minimum resistance shapes in linear continuum mechanics a necessary optimality condition for the problem of the minimum resistance shape for a rigid three dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint o

3d shape optimization in viscous incompressible fluid under oseen approximation the integral equation constrained optimization approach to finding three dimensional 3d minimum drag shapes for bodies translating in viscous incompressible fluid under the oseen approximation of the navier stokes equations is presented the approach formulates the oseen flow problem as a boundary integral equation and finds solutions to this equation and its adjoint in the form of function series minimum drag shapes being also represented by function series are then found by the adjoint equation based method with a gradient based algorithm in which the gradient for shape series coefficients is determined analytically compared to partial differential equation pde constrained optimization coupled with the finite element method fem the approach reduces dimensionality of the flow problem solves the issue with region truncation in exterior problems finds minimum drag shapes in semianalytical form and has fast convergence the approach is demonstrated in three drag minimization problems for different reynolds numbers for i a body of constant volume ii a torpedo with only fore and aft noses being optimized and iii a body of constant volume following another body of fixed shape the minimum drag shapes in problem i are in good agreement with the existing optimality conditions and conform to those obtained by pde constrained optimization

cauchy integral formula for generalized analytic functions in hydrodynamics it is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics the cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion if hydrodynamic fields velocity pressure and vorticity admit representations in terms of corresponding generalized analytic functions those representations and the cauchy integral formulae form two essential parts of the generalized analytic function approach which readily yields either closed form solutions or boundary integral equations this approach is demonstrated for problems of axisymmetric and asymmetric stokes flows two phase axisymmetric stokes flows two dimensional and axisymmetric oseen flows

neumann series and lattice sums we consider sums over the square lattice which depend only on radial distance and provide formulas which enable sums of functions with neumann series to be reexpressed as combinations of hypergeometric series we illustrate the procedure using trigonometric sums previously studied by borwein and borwein sums combining logarithms bessel functions j and powers of distance and sums of neumann functions we also exhibit sums which may be evaluated analytically and recurrence formulas linking sums

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

sandwiched renyi divergence satisfies data processing inequality sandwiched quantum renyi divergence has been recently defined in the independent works of wilde et al strong converse for the classical capacity of entanglement breaking channels preprint arxiv 1306 1586 2013 and muller lennert et al on quantum renyi entropies a new definition some properties and several conjectures preprint arxiv 1306 3142v1 2013 this new quantum divergence has already found applications in quantum information theory here we further investigate properties of this new quantum divergence in particular we show that sandwiched renyi divergence satisfies the data processing inequality for all values of 1 moreover we prove that holevo information a variant of holevo information defined in terms of sandwiched renyi divergence is super additive our results are based on holder s inequality the riesz thorin theorem and ideas from the theory of complex interpolation we also employ sion s minimax theorem

generalized analytic functions in magnetohydrodynamics an approach of generalized analytic functions to the magnetohydrodynamic mhd problem of an electrically conducting viscous incompressible flow past a solid non magnetic body of revolution is pres

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

quantum work statistics and resource theories bridging the gap through renyi divergences the work performed on or extracted from a nonautonomous quantum system described by means of a two point projective measurement approach is a stochastic variable we show that the cumulant generating function of work can be recast in the form of quantum r enyi ensuremath alpha divergences and by exploiting the convexity of this cumulant generating function derive a single parameter family of bounds for the first moment of work higher order moments of work can also be obtained from this result in this way we establish a link between quantum work statistics in stochastic approaches and resource theories for quantum thermodynamics a theory in which r enyi ensuremath alpha divergences take a central role to explore this connection further we consider an extended framework involving a control switch and an auxiliary battery which is instrumental to reconstructing the work statistics of the system we compare and discuss our bounds on the work distribution to findings on deterministic work studied in resource theoretic settings

renyi divergence and kullback leibler divergence renyi divergence is related to renyi entropy much like kullback leibler divergence is related to shannon s entropy and comes up in many settings it was introduced by renyi as a measure of information that satisfies almost the same axioms as kullback leibler divergence and depends on a parameter that is called its order in particular the renyi divergence of order 1 equals the kullback leibler divergence we review and extend the most important properties of renyi divergence and kullback leibler divergence including convexity continuity limits of sigma algebras and the relation of the special order 0 to the gaussian dichotomy and contiguity we also show how to generalize the pythagorean inequality to orders different from 1 and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs for all orders and present several other minimax results

