{"id":"https://openalex.org/W2076940986","doi":"https://doi.org/10.3390/sym2031461","title":"SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms","display_name":"SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms","publication_year":2010,"publication_date":"2010-07-12","ids":{"openalex":"https://openalex.org/W2076940986","doi":"https://doi.org/10.3390/sym2031461","mag":"2076940986"},"language":"en","primary_location":{"id":"doi:10.3390/sym2031461","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym2031461","pdf_url":"https://www.mdpi.com/2073-8994/2/3/1461/pdf?version=1278926309","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/2/3/1461/pdf?version=1278926309","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Natig M. Atakishiyev","orcid":null},"institutions":[{"id":"https://openalex.org/I8961855","display_name":"Universidad Nacional Aut\u00f3noma de M\u00e9xico","ror":"https://ror.org/01tmp8f25","country_code":"MX","type":"education","lineage":["https://openalex.org/I8961855"]}],"countries":["MX"],"is_corresponding":false,"raw_author_name":"Natig M. Atakishiyev","raw_affiliation_strings":["Instituto de Matem\u00e1ticas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Instituto de Matem\u00e1ticas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico","institution_ids":["https://openalex.org/I8961855"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Maurice R. Kibler","orcid":null},"institutions":[{"id":"https://openalex.org/I100532134","display_name":"Universit\u00e9 Claude Bernard Lyon 1","ror":"https://ror.org/029brtt94","country_code":"FR","type":"education","lineage":["https://openalex.org/I100532134","https://openalex.org/I203339264"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210098836","display_name":"Institut de Physique","ror":"https://ror.org/00z54nq84","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210098836"]},{"id":"https://openalex.org/I4210133362","display_name":"Institut National de Physique Nucl\u00e9aire et de Physique des Particules","ror":"https://ror.org/03fd77x13","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210133362"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Maurice R. Kibler","raw_affiliation_strings":["Institut de Physique Nucl\u00e9aire, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France","Universit\u00e9 Claude Bernard and CNRS/IN2P3, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France","Universit\u00e9 de Lyon, 37 rue du repos, 69361 Lyon, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut de Physique Nucl\u00e9aire, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France","institution_ids":["https://openalex.org/I4210098836"]},{"raw_affiliation_string":"Universit\u00e9 Claude Bernard and CNRS/IN2P3, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne, France","institution_ids":["https://openalex.org/I100532134","https://openalex.org/I1294671590","https://openalex.org/I4210133362"]},{"raw_affiliation_string":"Universit\u00e9 de Lyon, 37 rue du repos, 69361 Lyon, France","institution_ids":["https://openalex.org/I100532134"]}]},{"author_position":"last","author":{"id":null,"display_name":"Kurt Bernardo Wolf","orcid":null},"institutions":[{"id":"https://openalex.org/I884181516","display_name":"Universidad Aut\u00f3noma del Estado de Morelos","ror":"https://ror.org/03rzb4f20","country_code":"MX","type":"education","lineage":["https://openalex.org/I884181516"]},{"id":"https://openalex.org/I8961855","display_name":"Universidad Nacional Aut\u00f3noma de M\u00e9xico","ror":"https://ror.org/01tmp8f25","country_code":"MX","type":"education","lineage":["https://openalex.org/I8961855"]}],"countries":["MX"],"is_corresponding":false,"raw_author_name":"Kurt Bernardo Wolf","raw_affiliation_strings":["Instituto de Ciencias F\u00edsicas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias F\u00edsicas, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico","institution_ids":["https://openalex.org/I884181516","https://openalex.org/I8961855"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":1.1565,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.7606914,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"2","issue":"3","first_page":"1461","last_page":"1484"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.5907999873161316,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.5907999873161316,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.10159999877214432,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.038600001484155655,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hadamard-transform","display_name":"Hadamard