the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

geometric mean flows and the cartan barycenter on the wasserstein space over positive definite matrices abstract we introduce a class of flows on the wasserstein space of probability measures with finite first moment on the cartan hadamard riemannian manifold of positive definite matrices and consider the problem of differentiability of the corresponding cartan barycentric trajectory as a consequence we have a version of lie trotter formula and a related unitarily invariant norm inequality furthermore a fixed point theorem related to the karcher equation and the cartan barycentric trajectory is also presented as an application

an araki lieb thirring inequality for geometrically concave and geometrically convex functions for positive definite matrices a and b the araki lieb thirring inequality amounts to an eigenvalue log submajorisation relation for fractional powers atbt w log t ab 0 t 1 while for t 1 the reversed inequality holds in this paper i generalise this inequality replacing the fractional powers xt by a larger class of functions i prove that a continuous non negative function f with domain dom f 0 x0 for some positive x0 possibly infinity satisfies f a f b w log f2 1 2 ab for all positive semidefinite a and b with spectrum in dom f if and only if f is geometrically concave and 0 xf x f x for all x dom f the reversed inequality holds for a continuous non negative function f if and only if f is geometrically convex and xf x f x for all x dom f as an application i obtain a complementary inequality to the golden thompson inequality

majorization relations for hadamard products abstract we settle affirmatively the conjecture of johnson and bapat on the hadamard product a b of positive definite matrices a b i k n i a b ab k 1 2 n

the green function of an infinite fluid loaded membrane abstract in this paper the response of a fluid loaded plane structure a membrane to a concentrated line force excitation is considered in great detail the normalized velocity response here called the green function g depends upon a dimensionless range x0 km x where km is the free wavenumber on the membrane in a vacuum on the mach number m k 0 k m the ratio of wave phase speed km on the unloaded membrane to the sound speed k0 and on a parameter which can be regarded as a measure of fluid loading at the coincidence condition m 1 in the analogous problem involving a thin elastic plate the corresponding parameter is independent of frequency and plate thickness and may be regarded as an intrinsic measure of fluid loading moreover in cases of common interest steel in water aluminium in air that parameter is small in the present paper the asymptotic structure of g x0 m is therefore sought in the limit 0 naturally no single asymptotic expansion can be expected to be valid throughout the x0 m plane and the programme therefore involves the delineation of regions of that plane in which distinct asymptotic results apply the construction and discussion of those results and the asymptotic matching according to the procedures of the method of matched asymptotic expansions of results holding in adjoining regions the fourier integral for g is broken into surface wave and acoustic components and the asymptotic structure obtained for each previously obtained results for the behaviour at large distances are recovered with a demonstration that very large distances indeed x0 2 may be needed for their validity for some ranges of m and the drive point behaviour of g x0 0 m as 0 qua function of m is shown to correspond to that already discussed in the literature elsewhere in the covering of the whole x0 m plane by different asymptotic expressions a wide variety of analytical results is found reflecting the achievement in different regions of different balances among the five competing physical mechanisms represented in the model namely structural stiffness structural inertia fluid pressures fluid compressibility and fluid inertia these different balances give rise to a wide variety of expressions for the phase and amplitude of the surface wave and acoustic components which can now be used to isolate the dominant structural and acoustic mechanisms at any point in the x0 m plane

matrix inequalities in statistical mechanics some matrix inequalities used in statistical mechanics are presented a straightforward proof of the thermodynamic inequality is given and its equivalence to the peierls bogoliubov inequality is shown

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

inequalities for quantum relative entropy some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log majorization result the thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the peierls bogoliubov inequality is obtained

applications of geometric means on symmetric cones let v be a euclidean jordan algebra and let omega be the corresponding symmetric cone the geometric mean a b of two elements a and b in omega is defined as a unique solution which belongs to omega of the quadratic equation p x a 1 b where p is the quadratic representation of v in this paper we show that for any a in omega the sequence of n mathrm th iterate f a n x of the function f a omega to omega defined by f a x 1 over 2 x p a x 1 converges to a as applications we obtain that the geometric mean a b of omega can be represented as a limit of successive iteration of arithmetic means and harmonic means and we derive the lowner heinz inequality on the symmetric cone omega 0 leq a leq b mathrm implies a p leq b p mathrm for 0 leq p leq 1 furthermore we obtain a formula mathrm exp x y lim n to infty mathrm exp frac 2 n x exp frac 2 n y n which leads a golden thompson type inequality mathrm exp x y leq p mathrm exp x over 2 mathrm exp y for the spectral norm on v

order automorphisms on positive definite operators and a few applications abstract we first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so called chaotic order we then apply those results to the following problems 1 description of all bijective transformations on the space of nonsingular density operators quantum states which preserve the umegaki or the belavkin staszewski relative entropy 2 characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

