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Correlation of – Cos θ Between Measurements in a Bell’s Inequality Experiment Simulation Calculated Using Local Hidden Variables

Austin J. Fearnley

Published Oct 1, 2016• 0 citations
OpenAlex

Abstract

This paper shows in Simulation 2 that aggregates of elementary particles’ hidden variable unit vector spin axes e, when projected onto appropriate detector angle vectors, give values which are in exact accordance with values given by quantum mechanics calculations. However, it is found that no method of calculation breaks the Bell Inequality AB’ + BC’ ≥ AC’: not the method in Simulation 2 and not the quantum mechanics calculation using projection operators.

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This paper shows in Simulation 2 that aggregates of elementary particles’ hidden variable unit vector spin axes e, when projected onto appropriate detector angle vectors, give values which are in exact accordance with values given by quantum mechanics calculations. However, it is found that no method of calculation breaks the Bell Inequality AB’ + BC’ ≥ AC’: not the method in Simulation 2 and not the quantum mechanics calculation using projection operators.
Abstract. For unit vectors a, b, entanglement requires that spin measurements satisfy those just given for a≠b. Bell’s hidden variable quantum mechanical local particle with a=b cannot provide the values for a≠b and, hence, the values for an entangled particle system. The perception that any hidden variable quantum mechanical description of an entangled system of particles is non-local depends on Bell’s inappropriate definition of locality, which excludes entanglement for a distant particle, and the condition that a=b, which does not give the values for a≠b as required to give the quantum mechanical expectation values, when entanglement occurs for systems that are close, not distant, and ignores that entanglement is a local phenomenon. Expecting the condition a=b and a local particle analysis with no distant entangled particle to give the quantum mechanical result is absurd. Any disentanglement of once entangled particles moving apart sets the spin measurements at those given for a≠b if a≠b and those given for a=b if a=b. Any subsequent measurement of the set values agrees with those for a≠b if a≠b and those given for a=b if a=b just as if the particles are entangled still. In examples where Bell’s inequality is supposedly violated, the conditions for which the inequality is true are not met. There is no contradiction.
The basic problem we study here is whether the two-spin correlation experiments together with the theoretical Bell inequalities have already excluded the possibility of introducing hidden variables into quantum theory, as is often concluded. This question is answered negatively by explicitly reproducing the quantum-mechanical two-spin correlation function by a classical model, where the spin is associated with a classical dipolemoment vector. We then study the behavior of single events in classical and quantum models and conclude that the detector efficiency may be a fundamental limitation rather than just a technical problem to be overcome by better techniques. We further show that the assumptions underlying the derivation of Bell inequalities involve statements about single events which are consistent with neither the classical nor quantum models. It is important therefore to work with explicit physical situations rather than with abstract assumptions.
Abstract Bell’s theorem (Bell 1964) establishes that local hidden variable theories are committed to Bell inequalities violated by quantum correlations. Since violations of Bell inequalities can be verified without requiring that the observables whose correlations figure in the inequalities be measured with arbitrarily high precision, Bell’s theorem yields a method of falsifying local hidden variable the­ ories. Moreover, the evidence for violations of Bell inequalities We thank J. Finklestein, N. D. Mermin, and an anonymous referee for helpful comments. A. K. was supported by a Royal Society University Research Fellow­ ship and thanks the Oxford Centre for Quantum Computation for much appreciated hospitality.
Justifying a local hidden variable theory requires an explanation of Bell’s in-equality violation. Ever since Bell derived the inequality to test the classical pre-diction on the correlation of two spin-12 particles, many experiments have observed the violation, and thus concluded against the local realism, while validating the non-locality of quantum entanglement. Still, many scientists remain unconvinced of quantum entanglement because the experiments have loopholes that could po-tentially allow a local realistic explanation. Upholding the local realism, this paper introduces how a precession of the spin would produce a cosine-like correlation function, and furthermore how it would also contribute to a fair sampling loophole. Simulating the precession in Monte Carlo method reveals that it can explain the observed Bell’s violation using only classical mechanics. 1
Bell’s inequality can be traced back to a classical probabilistic formula that is invalid clearly, while spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Bell’s two misunderstandings about Bell’s theorem are pointed out: Firstly, the hidden variable theory Bell’s used, which is a special form of such theories, is regarded as a general one. Secondly, there are two theses obtainable from Bell’s hypotheses; one is compatible with quantum mechanics, the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G.Lochak has revealed the first thesis, and thereby he pointed out that Bell’s inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually without regard to Bell’s inequality.
Bell's theorem supposedly demonstrates an irreconcilable conflict between quantum mechanics and local realistic hidden-variable theories. Most proofs of Bell's theorem, are based on inequalities. In this paper we present an alternative proof which does not involve inequalities, but only a direct comparison between correlation functions calculated using quantum mechanics on the one hand, and those calculated according to local realistic hidden-variable theories on the other. Our proof is based on a physically motivated use of Fourier series for periodic functions, and confirms that local realistic hidden-variable theories are incompatible with quantum mechanics.