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

the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems for any function f from mathbb r to mathbb r one can define a corresponding function on the space of n n block diagonal real symmetric matrices by applying f to the eigenvalues of the spectral decomposition we show that this matrix valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as rho order semismoothness our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices we also apply our results to the semidefinite complementarity problem addressing some basic issues in the analysis of smoothing semismooth newton methods for solving this problem

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

additive bounds of minimum output entropies for unital channels and an exact qubit formula we find an upper bound for the minimum output entropy of a unital quantum channel and obtain an exact formula for general qubit channels our techniques incorporate the renyi entropies particularly with renyi parameter alpha 2 moreover since our upper bound is additive under tensor product we get as a corollary an upper bound for the classical capacity of unital quantum channels interestingly our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless hermitian operators and is tight in the sense that it gives the precise quantity of classical capacity of the werner holevo channel as an example we study quantum channels with operator sum representation that is made of the discrete weyl operators generalized pauli operators and explain how our formula works in this case finally we find new examples for which the minimum output renyi 2 entropy is additive

lp theory for the multidimensional aggregation equation we consider well posedness of the aggregation equation tu div uv 0 v k u with initial data in input amssym cal p 2 rm bbb r d rm cap l p bbb r d in dimensions 2 and higher we consider radially symmetric kernels where the singularity at the origin is of order x 2 d and prove local well posedness in input amssym cal p 2 bbb r d rm cap l p bbb r d for sufficiently large p ps in the special case of k x x the exponent ps d d 1 is sharp for local well posedness in that solutions can instantaneously concentrate mass for initial data in input amssym cal p 2 bbb r d rm cap l p bbb r d with p ps we also give an osgood condition on the potential k x that guarantees global existence and uniqueness in input amssym cal p 2 bbb r d rm cap l p bbb r d 2010 wiley periodicals inc

derivatives of compound matrix valued functions abstract primary matrix functions and spectral functions are two classes of orthogonally invariant functions on a symmetric matrix argument many of their properties have been investigated thoroughly and find numerous applications both theoretical and applied in areas ranging from engineering image processing optimization and physics we propose a family of maps that provide a natural connection and generalization of these two classes of functions the family of maps also contains the well known multiplicative and additive compound matrices we explain when each member of this family is a differentiable function and exhibit a formula for its derivative

self organization in systems of self propelled particles we investigate a discrete model consisting of self propelled particles that obey simple interaction rules we show that this model can self organize and exhibit coherent localized solutions in one and in two dimensions in one dimension the self organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges in two dimensions we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions even in the absence of a confining boundary furthermore we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent

cardinality estimation of support of the global minimizer for the interaction energy with mildly repulsive potentials abstract in this work we give a cardinality estimation of the support of any global minimizer for the interaction energy with mildly repulsive potentials for any global minimizer the fact that supp consists of finitely many points was proved in carrillo et al 2017 moreover the authors found a cardinality estimation of the support of on r 1 in order to advance some improvement of this result we give an upper estimate of card supp in arbitrary finite dimension

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

additivity for unital qubit channels additivity of the holevo capacity is proved for product channels under the condition that one of the channels is a unital qubit channel with the other completely arbitrary as a byproduct this proves that the holevo bound is the classical information capacity of such qubit channels and provides an explicit formula for this classical capacity additivity of minimal entropy and multiplicativity of p norms are also proved under the same assumptions the proof relies on a new bound for the p norm of an output state from the half noisy phase damping channel

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

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

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

the capacity of the quantum depolarizing channel the information carrying capacity of the d dimensional depolarizing channel is computed it is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space and using measurements which are products of projections onto this same orthonormal basis in other words neither entangled signal states nor entangled measurements give any advantage for information capacity the result follows from an additivity theorem for the product channel spl delta spl ominus spl psi where spl delta is the depolarizing channel and spl psi is a completely arbitrary channel we establish the amosov holevo werner see probl inform transm vol 36 p 305 313 2000 p norm conjecture for this product channel for all p spl ges 1 and deduce from this the additivity of the minimal entropy and of the holevo quantity spl chi sup

a new derivation of a formula by kato abstract if the m th largest eigenvalue m a of a real symmetric matrix a is simple then m is an analytic function in a neighbourhood of a in this note we provide a new derivation of the classical formulae for the coefficients in the power series expansion of t m a te for any real symmetric matrix e and t close to 0 kato s classical derivation of that formula uses a complex analytic approach involving properties of the resolvent of a te our derivation uses simple real analytic and combinatorial arguments in particular we derive and utilize a formula for the derivative of the moore penrose generalized inverse of the map x m x i x in direction e at real symmetric matrix a for any real symmetric matrix e

