Beating one bit of communication with and without quantum pseudo-telepathy

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.