joint estimation of real squeezing and displacement we study the problem of joint estimation of real squeezing and amplitude of the radiation field deriving the measurement that maximizes the probability density of detecting the true value of the unknown parameters more generally we provide a solution for the problem of estimating the unknown unitary action of a nonunimodular group in the maximum likelihood approach remarkably in this case the optimal measurements do not coincide with the so called square root measurements in the case of squeezing and displacement we analyse in detail the sensitivity of estimation for coherent states and displaced squeezed states deriving the asymptotic relation between the uncertainties in the joint estimation and the corresponding uncertainties in the optimal separate measurements of squeezing and displacement a two mode set up is also analysed showing how entanglement between optical modes can be used to approximate perfect estimation

on quantum detection and the square root measurement we consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states we consider a collection of pure states and seek a positive operator valued measure povm consisting of rank one operators with measurement vectors closest in squared norm to the given states we compare our results to previous measurements suggested by peres and wootters 1991 and hausladen et al 1996 where we refer to the latter as the square root measurement srm we obtain a new characterization of the srm and prove that it is optimal in a least squares sense in addition we show that for a geometrically uniform state set the srm minimizes the probability of a detection error this generalizes a similar result of ban et al see int j theor phys vol 36 p 1269 88 1997

asymptotic solution of the wang uhlenbeck recurrence time problem a langevin particle is initiated at the origin with positive velocity its trajectory is terminated when it returns to the origin in 1945 wang and uhlenbeck posed the problem of finding the joint probability density function pdf of the recurrence time and velocity naming it the recurrence time problem we show that the short time asymptotics of the recurrence pdf is similar to that of the integrated brownian motion solved in 1963 by mckean we recover the long time t ensuremath 3 2 decay of the first arrival pdf of diffusion by solving asymptotically an appropriate variant of mckean s integral equation

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 posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction diffusion problem on anisotropic meshes abstract based on conservative flux reconstruction we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction diffusion problem on anisotropic meshes since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes without the assumption that the meshes are shape regular our estimates give the upper bounds on the error only containing the alignment measure factor and therefore could provide actual numerical bounds if the alignment measure was approximated very well two resulting error estimators are presented to be equivalent to each other up to a data oscillation one of which can be directly constructed without solving local neumann problems and provide computable error bound numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure

optimal tight frames and quantum measurement tight frames and rank one quantum measurements are shown to be intimately related in fact the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank one generalized quantum measurements on that space using this relationship frame theoretical analogs of various quantum mechanical concepts and results are developed the analog of a least squares quantum measurement is a tight frame that is closest in a least squares sense to a given set of vectors the least squares tight frame is found for both the case in which the scaling of the frame is specified constrained least squares frame clsf and the case in which the scaling is chosen to minimize the least squares error unconstrained least squares frame ulsf the well known canonical frame is shown to be proportional to the ulsf and to coincide with the clsf with a certain scaling

optimal distinction between non orthogonal quantum states given a finite set of linearly independent quantum states an observer who examines a single quantum system may sometimes identify its state with certainty however unless these quantum states are orthogonal there is a finite probability of failure a complete solution is given to the problem of optimal distinction of three states having arbitrary prior probabilities and arbitrary detection values a generalization to more than three states is outlined

implementation of group covariant positive operator valued measures by orthogonal measurements we consider group covariant positive operator valued measures povms on a finite dimensional quantum system following neumark s theorem a povm can be implemented by an orthogonal measurement on a larger system accordingly our goal is to find a quantum circuit implementation of a given group covariant povm which uses the symmetry of the povm based on representation theory of the symmetry group we develop a general approach for the implementation of group covariant povms which consist of rank one operators the construction relies on a method to decompose matrices that intertwine two representations of a finite group we give several examples for which the resulting quantum circuits are efficient in particular we obtain efficient quantum circuits for a class of povms generated by weyl heisenberg groups these circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

cutoff rate for the m ary psk modulation channel with optimal quantum detection the cutoff rate r sub 0 of an optical channel transmitting m ary phase shift keyed coherent states is calculated for a receiver using the optimum quantum detector this is shown to be nearly equivalent when m 1 to a system that measures the phase of the received signal a closed form expression yielding a good approximation to r sub 0 is derived

a semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

multiaccess quantum channels shared communication channels are subject to multiple access interference transmitter and receiver design techniques that explicitly deal with this interference have been shown to improve substantially the performance of communication systems over radio frequency and other classical channels quantum multiple access communication channels on the other hand have received comparatively little attention in this paper an input output model for multiple access quantum channels relevant to optical communications is proposed the model accounts for multiaccess interference signal attenuation and random noise and can be used in the analysis and design of communication systems using a result from optimization a perturbation method is developed to find the minimum achievable error probability in small interference channels it is shown that the quantum measurement that minimizes the error probability in a no interference channel is robust in the presence of small multiaccess interference the results are illustrated with numerical examples which show that optimal quantum detectors can significantly outperform conventional detectors even for moderate levels of crosstalk