α z renyi relative entropies we consider a two parameter family of renyi relative entropies d z that are quantum generalisations of the classical renyi divergence d p q this family includes many known relative entropies or divergences such as the quantum relative entropy the recently defined quantum renyi divergences as well as the quantum renyi relative entropies all its members satisfy the quantum generalizations of renyi s axioms for a divergence we consider the range of the parameters z for which the data processing inequality holds we also investigate a variety of limiting cases for the two parameters obtaining explicit formulas for each one of them

two reverse inequalities associated with tsallis relative operator entropy via generalized kantorovich constant and their applications recently tsallis relative operator entropy tp a b and tsallis relative entropy dp a b are discussed by furuichi yanagi kuriyama we shall show two reverse inequalities involving tsallis relative operator entropy tp a b via generalized kantorovich constant k p as some applications of two reverse inequalities we shall show two trace reverse inequalities involving tr tp a b and dp a b and also a known reverse trace inequality involving the relative operator entropy s a b by fujii kamei and the umegaki relative entropy s a b is shown as a simple corollary we show the following result let a and b be strictly positive operators on a hilbert space h such that m1 i a m1 i 0 and m2 i b m2 i 0 put m m2m1 m m2m1 h mm m1m2m1m2 1 and p 0 1 let be normalized positive linear map on b h then the following inequalities hold i 1 k p p a p b tp a b tp a b tp a b and ii f p a tp a b tp a b tp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and k p 0 1 and f p mpphp hh 11 k p 1p 1 0 in addition let a and b be strictly positive definite matrices iii 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b and iv f p tr a dp a b tr tp a b dp a b in particular both iii and iv yield the following known result logs 1 tr a s a b tr s a b s a b where s 1 h1h 1elogh1h 1 is said to be the specht ratio and s 1 1

ribbed elastic structures under a mean flow the problem of a ribbed membrane or plate submerged in a fluid with mean flow is studied we first derive a method which can be used to reduce this and similar problems to a band matrix inversion we then find the pass and stop band structure found in the case of static fluid persists when a mean flow is introduced and we give an explanation in terms of the eigenvalues of the transfer matrix of the system we then study disordered structures and observe the phenomenon of anderson localization in some parameter regimes the addition of disorder causes significant delocalization

lie trotter means of positive definite operators abstract as extension of the lie trotter product formula we define the two variable and multivariate lie trotter means with several examples including the sagae tanabe and hansen inductive means we show that the weighted means satisfying the arithmetic geometric harmonic mean inequalities are the multivariate lie trotter means we finally give a few interesting versions of the extended lie trotter formula derived from multivariate lie trotter means

convexity inequalities for estimating free energy and relative entropy for quantum systems it is shown that the relative entropy s p q tr p log p tr p log q of two positive semi definite operators p and q satisfies gamma 1 tr p p1 gamma q gamma or s p q or gamma 1 tr p1 gamma qgamma p for 0 gamma or 1 and that these bounds become exact in the limit gamma to phi analogous inequalities hold for states in classical statistical mechanics or information theory with trace replaced by integration or summation furthermore the average of these bounds is in general a better approximation to s p q than either bound alone and the average is amenable to further improvement via repeated richardson extrapolation if p and q are gibbs equilibrium states then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation

a variational expression for the relative entropy we prove that for the relative entropy of faithful normal states and on the von neumann algebram the formula s varphi omega sup omega h log varphi h i h h in m holds

on upper bound for the quantum entropy abstract a new upper bound for the von neumann entropy of a state of a compound quantum system is given this leads to a log sobolev inequality on the matrix algebra

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

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

hypercontractivity in noncommutative lpspaces we introduce and study a setup for hypercontractivity in the interpolating family of noncommutative lpspaces associated to a locally normal state on an inductive limitc algebra

trace inequalities for logarithms and powers of j hermitian matrices abstract spectral inequalities are stated for the trace of the exponential or the logarithmic of certain j hermitian matrices j i r i n r 0 r n the obtained inequalities are established in the context of indefinite inner product spaces and they are known to be valid for hilbert space operators or operator algebras

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

convex multivariate operator means abstract we prove that a trace function generated by the functional calculus is geodesically convex in the riemannian manifold of positive definite matrices if and only if it is geodesically convex in positive numbers the analysis of multivariate operator means is facilitated by the study of classes of means that are fix points under a contraction with respect to the thompson metric although such methods are powerful they crucially depend on monotonicity we develop new techniques to prove existence of multivariate operator means that are not necessarily monotone

