Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1312.2999 · Journal of Physics A Mathematical and Theoretical

A robust mathematical model for a loophole-free Clauser–Horne experiment

Peter Bierhorst

Published Apr 17, 2015• 67 citations
arXivHTMLPDF

Abstract

Recent experiments (Giustina et al 2013 Nature 497 227–30 ; Christensen et al 2013 Phys. Rev. Lett. 111 130406 ) have reached detection efficiencies sufficient to close the detection loophole, testing the Clauser–Horne version of Bell's inequality. For a similar future experiment to be completely loophole-free, it will be important to have discrete experimental trials with randomized measurement settings for each trial, and the statistical analysis should not overlook the possibility of a local state varying over time with possible dependence on earlier trials (the ‘memory loophole’). In this paper, a mathematical model for such an experiment is presented, and a method for statistical analysis that is robust to memory effects is introduced. Additionally, a new method for calculating exact p -values for martingale-based statistics is described; previously, only non-sharp upper bounds derived from the Azuma–Hoeffding inequality have been available for such statistics. This improvement decreases the required number of experimental trials to demonstrate non-locality. The statistical techniques are applied to the data of Giustina et al (2013 Nature 497 227–30 ) and Christensen et al (2013 Phys. Rev. Lett. 111 130406 ) and found to perform well.

DOI
10.1088/1751-8113/48/19/195302

Related papers

Clauser-Horne/Eberhard Inequality Violation by a Local Model

Donald A. Graft

Thanks to its immunity to the detection loophole, the Clauser-Horne/Eberhard inequality plays an important role in tests of locality and in certification of quantum information protocols based on entanglement. I describe a local model that violates the inequality using a plausible mechanism relying upon a parameter of the apparatus, the source emission rate. The effect is generated through the analysis of time-tagged data using a standard windowed coincidence counting method. Significantly, the detection times here are not functions of the measurement settings, i.e., the fair coincidences assumption is satisfied. This finding has implications for the design and interpretation of experiments and for quantum information protocols, as it shows that the coincidence window mechanism cannot be eliminated by a demonstration of independence of the detection times and settings. The paper describes a reliable coincidence counting method and shows that it delivers an accurate count of true coincidences. Recent experimental tests of local realism based on the Clauser-Horne/Eberhard inequality are considered and it is shown that in one case (Christensen et al.) the emission rate is appropriately limited to ensure valid counting, and the data supports locality; in a second case (Giustina et al.) the experiment neglects to appropriately limit the emission rate, and the claimed violation can be accounted for locally.

arXiv:1507.06231•Advanced Science Engineering and Medicine•Jun 1, 2016•4 citations

A New Loophole in Recent Bell Test Experiments

Peter Bierhorst

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.

arXiv:1311.4488•Nov 18, 2013•2 citations

Bell-inequality violation with entangled photons, free of the coincidence-time loophole

Jan-Åke Larsson, Marissa Giustina, Johannes Kofler +3 more

In a local realist model, physical properties are defined prior to and independent of measurement and no physical influence can propagate faster than the speed of light. Proper experimental violation of a Bell inequality would show that the world cannot be described with such a model. Experiments intended to demonstrate a violation usually require additional assumptions that make them vulnerable to a number of ``loopholes.'' In both pulsed and continuously pumped photonic experiments, an experimenter needs to identify which detected photons belong to the same pair, giving rise to the coincidence-time loophole. Here, via two different methods, we derive Clauser-Horne- and Eberhard-type inequalities that are not only free of the fair-sampling assumption (thus not being vulnerable to the detection loophole), but also free of the fair-coincidence assumption (thus not being vulnerable to the coincidence-time loophole). Both approaches can be used for pulsed as well as for continuously pumped experiments. Moreover, as they can also be applied to already existing experimental data, we finally show that a recent experiment [Giustina et al., Nature (London) 497, 227 (2013)] violated local realism without requiring the fair-coincidence assumption.

