{"id":"https://openalex.org/W2099745078","doi":"https://doi.org/10.1109/tit.2009.2021373","title":"Squashed Entanglement for Multipartite States and Entanglement Measures Based on the Mixed Convex Roof","display_name":"Squashed Entanglement for Multipartite States and Entanglement Measures Based on the Mixed Convex Roof","publication_year":2009,"publication_date":"2009-06-16","ids":{"openalex":"https://openalex.org/W2099745078","doi":"https://doi.org/10.1109/tit.2009.2021373","mag":"2099745078"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2009.2021373","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2009.2021373","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0704.2236","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Dong Yang","orcid":null},"institutions":[{"id":"https://openalex.org/I55538621","display_name":"China Jiliang University","ror":"https://ror.org/05v1y0t93","country_code":"CN","type":"education","lineage":["https://openalex.org/I55538621"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dong Yang","raw_affiliation_strings":["Laboratory for Quantum Information, China Jiliang University, Hangzhou, Zhejiang, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory for Quantum Information, China Jiliang University, Hangzhou, Zhejiang, China","institution_ids":["https://openalex.org/I55538621"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Karol Horodecki","orcid":null},"institutions":[{"id":"https://openalex.org/I40413290","display_name":"University of Gda\u0144sk","ror":"https://ror.org/011dv8m48","country_code":"PL","type":"education","lineage":["https://openalex.org/I40413290"]},{"id":"https://openalex.org/I47996466","display_name":"University of Warmia and Mazury in Olsztyn","ror":"https://ror.org/05s4feg49","country_code":"PL","type":"education","lineage":["https://openalex.org/I47996466"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Karol Horodecki","raw_affiliation_strings":["Faculty of Mathematics, Physics and Computer Science, University of Gda\u0144sk, GdaDsk, Poland","Institute of Informatics, University of Warmia and Mazury, Warsaw, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, Physics and Computer Science, University of Gda\u0144sk, GdaDsk, Poland","institution_ids":["https://openalex.org/I40413290"]},{"raw_affiliation_string":"Institute of Informatics, University of Warmia and Mazury, Warsaw, Poland","institution_ids":["https://openalex.org/I47996466"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Michal Horodecki","orcid":null},"institutions":[{"id":"https://openalex.org/I40413290","display_name":"University of Gda\u0144sk","ror":"https://ror.org/011dv8m48","country_code":"PL","type":"education","lineage":["https://openalex.org/I40413290"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Michal Horodecki","raw_affiliation_strings":["Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, GdaDsk, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, GdaDsk, Poland","institution_ids":["https://openalex.org/I40413290"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Pawel Horodecki","orcid":null},"institutions":[{"id":"https://openalex.org/I40413290","display_name":"University of Gda\u0144sk","ror":"https://ror.org/011dv8m48","country_code":"PL","type":"education","lineage":["https://openalex.org/I40413290"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Pawel Horodecki","raw_affiliation_strings":["Faculty of Applied Physics and Mathematics, Technical University of Gda\u0144sk, GdaDsk, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Applied Physics and Mathematics, Technical University of Gda\u0144sk, GdaDsk, Poland","institution_ids":["https://openalex.org/I40413290"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Jonathan Oppenheim","orcid":null},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Jonathan Oppenheim","raw_affiliation_strings":["Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Cambridge, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Cambridge, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"last","author":{"id":null,"display_name":"Wei Song","orcid":null},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]},{"id":"https://openalex.org/I187400657","display_name":"South China Normal University","ror":"https://ror.org/01kq0pv72","country_code":"CN","type":"education","lineage":["https://openalex.org/I187400657"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wei Song","raw_affiliation_strings":["Department of Modern Physics, University of Science and Technology, Hefei, Anhui, China","School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Modern Physics, University of Science and Technology, Hefei, Anhui, China","institution_ids":["https://openalex.org/I126520041"]},{"raw_affiliation_string":"School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China","institution_ids":["https://openalex.org/I187400657"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.2207,"has_fulltext":false,"cited_by_count":88,"citation_normalized_percentile":{"value":0.89387669,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"55","issue":"7","first_page":"3375","last_page":"3387"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.9438999891281128,"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"}},"topics":[{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.9438999891281128,"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/T10622","display_name":"Quantum Mechanics and Applications","score":0.02370000071823597,"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/T11716","display_name":"Random Matrices and Applications","score":0.0035000001080334187,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multipartite","display_name":"Multipartite","score":0.9083999991416931},{"id":"https://openalex.org/keywords/multipartite-entanglement","display_name":"Multipartite entanglement","score":0.7447999715805054},{"id":"https://openalex.org/keywords/quantum-entanglement","display_name":"Quantum entanglement","score":0.6927000284194946},{"id":"https://openalex.org/keywords/squashed-entanglement","display_name":"Squashed entanglement","score":0.5708000063896179},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4860000014305115},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.45649999380111694},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.41119998693466187},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.40610000491142273},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.3991999924182892},{"id":"https://openalex.org/keywords/w-state","display_name":"W