The quadratic decaying property of the information rate function states that given a fixed conditional distribution $p_{\mathsf{Y}|\mathsf{X}}$, the mutual information between the (finite) discrete random variables $\mathsf{X}$ and $\mathsf{Y}$ decreases at least quadratically in the Euclidean distance as $p_\mathsf{X}$ moves away from the capacity-achieving input distributions. It is a property of the information rate function that is particularly useful in the study of higher order asymptotics and finite blocklength information theory, where it was already implicitly used by Strassen [1] and later, more explicitly, by Polyanskiy-Poor-Verd\'u [2]. However, the proofs outlined in both works contain gaps that are nontrivial to close. This comment provides an alternative, complete proof of this property.

翻译：信息率函数的二次衰减性质指出，在给定条件分布 $p_{\mathsf{Y}|\mathsf{X}}$ 的情况下，当 (有限) 离散随机变量 $\mathsf{X}$ 与 $\mathsf{Y}$ 之间的互信息量随 $p_\mathsf{X}$ 偏离容量可达输入分布的欧氏距离增加时，其衰减至少呈二次速率。该性质是信息率函数的一种特性，在高阶渐近分析和有限块长信息论研究中尤为重要。Strassen [1] 曾在其工作中隐含使用此性质，而 Polyanskiy-Poor-Verdú [2] 则更明确地加以运用。然而，两篇文献中概述的证明均存在难以填补的漏洞。本评注为该性质提供了一种替代性的完整证明。