According to Bell’s theorem, certain entangled states cannot be simulated classically using local hidden variables (LHV). Suppose that we can augment LHV by some amount of classical communication. The question then arises as to how many bits are needed to simulate entangled states? There is very strong evidence that a single bit of communication is powerful enough to simulate projective measurements on any two-qubit entangled state. However, the problem of simulating measurements on higher-dimensional systems remains largely unexplored. In this study, we present Bell-like scenarios, even with three inputs per party, in which bipartite correlations resulting from measurements on higher-dimensional states cannot be simulated with a single bit of communication. We consider the case where the communication direction is fixed and the case where it is bidirectional. To this end, we introduce constructions based on parallel repetition of pseudo-telepathy games and an original algorithm based on branch-and-bound technique to compute the one-bit classical bound. Two copies of emblematic Bell expressions, such as the Magic square pseudo-telepathy game, prove to be particularly powerful, requiring a 16 × 16 state to beat the bidirectional one-bit classical bound, and look a promising candidate for implementation on an optical platform.

根据贝尔定理，某些纠缠态不能使用局部隐变量（LHV）进行经典模拟。假设我们可以通过一定量的经典交流来增强 LHV。那么问题来了，需要多少比特来模拟纠缠态？有非常有力的证据表明，单个比特的通信足以模拟任何两个量子比特纠缠态的投影测量。然而，在高维系统上模拟测量的问题在很大程度上仍未得到探索。在这项研究中，我们提出了类似贝尔的场景，即使每方有三个输入，其中由高维状态测量产生的二方相关性也无法用单比特通信来模拟。我们考虑通信方向固定的情况和双向的情况。为此，我们引入了基于伪心灵感应游戏的并行重复的构造以及基于分支定界技术来计算一位经典界限的原始算法。象征性贝尔表达式的两个副本，例如魔方伪心灵感应游戏，被证明特别强大，需要 16 × 16 状态来击败双向一位经典界限，并且看起来是在光学平台上实现的有希望的候选者。