state","score":0.3937000036239624}],"concepts":[{"id":"https://openalex.org/C166733315","wikidata":"https://www.wikidata.org/wiki/Q6934831","display_name":"Multipartite","level":4,"score":0.9083999991416931},{"id":"https://openalex.org/C71799126","wikidata":"https://www.wikidata.org/wiki/Q6934832","display_name":"Multipartite entanglement","level":5,"score":0.7447999715805054},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.6927000284194946},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.669700026512146},{"id":"https://openalex.org/C32270708","wikidata":"https://www.wikidata.org/wiki/Q7582152","display_name":"Squashed entanglement","level":4,"score":0.5708000063896179},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4860000014305115},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.45649999380111694},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43320000171661377},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.41119998693466187},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.40610000491142273},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.3991999924182892},{"id":"https://openalex.org/C20593774","wikidata":"https://www.wikidata.org/wiki/Q7958522","display_name":"W state","level":4,"score":0.3937000036239624},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.3765999972820282},{"id":"https://openalex.org/C105141246","wikidata":"https://www.wikidata.org/wiki/Q6459475","display_name":"LOCC","level":4,"score":0.36719998717308044},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3440999984741211},{"id":"https://openalex.org/C171752962","wikidata":"https://www.wikidata.org/wiki/Q255166","display_name":"Kullback\u2013Leibler divergence","level":2,"score":0.33169999718666077},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.33000001311302185},{"id":"https://openalex.org/C73361133","wikidata":"https://www.wikidata.org/wiki/Q4810150","display_name":"Association scheme","level":2,"score":0.3203999996185303},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.3190000057220459},{"id":"https://openalex.org/C47191418","wikidata":"https://www.wikidata.org/wiki/Q3510462","display_name":"Information-theoretic security","level":3,"score":0.3124000132083893},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.29899999499320984},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.2865000069141388},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.2809000015258789},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.2612999975681305},{"id":"https://openalex.org/C111110010","wikidata":"https://www.wikidata.org/wiki/Q2627315","display_name":"Convex combination","level":4,"score":0.2612000107765198},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.2587999999523163},{"id":"https://openalex.org/C56911000","wikidata":"https://www.wikidata.org/wiki/Q831774","display_name":"Density matrix","level":3,"score":0.2574999928474426},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.251800000667572},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.2506999969482422}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tit.2009.2021373","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2009.2021373","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0704.2236","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0704.2236","pdf_url":"https://arxiv.org/pdf/0704.2236","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0704.2236","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0704.2236","pdf_url":"https://arxiv.org/pdf/0704.2236","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1489319061","https://openalex.org/W1725167044","https://openalex.org/W1760557826","https://openalex.org/W1966698790","https://openalex.org/W1982305921","https://openalex.org/W1983185363","https://openalex.org/W1990972698","https://openalex.org/W1993574974","https://openalex.org/W1993611246","https://openalex.org/W2000936997","https://openalex.org/W2010138343","https://openalex.org/W2011008529","https://openalex.org/W2024334792","https://openalex.org/W2027561650","https://openalex.org/W2043769961","https://openalex.org/W2044704854","https://openalex.org/W2052522564","https://openalex.org/W2057350190","https://openalex.org/W2069217971","https://openalex.org/W2076249063","https://openalex.org/W2077089904","https://openalex.org/W2099111195","https://openalex.org/W2106833918","https://openalex.org/W2133129288","https://openalex.org/W2136522857","https://openalex.org/W2144007657","https://openalex.org/W2155028760","https://openalex.org/W4239526858"],"related_works":[],"abstract_inverted_index":{"<para":[0],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[1],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[2],"New":[3],"measures":[4,93],"of":[5,14,20,24,31,41,56,66,94,102,119,133,137,153],"multipartite":[6,15,58,68,144,154],"entanglement":[7,34,60,95],"are":[8,159],"constructed":[9],"based":[10,80,96],"on":[11,44,81,97,150],"two":[12],"definitions":[13],"information":[16,40,69],"and":[17,48],"different":[18],"methods":[19],"optimizing":[21],"over":[22,52,116],"extensions":[23],"the":[25,32,38,45,50,57,67,98,109,126,134,151],"states.":[26,130],"One":[27],"is":[28,61,79,87,146],"a":[29,103,120],"generalization":[30],"squashed":[33,59],"where":[35],"one":[36,78,132],"takes":[37,49],"mutual":[39],"parties":[42],"conditioned":[43],"state's":[46],"extension":[47],"infimum":[51],"such":[53],"extensions.":[54,84],"Additivity":[55],"proved":[62],"for":[63],"both":[64],"versions":[65],"which":[70,89,105],"turn":[71],"out":[72],"to":[73,91,108,162],"be":[74],"related.":[75],"The":[76,157],"second":[77],"taking":[82],"classical":[83,166],"This":[85],"scheme":[86],"generalized,":[88],"enables":[90],"construct":[92],"mixed":[99],"convex":[100,111],"roof":[101,112],"quantity,":[104],"in":[106,165],"contrast":[107],"standard":[110],"method":[113],"involves":[114],"optimization":[115],"all":[117],"decompositions":[118,127],"density":[121],"matrix":[122],"rather":[123],"than":[124],"just":[125],"into":[128],"pure":[129],"As":[131],"possible":[135],"applications":[136],"these":[138],"results":[139,164],"we":[140],"prove":[141],"that":[142],"any":[143],"monotone":[145],"an":[147],"upper":[148],"bound":[149],"amount":[152],"distillable":[155],"key.":[156],"findings":[158],"finally":[160],"related":[161],"analogous":[163],"key":[167],"agreement.":[168],"</para>":[169]},"counts_by_year":[{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":9},{"year":2022,"cited_by_count":11},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":10},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2016-06-24T00:00:00"}