developed matrix inequalities via positive multilinear mappings abstract utilizing the notion of positive multilinear mappings we present some matrix inequalities in particular the choi davis jensen inequality f a b f a f b does not hold in general for a matrix convex function f and a positive multilinear mapping we prove this inequality under certain condition on f moreover some kantorovich and convexity type inequalities including positive multilinear mappings are presented

all the stabilizer codes of distance 3 we give necessary and sufficient conditions for the existence of stabilizer codes n k 3 of distance 3 for qubits n k log2 3n 1 n where n 1 if n 84m 1 3 1 2 or n 4m 2 1 3 1 2 3 for some integer m 1 and n 0 otherwise or equivalently a code n n r 3 exists if and only if n 4r 1 3 4r 1 3 n 1 2 3 for even r and n 8 4r 3 1 3 8 4 3 1 3 n 1 for odd r given an arbitrary length n we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound

duality in entanglement assisted quantum error correction the dual of an entanglement assisted quantum error correcting eaqec code is defined from the orthogonal group of a simplified stabilizer group from the poisson summation formula this duality leads to the macwilliams identities and linear programming bounds for eaqec codes we establish a table of upper and lower bounds on the minimum distance of any maximal entanglement eaqec code with length up to 15 channel qubits

operator means of probability measures abstract let p be the complete metric space consisting of positive invertible operators on an infinite dimensional hilbert space with the thompson metric we introduce the notion of operator means of probability measures on p in parallel with kubo and ando s definition of two variable operator means and show that every operator mean is contractive for the wasserstein distance by means of a fixed point method we consider deformation of such operator means and show that the deformation of any operator mean becomes again an operator mean in our sense based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

the minimum distance problem for two way entanglement purification entanglement purification takes a number of noisy epr pairs 00 11 and processes them to produce a smaller number of more reliable pairs if this is done with only a forward classical side channel the procedure is equivalent to using a quantum error correcting code qecc we instead investigate entanglement purification protocols with two way classical side channels 2 epps for finite block sizes in particular we consider the analog of the minimum distance problem for qeccs and show that 2 epps can exceed the quantum hamming bound and the quantum singleton bound we also show that 2 epps can achieve the rate k n 1 t n log sub 2 3 h t n o 1 n asymptotically reaching the quantum hamming bound where the epp produces at least k good pairs out of n total pairs with up to t arbitrary errors and h x xlog sub 2 x 1 x log sub 2 1 x is the usual binary entropy in contrast the best known lower bound on the rate of qeccs is the quantum gilbert varshamov bound k n spl ges 1 2t n log sub 2 3 h 2t n indeed in some regimes the known upper bound on the asymptotic rate of good qeccs is strictly below our lower bound on the achievable rate of 2 epps

diagonality measures of hermitian positive definite matrices with application to the approximate joint diagonalization problem in this paper we introduce properly invariant diagonality measures of hermitian positive definite matrices these diagonality measures are defined as distances or divergences between a given positive definite matrix and its diagonal part we then give closed form expressions of these diagonality measures and discuss their invariance properties the diagonality measure based on the log determinant divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case these diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of hermitian positive definite matrices numerical computations based on a modified newton method are presented and commented

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

a semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

geometry of midpoint sets for thompson s metric abstract it is well known that the convex cone p m of m m positive definite matrices is a cartan hadamard riemannian manifold with respect to the riemannian trace metric where the geometric mean of two positive definite matrices coincides with the unique metric midpoint between them in this paper we consider the thompson metric on p m inherited from the spectral norm and study some geometric structures of the thompson midpoints we prove that there is a unique midpoint minimal geodesic between a and b if and only if the spectrum of a 1 b is contained in a a 1 for some a 0 and the set of thompson midpoints between a and b is compact and is convex in both riemannian and euclidean sense it is further shown that the set of all weighted midpoints between a and b is compact and convex in riemannian sense

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

an order inequality characterizing invariant barycenters on symmetric cones abstract this paper is concerned with invariant and contractive barycenters on the wasserstein space of probability measures on metric spaces of non positive curvature where the center of gravity also called the cartan barycenter is the canonical barycenter on hadamard spaces we establish an order inequality of probability measures on partially ordered symmetric spaces of non compact type namely symmetric cones self dual homogeneous cones characterizing the cartan barycenter among other invariant and contractive barycenters the derived inequality and partially ordered structures on the probability measure space lead also to significant results on norm inequalities including the ando hiai inequality for probability measures on symmetric cones

