotone riemannian metrics and dynamic structure factor in condensed matter physics an analytical approach is developed to the problem of computation of monotone riemannian metrics e g bogoliubov kubo mori bures chernoff etc on the set of quantum states the obtained expressions originate from the morozova c encov and petz correspondence of monotone metrics to operator monotone functions the used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated nested commutators of a model hamiltonian t with the operator s involved through the control parameter h due to the sum rules for the frequency moments of the dynamic structure factor new presentations for the monotone riemannian metrics are obtained particularly relations between any monotone riemannian metric and the usual thermodynamic susceptibility or the variance of the operator s are discussed if the symmetry properties of the hamiltonian are given in terms of generators of some lie algebra the obtained expansions may be evaluated in a closed form these issues are tested on a class of model systems studied in condensed matter physics

quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

a priori probabilities of separable quantum states zyczkowski horodecki sanpera and lewenstein zhsl recently proposed a natural measure on the n dimensional quantum systems but expressed surprise when it led them to conclude that for n 2 2 disentangled separable systems are more probable 0 632 0 002 in nature than entangled ones we contend however that zhsl s rejected intuition has in fact a sound theoretical basis and that the a priori probability of disentangled 2 2 systems should more properly be viewed as considerably less than 0 5 we arrive at this conclusion in two quite distinct ways the first based on classical and the second quantum considerations both approaches however replace in whole or part the zhsl product measure by ones based on the volume elements of monotone metrics which in the classical case amounts to adopting the jeffreys prior of bayesian theory only the quantum theoretic analysis which yields the smallest probabilities of disentanglement uses the minimum number of parameters possible that is n2 1 as opposed to n2 n 1 although this over parametrization as recently indicated by byrd should be avoidable however despite substantial computation we are not able to obtain precise estimates of these probabilities and the need for additional possibly supercomputer analyses is indicated particularly so for higher dimensional quantum systems such as the 2 3 ones which we also study here

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

numerical solution of differential algebraic equations for constrained mechanical motion the two most popular formulations of the equations of constrained mechanical motion thedescriptor andstate space forms each have severe practical limitations in this paper we discuss and relate some proposed reformulations of the equations which have improved numerical properties

monotone riemannian metrics on density matrices with non monotone scalar curvature the theory of monotone riemannian metrics on the state space of a quantum system was established by denes petz in 1996 in a recent paper he argued that the scalar curvature of a statistically relevant monotone metric can be interpreted as an average statistical uncertainty the present paper contributes to this subject it is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed the manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state we show that not every monotone metric fulfils this expectation some of them behave in a very different way a mathematical condition is given for monotone riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics

geometry of canonical correlation on the state space of a quantum system a riemannian metric is defined on the state space of a finite quantum system by the canonical correlation or kubo mori bogoliubov scalar product this metric is infinitesimally induced by the nonsymmetric relative entropy functional or the von neumann entropy of density matrices hence its geometry expresses maximal uncertainty it is proven that the metric is monotone under stochastic mappings however an example shows that it is not the only such riemannian metric this fact is remarkable because in the probabilistic case the markovian monotonicity property characterizes the fisher information metric the essential difference appears in the curvatures of a classical state space and a quantum one a conjecture is made that the scalar curvature is monotone with respect to the more mixed statistical partial order of density matrices furthermore an information inequality resembling the cramer rao inequality of classical statistics is established the inequality provides a lower bound for the c

quantum and fisher information from the husimi and related distributions the two principal immediate influences which we seek to interrelate here upon the undertaking of this study are papers of yczkowski and slomczynski j phys a 34 6689 2001 and of petz and sudar j math phys 37 2262 1996 in the former work a metric the monge one specifically over generalized husimi distributions was employed to define a distance between two arbitrary density matrices in the petz sudar work completing a program of chentsov the quantum analog of the classically unique fisher information monotone metric of a probability simplex was extended to define an uncountable infinitude of riemannian also monotone metrics on the set of positive definite density matrices we pose here the questions of what is the specific unique fisher information metric for the classically defined husimi distributions and how does it relate to the infinitude of quantum metrics over the density matrices of petz and sudar we find a highly proximate small relative entropy relationship bet

