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arXiv:0808.0052 · Physical Review A

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

Daniel Braun, Mahn‐Soo Choi

Published Sep 19, 2008• 13 citations

Abstract

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%.

DOI: 10.1103/physreva.78.032114

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•Dec 28, 2014•0 citations

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%.

We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between two measurement settings, with each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the four total measurements are carried out by homodyne detection. The largest violation of the CHSH inequality is obtained when all four measurements are squeezed-and-displaced on-off detections. We test the robustness of violations against imperfections in on-off detectors and single-photon sources, finding that the squeezed-and-displaced measurement schemes perform better than the displacement-only measurement schemes.

Bellʼs inequality fundamentally changed our understanding of quantum mechanics. Bellʼs insight that non-local correlations between quantum systems cannot be explained classically can be verified experimentally, and has numerous applications in modern quantum information. Today, the Clauser–Horne–Shimony–Holt (CHSH) inequality is probably the most well-known Bell inequality and it has given us a wealth of understanding in what differentiates the classical from the quantum world. Yet, there are certainly other means of quantifying 'Bell non-locality without inequalities' such as the famous Hardyʼs paradox. As such, one may wonder whether these are entirely different approaches to non-locality. For this anniversary issue, we unify the perspective of the CHSH inequality and Hardy's paradox into one family of non-local games which include both as special cases. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

In this paper we show a Clauser-Horne (CH) inequality for two three-level quantum systems or qutrits, alternative to the CH inequality given by Kaszlikowski et al. [Phys. Rev. A65, 032118 (2002)]. In contrast to this latter CH inequality, the new one is shown to be equivalent to the Clauser-Horne-Shimony-Holt (CHSH) inequality for two qutrits given by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]. Both the CH and CHSH inequalities exhibit the strongest resistance to noise for a nonmaximally entangled state for the case of two von Neumann measurements per site, as first shown by Acin et al. [Phys. Rev. A65, 052325 (2002)]. This equivalence, however, breaks down when one takes into account the less-than-perfect quantum efficiency of detectors. Indeed, for the noiseless case, the threshold quantum efficiency above which there is no local and realistic description of the experiment for the optimal choice of measurements is found to be [Formula: see text] for the CH inequality, whereas it is equal to [Formula: see text] for the CHSH inequality.

The Clauser–Horne–Shimony–Holt (CHSH) inequality and its permutations are necessary and sufficient criteria for Bell nonlocality in the simplest Bell-nonlocality scenario: two parties, two measurements per party and two outcomes per measurement. Here we derive an inequality for Einstein–Podolsky–Rosen (EPR)-steering that is an analog of the CHSH, in that it is necessary and sufficient in this same scenario. However, since in the case of steering the device at Bob’s site must be specified (as opposed to the Bell case, in which it is a black box), the scenario we consider is that where Alice performs two (black-box) dichotomic measurements, and Bob performs two mutually unbiased qubit measurements. We show that this inequality is strictly weaker than the CHSH, as expected, and use it to decide whether a recent experiment [Phys. Rev. Lett.110, 130401 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.130401] involving a single-photon split between two parties has demonstrated EPR-steering.

The Clauser-Horne-Shimony-Holt (CHSH) inequality (and its permutations), are necessary and sufficient criteria for Bell nonlocality in the simplest Bell-nonlocality scenario: 2 parties, 2 measurements per party and 2 outcomes per measurement. Here we derive an inequality for EPR-steering that is an analogue of the CHSH, in that it is necessary and sufficient in this same scenario. However, since in the case of steering the device at Bob's site must be specified (as opposed to the Bell case in which it is a black box), the scenario we consider is that where Alice performs two (black-box) dichotomic measurements, and Bob performs two mutually unbiased qubit measurements. We show that this inequality is strictly weaker than the CHSH, as expected, and use it to decide whether a recent experiment [Phys. Rev. Lett. 110, 130401 (2013).] involving a single-photon split between two parties has demonstrated EPR-steering.