{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:53:38Z","timestamp":1760147618239,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Division of Thermal Engineering &amp; Instrumentation (IDeTIC). Universidad de Las Palmas de Gran Canaria"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, the nexus between the Bell-state measurement and extracting phase information from the zeropoint field is investigated. For this purpose, the Wigner representation in the Heisenberg picture is applied in a Bell-type experiment in which the polarisation-entangled photon pairs generated in a type-II parametric down-conversion do not overlap. The signal intensities at the detectors are calculated in a four-mode approximation, being expressed as functions of the modules and phases of the four zeropoint amplitudes entering the crystal. A general criterion for identifying the correlated detectors is proposed based on the equality of the signal intensities, and without involving the calculation of the joint detection probabilities. In addition, from the analyses in the rectilinear and diagonal basis, it is shown that the distinguishability of the polarisation Bell states, which is in direct correspondence with the joint detection events in each experiment, can be related to the knowledge of the phases of the vacuum field entering the entanglement source, and giving rise to correlated detections. To this purpose, it is conjectured that a detection event is associated with a maximum value of the signal intensity averaged in the modules of the zeropoint amplitudes, as a function of the vacuum phases.<\/jats:p>","DOI":"10.3390\/e25030393","type":"journal-article","created":{"date-parts":[[2023,2,22]],"date-time":"2023-02-22T02:57:38Z","timestamp":1677034658000},"page":"393","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Partial Bell-State Measurement with Type-II Parametric Down Conversion: Extracting Phase Information from the Zeropoint Field (I)"],"prefix":"10.3390","volume":"25","author":[{"given":"Alberto","family":"Casado","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica Aplicada III, Escuela T\u00e9cnica Superior de Ingenier\u00eda, Universidad de Sevilla, 41092 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Santiago","family":"Guerra","sequence":"additional","affiliation":[{"name":"Thermal Engineering & Instrumentation Division (IDeTIC), Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Casado, A., Guerra, S., and Pl\u00e1cido, J. (2019). From stochastic optics to the Wigner formalism: The role of the vacuum field in optical quantum communication experiments. Atoms, 7.","DOI":"10.3390\/atoms7030076"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1364\/JOSAB.15.001572","article-title":"Type-II parametric downconversion in the Wigner function formalism: Entanglement and Bell\u2019s inequalities","volume":"15","author":"Casado","year":"1998","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"045501","DOI":"10.1088\/0953-4075\/41\/4\/045501","article-title":"Wigner representation for experiments on quantum cryptography using two-photon polarization entanglement produced in parametric down-conversion","volume":"41","author":"Casado","year":"2008","journal-title":"J. Phys. B At. Mol. Opt.Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1140\/epjd\/e2014-50368-y","article-title":"Wigner representation for polarization-momentum hyperentanglement generated in parametric down-conversion, and its application to complete Bell-state measurement","volume":"68","author":"Casado","year":"2014","journal-title":"Eur. Phys. J. D"},{"key":"ref_5","first-page":"115","article-title":"Innsbruck teleportation experiment in the Wigner formalism: A realistic description based on the role of the zero-point field","volume":"50","author":"Casado","year":"2020","journal-title":"Front.-Phys.-Front. eBook"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4337","DOI":"10.1103\/PhysRevLett.75.4337","article-title":"New high-intensity source of polarization entangled photon pairs","volume":"75","author":"Kwiat","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3893","DOI":"10.1103\/PhysRevLett.71.3893","article-title":"Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by type-II parametric down conversion","volume":"71","author":"Kiess","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1923","DOI":"10.1103\/PhysRevLett.73.1923","article-title":"Violation of Bell\u2019s inequality over 4 km of optical fiber","volume":"71","author":"Tapster","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5039","DOI":"10.1103\/PhysRevLett.81.5039","article-title":"Violation of Bell\u2019s inequality under strict Einstein locality conditions","volume":"81","author":"Weihs","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"130406","DOI":"10.1103\/PhysRevLett.111.130406","article-title":"Detection-loophole-free test of quantum nonlocality, and applications","volume":"111","author":"Christensen","year":"2013","journal-title":"Phys Rev Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"25041","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. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"250402","DOI":"10.1103\/PhysRevLett.115.250402","article-title":"Strong loophole-free test of local realism","volume":"115","author":"Shalm","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1038\/scientificamerican1092-50","article-title":"Quantum cryptography","volume":"267","author":"Bennett","year":"1992","journal-title":"Sci. Am."