comments on hastings additivity counterexamples hastings 12 recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy in this paper we present an expanded version of hastings proof in addition to a careful elucidation of the details of the proof we also present bounds for the minimal dimensions needed to obtain a counterexample

statistical mechanics and phase transitions in clustering a new approach to clustering based on statistical physics is presented the problem is formulated as fuzzy clustering and the association probability distribution is obtained by maximizing the entropy at a given average variance the corresponding lagrange multiplier is related to the temperature and motivates a deterministic annealing process where the free energy is minimized at each temperature critical temperatures are derived for phase transitions when existing clusters split it is a hierarchical clustering estimating the most probable cluster parameters at various average variances

atoms and regularity for measures in a partially defined free convolution semigroup consider a borel probability measure on the real line and denote by t t 1 the free additive convolution semigroup defined by nica and speicher we show that the singular part of t is purely atomic and the density of t is locally analytic provided that t 1 the main ingredient is a global inversion theorem for analytic functions on a half plane

hastings s additivity counterexample via dvoretzky s theorem the goal of this note is to show that hastings s counterexample to the additivity of minimal output von neumann entropy can be readily deduced from a sharp version of dvoretzky s theorem

rough set theoretic fuzzy cues based object tracking under improved particle filter framework object tracking from video sequences although well researched still faces difficulties under certain challenging situations we propose a new particle filter based tracking where the fuzzy observation model exploits spatial correlation in a rough set theoretic framework we further improved certain critical steps of the particle filter such as mean state estimation and resampling in our approach mean state of the object at each time step is estimated with an adaptive number of high weighted samples these high weighted samples are propagated along with the fresh samples generated around the current mean state to maintain sample diversity and avoid sample impoverishment we later extended our approach to multi cues given the wide applicability of statistical texture in various vision applications due to its robustness under various challenging situations we propose to use fuzzy statistical texture cues along with fuzzy roughness color cues and the weight for each cue is adaptively determined using the discriminating ability of the cue results show that our proposed particle filtering approach has less error variance across simulations and can perform tracking under challenging situations better than the contemporary ones

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

c ondensation conditional density propagation forvisual tracking the problem of tracking curves in dense visual clutter is challenging kalman filtering is inadequate because it is based on gaussian densities which being unimo dal cannot represent simultaneous alternative hypotheses the condensation algorithm uses factored sampling previously applied to the interpretation of static images in which the probability distribution of possible interpretations is represented by a randomly generated set condensation uses learned dynamical models together with visual observations to propagate the random set over time the result is highly robust tracking of agile motion notwithstanding the use of stochastic methods the algorithm runs in near real time

stein extensions of real symmetric spaces and the geometry of the flag manifold let g be a connected semisimple lie group contained in its simply connected complexification g c let k g k c be a maximal compact subgroup of g denote by x o the unique closed g orbit in the full flag manifold and by the unique open k c orbit in the set consisting of the elements gk c so that gx o is an stein extension of g k g c k c there is a universal domain a scr subsetg scr scr scr bf scr scr c scr scr scr k scr scr scr bf scr scr c scr scr natural form the point of view of group actions which has been conjectured to be stein the main result of this paper is the inclusion in the second part of the paper i show under some dominance condition in the parameter that representations in dolbeault cohomology can be realized as holomorphic sections of vector bundles over

entanglement of random subspaces via the hastings bound recently hastings a counterexample to additivity of minimum output entropy nat phys 5 255 2009 e print arxiv 0809 3972v3 proved the existence of random unitary channels which violate the additivity conjecture in this paper we use hastings method to derive new bounds for the entanglement of random subspaces of bipartite systems as an application we use these bounds to prove the existence of nonunital channels which violate additivity of minimal output entropy

stochastic global optimization methods part 1 clustering methods in this stochastic approach to global optimization clustering techniques are applied to identify local minima of a real valued objective function that are potentially global three different methods of this type are described their accuracy and efficiency are analyzed in detail

eigenvectors and eigenvalues in a random subspace of a tensor product given two positive integers n and k and a parameter t 0 1 we choose at random a vector subspace v n k n of dimension n tnk we show that the set of k tuples of singular values of all unit vectors in v n fills asymptotically as n tends to infinity a deterministic convex set k k t that we describe using a new norm in k