hilbert formulas for r analytic functions in the domain exterior to spindle hilbert formulas for an r analytic function defined by a generalized cauchy riemann system in the domain exterior to the contour of a spindle in the meridional cross section plane have been derived the derivation is based on the theory of riemann boundary value problems for analytic functions for numerical calculations fourier integrals with hilbert formulas representing the real and imaginary parts of the r analytic function have been reduced to the form of regular integrals the problem of the axially symmetric steady motion of a rigid spindle shaped body in a stokes fluid has been solved and the pressure in the fluid has been expressed analytically based on a hilbert formula as an illustration streamlines about the body vortex and pressure functions at the contour of the body and the drag force exerted on the body by the fluid have been calculated

an efficient numerical evaluation of the green s function for the helmholtz operator on periodic structures abstract numerical methods which are based on boundary integral formulations require the numerical evaluation of the green s function associated with the problem in the case of periodic structures the green s function is often an infinite series this infinite series may converge slowly making numerical evaluation expensive here we present a practical computer implementation of a technique which dramatically speeds up the convergence of the infinite series green s function associated with the helmholtz operator to show the consequences of using this technique we include some numerical examples

the framework of k harmonically analytic functions for three dimensional stokes flow problems part ii the framework of generalized analytic functions arising from the related potentials so called k harmonically analytic functions has been developed in application to three dimensional 3d axially symmetric stokes flow problems cauchy s integral formula for the class of k harmonically analytic functions has been obtained and series representations for k harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived as the central result in the developed framework a solution form representing the velocity field and pressure for 3d axially symmetric stokes flows has been constructed in terms of two 0 harmonically analytic functions it has also been shown that it uniquely determines an external velocity field vanishing at infinity with the obtained solution form the problem of 3d stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary value problem for two 0 harmonically analytic function

the second laws of quantum thermodynamics the second law of thermodynamics places constraints on state transformations it applies to systems composed of many particles however we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath is there a second law of thermodynamics in this regime here we find that for processes which are approximately cyclic the second law for microscopic systems takes on a different form compared to the macroscopic scale imposing not just one constraint on state transformations but an entire family of constraints we find a family of free energies which generalize the traditional one and show that they can never increase the ordinary second law relates to one of these with the remainder imposing additional constraints on thermodynamic transitions we find three regimes which determine which family of second laws govern state transitions depending on how cyclic the process is in one regime one can cause an apparent violation of the usual second law through a process of embezzling work from a large system which remains arbitrarily close to its original state these second laws are relevant for small systems and also apply to individual macroscopic systems interacting via long range interactions by making precise the definition of thermal operations the laws of thermodynamics are unified in this framework with the first law defining the class of operations the zeroth law emerging as an equivalence relation between thermal states and the remaining laws being monotonicity of our generalized free energies

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

on a transplant operator and explicit construction of cauchy type integral representations for p analytic functions we present a new technique for explicit construction of cauchy kernels and cauchy integral representations for a class of generalized analytic functions and p analytic functions

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

relating different quantum generalizations of the conditional renyi entropy recently a new quantum generalization of the renyi divergence and the corresponding conditional renyi entropies was proposed here we report on a surprising relation between conditional renyi entropies based on this new generalization and conditional renyi entropies based on the quantum relative renyi entropy that was used in previous literature our result generalizes the well known duality relation h a b h a c 0 of the conditional von neumann entropy for tripartite pure states to renyi entropies of two different kinds as a direct application we prove a collection of inequalities that relate different conditional renyi entropies and derive a new entropic uncertainty relation

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

fast algorithms for helmholtz green s functions the formal representation of the quasi periodic helmholtz green s function obtained by the method of images is only conditionally convergent and thus requires an appropriate summation convention for its evaluation instead of using this formal sum we derive a candidate green s function as a sum of two rapidly convergent series one to be applied in the spatial domain and the other in the fourier domain as in ewald s method we prove that this representation of green s function satisfies the helmholtz equation with the quasi periodic condition and furthermore leads to a fast algorithm for its application as an operator we approximate the spatial series by a short sum of separable functions given by gaussians in each variable for the series in the fourier domain we exploit the exponential decay of its terms to truncate it we use fast and accurate algorithms for convolving functions with this approximation of the quasi periodic green s function the resulting method yields a fast solver for the helmholtz equation with the quasi periodic boundary condition the algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when green s function is applied to discontinuous functions or potentials with singularities we also construct helmholtz green s functions with dirichlet neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi periodic green s function to apply them the complexity in dimension d 2 of these algorithms is graphic 1 d log c log 1 d where is the desired accuracy is proportional to the number of wavelengths contained in the computational domain and c is a constant we illustrate our approach with examples 1 embed inline graphic 1 gif

