{"title":"不可区分量子粒子的熵关系","authors":"Marius Lemm","doi":"10.1088/1742-5468/ad343a","DOIUrl":null,"url":null,"abstract":"The von Neumann entropy of a <italic toggle=\"yes\">k</italic>-body-reduced density matrix <italic toggle=\"yes\">γ</italic>\n<sub>\n<italic toggle=\"yes\">k</italic>\n</sub> quantifies the entanglement between <italic toggle=\"yes\">k</italic> quantum particles and the remaining ones. In this paper, we rigorously prove general properties of this entanglement entropy as a function of <italic toggle=\"yes\">k</italic>; it is concave for all <inline-formula>\n<tex-math><?CDATA $1\\unicode{x2A7D} k\\unicode{x2A7D} N$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mn>1</mml:mn><mml:mtext>⩽</mml:mtext><mml:mi>k</mml:mi><mml:mtext>⩽</mml:mtext><mml:mi>N</mml:mi></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad343aieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> and non-decreasing until the midpoint <inline-formula>\n<tex-math><?CDATA $k\\unicode{x2A7D} \\lfloor{N/2} \\rfloor$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>k</mml:mi><mml:mtext>⩽</mml:mtext><mml:mo fence=\"false\" stretchy=\"false\">⌊</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⌋</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"jstatad343aieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. The results hold for indistinguishable quantum particles and are independent of the statistics.","PeriodicalId":17207,"journal":{"name":"Journal of Statistical Mechanics: Theory and Experiment","volume":"61 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Entropic relations for indistinguishable quantum particles\",\"authors\":\"Marius Lemm\",\"doi\":\"10.1088/1742-5468/ad343a\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The von Neumann entropy of a <italic toggle=\\\"yes\\\">k</italic>-body-reduced density matrix <italic toggle=\\\"yes\\\">γ</italic>\\n<sub>\\n<italic toggle=\\\"yes\\\">k</italic>\\n</sub> quantifies the entanglement between <italic toggle=\\\"yes\\\">k</italic> quantum particles and the remaining ones. In this paper, we rigorously prove general properties of this entanglement entropy as a function of <italic toggle=\\\"yes\\\">k</italic>; it is concave for all <inline-formula>\\n<tex-math><?CDATA $1\\\\unicode{x2A7D} k\\\\unicode{x2A7D} N$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mn>1</mml:mn><mml:mtext>⩽</mml:mtext><mml:mi>k</mml:mi><mml:mtext>⩽</mml:mtext><mml:mi>N</mml:mi></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad343aieqn1.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> and non-decreasing until the midpoint <inline-formula>\\n<tex-math><?CDATA $k\\\\unicode{x2A7D} \\\\lfloor{N/2} \\\\rfloor$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mi>k</mml:mi><mml:mtext>⩽</mml:mtext><mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">⌊</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">⌋</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"jstatad343aieqn2.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>. The results hold for indistinguishable quantum particles and are independent of the statistics.\",\"PeriodicalId\":17207,\"journal\":{\"name\":\"Journal of Statistical Mechanics: Theory and Experiment\",\"volume\":\"61 1\",\"pages\":\"\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-04-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Statistical Mechanics: Theory and Experiment\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1088/1742-5468/ad343a\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Mechanics: Theory and Experiment","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1742-5468/ad343a","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
摘要
一个 k 体还原密度矩阵 γk 的冯-诺依曼熵量化了 k 个量子粒子与其余粒子之间的纠缠。在本文中，我们严格证明了这种纠缠熵作为 k 的函数的一般性质；它对所有 1⩽k⩽N 都是凹的，并且在中点 k⩽⌊N/2⌋ 之前是不递减的。这些结果适用于不可区分的量子粒子，并且与统计量无关。
Entropic relations for indistinguishable quantum particles
The von Neumann entropy of a k-body-reduced density matrix γk quantifies the entanglement between k quantum particles and the remaining ones. In this paper, we rigorously prove general properties of this entanglement entropy as a function of k; it is concave for all 1⩽k⩽N and non-decreasing until the midpoint k⩽⌊N/2⌋. The results hold for indistinguishable quantum particles and are independent of the statistics.
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