local channels preserving the states without measurement induced nonlocality measurement induced nonlocality is a measure of nonlocality introduced recently by luo and fu we present here sufficient and necessary conditions for a quantum state for which this quantity is equal to zero furthermore it is shown that for such a state ab with any local channel acting on ha cannot create measurement induced nonlocality if and only if either it is a completely contractive channel or it is a nontrivial isotropic channel for the qubit case this property is an additional characteristic of the completely contractive channel or the commutativity preserving unital channel

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

entanglement breaking channels with general outcome operator algebras a unit preserving and completely positive linear map or a channel a ain between c algebras a and ain is called entanglement breaking eb if idb is a separable state for any c algebra b and any state on the injective c tensor product ain b in this paper we establish the equivalence of the following conditions for a channel with a quantum input space and with a general outcome c algebra generalizing the known results in finite dimensions i is eb ii has a measurement prepare form holevo form iii n copies of are compatible for all 2 n iv countably infinite copies of are compatible by using this equivalence we also show that the set of randomization equivalence classes of normal eb channels with a fixed input von neumann algebra is upper and lower dedekind closed i e the supremum or infimum of any randomization increasing or decreasing net of eb channels is also eb as an example we construct an injective normal eb channel with an arbitrary outcome operator algebra m acting on an infinite dimensional separable hilbert space by using the coherent states and the bargmann measure

orthogonal polynomials and measures on the unit circle the geronimus transformations in this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle this perturbation is the inverse of the christoffel transformation and is called the geronimus transformation we study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated hessenberg matrices finally we show an example of this kind of transformation

a class of separable quantum states we present a sufficient condition for a bipartite state to be separable via the strong positive partial transposition structure introduced in chru ci ski et al 2008 phys rev a 77 022113 from which we obtain a class of separable states in particular we prove that any classical quantum state is a state of our class but not vice versa both finite and infinite dimensional systems are considered

on orthogonal polynomials transformed by the qr algorithm abstract in this paper we study two equivalent ways of transforming a system of orthogonal polynomials one is the so called lifting of the recurrence relation associated with an orthogonal polynomial system ops and the other consists of applying the qr algorithm to the jacobi matrix of the ops these transformations are useful for among other things identifying the sobolev orthogonal polynomials with respect to certain measures which were previously investigated by iserles et al 1991 as a by product we obtain a new class of modified lommel polynomials

ramanujan s series for 1 π arising from his cubic and quartic theories of elliptic functions using certain representations for eisenstein series we derive several of ramanujan s series for 1 arising from his cubic and quartic theories of elliptic functions

eventually entanglement breaking maps we analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves this means that the choi matrix of the iterated linear map becomes separable in the tensor product space if a linear map becomes entanglement breaking after finitely many iterations we say that the map has a finite index of separability in particular we show that every unital positive partial transpose ppt channel has a finite index of separability and that the class of unital channels that have a finite index of separability is a dense subset of the unital channels we construct concrete examples of maps which are not ppt but have a finite index of separability we prove that there is a large class of unital channels that are asymptotically entanglement breaking our work is motivated by christandl s ppt squared conjecture this conjecture states that every ppt channel when composed with itself becomes entanglement breaking

calculation of gauss quadratures with multiple free and fixed knots algorithms are derived for the evaluation of gauss knots in the presence of fixed knots by modification of the jacobi matrix for the weight function of the integral simple gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process based on the merging of free and fixed knots of quadratic convergence is presented for multiple gauss knots the procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple gauss knots a new characterization of simple gauss knots as a solution of a partial inverse eigenvalue problem is derived

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

finding decompositions of a class of separable states abstract we consider the class of separable states which admit a decomposition i a i b i with the b i s having independent images we give a simple intrinsic characterization of this class of states given a density matrix in this class we construct such a decomposition which can be chosen so that the a i s are distinct with unit trace and then the decomposition is unique we relate this to the facial structure of the set of separable states the states investigated include a class that corresponds under the choi jamiolkowski isomorphism to the quantum channels called quantum classical and classical quantum by holevo

class number three ramanujan type series for 1 p abstract there is a class of remarkable series for 1 of the form c 3 n 0 a nb c 3n 6n 3n n 3 where a b c are certain algebraic numbers these were first examined by ramanujan these examples arise by computing singular invariants for j and the constants involved will have degree equal to the class number of the associated imaginary quadratic field our intention is compute all such class number three examples the largest such example with discriminant 907 adds 37 additional digit accuracy per term a class number four example with discriminant 1555 gives 50 digits per term

multiple geronimus transformations abstract we consider multiple geronimus transformations and show that they lead to discrete non diagonal sobolev type inner products moreover it is shown that every discrete sobolev inner product can be obtained as a multiple geronimus transformation a connection with geronimus spectral transformations for matrix orthogonal polynomials is also considered

spectral properties of entanglement witnesses entanglement witnesses are observables which when measured detect entanglement in a measured composed system it is shown what kind of relations between eigenvectors of an observable should be fulfilled to allow an observable to be an entanglement witness some restrictions on the signature of entanglement witnesses based on an algebraic geometrical theorem will be given the set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive but not completely positive a translation of the results to the language of positive maps is also given the properties of entanglement witnesses and positive maps are expressed as special cases of general theorems for k schmidt witnesses and k positive maps the results are therefore presented in a general framework

