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Journal of Mathematical Physics

A perturbation of CHSH inequality induced by fluctuations of ensemble distributions

Andrei Yu. Khrennikov

Published Sep 1, 2000• 36 citations
Publisher

Abstract

We reconsider the theory of hidden variables under the assumption that the conjecture on the ensemble (experiment run) independence of the distribution of hidden variables (which was indirectly used by J. Bell and his followers) is violated. Ensemble fluctuations imply perturbations of Bell’s inequality and its generalizations. We study (by experimental reasons) CHSH (Clauser, Horne, Shimony, Holt) inequality and obtain its modification. This modified inequality is not in disaccord with the predictions of quantum formalism. The deviation from the standard CHSH inequality depends on the magnitude of ensemble fluctuations. We find these magnitude for fluctuating families of Gaussian distributions. We found that if the dimension of the space of hidden variables is very high, then to obtain a contradiction between the local realism and quantum formalism, we must be sure there is no even negligibly small deviations in probability distributions of hidden variables corresponding to different runs of the experiment (in particular, the efficiency of detectors must be equal to one).

DOI
10.1063/1.1287917

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We reconsider the theory of hidden variables under the assumption that the conjecture on the ensemble (experiment run) independence of the distribution of hidden variables (which was indirectly used by J. Bell and his followers) is violated. Ensemble fluctuations imply perturbations of Bell’s inequality and its generalizations. We study (by experimental reasons) CHSH (Clauser, Horne, Shimony, Holt) inequality and obtain its modification. This modified inequality is not in disaccord with the predictions of quantum formalism. The deviation from the standard CHSH inequality depends on the magnitude of ensemble fluctuations. We find these magnitude for fluctuating families of Gaussian distributions. We found that if the dimension of the space of hidden variables is very high, then to obtain a contradiction between the local realism and quantum formalism, we must be sure there is no even negligibly small deviations in probability distributions of hidden variables corresponding to different runs of the experiment (in particular, the efficiency of detectors must be equal to one).
In this work we aim to analyze the Clauser-Horne-Shimony-Holt CHSH inequality strictly in the context of probability theory. In the course of assembling inequality we have to take care not to produce assumptions a priori, that is, physically or intuitively accepted suppositions. Of course, this does not mean that after these considerations, we put the adequate physical conditions suitable and generally accepted in these contexts. This allows us to clearly visualize the possibility of finding a greater inequality than that of CHSH in which it is included. This inequality does not contradict the CHSH inequality. This result is suported by using a robust computational simulation, showing the possibility for obtaining an inequality for quantum mechanics that is not violated and allows random hidden variables without any conflict with Bell's inequality.
A number of papers have suggested that it is inappropriate to combine data from different experiments when undertaking experimental tests of Bell's inequalities. It has been suggested that a correct analysis, using a single probability space, leads to inequalities which are not violated by experiment. If correct, this would be contrary to the normal interpretation of such experimental data. However, in this note, a generalised Clauser-Horne-Shimony-Holt (CHSH) inequality is derived for a system of four experiments constructed on a single probability space which combines the data from the four experiments. It is shown that this leads to the standard CHSH inequality which is normally used to interpret experimental data. Thus the commonly accepted conclusion that experimental violations of Bell's inequality imply that local realistic models are inconsistent with the predictions of quantum mechanics has not been challenged.
In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nthn\mathrm{th}nth particle pair would be independent of measurement choices and outcomes for the first $(n\ensuremath{-}1)$ pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the two-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the two-sided memory loophole allows a systematic violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality when the data are analyzed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in the existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearized CHSH inequality as well as other Bell-type inequalities.
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.
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 validity of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for hidden-variable models of the Bell type and the quantum prediction that said inequality can be violated are two indisputable assertions. However, one can prove wrong the assertion that no hidden-variable model is able to reproduce quantum predictions about the Bell-CHSH test. Here, we reproduce all quantum predictions of Bell-CHSH tests by means of the Bell-Kochen-Specker hidden-variable model for a qubit plus Einstein-Podolsky-Rosen “criterion of physical reality”.