the χ2 divergence and mixing times of quantum markov processes we introduce quantum versions of the 2 divergence provide a detailed analysis of their properties and apply them in the investigation of mixing times of quantum markov processes an approach similar to the one presented in the literature for classical markov chains is taken to bound the trace distance from the steady state of a quantum processes a strict spectral bound to the convergence rate can be given for time discrete as well as for time continuous quantum markov processes furthermore the contractive behavior of the 2 divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm in this context we analyze different versions of quantum detailed balance and finally give a geometric conductance bound to the convergence rate for unital quantum markov processes

covariance subspace and intrinsic crame spl acute r rao bounds crame spl acute r rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists the frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail the set of subspaces called the grassmann manifold and the set of covariance positive definite hermitian matrices have no fixed coordinate system associated with them and do not possess a vector space structure both of which are required for deriving classical crame spl acute r rao bounds intrinsic versions of the crame spl acute r rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators in the example of covariance matrix estimation closed form expressions for both the intrinsic and flat bounds are derived and compared with the root mean square error rmse of the sample covariance matrix scm estimator for varying sample support k the accuracy bound on unbiased covariance matrix estimators is shown to be about 10 log 10 n k sup 1 2 db where n is the matrix order remarkably it is shown that from an intrinsic perspective the scm is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice the rmse of the standard method of estimating subspaces using the singular value decomposition svd is compared with the intrinsic subspace crame spl acute r rao bound derived in closed form by varying both the signal to noise ratio snr of the unknown p dimensional subspace and the sample support in the simplest case the crame spl acute r rao bound on subspace estimation accuracy is shown to be about p n p sup 1 2 1 2 snr sup 1 2 rad for p dimensional subspaces it is seen that the svd based method yields accuracies very close to the crame spl acute r rao bound establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace the analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

out of time order fluctuation dissipation theorem we prove a generalized fluctuation dissipation theorem for a certain class of out of time ordered correlators otocs with a modified statistical average which we call bipartite otocs for general quantum systems in thermal equilibrium the difference between the bipartite and physical otocs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the wigner yanase skew information within this difference the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear response function that involves a time reversed process we show that the theorem can be generalized to higher order n partite otocs as well as in the form of generalized covariance

metric adjusted skew information we extend the concept of wigner yanase dyson skew information to something we call metric adjusted skew information of a state with respect to a conserved observable this skew information is intended to be a non negative quantity bounded by the variance of an observable in a state that vanishes for observables commuting with the state we show that the skew information is a convex function on the manifold of states it also satisfies other requirements proposed by wigner and yanase for an effective measure of information content of a state relative to a conserved observable we establish a connection between the geometrical formulation of quantum statistics as proposed by chentsov and morozova and measures of quantum information as introduced by wigner and yanase and extended in this article we show that the set of normalized morozova chentsov functions describing the possible quantum statistics is a bauer simplex and determine its extreme points we determine a particularly simple skew information the skew information parametrized by a 0 1 and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations

an efficient method for solving implicit and explicit stiff differential equations when the s stage fully implicit runge kutta rk method is used to solve a system of n ordinary differential equations ode the resulting algebraic system has a dimension ns its solution by gauss elimination is expensive and requires 2s3n3 3 operations in this paper we present an efficient algorithm which differs from the traditional rk method the formal procedure for uncoupling the algebraic system into a block diagonal matrix with s blocks of size n is derived for any s its solution is s2 2 times faster than the original nondiagonalized system for s even and s3 s 1 for s odd in terms of number of multiplications as well as s2 times in terms of number of additions multiplications in particular for s 3 the method has the same precision and stability properties as the well known rk based radauiia quadrature of ehle implemented by hairer and wanner in radau5 algorithm unlike radau5 however the method is applicable with any s and not only to the explicit odes my f x y where m const but also to the general implicit odes of the form f x y y 0 the block diagonal form of the algebraic system allows parallel processing the algorithm formally differs from the implicit rk methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coefficients are found by enforcing y to satisfy the differential equation at the collocation points locations of those points are found from the derived stability function such as to guarantee either a or l stability properties as well as a superior precision of the algorithm if constructed such as to be l stable the method is a good candidate for solving differential algebraic equations daes although not limited to any specific field the application of the method is illustrated by its implementation in the multibody dynamics described by both odes and daes copyright 2000 john wiley sons ltd

