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:2501.14027 · Physical Review Letters

Bell Nonlocality in Quantum Networks with Unreliable Sources: Loophole-Free Postelection via Self-Testing

Sadra Boreiri, Nicolas Brunner, Pavel Sekatski

Published Sep 2, 2025• 3 citations
arXivHTMLPDF

Abstract

We discuss Bell nonlocality in quantum networks with unreliable sources. Our main result is a condition on the observed data that ensures that inconclusive events can be safely discarded, without introducing any loophole. More formally, we characterize the fair-sampling property for measurements in a network. When all measurements are fair-sampling, we show that the postselection of conclusive outcomes does not compromise the assumption of source independence, hence avoiding the detection loophole. Furthermore, we show that in some cases, the fair-sampling property can in fact be certified directly from observed data. To show this, we prove that saturation of the Finner inequality provides a self-test of the underlying quantum model. We illustrate the relevance of our results by demonstrating an improvement in device-independent randomness generation for a photonic Bell test with a probabilistic source and for the triangle network.

DOI
10.1103/bqwm-ll5y

Related papers

Bell experiments with random destination sources

Fabio Sciarrino, Giuseppe Vallone, Adán Cabello +1 more

It is generally assumed that sources randomly sending two particles to one or two different observers, random destination sources (RDSs), cannot be used for genuine quantum nonlocality tests because of the postselection loophole. We demonstrate that Bell experiments not affected by the postselection loophole may be performed with (i) an RDS and local postselection using perfect detectors, (ii) an RDS, local postselection, and fair sampling assumption with any detection efficiency, and (iii) an RDS and a threshold detection efficiency required to avoid the detection loophole. These results allow the adoption of RDS setups which are simpler and more efficient for long-distance free-space Bell tests, and extend the range of physical systems which can be used for loophole-free Bell tests.

arXiv:1009.4854•Physical Review A•Mar 14, 2011•13 citations

Partial Self-Testing and Randomness Certification in the Triangle Network

Pavel Sekatski, Sadra Boreiri, Nicolas Brunner

Quantum nonlocality can be demonstrated without inputs (i.e., each party using a fixed measurement setting) in a network with independent sources. Here we consider this effect on ring networks, and show that the underlying quantum strategy can be partially characterized, or self-tested, from observed correlations. Applying these results to the triangle network allows us to show that the nonlocal distribution of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] requires that (i) all sources produce a minimal amount of entanglement, (ii) all local measurements are entangled, and (iii) each local outcome features a minimal entropy. Hence we show that the triangle network allows for genuine network quantum nonlocality and certifiable randomness.

arXiv:2209.09921•Physical Review Letters•Sep 5, 2023•22 citations

Detection-Loophole-Free Test of Quantum Nonlocality, and Applications

Bradley Christensen, Kevin T. McCusker, Joseph B. Altepeter +11 more

We present a source of entangled photons that violates a Bell inequality free of the "fair-sampling" assumption, by over 7 standard deviations. This violation is the first reported experiment with photons to close the detection loophole, and we demonstrate enough "efficiency" overhead to eventually perform a fully loophole-free test of local realism. The entanglement quality is verified by maximally violating additional Bell tests, testing the upper limit of quantum correlations. Finally, we use the source to generate "device-independent" private quantum random numbers at rates over 4 orders of magnitude beyond previous experiments.

arXiv:1306.5772•Physical Review Letters•Sep 26, 2013•477 citations

How post-selection affects device-independent claims under the fair sampling assumption

