  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.
