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 axiomatic 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 ALAAF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

翻译：本文证明了与夹层Rényi散度（如夹层Rényi条件熵）相关的熵量的均匀连续性界。我们采用三种不同的方法：第一种是公理化方法，利用散度指数的次可加性/超可加性以及联合凹性/凸性；第二种方法称为“算子空间方法”，将熵测度表示为范数并利用其性质建立界。这些范数源于插值空间范数。我们不仅仅依赖矩阵分析工具证明范数性质，还将适用性扩展到资源理论相关情境。通过此方法，扩展了Marwah和Dupuis以及Beigi和Goodarzi在夹层Rényi条件熵背景下采用的策略。最后，我们将这些方法合并为具有有利性质的混合方法，并讨论各方法在不同场景下的最优表现。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。在独立贡献中，我们使用部分作者先前文章开发的ALAAF方法研究近似量子马尔可夫链的稳定性。