on weyl channels being covariant with respect to the maximum commutative group of unitaries we investigate the weyl channels being covariant with respect to the maximum commutative group of unitary operators this class includes the quantum depolarizing channel and the two pauli channel as well then we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the two pauli channel in the qubit case

on automatic boundary corrections many popular curve estimators based on smoothing have difficulties caused by boundary effects these effects are visually disturbing in practice and can play a dominant role in theoretical analysis local polynomial regression smoothers are known to correct boundary effects automatically some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections in both settings of density and regression estimation we investigate best weight functions for local polynomial fitting at the endpoints and find a simple solution the solution is universal for general degree of local polynomial fitting and general order of estimated derivative furthermore such local polynomial estimators are best among all linear estimators in a weak minimax sense and they are highly efficient even in the usual linear minimax sense

time domain finite difference and finite element methods for maxwell equations in complex media extensions of finite difference time domain fdtd and finite element time domain fetd algorithms are reviewed for solving transient maxwell equations in complex media also provided are a few representative examples to illustrate the modeling capabilities of fdtd and fetd for complex media the term complex media refers here to media with dispersive bi anisotropic inhomogeneous and or nonlinear properties present in the constitutive tensors

additive bounds of minimum output entropies for unital channels and an exact qubit formula we find an upper bound for the minimum output entropy of a unital quantum channel and obtain an exact formula for general qubit channels our techniques incorporate the renyi entropies particularly with renyi parameter alpha 2 moreover since our upper bound is additive under tensor product we get as a corollary an upper bound for the classical capacity of unital quantum channels interestingly our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless hermitian operators and is tight in the sense that it gives the precise quantity of classical capacity of the werner holevo channel as an example we study quantum channels with operator sum representation that is made of the discrete weyl operators generalized pauli operators and explain how our formula works in this case finally we find new examples for which the minimum output renyi 2 entropy is additive

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

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

stability and dispersion analysis for adi fdtd method in lossy media the stability and dispersion analysis for the alternating direction implicit finite difference time domain adi fdtd method in lossy media is presented although the stability and numerical dispersion have been analyzed for the adi fdtd method most of the analysis is dedicated to the cases of lossless media here the stability and dispersion analysis is performed for the method in lossy media the stability analysis theoretically proves the unconditional stability of the adi fdtd method in lossy media meanwhile the dispersion analysis reveals the numerical loss and dispersion characteristics of this method this will be meaningful for the evaluation and further development of the adi fdtd method in lossy media

a stable fdtd algorithm for non diagonal anisotropic dielectrics a stable fdtd algorithm is developed for simulating maxwell s equations in anisotropic dielectric materials with principal axes not aligned with the grid the algorithm is stable because the finite difference operator that converts d to e is symmetric and positive semidefinite for contrast a previously developed asymmetric algorithm is shown to suffer from late time instabilities the presented algorithm has second order error for continuous dielectric materials and the error can be reduced to third order by richardson extrapolation applied to dielectrics with sharp interfaces the algorithm has first order error even when averaging the dielectric within partially filled grid cells however averaging the dielectric permits richardson extrapolation to obtain second order error

additivity for unital qubit channels additivity of the holevo capacity is proved for product channels under the condition that one of the channels is a unital qubit channel with the other completely arbitrary as a byproduct this proves that the holevo bound is the classical information capacity of such qubit channels and provides an explicit formula for this classical capacity additivity of minimal entropy and multiplicativity of p norms are also proved under the same assumptions the proof relies on a new bound for the p norm of an output state from the half noisy phase damping channel

quantum channels irreducibly covariant with respect to the finite group generated by the weyl operators in matrix algebras we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the weyl operators in particular we analyze the irreducibly covariant quantum channels that is the completely positive and trace preserving linear maps interestingly imposing additional symmetries leads to the so called generalized pauli channels which were recently considered in the context of the non markovian quantum evolution finally we provide examples of irreducibly covariant positive but not necessarily completely positive maps

a split step fdtd method for 3 d maxwell s equations in general anisotropic media a split step finite difference time domain fdtd method is presented for 3 d maxwell s equations in general anisotropic media the general anisotropic media can be characterized by full permittivity and permeability tensors stability analysis of the proposed split step fdtd method using the fourier method is presented the eigenvalues of the fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the courant limit time step thereby illustrating the stable and non dissipative nature of the split step fdtd method in general anisotropic media numerical results are presented to further validate the accuracy and stability of the proposed split step fdtd method in general anisotropic media

