On small neighborhoods of the capacity-achieving input distributions, the decrease of the mutual information with the distance to the capacity-achieving input distributions is bounded below by a linear function of the square of the distance to the capacity-achieving input distributions for all channels with (possibly multiple) linear constraints and finite input sets using an identity due to Tops{\o}e and Pinsker's inequality. Counter examples demonstrating non-existence of such a quadratic bound are provided for the case of infinite many linear constraints and the case of infinite input sets. Using a Taylor series approximation, rather than Pinsker's inequality, the exact characterization of the slowest decrease of the mutual information with the distance to the capacity-achieving input distributions is determined on small neighborhoods of the capacity-achieving input distributions. Analogous results are established for classical-quantum channels whose output density operators are defined on a separable Hilbert spaces. Implications of these observations for the channel coding problem and applications of the proof technique to related problems are discussed.

翻译：对于具有（可能多个）线性约束和有限输入集的所有信道，利用Tops{\o}e恒等式和Pinsker不等式，在容量可达输入分布的小邻域内，互信息随到容量可达输入分布距离的下降被该距离平方的线性函数界定为下界。针对无限多个线性约束和无限输入集的情形，提供了此类二次型界不存在的反例。通过使用泰勒级数近似（而非Pinsker不等式），在容量可达输入分布的小邻域内，确定了互信息随距离下降最慢程度的精确刻画。对于输出密度算子定义在可分离希尔伯特空间上的经典-量子信道，建立了类似的结果。讨论了这些观察结果对信道编码问题的启示，以及该证明技术在相关问题中的应用。