In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched R\'enyi divergences such as the sandwiched R\'enyi conditional entropy. We follow three different approaches: The first one is the "almost additive approach", which exploits the sub-/ superadditivity and joint concavity/ convexity of the exponential of the divergence. In our second approach, termed the "operator space approach", we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched R\'enyi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAFF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

翻译：在本工作中，我们证明了与夹层Rényi散度相关的熵度量（如夹层Rényi条件熵）的一致连续性界。我们遵循三种不同方法：第一种是“近似可加性方法”，该方法利用散度指数函数的次/超可加性及联合凹/凸性。第二种称为“算子空间方法”，我们将熵度量表示为范数，并利用其性质建立界。这些范数的构造受到插值空间范数的启发。我们不仅完全依赖矩阵分析工具证明了范数性质，还将其适用性扩展到资源理论相关的语境中。通过这种方式，我们拓展了Marwah与Dupuis以及Beigi与Goodarzi在夹层Rényi条件熵研究中采用的策略。最后，我们将这些方法融合为具有若干优势特性的混合方法，进而讨论每种界在不同参数区域的最优表现。我们的结果改进了先前的最佳连续性界，有时甚至给出了首个可用的连续性界。在另一项独立贡献中，我们采用部分作者先前提出的ALAFF方法，研究了近似量子马尔可夫链的稳定性。