Bell nonlocality with a single shot

TL;DR

This paper introduces a method to optimize Bell inequalities for single-shot tests of nonlocality, enabling the rejection of local hidden variables with minimal p-value without statistical accumulation.

Contribution

It develops algorithms to transform Bell inequalities into nonlocal games with maximal gaps and efficiently compute local bounds, enhancing single-shot nonlocality tests.

Findings

Largest possible gap approaches one for certain inequalities

Single-shot rejection of local hidden variables demonstrated

New algorithm for faster local bound calculations

Abstract

In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a p-value it will give for a physical experiment. Here we show that to obtain a small expected p-value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and $I_{nn22}$ inequalities. We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small p-value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the na\"ive approach, which may be of independent interest.