arXiv subjects

Fumio Hiai

Publications and source records attributed to Fumio Hiai.

At least 19 recordsLinked to original sources

arXiv2026

Hockey stick $f$-divergences

In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum $f$-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick $f$-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick $f$-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi $α$-divergences coincide with the Petz-type Rényi divergences for $α\in(0,1)$ and with the sandwiched Rényi divergences for $α>1$. Moreover, we give some partial results on the characterization of when different notions of quantum $f$-divergences give the same value on a pair of quantum states.

quant-ph↗

arXiv2026

Log-majorizations between quasi-geometric type means for matrices

In this paper, for $α\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{α,p}(A,B):=\mathcal{M}_α(A^p,B^p)^{1/p}$ of several $α$-weighted geometric type matrix means $\mathcal{M}_α(A,B)$ such as the $α$-weighted geometric mean in Kubo--Ando's sense, the Rényi mean, etc. The log-majorization $\mathcal{M}_{α,p}(A,B)\prec_{\log}\mathcal{N}_{α,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $α$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{α,p}$ is also discussed based on theory of quantum divergences.

math.FA↗

arXiv2025

Various inequalities between quasi-arithmetic mean and quasi-geometric type means for matrices

In this paper, for $0<α<1$, $p>0$ and positive semidefinite matrices $A,B\ge0$, we consider the quasi-extension $\mathcal{A}_{α,p}(A,B):=((1-α)A^p+αB^p)^{1/p}$ of the $α$-weighted arithmetic matrix mean, and the quasi-extensions $\mathcal{M}_{α,p}(A,B):=\mathcal{M}_α(A^p,B^p)^{1/p}$ of several different $α$-weighted geometric-type matrix means $\mathcal{M}_α(A,B)$ such as the $α$-weighted geometric mean in Kubo and Ando's sense and two types of $α$-weighted version of Fiedler and Pták's spectral geometric mean, as well as the Rényi mean and the $α$-weighted Log-Euclidean mean. For these we examine the inequalities $\mathcal{A}_{α,p}(A,B)\triangleleft\mathcal{A}_{α,q}(A,B)$ and $\mathcal{M}_{α,p}(A,B)\triangleleft\mathcal{A}_{α,q}(A,B)$ of arithmetic-geometric type, where $\triangleleft$ is one of several different matrix orderings varying from the strongest Loewner order to the weakest order determined by trace inequality. For each choice of the above inequalities, our goal is to hopefully obtain the necessary and sufficient condition on $p,q,α$ under which the inequality holds for all $A,B\ge0$.

math.FA↗

arXiv2024

$α$-$z$-Rényi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in $α,z$

We study the $α$-$z$-Rényi divergences $D_{α,z}(ψ\|φ)$ where $α,z>0$ ($α\ne1$) for normal positive functionals $ψ,φ$ on general von Neumann algebras, introduced in [S.~Kato and Y.~Ueda, arXiv:2307.01790] and [S.~Kato, arXiv:2311.01748]. We prove the variational expressions and the data processing inequality (DPI) for the $α$-$z$-Rényi divergences. We establish the sufficiency theorem for $D_{α,z}(ψ\|φ)$, saying that for $(α,z)$ inside the DPI bounds, the equality $D_{α,z}(ψ\circγ\|φ\circγ)=D_{α,z}(ψ\|φ)<\infty$ in the DPI under a quantum channel (or a normal $2$-positive unital map) $γ$ implies the reversibility of $γ$ with respect to $ψ,φ$. Moreover, we show the monotonicity properties of $D_{α,z}(ψ\|φ)$ in the parameters $α,z$ and their limits to the normalized relative entropy as $α\nearrow1$ and $α\searrow1$.

quant-ph↗

arXiv2024

Log-majorization and matrix norm inequalities with application to quantum information

We are concerned with log-majorization for matrices in connection with the multivariate Golden--Thompson trace inequality and the Karcher mean (i.e., a multivariate extension of the weighted geometric mean). We show an extension of Araki's log-majorization and apply it to the $α$-$z$-Rényi divergence in quantum information. We discuss the equality cases in the multivariate trace inequality of Golden--Thompson type and in the norm inequality for the Karcher mean. The paper includes an appendix to correct the proof of the author's old result on the equality case in the norm inequality for the weighted geometric mean.

math.FA↗

arXiv2023

Equality cases in monotonicity of quasi-entropies, Lieb's concavity and Ando's convexity

We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let $Φ:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$ be a trace-preserving map such that $Φ^*$ is a Schwarz map. When $f$ is an operator monotone or operator convex function on $[0,\infty)$, we present several equivalent conditions for the equality $S_f^K(Φ(ρ)\|Φ(σ))=S_f^{Φ^*(K)}(ρ\|σ)$ to hold for given positive operators $ρ,σ$ on $\mathcal{H}$ and $K\in\mathcal{B}(\mathcal{K})$. The conditions include equality cases in the monotonicity versions of Lieb's concavity and Ando's convexity theorems. Specializing the map $Φ$ we have equivalent conditions for equality cases in Lieb's concavity and Ando's convexity. Similar equality conditions are discussed also for monotone metrics and $χ^2$-divergences. We further consider some types of linear preserver problems for those quantum information quantities.

