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arXiv:2205.06606

The geometric link between Hardy and Clauser-Horne-Shimony-Holt

Johannes Seiler, Thomas Strohm, Wolfgang P. Schleich

Published May 13, 2022• 0 citations
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Abstract

We show that the Hardy nonlocality condition is equivalent to the violation of the CHSH inequality with additional constraints. We adapt the geometrical optimization of the violation of the CHSH inequality to these additional constraints and show that the Hardy condition is equivalent to optimizing the length difference of two sides in a triangle. Furthermore, we discuss the effects of the different constraints.

DOI
10.48550/arxiv.2205.06606

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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 study a recently proposed Einstein–Podolsky–Rosen steering inequality (Cavalcanti et al 2015 J. Opt. Soc. Am. B 32 A74–A81 ). Analogous to Clauser–Horne–Shimony–Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., two parties, two measurements per party and two outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation amount matches with optimal quantum violation of CHSH inequality, i.e., Cirel'son quantity. We further study the optimal violation of this inequality for different classes of 2-qubit quantum states.
A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of 2\sqrt 22​, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.
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'.