comments on hastings additivity counterexamples hastings 12 recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy in this paper we present an expanded version of hastings proof in addition to a careful elucidation of the details of the proof we also present bounds for the minimal dimensions needed to obtain a counterexample

on weyl channels being covariant with respect to the maximum commutative group of unitaries we investigate the weyl channels being covariant with respect to the maximum commutative group of unitary operators this class includes the quantum depolarizing channel and the two pauli channel as well then we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the two pauli channel in the qubit case

the clifford algebra of an infinite dimensional space the clifford algebra of a real infinite dimensional vector space is constructed this algebra contains all finite and infinite dimensional clifford algebras over the real and complex numbers some of the mathematical properties of these algebras are discussed the automorphisms and involutions are specifically addressed the special case with metric g 1 2 follows as a corollary to the more general results

multi stage robust chinese remainder theorem it is well known that the traditional chinese remainder theorem crt is not robust in the sense that a small error in a remainder may cause a large reconstruction error a robust crt was recently proposed for a special case when the greatest common divisor gcd of all the moduli is more than 1 and the remaining integers factorized by the gcd are co prime it basically says that the reconstruction error is upper bounded by the remainder error level tau if tau is smaller than a quarter of the gcd of all the moduli in this paper we consider the robust reconstruction problem for a general set of moduli we first present a necessary and sufficient condition on the remainder errors with a general set of moduli and also a corresponding robust reconstruction method this can be thought of as a single stage robust crt we then propose a two stage robust crt by grouping the moduli into several groups as follows first the single stage robust crt is applied to each group then with these robust reconstructions from all the groups the single stage robust crt is applied again across the groups this is easily generalized to multi stage robust crt with this two stage robust crt the robust reconstruction holds even when the remainder error level tau is above the quarter of the gcd of all the moduli and an algorithm on how to group a set of moduli for a better reconstruction robustness is proposed in some special cases

harmonic operators the dual perspective the study of harmonic functions on a locally compactgroupghas recently been transferred to a non commutative setting in two different directions chu and lau replaced the algebra l g by the group von neumann algebra vn g and the convolution action of a probability measure on l g by the canonical action of a positive definite function on vn g on the other hand jaworski and the first author replaced l g by b l 2 g to which the convolution action by can be extended in a natural way we establish a link between both approaches the action of on vn g can be extended to b l 2 g we study the corresponding space h

the fidelity of density operators in an operator algebraic framework josza s definition of fidelity r jozsa j mod opt 41 12 2315 2323 1994 for a pair of mixed quantum states is studied in the context of two types of operator algebras the first setting is mainly algebraic in that it involves unital c algebras a that possess a faithful trace functional in this context the role of quantum states that is density operators in the classical quantum mechanical framework is assumed by positive elements a for which 1 the second setting is more operator theoretic by fixing a faithful normal semifinite trace on a semifinite von neumann algebra m we define and consider the fidelity of pairs of positive operators in m of unit trace the main results of this paper address monotonicity and preservation of fidelity under the action of certain trace preserving positive linear maps of a or of the predual m our results in the von neumann algebra setting are novel in that we focus on the schrodinger picture rather than the heisenberg picture and they a

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

algebraic spinor solution to the kahler equation the kahler equation for an inhomogeneous differential form is analyzed in some detail algebraic spinorial solution to the kahler equation lying in a minimal left ideal characterized by a certain idempotent projector has been examined

noncommutative poisson boundaries of unital quantum operations in this paper poisson boundaries of unital quantum operations also called markov operators are investigated in the case of unital quantum channels compact operators belonging to poisson boundaries are characterized using the characterization of amenable groups by the injectivity of their von neumann algebras we will answer negatively some conjectures appearing in the work of arias et al fixed points of quantum operations j math phys 43 5872 2002 about injectivity of the commuting algebra of the kraus operators of unital quantum operations and their injective envelopes

