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 and 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 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.

翻译：通信复杂度量化了两台远程计算机在尽可能少交换比特的约束下，计算函数f(X,Y)的难度，其中字符串X和Y分别分配给第一台和第二台计算机。令人惊讶的是，某些非局域盒子（两台计算机共享的资源）具有强大的能力，以至于它们能够导致通信复杂度的坍塌——即任何布尔函数f仅通过交换一个比特的通信量就能被正确估计。波佩斯库-罗里希（PR）盒子便是这种坍塌性资源的一个例子，但关于坍塌性非局域盒子集合的全面描述仍难以捉摸。在本工作中，我们对连接非局域盒子的布线结构进行了代数研究，从而定义了"盒子乘积"的概念$P\boxtimes Q$，并证明了相关的结合律与交换律性质。这催生了"盒子轨道"的概念，揭示了关于蒸馏盒子对齐性与平行性的惊人几何性质。这一新框架的强大之处在于，它能够证明先前文献中关于连续盒子最优布线方式的数值观察结果，并能数值化与解析化地重现近期发现的、在不同噪声模型下导致通信复杂度坍塌的含噪PR盒子。