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

random quantum channels ii entanglement of random subspaces renyi entropy estimates and additivity problems in this paper we obtain new bounds for the minimum output entropies of random quantum channels these bounds rely on random matrix techniques arising from free probability theory we then revisit the counterexamples developed by hayden and winter to get violations of the additivity equalities for minimum output renyi entropies we show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the p renyi entropy for some sequences of random quantum channels we compute almost surely the limit of their schatten s 1 to s p norms

kernel based object tracking a new approach toward target representation and localization the central component in visual tracking of nonrigid objects is proposed the feature histogram based target representations are regularized by spatial masking with an isotropic kernel the masking induces spatially smooth similarity functions suitable for gradient based optimization hence the target localization problem can be formulated using the basin of attraction of the local maxima we employ a metric derived from the bhattacharyya coefficient as similarity measure and use the mean shift procedure to perform the optimization in the presented tracking examples the new method successfully coped with camera motion partial occlusions clutter and target scale variations integration with motion filters and data association techniques is also discussed we describe only a few of the potential applications exploitation of background information kalman tracking using motion models and face tracking

constructive counterexamples to the additivity of the minimum output renyi entropy of quantum channels for all p 2 we present a constructive example of the violation of the additivity of minimum output renyi entropy for each p 2 the example is provided by an antisymmetric subspace of a suitable dimension we discuss the possibility of extension of the result to go beyond p 2 and obtain additivity for p 0 for a class of entanglement breaking channels

analytic continuation of representations and estimates of automorphic forms 0 1 analytic vectors and their analytic continuation let g be a lie group and g v a continuous representation of g in a topological vector space v a vector v v is called analytic if the functionv g 7 g v is a real analytic function on g with values in v this means that there exists a neighborhood u of g in its complexification gc such thatv extends to a holomorphic function on u in other words for each element g u we can unambiguously define the vector g v asv g i e we can extend the action of g to a somewhat larger set in this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates unless otherwise stated g sl 2 r so gc sl 2 c we consider a typical representation ofg i e a representation of the principal series namely fix c and consider the space dof smooth homogeneous functions of degree 1 on r 2 0 i e d c r 2 0 ax ay a 1 x y we

revisiting additivity violation of quantum channels we prove additivity violation of minimum output entropy of quantum channels by straightforward application of epsilon net argument and levy s lemma the additivity conjecture was disproved initially by hastings later a proof via asymptotic geometric analysis was presented by aubrun szarek and werner which uses dudley s bound on gaussian process or dvoretzky s theorem with schechtman s improvement in this paper we develop another proof along dvoretzky s theorem in milman s view showing additivity violation in broader regimes than the existing proofs importantly dvoretzky s theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm like structures positive homogeneity and triangle inequality then a connection between hastings method and ours is also discussed in addition we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels

equivalence of domains arising from duality of orbits on flag manifolds iii in gindikin and matsuki 2003 we defined a invariant subset of for each orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the akhiezer gindikin domain if is of nonholomorphic type this conjecture was proved for closed in wolf and zierau 2000 and 2003 fels and huckleberry 2005 and matsuki 2006 and for open in matsuki 2006 it was proved for the other orbits in matsuki 2006 when is of non hermitian type in this paper we prove the conjecture for an arbitrary non closed orbit when is of hermitian type thus the conjecture is completely solved affirmatively

a convexity property for the so 2 ℂ double coset decomposition of sl 2 ℂ and applications to spherical functions we make a fine study of the so 2 double coset decomposition in sl 2 and give a full description of the intersection of the various cells with the complex crown of sl 2 so 2 a non linear convexity theorem is proved and applications to analytically continued spherical functions are given

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

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

counterexamples to additivity of minimum output p renyi entropy for p close to 0 complementing recent progress on the additivity conjecture of quantum information theory showing that the minimum output p renyi entropies of channels are not generally additive for p 1 we demonstrate here by a careful random selection argument that also at p 0 and consequently for sufficiently small p there exist counterexamples

schubert varieties and cycle spaces let g0 be a real semisimple lie group it acts naturally on every complex flag manifold z g q of its complexification given an iwasawa decomposition g0 k0a0n0 a g0 orbitz and the dual k orbit z schubert varieties are studied and a theory of schubert slices for arbitrary g0 orbits is developed for this certain geometric properties of dual pairs are underlined canonical complex analytic slices contained in a given g0 orbit which are transversal to the dual k0 orbit are constructed and analyzed associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains in particular it is shown that the linear cycle space w d is a stein domain that contains the universally defined iwasawa domain i this is one of the main ingredients in the proof that w d ag for all but a few hermitian exceptions in the hermitian case w d is concretely described in terms of the associated bounded symmetric domain