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

some log majorizations and an extension of a determinantal inequality abstract an eigenvalue inequality involving a matrix connection and its dual is established and some log majorization type results are obtained in particular some eigenvalues inequalities considered by f hiai and m lin 9 an associated conjecture and a singular values inequality by l zou 20 are revisited a reformulation of the inequality det a u b det a b for positive semidefinite matrices a b with u a unitary matrix that appears in the polar decomposition of b a is also extended using some known norm inequalities associated to furuta inequality and araki cordes inequality

eigenvalue inequalities for products of matrix exponentials motivated by models from stochastic population biology and statistical mechanics we prove new inequalities of the form cp eae 2 cp eatr where a and b are n x n complex matrices 1 n 00 and cp is a real valued continuous function of the eigenvalues of its matrix argument for example if a is essentially nonnegative b is diagonal real and cp is the spectral radius then holds if in addition a is irreducible and b has at least two different diagonal elements then the inequality is strict the proof uses kingman s theorem on the logconvexity of the spectral radius lie s product formula and perturbation theory we conclude with conjectures

les espaces lp d une algebre de von neumann definies par la derivee spatiale abstract this work answers a question raised by a connes on the spatial theory of von neumann algebras preprint inst hautes etudes sci france and generalizes for a general von neumann algebra the theory of non commutative integration of j dixmier bull soc math france 81 1953 and i segal ann of math 57 1973

the vibration modes of concentrically supported free circular plates abstract vibration of a circular plate internally concentrically supported is basic in structural mechanics the complete first six frequencies and mode shapes are determined for clamped simply supported sliding and ring supports using exact characteristic frequency equations complex mode switches occur as the support radius is varied asymptotic formulas for large support radii are derived different types of singularities as the support radius shrinks to zero are distinguished and classified

vibration of annular plate concentrators with conical cross section in this paper the torsional vibration and radial vibration of isotropic metal thin annular plates with conical cross section are deduced by using bessel functions and according to electro mechanical analogy the electro mechanical equivalent circuits are obtained based on the electro mechanical equivalent circuits the resonance frequency equations and the expressions of angular and radial displacement amplitude magnifications are derived and the resonance frequencies and magnifications are solved by analytical method the relationships between torsional and radial resonance frequencies and geometrical dimensions are discussed and the relationships between angular and radial displacement amplitude magnifications and the ratio of outer radius over inner radius are analyzed modal shapes are analyzed by finite element method ansys it is shown that the values calculated by using the theory of this paper are in good agreement with the results simulated by ansys the method can serve as benchmark values for researchers to validate other variable thickness thin annular plates

transverse vibration of annular circular and elliptic plates using the characteristic orthogonal polynomials in two dimensions abstract characteristic orthogonal polynomials in two variables have been generated over the annular region occupied by a circular or elliptic plate satisfying the essential boundary conditions these are used to express the displacement in the transverse vibration of the plate the rayleigh ritz method is then applied to study the fundamental mode of vibration comparison has been made with the known results for circular annular plates in some cases the results for the elliptic annular plate are entirely new and are not available elsewhere all the results are summarized in tables covering various combinations of the boundary conditions

trace inequalities in nonextensive statistical mechanics abstract in this short paper we establish a variational expression of the tsallis relative entropy in addition we derive a generalized thermodynamic inequality and a generalized peierls bogoliubov inequality finally we give a generalized golden thompson inequality

tsallis entropies and matrix trace inequalities in quantum statistical mechanics extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the tsallis entropy namely one parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized peierls bogoliubov inequality is stated in the sense that one may be obtained from the other and vice versa

reverse inequalities involving two relative operator entropies and two relative entropies we shall discuss relation among tsallis relative operator entropy tp a b the relative operator entropy s a b by j i fujii kamei the tsallis relative entropy dp a b by furuichi yanagi kuriyama and the umegaki relative entropy s a b we show the following result let a and b be strictly positive definite matrices such that m1i a m1i 0 and m2i b m2i 0 put h m1m2m1m2 1 and p 0 1 then the following inequalities hold 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and the first inequality is the reverse one of the second known inequalty in particular logs 1 tr a s a b tr s a b s a b where s 1 is the specht ratio defined by s 1 h1h 1elogh1h 1 and the first inequality is the reverse one of the second known inequalty it is known that k p 0 1 for p 0 1 and s 1 1