arXiv:1309.0712•Physical Review A•Sep 16, 2014•53 citations

Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis

Bas Hensen, Norbert Kalb, Machiel Blok +11 more

The recently reported violation of a Bell inequality using entangled electronic spins in diamonds (Hensen et al., Nature 526, 682-686) provided the first loophole-free evidence against local-realist theories of nature. Here we report on data from a second Bell experiment using the same experimental setup with minor modifications. We find a violation of the CHSH-Bell inequality of 2.35 ± 0.18, in agreement with the first run, yielding an overall value of S = 2.38 ± 0.14. We calculate the resulting P-values of the second experiment and of the combined Bell tests. We provide an additional analysis of the distribution of settings choices recorded during the two tests, finding that the observed distributions are consistent with uniform settings for both tests. Finally, we analytically study the effect of particular models of random number generator (RNG) imperfection on our hypothesis test. We find that the winning probability per trial in the CHSH game can be bounded knowing only the mean of the RNG bias. This implies that our experimental result is robust for any model underlying the estimated average RNG bias, for random bits produced up to 690 ns too early by the random number generator.

arXiv:1603.05705•Scientific Reports•Aug 11, 2016•95 citations

Consequences of recent loophole-free experiments on a relaxation of measurement independence

Alejandro Andrés Hnilo

Recent experiments using innovative optical detectors and techniques have strongly increased the capacity of testing the violation of the Bell's inequalities in Nature. Most of them have used the Eberhard's inequality (EI) to close the ``detection'' loophole. Closing the ``locality'' loophole has been attempted by spacelike separated detections and fast changes of the bases of observation, driven by random number generators of new design. Also, pulsed pumping and time-resolved data recording to close the ``time-coincidence'' loophole, and sophisticated statistical methods to close the ``memory'' loophole, have been used. In this paper, the meaning of the EI is reviewed. A simple hidden variables theory based on a relaxation of the condition of ``measurement independence,'' which was devised long ago for the Clauser-Horne-Shimony and Holt inequality, is adapted to the EI case. It is used here to evaluate the significance of the results of the recent experiments, which are briefly described. A table summarizes the main results.

arXiv:1607.04177•Physical Review A•Feb 2, 2017•7 citations

On the equivalence of the Clauser–Horne and Eberhard inequality based tests

Andrei Yu. Khrennikov, Sven Ramelow, Rupert Ursin +3 more

Recently, the results of the first experimental test for entangled photons closing the detection loophole (also referred to as the fair sampling loophole) were published (Vienna, 2013). From the theoretical viewpoint the main distinguishing feature of this long-aspired to experiment was that the Eberhard inequality was used. Almost simultaneously another experiment closing this loophole was performed (Urbana-Champaign, 2013) and it was based on the Clauser–Horne inequality (for probabilities). The aim of this note is to analyze the mathematical and experimental equivalence of tests based on the Eberhard inequality and various forms of the Clauser–Horne inequality. The structure of the mathematical equivalence is nontrivial. In particular, it is necessary to distinguish between algebraic and statistical equivalence. Although the tests based on these inequalities are algebraically equivalent, they need not be equivalent statistically, i.e., theoretically the level of statistical significance can drop under transition from one test to another (at least for finite samples). Nevertheless, the data collected in the Vienna test implies not only a statistically significant violation of the Eberhard inequality, but also of the Clauser–Horne inequality (in the ratio-rate form): for both a violation .

