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On supplementary variables in spin-measuring equipments in EPR-Bell experiments

Won-Young Hwang, In Gyu Koh, Yeong Deok Han

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

We discuss about supplementary ( or hidden) variables in spin-measuring equipments in EPR-Bell experiment. We show that why the original supplementary variable λ is not to be regarded to include supplementary variables in spin-measuring equipments. Thus, supplementary variables should be introduced additionally in spin-measuring equipments. When the supplementary variables introduced in spin-measuring equipments have local correlations, the Bell inequality is recovered. On the other hand, when they have nonlocal correlations, the Bell inequality is not recovered. This fact means that the nonlocality can be ascribed to that of spin-measuring equipments. PACS number: 03.65.Bz Key words: Bell inequality, local realistic models, supplementary variables in spin-measuring

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We discuss supplementary (or hidden) variables in spin-measuring equipments in EPR-Bell experiment. This theme was considered in a Bell's later work. We generalize it. First, we show why the original supplementary variable $λ$ is not to be regarded to include supplementary variables in spin-measuring equipments (why supplementary variables should be introduced additionally in spin-measuring equipments) Next, we show the followings. When the supplementary variables introduced in spin-measuring equipments have local correlations, the Bell inequality is recovered. On the other hand, when they have nonlocal correlations, the Bell inequality is not recovered. This fact is in accord with the fact that the Bell inequality is derived for local realistic models.

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We discuss about supplementary ( or hidden) variables in spin-measuring equipments in EPR-Bell experiment. We show that why the original supplementary variable λ is not to be regarded to include supplementary variables in spin-measuring equipments. Thus, supplementary variables should be introduced additionally in spin-measuring equipments. When the supplementary variables introduced in spin-measuring equipments have local correlations, the Bell inequality is recovered. On the other hand, when they have nonlocal correlations, the Bell inequality is not recovered. This fact means that the nonlocality can be ascribed to that of spin-measuring equipments. PACS number: 03.65.Bz Key words: Bell inequality, local realistic models, supplementary variables in spin-measuring
We discuss supplementary (or hidden) variables in spin-measuring equipments in EPR-Bell experiment. This theme was considered in a Bell's later work. We generalize it. First, we show why the original supplementary variable λλλ is not to be regarded to include supplementary variables in spin-measuring equipments (why supplementary variables should be introduced additionally in spin-measuring equipments) Next, we show the followings. When the supplementary variables introduced in spin-measuring equipments have local correlations, the Bell inequality is recovered. On the other hand, when they have nonlocal correlations, the Bell inequality is not recovered. This fact is in accord with the fact that the Bell inequality is derived for local realistic models.
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.
In derivating of Bell’s inequalities, the probability distribution is supposed to be a function only of a hidden variable. We point out that the true implication of the probability distribution of Bell’s correlation function is the distribution of joint measurement outcomes on the two sides. It is therefore a function of both the hidden variable and the settings. In this case, Bell’s inequalities fail. Our further analysis shows that Bell’s locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, and hence violation of Bell’s inequalities cannot rule out the existence of local hidden variable. To explain the results of EPR-type experiments, we suppose that a polarization-entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any nonlocal influence on it. Then several experiments are suggested to reveal the components of the polarization-entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with the same quantum state behave the same under the same measuring conditions.
By implicitly assuming that all possible Bell-measurements occur simultaneously, all proofs of Bell's Theorem violate Heisenberg's Uncertainty Principle. This assumption is made in the original form of Bell's inequality, in Wigner's probability inequalities, and in the ``nonlocality without inequalities'' arguments. The introduction of time into derivations of these variants of Bell's theorem results in extra terms related to the time order of the measurements used in constructing correlation coefficients. Since the same locality assumptions are made in the Heisenberg-compliant derivations of this paper, only time-independent classical local hidden variable theories are forbidden by violations of the original Bell inequalities; time-dependent quantum local hidden variable theories can satisfy this new bound and agree with experiment. We further point out that factorizable wavefunctions have been used to describe some EPR experiments and can be used to describe others. These will generate local de Broglie-Bohm trajectories in the description of the data. This second, independent, line of argument also shows that violation of Bell's inequality is only evidence that Heisenberg's Uncertainty Principle cannot be ignored.
In the derivation of Bell's inequalities, probability distribution is supposed to be a function of only hidden variable. We point out that the true implication of the probability distribution of Bell's correlation function is the distribution of the joint measurement outcomes on the two sides. So it is a function of both hidden variable and settings. In this case, Bell's inequalities fail. Our further analysis shows that Bell's locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, thus violation of Bell's inequalities cannot rule out the existence of local hidden variable. In order to explain the results of EPR-type experiments, we suppose that polarization entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any non-local influence on it. Then several experiments are suggested to reveal the components of polarization entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with a same quantum state will behave in the same way under the same measuring condition.
We derive a Bell-like inequality involving all correlations in local observables with uncertainty free states and show that the inequality is violated in quantum mechanics for EPR and GHZ states. If the uncertainties are allowed in local observables then the statistical predictions of hidden variable theory is well respected in quantum world. We argue that the uncertainties play a key role in understanding the non-locality issues in quantum world. Thus we can not rule out the possibility that a local, realistic hidden variable theory with statistical uncertainties in the observables might reproduce all the results of quantum theory.