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

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

on the existence of darboux transformations for banded matrices the darboux transformations provide joint with other applications a method for obtaining solutions of some integrable systems in this work the concepts of darboux factorization and darboux transformations for arbitrary hessenberg banded matrices are analyzed specifically the existence of this kind of factorization is studied and some sufficient conditions for the uniqueness are determined

domb s numbers and ramanujan sato type series for 1 π abstract in this article we construct a general series for 1 we indicate that ramanujan s 1 series are all special cases of this general series and we end the paper with a new class of 1 series our work is motivated by series recently discovered by takeshi sato

series expansions for 1 πm and πm abstract in terms of the telescoping method we establish a general identity then it is used to derive several interesting series expansions for 1 m and m with m n

spectral transformations for hermitian toeplitz matrices in this paper we deal with some perturbations of probability measures supported on the unit circle as well as in a more general framework with hermitian linear functionals we focus our attention in the hessenberg matrix associated with the multiplication operator in terms of an orthogonal basis in the linear space of polynomials with complex coefficients the lu and qr factorizations of such a matrix are introduced then the connection between the above mentioned perturbations and such factorizations is presented

constructing positive maps from block matrices positive maps are useful for detecting entanglement in quantum information theory any entangled state can be detected by some positive map in this paper the relation between positive block matrices and completely positive trace preserving maps is characterized consequently a new method for constructing decomposable maps from positive block matrices is derived in addition a method for constructing positive but not completely positive maps from hermitian block matrices is also obtained

geronimus transformations of bivariate linear functionals abstract given a linear functional u in the linear space of polynomials in two variables with real coefficients and a polynomial x y in this contribution we deal with geronimus transformations of u i e those linear functionals v such that u x y v the connection formulas between the sequences of bivariate orthogonal polynomials with respect to u and v are given a matrix interpretation of such transformations by using lu and ul factorizations of the block jacobi matrices associated with such polynomials is given finally some illustrative examples of geronimus transformations of weight function supported in domains of r 2 are discussed

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

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

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

a characterization of positive linear maps and criteria of entanglement for quantum states let h and k be finite or infinite dimensional complex hilbert spaces a characterization of positive completely bounded normal linear maps from into is given which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras this characterization is then applied to give a representation of quantum channels operations between infinite dimensional systems a necessary and sufficient criterion of separability is given which shows that a state on h k is separable if and only if i 0 for all positive finite rank elementary operators examples of ncp and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the ppt criterion and the realignment criterion

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

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

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

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

spectral conditions for positive maps we provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions this construction generalizes the celebrated choi example of a map which is positive but not completely positive it is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements such maps provide basic tools to study quantum entanglement in multipartite systems

tensor powers of 2 positive maps by studying tensor powers n of a positive map of b k into b h for finite dimensional hilbert spaces k and h we give conditions for a map such that n 2 positive for all n to be nondistillable and completely positive

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

darboux transformation and perturbation of linear functionals abstract let l be a quasi definite linear functional defined on the linear space of polynomials with real coefficients in the literature three canonical transformations of this functional are studied xl l c x and 1 x l c x where x denotes the linear functional x xk k 0 and k 0 is the kronecker symbol let us consider the sequence of monic polynomials orthogonal with respect to l this sequence satisfies a three term recurrence relation whose coefficients are the entries of the so called monic jacobi matrix in this paper we show how to find the monic jacobi matrix associated with the three canonical perturbations in terms of the monic jacobi matrix associated with l the main tools are darboux transformations in the case that the lu factorization of the monic jacobi matrix associated with xl does not exist and darboux transformation does not work we show how to obtain the monic jacobi matrix associated with x 2 l as a limit case we also study perturbations of the functional l that are obtained by combining the canonical cases finally we present explicit algebraic relations between the polynomials orthogonal with respect to l and orthogonal with respect to the perturbed functionals

