on a family of linear maps from mn c to mn2 c abstract bhat 1 characterizes the family of linear maps defined on b h which preserve unitary conjugation we generalize this idea and study the maps with a similar equivariance property on finite dimensional matrix algebras we show that the maps with equivariance property are significant to study k positivity of linear maps defined on finite dimensional matrix algebras in 5 choi showed that n positivity is different from n 1 positivity for the linear maps defined on n by n matrix algebras in this paper we present a parametric family of linear maps n m n c m n 2 c and study the properties of positivity completely positivity decomposability etc we determine values of parameters and for which the family of maps n is positive for any natural number n 3 we focus on the case of n 3 that is 3 and study the properties of 2 positivity completely positivity and decomposability in particular we give values of parameters and for which the family of maps 3 is 2 positive and not completely positive

separable states with unique decompositions we search for faces of the convex set consisting of all separable states which are affinely isomorphic to simplices to get separable states with unique decompositions in the two qutrit case we found that six product vectors spanning a five dimensional space give rise to a face isomorphic to the 5 dimensional simplex with six vertices under a suitable linear independence assumption if the partial conjugates of six product vectors also span a 5 dimensional space then this face is inscribed in the face for ppt states whose boundary shares the fifteen 3 simplices on the boundary of the 5 simplex the remaining boundary points consist of ppt entangled edge states of rank four we also show that every edge state of rank four arises in this way if the partial conjugates of the above six product vectors span a 6 dimensional space then we have a face isomorphic to 5 simplex whose interior consists of separable states with unique decompositions but with non symmetric ranks we also construct a face isomorphic to the 9 simplex as applications we give answers to questions in the literature chen and djokovic j math phys 54 022201 2013 and chen and djokovic commun math phys 323 241 284 2013 and construct 3 3ppt states of type 9 5 for the qubit qudit cases with d 3 we also show that d 1 dimensional subspaces give rise to faces isomorphic to the d simplices in most cases

optimal indecomposable witnesses without extremality or the spanning property one of the interesting problems of optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable entanglement witness which neither has the spanning property nor is associated with an extremal positive linear map here we answer this question in the negative by examining the extremality of the positive linear maps constructed by qi and hou 2011 j phys a math theor 44 215305

on the geometry of positive maps in matrix algebras ii abstract geometric relations among the identity map the transpose map and the map x 1 n x tr x 1 n in the matrix algebra mn are illustrated providing parametric examples of maps which are not k 1 positive but which satisfy the matrix schwartz inequality of order k for 1 k n 1

construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension given a blow up solution with a characteristic point we refine the blow up behavior first derived by merle and zaag we also refine the geometry of the blow up set near a charac teristic point and show that except may be for one exceptional situation it is never symmetric with the respect to the characteristic point then we show that all blow up modalities predicted by those authors do occur more precisely given any integer k 2 and 0 we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times

a semidefinite hierarchy for containment of spectrahedra a spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program we propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one this approach comes from applying a moment relaxation to a suitable polynomial optimization formulation the hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by helton klep and mccullough as well as by the authors moreover several exactness results for the solitary criterion can be brought forward to the hierarchical approach the hierarchy also applies to the equivalent question of checking whether a map between matrix sub spaces is positive in this context the solitary criterion checks whether the map is completely positive and thus our results provide a hierarchy between positivity and complete positivity

on the profile of energy concentration at blow up points for subconformal focusing nonlinear waves we consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it at noncharacteristic points on the singularity surface merle and zaag have identified the rate of blow up of the h1 norm of the solution inside cones that terminate at the singularity we derive bounds that restrict how this h1 energy can be distributed inside such cones our proof relies on new localized estimates obtained using carlemantype inequalities for such nonlinear waves these bound the lp 1 norm in the interior of timelike cones by their h1 norm near the boundary of the cones such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations in the interior of time cones