quant-ph↗

arXiv2023

Some continuity properties of quantum Rényi divergences

In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel Rényi $α$-divergences over all $α>1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched Rényi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) Rényi divergences, which may be of independent interest.

quant-ph↗

arXiv2023

Quantum Rényi divergences and the strong converse exponent of state discrimination in operator algebras

The sandwiched Rényi $α$-divergences of two finite-dimensional quantum states play a distinguished role among the many quantum versions of Rényi divergences as the tight quantifiers of the trade-off between the two error probabilities in the strong converse domain of state discrimination. In this paper we show the same for the sandwiched Rényi divergences of two normal states on an injective von Neumann algebra, thereby establishing the operational significance of these quantities. Moreover, we show that in this setting, again similarly to the finite-dimensional case, the sandwiched Rényi divergences coincide with the regularized measured Rényi divergences, another distinctive feature of the former quantities. Our main tool is an approximation theorem (martingale convergence) for the sandwiched Rényi divergences, which may be used for the extension of various further results from the finite-dimensional to the von Neumann algebra setting. We also initiate the study of the sandwiched Rényi divergences of pairs of states on a $C^*$-algebra, and show that the above operational interpretation, as well as the equality to the regularized measured Rényi divergence, holds more generally for pairs of states on a nuclear $C^*$-algebra.

quant-ph↗

arXiv2022

Test-measured Rényi divergences

One possibility of defining a quantum Rényi $α$-divergence of two quantum states is to optimize the classical Rényi $α$-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi $α$-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi $α$-divergence coincides with the sandwiched Rényi $α$-divergence when $α>1$. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider $2$-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi $α$-divergence for $n$ copies might require a number of measurement outcomes that diverges in $n$, in general). In view of this, it seems natural to expect the same when $α<1$; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using $2$-outcome measurements is in general strictly smaller than the Rényi $α$-divergence (which is unique in the classical case). In the general quantum case this shows that the above "regularized test-measured" Rényi $α$-divergence is not even a quantum extension of the classical Rényi divergence when $α<1$, in sharp contrast to the $α>1$ case.

quant-ph↗

arXiv2021

Pusz--Woronowicz functional calculus and extended operator convex perspectives

In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,\infty)^2$. Our construction has special features that functions on $[0,\infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $(-\infty,\infty]$. Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $(0,\infty)$. We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Ando's theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.

math.FA↗

arXiv2021

Connections of unbounded operators and some related topics: von Neumann algebra case

The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $τ$-measurable operators (affiliated with a semi-finite von Neumann algebra equipped with a trace $τ$), (ii) positive elements in Haagerup's $L^p$-spaces, (iii) semi-finite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of non-commutative (Hilsum) $L^p$-spaces.

math.OA↗

arXiv2020

Concise lectures on selected topics of von Neumann algebras

A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970's by Araki, Connes, Haagerup, Takesaki and others, which are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. The course was aimed at giving a fast track study of those main classics of the theory, from which people gain an enough background knowledge so that they can consult suitable volumes when more details are needed.

math.OA↗

arXiv2019

Ando-Hiai type inequalities for operator means and operator perspectives

We improve the existing Ando-Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie-Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.

math.FA↗

arXiv2019

Operator means of probability measures

Let $\mathbb{P}$ 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 $\mathbb{P}$, in parallel with Kubo and Ando's definition of two-variable operator means, and show that every operator mean is contractive for the $\infty$-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.

math.FA↗

arXiv2018

Matrix limit theorems of Kato type related to positive linear maps and operator means

We obtain limit theorems for $Φ(A^p)^{1/p}$ and $(A^pσB)^{1/p}$ as $p\to\infty$ for positive matrices $A,B$, where $Φ$ is a positive linear map between matrix algebras (in particular, $Φ(A)=KAK^*$) and $σ$ is an operator mean (in particular, the weighted geometric mean), which are considered as certain reciprocal Lie-Trotter formulas and also a generalization of Kato's limit to the supremum $A\vee B$ with respect to the spectral order.

math.FA↗

arXiv2018

Log-majorization related to Rényi divergences

For $α,z>0$ with $α\ne1$, motivated by comparison between different kinds of Rényi divergences in quantum information, we consider log-majorization between the matrix functions \begin{align*} P_α(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^αB^{1/2}, \\ Q_{α,z}(A,B)&:=(B^{1-α\over2z}A^{α\over z}B^{1-α\over2z})^z \end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $α,z$ for which $P_α(A,B)\prec_{\log}Q_{α,z}(A,B)$ and $Q_{α,z}(A,B)\prec_{\log}P_α(A,B)$ holds, respectively.

math.FA↗

arXiv2018

Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences

As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerup's $L^1$-space, we present the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.

math-ph↗

arXiv2018

Convergence theorems for barycentric maps

We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $β$, and then give convergence theorems for martingales of $β$-conditional expectations. We give the Birkhoff ergodic theorem for $β$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $β$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on $M$. Finally, the large derivation property of $β$-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

math.PR↗