NPA Hierarchy and Extremal Criterion in the Simplest Bell Scenario

TL;DR

This paper explores the boundaries of quantum correlations in a basic Bell scenario using the NPA hierarchy and confirms a plausible analytical criterion.

Contribution

The paper verifies a plausible criterion for quantum correlations by analyzing the NPA hierarchy in the simplest Bell scenario.

Findings

The 1+AB level and second level for correlations were found to be equal in the simplest Bell scenario.

The extremal condition was confirmed to be equivalent to the 1+AB level.

The plausible criterion was verified as correct and explained its simplicity.

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-Acín (NPA) hierarchy to study the algebraic structure and found that the problem could not be simplified using the 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.

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