contraction coefficients for noisy quantum channels generalized relative entropy monotone riemannian metrics geodesic distance and trace distance are all known to decrease under the action of quantum channels we give some new bounds on and relationships between the maximal contraction for these quantities

wigner yanase information on quantum state space the geometric approach in the search of appropriate riemannian metrics on quantum state space the concept of statistical monotonicity or contraction under coarse graining has been proposed by chentsov the metrics with this property have been classified by petz all the elements of this family of geometries can be seen as quantum analogs of fisher information although there exists a number of general theorems shedding light on this subject many natural questions also stemming from applications are still open in this paper we discuss a particular member of the family the wigner yanase information using a well known approach that mimics the classical pull back approach to fisher information we are able to give explicit formulas for the geodesic distance the geodesic path the sectional and scalar curvatures associated to wigner yanase information moreover we show that this is the only monotone metric for which such an approach is possible

on quantum statistical inference interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially in step with dramatic new developments in experimental techniques for studying small quantum systems furthermore developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory the present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

non orthogonal domain parabolic equation and its tilted gaussian beam solutions a non orthogonal coordinate system which is a priori matched to localized initial field distributions for time harmonic wave propagation is presented applying in addition a rigorous paraxial asymptotic approximation results in a novel parabolic wave equation for beam type field propagation in 3d homogeneous media localized solutions to this equation that exactly match linearly phased gaussian aperture distributions are termed tilted gaussian beams these beams serve as the building blocks for various beam type expansion schemes application of the scalar waveobjects to electromagnetic field beam type expansion as well as reflection and transmission of these waveobjects by planar velocity dielectric discontinuity are presented a numerical example which demonstrates the enhanced accuracy of the tilted gaussian beams over the conventional ones concludes the paper

volume of the quantum mechanical state space the volume of the quantum mechanical state space over n dimensional real complex and quaternionic hilbert spaces with respect to the canonical euclidean measure is computed and explicit formulae are presented for the expected value of the determinant also in the general setting the case when the state space is endowed with a monotone metric or a pull back metric is also considered we give formulae for the volume of the state space with respect to the given riemannian metric we present the volume of the space of qubits with respect to various monotone metrics it turns out that the volume of the space of qubits can also be infinite we characterize those monotone metrics which generate infinite volume

on the curvature of the quantum state space with pull back metrics abstract the aim of the paper is to extend the notion of geometry in the classical and in the noncommutative case by introducing a more general class of pull back metrics and to give concrete formulas for the scalar curvature of these riemannian manifolds we introduce a more general class of pull back metrics of the noncommutative state spaces we pull back the euclidean riemannian metric of the space of self adjoint matrices with functions which have an analytic extension to a neighborhood of the interval 0 1 and whose derivative are nowhere zero we compute the scalar curvature in this setting and as a corollary we have the scalar curvature of the classical probability space when it is endowed with such a general pull back metric in the noncommutative setting we consider real and complex state spaces too we give a simplification of gibilisco isola conjecture for the first nontrivial classical probability space and we present the result of a numerical computation which indicate that the conjecture may be true for the space of real and complex qubits and for 3 3 density matrices too

quantum covariance quantum fisher information and the uncertainty relations in this paper the relation between quantum covariances and quantum fisher informations is studied this study is applied to generalize a recently proved uncertainty relation based on quantum fisher information the proof given here considerably simplifies the previously proposed proofs and leads to more general inequalities

families of completely positive maps associated with monotone metrics abstract an operator convex function on 0 which satisfies the symmetry condition k x 1 x k x can be used to define a type of non commutative multiplication by a positive definite matrix or its inverse using the primitive concepts of left and right multiplication and the functional calculus the operators for the inverse can be used to define quadratic forms associated with riemannian metrics which contract under the action of completely positive trace preserving maps we study the question of when these operators define maps which are also completely positive cp although a d 1 2 a d 1 2 is the only case for which both the map and its inverse are cp there are several well known one parameter families for which either the map or its inverse is cp we present a complete analysis of the behavior of these families as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points our primary tool is an order relation based on the concept of positive definite functions although some results can be obtained from known properties we also prove new results based on the positivity of the fourier transforms of certain functions concrete computations of certain fourier transforms not only yield new examples of positive definite functions but also examples in the much stronger class of infinitely divisible functions

