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A Statistical Approach to Two-Particle Bell Tests

Xianming Meng

Published May 1, 2021• 0 citations
OpenAlex

Abstract

Extensive experimental tests of the Bell inequality have been conducted over time and the test results are viewed as a testimony to quantum mechanics. In considering the close tie between quantum mechanics and statistical theory, this paper identifies the mistake in previous statistical explanation and uses an elegant statistical approach to derive general formulas for two-particle Bell tests, without invoking any wavefunctions. The results show that, for the special case where the spins/polarizations are in the same, opposite, or perpendicular directions, the general formulas derived in this paper convert to quantum predictions, which are confirmed by numerous experiments. The paper also investigates the linkages between the statistical and quantum predictions and finds that vector decomposition and probability law are at the heart of both approaches. Based on this finding, the paper explains statistically why the local hidden variable theory fails the Bell tests. The paper has important implications for quantum computing, quantum theory in general, and the role of randomism and realism in physics.

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Extensive experimental tests of the Bell inequality have been conducted over time and the test results are viewed as a testimony to quantum mechanics. In considering the close tie between quantum mechanics and statistical theory, this paper identifies the mistake in previous statistical explanation and uses an elegant statistical approach to derive general formulas for two-particle Bell tests, without invoking any wavefunctions. The results show that, for the special case where the spins/polarizations are in the same, opposite, or perpendicular directions, the general formulas derived in this paper convert to quantum predictions, which are confirmed by numerous experiments. The paper also investigates the linkages between the statistical and quantum predictions and finds that vector decomposition and probability law are at the heart of both approaches. Based on this finding, the paper explains statistically why the local hidden variable theory fails the Bell tests. The paper has important implications for quantum computing, quantum theory in general, and the role of randomism and realism in physics.
It is generally believed that Bell's inequality holds for the case of entangled states, including two correlated particles or special states of a single particle. Here, we derive a single-particle Bell's inequality for two correlated spin states at two successive times, appealing to the statistical independence condition in an ideal experiment, for a locally causal hidden variables theory. We show that regardless of the locality assumption, the inequality can be violated by some quantum predictions.
This thesis presents a study of Bell’s inequality and its experimental tests, with a focus on their implications for the concepts of locality and realism in quantum mechanics. The work reviews the theoretical framework underlying Bell’s inequality, the assumptions of hidden variable theories, and the contrasting predictions of quantum mechanics. It further provides an overview of landmark experimental tests of Bell’s inequality, emphasizing their role in shaping the modern understanding of quantum entanglement and nonlocality. The aim of the study is to offer a clear and pedagogical introduction to the subject, bridging the mathematical formalism with experimental verification of one of the most fundamental results in the foundations of quantum theory.
Abstract The experimentally verified violation of Bell’s inequalities apparently implies that at least one of two beliefs, which are accepted as true in almost all scientific practice, must be false. These beliefs are: that effects propagating at infinite velocity do not exist, and that natural phenomena occur independently of being observed. Giving up any one of these two beliefs (usually known together as Local Realism) is controversial. For many years, many theories have been proposed to reconcile the violation of Bell’s inequalities with Local Realism, but none has been fully successful. In this paper, it is recalled that Bell’s inequalities are equivalent to the conditions to decide the completeness of a theory according to Boolean logic. Therefore, any theory aimed to violate Bell’s inequalities must start by giving up Boolean logic. Two problems are therefore defined: the “soft” one is to explain the violation of Bell’s inequalities without violating (non-Boolean) Local Realism; the “hard” one is to predict the time values when single particles are detected, in such a way that the resulting number of coincidences violates Bell’s inequalities. A simple hidden variables model is introduced, which solves the “soft” problem even in an ideal setup. This is possible thanks to the use of vectors in real space as the hidden variables and the corresponding operation (projection), which do not hold to Boolean logic. This model reconciles the violation of Bell’s inequalities with Local Realism and should end a controversy that has lasted for decades. Regarding the “hard” problem, the introduced model is as incomplete as Quantum Mechanics is. It is claimed that solving the “hard” problem requires devising a new kind of quantum computer, which should be able to accept (non-Boolean) hidden variable values as input data and to replace the statistical Born’s rule of Quantum Mechanics with a deterministic threshold condition.
Bell's inequalities(1964) provides a quantitative criterion to test the local hidden variable theory(LHVT) versus quantum mechanics(QM).For the first time,it leads to the possibility to experimentally distinguish LHVT and QM.So far,there have been tens of experiments ever since the first experiment by Clauser et al(1972).An overwhelming majority of these experiments are in good agreement with the predictions of QM yet violate those of LHVT.Some loopholes in these experiments are discussed in Section I,followed by a respective summary to 12 experiments between 1972 to 1982 and 31 experiments from 1985 and on in Sections II and III.It is finally pointed out that so far there is not a single experiment to close both the locality loophole and the detection loophole at the same time,but the experimental outcomes are irreversible.Accordingly,theoretical physicists must admit that the present physics foundation is broken,i.e.quantum mechanics is contradiction with special relativity.
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.