{"id":"https://openalex.org/W2302010010","doi":"https://doi.org/10.1103/physreva.93.052118","title":"Products of weak values: Uncertainty relations, complementarity, and incompatibility","display_name":"Products of weak values: Uncertainty relations, complementarity, and incompatibility","publication_year":2016,"publication_date":"2016-05-27","ids":{"openalex":"https://openalex.org/W2302010010","doi":"https://doi.org/10.1103/physreva.93.052118","mag":"2302010010"},"language":"en","primary_location":{"id":"doi:10.1103/physreva.93.052118","doi":"https://doi.org/10.1103/physreva.93.052118","is_oa":false,"landing_page_url":"https://doi.org/10.1103/physreva.93.052118","pdf_url":null,"source":{"id":"https://openalex.org/S164566984","display_name":"Physical Review A","issn_l":"0556-2791","issn":["0556-2791","1050-2947","1094-1622","1538-4446","2469-9926","2469-9934","2469-9942"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-a","jufo-2","ki-jl-1","norway-2","poland-100","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320261","host_organization_name":"American Physical Society","host_organization_lineage":["https://openalex.org/P4310320261"],"host_organization_lineage_names":["American Physical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physical Review A","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1603.05776","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029817007","display_name":"Michael J. W. Hall","orcid":"https://orcid.org/0000-0001-7809-4463","observed_orcids":["https://orcid.org/0000-0001-7809-4463"]},"institutions":[{"id":"https://openalex.org/I11701301","display_name":"Griffith University","ror":"https://ror.org/02sc3r913","country_code":"AU","type":"education","lineage":["https://openalex.org/I11701301"]},{"id":"https://openalex.org/I4210096034","display_name":"Centre for Quantum Computation and Communication Technology","ror":"https://ror.org/00rnbty21","country_code":"AU","type":"facility","lineage":["https://openalex.org/I1337719021","https://openalex.org/I165143802","https://openalex.org/I2801453606","https://openalex.org/I4210096034","https://openalex.org/I4210132349"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Michael J. W. Hall","raw_affiliation_strings":["Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia","institution_ids":["https://openalex.org/I11701301","https://openalex.org/I4210096034"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067671236","display_name":"Arun Kumar Pati","orcid":"https://orcid.org/0000-0002-7761-9149","observed_orcids":["https://orcid.org/0000-0002-7761-9149"]},"institutions":[{"id":"https://openalex.org/I35409948","display_name":"Harish-Chandra Research Institute","ror":"https://ror.org/0165d9303","country_code":"IN","type":"education","lineage":["https://openalex.org/I35409948"]},{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN","IN"],"is_corresponding":false,"raw_author_name":"Arun Kumar Pati","raw_affiliation_strings":["Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China","Quantum Information and Computation Group, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China","institution_ids":["https://openalex.org/I76130692"]},{"raw_affiliation_string":"Quantum Information and Computation Group, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India","institution_ids":["https://openalex.org/I35409948"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042162337","display_name":"Junde Wu","orcid":"https://orcid.org/0000-0001-5334-8391","observed_orcids":["https://orcid.org/0000-0001-5334-8391"]},"institutions":[{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Junde Wu","raw_affiliation_strings":["Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China","institution_ids":["https://openalex.org/I76130692"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2675,"currency":"USD","value_usd":2675},"apc_paid":null,"fwci":1.7346,"has_fulltext":false,"cited_by_count":37,"citation_normalized_percentile":{"value":0.843421,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"93","issue":"5","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.9990000128746033,"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":9.999999747378752e-05,"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"}},{"id":"https://openalex.org/T11415","display_name":"Noncommutative and Quantum Gravity