minimum error discrimination between self symmetric mixed quantum state signals we derive some properties of minimum error measurements for mixed quantum state signals where a density operator itself has a certain symmetry in the first case we show that if there exists a regular normal operator that commutes with each density operator of mixed state signals then there exists an optimal measurement that consists of detection operators expressed as the direct sum of positive operators with supports in the eigenspaces of the regular normal operator in the second case we show that for abelian geometrically uniform state signals if a matrix which represents one of the signals in a certain basis has all real elements then the matrix which represents the corresponding detection operator of an optimal measurement in that basis also has all real elements in addition we discuss some examples of applications of these properties and derive analytical solutions of optimal measurements for special cases

a local problem error estimator for anisotropic tetrahedral finite element meshes the poisson problem is solved by the finite element method on anisotropic tetrahedral or triangular meshes the focus is on adaptive algorithms and in particular on a posteriori error estimators based on the solution of a local problem on anisotropic meshes such estimators cannot be analyzed in the common way known from isotropic meshes the first estimator proposed here is a slight modification of a familiar isotropic counterpart by a rigorous analysis it is proven that this estimator is equivalent to a known anisotropic residual error estimator hence the local problem error estimator also yields reliable upper and lower error bounds on anisotropic meshes the local problems are shown to be well conditioned two further local problem error estimators originally defined for isotropic meshes are investigated in the anisotropic context here the results reveal the significance of the proper choice of local problem furthermore the analysis covers dirichlet neumann and robin boundary conditions particular attention is paid to the robin boundary conditions since they require a different treatment than the neumann boundary conditions a numerical example supports the error analysis

least squares orthogonalization using semidefinite programming we consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real hilbert space the vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors we show that the design of the optimal vectors referred to as the least squares ls orthogonal vectors can be formulated as a semidefinite programming sdp problem using the many well known algorithms for solving sdps which are guaranteed to converge to the global optimum the ls vectors can be computed very efficiently in polynomial time within any desired accuracy by exploiting the connection between our problem and a quantum detection problem we derive a closed form analytical expression for the ls orthogonal vectors for vector sets with a broad class of symmetry properties specifically we consider geometrically uniform gu sets with a possibly non abelian generating group and compound gu sets which consist of subsets that are gu

evaluation of relative entropy of entanglement and derivation of optimal lewenstein sanpera decomposition of bell decomposable states via convex optimization abstract we provide an analytical expression for optimal lewenstein sanpera decomposition of bell decomposable states by using semi definite programming also using the karush kuhn tucker optimization method the minimum relative entropy of entanglement of bell decomposable states has been evaluated and it is shown that the same separable bell decomposable state lying at the boundary of convex set of separable bell decomposable states optimizes both lewenstein sanpera decomposition and relative entropy of entanglement

maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction diffusion problem a singularly perturbed semilinear reaction diffusion equation posed in the unit square is discretized on arbitrary nonuniform tensor product meshes we establish a second order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter no mesh aspect ratio assumption is made numerical results are presented that support our theoretical estimate

on the distinguishability of random quantum states we develop two analytic lower bounds on the probability of success p of identifying a state picked from a known ensemble of pure states a bound based on the pairwise inner products of the states and a bound based on the eigenvalues of their gram matrix we use the latter and results from random matrix theory to lower bound the asymptotic distinguishability of ensembles of n random quantum states in d dimensions where n d approaches a constant in particular for almost all ensembles of n states in n dimensions p 0 72 an application to distinguishing boolean functions the oracle identification problem in quantum computation is given

a generalized wave particle duality relation for finite groups wave particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave like properties in particular its ability to generate an interference pattern recently duality relations in which the wave like properties are quantified by using measures of quantum coherence have been proposed quantum coherence can be generalized to a property called group asymmetry here we derive a generalized duality relation involving group asymmetry which is closely related to the success probability of discriminating between the actions of the elements of a group the second quantity in the duality relation the one generalizing which path information is related to information about the irreducible representations that make up the group representation

escape by diffusion from a double well potential across a barrier abstract escape by diffusion from a double well potential across a barrier is studied on the basis of the smoluchowski equation in one dimension by comparison with exact results for a piecewise parabolic potential a reduced description is constructed in terms of a set of rate equations for the occupation probabilities of the two wells the rate equations contain memory terms and a source term for the rate of return from the outer space the reduced description yields a quite accurate approximation for the two example potentials studied

