geometry of sets of quantum maps a generic positive map acting on a high dimensional system is not completely positive we investigate the set a of positive trace preserving maps acting on density matrices of size n and a sequence of its nested subsets the sets of maps which are b decomposable c completely positive and d extended by identity impose positive partial transpose and e are superpositive working with the hilbert schmidt euclidean measure we derive tight explicit two sided bounds for the volumes of all five sets a sample consequence is the fact that as n increases a generic positive map becomes not decomposable and a fortiori not completely positive due to the jamiolkowski isomorphism the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets such as states with positive partial transpose or separable states or supersets our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity high dimensional probability and geometry of banach spaces some of which are not standard

degree product rule tempers explosive percolation in the absence of global information we introduce a guided network growth model which we call the degree product rule process that uses solely local information when adding new edges for small numbers of candidate edges our process gives rise to a second order phase transition but becomes first order in the limit of global choice we provide the set of critical exponents required to characterize the nature of this percolation transition such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes

on the structure of the body of states with positive partial transpose we show that the convex set of separable mixed states of the 2 2s ystem is a body of a constant height this fact is used to prove that the probability of finding a random state to be separable equals twice the probability of finding a random boundary state to be separable provided that the random states are generated uniformly with respect to the hilbert schmidt euclidean measure an analogous property holds for the set of positive partial transpose states for an arbitrary bipartite system

separability of very noisy mixed states and implications for nmr quantum computing we give a constructive proof that all mixed states of n qubits in a sufficiently small neighborhood of the maximally mixed state are separable unentangled the construction provides an explicit representation of any such state as a mixture of product states we give upper and lower bounds on the size of the neighborhood which show that its extent decreases exponentially with the number of qubits the bounds show that no entanglement appears in the physical states at any stage of present nmr experiments though this result raises questions about nmr quantum computation further analysis would be necessary to assess the power of the general unitary transformations which are indeed implemented in these experiments in their action on separable states

an analysis of completely positive trace preserving maps on m2 abstract we give a useful new characterization of the set of all completely positive trace preserving maps m 2 m 2 from which one can easily check any trace preserving map for complete positivity we also determine explicitly all extreme points of this set and give a useful parameterization after reduction to a certain canonical form this allows a detailed examination of an important class of non unital extreme points that can be characterized as having exactly two images on the bloch sphere we also discuss a number of related issues about the images and the geometry of the set of stochastic maps and show that any stochastic map on m 2 can be written as a convex combination of two generalized extreme points

entropy and asymptotic geometry of non symmetric convex bodies abstract we extend to the general not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs some of these extensions look surprising the main additional tool is a study of volume behavior around the centroid of the body

about the fastest growth of the order parameter in models of percolation abstract the growth of the average size s max of the largest component at the percolation threshold p c n on a graph of size n has been defined as s max p c n n n here we argue that the precise value of the growth exponent indicates the nature of percolation transition 1 or 1 determines if the transition is continuous or discontinuous we show that a related exponent 1 which describes how the average maximal jump sizes in the order parameter decays on increasing the system size is the single exponent that describes the finite size scaling of a number of distributions related to the fastest growth of the order parameter in these problems excellent quality scaling analysis are presented for the two single peak distributions corresponding to the order parameters at the two ends of the maximal jump the bimodal distribution constructed by the weighted average of these distributions and for the distribution of the maximal jump in the order parameter

approximating the set of separable states using the positive partial transpose test the positive partial transpose test is one of the main criteria for detecting entanglement and the set of states with positive partial transpose is considered as an approximation of the set of separable states however we do not know to what extent this criterion as well as the approximation is efficient in this paper we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states more precisely we prove that as the dimension of the space tends to infinity the maximum trace distance of a positive partial transpose state from separable states tends to 1 using similar techniques we show that the same result holds for other well known separability criteria such as reduction criterion majorization criterion and symmetric extension criterion we also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes

rogers shephard inequality for log concave functions abstract in this paper we prove different functional inequalities extending the classical rogers shephard inequalities for convex bodies the original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body such as its difference body we characterize the equality cases in all these inequalities our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log concave functions and the study of some geometrical properties of these new sets

hilbert schmidt volume of the set of mixed quantum states we compute the volume of the convex n2 1 dimensional set n of density matrices of size n with respect to the hilbert schmidt measure the hyper area of the boundary of this set is also found and its ratio to the volume provides information about the structure of n similar investigations are also performed for the smaller set of all real density matrices as an intermediate step we analyse volumes of the unitary and orthogonal groups and of the flag manifolds

