{"id":"https://openalex.org/W2116898717","doi":"https://doi.org/10.1007/s11080-006-7271-8","title":"Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria","display_name":"Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria","publication_year":2006,"publication_date":"2006-03-01","ids":{"openalex":"https://openalex.org/W2116898717","doi":"https://doi.org/10.1007/s11080-006-7271-8","mag":"2116898717"},"language":"en","primary_location":{"id":"doi:10.1007/s11080-006-7271-8","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s11080-006-7271-8","pdf_url":null,"source":{"id":"https://openalex.org/S27332288","display_name":"Open Systems & Information Dynamics","issn_l":"1230-1612","issn":["1230-1612","1573-1324","1793-7191"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Open Systems &amp; Information Dynamics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5033198961","display_name":"Micha\u0142 Horodecki","orcid":"https://orcid.org/0000-0002-0446-3059"},"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":true,"raw_author_name":"Micha\u0142 Horodecki","raw_affiliation_strings":["Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-952 Gda\u0144sk, Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-952 Gda\u0144sk, Poland","institution_ids":["https://openalex.org/I40413290"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081239197","display_name":"Pawe\u0142 Horodecki","orcid":"https://orcid.org/0000-0002-3233-1336"},"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":"Pawe\u0142 Horodecki","raw_affiliation_strings":["Faculty of Applied Physics and Mathematics, Technical University of Gda\u0144sk, Poland"],"affiliations":[{"raw_affiliation_string":"Faculty of Applied Physics and Mathematics, Technical University of Gda\u0144sk, Poland","institution_ids":["https://openalex.org/I40413290"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5012662473","display_name":"Ryszard Horodecki","orcid":"https://orcid.org/0000-0003-2935-290X"},"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":"Ryszard Horodecki","raw_affiliation_strings":["Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-952 Gda\u0144sk, Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-952 Gda\u0144sk, Poland","institution_ids":["https://openalex.org/I40413290"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5033198961"],"corresponding_institution_ids":["https://openalex.org/I40413290"],"apc_list":null,"apc_paid":null,"fwci":12.0954,"has_fulltext":false,"cited_by_count":135,"citation_normalized_percentile":{"value":0.98539657,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"13","issue":"01","first_page":"103","last_page":"111"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9991000294685364,"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/T11414","display_name":"Quantum optics and atomic interactions","score":0.9987000226974487,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/multipartite","display_name":"Multipartite","score":0.8651256561279297},{"id":"https://openalex.org/keywords/transpose","display_name":"Transpose","score":0.7915627956390381},{"id":"https://openalex.org/keywords/peres\u2013horodecki-criterion","display_name":"Peres\u2013Horodecki criterion","score":0.6623952388763428},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6522799134254456},{"id":"https://openalex.org/keywords/permutation-matrix","display_name":"Permutation matrix","score":0.6294003129005432},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.6028500199317932},{"id":"https://openalex.org/keywords/separable-state","display_name":"Separable state","score":0.5393422245979309},{"id":"https://openalex.org/keywords/transposition","display_name":"Transposition (logic)","score":0.5337002873420715},{"id":"https://openalex.org/keywords/quantum-entanglement","display_name":"Quantum entanglement","score":0.5329651832580566},{"id":"https://openalex.org/keywords/density-matrix","display_name":"Density matrix","score":0.5018861293792725},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.500413179397583},{"id":"https://openalex.org/keywords/multipartite-entanglement","display_name":"Multipartite entanglement","score":0.4875330626964569},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.4723856449127197},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4613625407218933},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4103480279445648},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4070233106613159},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35431110858917236},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.3539976477622986},{"id":"https://openalex.org/keywords/squashed-entanglement","display_name":"Squashed entanglement","score":0.20770016312599182},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.1381622552871704},{"id":"https://openalex.org/keywords/quantum-discord","display_name":"Quantum discord","score":0.1111239492893219},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08592039346694946},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07771790027618408}],"concepts":[{"id":"https://openalex.org/C166733315","wikidata":"https://www.wikidata.org/wiki/Q6934831","display_name":"Multipartite","level":4,"score":0.8651256561279297},{"id":"https://openalex.org/C200106649","wikidata":"https://www.wikidata.org/wiki/Q223683","display_name":"Transpose","level":3,"score":0.7915627956390381},{"id":"https://openalex.org/C4119822","wikidata":"https://www.wikidata.org/wiki/Q7167895","display_name":"Peres\u2013Horodecki