相关论文: On the Existence of the Augustin Mean
R\'enyi and Augustin information are generalizations of mutual information defined via the R\'enyi divergence, playing a significant role in evaluating the performance of information processing tasks by virtue of its connection to the error…
We propose a Riemannian gradient descent with the Poincar\'e metric to compute the order-$\alpha$ Augustin information, a widely used quantity for characterizing exponential error behaviors in information theory. We prove that the algorithm…
R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by R\'enyi as a measure of information that satisfies almost the same…
The sandwiched R\'enyi $\alpha$-divergences of two finite-dimensional quantum states play a distinguished role among the many quantum versions of R\'enyi divergences as the tight quantifiers of the trade-off between the two error…
In this article, we impose a new class of fractional analytic functions in the open unit disk. By considering this class, we define a fractional operator, which is generalized Salagean and Ruscheweyh differential operators. Moreover, by…
Consider the problems of computing the Augustin information and a R\'{e}nyi information measure of statistical independence, previously explored by Lapidoth and Pfister (IEEE Information Theory Workshop, 2018) and Tomamichel and Hayashi…
Extending M.\ Daws' definition of ultra-amenable Banach algebras, we introduce the notion of operator ultra-amenability for completely contractive Banach algebras. For a locally compact group $G$, we show that the operator ultra-amenability…
We study the computation of the Petz-Augustin mean of order $\alpha \in (0,1) \cup (1,\infty)$, defined as the minimizer of a weighted sum of $n$ Petz-R\'enyi divergences of order $\alpha$ over the set of $d$-by-$d$ quantum states, where…
The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers…
We introduce a new quantum R\'enyi divergence $D^{\#}_{\alpha}$ for $\alpha \in (1,\infty)$ defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an…
Abstract convexity generalises classical convexity by considering the suprema of functions taken from an arbitrarily defined set of functions. These are called the abstract linear (abstract affine) functions. The purpose of this paper is to…
The sandwiched R\'enyi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched R\'enyi divergences as the operationally relevant…
We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is…
Uhlmann's theorem is a cornerstone of quantum information theory, stating that for any quantum state $\rho_{AB}$ and any state $\sigma_A$, there exists an extension $\sigma_{AB}$ of $\sigma_A$ such that the fidelity between $\rho_{AB}$ and…
We obtain uniqueness theorems for harmonic and subharmonic functions of a new type. They lead to new analytic extension criteria and new conditions for stability of operator semigroups in Banach spaces with Fourier type.
We consider a monotone increasing operator in an ordered Banach space having $u_-$ and $u_+$ as a strong super- and subsolution, respectively. In contrast with the well studied case $u_+ < u_-$, we suppose that $u_- < u_+$. Under the…
It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }…
In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $\left(\varphi_1, \varphi_2\right)-$convex function $g, $ with arbitrarily small norm, such that $f + g…
Classical Banach principle is an essential tool for the investigation of the ergodic properties of Cesaro subsequences. The aim of this work is to extend Banach principle to the case of the stochastic convergence in the operator algebras.…
This paper introduces statistical order convergence and its pointwise variant for sequences of order bounded operators between Riesz spaces. We establish fundamental properties: uniqueness of the limit, stability under lattice operations,…