  Communication complexity quantifies how difficult it is for two distant
computers to evaluate a function f(X,Y), where the strings X and Y are
distributed to the first and second computer respectively, under the constraint
of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which
are resources shared by the two computers, are so powerful that they allow to
collapse communication complexity, in the sense that any Boolean function f can
be correctly estimated with the exchange of only one bit of communication. The
Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a
comprehensive description of the set of collapsing nonlocal boxes remains
elusive.
  In this work, we carry out an algebraic study of the structure of wirings
connecting nonlocal boxes, thus defining the notion of the "product of boxes"
$P\boxtimes Q$, and we show related associativity and commutativity results.
This gives rise to the notion of the "orbit of a box", unveiling surprising
geometrical properties about the alignment and parallelism of distilled boxes.
The power of this new framework is that it allows us to prove
previously-reported numerical observations concerning the best way to wire
consecutive boxes, and to numerically and analytically recover
recently-identified noisy PR boxes that collapse communication complexity for
different types of noise models.
