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A key postulate is the purification postulate, which we propose to replace by a more generally applicable postulate that we call the complete extension postulate (CEP), i.e., the existence of an extension of a physical system from which one can generate any other extension. This new concept leads to a plethora of open questions and research directions in the study of general theories satisfying the CEP (which may include a theory that hyper-decoheres to quantum theory). For example, we show that the CEP implies the impossibility of bit-commitment. This is exemplified by a case study of the theory of non-signalling behaviors which we show satisfies the CEP. We moreover show that in certain cases the complete extension will not be pure, highlighting the key divergence from the purification postulate.<\/jats:p>","DOI":"10.22331\/q-2023-11-03-1159","type":"journal-article","created":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T12:06:59Z","timestamp":1699013219000},"page":"1159","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Complete extension: the non-signaling analog of quantum purification"],"prefix":"10.22331","volume":"7","author":[{"given":"Marek","family":"Winczewski","sequence":"first","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, Wita Stwosza 63, 80-308 Gda\u0144sk, Poland"},{"name":"Institute of Theoretical Physics and Astrophysics and National Quantum Information Centre in Gda\u0144sk, University of Gda\u0144sk, 80\u2013952 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tamoghna","family":"Das","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, Wita Stwosza 63, 80-308 Gda\u0144sk, Poland"},{"name":"Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John H.","family":"Selby","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, Wita Stwosza 63, 80-308 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karol","family":"Horodecki","sequence":"additional","affiliation":[{"name":"Institute of Informatics and National Quantum Information Centre in Gda\u0144sk, Faculty of Mathematics, Physics and Informatics, University of Gda\u0144sk, 80\u2013952 Gda\u0144sk, Poland"},{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, Wita Stwosza 63, 80-308 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pawe\u0142","family":"Horodecki","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, Wita Stwosza 63, 80-308 Gda\u0144sk, Poland"},{"name":"Faculty of Applied Physics and Mathematics, Gda\u0144sk University of Technology, 80\u2013233 Gda\u0144sk, Poland"},{"name":"National Quantum Information Centre, University of Gda\u0144sk, ul. 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