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2881","DOI":"10.1103\/PhysRevLett.69.2881","article-title":"Communication via one and two particle operators on Einstein-Podolsky-Rosen states","volume":"69","author":"Bennett","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1895","DOI":"10.1103\/PhysRevLett.70.1895","article-title":"Teleporting an unknown quantum state via dual classical and EPR channels","volume":"70","author":"Bennett","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2000132","DOI":"10.1002\/qute.202000132","article-title":"Spontaneous parametric down-conversion sources for multiphoton experiments","volume":"4","author":"Zhang","year":"2021","journal-title":"Adv. Quantum Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"064048","DOI":"10.1103\/PhysRevApplied.16.064048","article-title":"Producing multiple qubits via spontaneous parametric down-conversion","volume":"16","author":"Heitert","year":"2021","journal-title":"Phys. Rev. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1364\/OPTICA.451115","article-title":"Inverse design of spontaneous parametric down-conversion for generation of high-dimensional qudits","volume":"9","author":"Rozenberg","year":"2022","journal-title":"Optica"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"R773(R)","DOI":"10.1103\/PhysRevA.60.R773","article-title":"Ultrabright source of polarization-entangled photon","volume":"60","author":"Kwiat","year":"1999","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"022340","DOI":"10.1103\/PhysRevA.84.022340","article-title":"Distinguishability of hyperentangled Bell states by linear evolution and local projective measurement","volume":"84","author":"Pisenti","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4656","DOI":"10.1103\/PhysRevLett.76.4656","article-title":"Dense Coding in Experimental Quantum Communication","volume":"76","author":"Mattle","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"R2623","DOI":"10.1103\/PhysRevA.58.R2623","article-title":"Embedded Bell-state analysis","volume":"58","author":"Kwiat","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"060305(R)","DOI":"10.1103\/PhysRevA.75.060305","article-title":"Hyperentangled Bell-state analysis","volume":"75","author":"Wei","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"042313","DOI":"10.1103\/PhysRevA.68.042313","article-title":"Hyperentanglement-assisted Bell-state analysis","volume":"68","author":"Walborn","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"409","DOI":"10.3390\/foundations2020028","article-title":"Stochastic interpretation of Quantum Mechanics assuming that vacuum fields are real","volume":"2","author":"Santos","year":"2022","journal-title":"Foundations"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"191","DOI":"10.3389\/fphy.2020.00191","article-title":"Local realistic interpretation of entangled photon pairs in the Weyl-Wigner formalism","volume":"8","author":"Santos","year":"2020","journal-title":"Front. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"240402","DOI":"10.1103\/PhysRevLett.111.240402","article-title":"Asking photons where they have been","volume":"111","author":"Danan","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Vaidman, L., and Tsutsui, I. (2018). When photons are lying about where they have been. Entropy, 20.","DOI":"10.3390\/e20070538"},{"key":"ref_29","unstructured":"Reznik, G., Versmold, C., Dziewior, J., Huber, F., Bagchi, S., Weinfurter, H., Dressel, J., and Vaidman, L. (2022). Photons are lying about where they have been, again. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1140\/epjp\/s13360-023-03677-z","article-title":"Photons can tell \u201ccontradictory\u201d answer about where they have been","volume":"138","author":"Yuan","year":"2023","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_31","first-page":"79","article-title":"Polarization correlation of entangled photons derived without using non-local interactions","volume":"8","author":"Jung","year":"2020","journal-title":"Front.-Phys.-Front. eBook"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2022). Contextuality, complementarity, signaling, and Bell tests. Entropy, 24.","DOI":"10.3390\/e24101380"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"045007","DOI":"10.1103\/RevModPhys.94.045007","article-title":"Kochen-Specker contextuality","volume":"94","author":"Budroni","year":"2022","journal-title":"Rev. Mod. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"De Barros, J.A., and Holik, F. (2020). Indistinguishability and negative probabilities. Entropy, 22.","DOI":"10.3390\/e22080829"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"De Barros, J.A., Holik, F., and Krause, D. (2017). Contextuality and indistinguishability. Entropy, 19.","DOI":"10.3390\/e19090435"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"360","DOI":"10.3389\/fphy.2020.00360","article-title":"Locality is dead! Long live locality!","volume":"8","author":"Sulis","year":"2020","journal-title":"Front. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/393\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:38:11Z","timestamp":1760121491000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/393"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,21]]},"references-count":36,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["e25030393"],"URL":"https:\/\/doi.org\/10.3390\/e25030393","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,2,21]]}}}