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

frequency constants for transverse vibration of annular disks frequency constants for the transverse vibration of thin annular disks are presented all nine combinations of the boundary conditions free simply supported and clamped are included for poisson s ratios of 0 0 0 3 and 0 5 many of these cases have been treated earlier in the literature but with occasional significant error consequently several tests of the validity of the present solutions are described

quantumness of bipartite states in terms of conditional entropies quantum discord as defined by olliver and zurek 2002 phys rev lett 88 017901 as the difference of two natural quantum extensions of the classical mutual information plays an interesting role in characterizing quantumness of correlations inspired by this idea we will study quantumness of bipartite states arising from different quantum analogs of the classical conditional entropy our approach is intrinsic in contrast to the olliver zurek method that involves extrinsic local measurements for this purpose we introduce two alternative variants of quantum conditional entropies via conditional density operators which in turn are intuitive quantum extensions of equivalent classical expressions for the conditional probability the significance of these quantum conditional entropies in characterizing quantumness of bipartite states is illustrated through several examples

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

hadamard products and golden thompson type inequalities abstract by using hadamard products we give some reasonable upper and lower bounds of golden thompson type for e h 1 h m where h i i 1 2 m are arbitrary hermitian matrices and is an arbitrary unitarily invariant norm

effect of an isolated irregularity on the transmission of energy down a periodically ribbed fluid loaded elastic structure the paper considers the problem of wave transmission in a configuration consisting of an infinite fluid loaded elastic membrane supported by a finite array of thin ribs the ribs are equally spaced except for one which is randomly displaced from its periodic location by a small amount one of the ribs not the displaced one is driven at a prescribed velocity by a time harmonic force and all the others are held at rest they are assumed to have infinite mechanical impedance so that fluid loading provides the only mechanism by which energy can be transmitted down the ribbed part of the structure an exact solution for the forces exerted on the membrane by the ribs is obtained in terms of the eigenvalues and eigenvectors of a 2 x 2 propagation matrix and the solution is shown to be consistent with energy conservation requirements it is found that the random displacement of one rib leaves the response in the stop bands almost unchanged including the response at the displaced rib but has a large effect in the pass bands there the energy transmitted to infinity beyond the displaced rib is dramatically reduced two effects contributing to this are first that the input power to the irregular configuration is generally less than that to the regular one and second that the array of regular ribs with one isolated irregularity is found to reflect most of the energy incident on it from the driven rib provided the degree of irregularity is not too small and provided that the frequency does not lie in one of a small number of narrow bands of high transmission

anderson localization effects in the transmission of energy down an irregularly ribbed fluid loaded structure the paper studies wave transmission along an infinite fluid loaded elastic membrane supported by a finite array of irregularly spaced thin ribs all having infinite mechanical impedance except for the first which is driven by a time harmonic force when the green s function of the unribbed structure is taken to consist only of its subsonic surface wave component the response of the structure is exponentially localized anderson localization a localization length is calculated as a function of frequency in both the small and the large disorder limits and in both of these limits the calculated localization length is found to agree extremely well with the results of numerical simulations when a weak acoustic component is included in the green s function it is shown that the exponential localization is short circuited the response now decaying algebraically with distance from the first rib and with negligible sensitivity to disorder even for rather large degrees of disorder

a generalization of araki s log majorization abstract we generalize araki s log majorization to the log convexity theorem for the eigenvalues of a p 1 2 b p a p 1 2 as a function of p 0 where a b are positive semidefinite matrices and are positive linear maps between matrix algebras similar generalizations of the log majorization of ando hiai type for the weighted geometric mean a b are also given including a lemma on general operator means a e of a and a projection e

disorder and localization in ribbed structures with fluid loading the paper considers the steady state harmonic response of an elastic fluid loaded membrane supported by irregularly spaced ribs under the assumption of subsonic wave coupling the solution is given exactly for any configuration as a product of 2 times 2 transfer matrices it is well known that the response of a periodically ribbed membrane exhibits a pass stop band structure although this structure is destroyed in the irregular case we find that two distinct regimes remain smooth and fluctuating exponential decay the transfer matrix solution is used to explain these regions the average transfer matrix is obtained exactly where the decay is smooth its eigenvalues approximately determine the localization length

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

a general approximate solution of the problem of free vibrations of annular plates of stepped thickness abstract an approximate solution to the title problem is proposed for the case in which the inner boundary of the circular plate is free and the outer edge is elastically restrained against rotation the displacement amplitude is approximated by means of a superposition of simple polynomial co ordinate functions which satisfy identically the boundary conditions at the outer edge only and which yield excellent accuracy in the case of simply connected circular plates the optimized rayleigh ritz method is then used to generate the frequency determinant minimizing the eigenvalues with respect to an exponential parameter contained in each polynomial co ordinate function a comparison with eigenvalues obtained using a finite element solution is also presented and good agreement is shown to exist

matrix trace inequalities on tsallis relative entropy of negative order abstract in this paper we show matrix trace inequalities on the quantum tsallis relative entropy of negative order which includes the quasi geometric mean of positive definite matrices moreover we show an estimate of the difference between two tsallis relative entropies of negative order