{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T14:02:57Z","timestamp":1768744977562,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T00:00:00Z","timestamp":1578009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CRC 1119 CROSSING"],"award-info":[{"award-number":["CRC 1119 CROSSING"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>Loophole-free violations of Bell inequalities are crucial for fundamental tests of quantum nonlocality. They are also important for future applications in quantum information processing, such as device-independent quantum key distribution. Based on a detector model which includes detector inefficiencies and dark counts, we estimate the minimal requirements on detectors needed for performing loophole-free bipartite and tripartite Bell tests. Our numerical investigation is based on a hierarchy of semidefinite programs for characterizing possible quantum correlations. We find that for bipartite setups with two measurement choices and our detector model, the optimal inequality for a Bell test is equivalent to the Clauser\u2013Horne inequality.<\/jats:p>","DOI":"10.3390\/cryptography4010002","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T11:55:07Z","timestamp":1578052507000},"page":"2","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quantum Bounds on Detector Efficiencies for Violating Bell Inequalities Using Semidefinite Programming"],"prefix":"10.3390","volume":"4","author":[{"given":"Alexander","family":"Sauer","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Angewandte Physik, Technische Universit\u00e4t Darmstadt, 64289 Darmstadt, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4676-0024","authenticated-orcid":false,"given":"Gernot","family":"Alber","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Physik, Technische Universit\u00e4t Darmstadt, 64289 Darmstadt, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195","article-title":"On the Einstein-Podolsky-Rosen paradox","volume":"1","author":"Bell","year":"1964","journal-title":"Physics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"250401","DOI":"10.1103\/PhysRevLett.115.250401","article-title":"Significant-loophole-free test of Bell\u2019s theorem with entangled photons","volume":"115","author":"Giustina","year":"2015","journal-title":"Phys. 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