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 Navascués-Pironio-Ac\'ın (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.

5 pages, 2 figures

References in corpus (5)

• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Bounding the set of quantum correlations
• The uncertainty principle determines the non-locality of quantum mechanics
• Bounding the set of finite dimensional quantum correlations
• Quantum Correlations in the Minimal Scenario