{"title":"冯诺依曼子代数的相对熵","authors":"Li Gao, M. Junge, Nicholas Laracuente","doi":"10.1142/s0129167x20500469","DOIUrl":null,"url":null,"abstract":"We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\\le p\\le \\infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy with respect to a subalgebra. These relative entropy generalizes subfactors index and has application in estimating decoherence time of quantum Markov semigroup.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"22","resultStr":"{\"title\":\"Relative entropy for von Neumann subalgebras\",\"authors\":\"Li Gao, M. Junge, Nicholas Laracuente\",\"doi\":\"10.1142/s0129167x20500469\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\\\\le p\\\\le \\\\infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy with respect to a subalgebra. These relative entropy generalizes subfactors index and has application in estimating decoherence time of quantum Markov semigroup.\",\"PeriodicalId\":351745,\"journal\":{\"name\":\"arXiv: Operator Algebras\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"22\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Operator Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1142/s0129167x20500469\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Operator Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0129167x20500469","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy with respect to a subalgebra. These relative entropy generalizes subfactors index and has application in estimating decoherence time of quantum Markov semigroup.
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