an ergodic theorem for the quantum relative entropy we prove the ergodic version of the quantum stein s lemma which was conjectured by hiai and petz the result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability and contains as a special case the quantum version of the shannon mcmillan theorem for ergodic states a version of the quantum relative asymptotic equipartition property aep is given

quantum hypothesis testing with group symmetry the asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system we consider various asymptotic error exponents in connection with the problems of the chernoff bound the hoeffding bound and stein s lemma and derive bounds on these quantities in terms of their corresponding statistical distance measures a special emphasis is put on the comparison of the performances of group invariant and unrestricted measurements

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

hierarchical image segmentation part i detection of regular curves in a vector graph the problem of edge detection is viewed as a hierarchy of detection problems where the geometric objects to be detected e g edge points curves regions have increasing complexity and spatial extent an early stage of the proposed hierarchy consists in detecting the regular portions of the visible edges the input to this stage is given by a graph whose vertices are tangent vectors representing local and uncertain information about the edges a model relating the input vector graph to the curves to be detected is proposed an algorithm with linear time complexity is described which solves the corresponding detection problem in a worst case scenario the stability of curve reconstruction in the presence of uncertain information and multiple responses to the same edge is analyzed and addressed explicitly by the proposed algorithm

multiresolution analysis of a class of nonstationary processes processing nonstationary signals is an important and challenging problem we focus on the class of nonstationary processes with stationary increments of an arbitrary order and place them in a multiscale framework unlike other related studies we concentrate on the discrete time analysis and derive a number of new results in addition to placing the related existing ones in the same framework we extend the study to various parametric models for which we derive the resulting multiresolution description we show that wide sense stationarity may be achieved by adequately selecting the analysis wavelet after generalizing the study to wavelet packet analysis we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities

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

local hypothesis testing between a pure bipartite state and the white noise state in this paper we treat a local discrimination problem in the framework of asymmetric hypothesis testing we choose a known bipartite pure state psi rangle as an alternative hypothesis and the completely mixed state as a null hypothesis as a result we analytically derive an optimal type 2 error and an optimal positive operator valued measure povm for one way local operations and classical communication locc povm and separable povm for two way locc povm we study a family of simple three step locc protocols and show that the best protocol in this family has strictly better performance than any one way locc protocol in low dimensional systems when there may exist differences between two way locc povm and one way locc povm

large deviations and chernoff bound for certain correlated states on a spin chain in this paper we extend the results of lenci and rey bellet j stat phys 119 715 2005 on the large deviation upper bound of the distribution measures of local hamiltonians with respect to a gibbs state in the setting of translation invariant finite range interactions we show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by gibbs states of the above kind as well as finitely correlated states as an application of the methods the chernoff bound for correlated states with factorization property is studied in the specific case of the distributions of the ergodic averages of a one site observable with respect to an ergodic finitely correlated state the spectral theory of positive maps is applied to prove the full large deviation principle

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

invariant lagrangians mechanical connections and the lagrange poincare equations we deal with lagrangian systems that are invariant under the action of a symmetry group the mechanical connection is a principal connection that is associated with lagrangians which have a kinetic energy function that is defined by a riemannian metric in this paper we extend this notion to arbitrary lagrangians we then derive the reduced lagrange poincare equations in a new fashion and we show how solutions of the euler lagrange equations can be reconstructed with the help of the mechanical connection illustrative examples confirm the theory

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

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

trace inference curvature consistency and curve detection an approach is described for curve inference that is based on curvature information the inference procedure is divided into two stages a trace inference stage which is the subject of the present work and a curve synthesis stage it is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time and that tangent and curvature information are sufficient these make it possible to specify powerful constraints between estimated tangents to a curve in terms of a neighborhood relationship called cocircularity and between curvature estimates in terms of a curvature consistency relation because all curve information is quantized special care must be taken to obtain accurate estimates of trace points tangents and curvatures this issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint the procedure is applied to two types of images artificial images designed to evaluate curvature and noise sensitivity and natural images

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

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

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

generalized relative entropies and the capacity of classical quantum channels we provide lower and upper bounds on the information transmission capacity of one single use of a classical quantum channel the lower bound is expressed in terms of the hoeffding capacity which we define similarly to the holevo capacity but replacing the relative entropy with the hoeffding distance similarly our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max relative entropy in the definition of the divergence radius of a channel

asymptotic error rates in quantum hypothesis testing we consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero this leads to the identification of the quantum generalisation of the classical chernoff distance which is the corresponding quantity in classical symmetric hypothesis testing

