Bell Inequalities with Communication Assistance

Abstract: In this paper we consider the possible correlations between two parties using local machines and shared randomness with an additional amount of classical communication. This is a continuation of the work initiated by Bacon and Toner in Ref. [\textit{Phys. Rev. Lett.} \textbf{90}, 157904 (2003)] who characterized the correlation polytope for $2\times 2$ measurement settings with binary outcomes plus one bit of communication. Here, we derive a complete set of Bell Inequalities for $3\times 2$ measurement settings and a shared bit of communication. When the communication direction is fixed, nine Bell Inequalities characterize the correlation polytope, whereas when the communication direction is bi-directional, 143 inequalities describe the correlations. We then prove a tight lower bound on the amount of communication needed to simulate all no-signaling correlations for a given number of measurement settings.