{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T05:04:39Z","timestamp":1777439079374,"version":"3.51.4"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T00:00:00Z","timestamp":1630454400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,9,21]],"date-time":"2021-09-21T00:00:00Z","timestamp":1632182400000},"content-version":"vor","delay-in-days":20,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100008431","name":"Consejer\u00eda de Educaci\u00f3n, Junta de Castilla y Le\u00f3n","doi-asserted-by":"publisher","award":["VA296P18"],"award-info":[{"award-number":["VA296P18"]}],"id":[{"id":"10.13039\/501100008431","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004834","name":"Universitat Jaume I","doi-asserted-by":"publisher","award":["UJI-B2018-3"],"award-info":[{"award-number":["UJI-B2018-3"]}],"id":[{"id":"10.13039\/501100004834","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004834","name":"Universitat Jaume I","doi-asserted-by":"publisher","award":["UJI-B2018-10"],"award-info":[{"award-number":["UJI-B2018-10"]}],"id":[{"id":"10.13039\/501100004834","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["MTM2017-84851-C2-2"],"award-info":[{"award-number":["MTM2017-84851-C2-2"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-096446-B-C22"],"award-info":[{"award-number":["PGC2018-096446-B-C22"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"ministerio de ciencia e innovaci\u00f3n","doi-asserted-by":"crossref","award":["PID2020-119418GB-I00"],"award-info":[{"award-number":["PID2020-119418GB-I00"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004837","name":"ministerio de ciencia e innovaci\u00f3n","doi-asserted-by":"crossref","award":["PID2020-115930GA-I00"],"award-info":[{"award-number":["PID2020-115930GA-I00"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Linear optical systems acting on photon number states produce many interesting evolutions, but cannot give all the allowed quantum operations on the input state. Using Toponogov\u2019s theorem from differential geometry, we propose an iterative method that, for any arbitrary quantum operator<jats:italic>U<\/jats:italic>acting on<jats:italic>n<\/jats:italic>photons in<jats:italic>m<\/jats:italic>modes, returns an operator<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\widetilde{U}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mover><mml:mi>U<\/mml:mi><mml:mo>~<\/mml:mo><\/mml:mover><\/mml:math><\/jats:alternatives><\/jats:inline-formula>which can be implemented with linear optics. The approximation method is locally optimal and converges. The resulting operator<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\widetilde{U}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mover><mml:mi>U<\/mml:mi><mml:mo>~<\/mml:mo><\/mml:mover><\/mml:math><\/jats:alternatives><\/jats:inline-formula>can be translated into an experimental optical setup using previous results.<\/jats:p>","DOI":"10.1007\/s11128-021-03254-2","type":"journal-article","created":{"date-parts":[[2021,9,21]],"date-time":"2021-09-21T08:03:27Z","timestamp":1632211407000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal approximation to unitary quantum operators with linear optics"],"prefix":"10.1007","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9813-5004","authenticated-orcid":false,"given":"Juan Carlos","family":"Garcia-Escartin","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7982-1677","authenticated-orcid":false,"given":"Vicent","family":"Gimeno","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9128-4488","authenticated-orcid":false,"given":"Julio Jos\u00e9","family":"Moyano-Fern\u00e1ndez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,9,21]]},"reference":[{"key":"3254_CR1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198501770.001.0001","volume-title":"The Quantum Theory of Light","author":"R Loudon","year":"2000","unstructured":"Loudon, R.: The Quantum Theory of Light, 3rd edn. Oxford University Press, Great Clarendon Street, Oxford, UK (2000)","edition":"3"},{"key":"3254_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511791239","volume-title":"Introductory quantum optics","author":"C Gerry","year":"2004","unstructured":"Gerry, C., Knight, P.: Introductory quantum optics. Cambridge University Press, Cambridge (2004)"},{"key":"3254_CR3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511806117","volume-title":"Essential quantum optics: from quantum measurements to black holes","author":"U Leonhardt","year":"2010","unstructured":"Leonhardt, U.: Essential quantum optics: from quantum measurements to black holes. Cambridge University Press, Cambridge (2010)"},{"issue":"1","key":"3254_CR4","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1103\/PhysRevLett.73.58","volume":"73","author":"M Reck","year":"1994","unstructured":"Reck, M., Zeilinger, A., Bernstein, H.J., Bertani, P.: Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73(1), 58 (1994)","journal-title":"Phys. Rev. Lett."