on a positive semidefinite relaxation of the cut polytope abstract we study the convex set l n defined by l n z x x xij a positive semidefinite n n matrix xii 1 for all i we describe several geometric properties of l n in particular we show that l n has 2n 1 vertices which are its rank one matrices corresponding to all bipartitions of the set 1 2 n our main motivation for investigating the convex set l n comes from combinatorial optimization namely from approximating the max cut problem an important property of l n is that due to the positive semidefinite constraints one can optimize over it in polynomial time on the other hand l n still inherits the difficult structure of the underlying combinatorial problem in particular it is np hard to decide whether the optimum of the problem min tr cx x ln is reached at a vertex this result follows from the complete characterization of the matrices c of the form c bbt for some vector b for which the optimum of the above program is reached at a vertex

decoding of redundant residue polynomial codes using euclid s algorithm a decoding method is proposed for the redundant residue polynomial codes a class that includes reed solomon codes the method is based on properties of euclid s algorithm the advantage of the method is that the computation of both the error locator polynomial and the error evaluator polynomial needed in conventional decoding methods can be avoided the method is suitable for decoding concatenated codes whose outer codes are redundant residue polynomial codes since they are easily decoded by ignoring erasures detected in the inner codes

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

fixed points of quantum operations quantum operations frequently occur in quantum measurement theory quantum probability quantum computation and quantum information theory if an operator a is invariant under a quantum operation we call a a fixed point physically the fixed points are the operators that are not disturbed by the action of our main purpose is to answer the following question if a is a fixed point is a compatible with the operation elements of we shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von neumann algebras

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

a new residue arithmetic error correction scheme automatic detection and correction of errors in the residue number system involves the conversion of residue representations to integers and base extension the residue number system is generally restricted to moduli that are pairwise relatively prime in this paper we consider error detection and correction using a moduli set with common factors a method to construct a moduli set that leads to simplified error detection and correction is presented error detection can now be performed by computing residues in parallel error correction does not involve base extension any more it is also shown that removing all restrictions on the moduli set leads to more complex error detection correction algorithms

error correction in polynomial remainder codes with non pairwise coprime moduli and robust chinese remainder theorem for polynomials this paper investigates polynomial remainder codes with non pairwise coprime moduli we first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small namely the robust chinese remainder theorem crt for polynomials it basically says that a polynomial can be reconstructed from its erroneous residues such that the degree of the reconstruction error is upper bounded by tau whenever the degrees of all residue errors are upper bounded by tau where a sufficient condition for tau and a reconstruction algorithm are obtained by relaxing the constraint that all residue errors have small degrees another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues we finally obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result some errors with small degrees can be also corrected in the residues with this newly obtained result improvements in uncorrected error probability and burst error correction capability in data transmission are illustrated

factorization and dilation problems for completely positive maps on von neumann algebras we study factorization and dilation properties of markov maps between von neumann algebras equipped with normal faithful states i e completely positive unital maps which preserve the given states and also intertwine their automorphism groups the starting point for our investigation has been the question of existence of non factorizable markov maps as formulated by c anantharaman delaroche we provide simple examples of non factorizable markov maps on m n mathbb c for all n 3 as well as an example of a one parameter semigroup t t t 0 of markov maps on m 4 mathbb c such that t t fails to be factorizable for all small values of t 0 as applications we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum birkhoff conjecture as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little grothendieck inequality

completely isometric representations of _ and let g be a locally compact group it is shown that there exists a natural completely isometric representation of the completely bounded fourier multiplier algebra m cb a g which is dual to the representation of the measure algebra m g on b l 2 g the image algebras of m g and m cb a g in cb b l 2 g are intrinsically characterized and some commutant theorems are proved it is also shown that for any amenable group g there is a natural completely isometric representation of ucb g on b l 2 g which can be regarded as a duality result of neufang s completely isometric representation theorem for luc g

