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arXiv:quant-ph/0001034 · Physical Review Letters

Inequalities for Dealing with Detector Inefficiencies in Greenberger-Horne-Zeilinger–Type Experiments

J. Acácio de Barros, Patrick Suppes

Published Jan 31, 2000• 65 citations
arXivHTMLPDF

Abstract

In this article we show that the three-particle Greenberger-Horne-Zeilinger theorem can be reformulated in terms of inequalities, allowing imperfect correlations due to detector inefficiencies. We show quantitatively that taking into account these inefficiencies, the published results of the Innsbruck experiment support the nonexistence of local hidden variables that explain the experimental results.

DOI
10.1103/physrevlett.84.793

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In this article we show that the three-particle Greenberger-Horne-Zeilinger theorem can be reformulated in terms of inequalities, allowing imperfect correlations due to detector inefficiencies. We show quantitatively that taking into account these inefficiencies, the published results of the Innsbruck experiment support the nonexistence of local hidden variables that explain the experimental results.
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In this paper detector efficiency conditions are derived for the Greenberger-Horne-Zeilinger (GHZ) paradox. The conditions will be necessary and sufficient, i.e., the GHZ paradox is explicable in terms of a local-variable model if the efficiency is below the bounds, and the GHZ prerequisites are inconsistent at higher efficiencies. The derivation does not make use of any of the symmetry assumptions usually made in the literature, most notably the assumption of independent errors. The errors in local-hidden-variable models are governed by the ``hidden variable'' and, therefore, one cannot in general assume that the errors are independent. It will be shown that this assumption is not necessary. Moreover, bounds are presented that do not need the emission rate of particle triples to be known. An example of such a bound is the ratio of the triple coincidence rate and the double coincidence rate at two detectors, which needs to be higher than 75% to yield a contradiction.
We again consider (as in a companion paper) an entangled two-particle state that is produced from two independent down-conversion sources by the process of "entanglement-swapping", so that the particles have never met. We show that there is a natural extension of the Einstein-Pololsky-Rosen discussion of "elements of reality" to include inefficient detectors. We consider inefficient deterministic, local, realistic models of quantum theory that are "robust", which we consider to be the minimum requirement for them to be taken seriously. By robust, we mean they satisfy the following three criteria: (a) they reproduce the quantum results for perfect correlations, if all particles are detected; (b) they produce some counts for every setting of the angles (so they don't describe some experiments that can easily be performed as "impossible"); (c) all their hidden variables are relevant (they must each produce a detectable result in some experiment). For such models we prove a Greenberger-Horne-Zeilinger (GHZ) type theorem for arbitrary detection efficiencies, showing that any such theory is inconsistent with the quantum mechanical perfect correlations. This theorem holds for individual events with no inequalities. As a result, the theorem is also independent of any random sampling hypothesis.
We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of superluminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in Einstein-Podolsky-Rosen (EPR) experiments in the presence of noise. We apply our method to an example involving nnn parties sharing a Greenberger-Horne-Zeilinger-type state on which they carry out local measurements. For this example, we show that for local hidden-variable theories to reproduce the quantum correlations, the amount of superluminal classical communication ccc and the detector efficiency $\ensuremath{\eta}$ are constrained by $\ensuremath{\eta}{2}^{\ensuremath{-}c∕n}\ensuremath{\leqslant}O({n}^{\ensuremath{-}1∕6})$. This result holds even if the classical models are allowed to make an error with constant probability.