wide angle shift map pe for a piecewise linear terrain a finite difference approach a wide angle parabolic wave equation solution is presented using shift map and finite difference techniques the corresponding split step fourier solution is well known the solution using finite difference technique where the standard parabolic wave equation is modified into the so called claerbout equation allowing propagation angles up to 45deg from the paraxial direction is also well known here we present an extension to that solution in which the shift map technique is incorporated into the finite difference scheme allowing a varying terrain to be considered the result is a solution that corresponds to the well known split step solution which is believed to perform well for terrain slopes up to 10deg 15deg and discontinuous slope changes on the order of 15deg 20deg this solution is a first order one with respect to the terrain slope however when using the finite difference technique it is also possible to find a second order solution with respect to the terrain slope this new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg which is an improvement

riemannian metrics on positive definite matrices related to means abstract the riemannian metric on the manifold of positive definite matrices is defined by a kernel function in the form k d h k i j i j 1 tr p i hp j k when i i p i is the spectral decomposition of the foot point d and the hermitian matrices h k are tangent vectors for such kernel metrics the tangent space has an orthogonal decomposition the pull back of a kernel metric under a mapping d g d is a kernel metric as well several riemannian geometries of the literature are particular cases for example the statistical metric for multivariate gaussian distributions and the quantum fisher information in the paper the case x y m x y is mostly studied when m x y is a mean of the positive numbers x and y there are results about the geodesic curves and geodesic distances the geometric mean the logarithmic mean and the root mean are important cases

propagation modeling over irregular terrain by the improved two way parabolic equation method the two way split step parabolic equation 2w sspe method is improved to be able to model both forward and backward waves propagation over irregular perfectly electrical conducting pec surfaces and irregular lossy surfaces with impedance boundary conditions ibc the discrete mixed fourier transform dmft is applied in both forward and backward direction to deal with ibc and the boundary shift bs method is used to dispose of the irregular terrain effects numerical results obtained by the improved 2w sspe algorithm for several typical scenarios with different ground types are compared with those got by the previous 2w sspe method and the one way parabolic equation method

topology of the cone of positive maps on qubit systems an alternative geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented the premise of the proof is the identification of positive maps with operators preserving the lorentz cone in four dimensions and it allows to decompose the positive maps with respect to those preserving the boundary of the cone in addition useful conditions implying complete positivity of a map of have been given together with a sufficient condition for complete positivity of an extremal schwarz map of lastly following the same geometrical approach a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

local asymptotic normality in quantum statistics the theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to le cam roughly speaking local asymptotic normality means that the family varphi theta 0 u sqrt n n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of gaussian state u of an algebra of canonical commutation relations the convergence holds for all local parameters u in mathbb r m such that theta theta 0 u sqrt n parametrizes a neighborhood of a fixed point theta 0 in theta subset mathbb r m

global boundary flattening transforms for acoustic propagation under rough sea surfaces this paper introduces a conformal transform of an acoustic domain under a one dimensional rough sea surface onto a domain with a flat top this non perturbative transform can include many hundreds of wavelengths of the surface variation the resulting two dimensional flat topped domain allows direct application of any existing acoustic propagation model of the helmholtz or wave equation using transformed sound speeds such a transform model combination applies where the surface particle velocity is much slower than sound speed such that the boundary motion can be neglected once the acoustic field is computed the bijective one to one and onto mapping permits the field interpolation in terms of the original coordinates the bergstrom method for inverse riemann maps determines the transform by iterated solution of an integral equation for a surface matching term rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the helmholt

weak superadditivity of skew information we revisit the superadditivity conjecture for skew information which has recently been disproved by hansen 2007 j stat phys 126 643 we establish two weak forms of superadditivity which are conjectured to be optimal our results show that in a certain sense the superadditivity is true with 50 off

covariance and fisher information in quantum mechanics variance and fisher information are ingredients of the cramer rao inequality we regard fisher information as a riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and fisher information in this approach we show that there is a kind of dual one to one correspondence between the candidates of the two concepts we emphasize that fisher information is obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold

