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Conference on Lasers and Electro-Optics

The CHSH-Bell inequality and Tsirelson’s bound with postselection

Dominic William Berry, Hyunseok Jeong, Magdalena Stobińska, Timothy C. Ralph

Published May 4, 2008• 0 citations
Publisher

Abstract

We show the necessary and sufficient conditions on loss for valid CHSH-Bell experiments, and propose an experiment with loss which should violate Tsirelsonpsilas bound for entangled states.
DOI
10.1109/qels.2008.4553020

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Almost all Bell inequality experiments to date have used postselection and therefore relied on the fair sampling assumption for their interpretation. The standard form of the fair sampling assumption is that the loss is independent of the measurement settings, so the ensemble of detected systems provides a fair statistical sample of the total ensemble. This is often assumed to be needed to interpret Bell inequality experiments as ruling out hidden-variable theories. Here we show that it is not necessary; the loss can depend on measurement settings, provided the detection efficiency factorizes as a function of the measurement settings and any hidden variable. This condition implies that Tsirelson's bound must be satisfied for entangled states. On the other hand, we show that it is possible for Tsirelson's bound to be violated while the Clauser-Horne-Shimony-Holt (CHSH)-Bell inequality still holds for unentangled states, and present an experimentally feasible example.
The aim of this paper is to attract the attention of experimenters to the original Bell (OB) inequality that was shadowed by the common consideration of the Clauser-Horne-Shimony-Holt (CHSH) inequality. There are two reasons to test the OB inequality and not the CHSH inequality. First of all, the OB inequality is a straightforward consequence to the Einstein-Podolsky-Rosen (EPR) argumentation. In addition, only this inequality is directly related to the EPR-Bohr debate. The second distinguishing feature of the OB inequality was emphasized by Itamar Pitowsky. He pointed out that the OB inequality provides a higher degree of violations of classicality than the CHSH inequality. For the CHSH inequality, the fraction of the quantum (Tsirelson) bound Q CHSH = 2 2 to the classical bound C CHSH = 2 , i.e., F CHSH = Q CHSH C CHSH = 2 is less than the fraction of the quantum bound for the OB inequality Q OB = 3 2 to the classical bound C OB = 1 , i.e., F OB = Q OB C OB = 3 2 . Thus, by violating the OB inequality, it is possible to approach a higher degree of deviation from classicality. The main problem is that the OB inequality is derived under the assumption of perfect (anti-) correlations. However, the last few years have been characterized by the amazing development of quantum technologies. Nowadays, there exist sources producing, with very high probability, the pairs of photons in the singlet state. Moreover, the efficiency of photon detectors was improved tremendously. In any event, one can start by proceeding with the fair sampling assumption. Another possibility is to use the scheme of the Hensen et al. experiment for entangled electrons. Here, the detection efficiency is very high.