{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T17:35:11Z","timestamp":1780594511677,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,18]],"date-time":"2021-01-18T00:00:00Z","timestamp":1610928000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["6163202"],"award-info":[{"award-number":["6163202"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>According to the relevant theories on duality relation, the summation of the extractable information of a quanton\u2019s wave and particle properties, which are characterized by interference visibility V and path distinguishability D, respectively, is limited. However, this relation is violated upon quantum superposition between the wave-state and particle-state of the quanton, which is caused by the quantum beamsplitter (QBS). Along another line, recent studies have considered quantum coherence C in the l1-norm measure as a candidate for the wave property. In this study, we propose an interferometer with a quantum which-path detector (QWPD) and examine the generalized duality relation based on C. We find that this relationship still holds under such a circumstance, but the interference between these two properties causes the full-particle property to be observed when the QWPD system is partially present. Using a pair of polarization-entangled photons, we experimentally verify our analysis in the two-path case. This study extends the duality relation between coherence and path information to the quantum case and reveals the effect of quantum superposition on the duality relation.<\/jats:p>","DOI":"10.3390\/e23010122","type":"journal-article","created":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T04:55:31Z","timestamp":1611032131000},"page":"122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Wave-Particle Duality Relation with a Quantum Which-Path Detector"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5005-7695","authenticated-orcid":false,"given":"Dongyang","family":"Wang","sequence":"first","affiliation":[{"name":"Institute for Quantum Information &amp; State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5951-8988","authenticated-orcid":false,"given":"Junjie","family":"Wu","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information &amp; State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiangfang","family":"Ding","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information &amp; State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yingwen","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information &amp; State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anqi","family":"Huang","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information &amp; State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuejun","family":"Yang","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information &amp; State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BF01504968","article-title":"Das quantenpostulat und die neuere entwicklung der atomistik","volume":"16","author":"Bohr","year":"1928","journal-title":"Naturwissenschaften"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1038\/121580a0","article-title":"The quantum postulate and the recent development of atomic theory","volume":"121","author":"Bohr","year":"1928","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.1103\/PhysRevLett.77.2154","article-title":"Fringe visibility and which-way information: An inequality","volume":"77","author":"Englert","year":"1996","journal-title":"Phys. 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