Bell’s Inequality

David K. Ferry

Published Jan 2, 2019• 0 citations

Abstract

In the early 1960s, John Bell decided to look once again at the hidden variable problem in quantum mechanics, through an examination of the A. Einstein, with B. Podolsky and N. Rosen (EPR) paradox. Proposals for proofs against the possibility of hidden variables had been advanced both before and after EPR. Bell developed both a “theorem” and an inequality, but this may be overstating the case. Most interpretations are that Bell is ruling out any “action-at-a-distance” type of interactions or variables, but would accept that such forces could violate the inequality. By triple of random variables, mentioned by A. Yu. Krennikov, one has to think once again about the experiment suggested by the Bell inequality. Bell has assumed that the set of hidden variables, and more importantly the probability function of the hidden variables, is unchanging. K. Hess and Walter Philipp began to worry about this assumption, because it has some implicit assumptions of its own.

DOI: 10.1201/9781351207232-8

Related papers

The EPR-B Paradox Resolution. Bell inequalities revisited.

Jaykov Foukzon, Менькова Елена Романовна, Alexander A Potapov

Abstract One of the Bell’s assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell’s inequality was derived in its traditional form, without resorting to the hypothesis of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell’s inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained under new EPR-B nonlocality postulate. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The revisited Bell type inequality in canonical notatons reads 〈AB〉 + 〈A’B〉 + 〈AB’〉 − 〈A’B’〉 ⩽ 6.

Journal of Physics Conference SeriesNov 1, 20190 citations

Several foundational and information theoretic implications of Bell’s theorem

Guruprasad Kar, Manik Banik

In 1935, Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics (QM). EPR also postulated the existence of more fundamental theory where physical reality of any system would be completely described by the variables/states of that fundamental theory. This variable is commonly called hidden variable and the theory is called hidden variable theory (HVT). In 1964, John Bell proposed an empirically verifiable criterion to test for the existence of these HVTs. He derived an inequality, which must be satisfied by any theory that fulfill the conditions of locality and reality. He also showed that QM, as it violates this inequality, is incompatible with any local-realistic theory. Later it has been shown that Bell’s inequality (BI) can be derived from different set of assumptions and it also find applications in useful information theoretic protocols. In this review, we will discuss various foundational as well as information theoretic implications of BI. We will also discuss about some restricted nonlocal feature of quantum nonlocality and elaborate the role of Uncertainty principle and Complementarity principle in explaining this feature.

arXiv:1408.0526International Journal of Quantum InformationAug 18, 20165 citations

Why Bell’s inequality is wrong?

WU Xin-zhong

This paper examines the history from EPR argument to Bell’s inequality and the influence of Bell’s inequality on the frontier exploration of quantum mechanics. It is found that Bell’s inequality has some hidden assumptions different from EPR argument: Following Schrodinger's assumption that the measurement process of EPR experiment undergos time, Karl Hess regards time as an implicit variable of time series, and Joy Christian obtains a local implicit variable inequality different from Bell's inequality. They extended the relativistic space-time geometric analysis method to the localized geometric morphological analysis of quantum states, and considered that the Bell’s inequality reasoning did not meet the requirements of the completeness of EPR argument. Quantum statistics given by identical particles is different from classical statistics. Innovative ensemble interpretation can also solve Bell's inequality and give a demonstration of quantum non-locality. The main research progress of quantum gravity is great, but there are many divergences. According to Einstein's local realism, the new explanation and expression of quantum mechanics may open up new ideas for the study of quantum gravity.

Sep 11, 20200 citations

Spin-1 photons, spin-1/2 electrons, Bell's inequalities, and Feynman's special perspective on quantum mechanics

Masud Mansuripur

The Einstein-Podolsky-Rosen (EPR) paradox that argues for the incompleteness of quantum mechanics as a description of physical reality has been put to rest by John Bell’s famous theorem, which inspired numerous experimental tests and brought about further affirmations of quantum reality. Nevertheless, in his writings and public presentations, Richard Feynman never acknowledged the significance of Bell’s contribution to the resolution of the EPR paradox. In this paper, we discuss several variants of the Bell inequalities (including one that was specifically espoused by Feynman), and explore the ways in which they demolish the arguments in favor of local hidden-variable theories. We also examine the roots of Feynman’s attitude toward Bell’s theorem in the context of Feynman’s special perspective on quantum mechanics.

arXiv:2210.03365SpintronicsOct 3, 20223 citations

The 1960s

Andrew Charles Whitaker

During the 1960s, Bell produced his seminal work on the foundations of quantum theory, first showing that, contrary to von Neumann’s argument, hidden variable theories were allowed but then that they had to be non-local. Thus, he showed that quantum theory did not respect local causality, and he outlined experiments that could be used to test whether the assumption of local realism was true in quantum theory: Bell’s theorem, or Bell’s inequality, or Bell’s inequalities. His work was followed up by John Clauser, Abner Shimony, Michael Horne, and Richard Holt, who produced the CHSH inequality. Bell had moved to CERN, where he worked on the theory of neutrino experiments, and on nuclear and elementary particle physics, making crucial suggestions concerning the use of gauge theory for each type of physical force and also producing the Adler–Bell–Jackiw anomaly, or the ABJ anomaly.

Oxford University Press eBooksJun 16, 20160 citations

The Bell Inequality and the Many-Worlds Interpretation

Lev Vaidman A

It is argued that the lesson we should learn from Bell's inequalities is not that quantum mechanics requires some kind of action at a distance, but that it leads us to believe in parallel worlds. Introduction Bell's work [1] led to a revolution in our understanding of nature. I remember attending my first physics conference, on “Microphysical Reality and Quantum Formalism,” in Urbino, 1985. Most of the talks were about Aspect's experiment [2] confirming the nonlocality of quantum mechanics based on experimental violations of Bell's inequalities. Although I did not share the skepticism of many speakers regarding the results of Aspect, I was not ready to accept that a local action in one place can instantaneously change anything at another place. So, while for the majority the lesson from Bell was that quantum mechanics requires some “spooky action at a distance”, I was led by Bell's result to an alternative revolutionary change in our view of nature. I saw no other way than accepting the manyworlds interpretation (MWI) of quantum mechanics [3, 4]. I shall start by presenting the Einstein–Podolsky–Rosen (EPR) [5] argument. Then I will present Bell's presented using the Greenberger–Horne–Zeilinger setup [6] in the form proposed by Mermin [7, 8]. The discussion of nonlocality will suggest that Bell's inequalities are the only manifestation of action at a distance in nature. The demonstration of the necessity of action at a distance will be done through a detailed analysis of the GHZ experiment. Then I shall show how multiple worlds resolve the problem of action at a distance. After discussing the issue of nonlocality in the MWI, I shall conclude by citing Bell's views on the MWI. EPR–Bell–GHZ The story of Bell cannot be told without first describing the EPR argument. Instead of following the historical route, I shall use the GHZ setup, which, in my view, is the clearest way to explain the EPR and Bell's discovery.

arXiv:1501.02691Cambridge University Press eBooksAug 31, 201613 citations