performance of quantum data transmission systems in the presence of thermal noise in the literature the performance of quantum data transmission systems is usually evaluated in the absence of thermal noise a more realistic approach taking into account the thermal noise is intrinsically more difficult because it requires dealing with glauber coherent states in an infinite dimensional space in particular the exact evaluation of the optimal measurement operators is a very difficult task and numerical approximation is unavoidable the paper faces the problem by approximating the p representation of the noisy quantum states with a large but finite numbers of terms and applying to them the square root measurement srm approach comparisons with cases where the exact solution are known show that the srm approach gives quite accurate results as application the performance of quadrature amplitude modulation qam and phase shift keying psk systems is considered in spite of the fact that the srm approach is not optimal and overestimates the error probability also in these cases the quantum detection maintains its superiority with respect to the classical homodyne detection

bayes cost reduction algorithm in quantum hypothesis testing corresp an iterative procedure is described for reducing the bayes cost in decisions among m 2 quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses the resulting decision strategy is a projection valued measure and yields an upper bound to the minimum attainable bayes cost from it is derived an algorithm for finding the optimum measurement states for choosing among m linearly independent pure states with minimum probability of error the method is also applied to decisions among m unimodal coherent quantum signals in thermal noise

time dependence and line shape of spontaneous quantum tunneling the time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time dependent schrodinger equation as an initial value problem by the method of laplace transform the wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion a concise expression is derived for the probability current into the outer space at long times the current oscillates about zero before decaying with a t 4 long time tail a comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states

diffusion in a bistable potential abstract the problem of diffusion of a particle in a bistable potential is studied on the basis of the one dimensional smoluchowski equation for the space and time dependent probability distribution the potential is modeled as two parabolic wells separated by a parabolic barrier for the model potential the smoluchowski equation is solved exactly by a laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in one of the wells in the limit of a high barrier the rate of transition to the other well is given by an asymptotic result due to kramers for a potential barrier of moderate height there are significant corrections to the asymptotic result

geometrically uniform frames we introduce a new class of finite dimensional frames with strong symmetry properties called geometrically uniform gu frames that are defined over a finite abelian group of unitary matrices and are generated by a single generating vector the notion of gu frames is then extended to compound gu cgu frames which are generated by a finite abelian group of unitary matrices using multiple generating vectors the dual frame vectors and canonical tight frame vectors associated with gu frames are shown to be gu and therefore also generated by a single generating vector which can be computed very efficiently using a fourier transform ft defined over the generating group of the frame similarly the dual frame vectors and canonical tight frame vectors associated with cgu frames are shown to be cgu the impact of removing single or multiple elements from a gu frame is considered a systematic method for constructing optimal gu frames from a given set of frame vectors that are not gu is also developed finally the euclidean distance properties of gu frames are discussed and conditions are derived on the abelian group of unitary matrices to yield gu frames with strictly positive distance spectrum irrespective of the generating vector

geometric quantum discord with bures distance the qubit case the minimal bures distance of a quantum state of a bipartite system ab to the set of classical states for subsystem a defines a geometric measure of quantum discord when a is a qubit we show that this geometric quantum discord is given in terms of the eigenvalues of a 2nb 2nb hermitian matrix nb being the hilbert space dimension of the other subsystem b as a first application we calculate the geometric discord for the output state of the dqc1 algorithm we find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin as a second application we derive an explicit formula for the geometric discord of a two qubit state with maximally mixed marginals and compare it with other measures of quantum correlations we also determine the closest classical states to

mmse whitening and subspace whitening this correspondence develops a linear whitening transformation that minimizes the mean squared error mse between the original and whitened data i e one that results in a white output that is as close as possible to the input in an mse sense when the covariance matrix of the data is not invertible the whitening transformation is designed to optimally whiten the data on a subspace in which it is contained the optimal whitening transformation is developed both for the case of finite length data vectors and infinite length signals

escape by diffusion from a square well across a square barrier abstract the problem of escape of a particle by diffusion from a square potential well across a square barrier is studied on the basis of the one dimensional smoluchowski equation for the space and time dependent probability distribution for the model potential the smoluchowski equation is solved exactly by a laplace transform with respect to time in the limit of a high barrier the rate of escape is given by an asymptotic result similar to that derived by kramers for a curved well and a curved barrier an approximate analytic formula is derived for the outward time dependent probability current in terms of the width and depth of the well and the width and height of the barrier a similar expression holds for the complete probability distribution

