  We introduce a new information-theoretic complexity measure $IC_\infty$ for
2-party functions which is a lower-bound on communication complexity, and has
the two leading lower-bounds on communication complexity as its natural
relaxations: (external) information complexity ($IC$) and logarithm of
partition complexity ($\text{prt}$), which have so far appeared conceptually
quite different from each other. $IC_\infty$ is an external information
complexity measure based on R\'enyi mutual information of order infinity. In
the definition of $IC_\infty$, relaxing the order of R\'enyi mutual information
from infinity to 1 yields $IC$, while $\log \text{prt}$ is obtained by
replacing protocol transcripts with what we term "pseudotranscripts," which
omits the interactive nature of a protocol, but only requires that the
probability of any transcript given the inputs $x$ and $y$ to the two parties,
factorizes into two terms which depend on $x$ and $y$ separately. Further
understanding $IC_\infty$ might have consequences for important direct-sum
problems in communication complexity, as it lies between communication
complexity and information complexity.
  We also show that applying both the above relaxations simultaneously to
$IC_\infty$ gives a complexity measure that is lower-bounded by the (log of)
relaxed partition complexity, a complexity measure introduced by Kerenidis et
al. (FOCS 2012). We obtain a sharper connection between (external) information
complexity and relaxed partition complexity than Kerenidis et al., using an
arguably more direct proof.
