Bell Inequalities with Communication Assistance

TL;DR

This paper derives Bell inequalities for scenarios with two parties, multiple measurement settings, shared randomness, and classical communication, providing a complete characterization of the correlation polytope.

Contribution

It introduces a complete set of Bell inequalities for 3x2 measurement settings with communication, extending previous work and analyzing communication bounds for simulating correlations.

Findings

Nine Bell inequalities for fixed communication direction

143 Bell inequalities for bi-directional communication

Tight lower bounds on communication for simulating no-signaling correlations

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.