the real positive semidefinite completion problem for series parallel graphs abstract we consider the partial real symmetric matrices x whose diagonal entries are equal to 1 and whose off diagonal entries are specified only on a subset of the positions the question is to determine whether x can be completed to a positive semidefinite matrix extending a result of barrett et al 3 we give a set of necessary conditions for x to be completable and show that these conditions are also sufficient if and only if the graph corresponding to the positions of the specified entries is series parallel i e has no k 4 minor

measurable schur multipliers and completely bounded multipliers of the fourier algebras let g be a locally compact group and a g the fourier algebra of g from 17 in 24 herz de ned the space of herz schur multipliers b2 g he claimed that this space coincided with the space he denoted v g in his article 23 but o2ered no proof of this fact the completely bounded multipliers of the fourier algebra mcba g were de ned by de canniere and haagerup in 8 in was shown by bo zejko and fendler 7 that mcba g 1 4 b2 g isometrically unfortunately this result relies on unpublished work of gilbert entitled convolution operators and banach space tensor products i ii and iii but not widely available however there is a proof of the fact that mcba g 1 4 b2 g due to jolissaint 27 which relies on the representation theorem for completely bounded maps applied to the reduced c algebra c r g in this paper we prove an analogue of the fact that mcba g 1 4 b2 g in fact our result obtains that fact in the case that g is a discrete group and improves upon it in the respect that we obtain a complete isometry where both mcba g and b2 g are given natural operator space structures moreover we show that the natural module action of mcba g on the von neumann algebra generated by the left regular representation vn g extends completely isometrically to a normal action of mcba g on all of b l g the space of bounded operators on l g in fact the image of mcba g in the completely bounded normal maps on b l g consists of l1 g bimodule maps and can be characterised as a certain intrinsic invariant part amongst those maps we feel that our approach is interesting in and of itself since it makes systematic use of modern operator space theoretic techniques pioneered by blecher e2ros haagerup paulsen ruan and smith amongst others it thus gives a starting point for obtaining representations of completely bounded multipliers of kac algebras after 29 or of multipliers of locally compact quantum groups after vaes et al see 47 to support our approach in x 3 we describe the space of measurable schur multipliers v1 x for any suitable measure space x we develop the theory of v1 x fully and systematically our main tools for doing this are the results of smith 41 and blecher and smith 5 which characterise the normal completely bounded maps on the space of all bounded operators on a hilbert space which are module maps over a von neumann algebra we make great use of the weak haagerup tensor product of two von neumann algebras which is

on the completely positive and positive semidefinite preserving cones abstract the cone cp n q of completely positive linear transformations from m n c m n to m q is shown to be isometrically isomorphic to p nq the cone of nq by nq positive semidefinite matrices generalizations of scalar and matrix results to cp n q hp n q l m n m q where hp n q represents the hermitian preserving linear transformations are discussed relationships among the completely positives the set of positive semidefinite preservers p n and its dual m n are given left and right facial ideals of cp are characterized properties of the joint angular field of values of a finite sequence of hermitian matrices h 1 h m are studied leading to a characterization of p q p n

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

special classes of positive and completely positive maps abstract characterizations are given for the positive and completely positive maps on n n complex matrices that leave invariant the diagonal entries or the k th elementary symmetric function of the diagonal entries 1 k n in addition it is shown that such a positive map is always decomposable if n 3 and this fails to hold if n 3 the real case is also considered

geometrical properties of the algebraic spinors for r3 1 the different geometrical properties of majorana even dirac and chevalley spinors in the clifford algebra approach are investigated

a connection between positive semidefinite and euclidean distance matrix completion problems abstract the positive semidefinite and euclidean distance matrix completion problems have received a lot of attention in the literature results have been obtained for these two problems that are very similar in their formulations although there is a strong relationship between positive semidefinite matrices and euclidean distance matrices it was not clear as noted by johnson et al how to link the two completion problems the purpose of this note is twofold first we show how the results for the euclidean distance matrix completion problem can be derived from the corresponding results for the positive semidefinite completion problem using a functional transform introduced by schoenberg second we introduce a new set of necessary conditions that are stronger than some previously known ones and we identify the graphs for which these conditions suffice for ensuring completability

