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arXiv:1802.08074

A non-local linear dynamical system and violation of Bell's inequality

Renzo Mosetti

Published Feb 21, 2018• 0 citations
arXivHTMLPDF

Abstract

A simple classical non-local dynamical system with random initial conditions and an output projecting the state variable on selected axes has been defined to mimic a two-channel quantum coincidence experiment. Non-locality is introduced by a parameter connecting the initial conditions to the selection of the projection axes. The statistics of the results shows violations up to 100% of the Bell's inequality, in the form of Clauser-Horne- Shimony-Holt (CHSH), strongly depending on the non-locality parameter. Discussions on the parallelism with Bohmian mechanics are given.

DOI
10.48550/arxiv.1802.08074

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A simple classical non-local dynamical system with random initial conditions and an output projecting the state variable on selected axes has been defined to mimic a two-channel quantum coincidence experiment. Non-locality is introduced by a parameter connecting the initial conditions to the selection of the projection axes. The statistics of the results shows violations up to 100% of the Bell's inequality, in the form of Clauser-Horne- Shimony-Holt (CHSH), strongly depending on the non-locality parameter. Discussions on the parallelism with Bohmian mechanics are given.
We present an extremely simple linear dynamical model which, through initial conditions modified as a function of the measurement settings, produces violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality. The model is not a classi- cal counterexample to Bell’s theorem, since it explicitly breaks the assumption of independence between hidden variables and measurement settings. Instead, it pro- vides a clear example of superdeterminism and can be used as an educational tool to illustrate the crucial role of the assumptions of locality, realism, and freedom of choice in Bell’s theorem.
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Quantum theory predicts violations of Bell’s inequality. This implies a contradiction with at least one of three statistical conditions, known as “Realism” (joint distribution of relevant observables), “Locality” (conditional separability of pairwise observations, given a set of hidden variables), and “Free Will” (statistical independence between hidden variables and measurement settings). Here we describe a classical system with “clockwork” properties that is unquestionably local. It produces maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It contradicts every one of the three listed formal conditions. Nonetheless, it expresses the qualitative markers of ontological Realism, ontological Locality, and ontological Free Will, all at the same time. In other words, the three listed conditions do not correspond to the three ontological concepts with the same names. Therefore, there is no conflict between quantum theory and Local Realism.
Bellʼs inequality fundamentally changed our understanding of quantum mechanics. Bellʼs insight that non-local correlations between quantum systems cannot be explained classically can be verified experimentally, and has numerous applications in modern quantum information. Today, the Clauser–Horne–Shimony–Holt (CHSH) inequality is probably the most well-known Bell inequality and it has given us a wealth of understanding in what differentiates the classical from the quantum world. Yet, there are certainly other means of quantifying 'Bell non-locality without inequalities' such as the famous Hardyʼs paradox. As such, one may wonder whether these are entirely different approaches to non-locality. For this anniversary issue, we unify the perspective of the CHSH inequality and Hardy's paradox into one family of non-local games which include both as special cases. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.