transform","score":0.677299976348877},{"id":"https://openalex.org/keywords/mutually-unbiased-bases","display_name":"Mutually unbiased bases","score":0.6395000219345093},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6208999752998352},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.5956000089645386},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.5852000117301941},{"id":"https://openalex.org/keywords/discrete-fourier-transform","display_name":"Discrete Fourier transform (general)","score":0.5056999921798706},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.49779999256134033},{"id":"https://openalex.org/keywords/harmonic-oscillator","display_name":"Harmonic oscillator","score":0.46639999747276306},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.43630000948905945}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6926000118255615},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.677299976348877},{"id":"https://openalex.org/C158587675","wikidata":"https://www.wikidata.org/wiki/Q17160413","display_name":"Mutually unbiased bases","level":3,"score":0.6395000219345093},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6208999752998352},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5956000089645386},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.5852000117301941},{"id":"https://openalex.org/C57733114","wikidata":"https://www.wikidata.org/wiki/Q1006032","display_name":"Discrete Fourier transform (general)","level":5,"score":0.5056999921798706},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.49779999256134033},{"id":"https://openalex.org/C40934718","wikidata":"https://www.wikidata.org/wiki/Q190070","display_name":"Harmonic oscillator","level":2,"score":0.46639999747276306},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.43630000948905945},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41359999775886536},{"id":"https://openalex.org/C122332657","wikidata":"https://www.wikidata.org/wiki/Q677864","display_name":"Quantum harmonic oscillator","level":3,"score":0.36399999260902405},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.33730000257492065},{"id":"https://openalex.org/C161794534","wikidata":"https://www.wikidata.org/wiki/Q177493","display_name":"Gauss","level":2,"score":0.3294000029563904},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.3118000030517578},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.29179999232292175},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.28780001401901245},{"id":"https://openalex.org/C131770355","wikidata":"https://www.wikidata.org/wiki/Q876215","display_name":"Harmonic analysis","level":2,"score":0.287200003862381},{"id":"https://openalex.org/C186348414","wikidata":"https://www.wikidata.org/wiki/Q526221","display_name":"Gauss sum","level":2,"score":0.2824000120162964},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27320000529289246},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.27149999141693115},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.26600000262260437},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2572999894618988},{"id":"https://openalex.org/C76563020","wikidata":"https://www.wikidata.org/wiki/Q4817582","display_name":"Fractional Fourier transform","level":4,"score":0.25049999356269836},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2500999867916107}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym2031461","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym2031461","pdf_url":"https://www.mdpi.com/2073-8994/2/3/1461/pdf?version=1278926309","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1008.2881","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1008.2881","pdf_url":"https://arxiv.org/pdf/1008.2881","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:in2p3-00510015v1","is_oa":false,"landing_page_url":"https://in2p3.hal.science/in2p3-00510015","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, 2010, 2, pp.1461. &#x27E8;10.3390/sym2031461&#x27E9;","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:42157c84cb874cae9df4a11505a6ca5b","is_oa":true,"landing_page_url":"https://doaj.org/article/42157c84cb874cae9df4a11505a6ca5b","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 