regular operator mappings and multivariate geometric means we introduce the notion of regular operator mappings of several variables generalising the notion of spectral function this setting is convenient for studying maps more general than what can be obtained from the functional calculus and it allows for jensen type inequalities and multivariate non commutative perspectives as a main application of the theory we consider geometric means of k operator variables extending the geometric mean of k commuting operators and the geometric mean of two arbitrary positive definite matrices we propose dierent types of updating conditions that seems natural in many applications and prove that each of these conditions together with a few other natural axioms uniquely defines the geometric mean for any number of operator variables the means defined in this way are given by explicit formulas and are computationally tractable 1 msc2010 classification 26b25 47a64

new asymptotic bounds for self dual codes and lattices we give an independent proof of the krasikov litsyn bound d n spl lsim 1 5 sup 1 4 2 on doubly even self dual binary codes the technique used a refinement of the mallows odlyzko sloane approach extends easily to other families of self dual codes modular lattices and quantum codes in particular we show that the krasikov litsyn bound applies to singly even binary codes and obtain an analogous bound for unimodular lattices we also show that in each case our bound differs from the true optimum by an amount growing faster than o spl radic n

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

codeword stabilized quantum codes we present a unifying approach to quantum error correcting code design that encompasses additive stabilizer codes as well as all known examples of nonadditive codes with good parameters we use this framework to generate new codes with superior parameters to any previously known in particular we find 10 18 3 and 10 20 3 codes we also show how to construct encoding circuits for all codes within our framework

operator power means due to lawson lim palfia for 1 t 2 abstract for 1 t 1 lim palfia defined a new family of operator power means of positive definite matrices and subsequently by lawson lim their notion and most of their results extend to the setting of positive invertible operators on a hilbert space each of these means except t 0 arises as a unique positive invertible solution of a non linear operator equation and satisfies all desirable properties of power arithmetic means of positive real numbers the purpose of this paper is to extend the range in which operator power means due to lawson lim palfia are defined we investigate some properties of operator power means for t 2 2 1 1

two infinite families of nonadditive quantum error correcting codes we construct explicitly two infinite families of genuine nonadditive 1 error correcting quantum codes and prove that their coding subspaces are 50 larger than those of the optimal stabilizer codes of the same parameters via the linear programming bound all these nonadditive codes can be characterized by a stabilizer like structure and thus their encoding circuits can be designed in a straightforward manner

quantum error detection i statement of the problem this paper is devoted to the problem of error detection with quantum codes we show that it is possible to give a consistent definition of the undetected error event to prove this we examine possible problem settings for quantum error detection our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes we discuss possible transmission protocols with stabilizer and unrestricted quantum codes the set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators we examine polynomial invariants of quantum codes and show that coefficients of rains s see ibid vol44 p 1388 94 1998 unitary weight enumerators are known for classical codes under the name of binomial moments of the distance distribution as in the classical situation these enumerators provide an alternative expression for the probability of undetected error

jensen and minkowski inequalities for operator means and anti norms abstract jensen inequalities for positive linear maps of choi and hansen pedersen type are established for a large class of operator matrix means such as some p means and some kubo ando means these results are also extensions of the minkowski determinantal inequality to this end we develop the study of anti norms a notion parallel to the symmetric norms in matrix analysis including functionals like schatten q norms for a parameter q 1 and the minkowski functional det 1 n a

a fixed point mean approximation to the cartan barycenter of positive definite matrices abstract in this paper we define a new weighted inductive geometric mean h n on the riemannian manifold of positive definite matrices of fixed size and present its fixed point mean approximation to the cartan barycenter karcher mean we show that for t 0 1 the weighted geometric mean equation x h n 1 t a 1 a n x where t t 1 t w 1 w n 1 t has a unique positive definite solution that approaches as t 0 to the weighted cartan mean of a 1 a n numerical computations for the stochastic convergence in terms of h n to derive a new contractive barycenter other than the cartan barycenter are presented

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

parallel quantum computation and quantum codes we study the class qnc of efficient parallel quantum circuits the quantum analog of nc we exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth including circuits for encoding and decoding standard quantum error correcting codes or more generally any circuit consisting of controlled not gates controlled pi shifts and hadamard gates finally while we note the exact quantum fourier transform can be parallelized to linear depth we conjecture that neither it nor a simpler staircase circuit can be parallelized to less than this

structure of hiai petz parametrized geometry for positive definite matrices abstract recently hiai petz 2009 10 discussed a parametrized geometry for positive definite matrices with a pull back metric for a diffeomorphism to the euclidean space though they also showed that the geodesic is a path of operator means their interest lies mainly in metrics of the geometry in this paper we reconstruct their geometry without metrics and then we show their metric for each unitarily invariant norm defines a finsler one also we discuss another type of geometry in hiai and petz 2009 10 which is a generalization of corach porta recht s one 3