do validation for kernel density estimation bandwidth selection in kernel density estimation is one of the fundamental model selection problems of mathematical statistics the study of this problem took major steps forward with the articles of hall and marron 1987 and hall and johnstone 1992 since then the focus seems to have been on various versions of implementing the so called plug in method aimed at estimating the minimum mean integrated squared error mise the most successful of these efforts still seems to be the plug in method of sheather and jones 1991 or park and marron 1990 that we also use as a benchmark in this article in this article we derive a new theorem deriving the asymptotic theory for linear combinations of bandwidths obtained from different selectors as for example direct and indirect cross validation and plug in where we take advantage of recent advances in the study of indirect cross validation see hart and yi 1998 hart and lee 2005 and savchuk hart and sheather 2008 2010 we conclude that the slow

the capacity of the quantum depolarizing channel the information carrying capacity of the d dimensional depolarizing channel is computed it is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space and using measurements which are products of projections onto this same orthonormal basis in other words neither entangled signal states nor entangled measurements give any advantage for information capacity the result follows from an additivity theorem for the product channel spl delta spl ominus spl psi where spl delta is the depolarizing channel and spl psi is a completely arbitrary channel we establish the amosov holevo werner see probl inform transm vol 36 p 305 313 2000 p norm conjecture for this product channel for all p spl ges 1 and deduce from this the additivity of the minimal entropy and of the holevo quantity spl chi sup

the spatial fourth order energy conserved s fdtd scheme for maxwell s equations abstract in this paper we develop a new spatial fourth order energy conserved splitting finite difference time domain method for maxwell s equations based on the staggered grids the splitting technique is applied to lead to a three stage energy conserved splitting scheme at each stage using the spatial fourth order difference operators on the strict interior nodes by a linear combination of two central differences one with a spatial step and the other with three spatial steps we first propose the spatial high order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth order accuracy in space step the proposed scheme has the important properties energy conserved unconditionally stable non dissipative high order accurate and computationally efficient we first prove that the scheme satisfies energy conversations and is in unconditional stability we then prove the optimal error estimates of fourth order in spatial step and second order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence free as well numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results

hypercontractivity in finite dimensional matrix algebras we obtain hypercontractivity estimates for a large class of semigroups defined on finite dimensional matrix algebras mn these semigroups arise from poisson like length functions on zn zn and provide new hypercontractive families of quantum channels when is conditionally negative we also study the optimality of our estimates

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

wavelets for the maxwell s equations an overview abstract in recent years wavelets decompositions have been widely used in computational maxwell s curl equations to effectively resolve complex problems in this paper we review different types of wavelets that we can consider the cohen daubechies feauveau biorthogonal wavelets the orthogonal daubechies wavelets and the deslauries dubuc interpolating wavelets we summarize the main features of these frameworks and we propose some possible future works

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 bandwidth selector for local linear density estimators local linear density estimators achieve automatic boundary corrections and enjoy some typical optimal properties proper choice of the smoothing parameters is crucial for their performance a data based bandwidth selector is developed in the spirit of plug in rules consistency and asymptotic normality of the selected bandwidth are demonstrated the bandwidth is very efficient regardless of whether there are non smooth boundaries in the support of the density or not

an improved transformation based kernel estimator of densities on the unit interval the kernel density estimator kde suffers boundary biases when applied to densities on bounded supports which are assumed to be the unit interval transformations mapping the unit interval to the real line can be used to remove boundary biases however this approach may induce erratic tail behaviors when the estimated density of transformed data is transformed back to its original scale we propose a modified transformation based kde that employs a tapered and tilted back transformation we derive the theoretical properties of the new estimator and show that it asymptotically dominates the naive transformation based estimator while maintains its simplicity we then propose three automatic methods of smoothing parameter selection our monte carlo simulations demonstrate the good finite sample performance of the proposed estimator especially for densities with poles near the boundaries an example with real data is provided

pauli diagonal channels constant on axes we define and study the properties of channels which are analogous to unital qubit channels in several ways a full treatment can be given only when the dimension d pm a prime power in which case each of the d 1 mutually unbiased bases mub defines an axis along each axis the channel looks like a depolarizing channel but the degree of depolarization depends on the axis when d is not a prime power some of our results still hold particularly in the case of channels with one symmetry axis we describe the convex structure of this class of channels and the subclass of entanglement breaking channels we find new bound entangled states for d 3 for these channels we show that the multiplicativity conjecture for maximal output p norm holds for p 2 we also find channels with behaviour not exhibited by unital qubit channels including two pairs of orthogonal bases with equal output entropy in the absence of symmetry this provides new numerical evidence for the additivity of minimal output entropy

maximization of capacity and lp norms for some product channels it is conjectured that the holevo capacity of a product channel is achieved when product states are used as input amosov holevo and werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states in this article we establish both of these conjectures in the case that is arbitrary and is a cq or qc channel as defined by holevo we also establish the amosov holevo and werner conjecture when is arbitrary and either is a qubit channel and p 2 or is a unital qubit channel and p is integer our proofs involve a new conjecture for the norm of an output state of the half noisy channel i when is a qubit channel we show that this conjecture in some cases also implies additivity of the holevo capacity