on the quantum renyi relative entropies and related capacity formulas following csiszar s approach in classical information theory it is shown that the quantum relative entropies with parameter 0 1 can be represented as generalized cutoff rates and hence a direct operational interpretation of the quantum relative entropies are provided it is also shown that various generalizations of the holevo capacity defined in terms of the relative entropies coincide for the parameter range 0 2 and an upper bound on the one shot e capacity of a classical quantum channel in terms of these capacities is given

fast convolution with the free space helmholtz green s function we construct an approximation of the free space green s function for the helmholtz equation that splits the application of this operator between the spatial and the fourier domains as in ewald s method for evaluating lattice sums in the spatial domain we convolve with a sum of decaying gaussians with positive coefficients and in the fourier domain we multiply by a band limited kernel as a part of our approach we develop new quadratures appropriate for the singularity of green s function in the fourier domain the approximation and quadratures yield a fast algorithm for computing volumetric convolutions with green s function in dimensions two and three the algorithmic complexity scales as where is selected accuracy is the number of wavelengths in the problem d is the dimension and c is a constant the algorithm maintains its efficiency when applied to functions with singularities in contrast to the fast multipole method as our approximation makes a transition to that of the free space green s function for the poisson equation we illustrate our approach with examples

efficient computation of the 2d periodic green s function using the ewald method an efficient computation of the periodic helmholtz green s function for a 2d array of point sources using the ewald method is presented limitations on the numerical accuracy when using the optimum e parameter which gives optimum asymptotic convergence at high frequency are discussed a best e parameter is then derived to overcome these limitations which allows for the fastest convergence while maintaining a specific level of accuracy loss of significant figures in the final result the actual loss of significant figures has been verified through numerical simulations formulas for the number of terms needed for convergence have also been derived for both the spectral and the spatial series that appear in the ewald method and are found to be accurate in almost all cases

discrete gradient flows for shape optimization and applications we present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation its full discretization and the resulting linear algebraic systems our approach hinges on the following essential features shape differential calculus a semi implicit time discretization and a finite element method for space discretization we use shape differential calculus to express variations of bulk and surface energies with respect to domain changes the semi implicit time discretization allows us to track the domain boundary without an explicit parametrization and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity we propose a schur complement approach to solve the resulting linear systems efficiently we discuss applications of this framework to image segmentation optimal shape design for pde and surface diffusion along with the choice of suitable scalar products in each case we illustrate the method with several numerical experiments some developing pinch off and topological changes in finite time

strong converse exponents for a quantum channel discrimination problem and quantum feedback assisted communication this paper studies the difficulty of discriminating between an arbitrary quantum channel and a replacer channel that discards its input and replaces it with a fixed state the results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination we show that in this particular setting the most general adaptive discrimination strategies provide no asymptotic advantage over non adaptive tensor power strategies this conclusion follows by proving a quantum stein s lemma for this channel discrimination setting showing that a constant bound on the type i error leads to the type ii error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels the strong converse part of the lemma states that any attempt to make the type ii error decay to zero at a rate faster than the channel relative entropy implies that the type i error necessarily converges to one we then refine this latter result by identifying the optimal strong converse exponent for this task as a consequence of these results we can establish a strong converse theorem for the quantum feedback assisted capacity of a channel sharpening a result due to bowen furthermore our channel discrimination result demonstrates the asymptotic optimality of a non adaptive tensor power strategy in the setting of quantum illumination as was used in prior work on the topic the sandwiched renyi relative entropy is a key tool in our analysis finally by combining our results with recent results of hayashi and tomamichel we find a novel operational interpretation of the mutual information of a quantum channel mathcal n as the optimal type ii error exponent when discriminating between a large number of independent instances of mathcal n and an arbitrary worst case replacer channel chosen from the set of all replacer channels

optimal sequence of quantum measurements in the sense of stein s lemma in quantum hypothesis testing we derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing we discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0 as an asymptotically optimal measurement we propose a projection measurement characterized by the irreducible representation theory of the special linear group sl especially in the spin 1 2 system it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction as a by product we obtain another proof of quantum stein s lemma in addition an asymptotically optimal measurement is constructed in the quantum gaussian case and it is physically meaningful