on the structure of positive maps finite dimensional case a natural and intrinsic characterization of the structure of the set c of positive unital maps is given i e it is shown that c is isometrically isomorphic to the subset d of bp positive density matrices endowed with the geometry given by the norm dual to the grothendieck projective norm the structure of d is determined by the set of its exposed points and finally a characterization of exposed points of d in terms of convex analysis is presented this seems to be an answer to an old open problem characterization of the structure of the set of positive maps studied both in quantum information and operator algebras our arguments are based on the concept of exposed points and links between tensor products and mapping spaces it should be emphasized that the concept of exposed point depends strongly on the geometry hence the role of the norm is crucial

openness of the set of non characteristic points and regularity of the blow up curve for the 1 d semilinear wave equation we consider here the 1 d semilinear wave equation with a power nonlinearity and with no restriction on initial data we first prove a liouville theorem for that equation then we consider a blow up solution its blow up curve x mapsto t x and i 0 subset mathbb r the set of non characteristic points we show that i0 is open and that t x is c1 on i0 all these results fundamentally use our previous result in 19 showing the convergence in selfsimilar variables for x in i 0

high dimensional entanglement in states with positive partial transposition genuine high dimensional entanglement i e the property of having a high schmidt number constitutes an instrumental resource in quantum communication overcoming limitations of low dimensional systems states with a positive partial transpose ppt are generally considered weakly entangled as they can never be distilled into pure entangled states this naturally raises the question of whether high schmidt numbers are possible for ppt states volume estimates suggest that optimal i e linear scaling in the local dimension should be possible albeit without providing insight into the possible slope we provide the first explicit construction of a family of ppt states that achieves linear scaling in the local dimension and we prove that random ppt states typically share this feature our construction also allows us to prove a recent conjecture of chen et al on the existence of ppt states whose schmidt number increases by an arbitrarily large amount upon partial transposition finally we link the schmidt number to entangled sub block matrices of a quantum state we use this connection to prove that quantum states that are either i invariant under partial transposition on the smallest of their two subsystems or ii absolutely ppt cannot have a maximal schmidt number overall our findings shed new light on some fundamental problems in entanglement theory

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

stable blowup for the cubic wave equation in higher dimensions abstract we consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data we prove that there exists an open set of initial data such that the corresponding solution exists in a backward light cone and approaches the ode blowup profile

formulas for calculating the extremum ranks and inertias of a four term quadratic matrix valued function and their applications abstract this paper studies the quadratic matrix valued function x dxax d dxb b x d c through some expansion formulas for ranks and inertias of hermitian matrices where a b c and d are given complex matrices with a and c hermitian x is a variable matrix and denotes the conjugate transpose of a complex matrix we first introduce an algebraic linearization method for studying this matrix valued function and establish a group of explicit formulas for calculating the global maximum and minimum ranks and inertias of this matrix valued function with respect to the variable matrix x we then use these rank and inertia formulas to derive i necessary and sufficient conditions for the matrix equation x 0 to have a solution as well as the four matrix inequalities x 0 in the lowner partial ordering to be feasible respectively ii necessary and sufficient conditions for the four matrix inequalities x 0 in the lowner partial ordering to hold for all matrices x respectively iii the two matrices x and x such that the inequalities x x and x x hold for all matrices x in the lowner partial ordering respectively an application of the quadratic matrix valued function in control theory is also presented

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

wave equations with concentrated nonlinearities in this paper we address the problem of wave dynamics in the presence of concentrated nonlinearities given a vector field v on an open subset of and a discrete set with n elements we define a nonlinear operator v y on which coincides with the free laplacian when restricted to regular functions vanishing at y and which reduces to the usual laplacian with point interactions placed at y when v is linear and represented by a hermitian matrix we then consider the nonlinear wave equation and study the corresponding cauchy problem giving an existence and uniqueness result when v is lipschitz the solution of such a problem is explicitly expressed in terms of the solutions of two cauchy problems one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part the main properties of the solution are given and when y is a singleton the mechanism and details of blow up are studied