criterion","level":5,"score":0.6623952388763428},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6522799134254456},{"id":"https://openalex.org/C84140500","wikidata":"https://www.wikidata.org/wiki/Q851512","display_name":"Permutation matrix","level":3,"score":0.6294003129005432},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.6028500199317932},{"id":"https://openalex.org/C55795843","wikidata":"https://www.wikidata.org/wiki/Q614178","display_name":"Separable state","level":5,"score":0.5393422245979309},{"id":"https://openalex.org/C12455157","wikidata":"https://www.wikidata.org/wiki/Q7835331","display_name":"Transposition (logic)","level":2,"score":0.5337002873420715},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.5329651832580566},{"id":"https://openalex.org/C56911000","wikidata":"https://www.wikidata.org/wiki/Q831774","display_name":"Density matrix","level":3,"score":0.5018861293792725},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.500413179397583},{"id":"https://openalex.org/C71799126","wikidata":"https://www.wikidata.org/wiki/Q6934832","display_name":"Multipartite entanglement","level":5,"score":0.4875330626964569},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.4723856449127197},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4613625407218933},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4103480279445648},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4070233106613159},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35431110858917236},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.3539976477622986},{"id":"https://openalex.org/C32270708","wikidata":"https://www.wikidata.org/wiki/Q7582152","display_name":"Squashed entanglement","level":4,"score":0.20770016312599182},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.1381622552871704},{"id":"https://openalex.org/C139356082","wikidata":"https://www.wikidata.org/wiki/Q7269038","display_name":"Quantum discord","level":4,"score":0.1111239492893219},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08592039346694946},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07771790027618408},{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.0},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11080-006-7271-8","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s11080-006-7271-8","pdf_url":null,"source":{"id":"https://openalex.org/S27332288","display_name":"Open Systems & Information Dynamics","issn_l":"1230-1612","issn":["1230-1612","1573-1324","1793-7191"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Open Systems &amp; Information Dynamics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1515055485","https://openalex.org/W1963952399","https://openalex.org/W1982192948","https://openalex.org/W1997082258","https://openalex.org/W2015775929","https://openalex.org/W2020910139","https://openalex.org/W2032083080","https://openalex.org/W2033909248","https://openalex.org/W2034920898","https://openalex.org/W2042951286","https://openalex.org/W2043293634","https://openalex.org/W2062292260","https://openalex.org/W2071394131","https://openalex.org/W2073205428","https://openalex.org/W2094092037","https://openalex.org/W2103713333","https://openalex.org/W2144126354","https://openalex.org/W3100928949","https://openalex.org/W3122850579"],"related_works":["https://openalex.org/W2379218716","https://openalex.org/W2962738610","https://openalex.org/W2036329837","https://openalex.org/W4250308684","https://openalex.org/W2138689422","https://openalex.org/W1978236786","https://openalex.org/W2062022662","https://openalex.org/W3143349443","https://openalex.org/W2253789365","https://openalex.org/W2116898717"],"abstract_inverted_index":{"Recently,":[0],"a":[1,50,94,119],"powerful":[2],"separability":[3,100,120],"criterion":[4,36,126],"was":[5],"introduced":[6],"by":[7,13,55,92],"O.":[8],"Rudolf":[9],"in":[10,18,43,114],"[5]":[11],"and":[12,37,40,74,124],"K.":[14],"Chen":[15],"et":[16],"al.":[17],"[6]":[19],"\u2014":[20],"basing":[21],"on":[22],"realignment":[23,125],"of":[24,26,45,52,57,96,108,110,130],"elements":[25],"density":[27,111],"matrix.":[28],"Composing":[29],"the":[30,34,38,69],"main":[31],"idea":[32,70],"behind":[33],"above":[35],"necessary":[39],"sufficient":[41],"condition":[42],"terms":[44],"positive":[46,65],"maps,":[47],"we":[48],"provide":[49],"characterization":[51],"separable":[53],"states":[54],"means":[56],"linear":[58],"contractions.":[59],"The":[60],"latter":[61],"need":[62],"not":[63],"be":[64],"maps.":[66],"We":[67,89],"extend":[68],"to":[71,118],"multipartite":[72,103],"systems,":[73],"find":[75],"that,":[76],"somewhat":[77],"suprisingly,":[78],"partial":[79,82],"realigment":[80],"(unlike":[81],"transposition)":[83],"can":[84],"detect":[85],"genuinely":[86],"tri-parite":[87],"entanglement.":[88],"generalize":[90],"it":[91],"introducing":[93],"family":[95],"so":[97],"called":[98],"permutation":[99,107,131],"criteria":[101],"for":[102],"states.":[104],"Namely,":[105],"any":[106],"indices":[109],"matrix":[112],"written":[113],"product":[115],"basis":[116],"leads":[117],"criterion.":[121],"Partial":[122],"transpose":[123],"are":[127],"special":[128],"cases":[129],"criteria.":[132]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":9},{"year":2023,"cited_by_count":39},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