a unified approach to operator monotone functions abstract the notion of operator monotonicity dates back to a work by lowner in 1934 a map f s n s m is called operator monotone if a b implies f a f b here s n is the space of symmetric matrices with the semidefinite partial order often the function f is defined in terms of an underlying simpler function f of main interest is to find the properties of f that characterize operator monotonicity of f in that case it is said that f is also operator monotone classical examples are the lowner operators and the spectral scalar valued isotropic functions operator monotonicity for these two classes of functions is characterized in seemingly very different ways this work extends the notion of operator monotonicity to symmetric functions f on k arguments the latter is used to define generated k isotropic maps f s n s n k for any n k necessary and sufficient conditions are given for f to generate an operator monotone k isotropic map f when k 1 the k isotropic map becomes a lowner operator and when k n it becomes a spectral function this allows us to reconcile and explain the differences between the conditions for monotonicity for the lowner operators and the spectral functions

the higher order derivatives of spectral functions in this paper we are interested in the higher order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument we describe the formula for the k th derivative of such functions in two general cases the flrst case concerns the derivatives of the composition of an arbitrary not necessarily symmetric k times difierentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues the second case describes the derivatives of the composition of a k times difierentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix we show that the formula signiflcantly simplifles when the separable symmetric function is k times continuously difierentiable as an application of the developed techniques we re derive the formula for the hessian of a general spectral function at an arbitrary symmetric matrix the new tools lead to a shorter cleaner derivation than the original one in 16 to make the exposition as self contained as possible we have included the necessary background results and deflnitions the proofs of the intermediate technical results are collected in the appendices

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

contour dynamics for the euler equations in two dimensions we present a contour dynamics algorithm for the euler equations of fluid dynamics in two dimensions this is applied to regions of piecewise constant vorticity withinfinite area vortexregions favrs essentially this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self consistently a leapfrog centered scheme is used for temporal advancement computer simulation results are given for two and four like signed interacting favrs in all cases wavelike surface deformations are observed if the distance between favrs is comparable to their extent diameter these surface deformations are large they play an essential role in the observed coalescence of favrs

from discrete cucker smale model to continuous cucker smale model in a temperature field we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results

secure key rate of the bb84 protocol using finite sample bits we improve the non asymptotic key rate shown by scarani and renner by proposing several methods to construct tighter conservative confidence intervals of the phase error rate than one previously shown in addition we show that the accurate channel estimation method non asymptotically increases the key rate over the amplitude damping channel as well as the asymptotic case in the bb84 protocol

inequalities for trace norms of 2 2 block matrices this paper derives an inequality relating the p norm of a positive 2 2 block matrix to the p norm of the 2 2 matrix obtained by replacing each block by its p norm the inequality had been known for integer values of p so the main contribution here is the extension to all values p 1 in a special case the result reproduces hanner s inequality a weaker inequality which applies also to non positive matrices is presented as an application in quantum information theory the inequality is used to obtain some results concerning maximal p norms of product channels

phase transitions in self driven many particle systems and related non equilibrium models a network approach we investigate the conditions that produce a phase transition from an ordered to a disordered state in a family of models of two dimensional elements with a ferromagnetic like interaction this family is defined to contain under the same framework among others the xy model and the self driven particles model introduced by vicsek et al each model is distinguished only by the rules that determine the set of elements with which each element interacts we propose a new member of the family the vectorial network model in which a given fraction of the elements interact through direct random connections this model is analogous to an xy system on a network and as such can be of interest for a wide range of problems it captures the main aspects of the interaction dynamics that produce the phase transition in other models of the family the network approach allows us to show analytically the existence of a phase transition in this vectorial network model and to compute its relevant parameters for the case in which all elements are randomly connected finally we study numerically the conditions required for a phase transition to exist for different members of the family our results show that a qualitatively equivalent phase transition appears whenever even a small amount of long range interactions are present or built over time regardless of other equilibrium or non equilibrium properties of the system

average output entropy for quantum channels we study the regularized average renyi output entropy s r reg of quantum channels this quantity gives information about the average noisiness of the channel output arising from a typical highly entangled input state in the limit of infinite dimensions we find a closed expression for r reg a quantity which we conjecture to be equal to s r reg we find an explicit form for r reg for some entanglement breaking channels and also for the qubit depolarizing channel as a function of the parameter we prove equality of the two quantities in some cases in particular we conclude that for both are non analytic functions of the variable