number of trials required to estimate a free energy difference using fluctuation relations the difference f between free energies has applications in biology chemistry and pharmacology the value of f can be estimated from experiments or simulations via fluctuation theorems developed in statistical mechanics calculating the error in a f estimate is difficult worse atypical trials dominate estimates how many trials one should perform was estimated roughly by jarzynski phys rev e 73 046105 2006 we enhance the approximation with the following information theoretic strategies we quantify dominance with a tolerance parameter chosen by the experimenter or simulator we bound the number of trials one should expect to perform using the order renyi entropy the bound can be estimated if one implements the good practice of bidirectionality known to improve estimates of f estimating f from this number of trials leads to an error that we bound approximately numerical experiments on a weakly interacting dilute classical gas support our analytical calculations

on certain properties of the relative entropy of states of operator algebras the aim of the present paper is to establish certain properties of the relative entropy functional in the general context of c algebras most of these properties have been observed under more restrictive conditions in earlier papers the celebrated strong subadditivity of the entropy gives such an example the general approach is applied to prove the existence of the mean relative entropy for a noncommutative stationary markov chain with conditional expectations

continuity and relative hamiltonians let n n 1 be a norm convergent sequence of normal states on a von neumann algebraa with n let kn n 1 be a strongly convergent sequence of self adjoint elements ofa withkn k it is shown that the sequence omega n k n n geqq 1 of perturbed states converges in norm to a related result holds forc algebras a counter example is provided to show that it is not sufficient to assume weak convergence of n n 1 even whenkn k for alln however conditions are given which together with weak convergence are sufficient relative entropy methods are used and a relative entropy inequality is proved

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

classical capacities of compound and averaged quantum channels we determine the capacity of compound classical quantum channels as a consequence we obtain the capacity formula for the averaged classical quantum channels the capacity result for compound channels demonstrates as in the classical setting the existence of reliable universal classical quantum codes in scenarios where the only a priori information about the channel used for the transmission of information is that it belongs to a given set of memoryless classical quantum channels our approach is based on a universal classical approximation of the quantum relative entropy which in turn relies on a universal hypothesis testing result

discriminating quantum states the multiple chernoff distance we consider the problem of testing multiple quantum hypotheses rho 1 otimes n ldots rho r otimes n where an arbitrary prior distribution is given and each of the r hypotheses is n copies of a quantum state it is known that the minimal average error probability p e decays exponentially to zero that is p e exp xi n o n however this error exponent xi is generally unknown except for the case that r 2 in this paper we solve the long standing open problem of identifying the above error exponent by proving nussbaum and szkola s conjecture that xi min i neq j c rho i rho j the right hand side of this equality is called the multiple quantum chernoff distance and c rho i rho j max 0 leq s leq1 log operatorname tr rho i s rho j 1 s has been previously identified as the optimal error exponent for testing two hypotheses rho i otimes n versus rho j otimes n the main ingredient of our proof is a new upper bound for the average error probability for testing an ensemble of finite dimensional but otherwise general quantum states this upper bound up to a states dependent factor matches the multiple state generalization of nussbaum and szkola s lower bound specialized to the case r 2 we give an alternative proof to the achievability of the binary hypothesis chernoff distance which was originally proved by audenaert et al

optimum profiles in two dimensional stokes flow we consider the problem of designing the section of a cylinder to minimize the drag per unit length it experiences when placed perpendicular to a uniform stream at low reynolds number we suppose the area of the cross section to be given and the flow to be two dimensional the relevant properties of a cylinder of general cross section in a particular orientation can conveniently be expressed in terms of its equivalent radius when the drag and flow at infinity are parallel this equivalent radius is the radius of the circular cylinder giving rise to the same drag per unit length we obtain a variational formula for this equivalent radius when the surface of the cylinder is perturbed this shows that the optimum profile we seek must be such that the flow past it has a vorticity of constant magnitude at its surface and this fact enables the optimum to be determined analytically the efficacy of a particular section may be measured by its effective radius this being the equivalent radius when the length scale is chosen to give the section an area thus a circular cylinder has an effective radius of 1 the minimum possible effective radius achieved by the optimum profile is 0 88876 to illustrate some of the arguments we exploit in a more familiar setting we also obtain a variational formula for the drag on a three dimensional body in stokes flow when its surface is perturbed

