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 exponent analysis. In quantum information theory, there are three generalizations of the classical R\'enyi divergence -- the Petz's, sandwiched, and log-Euclidean versions, that possess meaningful operational interpretation. However, the associated quantum R\'enyi and Augustin information are much less explored compared with their classical counterpart, and lacking crucial properties hinders applications of these quantities to error exponent analysis in the quantum regime. The goal of this paper is to analyze fundamental properties of the R\'enyi and Augustin information from a noncommutative measure-theoretic perspective. Firstly, we prove the uniform equicontinuity for all three quantum versions of R\'enyi and Augustin information, and it hence yields the joint continuity of these quantities in order and prior input distributions. Secondly, we establish the concavity of the scaled R\'enyi and Augustin information in the region of $s\in(-1,0)$ for both Petz's and the sandwiched versions. This completes the open questions raised by Holevo [IEEE Trans.~Inf.~Theory, 46(6):2256--2261, 2000], and Mosonyi and Ogawa [Commun.~Math.~Phys., 355(1):373--426, 2017]. For the applications, we show that the strong converse exponent in classical-quantum channel coding satisfies a minimax identity, which means that the strong converse exponent can be attained by the best constant composition code. The established concavity is further employed to prove an entropic duality between classical data compression with quantum side information and classical-quantum channel coding, and a Fenchel duality in joint source-channel coding with quantum side information.

翻译：R\'enyi 和 Augustin 信息是通过 R\'enyi 差异定义的相互信息的一般化。 由于缺乏关键特性, 无法在量子系统中应用这些数量来评估信息处理任务的表现。 在量子信息理论中, 有三种经典R\'enyi 差异的概括化R\'enyi 和 log- Euclidean 版本, 它们都有有意义的操作解释。 然而, 相关的量R\' enyi 和 Augustin 信息与其古典对应方相比, 缺乏关键特性, 从而妨碍这些数量应用在量子系统中用于错误的正数分析。 本文的目标是从非对等测量度的度- enyi 和 Augustin 信息的基本特性分析 R\'enyyi 和 Exmical 信息。 首先, 我们证明了所有三种量子的量组、 eqiventi 和 A. frecialal 的 comnal 和 a.