{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:46:53Z","timestamp":1767340013388},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2012,11,9]],"date-time":"2012-11-09T00:00:00Z","timestamp":1352419200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2013,6]]},"abstract":"<jats:p>We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative \u2020-Frobenius monoid in the category <jats:bold>FdHilb<\/jats:bold>, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative \u2020-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the \u2020-functor. Moreover, this characterisation can be interpreted operationally, since the \u2020-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.<\/jats:p>","DOI":"10.1017\/s0960129512000047","type":"journal-article","created":{"date-parts":[[2012,11,9]],"date-time":"2012-11-09T09:33:27Z","timestamp":1352453607000},"page":"555-567","source":"Crossref","is-referenced-by-count":56,"title":["A new description of orthogonal bases"],"prefix":"10.1017","volume":"23","author":[{"given":"BOB","family":"COECKE","sequence":"first","affiliation":[]},{"given":"DUSKO","family":"PAVLOVIC","sequence":"additional","affiliation":[]},{"given":"JAMIE","family":"VICARY","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2012,11,9]]},"reference":[{"key":"S0960129512000047_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0063104"},{"key":"S0960129512000047_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2006.12.018"},{"key":"S0960129512000047_ref17","unstructured":"Vicary J. (2008) Categorical formulation of finite-dimensional quantum algebras. (Available at arXiv:0805.0432.)"},{"key":"S0960129512000047_ref5","first-page":"567","volume-title":"Mathematics of Quantum Computing and Technology","author":"Coecke","year":"2007"},{"key":"S0960129512000047_ref6","first-page":"29","volume-title":"Semantic Techniques for Quantum Computation","author":"Coecke","year":"2008"},{"key":"S0960129512000047_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-12821-9_4"},{"key":"S0960129512000047_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90121-6"},{"key":"S0960129512000047_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-70583-3_25"},{"key":"S0960129512000047_ref1","doi-asserted-by":"crossref","unstructured":"Abramsky S. and Coecke B. (2004) A categorical semantics of quantum protocols. In: Proceedings of 19th IEEE conference on Logic in Computer Science 415\u2013425. (Available at arXiv:quant-ph\/0402130 & arXiv:0808.1023.)","DOI":"10.1109\/LICS.2004.1319636"},{"key":"S0960129512000047_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0083085"},{"key":"S0960129512000047_ref12","volume-title":"C*-Algebras and Operator Theory","author":"Murphy","year":"1990"},{"key":"S0960129512000047_ref2","first-page":"51","article-title":"A note on strongly separable algebras","volume":"65","author":"Aguiar","year":"2000","journal-title":"Bolet\u00edn de la Academia Nacional de Ciencias (C\u00f3rdoba, Argentina)"},{"key":"S0960129512000047_ref14","volume-title":"The Road to Reality: A Complete Guide to the Laws of the Universe","author":"Penrose","year":"2005"},{"key":"S0960129512000047_ref13","doi-asserted-by":"publisher","DOI":"10.1038\/404130b0"},{"key":"S0960129512000047_ref18","doi-asserted-by":"publisher","DOI":"10.1038\/299802a0"},{"key":"S0960129512000047_ref10","volume-title":"Frobenius Algebras and 2D Topological Quantum Field Theories","author":"Kock","year":"2004"},{"key":"S0960129512000047_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(91)90003-P"},{"key":"S0960129512000047_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(82)90084-6"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129512000047","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T21:26:54Z","timestamp":1556054814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129512000047\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,9]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,6]]}},"alternative-id":["S0960129512000047"],"URL":"https:\/\/doi.org\/10.1017\/s0960129512000047","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,11,9]]}}}