asymptotics of quantum relative entropy from a representation theoretical viewpoint in this paper it is proved that the quantum relative entropy d can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive operator valued measures povms the sequence of povms depends on but is independent of the choice of

minimum drag shapes in magnetohydrodynamics a necessary optimality condition for the minimum drag shape for a non magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained it is assumed that the flow and magnetic field are uniform and parallel at infinity and that the body and fluid have the same magnetic permeability the condition is derived based on the linearized magnetohydrodynamic mhd equations subject to a constraint on the body s volume and generalizes the existing optimality conditions for the minimum drag shapes for the body in the stokes and oseen flows of a non conducting fluid it is shown that for any hartmann number m reynolds number re and magnetic reynolds number re m the minimum drag shapes are fore and aft symmetric and have conic vertices with an angle of 2 3 the minimum drag shapes are represented in a function series form and the series coefficients are found iteratively with the derived optimality condition at each iteration the mhd problem is solved via the boundary integral equations obtained based on the cauchy integral formula for generalized analytic functions with respect to the equal volume sphere drag reduction as a function of the cowling number s m 2 re m re is smallest at s 1 also in the considered examples the drag values for the minimum drag shapes and equal volume minimum drag spheroids are sufficiently close

a new integral representation for quasi periodic fields and its application to two dimensional band structure calculations in this paper we consider band structure calculations governed by the helmholtz or maxwell equations in piecewise homogeneous periodic materials methods based on boundary integral equations are natural in this context since they discretize the interface alone and can achieve high order accuracy in complicated geometries in order to handle the quasi periodic conditions which are imposed on the unit cell the free space green s function is typically replaced by its quasi periodic cousin unfortunately the quasi periodic green s function diverges for families of parameter values that correspond to resonances of the empty unit cell here we bypass this problem by means of a new integral representation that relies on the free space green s function alone adding auxiliary layer potentials on the boundary of the unit cell itself an important aspect of our method is that by carefully including a few neighboring images the densities may be kept smooth and convergence rapid this framework results in an integral equation of the second kind avoids spurious resonances and achieves spectral accuracy because of our image structure inclusions which intersect the unit cell walls may be handled easily and automatically our approach is compatible with fast multipole acceleration generalizes easily to three dimensions and avoids the complication of divergent lattice sums

maximum one shot dissipated work from renyi divergences thermodynamics describes large scale slowly evolving systems two modern approaches generalize thermodynamics fluctuation theorems which concern finite time nonequilibrium processes and one shot statistical mechanics which concerns small scales and finite numbers of trials combining these approaches we calculate a one shot analog of the average dissipated work defined in fluctuation contexts the cost of performing a protocol in finite time instead of quasistatically the average dissipated work has been shown to be proportional to a relative entropy between phase space densities to a relative entropy between quantum states and to a relative entropy between probability distributions over possible values of work we derive one shot analogs of all three equations demonstrating that the order infinity renyi divergence is proportional to the maximum possible dissipated work in each case these one shot analogs of fluctuation theorem results contribute to the unification of these two toolkits for small scale nonequilibrium statistical physics

asymptotics of varadhan type and the gibbs variational principle for a large class of quantum models of mean field type the thermodynamic limit of the free energy density is proved to be given by the gibbs variational principle the latter is shown to be equivalent to a non commutative version of varadhan s asymptotic formula

a study of locc detection of a maximally entangled state using hypothesis testing we study how well we can answer the question is the given quantum state equal to a certain maximally entangled state using locc in the context of hypothesis testing under several locality and invariance conditions optimal tests will be derived for several special cases by using basic theory of group representations some optimal tests are realized by performing quantum teleportation and checking whether the state is teleported we will also give a finite process for realizing some optimal tests the performance of the tests will be numerically compared

strengthened monotonicity of relative entropy via pinched petz recovery map the quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy it is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states in this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version the recovery map is a convex combination of rotated petz recovery maps and perfectly reverses the quantum channel on the second state as a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved by fawzi and renner our proof only relies on the elementary properties of pinching maps and the operator logarithm