},{"key":"3254_CR5","doi-asserted-by":"publisher","first-page":"062316","DOI":"10.1103\/PhysRevA.89.062316","volume":"89","author":"A Bouland","year":"2014","unstructured":"Bouland, A., Aaronson, S.: Generation of universal linear optics by any beam splitter. Phys. Rev. A 89, 062316 (2014)","journal-title":"Phys. Rev. A"},{"issue":"3 & 4","key":"3254_CR6","first-page":"0291","volume":"16","author":"A Sawicki","year":"2016","unstructured":"Sawicki, A.: Universality of beamsplitters. Quantum Inf. Comput. 16(3 & 4), 0291\u20130312 (2016)","journal-title":"Quantum Inf. Comput."},{"issue":"12","key":"3254_CR7","doi-asserted-by":"publisher","first-page":"1460","DOI":"10.1364\/OPTICA.3.001460","volume":"3","author":"WR Clements","year":"2016","unstructured":"Clements, W.R., Humphreys, P.C., Metcalf, B.J., Kolthammer, W.S., Walmsley, I.A.: Optimal design for universal multiport interferometers. Optica 3(12), 1460\u20131465 (2016)","journal-title":"Optica"},{"key":"3254_CR8","doi-asserted-by":"publisher","first-page":"022328","DOI":"10.1103\/PhysRevA.97.022328","volume":"97","author":"H de Guise","year":"2018","unstructured":"de Guise, H., Di Matteo, O., S\u00e1nchez-Soto, L.L.: Simple factorization of unitary transformations. Phys. Rev. A 97, 022328 (2018)","journal-title":"Phys. Rev. A"},{"key":"3254_CR9","doi-asserted-by":"crossref","unstructured":"Aaronson,S., Arkhipov,A.: The computational complexity of linear optics. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, STOC \u201911, pp. 333\u2013342, New York, NY, USA, (2011). ACM","DOI":"10.1145\/1993636.1993682"},{"issue":"12","key":"3254_CR10","doi-asserted-by":"publisher","first-page":"1634","DOI":"10.1007\/BF02781659","volume":"10","author":"ER Caianiello","year":"1953","unstructured":"Caianiello, E.R.: On quantum field theory \u2013 I: Explicit solution of Dysons equation in electrodynamics without use of Feynman graphs. Il Nuovo Cimento (1943-1954) 10(12), 1634\u20131652 (1953)","journal-title":"Il Nuovo Cimento (1943-1954)"},{"key":"3254_CR11","unstructured":"Scheel,S.: Permanents in linear optical networks. quant-ph\/0406127, (2004)"},{"issue":"11","key":"3254_CR12","doi-asserted-by":"publisher","first-page":"1385","DOI":"10.1119\/1.1775241","volume":"72","author":"J Skaar","year":"2004","unstructured":"Skaar, J., Garc\u00eda Escart\u00edn, J.C., Landro, H.: Quantum mechanical description of linear optics. Am. J. Phys. 72(11), 1385\u20131391 (2004)","journal-title":"Am. J. Phys."},{"issue":"1","key":"3254_CR13","doi-asserted-by":"publisher","first-page":"L1","DOI":"10.1088\/1464-4266\/6\/1\/L01","volume":"6","author":"U Leonhardt","year":"2004","unstructured":"Leonhardt, U., Neumaier, A.: Explicit effective hamiltonians for general linear quantum-optical networks. J. Opt. B: Quantum Semiclass. Opt. 6(1), L1 (2004)","journal-title":"J. Opt. B: Quantum Semiclass. Opt."},{"issue":"1","key":"3254_CR14","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1088\/1355-5111\/9\/1\/004","volume":"9","author":"HY Fan","year":"1997","unstructured":"Fan, H.Y., Xiao, M.: Construction of optical networks by virture of the IWOP technique. Quantum Semiclass. Opt.: J. Eur. Opt. Soc. Part B 9(1), 53\u201358 (1997)","journal-title":"Quantum Semiclass. Opt.: J. Eur. Opt. Soc. Part B"},{"issue":"12","key":"3254_CR15","doi-asserted-by":"publisher","first-page":"S711","DOI":"10.1088\/1464-4266\/7\/12\/038","volume":"7","author":"P Aniello","year":"2005","unstructured":"Aniello, P., Cagli, R.C.: An algebraic approach to linear-optical schemes for deterministic quantum computing. J. Opt. B: Quantum Semiclass. Opt. 7(12), S711\u2013S720 (2005)","journal-title":"J. Opt. B: Quantum Semiclass. Opt."},{"key":"3254_CR16","unstructured":"Biedenharn ,L.C., Louck,J.D.: Angular momentum in quantum physics. Theory and application, volume\u00a08 of Encycl. Math. Appl. Addison-Wesley, (1981)"},{"issue":"4","key":"3254_CR17","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/s11080-006-9023-1","volume":"13","author":"P Aniello","year":"2006","unstructured":"Aniello, P., Lupo, C., Napolitano, M.: Exploring representation theory of unitary groups via linear optical passive devices. Open Syst. Inf. Dyn. 13(4), 415\u2013426 (2006)","journal-title":"Open Syst. Inf. Dyn."