bounds on the distance between a unital quantum channel and the convex hull of unitary channels motivated by the recent resolution of asymptotic quantum birkhoff conjecture aqbc we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels we provide two lower bounds on this distance by employing techniques from quantum information and operator algebras respectively we then show how to apply these results to construct some explicit counterexamples to aqbc we also point out an interesting connection between the grothendieck s inequality and aqbc

the minimum entropy output of a quantum channel is locally additive we show that the minimum von neumann entropy output of a quantum channel is locally additive hastings counterexample for the additivity conjecture makes this result quite surprising in particular it indicates that the nonadditivity of the minimum entropy output is a global effect of quantum channels

capacity bounds via operator space methods we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity

systematic redundant residue number system codes analytical upper bound and iterative decoding performance over awgn and rayleigh channels the novel family of redundant residue number system rrns codes is studied rrns codes constitute maximum minimum distance block codes exhibiting identical distance properties to reed solomon codes binary to rrns symbol mapping methods are proposed in order to implement both systematic and nonsystematic rrns codes furthermore the upper bound performance of systematic rrns codes is investigated when maximum likelihood ml soft decoding is invoked the classic chase algorithm achieving near ml soft decoding is introduced for the first time for rrns codes in order to decrease the complexity of the ml soft decoding furthermore the modified chase algorithm is employed to accept soft inputs as well as to provide soft outputs assisting in the turbo decoding of rrns codes by using the soft input soft output chase algorithm

positive semidefinite quadratic forms on unitary matrices abstract let f x 1 x n i j 1 n ij x i x j a ij a ji r be a real quadratic form such that the trace of the hermitian matrix f v 1 v n i j 1 n ij v i v j is nonnegative for all unitary 2 n 2 n matrices v 1 v n we prove that f u 1 u n is positive semidefinite for all unitary matrices u 1 u n of arbitrary size m m

on the extreme points of quantum channels abstract let l m n denote the convex set of completely positive trace preserving operators from c m m to c n n i e quantum channels we give a necessary condition for l l m n to be an extreme point we show that generically this condition is also sufficient we characterize completely the extreme points of l 2 2 and l 3 2 i e quantum channels from qubits to qubits and from qutrits to qubits

factorizations of quantum channels abstract we find a connection between the existence of a factorization of a quantum channel and the existence of low rank solutions to certain linear matrix equations using this we show that if a quantum channel is factorized by a direct integral of factors it must lie in the convex hull of quantum channels which are factorized respectively by the factors in the direct integral we use this to characterize some non trivial extreme points in the set of factorizable quantum channels and give an example

fast fault tolerant digital convolution using a polynomial residue number system a fault tolerant convolution algorithm that is an extension of residue number system fault tolerance schemes applied to polynomial rings is described the algorithm is suitable for implementation on multiprocessor systems and is able to concurrently mask processor failures a fast algorithm based on long division for detecting and correcting multiple processor failures is presented moduli polynomials that yield an efficient and robust fast fourier transform fft based algorithm are selected for this implementation a single fault detection and correction is studied and a generalized likelihood ratio test is applied to optimally detect system failures in the presence of computational noise the coding scheme is capable of protecting over 90 of the computation involved in convolution parts not covered by the scheme are assumed to be protected via triple modular redundancy this hybrid approach can detect and correct any single system failure with as little as 70 overhead compared with 200 needed for a system fully protected via modular redundancy