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

entropy and quantum kolmogorov complexity a quantum brudno s theorem in classical information theory entropy rate and algorithmic complexity per symbol are related by a theorem of brudno in this paper we prove a quantum version of this theorem connecting the von neumann entropy rate and two notions of quantum kolmogorov complexity both based on the shortest qubit descriptions of qubit strings that run by a universal quantum turing machine reproduce them as outputs

coding theorems for compound problems via quantum renyi divergences recently a new notion of quantum renyi divergences has been introduced by muller lennert dupuis szehr fehr and tomamichel and wilde winter and yang which found a number of applications in strong converse theorems here we show that these new renyi divergences are also useful tools to obtain coding theorems in the direct domain of various problems we demonstrate this by giving new and considerably simplified proofs for the achievability parts of stein s lemma with composite null hypothesis universal state compression and the classical capacity of compound classical quantum channels based on single shot error bounds already available in the literature and simple properties of the quantum renyi divergences the novelty of our proofs is that the composite compound coding theorems can be almost directly obtained from the single shot error bounds essentially with the same effort as for the case of simple null hypothesis single source single channel

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

analysis of shape optimization for magnetic microswimmers we analyze an infinite dimensional geometrically constrained shape optimization problem for magnetically driven microswimmers locomotors in three dimensional 3 d stokes flow and give a well posed descent scheme for computing optimal shapes the problem is inspired by recent experimental work in this area we show the existence of a minimizer of the optimization problem using analytical tools for elastic rods that respect the excluded volume constraint we derive a variational gradient descent method for computing optimal locomotor shapes using the tools of shape differential calculus the descent direction is obtained by solving a saddle point system which we prove is well posed we also introduce a finite element approximation of the gradient descent method and prove its stability we present numerical results illustrating our method and the effect that finite aspect ratio and external cargo can have on the optimal shape the 3 d stokes equations are solved by a boundary integral method

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

asymmetric creeping motion of a rigid spindle shaped body in a viscous fluid exact solutions to the three dimensional problems of asymmetric creeping translation and rotation of a rigid spindle shaped body in a viscous incompressible fluid have been obtained in both problems the velocity field has been represented in the form of dean and o neill and under certain conditions the equation of continuity has been reduced to a three contour equation for an analytic function related to the density in a fourier integral representing the pressure in bispherical coordinates then the three contour equation has been reduced to a fredholm integral equation of the second kind with a quasi difference kernel by the complex fourier transform as an illustration for the obtained solutions the pressure at the surface of the body has been calculated and analyzed the resisting force and torque have been obtained for an arbitrary body of revolution via limits of certain harmonic functions at infinity and as an example have been computed for various values of a geometrical parameter of the s

generalized analytic functions in axially symmetric oseen flows a class of generalized analytic functions arising from the oseen equations in the axially symmetric case has been identified for this class of functions the generalized cauchy integral formula has been obtained and a series representation for the region exterior to a sphere has been constructed the velocity field of the axially symmetric oseen flow has been represented in terms of two generalized analytic functions and it has been shown that for an exterior oseen flow problem those functions are uniquely determined provided that they both vanish at infinity also the pressure and vorticity have been determined as the real and imaginary parts of the two functions representing the velocity field and the drag exerted on a solid body of revolution in the axially symmetric oseen flow has been expressed in terms of one of the involved generalized analytic functions the problem of the axially symmetric oseen flow past a solid body of revolution has been reduced to an integral equation based on the gener

finite blocklength and moderate deviation analysis of hypothesis testing of correlated quantum states and application to classical quantum channels with memory martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning here we apply techniques borrowed from this field to quantum hypothesis testing which is the problem of discriminating quantum states belonging to two different sequences rho n n and sigma n n we obtain achievability bounds on the finite blocklength type ii stein and hoeffding errors which for i i d states are in general tighter than the corresponding bounds obtained by audenaert mosonyi and verstraete we also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a non homogeneous factorization property examples of such sequences include gibbs states of spin chains with translation invariant finite range interaction as well as finitely correlated quantum states we apply our results to find bounds on the capacity of a certain class of classical quantum channels with memory which satisfy a so called channel factorization property both in the finite blocklength and moderate deviation regimes

short note the type 3 nonuniform fft and its applications the nonequispaced or nonuniform fast fourier transform nufft arises in a variety of application areas including imaging processing and the numerical solution of partial differential equations in its most general form it takes as input an irregular sampling of a function and seeks to compute its fourier transform at a nonuniform sampling of frequency locations this is sometimes referred to as the nufft of type 3 like the fast fourier transform the amount of work required is of the order o nlogn where n denotes the number of sampling points in both the physical and spectral domains in this short note we present the essential ideas underlying the algorithm in simple terms we also illustrate its utility with application to problems in magnetic resonance imagin and heat flow

lattice sums and the two dimensional periodic green s function for the helmholtz equation many algorithms that are currently used for the solution of the helmholtz equation in periodic domains require the evaluation of the green s function g x x0 the fact that the natural representat