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

fundamental schemes for efficient unconditionally stable implicit finite difference time domain methods generalized formulations of fundamental schemes for efficient unconditionally stable implicit finite difference time domain fdtd methods are presented the fundamental schemes constitute a family of implicit schemes that feature similar fundamental updating structures which are in simplest forms with most efficient right hand sides the formulations of fundamental schemes are presented in terms of generalized matrix operator equations pertaining to some classical splitting formulae including those of alternating direction implicit locally one dimensional and split step schemes to provide further insights into the implications and significance of fundamental schemes the analyses are also extended to many other schemes with distinctive splitting formulae detailed algorithms are described for new efficient implementations of the unconditionally stable implicit fdtd methods based on the fundamental schemes a comparative study of various implicit schemes in their original and new implementations is carried out which includes comparisons of their computation costs and efficiency gains

a split step approach for the 3 d maxwell s equations split step procedures have previously been used successfully in a number of situations e g for hamiltonian systems such as certain nonlinear wave equations in this study we note that one particular way to write the 3 d maxwell s equations separates these into two parts requiring only the solution of six uncoupled 1 d wave equations the approach allows arbitrary orders of accuracy in both time and space and features in many cases unconditional stability

development of the three dimensional unconditionally stable lod fdtd method a three dimensional unconditionally stable locally one dimensional finite difference time domain lod fdtd method is proposed and is proved unconditionally stable analytically in it the number of equations to be computed is the same as that with the conventional three dimensional alternating direction implicit fdtd adi fdtd but with reduced arithmetic operations the reduction in arithmetic operations leads to approximately 20 less computational time in comparisons with the adi fdtd method

maximal output purity and capacity for asymmetric unital qudit channels we consider generalizations of depolarizing channels to maps of the form with vk being unitary and kak a 1 we show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique we give conditions on vk under which multiplicativity of the maximal p norm and additivity of the minimal output entropy can be proved for with arbitrary we also show that the holevo capacity need not equal log d smin as one might expect for a convex combination of unitary conjugations

counterexample to an additivity conjecture for output purity of quantum channels a conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel as measured by the p norm should be multiplicative with respect to the tensor product of channels we disprove this conjecture for p 4 79 the same example with p also disproves a conjecture for the multiplicativity of the injective norm of hilbert space tensor products

boundary aware estimators of integrated density derivative products integrated squared density derivatives are important to the plug in type of bandwidth selector for kernel density estimation conventional estimators of these quantities are inefficient when there is a non smooth boundary in the support of the density we introduce estimators that utilize density derivative estimators obtained from local polynomial fitting they retain the rates of convergence in mean squared error that are familiar from non boundary cases and the constant coefficients have similar forms the estimators and the formula for their asymptotically optimal bandwidths which depend on integrated products of density derivatives are applied to automatic bandwidth selection for local linear density estimation simulation studies show that the constructed bandwidth rule and the sheather jones bandwidth are competitive in non boundary cases but the former overcomes boundary problems whereas the latter does not

an application of decomposable maps in proving multiplicativity of low dimensional maps in this paper we present a class of maps for which the multiplicativity of the maximal output p norm holds for p 2 and p 4 the class includes all positive trace preserving maps from b c3 to b c2 in this sense the result is a generalization of the corresponding result in the work of king and koldan new multiplicativity results for qubit maps j math phys 47 042106 2006 where the multiplicativity was proved for all positive trace preserving maps from b c2 to b c2 with p 2 and p 4 interestingly by contrast the multiplicativity of p norm was investigated in the context of quantum information theory and shown not to hold in general for high dimensional quantum channels hayden p and winter a counterexamples to the maximal p norm multiplicativity conjecture for all p 1 commun math phys 284 263 2008 moreover the werner holevo channel which is a map from b c3 to b c3 is a counterexample for p 4 79 werner and holevo j math phys 43 4353 2002

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

quantum channels arising from abstract harmonic analysis we present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups g beginning with a representation of the measure algebra m g we unify and elaborate on recent counter examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions as well as present an application to the noiseless subsystems method of quantum error correction using a representation of the completely bounded fourier multiplier algebra mcba g we provide a new class of counter examples to the recently solved asymptotic quantum birkhoff conjecture along with a systematic method of producing the examples using a geometric representation of schur maps further properties of our channels including duality quantum capacity and entanglement preservation are discussed along with potential applications to additivity conjectures