Chain Rules for Renyi Information Combining

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.