Better bounds on finite-order Grothendieck constants

TL;DR

This paper improves bounds on finite-order Grothendieck constants using a Frank-Wolfe approach, solving complex optimization problems to enhance understanding of quantum mechanics and Bell inequalities.

Contribution

It introduces new methods to lower bound Grothendieck constants for dimensions up to 9, including symmetric instances and heuristic solutions, advancing the state of the art.

Findings

New lower bounds for $K_G(d)$ for $d\,\leq 9$

Construction of symmetric instances achieving higher bounds

Improved bounds on $K_G(d\mapsto 2)$ related to quantum mechanics

Abstract

Grothendieck constants $K_G(d)$ bound the advantage of $d$-dimensional strategies over $1$-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from $d=2$, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For $d\in\{3,4,5\}$, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for $d\leqslant9$. For $d\in\{4,7,8\}$, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants $K_G(d\mapsto2)$ as the advantage of complex $d$-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on $K_G(d\mapsto2)$.