Hardy's Test versus the CHSH Test of Quantum Non-Locality: Fundamental and Practical Aspects

TL;DR

This paper compares Hardy's and CHSH tests of quantum non-locality, analyzing their theoretical foundations and proposing a robust mesoscopic circuit implementation with practical considerations for experimental imperfections.

Contribution

It clarifies the geometric relationship between Hardy's equations and the CHSH inequality and proposes a feasible mesoscopic circuit for testing quantum non-locality.

Findings

Hardy's test equations generalize to the CHSH inequality with imperfections.

The proposed circuit is robust against fluctuations and particle loss.

Both tests can be implemented with simple gate voltage adjustments.

Abstract

We compare two different tests of quantum non-locality, both in theoretical terms and with respect to a possible implementation in a mesoscopic circuit: Hardy's test [Hardy, Phys. Rev. Lett. \textbf{68}, 2981 (1992)] and the CHSH test, the latter including a recently discovered inequality relevant for experiments with three possible outcomes [Collins and Gisin, J. Phys. A \textbf{37}, 1775 (2004)]. We clarify the geometry of the correlations defined by Hardy's equations with respect to the polytope of causal correlations, and show that these equations generalize to the CHSH inequality if the slightest imperfections in the setup need to be taken into account. We propose a mesoscopic circuit consisting of two interacting Mach-Zehnder interferometers in a Hall bar system for which both Hardy's test and the CHSH test can be realized with a simple change of gate voltages, and evaluate the robustness of the two tests in the case of fluctuating experimental parameters. The proposed setup is remarkably robust and should work for fluctuations of beam splitter angles or phases up to the order of one radian, or single particle loss rates up to about 15%.