  Non-local correlations are one of the most fascinating consequences of
quantum physics from the point of view of information: Such correlations,
although not allowing for signaling, are unexplainable by pre-shared
information. The correlations have applications in cryptography, communication
complexity, and sit at the very heart of many attempts of understanding quantum
theory -- and its limits -- better in terms of classical information. In these
contexts, the question is crucial whether such correlations can be distilled,
i.e., whether weak correlations can be used for generating (a smaller amount
of) stronger. Whereas the question has been studied quite extensively for
bipartite correlations (yielding both pessimistic and optimistic results), only
little is known in the multi-partite case. We show that a natural
generalization of the well-known Popsecu-Rohrlich box can be distilled, by an
adaptive protocol, to the algebraic maximum. We use this result further to show
that a much bigger class of correlations, including all purely three-partite
correlations, can be distilled from arbitrarily weak to maximal strength with
partial communication, i.e., using only a subset of the channels required for
the creation of the same correlation from scratch. In other words, we show that
arbitrarily weak non-local correlations can have a "communication value" in the
context of the generation of maximal non-locality.
