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arXiv:quant-ph/9810010

Maximal violation of Bell's inequality in the case of real experiments

M. Ardehali

Published Oct 4, 1998• 0 citations
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Abstract

Einstein's locality is invoked to derive a correlation inequality. In the case of ideal experiments, this inequality is equivalent to Bell's original inequality of 1965 which, as is well known, is violated by a maximum factor of 1.5. The crucial point is that even in the case of real experiments where polarizers and detectors are non-ideal, the present inequality is violated by a factor of 1.5, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of $\sqrt 2$. The larger magnitude of violation can be of importance for the experimental test of locality. Moreover, the supplementary assumption used to derive this inequality is weaker than Garuccio-Rapisarda assumption. Thus an experiment based on this inequality refutes a larger family of hidden variable theories than an experiment based on Garuccio-Rapisarda inequality.

DOI
10.48550/arxiv.quant-ph/9810010

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Einstein's locality is invoked to derive a correlation inequality. In the case of ideal experiments, this inequality is equivalent to Bell's original inequality of 1965 which, as is well known, is violated by a maximum factor of 1.5. The crucial point is that even in the case of real experiments where polarizers and detectors are non-ideal, the present inequality is violated by a factor of 1.5, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of 2\sqrt 22​. The larger magnitude of violation can be of importance for the experimental test of locality. Moreover, the supplementary assumption used to derive this inequality is weaker than Garuccio-Rapisarda assumption. Thus an experiment based on this inequality refutes a larger family of hidden variable theories than an experiment based on Garuccio-Rapisarda inequality.
An inequality is deduced from Einstein's locality and a supplementary assumption. This inequality defines an experiment that can actually be performed with present technology to test local realism. Quantum mechanics violates this inequality a factor of 1.5. In contrast, quantum mechanics violates previous inequalities [for example, those of J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969); S. J. Freedman and I. F. Clauser ibid. 28, 938 (1972); and J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974)] by a factor of 2.\sqrt{2}.2​. Thus the magnitude of violation of the inequality derived in this paper is approximately 20.7% larger than the magnitude of violation of previous inequalities. This result can be particularly important for the experimental test of locality.
A correlation inequality is derived from local realism and a supplementary assumption. This inequality is violated by a factor of 1.5 in the case of real experiments, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of 2\sqrt 22​. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, the present inequality can be used to test locality very simply because it requires the measurements of only two detection probabilities. In contrast, Clauser-Horne inequality requires the measurements of five detection probabilities.
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.
Bell's theorem of 1965 is a proof that all realistic interpretations of quantum mechanics must be non-local. Bell's theorem consists of two parts: first a correlation inequality is derived that must be satisfied by all local realistic theories; second it is demonstrated that quantum mechanical probabilities violate this inequality in certain cases. In the case of ideal experiments, Bell's theorem has been proven. However, in the case of real experiments where polarizers and detectors are non-ideal, the theorem has not yet been proven since the proof always requires some arbitrary and {\em ad hoc} supplementary assumptions. In this paper, we state a new and rather weak supplementary assumption for the ensemble of photons that emerge from the polarizers, and we show that the conjunction of Einstein's locality with this assumption leads to validity of an inequality that is violated by a factor as large as 1.5 in the case of real experiments. Moreover, the present supplementary assumption is considerably weaker and more general than Clauser, Horne, Shimony, Holt supplementary assumption.
The Bell and the Clauser-Horne-Shimony-Holt inequalities are shown to hold for both the cases of complex and real analytic nonlocality in the setting parameters of Einstein-Podolsky-Rosen-Bohm experiments for spin 1/2 particles and photons, in both the deterministic and stochastic cases. Therefore, the theoretical and experimental violation of the inequalities by quantum mechanics excludes all hidden variables theories with that kind of nonlocality. In particular, real analyticity leads to negative definite correlations, in contradiction with quantum mechanics.
It is demonstrated that the conjunction of the premises of Einstein-Podolsky-Rosen paper with a supplementary assumption lead to validity of an inequality that is sometimes grossly violated by the quantum-mechanical probabilities in the case of real experiments. The supplementary assumption is as follows: For every photon in the state \ensuremath{\lambda}, the sum of the detection probabilities in the ordinary and in the extraordinary beams emerging from a two-way polarizer is less than or equal to the detection probability with the polarizer removed. This assumption, which always holds for an ensemble of photons, is weaker and more general than the assumption of Garuccio and Rapisarda [Nuovo Cimento A 65, 269 (1981)] and the assumption of Clauser and Horne [Phys. Rev. D 10, 526 (1974)]. Hence an experiment based on this assumption refutes a larger family of hidden-variable series than an experiment based on the assumption of Garuccio and Rapisardo or on the assumption of Clauser and Horne.