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arXiv:quant-ph/0406042 · Fortschritte der Physik

Complete correlation, detection loophole and Bell's theorem

Afshin Shafiee, M. Golshani

Published Dec 1, 2004• 2 citations
arXivHTMLPDF

Abstract

Abstract Two new formulations of Bell's theorem are given here. First, we consider a definite set of two entangled photons with only two polarization directions, for which Bell's locality assumption is violated for the case of perfect correlation. Then, using a different approach, we prove an efficient Bell‐type inequality which is violated by some quantum mechanical predictions, independent of the efficiency factors.

DOI
10.1002/prop.200410191

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Abstract Two new formulations of Bell's theorem are given here. First, we consider a definite set of two entangled photons with only two polarization directions, for which Bell's locality assumption is violated for the case of perfect correlation. Then, using a different approach, we prove an efficient Bell‐type inequality which is violated by some quantum mechanical predictions, independent of the efficiency factors.
John Bell’s theorem of 1964 states that local elements of physical reality, existing independent of measurement, are inconsistent with the predictions of quantum mechanics (Bell, J. S. (1964), Physics (College. Park. Md). Specifically, correlations between measurement results from distant entangled systems would be smaller than predicted by quantum physics. This is expressed in Bell’s inequalities. Employing modifications of Bell’s inequalities, many experiments have been performed that convincingly support the quantum predictions. Yet, all experiments rely on assumptions, which provide loopholes for a local realist explanation of the measurement. Here we report an experiment with polarization-entangled photons that simultaneously closes the most significant of these loopholes. We use a highly efficient source of entangled photons, distributed these over a distance of 58.5 meters, and implemented rapid random setting generation and high-efficiency detection to observe a violation of a Bell inequality with high statistical significance. The merely statistical probability of our results to occur under local realism is less than 3.74×10-31, corresponding to an 11.5 standard deviation effect.
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.
Bell's theorem of 1965 is a proof that all realistic interpretations of quantum mechanics must be non-local. Bell's theorem consists of two parts: first a correlation inequality is derived that must be satisfied by all local realistic theories; second it is demonstrated that quantum mechanical probabilities violate this inequality in certain cases. In the case of ideal experiments, Bell's theorem has been proven. However, in the case of real experiments where polarizers and detectors are non-ideal, the theorem has not yet been proven since the proof always requires some arbitrary and {\em ad hoc} supplementary assumptions. In this paper, we state a new and rather weak supplementary assumption for the ensemble of photons that emerge from the polarizers, and we show that the conjunction of Einstein's locality with this assumption leads to validity of an inequality that is violated by a factor as large as 1.5 in the case of real experiments. Moreover, the present supplementary assumption is considerably weaker and more general than Clauser, Horne, Shimony, Holt supplementary assumption.
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.
The experimental test of the CHSH inequality confirms the quantum-mechanically predicted perfect correlation between two spatially separated photons. By interpreting the correlation non-locally as the so-called entanglement, Bell's theorem denies the existence of definite polarizations possessed by photons in the absence of measurement. The denial amounts to refuting Einstein's argument for defending his local-realist world view based on his separability principle and his ensemble interpretation of ψ-functions in the Einstein-Bohr debate on the conceptual foundations of quantum mechanics. This paper shows that Bell's theorem is incorrect and presents an explanation of the correlation based on Einstein's argument. The correlation is deterministic. A formal analysis supporting Einstein's explanation is also presented.
We introduce an extended version of a previous all-versus-nothing proof of impossibility of Einstein-Podolsky-Rosen's local elements of reality for two photons entangled both in polarization and path degrees of freedom (A. Cabello, quant-ph/0507259), which leads to a Bell's inequality where the classical bound is 8 and the quantum prediction is 16. A simple estimation of the detection efficiency required to close the detection loophole using this extended version gives $\ensuremath{\eta}>0.69$. This efficiency is lower than that required for previous proposals.