geometric mean block matrices abstract we consider an m m block matrix g with entries a i a j where a 1 a m are positive definite matrices of fixed size and a b is the geometric mean of positive definite matrix a and b we show that g is positive semidefinite if and only if the family of a 1 a m is commuting it can be transformed to a commuting family of positive definite matrices by a congruence transformation this result via commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices

monotonicity of the quantum linear programming bound the most powerful technique known at present for bounding the size of quantum codes of prescribed minimum distance is the quantum linear programming bound unlike the classical linear programming bound it is not immediately obvious that if the quantum linear programming constraints are satisfiable for dimension k then the constraints can be satisfied for all lower dimensions we show that the quantum linear programming bound is monotonic in this sense and give an explicitly monotonic reformulation

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

a nonadditive quantum code every good quantum error correcting code discovered thus far such as those known as stabilizer or additive codes has had the structure of an eigenspace of an abelian group generated by tensor products of pauli matrices in this letter we present the first example of a code that is better than any code of this type it encodes six states in five qubits and can correct the erasure of any single qubit

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

quantum codes of minimum distance two it is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2 thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize with this in mind we present a number of results on codes of minimum distance 2 we first compute the linear programming bound on the dimension of such a code then show that this bound can only be attained when the code either is of even length or is of length 3 or 5 we next consider questions of uniqueness showing that the optimal code of length 2 or 1 is unique implying that the well known one qubit in five single error correcting code is unique and presenting nonadditive optimal codes of all greater even lengths finally we compute the full automorphism group of the more important distance 2 codes allowing us to determine the full automorphism group of any gf 4 linear code

the least squares mean of positive hilbert schmidt operators abstract we show that the least squares mean on the riemannian manifold of positive operators in the extended hilbert schmidt algebra of linear operators on a hilbert space equipped with the canonical trace metric is the unique solution of the corresponding karcher equation this allows us to conclude that the least squares mean is the restriction of the karcher mean on the open cone of all bounded positive definite operators and hence inherits the basic properties of that mean conversely the karcher mean on the positive definite operators is shown to be the unique monotonically strongly continuous extension of the least squares mean on

inequalities for the wasserstein mean of positive definite matrices abstract we prove majorization inequalities for different means of positive definite matrices these include the cartan mean the karcher mean the log euclidean mean the wasserstein mean and the power mean

operator means of probability measures and generalized karcher equations abstract in this article we consider means of positive bounded linear operators on a hilbert space we present a complete theory that provides a framework which extends the theory of the karcher mean its approximating matrix power means and a large part of kubo ando theory to arbitrary many variables in fact to the case of probability measures with bounded support on the cone of positive definite operators this framework characterizes each operator mean extrinsically as unique solutions of generalized karcher equations which are obtained by exchanging the matrix logarithm function in the karcher equation to arbitrary operator monotone functions over the positive real half line if the underlying hilbert space is finite dimensional then these generalized karcher equations are riemannian gradients of convex combinations of strictly geodesically convex log determinant divergence functions hence these new means are their global minimizers in analogy to the case of the karcher mean as pointed out our framework is based on fundamental contraction results with respect to the thompson metric which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

γ commuting triples of positive definite matrices and midpoint equations on unitary matrices triples of positive definite matrices that can be transformed to a commuting triple by a congruence transformation are called commuting in this paper we give various descriptions and characterizations of commuting triples and particularly focus on those involving the geometric mean and midpoint equations on a generating set of the special unitary group

means in metric spaces and the center of mass in this article k convex metric spaces are considered where a several variable mapping is provided as a limit point of an iteration scheme based on the midpoint map in the metric space itself this mapping considered as a mean of its variables has some properties which relates it to the center of mass of these variables in the metric space sufficient conditions are given here for the two points to be identical as well as upper bounds on their distances from one another the asymptotic rate of convergence of the iterative process defining the mean is also determined here the case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied here as examples of k convex metric spaces

quantum shadow enumerators in a previous paper shor and laflamme see phys rev lett vol 78 p 1600 02 1997 define two weight enumerators for quantum error correcting codes connected by a macwilliams 1977 transform and use them to give a linear programming bound for quantum codes we extend their work by introducing another enumerator based on the classical theory of shadow codes that tightens their bounds significantly in particular nearly all of the codes known to be optimal among additive quantum codes codes derived from orthogonal geometry can be shown to be optimal among all quantum codes we also use the shadow machinery to extend a bound on additive codes to general codes obtaining as a consequence that any code of length can correct at most n 1 6 errors