quantum parameter estimation of a generalized pauli channel we present a quantum parameter estimation theory for a generalized pauli channel d d where the parameter is regarded as a coordinate system of the probability simplex d2 1 we show that for each degree n of extension id n d d n d d n the sld fisher information matrix for the output states takes the maximum when the input state is an n tensor product of a maximally entangled state me d d we further prove that for the corresponding quantum cramer rao inequality there is an efficient estimator if and only if the parameter is m affine in d2 1 these results rely on the fact that the family id me of output states can be identified with d2 1 in the sense of quantum information geometry this fact further allows us to investigate submodels of generalized pauli channels in a unified manner

propagation modeling over terrain using the parabolic wave equation we address the numerical solution of the parabolic wave equation over terrain using the fourier split step approach the method referred to as a shift map generalizes that of beilis and tappert 1979 who introduced a coordinate transformation technique to flatten the boundary this technique is extended to a wide angle form allowing larger propagation angles with respect to the horizon a new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave pwe using the previously developed mixed fourier transform it is also shown by example that in many cases of interest the boundary may be approximated by discrete piecewise linear segments without affecting the field solution a more accurate shift map solution of the pwe for a piecewise linear boundary is therefore developed for modeling propagation over digitally sampled terrain data the shift map solution is applied to various surface types including ramps wedges curved obstacles and actual terrain where possible comparisons are made between the numerical solution and an exact analytical form the examples demonstrate that the shift map performs well for surface slopes as large as 10 15 spl deg and discontinuous slope changes on the order of 15 20 spl deg to accommodate a larger range of slopes it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well known terrain masking knife edge diffraction approximation

minimal entropy of states emerging from noisy quantum channels we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel which is modeled by the action of a completely positive trace preserving or stochastic map we provide strong support for the conjecture that this minimal entropy is additive namely that the minimum entropy can be achieved when product states are transmitted explicitly we prove that for a tensor product of two unital stochastic maps on qubit states using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states we give a separate argument based on the geometry of the image of the set of density matrices under stochastic maps which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps we also show that the maximal norm of the output states is multiplicative for most product maps on n qubit states including all those for which at least one map is unital for the class of unital channels on c sup 2 we show that additivity of minimal entropy implies that the holevo see ieee trans inform theory vol 44 p 269 73 1998 and russian math surv p 1295 1331 1999 capacity of the channel is additive over two inputs achievable with orthogonal states and equal to the shannon capacity this implies that superadditivity of the capacity is possible only for nonunital channels

a new split step fourier algorithm for solving the parabolic wave equation with rough surface scattering a novel technique based on a generalized method of images has been developed to solve the parabolic wave equation with irregular ocean surface treated as a pressure release boundary this exact technique exploits the fact that full fast fourier transforms performed on both real and image oceans have extra degrees of freedom not available when fast sine transforms are used and makes possible a split step fourier algorithm that ensures that the pressure field vanishes at the prescribed surface elevation as a function of range the new algorithm is nearly as fast per range step as the standard flat surface split step fourier algorithm and retains the other pe model advantages of being fully diffractive and fully range dependent examples obtained using the special purpose compact microcomputer system pesogen pe solution generator will be presented

a dynamical uncertainty principle in von neumann algebras by operator monotone functions suppose that a1 an are observables selfadjoint matrices and is a state density matrix in this case the standard uncertainty principle proved by robertson gives a bound for the quantum generalized variance namely for det cov aj ak using the commutators aj ak this bound is trivial when n is odd recently a dierent inequality of robertson type has been proved by the authors with the help of the theory of operator monotone functions in this case the bound makes use of the commutators a j and is non trivial for any n in the present paper we generalize this new result to the von neumann algebra case nevertheless the proof appears to simplify all the existing ones

operator monotonicity of some functions abstract we investigate the operator monotonicity of the following functions f t t t 1 1 t 2 1 t n 1 t 1 1 t 2 1 t n 1 t 0 where r and i j 0 with i j i j 1 2 n this property for these functions has been considered by v e s szabo 14

a comparison of forward boundary integral and parabolic wave equation propagation models the parabolic wave equation and its variants have provided the theoretical framework for most practical forward propagation models split step integration generates an easily obtained robust solution for most applications irregular boundaries can be incorporated by using a conformal mapping technique introduced by beilis and tappert 1979 and refined by donohue and kuttler see ibid vol 48 p 260 77 2000 in an earlier paper we demonstrated an alternative method that incorporates a numerical solution to the forward boundary integral equation within each split step cycle this paper compares predictions of forward propagation obtained by these two distinctly different methods the results confirm that the pwe based method is very accurate for smoothly varying surfaces and that it captures the primary forward structure even in the presence of unresolved surface detail the moderate loss of fidelity is often an acceptable trade for increased computational efficiency there are situations however where the details of the surface structure are important furthermore the induced surface currents are unique to the forward boundary integral method we illustrate their use by calculating the bistatic scatter that would he measured from an isolated surface segment we show that the scattered field measured in this way can be normalized to form a bistatic scatter function only when the illuminating beam is tilted slightly toward the surface we interpret this disparity as a breakdown in concept that underlies a local scattering function