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

diffusion and convection after escape from a potential well abstract the escape by diffusion of a particle from a potential well in one dimension is strongly influenced by the application of a field in the adjacent half space at long times the probability distribution becomes a uniformly moving and steadily broadening gaussian in this half space the mean time of escape from the well is given by a simple expression in terms of the mean first passage time and the coefficient of the long time tail in the occupation probability of the well in the absence of the field transient effects in space and time are studied in explicit form for a parabolic potential well

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

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

no signaling principle can determine optimal quantum state discrimination we provide a general framework of utilizing the no signaling principle in derivation of the guessing probability in the minimum error quantum state discrimination we show that remarkably the guessing probability can be determined by the no signaling principle this is shown by proving that in the semidefinite programing for the discrimination the optimality condition corresponds to the constraint that quantum theory cannot be used for a superluminal communication finally a general bound to the guessing probability is presented in a closed form

svd iterative decision feedback demodulation and detection of coded space time unitary differential modulation a new suboptimal demodulator based on iterative decision feedback demodulation dfd and a singular value decomposition svd for estimation of unitary matrices is introduced noncoherent communication over the rayleigh flat fading channel with multiple transmit and receive antennas where no channel state information csi is available at the receiver is investigated with four transmit antennas codes achieving bit error rate ber lower than 10 sup 4 at bit energy over the noise spectral density ratio e sub b n sub o of 0 25 db up to 3 5 db with coding rates of 1 6875 to 5 06 bits per channel use were found the performance is compared to the mutual information upper bound of the capacity attaining isotropically random ir unitary transmit matrices the codes achieve ber lower than 10 sup 4 at e sub b n sub o of 3 2 db to 5 8 db from this bound system performance including the iterative dfd algorithm is compared to the one using euclidean distance as a reliability measure for demodulation the dfd system presents a performance gain of up to 1 5 db uncoded systems doing iterative dfd demodulation and idealized pilot sequence assisted modulation psam detection are compared iterative dfd introduces a gain of more than 1 2 db the coded system comprises a serial concatenation of turbo code and a unitary matrix differential modulation code the receiver employs the high performance coupled iterative decoding of the turbo code and the modulation code information theoretic arguments are harnessed to form guidelines for code design and to evaluate performance of the iterative decoder

on a posteriori error estimator for primal equilibrium and mixed approximation of diffusion equations in this paper we present an a posteriori error estimator for diffusion equation with the solution obtained by black box solver the estimator is valid for classical primal equilibrium and mixed finite element approximations with or without numerical integration

a pde approach to fractional diffusion we derive a computable a posteriori error estimator for the harmonic extension problem which localizes the fractional powers of elliptic operators supplemented with dirichlet boundary conditions our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation under suitable assumptions we design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance

a semidefinite programming approach to optimal unambiguous discrimination of quantum states we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states we show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem based on this formulation we develop a set of necessary and sufficient conditions for an optimal quantum measurement we show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well known algorithms for solving semidefinite programs which are guaranteed to converge to the global optimum using the general conditions for optimality we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal we refer to this measurement as the equal probability measurement epm we then show that for any state set the prior probabilities of the states can be chosen such that the epm is optimal finally we consider state sets with strong symmetry properties and equal prior probabilities for which the epm is optimal we first consider geometrically uniform gu state sets that are defined over a group of unitary matrices and are generated by a single generating vector we then consider compound gu state sets which are generated by a group of unitary matrices using multiple generating vectors where the generating vectors satisfy a certain weighted norm constraint

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

square root measurement for quantum symmetric mixed state signals a certain class of mixed states artificially constructed that is used in quantum cryptography is crucial for representation of signals this correspondence shows that the square root measurement gives the optimal measurement for such a class of mixed states

svd iterative detection of turbo coded multiantenna unitary differential modulation a new suboptimal demodulator based on a singular value decomposition for estimation of unitary matrices is introduced noncoherent communication over the rayleigh flat fading channel with multiple transmit and receive antennas where no channel state information is available at the receiver is investigated codes achieving bit error rate ber lower than 10 sup 4 at bit energy over the noise spectral density ratio e sub b n sub 0 of 1 6 1 9 db from code restricted capacity limit were found at higher data rates computation of code restricted capacity is impractical therefore the mutual information upper bound of the capacity attaining isotropically random unitary transmit matrices was used the codes achieve ber lower than 10 sup 4 at e sub b n sub 0 of 3 2 6 db from this bound with coding rates of 1 125 5 06 bits per channel use and different modulation decoding complexities the codes comprise a serial concatenation of turbo code and a unitary matrix differential modulation code the receiver employs the high performance coupled iterative decoding of the turbo code and the modulation code information theoretic arguments are harnessed to form guidelines for code design and to evaluate performance of the iterative decoder

