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Communication complexity quantifies how difficult it is for two distantcomputers to evaluate a function f(X,Y) where the strings X and Y aredistributed to the first and second computer, respectively and under theconstraint of exchanging as few bits as possible. Surprisingly, some nonlocalboxes, which are resources shared by the two computers, are so powerful thatthey allow to collapse communication complexity, in the sense that any Booleanfunction f can be correctly estimated with the exchange of only one bit ofcommunication. The Popescu-Rohrlich (PR) box is an example of such a collapsingresource, but a comprehensive description of the set of collapsing nonlocalboxes remains elusive.  In this work, we carry out an algebraic study of the structure of wiringsconnecting 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 surprisinggeometrical properties about the alignment and parallelism of distilled boxes.The power of this new framework is that it allows to prove previously-reportednumerical observations concerning the best way to wire consecutive boxes, andto numerically and analytically recover recently-identified noisy PR boxes thatcollapse communication complexity for different types of noise models.

Algebra of Nonlocal Boxes and the Collapse of Communication Complexity