  Non-local correlations are among the most fascinating features of quantum
theory 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 -- in terms of classical information. In these contexts, the
question is crucial whether such correlations can be amplified or distilled,
i.e., whether and how 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 introduce a general
framework of reductions between multi-party input-output systems. Within this
formalism, we show that a natural n-party generalization of the well-known
Popescu-Rohrlich box can be distilled, by an adaptive protocol, to the
algebraic maximum.We use this result further to show that a much broader class
of correlations, including all purely threepartite correlations, can be
distilled from arbitrarily weak to almost maximal strength with partial
communication, i.e., using only a subset of the channels required for the
creation of the same correlation from scratch. Alternatively, this means that
arbitrarily weak non-local correlations can have a "communication value" in the
context of the generation of maximal non-locality.
