NPA Hierarchy and Extremal Criterion in the Simplest Bell Scenario

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.