a clifford algebra quantization of dirac s electron positron field the quantum field theory of free dirac particles four component massive spin 1 2 particles is derived by a segal quantization procedure first details are given on how the spinor space of dirac is actually a minimal left ideal of the clifford algebra associated with a lorentz inner product space and how the homogeneous group actions break the natural two component quaternion structure to give familiar four component complex spinors second wigner s procedure for constructing unitary representations of the poincare group is used to construct the appropriately induced infinite dimensional representation of the inhomogeneous group starting from the above four dimensional nonunitary representation third and finally segal s procedure for quantizing classical fermion fields is adapted to this infinite dimensional hilbert space to obtain the clifford algebra of annihilation creation operators for spin 1 2 particles the familiar fock space appears as a minimal left ideal in this second clifford algebra

fault tolerant convolution via chinese remainder codes constructed from non coprime moduli this paper develops a framework for performing fault tolerant convolution via error correcting codes based on the chinese remainder theorem crt with non coprime moduli in contrast to convolution that is protected through crt codes with coprime moduli our scheme allows errors to be detected and located in a highly parallel manner and has the advantage of being robust to faults that occur during the error handling stage in addition for certain important classes of errors the codes developed in this paper require less redundancy than crt codes constructed from coprime moduli we demonstrate the applicability and advantages of our codes by comparing them to both crt codes with coprime moduli and repetition codes we focus on codes over polynomial rings but our constructions apply naturally to integer codes we discuss this extension and show that it generalizes the distance properties and error correction strategies associated with previously developed approaches for codes based on non coprime integer moduli

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

smooth manifold structure for extreme channels a quantum channel from a system a of dimension da to a system b of dimension db is a completely positive trace preserving map from complex da da to db db matrices and the set of all such maps with kraus rank r has the structure of a smooth manifold we describe this set in two ways first as a quotient space of a subset of the rdb da dimensional stiefel manifold second as the set of all choi states of a fixed rank r these two descriptions are topologically equivalent this allows us to show that the set of all choi states corresponding to extreme channels from system a to system b of a fixed kraus rank r is a smooth submanifold of dimension 2rdadb da2 r2 of the set of all choi states of rank r as an application we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from a to b arbitrarily well

completely positive semidefinite rank an n times n matrix x is called completely positive semidefinite cpsd if there exist d times d hermitian positive semidefinite matrices p i i 1 n for some d ge 1 such that x ij mathrm tr p ip j for all i j in 1 ldots n the cpsd rank of a cpsd matrix is the smallest d ge 1 for which such a representation is possible in this work we initiate the study of the cpsd rank which we motivate in two ways first the cpsd rank is a natural non commutative analogue of the completely positive rank of a completely positive matrix second we show that the cpsd rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior in this work we present several properties of the cpsd rank unlike the completely positive rank which is at most quadratic in the size of the matrix no general upper bound is known on the cpsd rank of a cpsd matrix in fact we show that the cpsd rank can be sub exponential in terms of the size specifically for any n ge 1 we construct a cpsd matrix of size 2n whose cpsd rank is 2 varomega sqrt n our construction is based on gram matrices of lorentz cone vectors which we show are cpsd the proof relies crucially on the connection between the cpsd rank and quantum behaviors in particular we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high rank extreme points of the n dimensional elliptope lastly we study cpsd graphs i e graphs g with the property that every doubly nonnegative matrix whose support is given by g is cpsd we show that a graph is cpsd if and only if it has no odd cycle of length at least 5 as a subgraph this coincides with the characterization of cp graphs

a closed form robust chinese remainder theorem and its performance analysis chinese remainder theorem crt reconstructs an integer from its multiple remainders that is well known not robust in the sense that a small error in a remainder may cause a large error in the reconstruction a robust crt has been recently proposed when all the moduli have a common factor and the robust crt is a searching based algorithm and no closed from is given in this paper a closed form robust crt is proposed and a necessary and sufficient condition on the remainder errors for the closed form robust crt to hold is obtained furthermore its performance analysis is given it is shown that the reason for the robustness is from the remainder differential process in both searching based and our proposed closed form robust crt algorithms which does no exist in the traditional crt we also propose an improved version of the closed form robust crt finally we compare the performances of the traditional crt the searching based robust crt and our proposed closed form robust crt and its improved version algorithms in terms of both theoretical analysis and numerical simulations the results demonstrate that the proposed closed form robust crt its improved version has the best performance has the same performance but much simpler form than the searching based robust crt