two sided estimates of minimum error distinguishability of mixed quantum states via generalized holevo curlander bounds we prove a concise factor of 2 estimate for the failure rate of optimally distinguishing an arbitrary ensemble of mixed quantum states generalizing work of holevo theor probab appl 23 411 1978 and curlander ph d thesis mit 1979 a modification to the minimal principle of cocha and poor proceedings of the 6th international conference on quantum communication measurement and computing rinton princeton nj 2003 is used to derive a suboptimal measurement which has an error rate within a factor of 2 of the optimal by construction this measurement is quadratically weighted and has appeared as the first iterate of a sequence of measurements proposed by je ek et al phys rev a 65 060301 2002 unlike the so called pretty good measurement it coincides with holevo s asymptotically optimal measurement in the case of nonequiprobable pure states a quadratically weighted version of the measurement bound by barnum and knill j math phys 43 2097 2002 is proven bounds on the distingu

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

least squares inner product shaping we develop methods that construct an optimal set of vectors with a specified inner product structure from a given set of vectors in a complex hilbert space the optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors four special cases are considered in the first the constructed vectors are orthonormal in the second they are orthogonal in the third the gram matrix of inner products of the constructed vectors is a circulant matrix as we show the vectors form a cyclic set in the fourth the gram matrix has the property that the rows are all permutations of each other the constructed vectors are shown to be geometrically uniform

unified adaptive variational multiscale method for two phase compressible incompressible flows in this paper we present a new stabilized finite element method to solve the two phase compressible incompressible fluid flow problems using the level set method an anisotropic mesh adaptation with a regularized interface is adopted to deal with the high discontinuity in the material properties the coupling between the pressure and the flow velocity is ensured by introducing mass conservation terms in the momentum equation therefore the same set of primitive unknowns and equations is described for both phases the unified system is solved using a new derived variational multiscale stabilized finite element method it is tested on three time dependent liquid gas interface problems the numerical results show good stability and accuracy properties permitting also to deal with important compressibility effect high density ratio and extremely stretched anisotropic elements

upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination we consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states 1 r by splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities these upper bounds allow us to deduce various bounds on the asymptotic error rate for which it has been hypothesized that it is given by the multi hypothesis quantum chernoff bound or chernoff divergence c 1 r as recently introduced by nussbaum and szkola in analogy with salikhov s classical multi hypothesis chernoff bound this quantity is defined as the minimum of the pairwise binary chernoff divergences minj kc j k it was known already that the optimal asymptotic rate must lie between c 3 and c and that for certain classes of sets of states the bound is actually achieved it was known to be achieved in particular when the state pair that is closest together in chernoff dive

escape by diffusion from a parabolic well across a parabolic barrier abstract the problem of escape of a particle by diffusion from a parabolic potential well across a parabolic barrier adjacent to free space is studied on the basis of the one dimensional smoluchowski equation for the space and time dependent probability distribution for the model potential the smoluchowski equation is solved exactly by a laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in the well in the limit of a high barrier the rate of escape is given by an asymptotic result due to kramers for a potential barrier of moderate height there are significant corrections to the asymptotic result the probability that the particle resides in the well barrier region at time t decays with a 1 t long time tail

an adaptive algorithm for the crank nicolson scheme applied to a time dependent convection diffusion problem an a posteriori upper bound is derived for the nonstationary convection diffusion problem using the crank nicolson scheme and continuous piecewise linear stabilized finite elements with large aspect ratio following lozinski et al 2009 13 a quadratic time reconstruction is used a space and time adaptive algorithm is developed to ensure the control of the relative error in the l 2 h 1 norm numerical experiments illustrating the efficiency of this approach are reported it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step even in the convection dominated regime and in the presence of boundary layers

a singular perturbation approach to kramers diffusion problem we consider kramers diffusion problem which seeks to calculate the rate of escape of a particle from one potential well over a barrier to another presumably deeper and therefore more stable well though kramers introduced the problem as a model for chemical reactions it applies to numerous rate processes including atomic migration and ionic conductivity in crystals and transitions due to noise between stable states of dynamical systems with multi stable states to name but a few we propose a new approach not based on a fokker planck equation but rather on the solution of a singularly perturbed boundary value problem specifically we relate the rate of escape to the first passage time from the domain of attraction of the stable point corresponding to the first well the first passage time is then characterized via the ito calculus as a solution of an elliptic partial differential equation of singular perturbation type finally this equation is solved asymptotically by methods previously develop

quantum detection and mutual information for qam and psk signals we analyze the performance of the error probability and the mutual information for quadrature amplitude modulation qam and phase shift keying systems based on quantum detection theory it is shown that the quantum receiver called square root measurement gives about 5 7 db improvement in power in comparison with the classical one furthermore we show that the quantum qam system can achieve the same reliability as the conventional intensity modulation direct detection im dd system with almost equal power compressing the required bandwidth while the reliability is degraded in general

