The Cramér-Rao approach and global quantum estimation of bosonic states

Masahito Hayashi; Yingkai Ouyang

doi:10.22331/q-2025-07-22-1806

The Cramér-Rao approach and global quantum estimation of bosonic states

Masahito Hayashi^1,2,3 and Yingkai Ouyang⁴

¹School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen, 518172, China
²International Quantum AcademyFutian District, Shenzhen 518048, China
³Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
⁴School of Mathematical and Physical Sciences, University of Sheffield, Sheffield, S3 7RH, United Kingdom

Published:	2025-07-22, volume 9, page 1806
Editor:	Anurag Anshu
Eprint:	arXiv:2409.11842v5
Doi:	https://doi.org/10.22331/q-2025-07-22-1806
Citation:	Quantum 9, 1806 (2025).

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Abstract

Quantum state estimation is a fundamental task in quantum information theory, where one estimates real parameters continuously embedded in a family of quantum states. In the theory of quantum state estimation, the widely used Cramér 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 Cramér-Rao approach is applicable for global estimation instead of local estimation. In this paper, we find situations where the Cramér-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 Cramér-Rao approach to extrapolate to the global estimation setting.

► BibTeX data

@article{Hayashi2025cramerraoapproach,
  doi = {10.22331/q-2025-07-22-1806},
  url = {https://doi.org/10.22331/q-2025-07-22-1806},
  title = {The {C}ram{\'{e}}r-{R}ao approach and global quantum estimation of bosonic states},
  author = {Hayashi, Masahito and Ouyang, Yingkai},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1806},
  month = jul,
  year = {2025}
}

► References

[1] Carl Wilhelm Helstrom. ``Quantum detection and estimation theory''. Academic press. (1976).
https://doi.org/10.1007/BF01007479

[2] Alexander S Holevo. ``Probabilistic and statistical aspects of quantum theory''. Edizioni della Normale. (2011).
https://doi.org/10.1007/978-88-7642-378-9

[3] Richard D. Gill and Serge Massar. ``State estimation for large ensembles''. Phys. Rev. A 61, 042312 (2000).
https://doi.org/10.1103/PhysRevA.61.042312

[4] Yuxiang Yang, Giulio Chiribella, and Masahito Hayashi. ``Attaining the ultimate precision limit in quantum state estimation''. Communications in Mathematical Physics 368, 223–293 (2019).
https://doi.org/10.1007/s00220-019-03433-4

[5] Masahito Hayashi and Keiji Matsumoto. ``Asymptotic performance of optimal state estimation in qubit system''. Journal of Mathematical Physics 49, 102101 (2008).
https://doi.org/10.1063/1.2988130

[6] Masahito Hayashi and Yingkai Ouyang. ``Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming''. Quantum 7, 1094 (2023).
https://doi.org/10.22331/q-2023-08-29-1094

[7] Wojciech Górecki, Rafał Demkowicz-Dobrzański, Howard M Wiseman, and Dominic W Berry. ``$\pi$-corrected heisenberg limit''. Physical review letters 124, 030501 (2020).
https://doi.org/10.1103/PhysRevLett.124.030501

[8] Johannes Jakob Meyer, Sumeet Khatri, Daniel Stilck França, Jens Eisert, and Philippe Faist. ``Quantum metrology in the finite-sample regime'' (2023). arXiv:2307.06370.
arXiv:2307.06370

[9] Wojciech Górecki and Rafał Demkowicz-Dobrzański. ``Multiparameter quantum metrology in the heisenberg limit regime: Many-repetition scenario versus full optimization''. Phys. Rev. A 106, 012424 (2022).
https://doi.org/10.1103/PhysRevA.106.012424

[10] Giovanni Chesi, Alberto Riccardi, Roberto Rubboli, Lorenzo Maccone, and Chiara Macchiavello. ``Protocol for global multiphase estimation''. Phys. Rev. A 108, 012613 (2023).
https://doi.org/10.1103/PhysRevA.108.012613

[11] Michael J W Hall and Howard M Wiseman. ``Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information''. New Journal of Physics 14, 033040 (2012).
https://doi.org/10.1088/1367-2630/14/3/033040

[12] Masahito Hayashi. ``Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation''. Commun. Math. Phys. 304, 689 – 709 (2011).
https://doi.org/10.1007/s00220-011-1239-4