an analytic approach to a permanent conjecture abstract this paper is concerned with the conjecture on the permanent of the hadamard product of two positive semidefinite matrices per a b pera perb we present some properties of correlation matrices and introduce an analytic approach of maximizing matrices for the permanent conjecture given an irreducible correlation matrix we show that its maximizing matrix is i singular ii irreducible and iii invariant by a row and column reduction

speedp an algorithm to compute sdp bounds for very large max cut instances we consider low rank semidefinite programming lrsdp relaxations of unconstrained 1 1 quadratic problems or equivalently of max cut problems that can be formulated as the non convex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints we prove the equivalence of the lrsdp problem with the unconstrained minimization of a new merit function and we define an efficient and globally convergent algorithm called speedp for finding critical points of the lrsdp problem we provide evidence of the effectiveness of speedp by comparing it with other existing codes on an extended set of instances of the max cut problem we further include speedp within a simply modified version of the goemans williamson algorithm and we show that the corresponding heuristic speedp mc can generate high quality cuts for very large sparse graphs of size up to a million nodes in a few hours

null spaces of correlation matrices abstract let r be the real numbers and r n the vector space of all column vectors of length n let c n be the convex set of all real correlation matrices of size n if v is a subspace of r n of dimension k we consider the face f v of c n consisting of all a c n such that v n a i e av 0 if f v is nonempty we say that v is realizable we give complete geometric descriptions of f v in the cases k 1 n 4 and k 2 n 5 for k 2 n 5 we provide a simple algebraic method for describing f v

on bipartite unitary matrices generating subalgebra preserving quantum operations we study the structure of bipartite unitary operators which generate via the stinespring dilation theorem quantum operations preserving some given matrix algebra independently of the ancilla state we characterize completely the unitary operators preserving diagonal block diagonal and tensor product algebras some unexpected connections with the theory of quantum latin squares are explored and we introduce and study a sinkhorn like algorithm used to randomly generate quantum latin squares

on birkhoff s theorem for doubly stochastic completely positive maps of matrix algebras abstract a study is made of the extreme points of the convex set of doubly stochastic completely positive maps of the matrix algebra m n if n 2 the extreme points are precisely the unitary maps but if n 3 there are nonunitary extreme points examples of which are exhibited a tilde operation is defined on the linear maps of m n and used to give an elementary derivation of a result of kummerer and maassen

fixed points for luders operations and commutators let b h be the algebra of all bounded linear operators on a complex hilbert space h and let tj j 1 b h be a sequence of operators such that j 1tj tj 1h in the weak operator topology let us define t b h b h by setting t y j 1t jytj for y b h we show that if x b h is an operator such that all the commutators xtj tjx as well as x itself are fixed points for t then xtj tjx 0 for all j 1 in the case when tj j 1 consists of commuting normal operators this shows that every fixed point for t commutes with all tj for j 1

extremal correlation matrices abstract let r n denote the convex compact set of all real n by n positive semidefinite matrices with main diagonal entries equal to 1 we examine the extreme points of r n focusing mainly on their rank the principal result is that r n contains extreme points of rank k if and only if k k 1 2 n

a note on infinite extreme correlation matrices abstract we give a characterization for the extreme points of the convex set of correlation matrices with a countable index set a hermitian matrix is called a correlation matrix if it is positive semidefinite with unit diagonal entries using the characterization we show that there exist extreme points of any rank