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

lpp elementary operator criterion of full separability for states in multipartite quantum systems let h1 h2 hn k n 2 be complex hilbert spaces of finite or infinite dimension a necessary and sufficient criterion of full separability for multipartite states is established which says that a state in h1 h2 hn k is fully separable if and only if id 0 for all finite rank lpp elementary operators from into furthermore a characterization of lpp elementary operators and a method of constructing lpp elementary operators by positive elementary operators are given an example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states

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

spectral properties of tensor products of channels abstract we investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps also known as quantum channels acting on matrix algebras this leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras we show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains consequently we fully describe the fixed points and peripheral eigen operators of the tensor product of channels through a structure theorem of maximal unital proper subalgebras mupsa of a matrix algebra we provide a non trivial upper bound of the recently introduced multiplicative index of a unital channel this bound gives a criteria on when a channel cannot be factored into a product of two different channels we construct examples of channels which cannot be realized as a tensor product of two channels in any way with these techniques and results we found some applications in quantum information theory

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

π formulas with free parameters abstract in terms of the hypergeometric method we establish ten general formulas with free parameters which include several known results as special cases

facial structures for unital positive linear maps in the two dimensional matrix algebra abstract we completely determine the lattice structure for the faces of the convex set of all unital positive linear maps between the 2 2 matrix algebras

continuous symmetrized sobolev inner products of order n i abstract given a symmetrized sobolev inner product of order n the corresponding sequence of monic orthogonal polynomials q n satisfies that q 2 n x p n x 2 q 2 n 1 x x r n x 2 for certain sequences of monic polynomials p n and r n in this paper we deduce the integral representation of the inner products such that p n and r n are the corresponding sequences of orthogonal polynomials moreover we state a relation between both inner products which extends the classical result for symmetric linear functionals

a summation formula and ramanujan type series abstract using some properties of the general rising shifted factorial and the gamma function we derive a variant form of dougall s f 4 5 summation for the classical hypergeometric functions this variant form allows us to derive easily many ramanujan type series for 1 and ramanujan type series for some other constants

spectral transformations self similar reductions and orthogonal polynomials we study spectral transformations in the theory of orthogonal polynomials which are similar to darboux transformations for the schr dinger equation linear transformations of the stieltjes function with coefficients that are rational in the argument are constructed as iterations of the christoffel and geronimus transformations we describe a characteristic property of semi classical orthogonal polynomials scop on the uniform and the exponential lattice namely that all these polynomials can be obtained through simple quasi periodic and q periodic self similar closures of the chain of linear spectral transformations in the self similar setting a characterization of the laguerre hahn polynomials on linear and q linear lattices is obtained by considering rational transformations of the stieltjes function generated by transitions to the associated polynomials

hypergeometric series summations and π formulas abstract by two slater s hypergeometric series identities we establish two general summation formulas with six free parameters specializing certain parameters in these formulas we obtain a list of ramanujan type series formulas for 2 and 3

a family of indecomposable positive linear maps based on entangled quantum states abstract we introduce a new family of indecomposable positive linear maps based on entangled quantum states central to our construction is the notion of an unextendible product basis the construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension

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

indecomposable positive maps in low dimensional matrix algebras abstract we classify a series of positive maps in low dimensional matrix algebras with respect to the degree of indecomposability and provide stronger indecomposable maps in m n we also present a basic result about the atomic property

geronimus spectral transforms and measures on the complex plane we analyze a special spectral transform of a measure m supported on a compact subset c of the complex plane a perturbation m 1 of m is said to be a geronimus spectral transform if d m 1 d m z a 2 where a c we focus our attention in the analysis of the hessenberg matrix associated with the multiplication operator in terms of the orthogonal polynomial basis defined by the measure m 1

polynomial perturbations of bilinear functionals and hessenberg matrices this paper deals with symmetric and non symmetric polynomial perturbations of symmetric quasi definite bilinear functionals we establish a relation between the hessenberg matrices associated with the initial and the perturbed functionals using lu and qr factorizations moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals

positivity of linear maps under tensor powers we investigate linear maps between matrix algebras that remain positive under tensor powers i e under tensoring with n copies of themselves completely positive and completely co positive maps are trivial examples of this kind we show that for every n there exist non trivial maps with this property and that for two dimensional hilbert spaces there is no non trivial map for which this holds for all n for higher dimensions we reduce the existence question of such non trivial tensor stable positive maps to a one parameter family of maps and show that an affirmative answer would imply the existence of non positive partial transpose bound entanglement as an application we show that any tensor stable positive map that is not completely positive yields an upper bound on the quantum channel capacity which for the transposition map gives the well known cb norm bound we furthermore show that the latter is an upper bound even for the local operations and classical communications assisted quantum capacity and that moreover it is a strong converse rate for this task