analysis of a multilevel inverse iteration procedure for eigenvalue problems we define and analyze a procedure for computing approximate eigenvalues and eigenfunctions for self adjoint elliptic operators using a combination of inverse iteration and a multilevel iterative technique this algorithm achieves the optimal order work estimates typical of multilevel techniques

construction of 3 3 entangled edge states with positive partial transposes we construct a class of 3 3 entangled edge states with positive partial transposes using indecomposable positive linear maps this class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes

a new criterion and a special class of k positive maps abstract we study k positive maps on operators we obtain a new criterion on k positivity in terms of the k numerical range and use it to improve and refine some earlier results on k positive maps related to the study of quantum information science we also consider a special class of positive maps extending the construction of choi on positive maps that are not completely positive some open questions on the decomposability of such maps are answered

generalized choi maps in three dimensional matrix algebra we consider a class of positive linear maps in the three dimensional matrix algebra which are generalizations of the positive linear map constructed by choi in the relation with positive semidefinite biquadratic forms we find conditions for which such maps are completely positive completely copositive decomposable and two positive

maximal rank of extremal marginal tracial states states on the coupled quantum system mn c mn c whose restrictions to each subsystem are the normalized traces are called marginal tracial states we investigate extremal marginal tracial states and compute their maximal rank diagonal marginal tracial states are also considered

solutions blowing up on any given compact set for the energy subcritical wave equation abstract we consider the focusing energy subcritical nonlinear wave equation t t u u u p 1 u in r n n 1 given any compact set k r n we construct finite energy solutions which blow up at t 0 exactly on k the construction is based on an appropriate ansatz the initial ansatz is simply u 0 t x t a x 2 p 1 where a 0 vanishes exactly on k which is a solution of the ode h h p we refine this first ansatz inductively using only ode techniques and taking advantage of the fact that for suitably chosen a space derivatives are negligible with respect to time derivatives we complete the proof by an energy argument and a compactness method

on the computation for blow up solutions of the nonlinear wave equation we consider in this paper the 1 dim nonlinear wave equation frac partial 2 u partial t 2 t x frac partial 2 u partial x 2 t x u 1 alpha t x alpha 0 and its finite difference analogue it is known that the solution of the current equation may become unbounded in finite time a phenomenon which is known as blow up moreover since the nonlinear wave equation enjoys finite speed of propagation even if the solution has become unbounded at certain points the solution continues to exist blows up at later times and forms the so called blow up curve up to the present however numerical approximation for blow up problems gave only the convergence up to the blow up time of vert u t cdot vert infty in fact adaptive time mesh strategy was used to numerically reproduce the phenomenon of finite time blow up nevertheless computation using such schemes can not be extended beyond the blow up time of vert u t cdot vert infty this is a fatal problem when computing blow up solutions for the nonlinear wave equation as a consequence we reconsider a finite difference scheme whose temporal grid size is given uniformly and propose an algorithm with rigorous convergence analysis to numerically reconstruct the blow up curve for the nonlinear wave equation

differentiability of the blow up curve for one dimensional nonlinear wave equations consider the cauchy problem rac artial 2 u artial t 2 umimits i 1 n rac artial 2 u artial x i 2 f u x n athbb r n t 0 ueft x 0 ight feft x ight eft x n athbb r n ight u t eft x 0 ight geft x ight eft x n athbb r n ight with n 3 it is well known that if f u is superlinear i e feft u ight a eft u ight p ifeft u ight o nfty eft a 0 p 1 ight 0 1 then solutions generally blow up in finite time see 2 6

positive linear maps between matrix algebras which fix diagonals abstract we consider a class of positive linar maps from the n dimensional matrix algebra into itself which fix diagonal entries we show that they are expressed by hadamard products and study their decompositions into the sums of completely positive linear maps and completely copositive linear maps in the three dimensional case we show that every positive linear map in this type is decomposable and give an intrinsic characterization for the positivity of these maps when the involving coefficients are real numbers