[13] Masahito Hayashi. ``Parallel treatment of estimation of SU(2) and phase estimation''. Physics Letters A 354, 183–189 (2006).
https://doi.org/10.1016/j.physleta.2006.01.043

[14] Shoukang Chang, Yunbin Yan, Lu Wang, Wei Ye, Xuan Rao, Huan Zhang, Liqing Huang, Mengmeng Luo, Yuetao Chen, Qiang Ma, and Shaoyan Gao. ``Global quantum thermometry based on the optimal biased bound''. Phys. Rev. Res. 6, 043171 (2024).
https://doi.org/10.1103/PhysRevResearch.6.043171

[15] Jesús Rubio, Janet Anders, and Luis A. Correa. ``Global quantum thermometry''. Phys. Rev. Lett. 127, 190402 (2021).
https://doi.org/10.1103/PhysRevLett.127.190402

[16] Masahito Hayashi. ``Quantum information theory: Mathematical foundation''. Springer. (2016).
https://doi.org/10.1007/978-3-662-49725-8

[17] Hiroshi Nagaoka. ``On fisher information of quantum statistical models''. In Masahito Hayashi, editor, Asymptotic Theory Of Quantum Statistical Inference: Selected Papers. Pages 113–124. World Scientific (2005).
https://doi.org/10.1142/9789812563071_0011

[18] Hiroshi Nagaoka. ``On the parameter estimation problem for quantum statistical models''. In Masahito Hayashi, editor, Asymptotic Theory Of Quantum Statistical Inference: Selected Papers. Pages 125–132. World Scientific (2005).
https://doi.org/10.1142/9789812563071_0011

[19] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum-enhanced measurements: Beating the standard quantum limit''. Science 306, 1330–1336 (2004).
https://doi.org/10.1126/science.1104149

[20] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum metrology''. Phys. Rev. Lett. 96, 010401 (2006).
https://doi.org/10.1103/PhysRevLett.96.010401

[21] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Advances in quantum metrology''. Nature Photonics 5, 222–229 (2011). url: https://doi.org/10.1038/nphoton.2011.35.
https://doi.org/10.1038/nphoton.2011.35

[22] Nicholas Thomas-Peter, Brian J. Smith, Animesh Datta, Lijian Zhang, Uwe Dorner, and Ian A. Walmsley. ``Real-world quantum sensors: Evaluating resources for precision measurement''. Phys. Rev. Lett. 107, 113603 (2011).
https://doi.org/10.1103/PhysRevLett.107.113603

[23] Jonathan A. Jones, Steven D. Karlen, Joseph Fitzsimons, Arzhang Ardavan, Simon C. Benjamin, G. Andrew D. Briggs, and John J. L. Morton. ``Magnetic field sensing beyond the standard quantum limit using 10-spin noon states''. Science 324, 1166–1168 (2009).
https://doi.org/10.1126/science.1170730

[24] Ryo Okamoto, Holger F Hofmann, Tomohisa Nagata, Jeremy L O'Brien, Keiji Sasaki, and Shigeki Takeuchi. ``Beating the standard quantum limit: phase super-sensitivity of n-photon interferometers''. New Journal of Physics 10, 073033 (2008).
https://doi.org/10.1088/1367-2630/10/7/073033

[25] Tomohisa Nagata, Ryo Okamoto, Jeremy L. O'Brien, Keiji Sasaki, and Shigeki Takeuchi. ``Beating the standard quantum limit with four-entangled photons''. Science 316, 726–729 (2007).
https://doi.org/10.1126/science.1138007

[26] Masahito Hayashi, Sai Vinjanampathy, and LC Kwek. ``Resolving unattainable cramer–rao bounds for quantum sensors''. Journal of Physics B: Atomic, Molecular and Optical Physics 52, 015503 (2018).
https://doi.org/10.1088/1361-6455/aaf348

[27] L. Paninski. ``Estimating entropy on m bins given fewer than m samples''. IEEE Transactions on Information Theory 50, 2200–2203 (2004).
https://doi.org/10.1109/TIT.2004.833360

[28] Gregory Valiant and Paul Valiant. ``Estimating the unseen: an n/log (n)-sample estimator for entropy and support size, shown optimal via new clts''. In Proceedings of the forty-third annual ACM symposium on Theory of computing. Pages 685–694. (2011).
https://doi.org/10.1145/1993636.1993727