},{"key":"3254_CR18","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1016\/j.optcom.2018.08.082","volume":"430","author":"JC Garcia-Escartin","year":"2019","unstructured":"Garcia-Escartin, J.C., Gimeno, V., Moyano-Fern\u00e1ndez, J.J.: Multiple photon effective Hamiltonians in linear quantum optical networks. Opt. Commun. 430, 434\u2013439 (2019)","journal-title":"Opt. Commun."},{"key":"3254_CR19","doi-asserted-by":"publisher","first-page":"022301","DOI":"10.1103\/PhysRevA.100.022301","volume":"100","author":"JC Garcia-Escartin","year":"2019","unstructured":"Garcia-Escartin, J.C., Gimeno, V., Moyano-Fern\u00e1ndez, J.J.: Method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe. Phys. Rev. A 100, 022301 (2019)","journal-title":"Phys. Rev. A"},{"key":"3254_CR20","doi-asserted-by":"crossref","unstructured":"Cheeger ,J., Ebin,D.\u00a0G.: Comparison theorems in Riemannian geometry, volume 365. American Mathematical Soc., (2008)","DOI":"10.1090\/chel\/365"},{"key":"3254_CR21","volume-title":"Microwave Engineering","author":"DM Pozar","year":"2004","unstructured":"Pozar, D.M.: Microwave Engineering, 4th edn. Wiley, New York (2004)","edition":"4"},{"key":"3254_CR22","first-page":"021017","volume":"8","author":"N Tischler","year":"2018","unstructured":"Tischler, N., Rockstuhl, C., S\u0142owik, K.: Quantum optical realization of arbitrary linear transformations allowing for loss and gain. Phys. Rev. X 8, 021017 (2018)","journal-title":"Phys. Rev. X"},{"key":"3254_CR23","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/j.optcom.2016.07.085","volume":"382","author":"JJ Moyano-Fern\u00e1ndez","year":"2017","unstructured":"Moyano-Fern\u00e1ndez, J.J., Garcia-Escartin, J.C.: Linear optics only allows every possible quantum operation for one photon or one port. Opt. Commun. 382, 237\u2013240 (2017)","journal-title":"Opt. Commun."},{"key":"3254_CR24","doi-asserted-by":"crossref","unstructured":"Carmo,M.P.\u00a0do.: Riemannian geometry. Birkh\u00e4user, (1992)","DOI":"10.1007\/978-1-4757-2201-7"},{"key":"3254_CR25","doi-asserted-by":"crossref","unstructured":"Sakai,T.: Riemannian geometry, volume 149. American Mathematical Society, (1996)","DOI":"10.1090\/mmono\/149"},{"issue":"3","key":"3254_CR26","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/S0001-8708(76)80002-3","volume":"21","author":"J Milnor","year":"1976","unstructured":"Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293\u2013329 (1976)","journal-title":"Adv. Math."},{"key":"3254_CR27","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix analysis","author":"RA Horn","year":"1985","unstructured":"Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1985)"},{"issue":"6","key":"3254_CR28","doi-asserted-by":"publisher","first-page":"744","DOI":"10.1002\/nla.1927","volume":"21","author":"TA Loring","year":"2014","unstructured":"Loring, T.A.: Computing a logarithm of a unitary matrix with general spectrum. Numer. Linear Algebra Appl. 21(6), 744\u2013760 (2014)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"5","key":"3254_CR29","first-page":"592","volume":"54","author":"F Mezzadri","year":"2007","unstructured":"Mezzadri, F.: How to generate random matrices from the classical compact groups. Not. Am. Math. Soc. 54(5), 592\u2013604 (2007)","journal-title":"Not. Am. Math. Soc."},{"issue":"6","key":"3254_CR30","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1016\/j.cpc.2008.03.007","volume":"179","author":"G T\u00f3th","year":"2008","unstructured":"T\u00f3th, G.: Qubit4Matlab v3.0: A program package for quantum information science and quantum optics for Matlab. Comput. Phys. Commun. 179(6), 430\u2013437 (2008)","journal-title":"Comput. Phys. Commun."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03254-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-021-03254-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03254-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,9]],"date-time":"2023-11-09T03:37:57Z","timestamp":1699501077000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-021-03254-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9]]},"references-count":30,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["3254"],"URL":"https:\/\/doi.org\/10.1007\/s11128-021-03254-2","relation":{},"ISSN":["1570-0755","1573-1332"],"issn-type":[{"value":"1570-0755","type":"print"},{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9]]},"assertion":[{"value":"6 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 September 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"314"}}