Extended Bell inequality and maximum violation

TL;DR

This paper extends Bell inequality to include entangled states with parallel spins, demonstrating that maximum violation depends on state parameters and measurement directions, with implications for photon pair experiments.

Contribution

It introduces an extended Bell inequality applicable to both parallel and antiparallel entangled spins, unifying local and non-local quantum correlations in a single formalism.

Findings

Extended Bell inequality valid for parallel and antiparallel spins

Maximum violation of Bell inequality can reach 2 depending on parameters

Violation depends on state parameters and measurement directions

Abstract

The original formula of Bell inequality (BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the state-density operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by $P_{B}$ is always less than or at most equal to one for the local realistic model ($P_{B}^{lc}\leq1$) regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as $P_{B}^{\max}$ $=2$, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.