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

a generalized pancharatnam geometric phase formula for three level quantum systems we describe a recently developed generalization of the poincare sphere method to represent pure states of a three level quantum system in a convenient geometrical manner the construction depends on the properties of the group su 3 and its generators in the defining representation and uses geometrical objects and operations in an eight dimensional real euclidean space this construction is then used to develop a generalization of the well known pancharatnam geometric phase formula for evolution of a three level system along a geodesic triangle in state space

entanglement thresholds for random induced states for a random quantum state on h c d otimes c d obtained by partial tracing a random pure state on h otimes c s we consider the whether it is typically separable or typically entangled for this problem we show the existence of a sharp threshold s 0 s 0 d of order roughly d 3 more precisely for any a 0 and for d large enough such a random state is entangled with very large probability when s 1 a s 0 one consequence of this result is as follows for a system of n identical particles in a random pure state there is a threshold k 0 k 0 n sim n 5 such that two subsystems of k particles each typically share entanglement if k k 0 and typically do not share entanglement if k k 0 our methods work also for multipartite systems and for unbalanced systems such as c d otimes c d d neq d the arguments rely on random matrices classical convexity high dimensional probability and geometry of banach spaces some of the auxiliary results may be of reference value a high level non technical overview of the results of this paper and of a related article arxiv 1011 0275 can be found in arxiv 1112 4582

spatio temporal multiscale denoising of fluoroscopic sequence in the past 20 years a wide range of complex fluoroscopically guided procedures have shown considerable growth biologic effects of the exposure radiation induced burn cancer lead to reduce the dose during the intervention for the safety of patients and medical staff however when the dose is reduced image quality decreases with a high level of noise and a very low contrast efficient restoration and denoising algorithms should overcome this drawback we propose a spatio temporal filter operating in a multi scales space this filter relies on a first order motion compensated recursive temporal denoising temporal high frequency content is first detected and then matched over time to allow for a strong denoising in the temporal axis we study this filter in the curvelet domain and in the dual tree complex wavelet domain and compare those results to state of the art methods quantitative and qualitative analysis on both synthetic and real fluoroscopic sequences demonstrate that the proposed filter allows a great dose reduction

cluster aggregation model for discontinuous percolation transitions the evolution of the erd ifmmode mbox h o else h o fi s r enyi er network by adding edges is a basis model for irreversible kinetic aggregation phenomena such er processes can be described by a rate equation for the evolution of the cluster size distribution with the connection kernel k ij ensuremath sim ij where ij is the product of the sizes of two merging clusters here we study that when the giant cluster is discouraged to develop by a sublinear kernel k ij ensuremath sim ij ensuremath omega with 0 ensuremath le ensuremath omega l1 2 the percolation transition pt is discontinuous such discontinuous pt can occur even when the er dynamics evolves from proper initial conditions the obtained evolutionary properties of the simple model sheds light on the origin of the discontinuous pt in other nonequilibrium kinetic systems

almost all quantum channels are equidistant in this work we analyze properties of generic quantum channels in the case of large system size we use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger as a measure of the distance we use the diamond norm in the case of a flat hilbert schmidt distribution on quantum channels we obtain that the distance converges to 12 2 giving also an estimate for the maximum success probability for distinguishing the channels we also consider the problem of distinguishing two random unitary rotations in this work we analyze properties of generic quantum channels in the case of large system size we use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger as a measure of the distance we use the diamond norm in the case of a flat hilbert schmidt distribution on quantum channels we obtain that the distance converges to 12 2 giving also an estimate for the maximum success probability for distinguishing the channels we also consider the problem of distinguishing two random unitary rotations

curvelet based contrast enhancement in fluoroscopic sequences image guided interventions have seen growing interest in recent years the use of x rays for the procedure impels limiting the dose over time image sequences obtained thereby exhibit high levels of noise and very low contrasts hence the development of efficient methods to enable optimal visualization of these sequences is crucial we propose an original denoising method based on the curvelet transform first we apply a recursive temporal filter to the curvelet coefficients as some residual noise remains a spatial filtering is performed in the second step which uses a magnitude based classification and a contextual comparison of curvelet coefficients this procedure allows to denoise the sequence while preserving low contrasted structures but does not improve their contrast finally a third step is carried out to enhance the features of interest for this we propose a line enhancement technique in the curvelet domain indeed thin structures are sparsely represented in that domain allowing a fast and efficient detection quantitative and qualitative evaluations performed on synthetic and real low dose sequences demonstrate that the proposed method enables a 50 dose reduction

