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arXiv:quant-ph/0608049

Locality and the Greenberger-Horne-Zeilinger Theorem

A. Fahmi, Golshani, M.

Published Aug 6, 2006• 1 citation
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Abstract

In all local realistic theories worked out till now, locality is considered as a basic assumption. Most people in the field consider the inconsistency between local realistic theories and quantum mechanics to be a result of non-local nature of quantum mechanics. In this Paper, we derive the Greenberger-Horne-Zeilinger type theorem for particles with instantaneous (non-local) interactions at the hidden-variable level. Then, we show that the previous contradiction still exists between quantum mechanics and non-local hidden variable models.

DOI
10.48550/arxiv.quant-ph/0608049

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In all local realistic theories worked out till now, locality is considered as a basic assumption. Most people in the field consider the inconsistency between local realistic theories and quantum mechanics to be a result of non-local nature of quantum mechanics. In this Paper, we derive the Greenberger-Horne-Zeilinger type theorem for particles with instantaneous (non-local) interactions at the hidden-variable level. Then, we show that the previous contradiction still exists between quantum mechanics and non-local hidden variable models.
In all local realistic theories worked out till now, locality is considered as a basic assumption. Most people in the field consider the inconsistency between local realistic theories and quantum mechanics to be a result of non-local nature of quantum mechanics. In this paper, we derive Bell's inequality for particles with instantaneous interactions, and show that the aforementioned contradiction still exists between quantum mechanics and non-local hidden variable models. Then, we use this non-locality to obtain the GHZ theorem. In what follows, we show that Bacon and Toner's protocol, for the simulation of Bell correlation, by using local hidden variables augmented by classical communication, have some inconsistency with quantum mechanics. Our approach can answer to Brassard questions from another viewpoint, we show that if we accept that our nature obeys quantum mechanical laws, then all of quantum mechanic results cannot be simulated by realistic theories augmented by classical communication or a single instance use of non-local box.
The Greenberger-Horne-Zeilinger~(GHZ) version of the Einstein-Podolsky-Rosen~(EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of a local and realistic model of hidden variables in which the observers are free to choose their own experimental settings. In this paper we show, however, that the GHZ argument implicitly relies on an additional crucial assumption, which is not required by fundamental physical principles and had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles as well as the orientation of the measurement apparatus that test the system can be defined. We further note that such an absolute frame of reference would not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping this unnecessary additional assumption we are able to build an explicitly local and realistic model of hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions, and thus completes the description of the system in the EPR sense.
It is widely appreciated, due to Bell's theorem, that quantum phenomena are inconsistent with local-realist models. In this context, locality refers to local causality, and there is thus an open possibility for reproducing the quantum predictions with models which internally violate the causal arrow of time, while otherwise adhering to the relevant locality condition. So far, this possibility has been demonstrated only at a toy-model level, and only for systems involving one or two spins (or photons). The present work extends one of these models to quantum correlations between three or more spins which are entangled in the Greenberger-Horne-Zeilinger state.
It is widely appreciated, due to Bell's theorem, that quantum phenomena are inconsistent with local-realist models. In this context, locality refers to local causality, and there is thus an open possibility for reproducing the quantum predictions with models which internally violate the causal arrow of time, while otherwise adhering to the relevant locality condition. So far, this possibility has been demonstrated only at a toy-model level, and only for systems involving one or two spins (or photons). The present work extends one of these models to quantum correlations between three or more spins which are entangled in the Greenberger-Horne-Zeilinger state.
The Greenberger, Horne, Zeilinger (GHZ) theorem is critically important to consideration of the possibility of hidden variables in quantum mechanics. Since it depends on predictions of single sets of measurements on three particles, it eliminates the sampling loophole encountered by the Bell theorem which requires a large number of observations to obtain a small number of useful joint measurements. In evading this problem, the GHZ theorem is believed to have confirmed Bell's historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem depends on predicting the results of sets of measurements of which only one may be performed, i.e., counterfactuals. In the present paper, the non-commutative aspects of these unperformed measurement sequences are critically examined. Three classical examples and two variations on the GHZ construction are analyzed to demonstrate that combined counter factual results of non-commuting operations are in general logically inconsistent with performable measurement sequences that take non-commutation into account. As a consequence, negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.
Bell's theorem revealed that a local hidden-variable model cannot completely reproduce the quantum mechanical predictions. Bell's inequality provides an upper bound under the locality and reality assumptions that can be violated by correlated measurement statistics of quantum mechanics. Greenberger, Horne, and Zeilinger (GHZ) gave a more compelling proof of Bell's theorem without inequalities by considering perfect correlations rather than statistical correlations. This work presents a temporal analog of the GHZ argument that establishes Bell's theorem in time without inequalities.