Reconsidering Local Hidden Variables: When One is Enough

TL;DR

This paper introduces a new Bell inequality based on single Local Hidden Variable models, which simplifies device-independent quantum key distribution and highlights differences between various quantum correlations.

Contribution

It derives a Bell inequality for s-LHV models, showing their sufficiency for device-independent QKD and entanglement witnessing, distinct from full LHV models.

Findings

s-LHV models can be ruled out by the inequality

Device-independent QKD is simplified with s-LHV models

Correlation requirements differ from Bell-nonlocality and EPR-steering

Abstract

In this Letter, we explore the possibility of developing Bell inequalities predicated on models using a single Local Hidden Variable (s-LHV), a strict subset of general LHV models. Because of the less strenuous constraints imposed by s-LHV models, we were able to derive a contingent Bell inequality in analogy to the CHSH inequality, but which does not require bounding of measurement statistics. Following this, we show by explicit example that there are cases of states that rule out s-LHV models by violating our inequality, but which nonetheless have a multivariate LHV model. Even so, we show how merely ruling out s-LHV models is still sufficient to allow for fully device independent quantum key distribution (QKD) and entanglement witnessing. This being the case, our inequality illustrates two things. First, it makes fully device-independent QKD on continuous variables substantially more straightforward. Second, it shows how the degree of correlation needed to demonstrate device-independent QKD is distinct from both Bell-nonlocality and EPR-steering. Although nonlocality in general requires ruling out all LHV models, s-LHV models are sufficiently useful to warrant further exploration.