Analytic quantification of the singlet nonlocality for the first Bell's inequality

TL;DR

This paper analytically calculates the volume of violation for the singlet state with respect to the first Bell's inequality, revealing that about one-third of measurement configurations violate local causality.

Contribution

It develops the concept of volume of violation and provides an analytical calculation for the singlet state, enhancing understanding of Bell nonlocality quantification.

Findings

Approximately 33.3% of configurations violate local causality.

Triples of measurement directions leading to violations constitute one-third of all possible configurations.

The work extends the quantification of Bell nonlocality through the volume of violation concept.

Abstract

Recently an alternative way to quantify Bell nonlocality has been proposed [Phys. Rev. A {\bf 92}, 030101(R) (2015)]. In this work we further develop this concept, the volume of violation, and analytically calculate its value for the spin-singlet state with respect to the settings of the first Bell's inequality. These settings correspond to three directions in space, or three arbitrary points on the unit sphere. It is shown that the triples of directions that lead to violations in local causality correspond to $1/3$ of all possible configurations. From the perspective of quantum communications, this means that two distant parties that were capable of align their measurements in one direction only (the remaining direction in each site being random), have a probability of about 33.3$\%$ to be able to certify their entanglement.