NPA Hierarchy and Extremal Criterion in the Simplest Bell Scenario

TL;DR

This paper investigates the extremal conditions of quantum correlations in the simplest Bell scenario using the NPA hierarchy, confirming the validity of a plausible analytical criterion and simplifying the extremal analysis.

Contribution

The study demonstrates that the extremal conditions can be verified at the $1+AB$ level of the NPA hierarchy, validating a previously plausible criterion for the simplest Bell scenario.

Findings

The $1+AB$ and second levels of the NPA hierarchy are equivalent for correlations.

The extremal condition can be replaced by the $1+AB$ level criterion.

The plausible criterion for extremality is confirmed and justified.

Abstract

It is difficult to establish an analytical criterion to identify the boundaries of quantum correlations, even for the simplest Bell scenario. Here, we briefly reviewed the plausible analytical criterion, and we found a way to confirm the extremal conditions from another direction. For that purpose, we analyzed the Navascu\'es-Pironio-Ac\'{\i}n (NPA) hierarchy to study the algebraic structure and found that the problem could not be simplified using $1\!+\!AB$ level. However, considering the plausible criterion, the $1\!+\!AB$ and second levels for correlations were equal, and the extremal condition in the simplest Bell scenario was replaced by that in the $1\!+\!AB$ level. Thus, the correctness of the plausible criterion was verified, and the results demonstrated that the plausible criterion held, thereby explaining its simplicity. It seemed plausible, but now it becomes more certain.