Computing quantum Bell inequalities

TL;DR

This paper investigates the boundary of quantum correlations by computing quantum Bell inequalities using quantifier elimination, focusing on a bipartite scenario with three measurement settings, and relates findings to the Tsirelson-Landau-Masanes inequality.

Contribution

It introduces a method to compute quantum Bell inequalities via quantifier elimination and characterizes a specific relaxation of the quantum set in a bipartite scenario with three measurements.

Findings

Characterization of a linear relaxation of the quantum set.

Quantum Bell inequalities equivalent to the Tsirelson-Landau-Masanes arcsin inequality.

Application of quantifier elimination to quantum correlation boundaries.

Abstract

Understanding the limits of quantum theory in terms of uncertainty and correlation has always been a topic of foundational interest. Surprisingly this pursuit can also bear interesting applications such as device-independent quantum cryptography and tomography or self-testing. Building upon a series of recent works on the geometry of quantum correlations, we are interested in the problem of computing quantum Bell inequalities or the boundary between quantum and post-quantum world. Better knowledge of this boundary will lead to more efficient device-independent quantum processing protocols. We show that computing quantum Bell inequalities is an instance of a quantifier elimination problem, and apply these techniques to the bipartite scenario in which each party can have three measurement settings. Due to heavy computational complexity, we are able to obtain the characterization of certain "linear" relaxation of the quantum set for this scenario. The resulting quantum Bell inequalities are shown to be equivalent to the Tsirelson-Landau-Masanes arcsin inequality, which is the only type of quantum Bell inequality found since 1987.