Position-Momentum Bell-Nonlocality with Entangled Photon Pairs

TL;DR

This paper explores the potential for demonstrating Bell nonlocality using entangled photon pairs' transverse positions and momenta, reimagining Bell's original approach with modern quantum optics techniques.

Contribution

It reformulates Bell's strategy for continuous-variable nonlocality using photon pairs' transverse degrees of freedom and estimates the maximum CHSH violation achievable with Bell's wavefunction.

Findings

Maximum violation for Bell's wavefunction is very small

Bell's wavefunction does not produce a large CHSH violation

Other states may achieve higher violations

Abstract

Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Though Bell nearly showed a violation using the CHSH inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he doesn't give a quantitative estimation of the maximum achievable violation for the wavefunction he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of $2\sqrt{2}$. Although Bell's wavefunction does not produce a large violation of the CHSH inequality, other states may yet do so.