arXiv:1403.2811•Physica Scripta•Dec 1, 2014•23 citations
Recent experiments (Giustina et al 2013 Nature 497 227–30 ; Christensen et al 2013 Phys. Rev. Lett. 111 130406 ) have reached detection efficiencies sufficient to close the detection loophole, testing the Clauser–Horne version of Bell's inequality. For a similar future experiment to be completely loophole-free, it will be important to have discrete experimental trials with randomized measurement settings for each trial, and the statistical analysis should not overlook the possibility of a local state varying over time with possible dependence on earlier trials (the ‘memory loophole’). In this paper, a mathematical model for such an experiment is presented, and a method for statistical analysis that is robust to memory effects is introduced. Additionally, a new method for calculating exact p -values for martingale-based statistics is described; previously, only non-sharp upper bounds derived from the Azuma–Hoeffding inequality have been available for such statistics. This improvement decreases the required number of experimental trials to demonstrate non-locality. The statistical techniques are applied to the data of Giustina et al (2013 Nature 497 227–30 ) and Christensen et al (2013 Phys. Rev. Lett. 111 130406 ) and found to perform well.
Thanks to its immunity to the detection loophole, the Clauser-Horne/Eberhard inequality plays an important role in tests of locality and in certification of quantum information protocols based on entanglement. I describe a local model that violates the inequality using a plausible mechanism relying upon a parameter of the apparatus, the source emission rate. The effect is generated through the analysis of time-tagged data using a standard windowed coincidence counting method. Significantly, the detection times here are not functions of the measurement settings, i.e., the fair coincidences assumption is satisfied. This finding has implications for the design and interpretation of experiments and for quantum information protocols, as it shows that the coincidence window mechanism cannot be eliminated by a demonstration of independence of the detection times and settings. The paper describes a reliable coincidence counting method and shows that it delivers an accurate count of true coincidences. Recent experimental tests of local realism based on the Clauser-Horne/Eberhard inequality are considered and it is shown that in one case (Christensen et al.) the emission rate is appropriately limited to ensure valid counting, and the data supports locality; in a second case (Giustina et al.) the experiment neglects to appropriately limit the emission rate, and the claimed violation can be accounted for locally.
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.
In a local realist model, physical properties are defined prior to and independent of measurement and no physical influence can propagate faster than the speed of light. Proper experimental violation of a Bell inequality would show that the world cannot be described with such a model. Experiments intended to demonstrate a violation usually require additional assumptions that make them vulnerable to a number of ``loopholes.'' In both pulsed and continuously pumped photonic experiments, an experimenter needs to identify which detected photons belong to the same pair, giving rise to the coincidence-time loophole. Here, via two different methods, we derive Clauser-Horne- and Eberhard-type inequalities that are not only free of the fair-sampling assumption (thus not being vulnerable to the detection loophole), but also free of the fair-coincidence assumption (thus not being vulnerable to the coincidence-time loophole). Both approaches can be used for pulsed as well as for continuously pumped experiments. Moreover, as they can also be applied to already existing experimental data, we finally show that a recent experiment [Giustina et al., Nature (London) 497, 227 (2013)] violated local realism without requiring the fair-coincidence assumption.
The recently reported violation of a Bell inequality using entangled electronic spins in diamonds (Hensen et al., Nature 526, 682-686) provided the first loophole-free evidence against local-realist theories of nature. Here we report on data from a second Bell experiment using the same experimental setup with minor modifications. We find a violation of the CHSH-Bell inequality of 2.35 ± 0.18, in agreement with the first run, yielding an overall value of S = 2.38 ± 0.14. We calculate the resulting P-values of the second experiment and of the combined Bell tests. We provide an additional analysis of the distribution of settings choices recorded during the two tests, finding that the observed distributions are consistent with uniform settings for both tests. Finally, we analytically study the effect of particular models of random number generator (RNG) imperfection on our hypothesis test. We find that the winning probability per trial in the CHSH game can be bounded knowing only the mean of the RNG bias. This implies that our experimental result is robust for any model underlying the estimated average RNG bias, for random bits produced up to 690 ns too early by the random number generator.
Recent experiments using innovative optical detectors and techniques have strongly increased the capacity of testing the violation of the Bell's inequalities in Nature. Most of them have used the Eberhard's inequality (EI) to close the ``detection'' loophole. Closing the ``locality'' loophole has been attempted by spacelike separated detections and fast changes of the bases of observation, driven by random number generators of new design. Also, pulsed pumping and time-resolved data recording to close the ``time-coincidence'' loophole, and sophisticated statistical methods to close the ``memory'' loophole, have been used. In this paper, the meaning of the EI is reviewed. A simple hidden variables theory based on a relaxation of the condition of ``measurement independence,'' which was devised long ago for the Clauser-Horne-Shimony and Holt inequality, is adapted to the EI case. It is used here to evaluate the significance of the results of the recent experiments, which are briefly described. A table summarizes the main results.
Recently, the results of the first experimental test for entangled photons closing the detection loophole (also referred to as the fair sampling loophole) were published (Vienna, 2013). From the theoretical viewpoint the main distinguishing feature of this long-aspired to experiment was that the Eberhard inequality was used. Almost simultaneously another experiment closing this loophole was performed (Urbana-Champaign, 2013) and it was based on the Clauser–Horne inequality (for probabilities). The aim of this note is to analyze the mathematical and experimental equivalence of tests based on the Eberhard inequality and various forms of the Clauser–Horne inequality. The structure of the mathematical equivalence is nontrivial. In particular, it is necessary to distinguish between algebraic and statistical equivalence. Although the tests based on these inequalities are algebraically equivalent, they need not be equivalent statistically, i.e., theoretically the level of statistical significance can drop under transition from one test to another (at least for finite samples). Nevertheless, the data collected in the Vienna test implies not only a statistically significant violation of the Eberhard inequality, but also of the Clauser–Horne inequality (in the ratio-rate form): for both a violation .