Chain rules for one-shot entropic quantities via operational methods

Abstract

We introduce a new operational technique for deriving chain rules for general information theoretic quantities. This technique is very different from the popular (and in some cases fairly involved) methods like SDP formulation and operator algebra or norm interpolation. Instead, our framework considers a simple information transmission task and obtains lower and upper bounds for it. The lower bounds are obtained by leveraging a successive cancellation encoding and decoding technique. Pitting the upper and lower bounds against each other gives us the desired chain rule. As a demonstration of this technique, we derive chain rules for the smooth max mutual information and the smooth-Hypothesis testing mutual information.