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

greene approximation wide angle parabolic equation for radio propagation this paper presents the greene second order parabolic equation pe solution to model radio propagation over irregular terrain this solution is implemented by finite difference and shift map technique the second order finite difference approach performs well for slopes up to about 15 and discontinuous slope changes up to about 30 which is better than can be achieved using the split step fourier approach the results prove that the greene pe solution has greater propagation angles than does the claerbout pe solution thus the greene second order solution can certainly give better results for large angle propagation over complicated terrain boundaries besides we treat the important problem of accuracy of the different approximations of the pe based propagation models and derive the general pe solutions with respect to the terrain slope for the different approximations it is useful to have collected in one paper the most important pe approximations for radio propagation

some operator monotone functions abstract we prove that the functions t t q 1 t p 1 1 are operator monotone in the positive half axis for 0 p q 1 and we calculate the two associated canonical representation formulae the result is used to find new monotone metrics quantum fisher information on the state space of quantum systems

uncertainty principle and quantum fisher information ii heisenberg and schrodinger uncertainty principles give lower bounds for the product of variances var a var b if the observables a b are not compatible namely if the commutator a b is not zero in this paper we prove an uncertainty principle in schrodinger form where the bound for the product of variances var a var b depends on the area spanned by the commutators i a and i b with respect to an arbitrary quantum version of the fisher information

a characterization of extended monotone metrics abstract classical information geometry has emerged from the study of geometrical aspect of the statistical estimation cencov characterized the fisher metric as a canonical metric on probability simplexes s n 1 x 1 x n r n x i 1 from a statistical point of view and campbell extended the characterization of the fisher metric from probability simplexes to positive cone r n in quantum information geometry quantum states which are represented by positive hermitian matrices with trace one are regarded as an extension of probability distributions a quantum version of the fisher metric is introduced and is called a monotone metric petz characterized the monotone metrics on the space of all quantum states in terms of operator monotone functions a purpose of the present paper is to extend a characterization of monotone metrics from the space of all states to the space of all positive hermitian matrices on finite dimensional hilbert space this characterization corresponds quantum modification of campbell s work

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

geometries of quantum states the quantum analog of the fisher information metric of a probability simplex is searched and several riemannian metrics on the set of positive definite density matrices are studied some of them appeared in the literature in connection with cramer rao type inequalities or the generalization of the berry phase to mixed states they are shown to be stochastically monotone here all stochastically monotone riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them a class of metrics can be extended to pure states and a constant multiple of the fubini study metric appears in the extension

bures volume of the set of mixed quantum states we compute the volume of the n2 1 dimensional set n of density matrices of size n with respect to the bures measure and show that it is equal to that of an n2 1 dimensional hyper hemisphere of radius 1 2 for n 2 we obtain the volume of the uhlmann hemisphere s3 4 we find also the area of the boundary of the set n and obtain analogous results for the smaller set of all real density matrices an explicit formula for the bures hall normalization constants is derived for an arbitrary n

explicit formulae for the bures metric the aim of this paper is to derive explicit formulae for the computation of the riemannian bures metric g on the manifold of finite dimensional nonsingular density matrices this riemannian metric introduced by uhlmann generalizes the fubini study metric to mixed states and is the infinitesimal version of the bures distance several formulae are known for computing the bures metric in low dimensions the formulae presented in this paper allow for computing in finite dimensions without any diagonalization procedures the first equations we give are essentially of the form where is a matrix of invariants of a further formula is adapted to the local orthogonal decomposition at generic points