[29] Jayadev Acharya, Alon Orlitsky, Ananda Theertha Suresh, and Himanshu Tyagi. ``The complexity of estimating rényi entropy''. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms. Pages 1855–1869. SIAM (2014).
https://doi.org/10.1137/1.9781611973730.124

[30] Jayadev Acharya, Ibrahim Issa, Nirmal V. Shende, and Aaron B. Wagner. ``Estimating quantum entropy''. IEEE Journal on Selected Areas in Information Theory 1, 454–468 (2020).
https://doi.org/10.1109/JSAIT.2020.3015235

[31] Qisheng Wang and Zhicheng Zhang. ``Time-Efficient Quantum Entropy Estimator via Samplizer''. In Timothy Chan, Johannes Fischer, John Iacono, and Grzegorz Herman, editors, 32nd Annual European Symposium on Algorithms (ESA 2024). Volume 308 of Leibniz International Proceedings in Informatics (LIPIcs), pages 101:1–101:15. Dagstuhl, Germany (2024). Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.ESA.2024.101

[32] Mankei Tsang, Francesco Albarelli, and Animesh Datta. ``Quantum semiparametric estimation''. Phys. Rev. X 10, 031023 (2020).
https://doi.org/10.1103/PhysRevX.10.031023

[33] Masahito Hayashi. ``Measuring quantum relative entropy with finite-size effect''. Quantum 9, 1725 (2025).
https://doi.org/10.22331/q-2025-05-05-1725

[34] Masahito Hayashi. ``Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing''. Journal of Physics A: Mathematical and General 35, 10759 (2002).
https://doi.org/10.1088/0305-4470/35/50/307

[35] Masahito Hayashi. ``Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation''. Commun. Math. Phys. 304, 689 – 709 (2011).
https://doi.org/10.1007/s00220-011-1239-4

[36] Mohammad Mehboudi, Mathias R. Jørgensen, Stella Seah, Jonatan B. Brask, Jan Kołodyński, and Martí Perarnau-Llobet. ``Fundamental limits in bayesian thermometry and attainability via adaptive strategies''. Phys. Rev. Lett. 128, 130502 (2022).
https://doi.org/10.1103/PhysRevLett.128.130502

[37] Masahito Hayashi and Shun Watanabe. ``Information geometry approach to parameter estimation in markov chains''. The Annals of Statistics 44, 1495–1535 (2016).
https://doi.org/10.1214/15-AOS1420

[38] Masahito Hayashi. ``Information geometry approach to parameter estimation in hidden Markov model''. Bernoulli 28, 307–342 (2022).
https://doi.org/10.3150/21-BEJ1344

[39] Mankei Tsang. ``Ziv-zakai error bounds for quantum parameter estimation''. Phys. Rev. Lett. 108, 230401 (2012).
https://doi.org/10.1103/PhysRevLett.108.230401

[40] Jesús Rubio, Paul Knott, and Jacob Dunningham. ``Non-asymptotic analysis of quantum metrology protocols beyond the cramér-rao bound''. Journal of Physics Communications 2, 015027 (2018).
https://doi.org/10.1088/2399-6528/aaa234

[41] S L Braunstein. ``How large a sample is needed for the maximum likelihood estimator to be approximately gaussian?''. Journal of Physics A: Mathematical and General 25, 3813 (1992).
https://doi.org/10.1088/0305-4470/25/13/027

[42] Zhao-Yi Zhou, Jing-Tao Qiu, and Da-Jian Zhang. ``Strict hierarchy of optimal strategies for global estimations: Linking global estimations with local ones''. Phys. Rev. Res. 6, L032048 (2024).
https://doi.org/10.1103/PhysRevResearch.6.L032048

[43] Julia Boeyens, Stella Seah, and Stefan Nimmrichter. ``Uninformed bayesian quantum thermometry''. Phys. Rev. A 104, 052214 (2021).
https://doi.org/10.1103/PhysRevA.104.052214

[44] Hiroshi Nagaoka and Akio Fujiwara. ``Autoparallelity of quantum statistical manifolds in the light of quantum estimation theory'' (2023). arXiv:2307.03431.
arXiv:2307.03431

[45] James R Anglin and Wolfgang Ketterle. ``Bose-Einstein condensation of atomic gases''. Nature 416, 211–218 (2002).
https://doi.org/10.1038/416211a

