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arXiv:2406.12184 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Explaining Bell locally

Charles Alexandre Bédard

Published Oct 1, 2025• 1 citation
arXivHTMLPDF

Abstract

In the Heisenberg picture of unitary quantum theory, Bell inequalities are violated with local elements of reality interacting locally. Here is how: Upon measuring her particle of the entangled pair, Alice—like other coupled systems—smoothly and locally evolves into two non-interacting versions of herself, each of which records a different outcome: she foliates . Everything that suitably interacts with the Alices foliates in turn, generating worlds which, for all practical purposes, remain distinct and autonomous. At spacelike separation, an analogous yet independent process occurs to Bob when he measures his particle, locally differentiating him and his surroundings into two non-interacting instances. To confirm the violation of Bell inequalities, Alice and Bob must further interact to produce a record of the joint outcomes. The record arises from the two local worlds of Alice, and those of Bob, and foliates into four instances: ‘ 00 ’, ‘01’, ‘10’ and ‘11’. The outcomes that win the Clauser–Horne–Shimony–Holt (CHSH) game sum to a measure of cos 2 ⁡ ( π / 8 ) .

DOI
10.1098/rspa.2025.0553

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In the Heisenberg picture of unitary quantum theory, Bell inequalities are violated with local elements of reality interacting locally. Here is how: Upon measuring her particle of the entangled pair, Alice—like other coupled systems—smoothly and locally evolves into two non-interacting versions of herself, each of which records a different outcome: she foliates . Everything that suitably interacts with the Alices foliates in turn, generating worlds which, for all practical purposes, remain distinct and autonomous. At spacelike separation, an analogous yet independent process occurs to Bob when he measures his particle, locally differentiating him and his surroundings into two non-interacting instances. To confirm the violation of Bell inequalities, Alice and Bob must further interact to produce a record of the joint outcomes. The record arises from the two local worlds of Alice, and those of Bob, and foliates into four instances: ‘ 00 ’, ‘01’, ‘10’ and ‘11’. The outcomes that win the Clauser–Horne–Shimony–Holt (CHSH) game sum to a measure of cos 2 ⁡ ( π / 8 ) .
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.
The problem of the existence of nonlocal effects in Quantum Mechanics is discussed. The problem is divided in two: the first ('soft') one is to explain the violation of Bell's inequalities as a statistical magnitude. This can be achieved by a simple model within non-Boolean Locality and Realism. This result shows that quantum non-Locality as a consequence of the statistical violation of Bell's inequalities is inexistent. The second ('hard') problem is to explain the violation as it is calculated from series of detection outcomes. L.Sica has demonstrated that, in order to violate Bell's inequalities, the series recorded at (say) Bob when the setting at station Alice is alfa, can be different from the series that would have been recorded at Bob if that setting had been alfa'instead. Therefore, non-Locality in the series of detection outcomes does exist. It cannot be experimentally verified because of its counterfactual nature, but is observed in computer simulations. An appropriate computer code is based on the simple model mentioned plus a contextual instruction. It explains 'how' (Sica's) non-Locality arises, and solves the hard problem. 'Why' the contextual instruction exists is explained by Hellwig and Kraus' postulate of covariant quantum state collapse. In consequence, (Sica's) non-Locality is not in contradiction with Relativity but, quite the opposite, it is implied by Relativistic covariance.
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.
Recently the authors of Phys. Rev. Lett. 125, 090401 (2020) considered the following scenario: Alice and Bob each have half of a pair of entangled qubit states. Bob measures his half and then passes his part to a second Bob who measures again, and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality with the single Alice. By taking the maximally entangled pure two-qubit state $|\ensuremath{\phi}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})$ as an example, it has been constructively proved that arbitrarily many independent Bobs can share the nonlocality with the single Alice. Here we demonstrate that arbitrarily many independent observers can share the nonlocality of a single arbitrary dimensional bipartite entangled but not necessary two-qubit entangled state. Furthermore, taking the generalized Greenberger-Horne-Zeilinger (GHZ) states as an example, we show that at most two Charlies can share the genuine nonlocality of a single generalized GHZ state with an Alice and a Bob.
Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.
On the 50th anniversary of Bell's monumental 1964 paper, there is still widespread misunderstanding about exactly what Bell proved. This misunderstanding derives in turn from a failure to appreciate the earlier argument of Einstein, Podolsky and Rosen. I retrace the history and logical structure of these arguments in order to clarify the proper conclusion, namely that any world that displays violations of Bell's inequality for experiments done far from one another must be non-local. Since the world we happen to live in displays such violations, actual physics is non-local. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.