exponential error rates in multiple state discrimination on a quantum spin chain we consider decision problems on finite sets of hypotheses represented by pairwise different shift invariant states on a quantum spin chain the decision in favor of one of the hypotheses is based on outcomes of generalized measurements performed on local states on blocks of finite size we assume the existence of the mean quantum chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum chernoff distance we establish that this minimum specifies an asymptotic bound on the exponential rate of decay of the averaged probability of rejecting the true state in increasing block size if the mean quantum chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem this assumption is in particular fulfilled by shift invariant product states independent and identically distributed states further we provide a constructive proof for the existence of a sequence of quan

designing optimal quantum detectors via semidefinite programming we consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing among a collection of quantum states represented by a set of density operators we show that the design of the optimal detector can be formulated as a semidefinite programming problem based on this formulation we derive a set of necessary and sufficient conditions for an optimal quantum measurement we then show that the optimal measurement can be found by solving a standard convex semidefinite program by exploiting the many well known algorithms for solving semidefinite programs which are guaranteed to converge to the global optimum the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy using the semidefinite programming formulation we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator in particular if the quantum state ensemble is a pure state ensemble consisting of not necessarily independent rank one density operators then we show that the optimal measurement is a pure state measurement consisting of rank one measurement operators

residual and destroyed accessible information after measurements when quantum states are used to send classical information the receiver performs a measurement on the signal states the amount of information extracted is often not optimal due to the receiver s measurement scheme and experimental apparatus for quantum nondemolition measurements there is potentially some residual information in the postmeasurement state while part of the information has been extracted and the rest is destroyed here we propose a framework to characterize a quantum measurement by how much information it extracts and destroys and how much information it leaves in the residual postmeasurement state the concept is illustrated for several receivers discriminating coherent states

two sided bounds on minimum error quantum measurement on the reversibility of quantum dynamics and on maximum overlap using directional iterates in a unified framework we estimate the following quantities of interest in quantum information theory 1 the minimum error distinguishability of arbitrary ensembles of mixed quantum states 2 the approximate reversibility of quantum dynamics in terms of entanglement fidelity this is referred to as channel adapted quantum error recovery when applied to the composition of an encoding operation and a noise channel 3 the maximum overlap between a bipartite pure quantum state and a bipartite mixed state that may be achieved by applying a local quantum operation to one part of the mixed state and 4 the conditional min entropy of bipartite quantum states a refined version of the author s techniques j tyson j math phys 50 032016 2009 for bounding the first quantity is employed to give two sided estimates of the remaining three quantities we obtain a closed form approximate reversal channel using a state dependent kraus decomposition our reversal may be interpreted as a quadratically

distinguishing separable and entangled states we show how to design families of operational criteria that distinguish entangled from separable quantum states the simplest of these tests corresponds to the well known peres horodecki positive partial transpose ppt criterion and the more complicated tests are strictly stronger the new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs we successfully applied the results to many low dimensional states from the literature where the ppt test fails as a by product of the criteria we provide an explicit construction of the corresponding entanglement witnesses

theory of quantum pulse position modulation and related numerical problems the paper deals with quantum pulse position modulation ppm both in the absence pure states and in the presence mixed states of thermal noise using the glauber representation of coherent laser radiation the objective is to find optimal or suboptimal measurement operators and to evaluate the corresponding error probability for ppm the correct formulation of quantum states is given by the tensorial product of m identical hilbert spaces where m is the ppm order the presence of mixed states due to thermal noise generates an optimization problem involving matrices of huge dimensions which already for 4 ppm are of the order of ten thousand to overcome this computational complexity the currently available methods of quantum detection which are based on explicit results convex linear programming and square root measurement are compared to find the computationally less expensive one in this paper a fundamental role is played by the geometrically uniform symmetry of the quantum ppm format the evaluation of error probability confirms the vast superiority of the quantum detection over its classical counterpart

