Bell theorem without inequalities for two particles. II. Inefficient detectors

Daniel M. Greenberger, Michael A. Horne, Anton Zeilinger, Marek Zukowski

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We again consider (as in a companion paper) an entangled two-particle state that is produced from two independent downconversion 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 do not 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 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, and we take it as a refutation of such realistic theories, free of the detection efficiency and random sampling “loopholes.” The hidden variable analysis depends crucially on the use of two independent laser sources for the downconversions. We also investigate the necessity of using two independent sources versus a single source for all particles. Finally, we argue that the state we use can legitimately be considered as a two-particle state, and used as such in experiments.
Original languageEnglish
Article number022111
Number of pages12
JournalPhysical Review A
Publication statusPublished - 2008

Austrian Fields of Science 2012

  • 103026 Quantum optics

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