{"title":"量子费雪信息的充分统计性和可恢复性","authors":"Li Gao, Haojian Li, Iman Marvian, Cambyse Rouzé","doi":"10.1007/s00220-024-05053-z","DOIUrl":null,"url":null,"abstract":"<p>We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output by some quantum operation, if and only if, the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger–Yanase–Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analogs of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying monotone metrics on the quantum state space, i.e. Riemannian metrics non-increasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum <span>\\(\\chi ^2\\)</span> divergence. We obtain an approximate recovery result in the sense that, if the quantum <span>\\(\\chi ^2\\)</span> divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the <span>\\(\\chi _{\\frac{1}{2}}\\)</span> divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.\n</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sufficient Statistic and Recoverability via Quantum Fisher Information\",\"authors\":\"Li Gao, Haojian Li, Iman Marvian, Cambyse Rouzé\",\"doi\":\"10.1007/s00220-024-05053-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output by some quantum operation, if and only if, the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger–Yanase–Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analogs of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying monotone metrics on the quantum state space, i.e. Riemannian metrics non-increasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum <span>\\\\(\\\\chi ^2\\\\)</span> divergence. We obtain an approximate recovery result in the sense that, if the quantum <span>\\\\(\\\\chi ^2\\\\)</span> divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the <span>\\\\(\\\\chi _{\\\\frac{1}{2}}\\\\)</span> divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.\\n</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-07-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s00220-024-05053-z\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-05053-z","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Sufficient Statistic and Recoverability via Quantum Fisher Information
We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output by some quantum operation, if and only if, the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger–Yanase–Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analogs of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying monotone metrics on the quantum state space, i.e. Riemannian metrics non-increasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum \(\chi ^2\) divergence. We obtain an approximate recovery result in the sense that, if the quantum \(\chi ^2\) divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the \(\chi _{\frac{1}{2}}\) divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.