quantum sphere packing bounds with polynomial prefactors we study lower bounds on the optimal error probability in classical coding over classical quantum channels at rates below the capacity commonly termed quantum sphere packing bounds winter and dalai have derived such bounds for classical quantum channels however the exponents in their bounds only coincide when the channel is classical in this paper we show that these two exponents admit a variational representation and are related by the golden thompson inequality reaffirming that dalai s expression is stronger in general classical quantum channels second we establish a finite blocklength sphere packing bound for classical quantum channels which significantly improves dalai s prefactor from the order of subexponential to polynomial furthermore the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o log n n indicating our sphere packing bound is almost exact in the high rate regime finally for a special class of symmetric classical quantum channels we can completely characterize its optimal error probability without the constant composition code assumption the main technical contributions are two converse hoeffding bounds for quantum hypothesis testing and the saddle point properties of error exponent functions

fast transform from an adaptive multi wavelet representation to a partial fourier representation we present a fast algorithm to compute the partial transformation of a function represented in an adaptive pseudo spectral multi wavelet representation to a partial fourier representation such fast transformations are useful in many contexts in physics and engineering where changes of representation from a piece wise polynomial basis to a fourier basis the algorithm is demonstrated for a gaussian in one and in three dimensions for 2d we apply this approach to a gaussian in a periodic domain the accuracy and the performance of this method is compared with direct summation

classical capacity of quantum channels with general markovian correlated noise the classical capacity of a quantum channel with arbitrary markovian correlated noise is evaluated for the general case of a channel with long term memory which corresponds to a markov chain which does not converge to equilibrium the capacity is expressed in terms of the communicating classes of the markov chain for an irreducible and aperiodic markov chain the channel is forgetful and one retrieves the known expression kretschmann and werner in phys rev a 72 062323 2005 for the capacity

quasi periodic green s functions of the helmholtz and laplace equations a classical problem of free space green s function g0 representations of the helmholtz equation is studied in various quasi periodic cases i e when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space exponentially convergent series for the free space quasi periodic g0 and for the expansion coefficients dl of g0 in the basis of regular cylindrical in two dimensions and spherical in three dimension 3d waves or lattice sums are reviewed and new results for the case of a one dimensional 1d periodicity in 3d are derived from a mathematical point of view a derivation of exponentially convergent representations for schlomilch series of cylindrical and spherical hankel functions of any integer order is accomplished exponentially convergent series for g0 and lattice sums dl hold for any value of the bloch momentum and allow g0 to be efficiently evaluated also in the periodicity plane the quasi periodic green s functions of the laplace equation are obtained from the corresponding representations of g0 of the helmholtz equation by taking the limit of the wave vector magnitude going to zero the derivation of relevant results in the case of a 1d periodicity in 3d highlights the common part which is universally applicable to any of remaining quasi periodic cases the results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics remote sensing of periodic surfaces periodic gratings and infinite arrays of resonators coupled to a waveguide in many contexts of simulating systems of charged particles in molecular dynamics for the description of quasi periodic arrays of point interactions in quantum mechanics and in various ab initio first principle multiple scattering theories for the analysis of diffraction of classical and quantum waves

efficient calculation of lattice sums for free space periodic green s function an efficient method to calculate the lattice sums is presented for a one dimensional 1 d periodic array of line sources the method is based on the recurrence relations for hankel functions and the fourier integral representation of the zeroth order hankel function the lattice sums of arbitrary high order are then expressed by an integral of elementary functions which is easily computed using a simple scheme of numerical integration the calculated lattice sums are used to evaluate the free space periodic green s function the numerical results show that the proposed method provides a highly accurate evaluation of the green s function with far less computation time even when the observation point is located near the plane of the array