Davide Orsucci, Jean-Daniel Bancal, Nicolas Sangouard +1 more

Device-independent certifications employ Bell tests to guarantee the proper functioning of an apparatus from the sole knowledge of observed measurement statistics, i.e. without assumptions on the internal functioning of the devices. When these Bell tests are implemented with devices having too low efficiency, one has to post-select the events that lead to successful detections and thus rely on a fair sampling assumption. The question that we address in this paper is what remains of a device-independent certification under fair sampling. We provide an intuitive description of post-selections in terms of filters and define the fair sampling assumption as a property of these filters, equivalent to the definition introduced in Ref. \cite{Berry10}. When this assumption is fulfilled, the post-selected data is reproduced by an ideal experiment where lossless devices measure a filtered state which can be obtained from the actual state via local probabilistic maps. Trusted conclusions can thus be obtained on the quantum properties of this filtered state and the corresponding measurement statistics can reliably be used, e.g., for randomness generation or quantum key distribution. We also explore a stronger notion of fair sampling leading to the conclusion that the post-selected data is a fair representation of the data that would be obtained with lossless detections. Furthermore, we show that our conclusions hold in cases of small deviations from exact fair sampling. Finally, we describe setups previously or potentially used in Bell-type experiments under fair sampling and identify the underlying device-specific assumptions.

arXiv:1908.11123•Quantum•Mar 2, 2020•19 citations

Extending the fair sampling assumption using causal diagrams

Valentin Gebhart, Augusto Smerzi

Discarding undesirable measurement results in Bell experiments opens the detection loophole that prevents a conclusive demonstration of nonlocality. As closing the detection loophole represents a major technical challenge for many practical Bell experiments, it is customary to assume the so-called fair sampling assumption (FSA) that, in its original form, states that the collectively postselected statistics are a fair sample of the ideal statistics. Here, we analyze the FSA from the viewpoint of causal inference: We derive a causal structure that must be present in any causal model that faithfully encapsulates the FSA. This provides an easy, intuitive, and unifying approach that includes different accepted forms of the FSA and underlines what is really assumed when using the FSA. We then show that the FSA can not only be applied in scenarios with non-ideal detectors or transmission losses, but also in ideal experiments where only parts of the correlations are postselected, e.g., when the particles' destinations are in a superposition state. Finally, we demonstrate that the FSA is also applicable in multipartite scenarios that test for (genuine) multipartite nonlocality.

arXiv:2207.09348•Quantum•Jan 13, 2023•3 citations

Experimental Full Network Nonlocality with Independent Sources and Strict Locality Constraints

Xuemei Gu, Liang Huang, Alejandro Pozas-Kerstjens +10 more

Nonlocality arising in networks composed of several independent sources gives rise to phenomena radically different from that in standard Bell scenarios. Over the years, the phenomenon of network nonlocality in the entanglement-swapping scenario has been well investigated and demonstrated. However, it is known that violations of the so-called bilocality inequality used in previous experimental demonstrations cannot be used to certify the nonclassicality of their sources. This has put forward a stronger concept for nonlocality in networks, called full network nonlocality. Here, we experimentally observe full network nonlocal correlations in a network where the source-independence, locality, and measurement-independence loopholes are closed. This is ensured by employing two independent sources, rapid setting generation, and spacelike separations of relevant events. Our experiment violates known inequalities characterizing nonfull network nonlocal correlations by over 5 standard deviations, certifying the absence of classical sources in the realization.

