{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T05:56:18Z","timestamp":1775282178968,"version":"3.50.1"},"reference-count":56,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>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<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>f<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>l<\/mml:mi><mml:mi>t<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>r<\/mml:mi><mml:mi>s<\/mml:mi><\/mml:math>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<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>f<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>l<\/mml:mi><mml:mi>t<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>r<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:math>state which can be obtained from the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>a<\/mml:mi><mml:mi>c<\/mml:mi><mml:mi>t<\/mml:mi><mml:mi>u<\/mml:mi><mml:mi>a<\/mml:mi><mml:mi>l<\/mml:mi><\/mml:math>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.<\/jats:p>","DOI":"10.22331\/q-2020-03-02-238","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T14:33:21Z","timestamp":1583159601000},"page":"238","source":"Crossref","is-referenced-by-count":15,"title":["How post-selection affects device-independent claims under the fair sampling assumption"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3087-8757","authenticated-orcid":false,"given":"Davide","family":"Orsucci","sequence":"first","affiliation":[{"name":"Quantum Optics Theory Group, Universit\u00e4t Basel, CH-4056 Basel, Switzerland"}]},{"given":"Jean-Daniel","family":"Bancal","sequence":"additional","affiliation":[{"name":"Quantum Optics Theory Group, Universit\u00e4t Basel, CH-4056 Basel, Switzerland"},{"name":"D\u00e9partement de Physique Appliqu\u00e9e, Universit\u00e9 de Gen\u00e8ve, CH-1211 Gen\u00e8ve, Switzerland"}]},{"given":"Nicolas","family":"Sangouard","sequence":"additional","affiliation":[{"name":"Quantum Optics Theory Group, Universit\u00e4t Basel, CH-4056 Basel, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8455-020X","authenticated-orcid":false,"given":"Pavel","family":"Sekatski","sequence":"additional","affiliation":[{"name":"Quantum Optics Theory Group, Universit\u00e4t Basel, CH-4056 Basel, Switzerland"}]}],"member":"9598","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"D. 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Jogenfors, Breaking the Unbreakable: Exploiting Loopholes in Bell\u2019s Theorem to Hack Quantum Cryptography, Link\u00f6ping University Electronic Press (2017).","DOI":"10.3384\/diss.diva-140912"},{"key":"12","doi-asserted-by":"publisher","unstructured":"B. Hensen et. al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015).","DOI":"10.1038\/nature15759"},{"key":"13","doi-asserted-by":"publisher","unstructured":"L. K. Shalm et. al., Strong loophole-free test of local realism, Physical Review Letters 115(25), 250402 (2015).","DOI":"10.1103\/PhysRevLett.115.250402"},{"key":"14","doi-asserted-by":"publisher","unstructured":"M. Giustina et. al., Significant-loophole-free test of Bell\u2019s theorem with entangled photons, Physical Review Letters 115(25), 250401 (2015).","DOI":"10.1103\/PhysRevLett.115.250401"},{"key":"15","doi-asserted-by":"publisher","unstructured":"W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. 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