a discrete phenomenon in the c and analytic regularity of the blow up curve of solutions to the liouville equation in one space dimension we give a complete discussion of the c or analytic regularity of blow up curves for cauchy problems or some mixed problems for the liouville equation in one space dimension in the case of mixed problems the regularity results depend on the boundary condition actually we show the existence of a sequence of boundary conditions for which the regularity of the blow up curve is better than in the general case

distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding we consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements let the bias of a povm be defined as the total variational distance between the outcome distributions for the two states to be distinguished the performance of a set of measurements can then be defined as the ratio of the bias of this povm and the largest bias achievable by any measurements we first provide lower bounds on the performance of various povms acting on a single system such as the isotropic povm and spherical 2 and 4 designs and show how these bounds can lead to certainty relations furthermore we prove lower bounds for several interesting povms acting on multipartite systems such as the set of local povms povms which can be implemented using local operations and classical communication locc separable povms and finally povms for which every bipartition results in a measurement having positive partial transpose ppt in particular our results show that a scheme of terhal et al for hiding data against local operations and classical communication 31 has the best possible dimensional dependence

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

the structural physical approximation conjecture it was conjectured that the structural physical approximation spa of an optimal entanglement witness is separable or equivalently that the spa of an optimal positive map is entanglement breaking this conjecture was disproved first for indecomposable maps and more recently for decomposable maps the arguments in both cases are sketched along with important related results this review includes background material on topics including entanglement witnesses optimality duality of cones decomposability and the statement and motivation for the spa conjecture so that it should be accessible for a broad audience

characterization of partial positive transposition states and measures of entanglement a detailed characterization of partial positive transposition ppt states both in the heisenberg and in the schrodinger picture is given measures of entanglement are defined and discussed in details illustrative examples are provided

the blow up boundary for nonlinear wave equations consider the cauchy problem for a nonlinear wave equation du f u in n space dimensions n x with x continuously difierentiable introduction consider the nonlinear wave equation 0 1 uu u t au f u for x 6 r i 0 with the initial data 0 2 u x 0 x ul x 0 g x for x g rn it is well known that if f u is nonnegative and superlinear then in general a solution cannot exist for all times furthermore if t is the supremum of all times s such that a classical solution exists for all 0 t s then sup w jc t oo if t t

cones with a mapping cone symmetry in the finite dimensional case abstract in the finite dimensional case we present a new approach to the theory of cones with a mapping cone symmetry first introduced by stormer our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the jamiolkowski choi isomorphism is an isometry we consider a slightly modified class of cones although not substantially different from the original mapping cones by stormer using the new approach several known results are proved faster and often in more generality than before for example the dual of a mapping cone turns out to be a mapping cone as well without any additional assumptions the main result of the paper is a characterization of cones with a mapping cone symmetry saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive a similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone our result is proved without the additional assumptions of symmetry and equality between the domain and the target space we show how it gives a number of older results as a corollary including an exemplary application

existence and classification of characteristic points at blow up for a semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension we first show the existence of a blow up solution with a characteristic point then we consider an arbitrary blow up solution u x t the graph x mapsto t x of its blow up points and scr s subset bbb r the set of all characteristic points and show that scr s has an empty interior finally given x 0 in scr s we show that in selfsimilar variables the solution decomposes into a decoupled sum of at least two solitons with alternate signs and that t x forms a corner of angle pi over 2 at x 0

extension of positive maps into b h soient 7b h et 7b k des espaces de hilbert complexes et soit a un sous espace lineaire auto adjoint norme clos d operateurs bornes b 7b k sur 7b k contenant l operateur identite si on note par b a 7b h les applications positives dans l ensemble b a 7b h des applications lineaires bornees de a dans b 7b h on etudie le probleme d extension pour des applications dans b a 7b h aux applications dans b b 7b k 7b h

