{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:04:00Z","timestamp":1760151840595,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,14]],"date-time":"2022-09-14T00:00:00Z","timestamp":1663113600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1439786","507536"],"award-info":[{"award-number":["DMS-1439786","507536"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"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>Two types of randomness are associated with a mixed quantum state: the uncertainty in the probability coefficients of the constituent pure states and the uncertainty in the value of each observable captured by the Born\u2019s rule probabilities. Entropy is a quantification of randomness, and we propose a spin-entropy for the observables of spin pure states based on the phase space of a spin as described by the geometric quantization method, and we also expand it to mixed quantum states. This proposed entropy overcomes the limitations of previously-proposed entropies such as von Neumann entropy which only quantifies the randomness of specifying the quantum state. As an example of a limitation, previously-proposed entropies are higher for Bell entangled spin states than for disentangled spin states, even though the spin observables are less constrained for a disentangled pair of spins than for an entangled pair. The proposed spin-entropy accurately quantifies the randomness of a quantum state, it never reaches zero value, and it is lower for entangled states than for disentangled states.<\/jats:p>","DOI":"10.3390\/e24091292","type":"journal-article","created":{"date-parts":[[2022,9,14]],"date-time":"2022-09-14T20:50:45Z","timestamp":1663188645000},"page":"1292","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Spin Entropy"],"prefix":"10.3390","volume":"24","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1353-4584","authenticated-orcid":false,"given":"Zvi M.","family":"Kedem","sequence":"additional","affiliation":[{"name":"Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF01725541","article-title":"The emergence of classical properties through interaction with the environment","volume":"59","author":"Joos","year":"1985","journal-title":"Z. 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