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Journal of Optics B Quantum and Semiclassical Optics

Correlation functions in EPR-type experiments: the low-detection- efficiency loophole

V. Berardi, Augusto Garuccio, Vito Luigi Lepore

Published Jul 4, 2000• 0 citations
Publisher

Abstract

In this paper we derive a general approach using correlation functions for interpreting the results of measurements on two-particle entangled states. In our analysis we explicitly take into account the detector efficiency. We show that all Bell-type inequalities have bounds depending on both the losses in the apparatus and the supplementary assumptions about the statistical distribution of the losses themselves. Moreover, our approach enables us to include in the causal and local theory the same assumptions as used in the quantum mechanical analysis of an experiment, and to deduce the inequality which has to be satisfied by any local realistic model incorporating these assumptions. We show that, although the most recent experiments on Bell's inequality can be considered to be a remarkable step in the direction of testing quantum mechanics predictions, the loophole of the low quantum efficiency of the detector is not yet solved. In fact, using the quantum mechanical assumption of random nondetection in the causal and local approach, we deduce a generalized Bell-type inequality which includes no-count events and whose upper bound can be violated by the quantum mechanical predictions only if the detector quantum efficiency is >0.811.

DOI
10.1088/1464-4266/2/4/304

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In this paper we derive a general approach using correlation functions for interpreting the results of measurements on two-particle entangled states. In our analysis we explicitly take into account the detector efficiency. We show that all Bell-type inequalities have bounds depending on both the losses in the apparatus and the supplementary assumptions about the statistical distribution of the losses themselves. Moreover, our approach enables us to include in the causal and local theory the same assumptions as used in the quantum mechanical analysis of an experiment, and to deduce the inequality which has to be satisfied by any local realistic model incorporating these assumptions. We show that, although the most recent experiments on Bell's inequality can be considered to be a remarkable step in the direction of testing quantum mechanics predictions, the loophole of the low quantum efficiency of the detector is not yet solved. In fact, using the quantum mechanical assumption of random nondetection in the causal and local approach, we deduce a generalized Bell-type inequality which includes no-count events and whose upper bound can be violated by the quantum mechanical predictions only if the detector quantum efficiency is >0.811.
The demonstration and use of nonlocality, as defined by Bell's theorem, rely strongly on dealing with nondetection events due to losses and detectors'inefficiencies. Otherwise, the so-called detection loophole could be exploited. The only way to avoid this is to have detection efficiencies that are above a certain threshold. We introduce the intermediate assumption of limited detection efficiency, that is, in each run of the experiment, the overall detection efficiency is lower bounded by η(min)>0. Hence, in an adversarial scenario, the adversaries have arbitrary large but not full control over the inefficiencies. We analyze the set of possible correlations that satisfy limited detection locality and show that they necessarily satisfy some linear Bell-like inequalities. We prove that quantum theory predicts the violation of one of these inequalities for all η(min)>0. Hence, nonlocality can be demonstrated with arbitrarily small limited detection efficiencies. We validate this assumption experimentally via a twin-photon implementation in which two users are provided with one photon each out of a partially entangled pair. We exploit on each side a passive switch followed by two measurement devices with fixed settings. Assuming the switches are not fully controlled by an adversary, nor by hypothetical local variables, we reveal the nonlocality of the established correlations despite a low overall detection efficiency.
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.
A general approach to Bell-type inequalities for trichotomic observables is presented. With this formalism, which explicitly includes no-counts events, we show that the limits of all Bell-type inequalities are functions of the supplementary assumptions. This approach enables us to include in the causal and local theory the same assumptions used in the quantum-mechanical analysis of an experiment, and to deduce the inequality that must be satisfied by any local realistic model incorporating these assumptions. We prove that, using the quantum-mechanical assumption of ``random nondetection'' in the causal and local approach, a generalized Bell-type inequality is deduced whose upper limit can be violated by the quantum-mechanical predictions for symmetrical experimental setups involving detectors with quantum efficiency greater than 0.811. \textcopyright{} 1996 The American Physical Society.
We analyze some aspects of recently performed Franson-type experiments with entangled photon pairs aimed to test Bell's inequalities. We point out that quantum theory leads to the coincidence rate between detectors which includes in fact a dependence on the distance. We study this dependence and obtain that for large distances the correlation function vanishes. Therefore with taking into account the space parts of wave functions of photons for large distances quantum mechanical predictions are consistent with Bell's inequalities. We propose an experimental study of space dependence of correlation functions in Bell-type experiments.
It is generally believed that Bell's inequality holds for the case of entangled states, including two correlated particles or special states of a single particle. Here, we derive a single-particle Bell's inequality for two correlated spin states at two successive times, appealing to the statistical independence condition in an ideal experiment, for a locally causal hidden variables theory. We show that regardless of the locality assumption, the inequality can be violated by some quantum predictions.
The empirical evidence tends to confirm the quantum - mechanically (QM) predicted violation of Bell's - type inequalities. The latter are widely accepted to be correct representation of the objective reality (OR) underlying EPR's gedankenexperiment. Contrariwise, we argue here that the starting premises leading to Bell's and CHSH's inequalities are not appropriate formulations of OR in the EPR context. Indeed, the essence of EPR argument is the correlation between pair of particles travelling with some well defined orientations. To measure such correlation, one requires detectors with high resolution, and some appropriately defined relative orientations. On the contrary, experimental tests of Bell's inequalities typically involve counting the number of particles with some value of spin (or polarisation) using low-resolution detectors and completely arbitrary orientations. Depending upon the relative orientation of good-resolution detectors, we obtain two families of CHSH-type inequalities. One of these families is the most often used version of CHSH inequality, that was derived by Clauser and Horne with the supplementary no - enhancement assumption: here, it is obtained without special provisions. The other family is completely new and refers to coincidence counts from multiple events. Experimental test to distinguish between the QM and OR descriptions of nature should take into account these new constraints.