arXiv:2302.02472•Physical Review Letters•May 11, 2023•24 citations
We discuss Bell nonlocality in quantum networks with unreliable sources. Our main result is a condition on the observed data that ensures that inconclusive events can be safely discarded, without introducing any loophole. More formally, we characterize the fair-sampling property for measurements in a network. When all measurements are fair-sampling, we show that the postselection of conclusive outcomes does not compromise the assumption of source independence, hence avoiding the detection loophole. Furthermore, we show that in some cases, the fair-sampling property can in fact be certified directly from observed data. To show this, we prove that saturation of the Finner inequality provides a self-test of the underlying quantum model. We illustrate the relevance of our results by demonstrating an improvement in device-independent randomness generation for a photonic Bell test with a probabilistic source and for the triangle network.
It is generally assumed that sources randomly sending two particles to one or two different observers, random destination sources (RDSs), cannot be used for genuine quantum nonlocality tests because of the postselection loophole. We demonstrate that Bell experiments not affected by the postselection loophole may be performed with (i) an RDS and local postselection using perfect detectors, (ii) an RDS, local postselection, and fair sampling assumption with any detection efficiency, and (iii) an RDS and a threshold detection efficiency required to avoid the detection loophole. These results allow the adoption of RDS setups which are simpler and more efficient for long-distance free-space Bell tests, and extend the range of physical systems which can be used for loophole-free Bell tests.
Quantum nonlocality can be demonstrated without inputs (i.e., each party using a fixed measurement setting) in a network with independent sources. Here we consider this effect on ring networks, and show that the underlying quantum strategy can be partially characterized, or self-tested, from observed correlations. Applying these results to the triangle network allows us to show that the nonlocal distribution of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.140401] requires that (i) all sources produce a minimal amount of entanglement, (ii) all local measurements are entangled, and (iii) each local outcome features a minimal entropy. Hence we show that the triangle network allows for genuine network quantum nonlocality and certifiable randomness.
We present a source of entangled photons that violates a Bell inequality free of the "fair-sampling" assumption, by over 7 standard deviations. This violation is the first reported experiment with photons to close the detection loophole, and we demonstrate enough "efficiency" overhead to eventually perform a fully loophole-free test of local realism. The entanglement quality is verified by maximally violating additional Bell tests, testing the upper limit of quantum correlations. Finally, we use the source to generate "device-independent" private quantum random numbers at rates over 4 orders of magnitude beyond previous experiments.
Device-independent certifications employ Bell tests to guarantee the proper functioning of an apparatus from the sole knowledge of observed measurement statistics, i.e. without assumptions on the internal functioning of the devices. When these Bell tests are implemented with devices having too low efficiency, one has to post-select the events that lead to successful detections and thus rely on a fair sampling assumption. The question that we address in this paper is what remains of a device-independent certification under fair sampling. We provide an intuitive description of post-selections in terms of filters and define the fair sampling assumption as a property of these filters, equivalent to the definition introduced in Ref. \cite{Berry10}. When this assumption is fulfilled, the post-selected data is reproduced by an ideal experiment where lossless devices measure a filtered state which can be obtained from the actual state via local probabilistic maps. Trusted conclusions can thus be obtained on the quantum properties of this filtered state and the corresponding measurement statistics can reliably be used, e.g., for randomness generation or quantum key distribution. We also explore a stronger notion of fair sampling leading to the conclusion that the post-selected data is a fair representation of the data that would be obtained with lossless detections. Furthermore, we show that our conclusions hold in cases of small deviations from exact fair sampling. Finally, we describe setups previously or potentially used in Bell-type experiments under fair sampling and identify the underlying device-specific assumptions.
Discarding undesirable measurement results in Bell experiments opens the detection loophole that prevents a conclusive demonstration of nonlocality. As closing the detection loophole represents a major technical challenge for many practical Bell experiments, it is customary to assume the so-called fair sampling assumption (FSA) that, in its original form, states that the collectively postselected statistics are a fair sample of the ideal statistics. Here, we analyze the FSA from the viewpoint of causal inference: We derive a causal structure that must be present in any causal model that faithfully encapsulates the FSA. This provides an easy, intuitive, and unifying approach that includes different accepted forms of the FSA and underlines what is really assumed when using the FSA. We then show that the FSA can not only be applied in scenarios with non-ideal detectors or transmission losses, but also in ideal experiments where only parts of the correlations are postselected, e.g., when the particles' destinations are in a superposition state. Finally, we demonstrate that the FSA is also applicable in multipartite scenarios that test for (genuine) multipartite nonlocality.
Nonlocality arising in networks composed of several independent sources gives rise to phenomena radically different from that in standard Bell scenarios. Over the years, the phenomenon of network nonlocality in the entanglement-swapping scenario has been well investigated and demonstrated. However, it is known that violations of the so-called bilocality inequality used in previous experimental demonstrations cannot be used to certify the nonclassicality of their sources. This has put forward a stronger concept for nonlocality in networks, called full network nonlocality. Here, we experimentally observe full network nonlocal correlations in a network where the source-independence, locality, and measurement-independence loopholes are closed. This is ensured by employing two independent sources, rapid setting generation, and spacelike separations of relevant events. Our experiment violates known inequalities characterizing nonfull network nonlocal correlations by over 5 standard deviations, certifying the absence of classical sources in the realization.