blow up theorem for semilinear wave equations with non zero initial position abstract one of the features of solutions of semilinear wave equations can be found in blow up results for non compactly supported data in spite of finite propagation speed of the linear wave we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak this was first observed by asakura 1986 2 finding out a critical decay to ensure the global existence of the solution but the blow up result is available only for zero initial position having positive speed in this paper the blow up theorem for non zero initial position by uesaka 2009 22 is extended to higher dimensional case and the assumption on the nonlinear term is relaxed to include an example u p 1 u moreover the critical decay of the initial position is clarified by example

positive maps from irreducibly covariant operators in this paper we discuss positive maps induced by irreducibly covariant linear operators for finite groups the application of group theory methods allows deriving some new results of a different kind in particular a family of necessary conditions for positivity for such objects is derived stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map in the low dimensional cases for the permutation group s 3 and the quaternion group q the necessary and sufficient conditions are given together with the discussion on their decomposability in higher dimensions we present positive maps induced by a three dimensional irreducible representation of the permutation group s 4 and d dimensional representation of the monomial unitary group mu d in the latter case we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction we show that the generalised choi map can be obtained by considered covariant maps induced by the monomial unitary group as an additional result a novel interpretation of the fujiwara algolet conditions for positivity and complete positivity is presented finally in the end a new form of an irreducible representation of the symmetric group s n is constructed allowing us to simplify the form of certain choi jamiolkowski images derived for irreducible representations of the symmetric group

the blow up surface for nonlinear wave equations with small spatial velocity on considere le probleme de cauchy pour u tt e 2 u f u a d dimensions d espace 3 ou f u est superlineaire et non negative la solution explose sur une surface t e x on note par t x la surface d explosion correspondant a v f v on demontre que e x x ce 2 e x x ce 2 dans un voisinage d un point x 0 ou x 0

multigrid methods for nearly singular linear equations and eigenvalue problems the purpose of this paper is to develop a convergence theory for multigrid methods applied to nearly singular linear elliptic partial differential equations of the type produced from a positive definite system by a shift with the identity one of the important aspects of this theory is that it allows such shifts to vary anywhere in the multigrid scheme enabling its application to a wider class of eigenproblem solvers the theory is first applied to a method for computing eigenvalues and eigenvectors that consists of multigrid iterations with zero right hand side and updating the shift from the rayleigh quotient before every cycle it is then applied to the rayleigh quotient multigrid rqmg method which is a more direct multigrid procedure for solving eigenproblems local convergence of the multigrid v cycle and global convergence for a full multigrid version of both methods is obtained

merging of positive maps a construction of various classes of positive maps on matrix algebras abstract for two positive maps i b k i b h i i 1 2 we construct a new linear map b h b k where k k 1 k 2 c h h 1 h 2 c by means of some additional ingredients such as operators and functionals we call it a merging of maps 1 and 2 the properties of this construction are discussed in particular conditions for positivity of as well as for 2 positivity complete positivity optimality and indecomposability are provided in particular we show that for a pair composed of 2 positive and 2 copositive maps there is an indecomposable merging of them one of our main results asserts that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps their canonical merging is an exposed positive map this result provides a wide class of new examples of exposed positive maps as an application new examples of entangled ppt states are described

choi matrices norms and entanglement associated with positive maps on matrix algebras abstract we study positive maps of b k into b h for finite dimensional hilbert spaces k and h our main emphasis is on how choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps special attention will be given to entanglement properties and k positive maps

a nonlinear nonlocal wave equation arising in combustion theory the initial value problem for the equation 2 t2 2 x2 t t 2 t 2 x2 et 1 is considered it is proved that under some restrictions on the initial data there is a curve denoted by t x which is positive lipschitz continuous and satisfies x 0 the case of 1 is also studied the solution for 1 blows up on t x and it is proved that under certain conditions the solutions for 1 converge to the one for 1 as 1 and lim inf 1 x x

