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Fumio Hiai

Publications and source records attributed to Fumio Hiai.

At least 37 records · Page 2Linked to original sources
arXiv2018

Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences

We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the Rényi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.

math-ph↗
arXiv2018

Ando-Hiai type inequalities for multivariate operator means

We present several Ando-Hiai type inequalities for $n$-variable operator means for positive invertible operators. Ando-Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant.

math.FA↗
arXiv2017

Generalized Log-Majorization and Multivariate Trace Inequalities

We show that recent multivariate generalizations of the Araki-Lieb-Thirring inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel, Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regards to logarithmic integral averages of vectors of singular values.

math-ph↗
arXiv2017

Operator means deformed by a fixed point method

By means of a fixed point method we discuss the deformation of operator means and multivariate means of positive definite matrices/operators. It is shown that the deformation of an operator mean becomes again an operator mean. The means deformed by the weighted power means are particularly examined.

math.FA↗
arXiv2017

Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan-Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.

math.FA↗
arXiv2017

The stochastic order of probability measures on ordered metric spaces

The general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first. Motivated by recent work on barycentric maps on spaces of probability measures on ordered Banach spaces, we introduce and study a stochastic order on the space of probability measures $\mathcal{P}(X)$, where $X$ is a metric space equipped with a closed partial order, and derive several useful equivalent versions of the definition. We establish the antisymmetry and closedness of the stochastic order (and hence that it is a closed partial order) for the case of a partial order on a Banach space induced by a closed normal cone with interior. We also consider order-completeness of the stochastic order for a cone of a finite-dimensional Banach space and derive a version of the arithmetic-geometric-harmonic mean inequalities in the setting of the associated probability space on positive matrices.

math.FA↗
arXiv2017

Different quantum f-divergences and the reversibility of quantum operations

The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $α$-$z$-Rényi divergences. In this paper we give a systematic overview of the various concepts of quantum $f$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum $f$-divergence by a quantum operation. In particular, we compare the standard and the maximal $f$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum $f$-divergences are strictly different for non-commuting operators unless $f$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard $f$-divergences. We also study the monotonicity of the $α$-$z$-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters $α,z$ where monotonicity holds, and where the preservation of the $α$-$z$-Rényi divergence implies the reversibility of the quantum operation.

math-ph↗
arXiv2017

Geometric mean flows and the Cartan barycenter on the Wasserstein space over positive definite matrices

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.

math.FA↗
arXiv2016

Log-majorization and Lie-Trotter formula for the Cartan barycenter on probability measure spaces

We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization.

math.FA↗
arXiv2016

A generalization of Araki's log-majorization

We generalize Araki's log-majorization to the log-convexity theorem for the eigenvalues of $Φ(A^p)^{1/2}Ψ(B^p)Φ(A^p)^{1/2}$ as a function of $p\ge0$, where $A,B$ are positive semidefinite matrices and $Φ,Ψ$ are positive linear maps between matrix algebras. A similar generalization of the log-majorization of Ando-Hiai type is given as well.

math.RA↗
arXiv2015

Concavity of certain matrix trace and norm functions. II

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{Tr}\,f(Φ(A^p)^{1/2}Ψ(B^q)Φ(A^p)^{1/2})$ and symmetric (anti-) norm functions of the form $\|f(Φ(A^p)\,σ\,Ψ(B^q))\|$, where $Φ$ and $Ψ$ are positive linear maps, $σ$ is an operator mean, and $f(x^γ)$ with a certain power $γ$ is an operator monotone function on $(0,\infty)$. Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on $(0,\infty)$ so that we prove joint concavity/convexity of more trace functions of Lieb type.

math.FA↗
arXiv2015

Operator $k$-tone functions and analytic functional calculus

Operator $k$-tone functions on an open interval of the real line, which are higher order extensions of operator monotone and convex functions, are characterized via certain inequalities for the real and imaginary parts of analytic functional calculus by those functions.

math.FA↗
arXiv2015

Anti-norms on finite von Neumann algebras

As the reversed version of usual symmetric norms, we introduce the notion of symmetric anti-norms $\|\cdot\|_!$ defined on the positive operators affiliated with a finite von Neumann algebra with a finite normal trace. Related to symmetric anti-norms, we develop majorization theory and superadditivity inequalities of the form $\|ψ(A+B)\|_!\ge\|ψ(A)\|_!+\|ψ(B)\|_!$ for a wide class of functions $ψ$.

math.OA↗
arXiv2014

Operator log-convex functions and operator means

We study operator log-convex functions on $(0,\infty)$, and prove that a continuous nonnegative function on $(0,\infty)$ 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.

math.FA↗
arXiv2014

Higher order extension of Löwner'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.

math.FA↗
arXiv2014

Orbital free pressure and its Legendre transform

Orbital counterparts of the free pressure and its Legendre transform (or $η$-entropy) are introduced and studied in comparison with other entropy quantities in free probability theory and in relation to random multi-matrix models.

math.OA↗
arXiv2013

Families of completely positive maps associated with monotone metrics

An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps. We study the question of when these operators define maps which are also completely positive (CP). Although A --> D^{-1/2} A D^{-1/2} is the only case for which both the map and its inverse are CP, there are several well-known one parameter families for which either the map or its inverse is CP. We present a complete analysis of the behavior of these families, as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points. Our primary tool is an order relation based on the concept of positive definite functions. Although some results can be obtained from known properties, we also prove new results based on the positivity of the Fourier transforms of certain functions. Concrete computations of certain Fourier transforms not only yield new examples of positive definite functions, but also examples in the much stronger class of infinitely divisible functions.

quant-ph↗