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

an optimal whitening approach to linear multiuser detection we propose a new linear multiuser receiver for synchronous code division multiple access cdma systems referred to as the orthogonal multiuser omu receiver unlike the linear minimum mean squared error mmse receiver the omu receiver depends only on the signature vectors and does not require knowledge of the received amplitudes or the channel signal to noise ratio snr several equivalent representations of the receiver are developed with different implications in terms of implementation in the first the receiver consists of a decorrelator demodulator followed by an optimal whitening transformation on a space formed by the signatures in the second the receiver consists of a bank of correlators with correlating vectors that are projections of a set of orthogonal vectors and are closest in a least squares sense to the decorrelator vectors and also closest in a least squares sense to the signature vectors in the third the receiver consists of a single user matched filter mf followed by an optimal whitening transformation on a space formed by the signatures we derive exact and approximate expressions for the probability of bit error as well as the asymptotic signal to interference noise ratio sinr in the large system limit the analysis suggests that over a wide range of channel parameters the omu receiver can outperform both the decorrelator and the single user mf and perform similarly to the linear mmse receiver despite not knowing the channel parameters

state discrimination with post measurement information we introduce a new state discrimination problem in which we are given additional information about the state after the measurement or more generally after a quantum memory bound applies the following special case plays an important role in quantum cryptographic protocols in the bounded storage model given a string x encoded in an unknown basis chosen from a set of mutually unbiased bases mubs you may perform any measurement but then store at most q qubits of quantum information and an unlimited amount of classical information later on you learn which basis was used how well can you compute a function f x of x given the initial measurement outcome the q qubits and the additional basis information we first show a lower bound on the success probability for any balanced function and any number of mutually unbiased bases beating the naive strategy of simply guessing the basis we then show that for two bases any boolean function f x can be computed perfectly if you are allowed to store just a single qubit independent of the number of possible input strings x however we show how to construct three bases such that you need to store all qubits in order to compute f x perfectly we then investigate how much advantage the additional basis information can give for a boolean function to this end we prove optimal bounds for the success probability for the and and the xor function for up to three mutually unbiased bases our result shows that the gap in success probability can be maximal without the basis information you can never do better than guessing the basis but with this information you can compute f x perfectly we also give an example where the extra information does not give any advantage at all

adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives an anisotropic error estimator involving only first order derivatives is proposed for the laplace problem and continuous piecewise linear finite elements upper and lower bounds are presented the involved constants being independent of the mesh aspect ratio provided the error gradient is equidistributed in the directions of maximum and minimum stretching an anisotropic adaptive algorithm is then proposed with aim to equidistribute the error gradient in the directions of maximum and minimum stretching numerical results in two and three space dimensions show that the effectivity index is aspect ratio independent on such adapted meshes

trapping of random walks on the line several features of the trapping of random walks on a one dimensional lattice are analyzed the results of this investigation are as follows 1 the correction term to the known asymptotic form for the survival probability ton steps is o 2n 1 3 where ln 1 c andc is the trap concentration 2 the short time form for the survival probability is found to be exp a c n1 2 wherea c is given in eq 21 3 the mean square displacement of a surviving random walker is found to go liken2 3for largen 4 when the distribution of trap free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn one such model is studied

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

finding optimal solutions for generalized quantum state discrimination problems we study an optimal quantum measurement for generalized quantum state discrimination problems which include the problem of finding an optimal measurement maximizing the average success probability with and without a fixed rate of inconclusive results and the problem of finding an optimal measurement in the neyman pearson strategy we show that for any generalized quantum state discrimination problem there is a corresponding minimum error discrimination problem sharing the same optimal measurement we propose an approach in which the optimal measurement is obtained by solving the corresponding minimum error discrimination problem and clarify the relationship between optimal solutions to the original and the corresponding problems with which one can obtain an optimal solution to the original problem in some cases moreover as an example of application of our approach we present an algorithm for numerically obtaining optimal solutions to generalized quantum state discrimination problems

optimal detection of symmetric mixed quantum states we develop a sufficient condition for the least squares measurement lsm or the square root measurement to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states using this condition we derive the optimal measurement for state sets with a broad class of symmetries we first consider geometrically uniform gu state sets with a possibly non abelian generating group and show that if the generator satisfies a weighted norm constraint then the lsm is optimal in particular for pure state gu ensembles the lsm is shown to be optimal for arbitrary gu state sets we show that the optimal measurement operators are gu with generator that can be computed very efficiently in polynomial time within any desired accuracy we then consider compound gu cgu state sets which consist of subsets that are gu when the generators satisfy a certain constraint the lsm is again optimal for arbitrary cgu state sets the optimal measurement operators are shown to be cgu with generators that can be computed efficiently in polynomial time

an approximation of anisotropic metrics from higher order interpolation error for triangular mesh adaptation we propose an efficient algorithm for the numerical approximation of metrics used for anisotropic mesh adaptation on triangular meshes with finite element computations we derive the metrics from interpolation error estimates expressed in terms of higher order derivatives for the p k lagrange finite element k 1 numerical examples of mesh adaptation done using metrics computed with our algorithm and derived from higher order derivatives as error estimates show that we obtain the right directions of anisotropy

