A Bell Theorem Without Inequalities for Two Particles, Using Inefficient Detectors

TL;DR

This paper presents a Bell theorem without inequalities for two entangled particles, demonstrating that certain local realistic models are incompatible with quantum correlations even with inefficient detectors.

Contribution

It extends the GHZ theorem to include inefficient detectors and robust local realistic models, without relying on inequalities or sampling assumptions.

Findings

Proves a GHZ-type contradiction for arbitrary detection efficiencies.

Shows local realistic models cannot reproduce quantum perfect correlations.

Demonstrates the theorem applies to individual events without inequalities.

Abstract

We again consider (as in a companion paper) an entangled two-particle state that is produced from two independent down-conversion sources by the process of "entanglement-swapping", so that the particles have never met. We show that there is a natural extension of the Einstein-Pololsky-Rosen discussion of "elements of reality" to include inefficient detectors. We consider inefficient deterministic, local, realistic models of quantum theory that are "robust", which we consider to be the minimum requirement for them to be taken seriously. By robust, we mean they satisfy the following three criteria: (a) they reproduce the quantum results for perfect correlations, if all particles are detected; (b) they produce some counts for every setting of the angles (so they don't describe some experiments that can easily be performed as "impossible"); (c) all their hidden variables are relevant (they must each produce a detectable result in some experiment). For such models we prove a Greenberger-Horne-Zeilinger (GHZ) type theorem for arbitrary detection efficiencies, showing that any such theory is inconsistent with the quantum mechanical perfect correlations. This theorem holds for individual events with no inequalities. As a result, the theorem is also independent of any random sampling hypothesis.