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arXiv:0807.2413 · Journal of Physics A Mathematical and Theoretical

A non-local hidden-variable model that violates Leggett-type inequalities

F. De Zela

Published Nov 3, 2008• 19 citations
arXivHTMLPDF

Abstract

Recent experiments of Gröblacher et al proved the violation of a Leggett-type inequality that was claimed to be valid for a broad class of non-local hidden-variable theories. The impossibility of constructing a non-local and realistic theory, unless it entails highly counterintuitive features, seems thus to have been experimentally proved. This would bring us close to a definite refutation of realism. Indeed, realism was proved to be also incompatible with locality, according to a series of experiments testing Bell inequalities. The present paper addresses the said experiments of Gröblacher et al and presents an explicit, contextual and realistic, model that reproduces the predictions of quantum mechanics. It thus violates the Leggett-type inequality that was established with the aim of ruling out a supposedly broad class of non-local models. We can thus conclude that plausible contextual, realistic, models are still tenable. This restates the possibility of a future completion of quantum mechanics by a realistic and contextual theory which is not in a class containing only highly counterintuitive models. The class that was ruled out by the experiments of Gröblacher et al is thus proved to be a limited one, arbitrarily separating models that physically belong in the same class.

DOI
10.1088/1751-8113/41/50/505301

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Recent experiments of Gröblacher et al proved the violation of a Leggett-type inequality that was claimed to be valid for a broad class of non-local hidden-variable theories. The impossibility of constructing a non-local and realistic theory, unless it entails highly counterintuitive features, seems thus to have been experimentally proved. This would bring us close to a definite refutation of realism. Indeed, realism was proved to be also incompatible with locality, according to a series of experiments testing Bell inequalities. The present paper addresses the said experiments of Gröblacher et al and presents an explicit, contextual and realistic, model that reproduces the predictions of quantum mechanics. It thus violates the Leggett-type inequality that was established with the aim of ruling out a supposedly broad class of non-local models. We can thus conclude that plausible contextual, realistic, models are still tenable. This restates the possibility of a future completion of quantum mechanics by a realistic and contextual theory which is not in a class containing only highly counterintuitive models. The class that was ruled out by the experiments of Gröblacher et al is thus proved to be a limited one, arbitrarily separating models that physically belong in the same class.
It is shown that the before-before (or Suarez-Scarani) experiment refutes hidden variable models with a deterministic ("realistic") nonlocal part, and the experimental violation of Leggett-type inequalities models with a random nonlocal part and a biased random local one. Therefore the claim that Groeblacher et al. present "an experimental test of nonlocal realism" [Nature, 446, 871-875 (2007)] is misleading, and Marek Zukowski's comment [arXiv:0809.0072v1] misses the point. A new experiment is proposed.
We consider the so called Legget inequalities which are deduced from the assumption of general (local or non-local) realism plus the arrow of time preservation. Then, instead of assuming cryto-nonlocal hidden variables, we assume any (local or non-local) realism compatible with the joint and non-joint expected values dictated by quantum mwechaanics. Hence, we prove that this double assumption is not consistent, since the corresponding general Leggett inequalities are violated by quantum mechanics. Thus, realism plus arrow of time preservation and quntum mechanics are not compatible. In other words, quantum mechanics cannot be completed with any (local or non-local) hidden variables, provide we assume the common sense of the arrow of time. The result would deserve to be experimentally tested and we discuss why it is not invalidated by hidden variable theories as the one from Bohm.
Bell's inequality is established based on local realism. The violation of Bell's inequality by quantum mechanics implies either locality or realism or both are untenable. Leggett's inequality is derived based on nonlocal realism. The violation of Leggett's inequality implies that quantum mechanics is neither local realistic nor nonlocal realistic. The incompatibility of nonlocal realism and quantum mechanics has been currently confirmed by photon experiments. In our work, we propose to test Leggett's inequality using the Aharonov-Casher effect. In our scheme, four entangled particles emitted from two sources manifest a two-qubit-typed correlation that may result in the violation of the Leggett inequality, while satisfying the no-signaling condition for spacelike separation. Our scheme is tolerant to some local inaccuracies due to the topological nature of the Aharonov-Casher phase. The experimental implementation of our scheme can be possibly realized by a calcium atomic polarization interferometer experiment.
