{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:38:14Z","timestamp":1773232694793,"version":"3.50.1"},"publisher-location":"Cham","reference-count":16,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783032039200","type":"print"},{"value":"9783032039217","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T00:00:00Z","timestamp":1760918400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T00:00:00Z","timestamp":1760918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"DOI":"10.1007\/978-3-032-03921-7_27","type":"book-chapter","created":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T22:04:30Z","timestamp":1760911470000},"page":"262-269","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Independent States are Orthogonal: A\u00a0Categorical Framework to\u00a0Treat Probability Geometrically"],"prefix":"10.1007","author":[{"given":"Matthew","family":"Di Meglio","sequence":"first","affiliation":[]},{"given":"Chris","family":"Heunen","sequence":"additional","affiliation":[]},{"given":"J. S.","family":"Lemay","sequence":"additional","affiliation":[]},{"given":"Paolo","family":"Perrone","sequence":"additional","affiliation":[]},{"given":"Dario","family":"Stein","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,20]]},"reference":[{"key":"27_CR1","doi-asserted-by":"publisher","unstructured":"Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Logic in Computer Science. IEEE (2004). https:\/\/doi.org\/10.1109\/LICS.2004.1319636","DOI":"10.1109\/LICS.2004.1319636"},{"key":"27_CR2","doi-asserted-by":"publisher","first-page":"938","DOI":"10.1017\/s0960129518000488","volume":"29","author":"K Cho","year":"2019","unstructured":"Cho, K., Jacobs, B.: Disintegration and Bayesian inversion via string diagrams. Math. Struct. Comput. Sci. 29, 938\u2013971 (2019). https:\/\/doi.org\/10.1017\/s0960129518000488","journal-title":"Math. Struct. Comput. Sci."},{"key":"27_CR3","doi-asserted-by":"publisher","unstructured":"Dahlqvist, F., Danos, V., Garnier, I., Silva, A.: Borel Kernels and their approximation, categorically. Math. Found. Programm. Semant. 341, 91\u2013119 (2018). https:\/\/doi.org\/10.1016\/j.entcs.2018.11.006","DOI":"10.1016\/j.entcs.2018.11.006"},{"key":"27_CR4","unstructured":"Di\u00a0Meglio, M.: R*-Categories: The Hilbert-Space Analogue of Abelian Categories (2024)"},{"key":"27_CR5","doi-asserted-by":"crossref","unstructured":"Di\u00a0Meglio, M., Heunen, C.: Dagger categories and the complex numbers: axioms for the category of finite-dimensional Hilbert spaces and linear contractions (2024)","DOI":"10.1007\/s10485-025-09803-5"},{"key":"27_CR6","unstructured":"Ensarguet, N., Perrone, P.: Categorical probability spaces, ergodic decompositions, and transitions to equilibrium (2024)"},{"key":"27_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2020.107239","volume":"370","author":"T Fritz","year":"2020","unstructured":"Fritz, T.: A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math. 370, 107239 (2020). https:\/\/doi.org\/10.1016\/j.aim.2020.107239","journal-title":"Adv. Math."},{"key":"27_CR8","doi-asserted-by":"publisher","DOI":"10.1112\/blms.13010","author":"C Heunen","year":"2024","unstructured":"Heunen, C., Kornell, A., van der Schaaf, N.: Axioms for the category of Hilbert spaces and linear contractions. Bull. Lond. Math. Soc. (2024). https:\/\/doi.org\/10.1112\/blms.13010","journal-title":"Bull. Lond. Math. Soc."},{"key":"27_CR9","unstructured":"Karvonen, M.: The way of the dagger. Ph.D. thesis, University of Edinburgh (2018)"},{"key":"27_CR10","doi-asserted-by":"publisher","unstructured":"Perrone, P.: Lifting couplings in Wasserstein spaces. Compositionality 7(2) (2025). https:\/\/doi.org\/10.46298\/compositionality-7-2","DOI":"10.46298\/compositionality-7-2"},{"key":"27_CR11","unstructured":"Perrone, P., Belle, R.V.: Convergence of martingales via enriched dagger categories (2024)"},{"key":"27_CR12","unstructured":"Sergeant-Perthuis, G.: Compositional statistical mechanics, entropy and variational inference (2024)"},{"key":"27_CR13","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/j.entcs.2018.03.028","volume":"336","author":"A Simpson","year":"2018","unstructured":"Simpson, A.: Category-theoretic structure for independence and conditional independence. Electron. Notes Theor. Comput. Sci. 336, 281\u2013297 (2018). https:\/\/doi.org\/10.1016\/j.entcs.2018.03.028","journal-title":"Electron. Notes Theor. Comput. Sci."},{"key":"27_CR14","doi-asserted-by":"publisher","unstructured":"Simpson, A.: Equivalence and conditional independence in atomic sheaf logic. In: Logic in Computer Science. IEEE (2024). https:\/\/doi.org\/10.1145\/3661814.3662132","DOI":"10.1145\/3661814.3662132"},{"key":"27_CR15","doi-asserted-by":"crossref","unstructured":"Stein, D.: Random variables, conditional independence and categories of abstract sample spaces. In: Logic in Computer Science. IEEE (2025)","DOI":"10.1109\/LICS65433.2025.00042"},{"key":"27_CR16","doi-asserted-by":"publisher","unstructured":"Villani, C.: Optimal transport: old and new, Grundlehren der mathematischen Wissenschaften, vol.\u00a0338. Springer, Heidelberg (2009). https:\/\/doi.org\/10.1007\/978-3-540-71050-9","DOI":"10.1007\/978-3-540-71050-9"}],"container-title":["Lecture Notes in Computer Science","Geometric Science of Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-03921-7_27","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T16:08:35Z","timestamp":1773158915000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-03921-7_27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,20]]},"ISBN":["9783032039200","9783032039217"],"references-count":16,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-03921-7_27","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,20]]},"assertion":[{"value":"20 October 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"The authors have no competing interests to declare that\u00a0are relevant to the content of this article.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Disclosure of Interests"}},{"value":"GSI","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Geometric Science of Information","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Saint-Malo","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"France","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"29 October 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"31 October 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"7","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"gsi2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/conference-gsi.org\/gsi-2025\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}