asymptotically well behaved input states do not violate additivity for conjugate pairs of random quantum channels it is now well known that with high probability the additivity of minimum output entropy does not hold for pairs of the random quantum channel and its complex conjugate we investigate asymptotic behavior of output states of r tensor powers of such pairs as the dimension of inputs grows we compute the limit output states for any sequence of well behaved inputs which consist of a large class of input states having a nice set of parameters then we show that among these input states tensor products of bell states give asymptotically the least output entropy giving positive mathematical evidence towards additivity of above pairs of channels

complex crowns of riemannian symmetric spaces and non compactly causal symmetric spaces in this paper we define a distinguished boundary for the complex crowns g c k c of non compact riemannian symmetric spaces g k the basic result is that affine symmetric spaces of g can appear as a component of this boundary if and only if they are non compactly causal symmetric spaces

almost one bit violation for the additivity of the minimum output entropy in a previous paper we proved that in the appropriate asymptotic regime the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic compact set kk t we also showed that the set kk t is obtained up to an intersection as the unit ball of the dual of a free compression norm in this paper we identify the maximum of ell p norms on the set kk t and prove that the maximum is attained on a vector of shape a b b where a b in particular we compute the precise limit value of the minimum output entropy of a single random quantum channel as a corollary we show that for any varepsilon 0 it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183 and that for appropriate choice of parameters the violation can be as large as log 2 varepsilon conversely our result implies that with probability one in the limit one does not obtain a violation of additivity using conjugate random quantum channels and the bell state in dimension 182 and less

density estimation with confidence sets exemplified by superclusters and voids in the galaxies abstract a method is presented for forming both a point estimate and a confidence set of semiparametric densities the final product is a three dimensional figure that displays a selection of density estimates for a plausible range of smoothing parameters the boundaries of the smoothing parameter are determined by a nonparametric goodness of fit test that is based on the sample spacings for each value of the smoothing parameter our estimator is selected by choosing the normal mixture that maximizes a function of the sample spacings a point estimate is selected from this confidence set by using the method of cross validation an algorithm to find the mixing distribution that maximizes the spacings functional is presented these methods are illustrated with a data set from the astronomy literature the measurements are velocities at which galaxies in the corona borealis region are moving away from our galaxy if the galaxies are clustered the velocity density will be multimodal with clusters correspond

mean shift mode seeking and clustering mean shift a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper this generalization makes some k means like clustering algorithms its special cases it is shown that mean shift is a mode seeking process on the surface constructed with a shadow kernal for gaussian kernels mean shift is a gradient mapping convergence is studied for mean shift iterations cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data applications in clustering and hough transform are demonstrated mean shift is also considered as an evolutionary strategy that performs multistart global optimization

on the minimum output entropy of random orthogonal quantum channels we consider the sequences of random quantum channels defined by using the stinespring formula with haar distributed random orthogonal matrices for any fixed sequence of input states we study the asymptotic eigenvalue distribution of the outputs through the tensor powers of random channels we show that the input states achieving minimum output entropy are tensor products of maximally entangled states bell states when the tensor power is even this phenomenon is completely different from the one for random quantum channels constructed from haar distributed random unitary matrices which leads us to formulate some conjectures about the regularized minimum output entropy

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

towards a state minimizing the output entropy of a tensor product of random quantum channels we consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels depending on natural assumptions that we make on the states the eigenvalues of their outputs have new properties which we describe our motivation is provided by the additivity questions in quantum information theory and we build on the idea that a bell state sent through a product of conjugated random channels has at least one large eigenvalue we generalize this setting in two directions first we investigate general entangled pure inputs and show that bell states give the least entropy among those inputs in the asymptotic limit we then study mixed input states and obtain new multi scale random matrix models that allow to quantify the difference of the outputs eigenvalues between a quantum channel and its complementary version in the case of a non pure input

counterexamples to the maximal p norm multiplicativity conjecture for all p 1 for all p 1 we demonstrate the existence of quantum channels with non multiplicative maximal output p norms equivalently for all p 1 the minimum output renyi entropy of order p of a quantum channel is not additive the violations found are large in all cases the minimum output renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors since p 1 corresponds to the von neumann entropy these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true it cannot be proved as a consequence of any channel independent guarantee of maximal p norm multiplicativity we also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p 2