blow up of the su 2 c yang mills fields it is remarked that if the gauge group is not compact finite time blowing up yang mills fields over the minkowski spaces may exist the example taken is g su 2 c it is shown that blow up can occur in an arbitrarily small time interval when the initial data of the fields are suitably chosen

on a characterization of positive maps drawing on results of choi stormer and woronowicz we present a nearly complete characterization of certain important classes of positive maps in particular we construct a general class of positive linear maps acting between two matrix algebras b h and b k where h and k are finite dimensional hilbert spaces it turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on b h k subsequently the relation between entanglements and positive maps is described finally a new characterization of the cone b h b k is given

decomposability of linear maps under tensor powers both completely positive and completely copositive maps stay decomposable under tensor powers i e under tensoring the linear map with itself but are there other examples of maps with this property we show that this is not the case any decomposable map that is neither completely positive nor completely copositive will lose decomposability eventually after taking enough tensor powers moreover we establish explicit bounds to quantify when this happens to prove these results we use a symmetrization technique from the theory of entanglement distillation and analyze when certain symmetric maps become non decomposable after taking tensor powers finally we apply our results to construct new examples of non decomposable positive maps and establish a connection to the positive partial transpose squared conjecture

squashed entanglement an additive entanglement measure in this paper we present a new entanglement monotone for bipartite quantum states its definition is inspired by the so called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions we derive certain properties of the new measure which we call squashed entanglement it is a lower bound on entanglement of formation and an upper bound on distillable entanglement furthermore it is convex additive on tensor products and superadditive in general continuity in the state is the only property of our entanglement measure which we cannot provide a proof for we present some evidence however that our quantity has this property the strongest indication being a conjectured fannes type inequality for the conditional von neumann entropy this inequality is proved in the classical case

separable states and positive maps abstract using the natural duality between linear functionals on tensor products of c algebras with the trace class operators on a hilbert space h and linear maps of the c algebra into b h we study the relationship between separability entanglement and the peres condition of states and positivity properties of the linear maps

generation of mapping cones from small sets we answer in the affirmative a recently posed question that asked if there exists an untypical convex mapping cone i e one that does not arise from the transpose map and the cones of k positive and k superpositive maps we explicitly construct such a cone based on atomic positive maps our general technique is to consider the smallest convex mapping cone generated by a single map and we derive several results on such mapping cones we use this technique to also present several other examples of untypical mapping cones including a family of cones generated by spin factors we also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality

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

a class of atomic positive linear maps in matrix algebras we find a large class of atomic positive linear maps between n dimensional matrix algebras and study their properties in connection with complete positivity 2 positivity and schwarz inequality

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

all 2 positive linear maps from m3 to m3 are decomposable abstract following an idea of choi we obtain a decomposition theorem for k positive linear maps from m m c to m n c where 2 k min m n as a consequence we give an affirmative answer to kye s conjecture also solved independently by choi that every 2 positive linear map from m 3 c to m 3 c is decomposable

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

positive groups on hn are completely positive we prove that an operator generates a positive group on the real space of real or complex hermitian matrices if and only if it is a lyapunov operator in particular it follows that every group of positive operators in fact is a group of completely positive operators

a multilevel variational method for au bu on composite grids abstract there are two existing general approaches to the numerical solution of differential eigen problems by multigrid one which uses a linear multigrid solver for the inner loop of a linearization technique e g inverse iteration and the other which integrates multigrid directly into the problem solution e g fas the present paper developes a third multigridlike approach rqmg that appeals directly to the variational problem of minimizing the rayleigh quotient applying easily to both global and local grid implementations numerical results are presented for a simplified single group diffusion problem