communicating over adversarial quantum channels using quantum list codes in this correspondence we study quantum communication in the presence of adversarial noise in this setting communicating with perfect fidelity requires a quantum code of bounded minimum distance for which the best known rates are given by the quantum gilbert varshamov qgv bound asking only for arbitrarily high fidelity and letting the sender and receiver use a secret key of length logarithmic in the number of qubits sent we find a dramatic improvement over the qgv rates in fact our protocols allow high fidelity transmission at noise levels for which perfect fidelity is impossible to achieve such rates we introduce fully quantum list codes which may be of independent interest

quantum weight enumerators in a recent paper shor and laflamme see phys rev lett vol 78 p 1600 2 1997 defined two weight enumerators for quantum error correcting codes connected by a macwilliams transform and used them to give a linear programming bound for quantum codes we introduce two new enumerators which while much less powerful at producing bounds are useful tools nonetheless the new enumerators are connected by a much simpler duality transform clarifying the duality between shor and laflamme s enumerators we also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance and to extend the enumerator theory to codes with block size greater than 2

generalized polya szego type inequalities for some non commutative geometric means in this paper we shall show generalized polya szego type inequalities of n positive invertible operators on a hilbert space for any integer n 3 in terms of the following two typical non commutative geometric means that is one is the higher order weighted geometric mean due to lawson lim which is an extension of the ando li mathias geometric mean and the other is the weighted chaotic geometric mean among others the specht ratio plays an important role in our discussion which is the upper bound of a ratio type reverse of the weighted arithmetic geometric mean inequality

on the semidefinite representation of real functions applied to symmetric matrices abstract we present a new semidefinite representation for the trace of a real function f applied to symmetric matrices when a semidefinite representation of the convex or concave function f is known our construction is intuitive and yields a representation that is more compact than the previously known one we also show with the help of matrix geometric means and a riemannian metric over the set of positive definite matrices that for a rational exponent p in the interval 0 1 the matrix x raised to p is the largest element of a set represented by linear matrix inequalities this result further generalizes to the case of the matrix a p b which is the point of coordinate p on the geodesic from a to b we give numerical results for a problem inspired from the theory of experimental designs which show that the new semidefinite programming formulation can yield an important speed up factor

matrix extremal problems and shift invariant means abstract we consider an m m real symmetric matrix m a x with a 1 a m on the main diagonal and x in all off diagonal positions where m 2 and a a 1 a m is a given m tuple of positive real numbers we study the extremal problem of finding the minimum and maximum of x where m a x is positive semidefinite we show that the polynomial det m a x in variable x has only real roots with a unique negative root and that m a x is positive semidefinite resp definite if and only if x lies in the closed resp open interval determined by the negative and smallest positive roots it is further shown that the negative and smallest positive root maps over m tuples of positive real numbers contract the thompson metric and induce new multivariate means of positive real numbers satisfying the monotonicity homogeneity joint concavity and super multiplicativity in particular the smallest positive root map extends to such a mean of infinite variable of positive real numbers that realizes the limits for decreasing sequences and it eventually gives rise to a shift invariant mean of bounded sequences

from the continuous pv to discrete painleve equations we study the discrete transformations that are associated with the auto backlund of the continuous pv equation we show that several two parameter discrete painleve equations can be obtained as contiguity relations of pv among them we find the asymmetric d pii equation which is a well known form of discrete piii the relation between the ternary pi previously obtained through the discrete dressing approach and pv is also established a new discrete painleve equation is also derived

linear programming bounds for entanglement assisted quantum error correcting codes by split weight enumerators linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes in this paper we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator after deriving the macwilliams identities for these enumerators we are able to prove algebraic linear programming bounds such as the singleton bound the hamming bound and the first linear programming bound our singleton bound and hamming bound are more general than the previous bounds for entanglement assisted quantum stabilizer codes in addition we show that the first linear programming bound improves the hamming bound when the relative distance is sufficiently large on the other hand we obtain additional constraints on the size of pauli subgroups for quantum codes which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length in particular we show that there is no 27 15 5 or 28 14 6 stabilizer code we also discuss the existence of some entanglement assisted quantum stabilizer codes with maximal entanglement as a result the upper and lower bounds on the minimum distance of maximal entanglement quantum stabilizer codes with length up to 20 are significantly improved

the relative operator entropy and the karcher mean abstract the karcher mean for positive invertible operators on a hilbert space which is the unique solution of the karcher operator equation was established by lawson lim based on the geometric considerations for positive invertible operators in this note we extend both this mean and the equation to the non invertible case the key concept is the relative operator entropy for operators whose ranges are closed we show that the karcher mean for the non invertible case is the unique solution of the extended karcher equation as a byproduct we can show the uniqueness of the karcher solution for the invertible case based only on that of the power means