Bounds on information combining are a fundamental tool in coding theory, in particular when analyzing polar codes and belief propagation. They usually bound the evolution of random variables with respect to their Shannon entropy. In recent work this approach was generalized to Renyi $\alpha$-entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of $\alpha=2$ we give the first optimal information combining bounds with quantum side information.

翻译：信息组合的界是编码理论中的基本工具，尤其在极化码和置信传播分析中至关重要。它们通常基于香农熵来刻画随机变量的演化过程。近期研究将该方法推广至Renyi $\alpha$-熵，然而由于Renyi熵缺乏传统的链式法则，相关理论框架尚不完整。本文通过建立连接Hayashi与Arimoto两种Renyi熵定义的Renyi链式法则，填补了这一缺失环节，从而提出了Arimoto Renyi熵的新型信息组合界。第二部分将链式法则推广至量子场景，并展示其如何使先前仅适用于冯·诺依曼熵的结果与猜想得到推广。特别地，针对$\alpha=2$的特例，我们首次给出了含量子边信息的最优信息组合界。