{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:35:03Z","timestamp":1790814903256,"version":"4.1.0"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,4,19]],"date-time":"2017-04-19T00:00:00Z","timestamp":1492560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We introduce the notion of a leak for general process theories and identify quantum theory as a theory with minimal leakage, while classical theory has maximal leakage. We provide a construction that adjoins leaks to theories, an instance of which describes the emergence of classical theory by adjoining decoherence leaks to quantum theory. Finally, we show that defining a notion of purity for processes in general process theories has to make reference to the leaks of that theory, a feature missing in standard definitions; hence, we propose a refined definition and study the resulting notion of purity for quantum, classical and intermediate theories.<\/jats:p>","DOI":"10.3390\/e19040174","type":"journal-article","created":{"date-parts":[[2017,4,19]],"date-time":"2017-04-19T10:22:01Z","timestamp":1492597321000},"page":"174","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Leaks: Quantum, Classical, Intermediate and More"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4596-7501","authenticated-orcid":false,"given":"John","family":"Selby","sequence":"first","affiliation":[{"name":"Department of Physics, Imperial College London, Kensington, London SW7 2AZ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bob","family":"Coecke","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3PA, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2818","DOI":"10.1103\/PhysRevLett.76.2818","article-title":"Noncommuting mixed states cannot be broadcast","volume":"76","author":"Barnum","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"240501","DOI":"10.1103\/PhysRevLett.99.240501","article-title":"A generalized no-broadcasting theorem","volume":"99","author":"Barnum","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","unstructured":"Landry, E. (2016). Categorical quantum mechanics I: Causal quantum processes. Categories for the Working Philosopher, Oxford University Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/BF02058098","article-title":"Quantum nonlocality as an axiom","volume":"24","author":"Popescu","year":"1994","journal-title":"Found. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Abramsky, S., and Coecke, B. (2004, January 13\u201317). A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS), Washington, DC, USA.","DOI":"10.1109\/LICS.2004.1319636"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Coecke, B., and Kissinger, A. (2016). Picturing Quantum Processes. A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press.","DOI":"10.1017\/9781316219317"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1109\/TIT.2003.811917","article-title":"The capacity of hybrid quantum memory","volume":"49","author":"Kuperberg","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1038\/nphys1202","article-title":"Quantum darwinism","volume":"5","author":"Zurek","year":"2009","journal-title":"Nat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Coecke, B., Selby, J., and Tull, S. (arXiv, 2017). Two roads to classicality, arXiv.","DOI":"10.4204\/EPTCS.266.7"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.entcs.2008.04.021","article-title":"Idempotents in Dagger Categories (Extended Abstract)","volume":"210","author":"Selinger","year":"2008","journal-title":"Electron. Notes Theor. Comput. Sci."},{"key":"ref_11","unstructured":"Heunen, C., Kissinger, A., and Selinger, P. (2013, January 17\u201319). Completely positive projections and biproducts. Proceedings of the 10th International Workshop on Quantum Physics and Logic, Barcelona, Spain."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Cunningham, O., and Heunen, C. (arXiv, 2015). Axiomatizing complete positivity, arXiv.","DOI":"10.4204\/EPTCS.195.11"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"032304","DOI":"10.1103\/PhysRevA.75.032304","article-title":"Information processing in generalized probabilistic theories","volume":"75","author":"Barrett","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_14","unstructured":"Richens, J., Selby, J., and Al-Safi, S. (arXiv, 2016). Entanglement is an inevitable feature of any non-classical theory, arXiv."},{"key":"ref_15","unstructured":"Mac Lane, S. (1998). Categories for the Working Mathematician, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"062348","DOI":"10.1103\/PhysRevA.81.062348","article-title":"Probabilistic theories with purification","volume":"81","author":"Chiribella","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Coecke, B. (arXiv, 2014). Terminality implies non-signalling, arXiv.","DOI":"10.4204\/EPTCS.172.3"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Coecke, B., and Kissinger, A. (arXiv, 2016). Categorical quantum mechanics II: Classical-quantum interaction, arXiv.","DOI":"10.1142\/S0219749916400207"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1017\/S0305004100074338","article-title":"Traced monoidal categories","volume":"Volume 119","author":"Joyal","year":"1996","journal-title":"Mathematical Proceedings of the Cambridge Philosophical Society"},{"key":"ref_20","first-page":"211","article-title":"Positive functions on C*-algebras","volume":"6","author":"Stinespring","year":"1955","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1103\/RevModPhys.79.555","article-title":"Reference frames, superselection rules, and quantum information","volume":"79","author":"Bartlett","year":"2007","journal-title":"Rev. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Chiribella, G., D\u2019Ariano, G.M., and Perinotti, P. (2016). Quantum from principles. Quantum Theory: Informational Foundations and Foils, Springer.","DOI":"10.1007\/978-94-017-7303-4"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Lee, C.M., and Selby, J.H. (arXiv, 2017). A no-go theorem for post-quantum theories that decohere to quantum theory, arXiv.","DOI":"10.1098\/rspa.2017.0732"},{"key":"ref_24","unstructured":"Selby, J., Scandolo, C.M., and Coecke, B. Quantum theory from diagrammatic postulates, Forthcoming submitted."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/174\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:32:58Z","timestamp":1760207578000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/174"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,19]]},"references-count":24,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,4]]}},"alternative-id":["e19040174"],"URL":"https:\/\/doi.org\/10.3390\/e19040174","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,4,19]]}}}