Theories","score":0.0,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/weak-values","display_name":"weak values","score":0.9860000014305115},{"id":"https://openalex.org/keywords/heisenberg-uncertainty-principle","display_name":"uncertainty principle","score":0.7620000243186951},{"id":"https://openalex.org/keywords/complementarity","display_name":"complementarity","score":0.7239999771118164},{"id":"https://openalex.org/keywords/strong-measurement","display_name":"strong measurement","score":0.6549999713897705},{"id":"https://openalex.org/keywords/weak-measurement","display_name":"weak measurement","score":0.7699999809265137},{"id":"https://openalex.org/keywords/joint-probability-distribution","display_name":"joint probability distribution","score":0.6769999861717224},{"id":"https://openalex.org/keywords/quantum-purity","display_name":"quantum purity","score":0.9200000166893005},{"id":"https://openalex.org/keywords/complex-random-variables","display_name":"complex random variables","score":0.6620000004768372}],"concepts":[{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.8029792308807373},{"id":"https://openalex.org/C202269582","wikidata":"https://www.wikidata.org/wiki/Q2644277","display_name":"Complementarity (molecular biology)","level":2,"score":0.7501505613327026},{"id":"https://openalex.org/C50150098","wikidata":"https://www.wikidata.org/wiki/Q2607127","display_name":"Weak measurement","level":3,"score":0.6797895431518555},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.573442816734314},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.5453188419342041},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5043045282363892},{"id":"https://openalex.org/C181543814","wikidata":"https://www.wikidata.org/wiki/Q44746","display_name":"Uncertainty principle","level":3,"score":0.4560345411300659},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.4219534993171692},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.29841989278793335},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.29675912857055664},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1103/physreva.93.052118","doi":"https://doi.org/10.1103/physreva.93.052118","is_oa":false,"landing_page_url":"https://doi.org/10.1103/physreva.93.052118","pdf_url":null,"source":{"id":"https://openalex.org/S164566984","display_name":"Physical Review A","issn_l":"0556-2791","issn":["0556-2791","1050-2947","1094-1622","1538-4446","2469-9926","2469-9934","2469-9942"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-a","jufo-2","ki-jl-1","norway-2","poland-100","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320261","host_organization_name":"American Physical Society","host_organization_lineage":["https://openalex.org/P4310320261"],"host_organization_lineage_names":["American Physical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physical Review A","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1603.05776","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1603.05776","pdf_url":"https://arxiv.org/pdf/1603.05776","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,"listed_in":[],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:research-repository.griffith.edu.au:10072/99566","doi":null,"is_oa":false,"landing_page_url":"http://hdl.handle.net/10072/99566","pdf_url":null,"source":{"id":"https://openalex.org/S7407064376","display_name":"Griffith Research Online","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Journal article"},{"id":"doi:10.48550/arxiv.1603.05776","doi":"https://doi.org/10.48550/arxiv.1603.05776","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1603.05776","pdf_url":null,"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,"listed_in":[],"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1603.05776","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1603.05776","pdf_url":"https://arxiv.org/pdf/1603.05776","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,"listed_in":[],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[{"score":0.4099999964237213,"display_name":"Decent work and economic