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" $\mathtt{P}\boxtimes\mathtt{Q}$, 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 intuitions 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交换尽可能少比特的约束下，评估函数$f(X,Y)$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）具有足够强大的能力，能够坍缩通信复杂度，即仅通过交换一个比特的通信即可正确估算任意布尔函数$f$。Popescu-Rohrlich（PR）盒子便是这种坍缩资源的典型例子，但坍缩非局域盒子集合的全面描述仍难以获得。在本研究中，我们对连接非局域盒子的连线结构进行代数分析，由此定义了"盒子乘积"$\mathtt{P}\boxtimes\mathtt{Q}$的概念，并给出了相关的结合律与交换律结果。这进一步引出了"盒子轨道"的概念，揭示了蒸馏盒子的对齐与平行性中令人惊讶的几何性质。这一新框架的强大之处在于：它既能证明此前关于串联连续盒子最佳方式的数值直觉，又能通过数值与解析方法重新获得最近发现的、能够针对不同类型噪声模型坍缩通信复杂度的含噪PR盒子。