convexity of quasi entropy type functions lieb s and ando s convexity theorems revisited given a positive function f on 0 and a non zero real parameter we consider a function if a b x tr x f larb 1 rb x in three matrices a b 0 and x this generalizes the notion of monotone metrics on positive definite matrices and in the literature 1 has been typical we investigate how operator monotony of f is sufficient and or necessary for joint convexity concavity of if a b x similar discussions are given for quasi entropies and quantum skew informations

trace functions as laplace transforms we study trace functions on the form t trf a tb where f is a real function defined on the positive half line and a and b are matrices such that a is positive definite and b is positive semidefinite if f is non negative and operator monotone decreasing then such a trace function can be written as the laplace transform of a positive measure the question is related to the bessis moussa villani conjecture

non commutative calculus optimal transport and functional inequalities in dissipative quantum systems we study dynamical optimal transport metrics between density matrices associated to symmetric dirichlet forms on finite dimensional c algebras our setting covers arbitrary skew derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for markov chains lindblad equations and the fermi ornstein uhlenbeck semigroup we develop a non nommutative differential calculus that allows us to obtain non commutative ricci curvature bounds logarithmic sobolev inequalities transport entropy inequalities and spectral gap estimates

d optimal 0 1 weighing designs for eight objects abstract let g m n max det w t w w m m n 0 1 a matrix w m m n 0 1 with det w t w g m n is called d optimal here we determine g m 8 for m large with the help of a variation of mitchell s detmax algorithm and describe the d optimal examples

a matrix convexity approach to some celebrated quantum inequalities abstract some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the hansen pedersen jensen inequalities a general argument is given using matrix perspectives of operator convex functions a matrix analogue of marechal s extended perspectives provides additional inequalities including a p q 1 result of lieb

the χ2 divergence and mixing times of quantum markov processes we introduce quantum versions of the 2 divergence provide a detailed analysis of their properties and apply them in the investigation of mixing times of quantum markov processes an approach similar to the one presented in the literature for classical markov chains is taken to bound the trace distance from the steady state of a quantum processes a strict spectral bound to the convergence rate can be given for time discrete as well as for time continuous quantum markov processes furthermore the contractive behavior of the 2 divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm in this context we analyze different versions of quantum detailed balance and finally give a geometric conductance bound to the convergence rate for unital quantum markov processes

operator log convex functions and operator means we study operator log convex functions on 0 and prove that a continuous nonnegative function on 0 is operator log convex if and only if it is operator monotone decreasing several equivalent conditions related to operator means are given for such functions operator log concave functions are also discussed

metric adjusted skew information we extend the concept of wigner yanase dyson skew information to something we call metric adjusted skew information of a state with respect to a conserved observable this skew information is intended to be a non negative quantity bounded by the variance of an observable in a state that vanishes for observables commuting with the state we show that the skew information is a convex function on the manifold of states it also satisfies other requirements proposed by wigner and yanase for an effective measure of information content of a state relative to a conserved observable we establish a connection between the geometrical formulation of quantum statistics as proposed by chentsov and morozova and measures of quantum information as introduced by wigner and yanase and extended in this article we show that the set of normalized morozova chentsov functions describing the possible quantum statistics is a bauer simplex and determine its extreme points we determine a particularly simple skew information the skew information parametrized by a 0 1 and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations

contraction coefficients for noisy quantum channels generalized relative entropy monotone riemannian metrics geodesic distance and trace distance are all known to decrease under the action of quantum channels we give some new bounds on and relationships between the maximal contraction for these quantities

notes on d optimal designs abstract the purpose of this paper is to exhibit new infinite families of d optimal 0 1 matrices we show that hadamard designs lead to d optimal matrices of size j mj and j 1 mj for certain integers j 3 mod 4 and all positive integers m for j a power of a prime and j 1 mod 4 supplementary difference sets lead to d optimal matrices of size j 2mj and j 1 2mj for all positive integers m we also show that for a given j and d sufficiently large about half of the entries in each column of a d optimal matrix are ones this leads to a new relationship between d optimality for 0 1 matrices and for 1 matrices known results about d optimal 1 matrices are then used to obtain new d optimal 0 1 matrices

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

existentially closed csa groups abstract we study existentially closed csa groups we prove that existentially closed csa groups without involutions are simple and divisible and that their maximal abelian subgroups are conjugate we also prove that every countable csa group without involutions embeds into a finitely generated one having up to conjugacy the same maximal abelian subgroups except maybe the infinite cyclic ones we deduce from this that there exist 2 0 countable existentially closed csa groups without involutions and that their first order theories have 2 0 types over

higher order extension of lowner s theory operator k tone functions the new notion of operator matrix k tone functions is introduced which is a higher order extension of operator matrix monotone and convex functions differential properties of matrix k tone functions are shown characterizations properties and examples of operator k tone functions are presented in particular integral representations of operator k tone functions are given generalizing familiar representations of operator monotone and convex functions

riemannian metrics on positive definite matrices related to means abstract the riemannian metric on the manifold of positive definite matrices is defined by a kernel function in the form k d h k i j i j 1 tr p i hp j k when i i p i is the spectral decomposition of the foot point d and the hermitian matrices h k are tangent vectors for such kernel metrics the tangent space has an orthogonal decomposition the pull back of a kernel metric under a mapping d g d is a kernel metric as well several riemannian geometries of the literature are particular cases for example the statistical metric for multivariate gaussian distributions and the quantum fisher information in the paper the case x y m x y is mostly studied when m x y is a mean of the positive numbers x and y there are results about the geodesic curves and geodesic distances the geometric mean the logarithmic mean and the root mean are important cases