cuts matrix completions and graph rigidity this paper brings together several topics arising in distinct areas polyhedral combinatorics in particular cut and metric polyhedra matrix theory and semidefinite programming in particular completion problems for positive semidefinite matrices and euclidean distance matrices distance geometry and structural topology in particular graph realization and rigidity problems cuts and metrics provide the unifying theme indeed cuts can be encoded as positive semidefinite matrices this fact underlies the approximative algorithm for max cut of goemans and williamson and both positive semidefinite and euclidean distance matrices yield points of the cut polytope or cone after applying the functions 1 arccos or when fixing the dimension in the euclidean distance matrix completion problem we find the graph realization problem and the related question of unicity of realization which leads to the question of graph rigidity our main objective here is to present in a unified setting a number of results and questions concerning matrix completion graph realization and rigidity problems these problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross fertilization between the various fields involved

chinese remaindering with errors the chinese remainder theorem states that a positive integer m is uniquely specified by its remainder module k relatively prime integers p sub 1 spl middot spl middot spl middot p sub k provided m spl pi sub i 1 sup k p sub i thus the residues of m module relatively prime integers p sub 1 p sub 2 spl middot spl middot spl middot p sub n form a redundant representation of m if m spl pi sub i 1 sup k p sub i and k n this gives a number theoretic construction of an error correcting code that has been considered often in the past in this code a message integer m spl pi sub i 1 sup k p sub i is encoded by the list of its residues module p sub 1 spl middot spl middot spl middot p sub n by the chinese remainder theorem if a codeword is corrupted in e n k 2 coordinates then there exists a unique integer m whose corresponding codesword differs from the corrupted word in at most e places furthermore mandelbaum 1976 1978 shows how m can be recovered efficiently given the corrupted word provided that the p sub i s are very close to one another to deal with arbitrary p sub i s we present a variant of his algorithm that runs in almost linear time and recovers from e log p sub 1 log p sub 1 log p sub n spl middot n k errors our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than n k 2 specifically given n residues r sub 1 spl middot spl middot spl middot r sub n and an agreement parameter t we find a list of all integers m spl pi sub i 1 sup k p sub i such that m mod p sub i r sub i for at least t values of i spl isin 1 spl middot spl middot spl middot n provided t spl omega spl radic kn log p sub n log p sub 1 for n spl gt k and p sub n spl les p sub 1 sup o 1 the fraction of error corrected by the algorithm is almost twice that corrected by the previous work more significantly the algorithm recovers the message even when the amount of agreement between the received word and the codeword is much smaller than the number of errors

multiple error detection and correction based on redundant residue number systems this paper presents some results on multiple error detection and correction based on the redundant residue number system rrns rrns is often used in parallel processing environments because of its ability to increase the robustness of information passing between the processors the proposed multiple error correction scheme utilizes the chinese remainder theorem crt together with a novel algorithm that significantly simplifies the error correcting process for integers an extension of the scheme further reduces the computational complexity without compromising its error correcting capability proofs and examples are provided for the coding technique

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

towards robustness in residue number systems the problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli namely the robust remaindering problem may occur in many applications including phase unwrapping frequency detection from several undersampled waveforms wireless sensor networks etc assuming that the dynamic range of the large number is the maximal possible one i e the least common multiple lcm of all the moduli a method called robust chinese remainder theorem for solving the robust remaindering problem has been recently proposed in this paper by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli a tradeoff between the dynamic range and the robustness bound for two modular systems is studied it basically says that a decrease in the dynamic range may lead to an increase of the robustness bound we first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed form formula for the robustness to hold we then propose simple closed form reconstruction algorithms furthermore the newly obtained two modular results are applied to the robust reconstruction for multimodular systems and generalized to real numbers finally some simulations are carried out to verify our proposed theoretical results

quantum channels arising from abstract harmonic analysis we present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups g beginning with a representation of the measure algebra m g we unify and elaborate on recent counter examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions as well as present an application to the noiseless subsystems method of quantum error correction using a representation of the completely bounded fourier multiplier algebra mcba g we provide a new class of counter examples to the recently solved asymptotic quantum birkhoff conjecture along with a systematic method of producing the examples using a geometric representation of schur maps further properties of our channels including duality quantum capacity and entanglement preservation are discussed along with potential applications to additivity conjectures