2, Iss 3, Pp 1461-1484 (2010)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym2031461","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym2031461","pdf_url":"https://www.mdpi.com/2073-8994/2/3/1461/pdf?version=1278926309","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2274830339","display_name":null,"funder_award_id":"SEP-CONACYT 79899","funder_id":"https://openalex.org/F4320321739","funder_display_name":"Consejo Nacional de Ciencia y Tecnolog\u00eda"}],"funders":[{"id":"https://openalex.org/F4320321739","display_name":"Consejo Nacional de Ciencia y Tecnolog\u00eda","ror":"https://ror.org/059ex5q34"},{"id":"https://openalex.org/F4320335100","display_name":"Direcci\u00f3n General de Asuntos del Personal Acad\u00e9mico, Universidad Nacional Aut\u00f3noma de M\u00e9xico","ror":"https://ror.org/01tmp8f25"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2076940986.pdf","grobid_xml":"https://content.openalex.org/works/W2076940986.grobid-xml"},"referenced_works_count":52,"referenced_works":["https://openalex.org/W1568450818","https://openalex.org/W1610274090","https://openalex.org/W1626617634","https://openalex.org/W1633483962","https://openalex.org/W1756227501","https://openalex.org/W1964691445","https://openalex.org/W1969304762","https://openalex.org/W1972171154","https://openalex.org/W1976693877","https://openalex.org/W1980296761","https://openalex.org/W1989697549","https://openalex.org/W1998169277","https://openalex.org/W2004996496","https://openalex.org/W2005758157","https://openalex.org/W2009377979","https://openalex.org/W2011425158","https://openalex.org/W2012101105","https://openalex.org/W2013175922","https://openalex.org/W2019388666","https://openalex.org/W2020406423","https://openalex.org/W2026092225","https://openalex.org/W2043438748","https://openalex.org/W2044540950","https://openalex.org/W2045490825","https://openalex.org/W2046938896","https://openalex.org/W2047425127","https://openalex.org/W2051087348","https://openalex.org/W2058124214","https://openalex.org/W2064549772","https://openalex.org/W2068601954","https://openalex.org/W2070976244","https://openalex.org/W2078471442","https://openalex.org/W2082685661","https://openalex.org/W2082899434","https://openalex.org/W2087407249","https://openalex.org/W2088572661","https://openalex.org/W2096438497","https://openalex.org/W2102435870","https://openalex.org/W2104277451","https://openalex.org/W2106340509","https://openalex.org/W2116983948","https://openalex.org/W2118323582","https://openalex.org/W2131402293","https://openalex.org/W2131730048","https://openalex.org/W2143572487","https://openalex.org/W2145576488","https://openalex.org/W2224594407","https://openalex.org/W2315209415","https://openalex.org/W3159989289","https://openalex.org/W4206509365","https://openalex.org/W6678405010","https://openalex.org/W6764359603"],"related_works":[],"abstract_inverted_index":{"We":[0,60],"propose":[1],"a":[2,102],"group-theoretical":[3,76],"approach":[4],"to":[5,28],"the":[6,29,35,39,43,51,57,62,66],"generalized":[7,110,115],"oscillator":[8,37],"algebra":[9],"A\u03ba":[10,73],"recently":[11],"investigated":[12],"in":[13,74,91,107],"J.":[14],"Phys.":[15],"A:":[16],"Math.":[17],"Theor.":[18],"2010,":[19],"43,":[20],"115303.":[21],"The":[22,78],"case":[23,44,80],"\u03ba":[24,45],"\u2265":[25],"0":[26,47],"corresponds":[27],"noncompact":[30],"group":[31,53],"SU(1,1)":[32],"(as":[33,55],"for":[34,56,72,83],"harmonic":[36],"and":[38,65,114],"P\u00f6schl-Teller":[40],"systems)":[41],"while":[42],"&lt;":[46],"is":[48,81],"described":[49],"by":[50],"compact":[52],"SU(2)":[54,79],"Morse":[58],"system).":[59],"construct":[61],"phase":[63,70],"operators":[64],"corresponding":[67],"temporally":[68],"stable":[69],"eigenstates":[71],"this":[75,95],"context.":[77],"exploited":[82],"deriving":[84],"families":[85],"of":[86,101],"mutually":[87],"unbiased":[88],"bases":[89],"used":[90],"quantum":[92],"information.":[93],"Along":[94],"vein,":[96],"we":[97],"examine":[98],"some":[99],"characteristics":[100],"quadratic":[103,111],"discrete":[104],"Fourier":[105],"transform":[106],"connection":[108],"with":[109],"Gauss":[112],"sums":[113],"Hadamard":[116],"matrices.":[117]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2016-06-24T00:00:00"}