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

the blow up rate for a non scaling invariant semilinear wave equations abstract we consider the semilinear wave equation 1 t 2 u u f u x t r n 0 t with f u u p 1 u log a 2 u 2 where p 1 and a r we show an upper bound for any blow up solution of 1 then in the one space dimensional case using this estimate and the logarithmic property we prove that the exact blow up rate of any singular solution of 1 is given by the ode solution associated with 1 namely u u p 1 u log a 2 u 2 unlike the pure power case g u u p 1 u the difficulties here are due to the fact that equation 1 is not scale invariant

quantum states with a positive partial transpose are useful for metrology we show that multipartite quantum states that have a positive partial transpose with respect to all bipartitions of the particles can outperform separable states in linear interferometers we introduce a powerful iterative method to find such states we present some examples for multipartite states and examine the scaling of the precision with the particle number some bipartite examples are also shown that possess an entanglement very robust to noise we also discuss the relation of metrological usefulness to bell inequality violation we find that quantum states that do not violate any bell inequality can outperform separable states metrologically we present such states with a positive partial transpose as well as with a nonpositive partial transpose

there is no direct generalization of positive partial transpose criterion to the three by three case we show that there cannot exist a straightforward generalization of the famous positive partial transpose criterion to three by three systems we call straightforward generalizations that use a finite set of positive maps and arbitrary local rotations of the tested two partite state in particular we show that a family of extreme positive maps discussed in a paper by ha and kye open syst inf dyn 18 323 337 2011 cannot be replaced by a finite set of witnesses in the task of entanglement detection in three by three systems in a more mathematically elegant parlance our result says that the convex cone of positive maps of the set of three dimensional matrices into itself is not finitely generated as a mapping cone

on the structure of positive maps ii low dimensional matrix algebras we use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on m2 c and m3 c our main tool stems from classical grothendieck theorem on tensor product of banach spaces and is an older and more general version of choi jamiolkowski isomorphism between positive maps and block positive choi matrices it takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps in this setting we show that in m2 c case a large class of nice positive maps can be generated from the small set of maps represented by self adjoint unitaries 2px with x maximally entangled vector and p 1 with p rank 1 projector we indicate problems with passing this result to m3 c case among similarities in both m2 c and m3 c cases any unital positive map represented by self adjoint unitary is unitarily equivalent to the transposition map consequently

the structural physical approximations and optimal entanglement witnesses we introduce the notions of positive and copositive types for entanglement witnesses depending on the distance to the positive part and copositive part an entanglement witness w is of positive type if and only if its partial transpose w is of copositive type we show that if the structural physical approximation spa of w is separable then w should be of copositive type and the spa of w is never separable unless w is of both positive and copositive types this shows that the spa conjecture is meaningful only for those of copositive type we provide examples to show that the spa conjecture fails even for the case of copositive types

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

classification of bi qutrit positive partial transpose entangled edge states by their ranks we construct 3 3 ppt entangled edge states with maximal ranks to complete the classification of 3 3 ppt entangled edge states by their types the ranks of the states and their partial transposes are 8 and 6 respectively these examples also disprove claims in the literature

the method of subspace corrections abstract this paper gives an overview for the method of subspace corrections the method is first motivated by a discussion on the local behavior of high frequency components in a solution to an elliptic problem a simple domain decomposition method is discussed as an illustrative example and multigrid methods are discussed in more detail brief discussions are also given to some non linear examples including eigenvalue problems obstacle problems and liquid crystal modelings the relationship between the method of subspace correction and the method of alternating projects is observed and discussed

incomplete blowup of solutions of quasilinear hyperbolic balance laws we study incomplete blowup of entropy solutions to first order quasilinear hyperbolic balance laws a general procedure to continue solutions beyond the blowup time which makes use of monotonicity methods is given the continuations thus obtained are possibly unbounded and satisfy suitable generalized entropy and rankine hugoniot conditions then the uniqueness of continuations satisfying such conditions is proved

a class of positive atomic maps we construct a new class of positive indecomposable maps in the algebra of d d complex matrices these maps are characterized by the weakest positivity property and for this reason they are called atomic this class provides a new rich family of atomic entanglement witnesses which define an important tool for investigating quantum entanglement it turns out that they are able to detect states with the weakest quantum entanglement