some weak flocking models and its application to target tracking abstract in this paper we introduce a new concept named the weak flocking behavior that is if the complex system has the weak flocking property the agents do not need have the same velocities to keep together when t this shows the biggest difference between weak flocking behavior and the flocking behavior the advantage of this concept is that its practical significance the agents don t need always keep in step in lots of the complex systems but they still keep together then we propose a weak flocking model with two agents and study two different mechanisms for this model moreover we also consider a system with n agents under the weak link mechanism the system with n agents can keep weak flocking

maximization of capacity and lp norms for some product channels it is conjectured that the holevo capacity of a product channel is achieved when product states are used as input amosov holevo and werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states in this article we establish both of these conjectures in the case that is arbitrary and is a cq or qc channel as defined by holevo we also establish the amosov holevo and werner conjecture when is arbitrary and either is a qubit channel and p 2 or is a unital qubit channel and p is integer our proofs involve a new conjecture for the norm of an output state of the half noisy channel i when is a qubit channel we show that this conjecture in some cases also implies additivity of the holevo capacity

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

maximal output purity and capacity for asymmetric unital qudit channels we consider generalizations of depolarizing channels to maps of the form with vk being unitary and kak a 1 we show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique we give conditions on vk under which multiplicativity of the maximal p norm and additivity of the minimal output entropy can be proved for with arbitrary we also show that the holevo capacity need not equal log d smin as one might expect for a convex combination of unitary conjugations

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

stochastic flocking dynamics of the cucker smale model with multiplicative white noises we present a strong asymptotic stochastic flocking estimate for the stochastically perturbed cucker smale model we characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states when the strength of multiplicative noises is sufficiently large we show that the strong stochastic flocking occurs even for negative communication weights

quadratic expansions of spectral functions a function f on the space of n n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument that is f a f uau t for every orthogonal u and symmetric a in its domain spectral functions are in one to one correspondence with the symmetric functions on r n those that are invariant under arbitrary swapping of their arguments in this paper we show that a spectral function has a quadratic expansion around a point a if and only if its corresponding symmetric function has quadratic expansion around a the vector of eigenvalues we also give a concise and easy to use formula for the hessian of the spectral function in the case of convex functions we show that a positive definite hessian of f implies positive definiteness of the hessian of f 2002 elsevier science inc all rights reserved

approximating functions on stratified sets we investigate smooth approximations of functions with prescribed gradient behavior on a distinguished stratified subset of the domain as an application we outline how our results yield important consequences for a recently introduced class of stochastic processes called the matrix valued bessel processes

potential output purity of completely positive maps the notion of potential output purity of a completely positive map is introduced as a generalization of the regularized output purity an upper bound is derived for this quantity and for several classes of maps including cq classical quantum qc quantum classical hadamard channels it is shown that potential purity does not exceed the standard output purity as an application the potential purity is used to bound the logarithmic sobolev constant of a product of depolarizing channel semigroups

multiplicativity properties of entrywise positive maps abstract multiplicativity of certain maximal p q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity cp is either augmented by or replaced by the requirement that the entries of a matrix representative of the map are non negative ep in particular for integer t multiplicativity holds for the maximal 2 2t norm of a product of two maps whenever one of the pair is ep for the maximal 1 t norm for pairs of cp maps when one of them is also ep and for the maximal 1 2t norm for the product of an ep and a 2 positive map similar results are shown in the infinite dimensional setting of convolution operators on l 2 r with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix these results apply in particular to gaussian bosonic channels

nonsmooth analysis of eigenvalues the eigenvalues of a symmetric matrix depend on the matrix nonsmoothly this paper describes the nonsmooth analysis of these eigenvalues in particular i present a simple formula for the approximate limiting frechet subdifferential of an arbitrary function of the eigenvalues subsuming earlier results on convex and clarke subgradients as an example i compute the subdifferential of the k th largest eigenvalue

