supersymmetric transformations for coupled channels with threshold differences the asymptotic behaviour of the superpotential of general susy transformations for a coupled channel hamiltonian with different thresholds is analysed it is shown that asymptotically the superpotential can tend to a diagonal matrix with an arbitrary number of positive and negative entries depending on the choice of the factorization solution the transformation of the jost matrix is generalized to non conservative susy transformations introduced in sparenberg et al 2006 j phys a math gen 39 l639 applied to the zero initial potential the method permits the construction of superpartners with a nontrivially coupled jost matrix illustrations are given for two and three channel cases

on effective computation of fundamental units ii introduction in part i of this paper we described the theoretical improvements of our method of computing fundamental units in algebraic number fields which we achieved in the last few years those improvements were of considerable influence on the corresponding computer program since its first 1976 2 implementation it was completely rewritten and changed in so many details that a new presentation cannot be avoided in sections 1 and 2 of this paper we describe the applications of part i 6 to the algorithm for constructing fundamental units section 3 contains a complete list of numerical examples concerning algebraic number fields of small degree and small absolute discriminant besides the fundamental units the tables contain much information about fields of degree five and six which was so far unknown the determination of those fields and their subfields is described in 3 besides the fundamental units we also listed the order of the torsion subgroup tuf of the unit group u whenever it is different from 2 it was computed by the methods of section 2 of 5 all computations were carried out on the control data cyber 76 of the computer center of the university of cologne

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

positive finite rank elementary operators and characterizing entanglement of states in this paper some indecomposable positive finite rank elementary operators of order n n are constructed this allows us to give a simple necessary and sufficient criterion for the separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order 2 2 and obtain some new mixed entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion

optimal entanglement witnesses from generalized reduction and robertson maps we provide a generalization of the reduction and robertson positive maps in matrix algebras they give rise to a new class of optimal entanglement witnesses their structural physical approximation is analyzed as a byproduct we provide new examples of ppt positive partial transpose entangled states

eigenphase preserving two channel susy transformations we propose a new kind of supersymmetric susy transformation in the case of the two channel scattering problem with equal thresholds for partial waves of the same parity this two fold transformation is based on two imaginary factorization energies with opposite signs and with mutually conjugated factorization solutions we call it an eigenphase preserving susy transformation as it relates two hamiltonians the scattering matrices of which have identical eigenphase shifts in contrast to known phase equivalent transformations the mixing parameter is modified by the eigenphase preserving transformation

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

on the computation of a table of complex cubic fields with discriminant d 10 6 a method for finding all the nonisomorphic complex cubic fields with discriminant d 106 is described three different methods were used to find the class number of each of these fields the speed of these techniques is discussed and several tables illustrating the computational results are presented these include tables of the distribution of the fields and the class numbers and the class group structures of these fields

detecting quantum entanglement we review the criteria for separability and quantum entanglement both in a bipartite as well as a multipartite setting we discuss bell inequalities entanglement witnesses entropic inequalities bound entanglement and several features of multipartite entanglement we indicate how these criteria bear on the experimental detection of quantum entanglement

singular matrix darboux transformations in the inverse scattering method singular darboux transformations in contrast to the conventional ones have a singular matrix as a coefficient before the derivative we incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain a determinant representation of the kohlhoff von geramb solution to the marchenko equation is given

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

coupled channel scattering and separation of coupled differential equations by generalized darboux transformations general methods for separation of coupled differential equations are applied in coupled channel scattering theory with appropriate asymptotic conditions it is shown that the problems with thresholds cannot be separated finally it is pointed out that separation of the coupled channel s wave problem without thresholds can be realized by the use of generalized darboux transformations

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

a series of absolutely indecomposable positive maps in matrix algebras abstract we provide examples of positive maps in m n c n 4 which cannot be decomposed into a sum of a 2 positive map and a 2 copositive map

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

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

on the equivalence of the integral and the differential exact solution generation methods for the one dimensional schrodinger equation it is demonstrated that the integral exact solution generation methods for the one dimensional schrodinger equation based on the gelfand levitan formalism are in some cases equivalent to the differential ones based on the n order darboux transformation some new exact solvable potentials are generated from the effective coulomb potential and the harmonic oscillator potential a new form of n soliton potential i e reflectionless potential with n discrete energy levels disposed in a desirable manner based on an explicit expression for an n order wronski determinant constructed from hyperbolic functions and its orthonormal discrete spectrum eigenfunctions are given

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

separable multipartite mixed states operational asymptotically necessary and sufficient conditions we introduce an operational procedure to determine with arbitrary probability and accuracy optimal entanglement witnesses for every multipartite entangled state this method provides an operational criterion for separability which is asymptotically necessary and sufficient our results are also generalized to detect all different types of multipartite entanglement

