Mutual information chain rules for security proofs robust against device imperfections

In this work we derive a number of chain rules for mutual information quantities, suitable for analyzing quantum cryptography with imperfect devices that leak additional information to an adversary. First, we derive a chain rule between smooth min-entropy and smooth max-information, which improves over previous chain rules for characterizing one-shot information leakage caused by an additional conditioning register. Second, we derive an ''information bounding theorem'' that bounds the R\'enyi mutual information of a state produced by a sequence of channels, in terms of the R\'enyi mutual information of the individual channel outputs, similar to entropy accumulation theorems. In particular, this yields simple bounds on the smooth max-information in the preceding chain rule. Third, we derive chain rules between R\'enyi entropies and R\'enyi mutual information, which can be used to modify the entropy accumulation theorem to accommodate leakage registers sent to the adversary in each round of a protocol. We show that these results can be used to handle some device imperfections in a variety of device-dependent and device-independent protocols, such as randomness generation and quantum key distribution.