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

rare events and the convergence of exponentially averaged work values equilibrium free energy differences are given by exponential averages of nonequilibrium work values such averages however often converge poorly as they are dominated by rare realizations i show that there is a simple and intuitively appealing description of these rare but dominant realizations this description is expressed as a duality between forward and reverse processes and provides both heuristic insights and quantitative estimates regarding the number of realizations needed for convergence of the exponential average analogous results apply to the equilibrium perturbation method of estimating free energy differences the pedagogical example of a piston and gas r c lua and a y grosberg j phys chem b 109 6805 2005 is used to illustrate the general discussion

generalized analytic functions in an extensional stokes flow with a deformable drop an approach of generalized analytic functions of a complex variable to the axially symmetric problem of a deformable drop in an extensional viscous incompressible flow under the zero reynolds number assumption is presented the velocity field and pressure in and out of the drop are represented in terms of four generalized analytic functions when the drop and ambient flow have equal viscosities integral representations for the velocity vorticity and pressure balancing the velocity and stress boundary conditions are obtained based on the generalized cauchy integral formula in this case steady shapes of the drop for different capillary numbers are found by minimizing the norm of the normal velocity over the interface and are shown to have rounded endpoints the obtained steady shapes satisfy the kinematic condition with high accuracy and are very close to spheroidal shapes the second part of this work constructs an analytical solution to the problem with the deformable spheroidal drop that balances th

min and max relative entropies and a new entanglement monotone two new relative entropy quantities called the min and max relative entropies are introduced and their properties are investigated the well known min and max entropies introduced by renner are obtained from these we define a new entanglement monotone which we refer to as the max relative entropy of entanglement and which is an upper bound to the relative entropy of entanglement we also generalize the min and max relative entropies to obtain smooth min and max relative entropies these act as parent quantities for the smooth renyi entropies eth zurich ph d dissertation 2005 and allow us to define the analogues of the mutual information in the smooth renyi entropy framework further the spectral divergence rates of the information spectrum approach are shown to be obtained from the smooth min and max relative entropies in the asymptotic limit

asymmetric three dimensional stokes flows about two fused equal spheres exact solutions to three dimensional stokes flow problems for asymmetric translation and rotation of two fused rigid spheres of equal size have been obtained in toroidal coordinates the problems have been reduced to three contour equations for meromorphic functions from a certain class and then the latter have been reduced to fredholm integral equations of the second kind by the mehler fock transform of order 1 for the specified class of functions the equivalence of the corresponding three contour and fredholm equations has been established in the framework of riemann boundary value problems for analytic functions as an illustration for the obtained solutions the pressure has been calculated at the surface of the body for both problems and resisting force and torque experienced by the body in asymmetric translation and rotation have been computed as functions of a geometrical parameter of the body

lattice sums for gratings and arrays lattice sums arising in quasiperiodic green s functions for the helmholtz equation over general two dimensional arrays are investigated the array sums are related to those over a single quasiperiodic line of sources and their difference is be expressed in terms of exponentially convergent series it is shown that our expressions can be used to generate the sums pertaining to the case of photonic gap states associated with complex quasiperiodicity bloch vectors the accuracy and computational speed of our expressions are illustrated

three dimensional shape optimization in stokes flow problems an integral equation constrained optimization approach to finding minimum drag shapes for solid bodies of revolution in stokes flows subject to constraints on body s volume and shape has been developed an axially symmetric stokes flow problem has been formulated in the form of an integral equation state equation and finding an optimal shape has been reduced to an integral equation constrained optimization problem the total variation of the lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation based method and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically it has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula which eliminates the need for solving the adjoint equation the approach has been illustrated in finding the minimum drag shape for a solid unit volume particle and in finding the

on converse bounds for classical communication over quantum channels we explore several new converse bounds for classical communication over quantum channels in both the one shot and asymptotic regimes first we show that the matthews wehner meta converse bound for entanglement assisted classical communication can be achieved by activated no signaling assisted codes suitably generalizing a result for classical channels second we derive a new efficiently computable meta converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel as applications we provide a finite resource analysis of classical communication over quantum erasure channels including the second order and moderate deviation asymptotics third we explore the asymptotic analogue of our new meta converse the upsilon information of the channel we show that its regularization is an upper bound on the classical capacity which is generally tighter than the entanglement assisted capacity and other known efficiently computable strong converse bounds for covariant channels we show that the upsilon information is a strong converse bound

a note on operator inequalities of tsallis relative operator entropy tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized shannon inequalities were shown in the previous paper after the review of some fundamental properties of tsallis relative operator entropy some operator inequalities related to tsallis relative operator entropy are shown in the present paper our inequalities give the upper and lower bounds of tsallis relative operator entropy the operator equality on tsallis relative operator entropy is also shown by considering the tensor product this relation generalizes the pseudoadditivity for tsallis entropy as a corollary of our operator equality derived from the tensor product manipulation we show several operator inequalities including the superadditivity and the subadditivity for tsallis relative operator entropy our results are generalizations of the superadditivity and the subadditivity for tsallis entropy