Algebra of Nonlocal Boxes and the Collapse of Communication Complexity
Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, Clément Pellegrini
TL;DR
The paper develops an algebraic framework for nonlocal boxes under wirings, introducing the box product $\mathtt{P} \boxtimes_\mathsf{W} \mathtt{Q}$ and embedding $\mathcal{N\!S}$ into a bilinear algebra. It defines the orbit of a box as all boxes obtainable by finite wirings, reveals striking alignment and parallelism in these orbits, and shows how right-multiplication structures can yield higher CHSH values. The authors provide both numerical algorithms (gradient descent with line-search and resets) and analytical proofs to identify new collapsing boxes and regions, recovering and extending known results such as the triangle PR–P0–P1 being collapsing and the existence of quantum voids. The work connects the algebraic properties of wirings to questions about the collapse of communication complexity, offering a unified lens on prior numerical and analytical results and highlighting open directions toward a more complete map of collapsing boxes. Overall, the study advances understanding of how distillation protocols and wiring order shape the landscape of nonlocal correlations and their computational power, while suggesting avenues for broader generalization beyond depth-2 wirings.
Abstract
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.