{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T19:28:04Z","timestamp":1778009284179,"version":"3.51.4"},"reference-count":36,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T00:00:00Z","timestamp":1655424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62171266"],"award-info":[{"award-number":["62171266"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U2001205"],"award-info":[{"award-number":["U2001205"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["16QNGG013"],"award-info":[{"award-number":["16QNGG013"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Funded Projects for the Academic Leaders and Academic Backbones, Shaanxi Normal University","award":["62171266"],"award-info":[{"award-number":["62171266"]}]},{"name":"Funded Projects for the Academic Leaders and Academic Backbones, Shaanxi Normal University","award":["U2001205"],"award-info":[{"award-number":["U2001205"]}]},{"name":"Funded Projects for the Academic Leaders and Academic Backbones, Shaanxi Normal University","award":["16QNGG013"],"award-info":[{"award-number":["16QNGG013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Belavkin\u2013Staszewski relative entropy can naturally characterize the effects of the possible noncommutativity of quantum states. In this paper, two new conditional entropy terms and four new mutual information terms are first defined by replacing quantum relative entropy with Belavkin\u2013Staszewski relative entropy. Next, their basic properties are investigated, especially in classical-quantum settings. In particular, we show the weak concavity of the Belavkin\u2013Staszewski conditional entropy and obtain the chain rule for the Belavkin\u2013Staszewski mutual information. Finally, the subadditivity of the Belavkin\u2013Staszewski relative entropy is established, i.e., the Belavkin\u2013Staszewski relative entropy of a joint system is less than the sum of that of its corresponding subsystems with the help of some multiplicative and additive factors. Meanwhile, we also provide a certain subadditivity of the geometric R\u00e9nyi relative entropy.<\/jats:p>","DOI":"10.3390\/e24060837","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T11:45:44Z","timestamp":1655466344000},"page":"837","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Belavkin\u2013Staszewski Relative Entropy, Conditional Entropy, and Mutual Information"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6179-8755","authenticated-orcid":false,"given":"Yuan","family":"Zhai","sequence":"first","affiliation":[{"name":"The College of Computer Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bo","family":"Yang","sequence":"additional","affiliation":[{"name":"The College of Computer Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhengjun","family":"Xi","sequence":"additional","affiliation":[{"name":"The College of Computer Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,17]]},"reference":[{"key":"ref_1","first-page":"547","article-title":"On measures of information and entropy","volume":"4","year":"1961","journal-title":"Proc. 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