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

on structural physical approximations and entanglement breaking maps very recently a conjecture saying that the so called structural physical approximations spas to optimal positive maps optimal entanglement witnesses give entanglement breaking eb maps separable states has been posed korbicz et al 2008 phys rev a 78 062105 the main purpose of this contribution is to explore this subject first we extend the set of entanglement witnesses supporting the conjecture then we ask whether spas constructed from other than the depolarizing channel maps also lead to eb maps and show that in general this is not the case on the other hand we prove an interesting fact that for any positive map there exists an eb channel such that the spa of constructed with the aid of is again an eb channel finally we ask similar questions in the case of continuous variable systems we provide a simple way of constructing spa and prove that in the case of the transposition map it gives the eb channel

distinguishing separable and entangled states we show how to design families of operational criteria that distinguish entangled from separable quantum states the simplest of these tests corresponds to the well known peres horodecki positive partial transpose ppt criterion and the more complicated tests are strictly stronger the new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs we successfully applied the results to many low dimensional states from the literature where the ppt test fails as a by product of the criteria we provide an explicit construction of the corresponding entanglement witnesses

on the computation of the class number of an algebraic number field it is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field f the method assumes the truth of the generalized riemann hypothesis in order to estimate the residue of the dedekind zeta function of f at s 1 sufficiently well that h can be determined unambiguously

a class of exposed indecomposable positive maps exposed positive maps in matrix algebras define a dense subset of extremal maps we provide a class of indecomposable positive maps in the algebra of 2n 2n complex matrices with n 2 it is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states

single and coupled channel radial inverse scattering with supersymmetric transformations the present status of the three dimensional inverse scattering method with supersymmetric transformations is reviewed for the coupled channel case we first revisit in a pedagogical way the single channel case where the supersymmetric approach is shown to provide a complete efficient and elegant solution to the inverse scattering problem for the radial schrodinger equation with short range interactions a special emphasis is put on the differences between conservative and non conservative transformations i e transformations that do or do not conserve the behaviour of solutions of the radial schrodinger equation at the origin in particular we show that for the zero initial potential a non conservative transformation is always equivalent to a pair of conservative transformations these single channel results are illustrated on the inversion of the neutron proton triplet eigenphase shifts for the s and d waves we then summarize and extend our previous works on the coupled channel case i e on systems of coupled radial schrodinger equations and stress remaining difficulties and open questions of this problem by putting it in perspective with the single channel case we mostly concentrate on two channel examples to illustrate general principles while keeping mathematics as simple as possible in particular we discuss the important difference between the equal threshold and different threshold problems for equal thresholds conservative transformations can provide non diagonal jost and scattering matrices iterations of such transformations in the two channel case are studied and shown to lead to practical algorithms for inversion a convenient particular technique where the mixing parameter can be fitted without modifying the eigenphases is developed with iterations of pairs of conjugate transformations this technique is applied to the neutron proton triplet s d scattering matrix for which exactly solvable matrix potential models are constructed for different thresholds conservative transformations do not seem to be able to provide a non trivial coupling between channels in contrast a single non conservative transformation can generate coupled channel potentials starting from the zero potential and is a promising first step towards a full solution to the coupled channel inverse problem with threshold differences

a note on class number one in certain real quadratic and pure cubic fields let p be any odd prime and let h p be the class number of the real quadratic field 2 f p the results of a computer run to determine the density of the field fp with h p 1 and p 108 are presented similar results are given for pure cubic fields 2 fp with p 106 1 introduction let p be any odd prime and let h h p be the class number of the quadratic field 2 p it is well known that h p is odd but the problem of how frequently h p 1 although it goes back to gauss is still unsolved if we let it a b x denote the number of primes of the form a bk less than or equal to x and f a b x denote the number of these primes p for which h p 1 we find see lakein 5 from a large table of kuroda 4 that r l 4 x l 4 x 77 l 4 x 7765 for x 2776817 that is over 77 of all the primes 1 mod4 up to 2776817 have h p 1 indeed according to the recent heuristic results of cohen and lenstra see cohen 1 we would expect that h p 1 with probability 75446 in order to test this heuristic we developed and ran a computer program which determined whether or not h p 1 for all primes p 108 in the next section of this note we give the results of this computer run in the following section we present some data for certain pure cubic fields p with p 106 2 the quadratic case in order to find h p we made use of the well known formula

how to construct indecomposable entanglement witnesses we present a very simple method for constructing indecomposable entanglement witnesses out of a given pair an entanglement witness w and the corresponding state detected by w this method may be used to produce new classes of atomic witnesses which are able to detect the weakest quantum entanglement actually it works perfectly in the multipartite case too moreover this method provides a powerful tool for constructing new examples of bound entangled states

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

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

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