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Detection Loophole in Bell Tests: Post-Selection with Outcome-Dependent Efficiencies

Aria Ghanbari

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

As children, many of us imagined our toys coming to life the moment we looked away, just like in Toy Story. Maybe they walked across the room and secretly planned adventures behind our backs. Surprisingly, physicists ask a very similar question about quantum particles: do they have definite properties before we measure them, or is reality decided at the moment of observation? Understanding the answer has led to technologies such as quantum computing, quantum teleportation, and quantum cryptography. In the quantum world, where particles such as electrons and positrons live, objects do not have definite positions or states but can exist in a superposition of possible states. Even stranger, two particles can become entangled, meaning measuring one, instantly predicts information about measuring the other, no matter how far apart they are. If our particles were Toy Story characters, Woody and Buzz, we could simply ask them questions in a courtroom trial and reveal the secret. In quantum mechanics, these questions are essentially measurements. However, Woody and Buzz can cheat in the trial, and so can our particles, by preplanning the outcome of any measurement in advance. This idea is known as a local hidden-variable theory, and for a long time, it was thought that there was no way to distinguish between such preplanned outcomes and outcomes determined only at the moment of measurement. In 1964, John Bell proposed a test, known as the Bell test, which can reveal this fact. This test can expose any hidden planned strategy. No matter how cleverly our particles coordinate, Bell’s test gives them no easy way to fake the results. His method is a statistical comparison between the results of measurements performed on a pair of particles, which consists of some inequalities that are satisfied if particles have pre-determined outcomes before they are separated, and violated for some of the quantum cases where particles live in a superposition of outcomes, and their outcomes are correlated to each other. But there is still a loophole. Remember the courtroom trial? Now imagine Woody and Buzz invoking the Fifth Amendment and refusing to answer questions. In physical terms, this means that the measurements are inefficient and detectors sometimes fail to produce a result. That missing information can mislead us if we perform our analysis based on the detected particles, a procedure known as post-selection, and this loophole is commonly referred to as the detection loophole. Many efforts have been made to close this loophole, both by increasing the detection efficiency of the detectors and by finding settings in which the minimum efficiency required for a reliable result is lower. In this thesis, we further investigated this loophole under the post-selection assumption. We considered a model in which the efficiency of each measurement depends on its outcome. We then studied how the Bell tests behave when the underlying theory of reality is either a local hidden-variable theory or a quantum-mechanical theory.

