{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:25:35Z","timestamp":1760149535597,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,17]],"date-time":"2023-08-17T00:00:00Z","timestamp":1692230400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation","award":["DMS-1439786","507536"],"award-info":[{"award-number":["DMS-1439786","507536"]}]},{"name":"Simons Foundation Institute","award":["DMS-1439786","507536"],"award-info":[{"award-number":["DMS-1439786","507536"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantum physics through the lens of Bayesian statistics considers probability to be a degree of belief and subjective. A Bayesian derivation of the probability density function in phase space is presented. Then, a Kullback\u2013Liebler divergence in phase space is introduced to define interference and entanglement. Comparisons between each of these two quantities and the entropy are made. A brief presentation of entanglement in phase space to the spin degree of freedom and an extension to mixed states completes the work.<\/jats:p>","DOI":"10.3390\/e25081227","type":"journal-article","created":{"date-parts":[[2023,8,17]],"date-time":"2023-08-17T10:42:29Z","timestamp":1692268949000},"page":"1227","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantum Knowledge in Phase Space"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2138-3517","authenticated-orcid":false,"given":"Davi","family":"Geiger","sequence":"first","affiliation":[{"name":"Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Avery, J. (2003). Information Theory and Evolution, World Scientific.","DOI":"10.1142\/5299"},{"key":"ref_3","first-page":"370","article-title":"An Essay toward solving a Problem in the Doctrine of Chances","volume":"53","author":"Bayes","year":"1764","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_4","unstructured":"Cournot, A.A. (1843). Exposition de la Th\u00e9orie des Chances et des Probabilit\u00e9s, L. Hachette."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Venn, J. (1888). The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability, with Especial Reference to Its Logical Bearings and Its Application to Moral and Social Science, and to Statistics, Macmillan & Co, London.","DOI":"10.5962\/bhl.title.19044"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","article-title":"Can Quantum-Mechanical Description of Physical Reality be Considered Complete?","volume":"47","author":"Einstein","year":"1935","journal-title":"Phys. Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1895","DOI":"10.1103\/PhysRevLett.70.1895","article-title":"Teleporting an unknown quantum state via dual classical and Einstein\u2013Podolsky-Rosen channels","volume":"70","author":"Bennett","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1038\/37539","article-title":"Experimental quantum teleportation","volume":"390","author":"Bouwmeester","year":"2006","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.1103\/PhysRevLett.80.1121","article-title":"Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein\u2013Podolsky-Rosen Channels","volume":"80","author":"Boschi","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1908","DOI":"10.1080\/09500340903159495","article-title":"Slow and fast light: Fundamentals and applications","volume":"56","author":"Boyd","year":"2009","journal-title":"J. Mod. Opt."},{"key":"ref_11","unstructured":"Ferrie, C. (2023). Quantum Entanglement Isn\u2019t All That Spooky After All, Scientific American."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Marage, P., and Wallenborn, G. (1999). The Solvay Councils and the Birth of Modern Physics, Birkh\u00e4user Basel.","DOI":"10.1007\/978-3-0348-7703-9"},{"key":"ref_13","unstructured":"Becker, A. (2018). What Is Real?: The Unfinished Quest for the Meaning of Quantum Physics, Basic Books."},{"key":"ref_14","first-page":"97","article-title":"Quantum theory, the Church\u2013Turing principle and the universal quantum computer","volume":"400","author":"Deutsch","year":"1985","journal-title":"Proc. R. Soc. Lond. Math. Phys. Sci."},{"key":"ref_15","unstructured":"Nielsen, M.A., and Chuang, I. (2010). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_16","unstructured":"Witten, E. (2018). A Mini-Introduction To Information Theory. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/JHEP02(2020)014","article-title":"Entanglement entropy and superselection sectors. Part I. Global symmetries","volume":"2020","author":"Casini","year":"2020","journal-title":"J. High Energy Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Weinberg, S. (1995). The Quantum Theory of Fields: Volume 1, (Foundations), Cambridge University Press.","DOI":"10.1017\/CBO9781139644167"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1103\/PhysRev.34.163","article-title":"The Uncertainty Principle","volume":"34","author":"Robertson","year":"1929","journal-title":"Phys. Rev."},{"key":"ref_20","unstructured":"Gibbs, J.W. (2014). Elementary Principles in Statistical Mechanics, Dover Publications."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the quantum correction for thermodynamic equilibrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_22","first-page":"264","article-title":"Some Formal Properties of the Density Matrix","volume":"22","author":"Husimi","year":"1940","journal-title":"Proceedings of the Physico-Mathematical Society of Japan, 3rd Series"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Geiger, D., and Kedem, Z.M. (2022). On Quantum Entropy. Entropy, 24.","DOI":"10.3390\/e24101341"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Geiger, D., and Kedem, Z.M. (2022). Spin Entropy. Entropy, 24.","DOI":"10.3390\/e24091292"},{"key":"ref_25","unstructured":"Woodhouse, N.M.J. (1997). Geometric Quantization, Oxford University Press."},{"key":"ref_26","unstructured":"Blau, M. (2023, August 14). Symplectic Geometry and Geometric Quantization. Lecture Notes. Available online: https:\/\/ncatlab.org\/nlab\/files\/BlauGeometricQuantization.pdf."},{"key":"ref_27","unstructured":"Nair, V.P. (2016). Elements of geometric quantization and applications to fields and fluids. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1227\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:36:14Z","timestamp":1760128574000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1227"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,17]]},"references-count":27,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["e25081227"],"URL":"https:\/\/doi.org\/10.3390\/e25081227","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,8,17]]}}}