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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Restrictions on Nonlocal Hidden-Variable Theory

J. H. McGuire, Edward S. Fry

Published Jan 15, 1973• 6 citations
Publisher

Abstract

In a recent experiment, Freedman and Clauser have found that Bell's inequality for local hidden-variable theories is violated by over six standard deviations. We derive an inequality similar to Bell's for nonlocal hidden-variable theories in which the hidden-variable density distribution of the particles measured is nonlocal, but where the detectors are contained within some finite volume. The measurement of Freedman and Clauser is used to put a lower limit on the nonlocality of the hidden-variable density distributions of the photons detected in their experiment.

DOI
10.1103/physrevd.7.555

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In a recent experiment, Freedman and Clauser have found that Bell's inequality for local hidden-variable theories is violated by over six standard deviations. We derive an inequality similar to Bell's for nonlocal hidden-variable theories in which the hidden-variable density distribution of the particles measured is nonlocal, but where the detectors are contained within some finite volume. The measurement of Freedman and Clauser is used to put a lower limit on the nonlocality of the hidden-variable density distributions of the photons detected in their experiment.
The assumption of local realism imposes constraints, such as Bell inequalities, on quantities obtained from measurements. In recent years, various tests of local realism have gained popularity in undergraduate laboratories, giving students the exciting opportunity to experimentally contradict this philosophical assumption. The standard test of the CHSH (Clauser–Horne–Shimony–Holt) Bell inequality requires 16 measurements, whereas a test of Freedman's inequality requires only three measurements. The calculations required to test Freedman's inequality are correspondingly simpler and the theory is less abstract. We suggest that students may benefit from testing Freedman's inequality before proceeding to the CHSH inequality and other more complicated experiments. Our measured data violated Freedman's inequality by more than six standard deviations.
Cooperative multiparticle quantum states in an Einstein-Podolsky-Rosen type of experiment are shown to violate a Bell inequality. Thus correlations can occur between wave packets each having NNN bosons which cannot be explained within local hidden-variable theories. This provides a way to test quantum measurement theory for multiparticle states. In these states, a local, positive probability distribution exists for all NNN. However, the representation of correlations with this probability distribution is different from that in standard (Bell-type) hidden-variable theories.
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.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
Standard Bell inequalities apply to correlations arising when two or more macroscopically separated systems are each subjected to a single ideal measurement. While these inequalities demonstrate that quantum mechanics and local hidden variable theories yield incompatible predictions for pure states, they fail for mixed states: For mixed states, correlations arising from a single ideal measurement on each system may obey standard Bell inequalities, yet when each system is subjected to a sequence of ideal measurements the correlations are nonlocal. For some situations even this last procedure fails, and we must consider more complex (``nonideal'') measurements.
The violation of the Bell inequalities implies that quantum mechanics cannot be interpreted using any local hidden variable theory. However, particular quantum states that can originally be described using a local hidden variable model surprisingly exhibit nonlocality by employing local filters before a standard Bell test is performed. This is referred to as hidden nonlocality. In this study, we provide the experimental demonstration of hidden nonlocality through linear optics towards the local states which are put forword by Hirsch et al., PRL 111, 160402 (2013). A class of local states is generated through a spontaneous parametric down-conversion process, and the violation of the Clauser-Horne-Shimony-Holt (CHSH)-Bell inequality is observed by applying local filters. Our experimental results confirm the superiority of local filters, and throw light on deep understanding the intriguing phenomena of hidden nonlocality and local states.