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In the theory of quantum state estimation, the widely used Cram\u00e9r Rao approach which considers local estimation gives the ultimate precision bound of quantum state estimation in terms of the quantum Fisher information. However practical scenarios need not offer much prior information about the parameters to be estimated, and the local estimation setting need not apply. In general, it is unclear whether the Cram\u00e9r-Rao approach is applicable for global estimation instead of local estimation. In this paper, we find situations where the Cram\u00e9r-Rao approach does and does not work for quantum state estimation problems involving a family of bosonic states in a non-IID setting, where we only use one copy of the bosonic quantum state in the large number of bosons setting. Our result highlights the importance of caution when using the results of the Cram\u00e9r-Rao approach to extrapolate to the global estimation setting.<\/jats:p>","DOI":"10.22331\/q-2025-07-22-1806","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T09:49:25Z","timestamp":1753177765000},"page":"1806","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["The Cram\u00e9r-Rao approach and global quantum estimation of bosonic states"],"prefix":"10.22331","volume":"9","author":[{"given":"Masahito","family":"Hayashi","sequence":"first","affiliation":[{"name":"School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen, 518172, China"},{"name":"International Quantum AcademyFutian District, Shenzhen 518048, China"},{"name":"Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingkai","family":"Ouyang","sequence":"additional","affiliation":[{"name":"School of Mathematical and Physical Sciences, University of Sheffield, Sheffield, S3 7RH, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Carl Wilhelm Helstrom. ``Quantum detection and estimation theory''. 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