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

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 location error of curved edges in low pass filtered 2 d and 3 d images the authors study the location error of curved edges in two and three dimensional images after analog and digital low pass filtering the zero crossing of a second derivative filter is a well known edge localization criterion the second derivative in gradient direction sdgd produces a predictable bias in edge location towards the centers of curvature while the linear laplace filter produces a shift in the opposite direction their sum called plus plus laplace sdgd leads to an edge detector that finds curved edges one order more accurately than its constituents this argument holds irrespective of the dimension the influence of commonly used low pass filters such as the psf originating from diffraction limited optics using incoherent light 2 d the gaussian filter with variable cutoff point d d and the isotropic uniform filter d d is studied

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

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

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

a bayesian multiple hypothesis approach to edge grouping and contour segmentation a contour segmentation algorithm is presented that takes an edge map and extracts continuous curves of arbitrary smoothness correctly handling curve intersections and capable of extrapolating over significant measurement gaps the algorithm incorporates noise models of the edge detection process and limited scene statistics it is based on an explicit contour model and employs a statistical distance measure to quantify the likelihood of each segmentation hypothesis a bayesian multiple hypothesis tree organizes possible segementations making it possible to postpone grouping decisions until a sufficient amount of information is available we have demonstrated its performance on real and synthetic images

fermionic quasifree states and maps in information theory this paper and the results therein are geared toward building a basic toolbox for calculations in quantum information theory of quasifree fermionic systems various entropy and relative entropy measures are discussed the main emphasis is on completely positive quasifree maps the set of quasifree affine maps on the state space is determined and fully characterized in terms of operations on one particle subspaces for a subclass of trace preserving completely positive maps and for their duals choi matrices and jamiolkowski states are discussed

asymptotic distinguishability measures for shift invariant quasifree states of fermionic lattice systems we apply the recent results of hiai et al j math phys 49 032112 2008 to the asymptotic hypothesis testing problem of locally faithful shift invariant quasifree states on a car algebra we use a multivariate extension of szeg s theorem to show the existence of the mean chernoff and hoeffding bounds and the mean relative entropy and show that these quantities arise as the optimal error exponents in suitable settings

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

entropy densities for algebraic states abstract algebraic or finitely correlated states are translation invariant states on an infinite tensor product c algebra whose construction is rather general including quantum markov chains for a strongly mixing algebraic state we obtain the relation between the mean entropy and another entropy density which means the macroscopic uniformity under the relevant state in the statistical mechanical sense for this purpose an outstanding property of approximately product type is shown for strongly mixing algebraic states we also obtain similar relations of the mean relative entropy to other entropy densities for two translation invariant states when the reference one is a strongly mixing algebraic state

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

routh s procedure for non abelian symmetry groups we extend routh s reduction procedure to an arbitrary lagrangian system that is one whose lagrangian is not necessarily the difference of kinetic and potential energies with a symmetry group which is not necessarily abelian to do so we analyze the restriction of the euler lagrange field to a level set of momentum in velocity phase space we present a new method of analysis based on the use of quasivelocities we discuss the reconstruction of solutions of the full euler lagrange equations from those of the reduced equations

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

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

second order asymptotics for quantum hypothesis testing in settings beyond i i d quantum lattice systems and more quantum stein s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state given the knowledge that it is one of two specific states or it was originally derived in the asymptotic i i d setting in which arbitrarily many say n identical copies of the state n or n are considered to be available in this setting the lemma states that for any given upper bound on the probability n of erroneously inferring the state to be the probability n of erroneously inferring the state to be decays exponentially in n with the rate of decay converging to the relative entropy of the two states the second order asymptotics for quantum hypothesis testing which establishes the speed of convergence of this rate of decay to its limiting value was derived in the i i d setting independently by tomamichel and hayashi and li we extend this result to settings beyond i i d examples of these include gibbs states of quantum spin systems with finite

asymptotic convertibility of entanglement an information spectrum approach to entanglement concentration and dilution we consider asymptotic convertibility of an arbitrary sequence of bipartite pure states into another by local operations and classical communication locc we adopt an information spectrum approach to address cases where each element of the sequences is not necessarily a tensor power of a bipartite pure state we derive necessary and sufficient conditions for the locc convertibility of one sequence to another in terms of spectral entropy rates of entanglement of the sequences based on these results we also provide simple proofs for previously known results on the optimal rates of entanglement concentration and dilution of general sequences of bipartite pure states

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

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

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

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

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

an information spectrum approach to classical and quantum hypothesis testing for simple hypotheses the information spectrum analysis made by han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing the results are also applied to fixed length source coding when loosening the normalizing condition for probability distributions and for quantum states we establish general formulas for several quantities relating to the asymptotic optimality of tests codes in terms of classical and quantum information spectra