covariance and fisher information in quantum mechanics variance and fisher information are ingredients of the cramer rao inequality we regard fisher information as a riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and fisher information in this approach we show that there is a kind of dual one to one correspondence between the candidates of the two concepts we emphasize that fisher information is obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold

the maximum determinant of 1 matrices we give a new proof for the bound on the value of the determinant of a 1 matrix of dimension n 1 mod 4 first given by barba adapting a construction of a e brouwer we give examples to show that the bound is sharp for infinitely many values of n this in turn gives an infinite family of examples which attain the bound given by h ehlich and by m wojtas for the determinant of a 1 matrix of dimension n 2 mod 4 for n 3 mod4 we construct an infinite family of 1 examples which attain slightly more than of the bound given by ehlich elsevier science inc 1997

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

largest j simplices in d cubes some relatives of the hadamard maximum determinant problem abstract this paper studies the computationally difficult problem of finding a largest j dimensional simplex in a given d dimensional cube the case in which j d is of special interest for it is equivalent to the hadamard maximum determinant problem it has been solved for infinitely many values of d but not for d 14 the subcase in which j d 3 mod 4 subsumes the famous problem on the existence of hadamard matrices the known results for the case j d are here summarized and used but the main focus is on fixed small values of j when j 1 the problem is trivial and when j 2 or j 3 it is here solved completely i e for all d beyond that the results are fragmentary but numerous and they lead to several attractive conjectures some other problems involving simplices in cubes are mentioned and the relationship of largest simplices to d optimal weighing designs is discussed

interpolation theory and the wigner yanase dyson lieb concavity the wigner yanase dyson lieb concavity is naturally captured in the frame of interpolation theory among other results a certain generalization involving operator monotone functions of this concavity in the context of general von neumann algebras is obtained also a close relationship between the above subjects and f hansen s inequality is clarified all results are proved by using simple variational expressions of involved quantities

a fully quantum asymptotic equipartition property the classical asymptotic equipartition property is the statement that in the limit of a large number of identical repetitions of a random experiment the output sequence is virtually certain to come from the typical set each member of which is almost equally likely in this paper a fully quantum generalization of this property is shown where both the output of the experiment and side information are quantum an explicit bound on the convergence is given which is independent of the dimensionality of the side information this naturally leads to a family of reacutenyi like quantum conditional entropies for which the von neumann entropy emerges as a special case

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

quantum state discrimination bounds for finite sample size in the problem of quantum state discrimination one has to determine by measurements the state of a quantum system based on the a priori side information that the true state is one of the two given and completely known states or in general it is not possible to decide the identity of the true state with certainty and the optimal measurement strategy depends on whether the two possible errors mistaking for or the other way around are treated as of equal importance or not results on the quantum chernoff and hoeffding bounds and the quantum stein s lemma show that if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies and the decay rate is given by a certain statistical distance between and the chernoff distance the hoeffding distances and the relative entropy respectively while these results provide a complete solution to the asymptotic problem they are not completely satisfying from a practical point of vi

concavity of certain matrix trace and norm functions ii abstract we refine epstein s method to prove joint concavity convexity of matrix trace functions of lieb type tr f a p 1 2 b q a p 1 2 and symmetric anti norm functions of the form f a p b q where and are positive linear maps is an operator mean and f x with a certain power is an operator monotone function on 0 moreover the variational method of carlen frank and lieb is extended to general non decreasing convex concave functions on 0 so that we prove joint concavity convexity of more trace functions of lieb type

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

d optimal weighing designs for n 1mod4 objects and a large number of weighings abstract let m m n 0 1 denote the set of all m n 0 1 matrices and let g m n max det x t x x m m n 0 1 inthis paper we exhibit some new formulas for g m n where n 1 mod 4 specifically for m nt r where 0 r n we show that for all sufficiently large t g nt r n is a polynomial in t of degree n that depends on the characteristic polynomial of the adjacency matrix of a certain regular graph thus the problem of finding g nt r n for large t is equivalent to finding a regular graph whose degree of regularity and number of vertices depend only on n and r with a certain trace minimal property in particular we determine the appropriate trace minimal graph and hence the formulas for g nt r n for n 11 15 all r and all sufficiently large t

trace minimal graphs and d optimal weighing designs let g v be the set of all regular graphs on v vertices certain graphs from among those in g v with maximum girth have a special property called trace minimality in particular all strongly regular graphs with no triangles and some cages are trace minimal these graphs play an important role in the statistical theory of d optimal weighing designs each weighing design can be associated with a 0 1 matrix let mm n 0 1 denote the set of all m n 0 1 matrices and let g m n maxdetxtx x mm n 0 1 a matrix x mm n 0 1 is a d optimal design matrix if det xtx g m n in this paper we exhibit some new formulas for g m n where n 1 mod 4 and m is sufficiently large these formulas depend on the congruence class of n mod n more precisely let m nt r where 0 r n for each pair n r there is a polynomial p n r t of degree n in t which depends only on n r such that g nt r n p n r t for all sufficiently large t the polynomial p n r t is computed from the characteristic polynomial of the adjacency matrix of a trace regular graph whose degree of regularity and number of vertices depend only on n and r we obtain explicit expressions for the polynomial p n r t for many pairs n r in particular we obtain formulas for g nt r n for n 19 23 and 27 all 0 r n and all sufficiently large t and we obtain families of formulas for p n r t from families of trace minimal graphs including bipartite graphs obtained from finite projective planes generalized quadrilaterals and generalized hexagons