Bounding Large-Scale Bell Inequalities

TL;DR

This paper introduces a fast, memory-efficient method for bounding quantum violations of large-scale Bell inequalities, enabling analysis of problems previously computationally infeasible.

Contribution

It combines the NPA hierarchy, alternating projections, and L-BFGS to efficiently compute upper bounds on quantum violations for large Bell inequalities.

Findings

Method is ~100x faster than MOSEK and SCS for large inequalities.

Provides bounds within ~2% of the optimal solution.

Applicable to inequalities with up to 130 inputs per side.

Abstract

Bell inequalities are an important tool for studying non-locality, however quickly become computationally intractable as the system size grows. We consider a novel method for finding an upper bound for the quantum violation of such inequalities by combining the NPA hierarchy, the method of alternating projections, and the memory-efficient optimisation algorithm L-BFGS. Whilst our method may not give the tightest upper bound possible, it often does so several orders of magnitude faster than state-of-the-art solvers, with minimal memory usage, thus allowing solutions to problems that would otherwise be intractable. We benchmark using the well-studied I3322 inequality as well as a more general large-scale randomized inequality RXX22. For randomized inequalities with 130 inputs either side (a first-level moment matrix of size 261x261), our method is ~100x faster than both MOSEK and SCS whilst giving a bound only ~2% above the optimum.