{"id":"https://openalex.org/W2156442048","doi":"https://doi.org/10.1007/978-3-642-40020-9_40","title":"Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information","display_name":"Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2156442048","doi":"https://doi.org/10.1007/978-3-642-40020-9_40","mag":"2156442048"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-642-40020-9_40","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-642-40020-9_40","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"book-chapter","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/A5006396647","display_name":"Roman V. Belavkin","orcid":"https://orcid.org/0000-0002-2356-1447"},"institutions":[{"id":"https://openalex.org/I60488453","display_name":"Middlesex University","ror":"https://ror.org/01rv4p989","country_code":"GB","type":"education","lineage":["https://openalex.org/I60488453"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Roman V. Belavkin","raw_affiliation_strings":["School of Engineering and Information Sciences, Middlesex University, London, NW4 4BT, UK"],"affiliations":[{"raw_affiliation_string":"School of Engineering and Information Sciences, Middlesex University, London, NW4 4BT, UK","institution_ids":["https://openalex.org/I60488453"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5006396647"],"corresponding_institution_ids":["https://openalex.org/I60488453"],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.20191109,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":null,"issue":null,"first_page":"369","last_page":"376"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.978600025177002,"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"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.978600025177002,"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/pythagorean-theorem","display_name":"Pythagorean theorem","score":0.9322977066040039},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.6726503372192383},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5027432441711426},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.4921385645866394},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.4779006242752075},{"id":"https://openalex.org/keywords/homomorphism","display_name":"Homomorphism","score":0.46305012702941895},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.4604955017566681},{"id":"https://openalex.org/keywords/information-theory","display_name":"Information theory","score":0.43485522270202637},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.43117833137512207},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3738830089569092},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3364826440811157},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3307075798511505},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23534205555915833},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11694568395614624},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09048113226890564}],"concepts":[{"id":"https://openalex.org/C100461760","wikidata":"https://www.wikidata.org/wiki/Q11518","display_name":"Pythagorean theorem","level":2,"score":0.9322977066040039},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.6726503372192383},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5027432441711426},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.4921385645866394},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4779006242752075},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.46305012702941895},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.4604955017566681},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.43485522270202637},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.43117833137512207},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3738830089569092},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3364826440811157},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3307075798511505},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23534205555915833},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11694568395614624},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09048113226890564},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/978-3-642-40020-9_40","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-642-40020-9_40","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.418.4550","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.418.4550","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.eis.mdx.ac.uk/staffpages/rvb/publications/rvb-gsi13.pdf","raw_type":"text"},{"id":"pmh:oai:eprints.mdx.ac.uk:13319","is_oa":false,"landing_page_url":"https://eprints.mdx.ac.uk/13319/","pdf_url":null,"source":{"id":"https://openalex.org/S4306400025","display_name":"Middlesex University Research Repository (Middlesex University Of London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I60488453","host_organization_name":"Middlesex University","host_organization_lineage":["https://openalex.org/I60488453"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Book Section"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.4399999976158142,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[{"id":"https://openalex.org/G8869259549","display_name":null,"funder_award_id":"EP/H031936/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W397736297","https://openalex.org/W592992621","https://openalex.org/W646945617","https://openalex.org/W1571989874","https://openalex.org/W1577338488","https://openalex.org/W1965555277","https://openalex.org/W1977076119","https://openalex.org/W2002109927","https://openalex.org/W2007737215","https://openalex.org/W2071902828","https://openalex.org/W2087171258","https://openalex.org/W2095337937","https://openalex.org/W2095674846","https://openalex.org/W2159628417","https://openalex.org/W2479395228","https://openalex.org/W2481729244","https://openalex.org/W2947000318","https://openalex.org/W4210458405","https://openalex.org/W4239588355","https://openalex.org/W4293820388"],"related_works":["https://openalex.org/W2995447873","https://openalex.org/W1802380781","https://openalex.org/W4378767041","https://openalex.org/W4210969441","https://openalex.org/W2050612014","https://openalex.org/W3166167233","https://openalex.org/W2387128994","https://openalex.org/W3093594498","https://openalex.org/W2947875480","https://openalex.org/W2157313229"],"abstract_inverted_index":null,"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}