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

two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states we present two general approaches to obtain the strong converse exponent of simple quantum hypothesis testing for correlated quantum states one approach requires that the states satisfy a certain factorization property typical examples of such states are the temperature states of translation invariant finite range interactions on a spin chain the other approach requires the differentiability of a regularized renyi alpha divergence in the parameter alpha typical examples of such states include temperature states of non interacting fermionic lattice systems and classical irreducible markov chains in all cases we get that the strong converse exponent is equal to the hoeffding anti divergence which in turn is obtained from the regularized renyi divergences of the two states

an efficient and provably correct algorithm for the multiscale estimation of image contours by means of polygonal lines a large portion of image contours is characterized by local properties such as sharp variations of the image intensity across the contour the integration of local image descriptors estimated by using these local properties into curvilinear descriptors is a difficult problem from a theoretical viewpoint because of the combinatorially large number of possible curvilinear descriptors to deal with this difficulty the notion of compressible graphs is introduced and a contour data model is defined leading to an efficient linear time algorithm which provably recovers contours with an upper bound on the approximation error

entropy growth of shift invariant states on a quantum spin chain we study the entropy of pure shift invariant states on a quantum spin chain unlike the classical case the local restrictions to intervals of length n are typically mixed and have therefore a nonzero entropy sn which is moreover monotonically increasing in n we are interested in the asymptotics of the total entropy we investigate in detail a class of states derived from quasi free states on a car algebra these are characterized by a measurable subset of the unit interval as the entropy density is known to vanish sn is sublinear in n for states corresponding to unions of finitely many intervals sn is shown to grow slower than log2 n numerical calculations suggest a log n behavior for the case with infinitely many intervals we present a class of states for which the entropy sn increases as n where can take any value in 0 1

hypothesis testing for gaussian states on bosonic lattices the asymptotic state discrimination problem with simple hypotheses is considered for a cubic lattice of bosons a complete solution is provided for the problems of the chernoff and the hoeffding bounds and stein s lemma in the case when both hypotheses are gaussian states with gauge and translation invariant quasifree parts

constructing simple stable descriptions for image partitioning a new formulation of the image partitioning problem is presented construct a complete and stable description of an image in terms of a specified descriptive language that is simplest in the sense of being shortest we show that a descriptive language limited to a low order polynomial description of the intensity variation within each region and a chain code like description of the region boundaries yields intuitively satisfying partitions for a wide class of images the advantage of this formulation is that it can be extended to deal with subsequent steps of the image understanding problem or to deal with other attributes such as texture in a natural way by augmenting the descriptive language experiments performed on a variety of both real and synthetic images demonstrate the superior performance of this approach over partitioning techniques based on clustering vectors of local image attributes and standard edge detection techniques

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

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

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

the structure of renyi entropic inequalities we investigate the universal inequalities relating the rnyi entropies of the marginals of a multi partite quantum state this is in analogy to the same question for the shannon and von neumann ent

quantum state discrimination bounds for finite sample size in the problem of quantum state discrimination one has to determine by measurements the state of a quantum system based on the a priori side information that the true state is one of the two given and completely known states or in general it is not possible to decide the identity of the true state with certainty and the optimal measurement strategy depends on whether the two possible errors mistaking for or the other way around are treated as of equal importance or not results on the quantum chernoff and hoeffding bounds and the quantum stein s lemma show that if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies and the decay rate is given by a certain statistical distance between and the chernoff distance the hoeffding distances and the relative entropy respectively while these results provide a complete solution to the asymptotic problem they are not completely satisfying from a practical point of vi

reduced dynamics and lagrangian submanifolds of symplectic manifolds in this paper we will see that the symplectic creed by weinstein everything is a lagrangian submanifold also holds for hamilton poincare and lagrange poincare reduction in fact we show that solutions of the hamilton poincare equations and of the lagrange poincare equations are in one to one correspondence with distinguished curves in a lagrangian submanifold of a symplectic manifold for this purpose we will combine the concept of a tulczyjew triple with marsden weinstein symplectic reduction

a generalization of quantum stein s lemma given many independent and identically distributed i i d copies of a quantum system described either by the state or called null and alternative hypotheses respectively what is the optimal measurement to learn the identity of the true state in asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying instead of while requiring that the probability that is identified in the place of is bounded by a small fixed number quantum stein s lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of and

error exponents in hypothesis testing for correlated states on a spin chain we study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum chernoff and hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain our results apply to states satisfying a certain factorization property typical examples are the global gibbs states of translation invariant finite range interactions as well as certain finitely correlated states

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

an asymptotic error bound for testing multiple quantum hypotheses we consider the problem of detecting the true quantum state among r possible ones based of measurements performed on n copies of a finite dimensional quantum system a special case is the problem of discriminating between r probability measures on a finite sample space using n i i d observations in this classical setting it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary chernoff bound between any possible pair of the r probability measures define analogously the multiple quantum chernoff bound considering all possible pairs of states recently it has been shown that this asymptotic error bound is attainable in the case of r pure states and that it is unimprovable in general here we extend the attainability result to a larger class of r tuples of states which are possibly mixed but pairwise linearly independent we also construct a quantum detector which universally attains the multiple quantum chernoff bound up to a factor 1 3

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