  A function is said to be additive if, similar to mutual information, expands
by a factor of $n$, when evaluated on $n$ i.i.d. repetitions of a source or
channel. On the other hand, a function is said to satisfy the tensorization
property if it remains unchanged when evaluated on i.i.d. repetitions. Additive
rate regions are of fundamental importance in network information theory,
serving as capacity regions or upper bounds thereof. Tensorizing measures of
correlation have also found applications in distributed source and channel
coding problems as well as the distribution simulation problem. Prior to our
work only two measures of correlation, namely the hypercontractivity ribbon and
maximal correlation (and their derivatives), were known to have the
tensorization property. In this paper, we provide a general framework to obtain
a region with the tensorization property from any additive rate region. We
observe that hypercontractivity ribbon indeed comes from the dual of the rate
region of the Gray-Wyner source coding problem, and generalize it to the
multipartite case. Then we define other measures of correlation with similar
properties from other source coding problems. We also present some applications
of our results.