peruse a numerical treatment of rough surface scattering for the parabolic wave equation scattering of underwater acoustic signals from real ocean surfaces often does not fit into any of the classical theoretical approaches to the problem thus the need for a numerical approach is clear a novel method is presented that uses a sequence of conformal mappings to locally flatten successive segments of the surface which is assumed piecewise linear and frozen in time each conformal mapping preserves the elliptic form of the reduced wave equation so that in each transformed space the parabolic approximation can be made and the solution advanced one range step using the split step fourier algorithm this peruse pe rough surface numerical method is validated by exhibiting classically predicted bragg angle peaks for single scatter of a plane wave from a sinusoidal surface for multiple rough surface scattering in surface ducts peruse results are analyzed by resolution in depth into the virtual modes of labianca j acoust soc am 53 1137 1147 1973 the resulting modal intensities are then

concrete examples of operator monotone functions obtained by an elementary method without appealing to lowner integral representation we study concrete examples of operator monotone functions obtained by an elementary method without appealing to lowner integral representation

an angle dependent impedance boundary condition for the split step parabolic equation method a formulation of the impedance boundary condition ibc is derived which can be used in split step parabolic wave equation solvers to accurately represent an interface with arbitrary dielectric properties the approach relies on ensuring that the plane wave decomposition of the field satisfies the appropriate ibc for each spectral component numerical experiments illustrating the robustness of the approach for low contrast interfaces and angles of incidence near the brewster angle are presented

comparison of various means for operators for hilbert space operatorsh k xwithh k 0 the norm inequality h1 2xk1 2 12 hx xk is known where is an arbitrary unitarily invariant norm a refinement of this arithmetic geometric mean inequality is studied similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean

an improved impedance boundary algorithm for fourier split step solutions of the parabolic wave equation a new implementation of the previously published mixed fourier transform mft method for including impedance boundaries in split step parabolic equation solutions is described and demonstrated the new algorithm is formulated entirely in the discrete domain which results in extended applicability and increased computation speed a brief review of the original mft solution is followed by a detailed description of the discrete formulation the performance of the new algorithm is then demonstrated with a few examples which rely heavily on the accuracy of the impedance boundary these examples include 10 mhz surface wave propagation over smooth and rough sea surfaces and 10 ghz calculations utilizing an effective rough surface impedance

on the geometry of mixed states and the fisher information tensor in this paper we will review the co adjoint orbit formulation of finite dimensional quantum mechanics and in this framework we will interpret the notion of quantum fisher information index and metric following previous work of part of the authors who introduced the definition of fisher information tensor we will show how its antisymmetric part is the pullback of the natural kostant kirillov souriau symplectic form along some natural diffeomorphism in order to do this we will need to understand the symmetric logarithmic derivative as a proper 1 form settling the issues about its very definition and explicit computation moreover the fibration of co adjoint orbits seen as spaces of mixed states is also discussed

how to distinguish quantum covariances using uncertainty relations abstract to each operator monotone function it is possible to associate a quantum version of the classical covariance we show that i only for regular quantum covariances one can prove non trivial uncertainty relations ii the usual quantum covariance gives the best inequalities in this setting

connections and metrics respecting purification of quantum states standard purification interlaces hermitian and riemannian metrics on the space of density operators with metrics and connections on the purifying hilbert schmidt space we discuss connections and metrics which are well adopted to purification and present a selected set of relations between them a connection as well as a metric on state space can be obtained from a metric on the purification space we include a condition with which this correspondence becomes one to one our methods are borrowed from elementary representation and fiber space theory we lift as an example solutions of a von neumann equation write down holonomy invariants for cyclic ones and add noise to a curve of pure states

a wide angle split step algorithm for the parabolic equation in this paper we describe a new wide angle parabolic equation based on an operator splitting that permits the use of a marching type fourier transform solution method the equation was first presented by feit and fleck appl opt 17 3990 3998 1978 for studying propagation within optical fibers existing computer codes which numerically solve the standard parabolic equation of ocean acoustics by the split step algorithm of tappert and hardin are easily modified to accommodate the wide angle capability of the new equation in addition since the new wide angle equation is less sensitive to the value of the reference wavenumber the effects of phase errors are greatly reduced the results of a simple error analysis indicate that improved accuracy can be achieved by the new wide angle equation for propagation conditions typical of deep ocean environments this is supported by our numerical experience a summary of which is presented in the paper for test cases where the variation of the acoustic index