geometry of quantum systems density states and entanglement various problems concerning the geometry of the space of hermitian operators on a hilbert space are addressed in particular we study the canonical poisson and riemann jordan tensors and the corresponding foliations into k hler submanifolds it is also shown that the space of density states on an n dimensional hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state thus the space of rank k states k 1 n is a smooth manifold of real dimension 2nk k2 1 and this stratification is maximal in the sense that every smooth curve in viewed as a subset of the dual to the lie algebra of the unitary group at every point must be tangent to the strata it crosses for a quantum composite system i e for a hilbert space decomposition an abstract criterion of entanglement is proved

subnormalized states and trace nonincreasing maps we investigate the set of completely positive trace nonincreasing linear maps acting on the set mn of mixed quantum states of size n extremal point of this set of maps are characterized and its volume with respect to the hilbert schmidt hs euclidean measure is computed explicitly for an arbitrary n the spectra of partially reduced rescaled dynamical matrices associated with trace nonincreasing completely positive maps belong to the n cube inscribed in the set of subnormalized states of size n as a by product we derive the measure in mn induced by partial trace of mixed quantum states distributed uniformly with respect to the hs measure in mn2

how often is a random quantum state k entangled the set of trace preserving positive maps acting on density matrices of size d forms a convex body we investigate its nested subsets consisting of kpositive maps where k 2 d working with the measure induced by the hilbert schmidt distance we derive asymptotically tight bounds for the volumes of these sets our results strongly suggest that the inner set of k 1 positive maps forms a small fraction of the outer set of k positive maps these results are related to analogous bounds for the relative volume of the sets of k entangled states describing a bipartite d d system

the inverse eigenvalue problem for entanglement witnesses abstract we consider the inverse eigenvalue problem for entanglement witnesses which asks for a characterization of their possible spectra or equivalently of the possible spectra resulting from positive linear maps of matrices we completely solve this problem in the two qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions we also establish a natural duality relationship with the set of absolutely separable states and we completely characterize witnesses i e separating hyperplanes of that set when one of the local dimensions is 2

correlated percolation and tricriticality the recent proliferation of correlated percolation models models where the addition of edges and or vertices is no longer independent of other edges and or vertices has been motivated by the quest to find discontinuous percolation transitions the leader in this proliferation is what is known as explosive percolation a recent proof demonstrates that a large class of explosive percolation type models does not in fact exhibit a discontinuous transition riordan and warnke science 333 322 2011 here we discuss two lesser known correlated percolation models the k 3 core model on random graphs and the counter balance model in two dimensions both exhibiting discontinuous transitions to search for tricriticality we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition using a powerful rate equation approach we demonstrate that a mixture of k 2 core and k 3 core vertices on the random graph exhibits a tricritical point however for a mixture of k core and counter balance vertices in two dimensions as the fraction of counter balance vertices is increased numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition i e when all vertices are counter balanced interestingly these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions in addition our results may have potential implications for glassy and jamming systems

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

cones and norms in the tensor product of matrix spaces abstract this paper is of expository nature in the tensor product of two matrix spaces m m and m n we consider three natural cones the first is the cone p 0 of positive semi definite matrices the second is the cone p generated by tensor products of positive semi definite matrices in the factor spaces and the third cone p is defined as the dual cone of p correspondingly in the tensor space identified with m mn besides the spectral and the trace norms we consider several tensor norms generated by sepectral and or trace norms in the factor spaces we will study relations among those norms as well as relations among the three kinds of order intervals in the tensor product

new examples of extremal positive linear maps abstract new families of nonnegative biquadratic forms that have 8 9 or 10 real zeros in p 2 p 2 are constructed these are the first examples with 8 9 or 10 real zeros it is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros our examples show that the upper bound is obtained such biquadratic forms define positive linear maps on real symmetric 3 3 matrices that are not completely positive our constructions are explicit and moreover we are able to determine which of the examples are extremal we extend the examples to positive maps on complex matrices and find families of extreme rays in the cone of positive maps