Hardy’s test versus the Clauser-Horne-Shimony-Holt test of quantum nonlocality: Fundamental and practical aspects

We compare two different tests of quantum nonlocality, both in theoretical terms and with respect to a possible implementation in a mesoscopic circuit: Hardy’s test [L. Hardy, Phys. Rev. Lett. 68, 2981 (1992)] and the Clauser-Horne-Shimony-Holt (CHSH) test, the latter including a recently discovered inequality relevant for experiments with three possible outcomes [D. Collins and N. Gisin, J. Phys. A 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 1 radian, or single particle loss rates up to about 15%.

Figure 1

(Color online) 3D cross sections of the Hardy's polytope $\mathcal{H}$ for $P(--\mid YY)=0$ (a), 0.25 (b), 0.5 (c), and 1 (d). All cross sections are cut once more at constant values $P(-+\mid YY)$ to reveal the triangular 2D cross sections.

Figure 2

(Color online) (a) A schematic of the mesoscopic circuit with two coupled Mach-Zehnder interferometers fabricated on a quantum Hall bar and (b) its equivalent diagram. The arrowed (black and red) lines are the edge channels of the quantum Hall liquid with filling factor 2. $S_j$ $(j=1,2)$ denote the electron sources, $A_j$ and $B_j$ the beam splitters (quantum point contacts), $D_j$ and $E_j$ the electron detectors, and $C$ the conditional phase shift. The coupling is realized by the Coulomb interaction between the two channels in the gray shaded region in (a).

Figure 3

(Color online) Boundaries of the regions where NCHV theories are expected to be falsified in Hardy’s test (lower surfaces) and the CHSH test (upper surfaces) for a selection of different errors. (a) Allowed fluctuations ${\delta }_1$ and ${\delta }_2$ in ${\theta }_1$ and ${\theta }_2$ of the beam splitters $A_1$ and $A_2$, respectively, when electrons are lost with probability $r$ per interferometer. (b) The same as (a) for the beam splitters $B_1$ and $B_2$. (c) The same as (a), but for the phase fluctuations $\delta {\varphi }_1$ and $\delta {\varphi }_2$ in the two Mach-Zehnder interferometers. (d) Same as (c), but with a phase fluctuation $\delta \varphi$ in the controlled phase shift $U_C\left(\varphi \right)$ instead of $r$ for $r=0$.