{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8986","name":"Supplementarity is Necessary for Quantum Diagram Reasoning","abstract":"The ZX-calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. We prove that its pi\/4-fragment is not complete, in other words the ZX-calculus is not complete for the so called \"Clifford+T quantum mechanics\". The completeness of this fragment was one of the main open problems in categorical quantum mechanics, a programme initiated by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum mechanics. On the other hand, its pi\/2-fragment is known to be complete, i.e. the ZX-calculus is complete for the so called \"stabilizer quantum mechanics\". Deciding whether its pi\/4-fragment is complete is a crucial step in the development of the ZX-calculus since this fragment is approximately universal for quantum mechanics, contrary to the pi\/2-fragment. \r\n\r\nTo establish our incompleteness result, we consider a fairly simple property of quantum states called supplementarity. We show that supplementarity can be derived in the ZX-calculus if and only if the angles involved in this equation are multiples of pi\/2. In particular, the impossibility to derive supplementarity for pi\/4 implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics. As a consequence, we propose to add the supplementarity to the set of rules of the ZX-calculus.\r\n\r\nWe also show that if a ZX-diagram involves antiphase twins, they can be merged when the ZX-calculus is augmented with the supplementarity rule. Merging antiphase twins makes diagrammatic reasoning much easier and provides a purely graphical meaning to the supplementarity rule.","keywords":["quantum diagram reasoning","completeness","ZX-calculus","quantum computing","categorical quantum mechanics"],"author":[{"@type":"Person","name":"Perdrix, Simon","givenName":"Simon","familyName":"Perdrix"},{"@type":"Person","name":"Wang, Quanlong","givenName":"Quanlong","familyName":"Wang"}],"position":76,"pageStart":"76:1","pageEnd":"76:14","dateCreated":"2016-08-19","datePublished":"2016-08-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Perdrix, Simon","givenName":"Simon","familyName":"Perdrix"},{"@type":"Person","name":"Wang, Quanlong","givenName":"Quanlong","familyName":"Wang"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.MFCS.2016.76","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6261","volumeNumber":58,"name":"41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)","dateCreated":"2016-08-19","datePublished":"2016-08-19","editor":[{"@type":"Person","name":"Faliszewski, Piotr","givenName":"Piotr","familyName":"Faliszewski"},{"@type":"Person","name":"Muscholl, Anca","givenName":"Anca","familyName":"Muscholl"},{"@type":"Person","name":"Niedermeier, Rolf","givenName":"Rolf","familyName":"Niedermeier"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8986","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6261"}}}