lower a posteriori error estimates on anisotropic meshes lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes a numerical example is given that clearly demonstrates that the short edge jump residual terms in such bounds are not sharp hence for linear finite element approximations of the laplace equation in polygonal domains a new approach is employed to obtain essentially sharper lower a posteriori error bounds and thus to show that the upper error estimator in the recent paper kopteva in numer math 137 607 642 2017 is efficient on partially structured anisotropic meshes

geometric coherence and quantum state discrimination the operational meaning of coherence measure lies at the very heart of the coherence theory in this paper we provide an operational interpretation for geometric coherence by proving that the geometric coherence of a quantum state is equal to the minimum error probability to discriminate a set of pure states with von neumann measurement on the other hand we also show that a task to ambiguously discriminate a set of linearly independent pure states can be regarded as a problem of calculating geometric coherence that is we reveal an equivalence relation between ambiguous quantum state discrimination and geometric coherence based on this equivalence moreover we improve the upper bound for geometric coherence and give the explicit expression of geometric coherence for a class of states besides we establish a complementarity relation of geometric coherence and path distinguishability with which the relationship between l 1 norm of coherence and geometric coherence is obtained finally with geometric coherence we study multiple copies of quantum state discrimination and give an example to show how to discriminate two pure states

anisotropic finite elements for the stokes problem a posteriori error estimator and adaptive mesh we propose an a posteriori error estimator for the stokes problem using the crouzeix raviart p0 pair its efficiency and reliability on highly stretched meshes are investigated the analysis is based on hierarchical space splitting whose main ingredients are the strengthened cauchy schwarz inequality and the saturation assumption we give a theoretical proof of a method to enrich the crouzeix raviart element so that the strengthened cauchy constant is always bounded away from unity independently of the aspect ratio an anisotropic self adaptive mesh refinement approach for which the saturation assumption is valid will be described our theory is confirmed by corroborative numerical tests which include an internal layer a boundary layer a re entrant corner and a crack simulation a comparison of the exact error and the a posteriori one with respect to the aspect ratio will be demonstrated

edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes summary both for the h 1 and l 2 norms we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes we also show that with a correct scaling edge residuals yield a robust error estimator for a singularly perturbed reaction diffusion equation

overview of theoretical models for reaction rates there is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences in this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the national institutes of health on may 6 8 1985 this review will focus on 1 the smoluchowski model for reaction rates together with its extension by onsager 2 first passage time formalism for discrete and continuous master equations and fokker planck equations 3 the kramers model and its extensions 4 diffusion in the presence of trapping centers

minimum error state discrimination constrained by the no signaling principle we provide a bound on the minimum error when discriminating among quantum states using the no signaling principle the bound is general in that it depends on neither dimensions nor specific structures of given quantum states to be discriminated among we show that the bound is tight for the minimum error state discrimination between symmetric both pure and mixed qubit states moreover the bound can be applied to a set of quantum states for which the minimum error state discrimination is not known yet finally our results strengthen the quantitative connection between two no go theorems the no signaling principle and the no perfect state estimation

slow dynamics of linear relaxation systems abstract linear relaxation occurring in dielectrics viscoelastic fluids and many other systems is often characterized by a broad continuous spectrum we show that the relaxation behavior may be analyzed effectively by means of n point pade approximants applied in the complex plane of the square root of frequency the method leads to a compact analytic expression for the laplace transform of the relaxation function characterized by a small number of poles and their residues in the square root of frequency plane we study the method in detail for a model of diffusion in three dimensions with a single or double radial potential barrier and demonstrate its use in the analysis of the viscoelastic relaxation spectrum of polyisobutylene

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

transitions between wells and escape by diffusion in a one dimensional potential landscape abstract the time dependence of the occupation probabilities of neighboring wells due to diffusion in one dimension is formulated in terms of a set of generalized rate equations describing transitions between neighboring wells and escape across a final barrier the equations contain rate coefficients memory coefficients and a long time coefficient characterizing the amplitude of long time decay on a more microscopic level the stochastic process is described by a smoluchowski equation for the one dimensional probability distribution a numerical procedure is presented which allows calculation of the transport coefficients in the set of generalized rate equations on the basis of the smoluchowski equation

anisotropic residual based a posteriori mesh adaptation in 2d element based approach abstract an element based adaptation method is developed for an anisotropic a posteriori error estimator the adaptation does not make use of a metric but instead equidistributes the error over elements using local mesh modifications numerical results are reported comparing with three popular anisotropic adaptation methods currently in use it was found that the new method gives favourable results for controlling the energy norm of the error in terms of degrees of freedom at the cost of increased cpu usage additionally we considered a new l 2 variant of the estimator the estimator is shown to be conditionally equivalent to the exact l 2 error we provide examples of adapted meshes with the l 2 estimator and show that it gives greater control of the l 2 error compared with the original estimator