  We study formal properties of smooth max-information, a generalization of von
Neumann mutual information derived from the max-relative entropy. Recent work
suggests that it is a useful quantity in one-shot channel coding, quantum rate
distortion theory and the physics of quantum many-body systems.
  Max-information can be defined in multiple ways. We demonstrate that
different smoothed definitions are essentially equivalent (up to logarithmic
terms in the smoothing parameters). These equivalence relations allow us to
derive new chain rules for the max-information in terms of min- and
max-entropies, thus extending the smooth entropy formalism to mutual
information.
