Self-testing tilted strategies for maximal loophole-free nonlocality

TL;DR

This paper identifies optimal quantum strategies for achieving maximal loophole-free nonlocality with inefficient detectors, demonstrating their uniqueness and robustness through analytical and numerical methods.

Contribution

It introduces a method to determine optimal tilted Bell strategies under detector inefficiencies and proves their uniqueness and self-testing properties.

Findings

Optimal strategies are those that maximally violate tilted Bell inequalities.

The doubly-tilted CHSH strategies are unique up to local isometries.

High levels of the NPA hierarchy cannot reach the maximum violation for these inequalities.

Abstract

The degree of experimentally attainable nonlocality, as gauged by the loophole-free or effective violation of Bell inequalities, remains severely limited due to inefficient detectors. We address an experimentally motivated question: Which quantum strategies attain the maximal loophole-free nonlocality in the presence of inefficient detectors? For any Bell inequality and any specification of detection efficiencies, the optimal strategies are those that maximally violate a tilted version of the Bell inequality in ideal conditions. In the simplest scenario, we demonstrate that the quantum strategies that maximally violate the doubly-tilted versions of Clauser-Horne-Shimony-Holt inequality are unique up to local isometries. We utilize a Jordan's lemma and Gr\"obner basis-based proof technique to analytically derive self-testing statements for the entire family of doubly-tilted CHSH inequalities and numerically demonstrate their robustness. These results enable us to reveal the insufficiency of even high levels of the Navascu\'es--Pironio--Ac\'in hierarchy to saturate the maximum quantum violation of these inequalities.