Quantum advantages of communication complexity from Bell nonlocality

Communication games are crucial tools for investigating the limitations of physical theories. The communication complexity (CC) problem is a typical example, for which several distributed parties attempt to jointly calculate a given function with limited classical communications. In this work, we present a method to construct CC problems from Bell tests in a graph-theoretic way. Starting from an experimental compatibility graph and the corresponding Bell test function, a target function which encodes the information of each edge can be constructed, then using this target function we could construct an CC function for which by pre-sharing entangled states, the success probability will exceed that for arbitrary classical strategy. The non-signaling protocol based on Popescu-Rohrlich box is also discussed, and the success probability in this case would reach one.