growth","id":"https://metadata.un.org/sdg/8"}],"study_designs":[],"awards":[{"id":"https://openalex.org/G2533383055","display_name":null,"funder_award_id":"CE110001027","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"},{"id":"https://openalex.org/G3318366576","display_name":null,"funder_award_id":"11571307","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4923916341","display_name":"\u91cf\u5b50\u6d4b\u91cf\u4e0e\u91cf\u5b50\u64cd\u4f5c\u7406\u8bba\u7684\u6570\u5b66\u57fa\u7840\u7814\u7a76","funder_award_id":"11171301","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":53,"referenced_works":["https://openalex.org/W204835608","https://openalex.org/W211974650","https://openalex.org/W745867715","https://openalex.org/W1521345913","https://openalex.org/W1561384205","https://openalex.org/W1626294324","https://openalex.org/W1760416935","https://openalex.org/W1966046470","https://openalex.org/W1968356682","https://openalex.org/W1975902009","https://openalex.org/W1976065586","https://openalex.org/W1980351059","https://openalex.org/W1980845541","https://openalex.org/W1983755817","https://openalex.org/W1984721431","https://openalex.org/W1986894959","https://openalex.org/W1988426340","https://openalex.org/W2012016549","https://openalex.org/W2013949436","https://openalex.org/W2018484468","https://openalex.org/W2023803206","https://openalex.org/W2026958063","https://openalex.org/W2030959471","https://openalex.org/W2034151643","https://openalex.org/W2035340268","https://openalex.org/W2037377988","https://openalex.org/W2037400514","https://openalex.org/W2041441518","https://openalex.org/W2042601507","https://openalex.org/W2052788241","https://openalex.org/W2053734218","https://openalex.org/W2057260579","https://openalex.org/W2058173071","https://openalex.org/W2059302706","https://openalex.org/W2061647257","https://openalex.org/W2066923915","https://openalex.org/W2067130070","https://openalex.org/W2072381571","https://openalex.org/W2072957404","https://openalex.org/W2074786363","https://openalex.org/W2075717983","https://openalex.org/W2079840239","https://openalex.org/W2080936681","https://openalex.org/W2090099030","https://openalex.org/W2090513440","https://openalex.org/W2094706601","https://openalex.org/W2098918476","https://openalex.org/W2115966878","https://openalex.org/W2140127475","https://openalex.org/W2141018846","https://openalex.org/W2341708862","https://openalex.org/W2741582595","https://openalex.org/W3101911436"],"related_works":["https://openalex.org/W1507686554","https://openalex.org/W4394649671","https://openalex.org/W1939627376","https://openalex.org/W7143733985","https://openalex.org/W3047796680","https://openalex.org/W3081099707","https://openalex.org/W2790426296","https://openalex.org/W1996988239","https://openalex.org/W2887735426","https://openalex.org/W1565006199"],"abstract_inverted_index":{"The":[0],"products":[1,146],"of":[2,5,12,68,75,103,109,116,147,152],"weak":[3,23,69,93,119,126,136,153,163],"values":[4,70,168],"quantum":[6,16,76,128,131],"observables":[7,84],"are":[8,86,142],"shown":[9],"to":[10,39,57,161],"be":[11,40],"value":[13],"in":[14,91,106,169],"deriving":[15],"uncertainty":[17,37,45,59],"and":[18,24,64,88,130,133,137,149],"complementarity":[19,98],"relations,":[20],"for":[21,47,61],"both":[22],"strong":[25,58,138],"measurement":[26,94],"statistics.":[27],"First,":[28],"a":[29,43,92,97,170],"``product":[30],"representation":[31],"formula''":[32],"allows":[33],"the":[34,101,107,114,117,159],"standard":[35],"Heisenberg":[36],"relation":[38,46,99],"derived":[41],"from":[42],"classical":[44],"complex":[48],"random":[49],"variables.":[50],"We":[51],"show":[52,80],"this":[53],"formula":[54],"also":[55,134],"leads":[56],"relations":[60,124,157],"unitary":[62],"operators":[63],"underlies":[65],"an":[66,110],"interpretation":[67],"as":[71],"optimal":[72],"(complex)":[73],"estimates":[74],"observables.":[77],"Furthermore,":[78],"we":[79],"that":[81,85],"two":[82],"incompatible":[83],"weakly":[87],"strongly":[89],"measured":[90],"context":[95],"obey":[96],"under":[100],"interchange":[102],"these":[104,156],"observables,":[105],"form":[108],"upper":[111],"bound":[112],"on":[113,145],"product":[115],"corresponding":[118],"values.":[120],"Moreover,":[121],"general":[122],"tradeoff":[123],"between":[125,135],"purity,":[127,129],"incompatibility,":[132],"joint":[139],"probability":[140],"distributions,":[141],"obtained":[143],"based":[144],"real":[148],"imaginary":[150],"components":[151],"values,":[154],"where":[155],"quantify":[158],"degree":[160],"which":[162],"probabilities":[164],"can":[165],"take":[166],"anomalous":[167],"given":[171],"context.":[172]},"counts_by_year":[{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2016-06-24T00:00:00"}