characterizations of certain means of positive operators abstract in this paper we present some characterizations for quasi arithmetic operator means among them the arithmetic and harmonic means on the positive definite cone of the full algebra of hilbert space operators and also for the kubo ando geometric mean on the positive definite cone of a general non commutative c algebra

two quantum analogues of fisher information from a large deviation viewpoint of quantum estimation we discuss two quantum analogues of the fisher information the symmetric logarithmic derivative fisher information and kubo mori bogoljubov fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators as another comparison it is shown that the difference between them is characterized by the change of the order of limits

second order efficiency of minimum contrast estimators in a curved exponential family this paper presents a sufficient condition for second order efficiency of an estimator the condition is easily checked in the case of minimum contrast estimators the a minimum contrast estimator is defined and proved to be second order efficient for every a 0 a 1 the fisher scoring method is also considered in the light of second order efficiency it is shown that a contrast function is associated with the second order tensor and the affine connection this fact leads us to prove the above assertions in the differential geometric framework due to amari

remarks on kim s strong subadditivity matrix inequality extensions and equality conditionsa we describe recent work of kim j math phys 53 122204 2012 to show that operator convex functions associated with quasi entropies can be used to prove a large class of new matrix inequalities in the tri partite and bi partite setting by taking a judiciously chosen partial trace over all but one of the spaces we give some additional examples in both settings furthermore we observe that the equality conditions for all the new inequalities are essentially the same as those for strong subadditivity

on skew information in this paper we show that skew information introduced by wigner and yanase which is a natural informational extension of variance for pure states can be interpreted as a measure of quantum uncertainty by virtue of skew information we establish a new uncertainty relation in the spirit of schrodinger which incorporates both incompatibility encoded in the commutator and correlations encoded in a new correlation measure related to skew information between observables and moreover is stronger than the conventional ones

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

completely positive mappings and mean matrices abstract some functions f r r induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices moreover such a function f can define a linear mapping j d f 1 m n m n on matrices which is basic in the constructions of monotone metrics the present subject is to check the complete positivity of j d f 1 in the case of a few concrete functions f this problem has been motivated by applications in quantum information

quantum monotone metrics induced from trace non increasing maps and additive noise quantum monotone metric was introduced by petz linear algebra appl 244 81 96 1996 and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions later these were extended to monotone metrics on the set of positive operators whose traces are not always the one based on completely positive trace preserving maps it was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators in this paper we introduce another extension of quantum monotone metrics that have monotonicity under completely positive trace non increasing maps and additive noise we prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics we show that our monotone metrics have some natural properties such as additivity of direct sum convexity and monotonicity with respect to positive operators

an alternative direction decomposition scheme and error analysis for parabolic equation model alternate direction implicit adi method has been widely used to analyze the tunnel propagation or electromagnetic scattering problem for the finite difference method in this paper a novel alternate direction decomposition add scheme is proposed to simplify the split step fourier transform ssft algorithm in parabolic equation pe model the spatial matrix of electric field is decoupled into two subarrays by add ssft algorithm which is used to decrease the computational burden for 3 d pe problems in a large scale environment after the integration order exchanged also the phase error of this approximation is analyzed to show the algorithm performance along with several numerical results including the tunnel cases with traditional modal theory and they are provided to validate and calibrate the pe model and demonstrate the efficiency and accuracy of the proposed algorithm moreover the scattering computation is investigated and compared with multilevel fast multipole method to show that add ssft is appropriate for solving the scattering problems for the electrical large target

an l stable rosenbrock method for step by step time integration in structural dynamics abstract a single step two stage l stable rosenbrock method is given in a form suitable for step by step integration of linear and non linear structural dynamics problems the method has second order accuracy and is a true single step method in that it does not require initial values of acceleration for each time step the method requires a single formation of the tangential stiffness matrix a single lu decomposition and two linear system substitutions but does not require iteration analysis and numerical examples are used to show the good stability and accuracy properties of the method and the importance of l stability when integrating numerically stiff problems in structural dynamics

on the monotonicity of scalar curvature in classical and quantum information geometry we study the monotonicity under mixing of the scalar curvature for the geometries on the simplex of probability vectors from the results obtained and from numerical data we are led to some conjectures about quantum geometries and wigner yanase dyson information finally we show that this last conjecture implies the truth of the petz conjecture about the monotonicity of the scalar curvature of the bogoliubov kubo mori monotone metric