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

a fast algorithm for convolution integrals with space and time variant kernels mathematical models in many fields of the physical sciences involve nonlocal terms which are formally similar to convolution integrals we show that it is possible to approximate a particular class of such integrals which by themselves are not convolutions as a linear combination of convolution integrals allowing for their efficient numerical computation as an o n log n process

an application of decomposable maps in proving multiplicativity of low dimensional maps in this paper we present a class of maps for which the multiplicativity of the maximal output p norm holds for p 2 and p 4 the class includes all positive trace preserving maps from b c3 to b c2 in this sense the result is a generalization of the corresponding result in the work of king and koldan new multiplicativity results for qubit maps j math phys 47 042106 2006 where the multiplicativity was proved for all positive trace preserving maps from b c2 to b c2 with p 2 and p 4 interestingly by contrast the multiplicativity of p norm was investigated in the context of quantum information theory and shown not to hold in general for high dimensional quantum channels hayden p and winter a counterexamples to the maximal p norm multiplicativity conjecture for all p 1 commun math phys 284 263 2008 moreover the werner holevo channel which is a map from b c3 to b c3 is a counterexample for p 4 79 werner and holevo j math phys 43 4353 2002

global in time weak measure solutions and finite time aggregation for nonlocal interaction equations in this paper we provide a well posedness theory for weak measure solutions of the cauchy problem for a family of nonlocal interaction equations these equations are continuum models for interacting particle systems with attractive repulsive pairwise interaction potentials the main phenomenon of interest is that even with smooth initial data the solutions can concentrate mass in finite time we develop an existence theory that enables one to go beyond the blow up time in classical norms and allows for solutions to form atomic parts of the measure in finite time the weak measure solutions are shown to be unique and exist globally in time moreover in the case of sufficiently attractive potentials we show the finite time total collapse of the solution onto a single point for compactly supported initial measures our approach is based on the theory of gradient flows in the space of probability measures endowed with the wasserstein metric in addition to classical tools we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

large scale binary quadratic optimization using semidefinite relaxation and applications in computer vision many problems can be formulated as binary quadratic programs bqps which are in general np hard finding a solution when the problem is of large size to be of practical interest typically requires relaxation semidefinite relaxation usually yields tight bounds but its computational complexity is high in this work we present a semidefinite programming sdp formulation for bqps with two desirable properties first it produces similar bounds to the standard sdp formulation second compared with the conventional sdp formulation the proposed sdp formulation leads to a considerably more efficient and scalable dual optimization approach we then propose two solvers namely quasi newton and smoothing newton methods for the simplified dual problem both of them are significantly more efficient than standard interior point methods empirically the smoothing newton solver is faster than the quasi newton solver for dense or medium sized problems while the quasi newton solver is preferable for large sparse structured problems

conditions for multiplicativity of maximal lp norms of channels for fixed integer p we introduce a condition for memoryless quantum channels which when satisfied guarantees the multiplicativity of the maximal lp norm with p a fixed integer by applying the condition to qubit channels it can be shown that it is not a necessary condition although some known results for qubits can be recovered when applied to the werner holevo channel which is known to violate multiplicativity when p is large relative to the dimension d the condition suggests that multiplicativity holds when d 2p 1 this conjecture is proved explicitly for p 2 3 4 finally a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix it is shown that these channels are multiplicative for p 2

models of collective cell motion for cell populations with different aspect ratio diffusion proliferation and travelling waves continuum partial differential equation models are often used to describe the collective motion of cell populations with various types of motility represented by the choice of diffusion coefficient and cell proliferation captured by the source terms previously the choice of diffusion coefficient has been largely arbitrary with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual level properties of the cell motility mechanism in this work we provide a new link between individual level models which account for important cell properties such as varying cell shape and volume exclusion and population level partial differential equation models we work in an exclusion process framework considering aligned elongated cells that may occupy more than one lattice site in order to represent populations of agents with different sizes three different idealisations of the individual level mechanism are proposed and these are connected to three different partial differential equations each with a different diffusion coefficient one linear one nonlinear and degenerate and one nonlinear and nondegenerate we test the ability of these three models to predict the population level response of a cell spreading problem for both proliferative and nonproliferative cases we also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two dimensional spreading and invasion our results show that each model can accurately predict density data for nonproliferative systems but that only one does so for proliferative systems hence great care must be taken to predict density data with varying cell shape

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

new multiplicativity results for qubit maps let be a trace preserving positivity preserving but not necessarily completely positive linear map on the algebra of complex 2 2 matrices and let be any finite dimensional completely positive map for p 2 and p 4 we prove that the maximal p norm of the product map is the product of the maximal p norms of and restricting to the class of completely positive maps this settles the multiplicativity question for all qubit channels in the range of values p 4