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Modeling the Singlet State with Local Variables

Larsson, J A

Published Jan 1, 1999• 26 citations
OpenAlex

Abstract

A local-variable model yielding the statistics from the singlet state is presented for the case of inefficient detectors and/or lowered visibility. It has independent errors and the highest efficiency at perfect visibility is 77.80%, while the highest visibility at perfect detector-efficiency is 63.66%. Thus, the model cannot be refuted by measurements made to date.

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Modeling the singlet state with local variables

Jan-Åke Larsson

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A local-variable model yielding the statistics from the singlet state is presented for the case of inefficient detectors and/or lowered visibility. It has independent errors and the highest efficiency at perfect visibility is 77.80%, while the highest visibility at perfect detector-efficiency is 63.66%. Thus, the model cannot be refuted by measurements made to date.
A local hidden variable model exploiting the detection loophole to reproduce exactly the quantum correlation of the singlet state is presented. The model is shown to be compatible with both the CHSH and the CH Bell inequalities. Moreover, it bears the same rotational symmetry as spins. The reason why the model can reproduce the quantum correlation without violating the Bell theorem is that in the model the efficiency of the detectors depends on the local hidden variable. On average the detector efficiency is limited to 75%.
The question of how to deal with inefficient detectors in actual experiments of the Einstein-Podolsky-Rosen type is studied. We derive the necessary and sufficient condition for compatibility with local realism of data collected in experiments with two settings of each detector, without making auxiliary assumptions about undetected events. For the conventional experiment with particles in the singlet state (or its photon analogue), the data predicted by the quantum theory do not violate this condition unless the quantum efficiency of the detectors exceeds 83%.
The essence of Bell’s theorem is that, in general, quantum statistics cannot be reproduced by a local hidden variable (LHV) model. This impossibility is strongly manifested when statistics collected by measuring certain local observables on a singlet state, violates the Bell inequality. In this work, we search for local POVMs with binary outcomes for which an LHV model can be constructed for a singlet state. We provide various subsets of observables for which an LHV model can be provided for singlet statistics.
Bell’s theorem teaches us that there are quantum correlations that can not be simulated by just shared randomness (Local Hidden variable). There are some recent results which simulate singlet correlation by using either 1 cbit or a binary (no-signaling) correlation which violate Bell’s inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, MJW Hall showed that the statistics of singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depend on measurement direction of both parties [Phys. Rev. Lett.105 250404 (2010)]. He also proved that for any model of singlet correlation, 86 % measurement independence is optimal. In this paper, we show that 59 % measurement independence is optimal for simulating singlet correlation when the underlying distribution of hidden variables depend only on measurements of one party. We also show that a distribution corresponding to this optimal lack of free will, already exists in literature which first appeared in the context of detection efficiency loophole. I.
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.