This article develops an expanded critical–propositional analysis of Simon Gröblacher et al.’s 2007 experiment, “An Experimental Test of Non-Local Realism”, published in Nature, in systematic dialogue with the Theory of Objectivity (TO) formulated by Vidamor Cabannas and Denivaldo Silva. Gröblacher et al. investigated whether abandoning locality is sufficient to preserve a particular form of physical realism in the face of quantum-mechanical correlations. Building upon A. J. Leggett’s class of crypto-nonlocal hidden-variable models, the experiment tested theories in which individual measurement outcomes arise from pre-existing properties, physical states consist of statistical mixtures of subensembles carrying definite polarizations, and the local averages of those subensembles obey Malus’ law. Using high-visibility polarization-entangled photon pairs generated through sponta- neous parametric down-conversion, Gröblacher et al. measured correlations in different planes of the Poincaré sphere and reported a violation of a generalized Leggett-type inequality. At a difference angle of 20◦ , the experiment yielded SNLHV = 3.8521 ± 0.0227, corresponding to a violation of approximately 3.2 standard deviations. The same data also gave SCHSH = 2.178 ± 0.0199, violating the local-realistic CHSH bound by approximately nine standard deviations (Gröblacher et al. 2007a). A subsequent corrigendum clarified the relevant experimental angle and other technical details while leaving the final inequality and scientific conclusions unchanged (Gröblacher et al. 2007b). The present analysis argues that these results must not be interpreted as an experimental refutation of objectivity, external reality, or realism in every conceivable sense. Instead, they constrain a particular class of realist ontologies in which maximally entangled systems can be decomposed into subensembles endowed with definite individual polarizations satisfying Malus’ law. This distinction is decisive for comparison with TO. Within TO, objectivity is grounded not in human observation but in modal conditions concerning elements, individual fields, boundaries, relations among elements, composition, and an informational or transcendent dimension. The Fifth Absolute Truth, in particular, defines observation non-anthropocentrically: an element acquires logical existence through relational inclusion within the observation or frequency of at least two other elements. The Seventh Absolute Truth identifies the transcendent element with knowledge or information generated in atomic relations and considers such information equivalent, within TO, to atomic radiation. The analysis identifies substantial conceptual compatibility between Gröblacher et al. and relational readings of the Second, Fourth, Fifth, and Seventh Absolute Truths. Quantum entanglement preserves operational distinguishability between subsystems while invalidating simple classical separability; experimental observation is physically imple- mented without requiring human consciousness; and experimentally relevant information resides crucially in correlations rather than solely in independently assigned classical properties. At the same time, the experiment imposes an important restriction upon any future operationalization of TO: an individual field, physical boundary, or objective existence cannot simply be identified with a catalogue of predetermined local polarization values of the Leggett type. If TO is to establish strong empirical contact with quantum physics, it must derive quantitative correlation functions that either reproduce existing quantum results or predict measurable deviations from them. The First Absolute Truth—the logically, mathematically, and eternally structured Nothingness preceding the universe—remains outside the empirical domain of Gröblacher’s laboratory test. The Third Absolute Truth, the TO cosmogonic theorem, the Inductive Effects, and the cosmological Eras are likewise not directly tested. Consequently, the experiment is classified here not as a proof of TO but as a case of indirect empirical corroboration of selected relational and non-anthropocentric features combined with a significant negative constraint on overly classical interpretations of objectivity. The article concludes that Gröblacher et al. merits a dialogue score of 8.3/10 in relation to TO and proposes a program for transforming the present interpretive convergence into genuine differential empirical tests. Keywords: Theory of Objectivity; non-local realism; Leggett inequality; Bell theo- rem; quantum entanglement; objectivity; modal ontology; quantum information; non- anthropocentric observation; relationality; Gröblacher; quantum foundations.