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arXiv:1611.03444•Nov 10, 2016•1 citation
As children, many of us imagined our toys coming to life the moment we looked away, just like in Toy Story. Maybe they walked across the room and secretly planned adventures behind our backs. Surprisingly, physicists ask a very similar question about quantum particles: do they have definite properties before we measure them, or is reality decided at the moment of observation? Understanding the answer has led to technologies such as quantum computing, quantum teleportation, and quantum cryptography. In the quantum world, where particles such as electrons and positrons live, objects do not have definite positions or states but can exist in a superposition of possible states. Even stranger, two particles can become entangled, meaning measuring one, instantly predicts information about measuring the other, no matter how far apart they are. If our particles were Toy Story characters, Woody and Buzz, we could simply ask them questions in a courtroom trial and reveal the secret. In quantum mechanics, these questions are essentially measurements. However, Woody and Buzz can cheat in the trial, and so can our particles, by preplanning the outcome of any measurement in advance. This idea is known as a local hidden-variable theory, and for a long time, it was thought that there was no way to distinguish between such preplanned outcomes and outcomes determined only at the moment of measurement. In 1964, John Bell proposed a test, known as the Bell test, which can reveal this fact. This test can expose any hidden planned strategy. No matter how cleverly our particles coordinate, Bell’s test gives them no easy way to fake the results. His method is a statistical comparison between the results of measurements performed on a pair of particles, which consists of some inequalities that are satisfied if particles have pre-determined outcomes before they are separated, and violated for some of the quantum cases where particles live in a superposition of outcomes, and their outcomes are correlated to each other. But there is still a loophole. Remember the courtroom trial? Now imagine Woody and Buzz invoking the Fifth Amendment and refusing to answer questions. In physical terms, this means that the measurements are inefficient and detectors sometimes fail to produce a result. That missing information can mislead us if we perform our analysis based on the detected particles, a procedure known as post-selection, and this loophole is commonly referred to as the detection loophole. Many efforts have been made to close this loophole, both by increasing the detection efficiency of the detectors and by finding settings in which the minimum efficiency required for a reliable result is lower. In this thesis, we further investigated this loophole under the post-selection assumption. We considered a model in which the efficiency of each measurement depends on its outcome. We then studied how the Bell tests behave when the underlying theory of reality is either a local hidden-variable theory or a quantum-mechanical theory.
Let us imagine an experimentalist, who has maximally entangled photons and wants to violate Bell’s inequality in his laboratory. Among many obstacles he has to overcome, two of them are the most significant. Firstly, he has to avoid exchanging information about the measurement settings between the particles. Secondly, he has to detect particles with a high enough probability, because if the detection efficiency is too low, one can reproduce quantum correlations using a LHV (Local Hidden Variables) model. These two problems are known in the literature, respectively, as the locality loophole [1] and the detection loophole [2, 3]. Fortunately, it is possible to close the locality loophole using photons in the experiment [4, 5, 6, 7, 8]. However, closing the detection loophole is still an object of study. The authors of [9] showed that for the CHSH inequality, the lower bound for detector efficiency needed to close the loophole is 82.8% for maximally entangled states. This bound was lowered to 66.7% by Eberhard [10]. Furthermore, the detector efficiency has been studied for other Bell inequalities [11, 12, 13, 14, 15, 16]. In our research, we are trying to answer the following question: If we prepare a Bell experiment and we have two detectors, having different efficiencies both smaller than 100%, which detector should each party receive? And also: Which representation of the chosen Bell inequality should we consider? These are very crucial questions, which arise in laboratories. Hence, we hope that our results will be useful for experimentalists. We focus on the I3322 inequality [17], which is a natural candidate for testing, because of its asymmetry [14, 18]. We consider the case of a post-selection strategy, which consists in rejecting all the rounds in the Bell experiment when one of the detectors does not click. We used numerical optimisation methods to find both the best classical strategy for Alice and Bob and the best detector clicking strategy. Thus, we have found minimal efficiencies of the detectors of Alice and Bob needed to violate I3322. We made the same studies also for symmetric representations of I3322 [19]. Our results show that the minimal efficiencies of Alice and Bob’s detectors of the asymmetric representation of I3322 is lower than in case of the symmetric one [20]. It is work in progress, and we are examining other post-selection methods as well as other Bell inequalities. We want to consider also the case when non-maximally entangled states are used in Bell experiments.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated \nmeasurements on entangled quantum systems violate a Bell inequality [Bell1964]. \nWe put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate \nparticular maps, or functions to particular theories. The violation of a Bell \ninequality then implies the ability to perform some functions, or computations \nthat classical, or more generally, local hidden variable (LHV) theories cannot. \nWe derive an in�finite class of Bell inequalities that establish a link to so-called "non-local games" [Cleve2004]. We then make the connection between \nRaussendorf and Briegel's formulation of Measurement-based Quantum Computing (MBQC) [Raussendorf2001], and these non-local games. Not only can we show that a quantum violation implies a computational advantage in this model, \nwe show that adaptive measurements are required to perform all quantum computations. \nFinally, we explore post-selection of data in Bell tests from both a practical and \nconceptual point-of-view, with particular consideration to so-called "loopholes". \nLoopholes allow LHV theories to simulate quantum correlations through post-selection. \nWe give a computational description of how loopholes can emerge in diff�erent post-selection scenarios. This motivates us to �find a form of post-selection \nthat does not lead to loopholes. Central again to this discussion is the \ndescription of LHV theories in terms of computations. \nInterestingly, quantum correlators can be made more "non-classical" with this \nloophole-free post-selection. This method of post-selection also can simulate \ninformation processing tasks, such as MBQC, that have time-like separated components. \nThis opens up new avenues for the study of time-like tasks studied \nwithin the space-like separated scenario of the Bell test.
Recent experiments have reached detection efficiencies sufficient to close the detection loophole with photons. Both experiments ran multiple successive trials in fixed measurement configurations, rather than randomly re-setting the measurement configurations before each measurement trial. This opens a new potential loophole for a local hidden variable theory. The loophole invalidates one proposed method of statistical analysis of the experimental results, as demonstrated in this note. Therefore a different analysis will be necessary to definitively assert that these experiments are subject only to the locality loophole.
Bell's inequalities(1964) provides a quantitative criterion to test the local hidden variable theory(LHVT) versus quantum mechanics(QM).For the first time,it leads to the possibility to experimentally distinguish LHVT and QM.So far,there have been tens of experiments ever since the first experiment by Clauser et al(1972).An overwhelming majority of these experiments are in good agreement with the predictions of QM yet violate those of LHVT.Some loopholes in these experiments are discussed in Section I,followed by a respective summary to 12 experiments between 1972 to 1982 and 31 experiments from 1985 and on in Sections II and III.It is finally pointed out that so far there is not a single experiment to close both the locality loophole and the detection loophole at the same time,but the experimental outcomes are irreversible.Accordingly,theoretical physicists must admit that the present physics foundation is broken,i.e.quantum mechanics is contradiction with special relativity.
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.
Contrary to counterfactual definiteness quantum theory teaches us that measuring instruments are not passively reading predetermined values of physical observables. Counterfactual definiteness allows proving Bell inequalities. If the contextual character of quantum measurements is correctly taken into account the proofs of these inequalities may not be done. In recent computer simulations of idealized Bell experiment predetermined successive outcomes of measurements for each setting and predetermined time delays of their registrations are calculated. Time windows and time delays are used to select various samples. Correlations, estimated using these selected samples are consistent with the predictions of quantum theory and the time window dependence is similar to the dependence observed in some real experiments. It is an important example how correlations can be explained without evoking quantum non-locality. However by using a suitable post-selection one may prove anything. Since before the post-selection generated samples may not violate Bell inequalities as significantly as finite samples generated using quantum predictions thus one may not conclude that counterfactual definiteness is not able to distinguish classical from quantum physics. Moreover we show that for each choice of a time window there exists a contextual hidden variable probabilistic model consistent with the post-selection procedure used by the authors what explains why they are able to reproduce quantum predictions.