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arXiv:quant-ph/0312035 · Europhysics Letters (EPL)

Bell's inequality and the coincidence-time loophole

Jan-Åke Larsson, Richard David Gill

Published Aug 25, 2004• 94 citations
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Abstract

This paper analyzes effects of time dependence in the Bell inequality. A generalized inequality is derived for the case when coincidence and non-coincidence (and hence whether or not a pair contributes to the actual data) is controlled by timing that depends on the detector settings. Needless to say, this inequality is violated by quantum mechanics and could be violated by experimental data provided that the loss of measurement pairs through failure of coincidence is small enough, but the quantitative bound is more restrictive in this case than in the previously analyzed "efficiency loophole".

DOI
10.1209/epl/i2004-10124-7

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This paper analyzes effects of time dependence in the Bell inequality. A generalized inequality is derived for the case when coincidence and non-coincidence (and hence whether or not a pair contributes to the actual data) is controlled by timing that depends on the detector settings. Needless to say, this inequality is violated by quantum mechanics and could be violated by experimental data provided that the loss of measurement pairs through failure of coincidence is small enough, but the quantitative bound is more restrictive in this case than in the previously analyzed "efficiency loophole".
This paper analyzes effects of time‐dependence in the Bell inequality. A generalized inequality is derived for the case when coincidence and non‐coincidence [and hence whether or not a pair contributes to the actual data] is controlled by timing that depends on the detector settings. Needless to say, this inequality is violated by quantum mechanics and could be violated by experimental data provided that the loss of measurement pairs through failure of coincidence is small enough, but the quantitative bound is more restrictive in this case than in the previously analyzed “efficiency loophole.”
We apply a distance-based Bell-test analysis method [E. Knill et al., Phys. Rev. A 91, 032105 (2015)] to three experimental data sets where conventional analyses failed or required additional assumptions. The first is produced from a classical source exploiting a ``coincidence-time loophole'' for which standard analysis falsely shows a Bell violation. The second is from a source previously shown to violate a Bell inequality; the distance-based analysis agrees with the previous results but with fewer assumptions. The third data set does not show a violation with standard analysis despite the high source quality, but is shown to have a strong violation with the distance-based analysis method.
In this paper, a method of generalizing the Bell inequality is presented that makes it possible to include detector inefficiency directly in the original Bell inequality. To enable this, the concept of ``change of ensemble'' will be presented, providing both qualitative and quantitative information on the nature of the ``loophole'' in the proof of the original Bell inequality. In a local hidden-variable model lacking change of ensemble, the generalized inequality reduces to an inequality that quantum mechanics violates as strongly as the original Bell inequality, irrespective of the level of efficiency of the detectors. A model that contains change of ensemble lowers the violation, and a bound for the level of change is obtained. The derivation of the bound in this paper is not dependent upon any symmetry assumptions such as constant efficiency, or even the assumption of independent errors.
Quantum causality violates classical intuitions of cause and effect and is a unique quantum feature different from other quantum phenomena such as entanglement and quantum nonlocality. In order to avoid the detection loophole in quantum causality, we initiate the study of the detection efficiency requirement for observing quantum causality. We first show that previous classical causal inequalities require detection efficiency at least 95.9795.97%95.97 (89.4489.44%89.44) to show violation with quantum (nonsignaling) correlations. Next we derive a classical causal inequality I222{I}_{222}I222​ and show that it requires lower detection efficiency to be violated, 92.3992.39%92.39 for quantum correlations and 81.6581.65%81.65 for nonsignaling correlations, hence substantially reducing the requirement on detection. Then we extend this causal inequality to the case of multiple measurement settings and analyze the corresponding detection efficiency. After that, we show that previous quantum causal inequalities require detection efficiency at least 94.2994.29%94.29 to violate with nonsignaling correlations. We subsequently derive a quantum causal bound J222{J}_{222}J222​ that has a lower detection efficiency requirement of 91.0291.02%91.02 for violation with nonsignaling correlations. Our work paves the way towards an experimental demonstration of quantum causality and shows that causal inequalities significantly differ from Bell inequalities in terms of the detection efficiency requirement.
In many quantum information applications, a minimum detection efficiency must be exceeded to ensure success. Protocols depending on the violation of a Bell inequality, for instance, may be subject to the so-called detection loophole: Imperfect detectors may yield spurious violations, which consequently cannot be used to ensure, say, quantum cryptographic security. Hence, we investigate the possibility of giving lower bounds on detector efficiency even if an adversary has full control over both the source and the detectors. To this end, we present a technique to systematically derive Bell inequalities free from the detection loophole using only the observed measurements statistics. The violation of these inequalities certifies that the detectors used exceed a certain minimal efficiency.
In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nthn\mathrm{th}nth particle pair would be independent of measurement choices and outcomes for the first $(n\ensuremath{-}1)$ pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the two-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the two-sided memory loophole allows a systematic violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality when the data are analyzed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in the existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearized CHSH inequality as well as other Bell-type inequalities.