Improved local models & new Bell inequalities via Frank-Wolfe algorithms

In Bell scenarios with two outcomes per party, we algorithmically consider the two sides of the membership problem for the local polytope: constructing local models and deriving separating hyperplanes, that is, Bell inequalities.

We take advantage of the recent developments in so-called Frank--Wolfe algorithms to significantly increase the convergence rate of existing methods. As an application, we study the threshold value for the nonlocality of two-qubit Werner states under projective measurements. Here, we improve on both the upper and lower bounds present in the literature, almost halving the length of the accompanying range. Importantly, our bounds are entirely analytical and yield refined bounds 1.4376⩽KG(3)⩽1.4546 on the value of the Grothendieck constant of order three. We also demonstrate the efficiency of our approach in multipartite Bell scenarios, and present the first local models for all projective measurements with visibilities noticeably higher than the entanglement threshold. We make our entire code accessible as a Julia library called BellPolytopes.jl.