[46] Mattias T. Johnsson, Nabomita Roy Mukty, Daniel Burgarth, Thomas Volz, and Gavin K. Brennen. ``Geometric pathway to scalable quantum sensing''. Phys. Rev. Lett. 125, 190403 (2020).
https://doi.org/10.1103/PhysRevLett.125.190403

[47] Yingkai Ouyang and Gavin K Brennen. ``Finite-round quantum error correction on symmetric quantum sensors'' (2022). arXiv:arxiv:2212.06285.
arXiv:2212.06285

[48] Philipp Hyllus, Wiesław Laskowski, Roland Krischek, Christian Schwemmer, Witlef Wieczorek, Harald Weinfurter, Luca Pezzé, and Augusto Smerzi. ``Fisher information and multiparticle entanglement''. Phys. Rev. A 85, 022321 (2012).
https://doi.org/10.1103/PhysRevA.85.022321

[49] Géza Tóth. ``Multipartite entanglement and high-precision metrology''. Phys. Rev. A 85, 022322 (2012).
https://doi.org/10.1103/PhysRevA.85.022322

[50] Bernd Lücke, Manuel Scherer, Jens Kruse, Luca Pezzé, Frank Deuretzbacher, Phillip Hyllus, Oliver Topic, Jan Peise, Wolfgang Ertmer, Jan Arlt, et al. ``Twin matter waves for interferometry beyond the classical limit''. Science 334, 773–776 (2011).
https://doi.org/10.1126/science.1208798

[51] Yi-Quan Zou, Ling-Na Wu, Qi Liu, Xin-Yu Luo, Shuai-Feng Guo, Jia-Hao Cao, Meng Khoon Tey, and Li You. ``Beating the classical precision limit with spin-1 Dicke states of more than 10,000 atoms''. Proceedings of the National Academy of Sciences 115, 6381–6385 (2018).
https://doi.org/10.1073/pnas.1715105115

[52] G.M D'Ariano, C Macchiavello, and M.F Sacchi. ``On the general problem of quantum phase estimation''. Physics Letters A 248, 103–108 (1998).
https://doi.org/10.1016/S0375-9601(98)00702-6

[53] Chiara Macchiavello. ``Optimal estimation of multiple phases''. Phys. Rev. A 67, 062302 (2003).
https://doi.org/10.1103/PhysRevA.67.062302

[54] Valentin Gebhart, Augusto Smerzi, and Luca Pezzè. ``Bayesian quantum multiphase estimation algorithm''. Phys. Rev. Appl. 16, 014035 (2021).
https://doi.org/10.1103/PhysRevApplied.16.014035

[55] Wojciech Górecki and Rafał Demkowicz-Dobrzański. ``Multiple-phase quantum interferometry: Real and apparent gains of measuring all the phases simultaneously''. Phys. Rev. Lett. 128, 040504 (2022).
https://doi.org/10.1103/PhysRevLett.128.040504

[56] Masahito Hayashi, Zi-Wen Liu, and Haidong Yuan. ``Global heisenberg scaling in noisy and practical phase estimation''. Quantum Science and Technology 7, 025030 (2022).
https://doi.org/10.1088/2058-9565/ac5d7e

[57] A. Luis and J. Peřina. ``Optimum phase-shift estimation and the quantum description of the phase difference''. Phys. Rev. A 54, 4564–4570 (1996).
https://doi.org/10.1103/PhysRevA.54.4564

[58] V. Bužek, R. Derka, and S. Massar. ``Optimal quantum clocks''. Phys. Rev. Lett. 82, 2207–2210 (1999).
https://doi.org/10.1103/PhysRevLett.82.2207

[59] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde. ``Entanglement-free heisenberg-limited phase estimation''. Nature 450, 393–396 (2007).
https://doi.org/10.1038/nature06257

[60] Alexei Y. Kitaev. ``Quantum measurements and the abelian stabilizer problem''. Electron. Colloquium Comput. Complex. TR96-003 (1996). arXiv:TR96-003.
arXiv:quant-ph/9511026

[61] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. ``Quantum algorithms revisited''. Proc. R. Soc. Lond. A. 454, 339–354 (1998). url: http://doi.org/10.1098/rspa.1998.0164.
https://doi.org/10.1098/rspa.1998.0164

[62] Hiroshi Imai and Masahito Hayashi. ``Fourier analytic approach to phase estimation in quantum systems''. New Journal of Physics 11, 043034 (2009).
https://doi.org/10.1088/1367-2630/11/4/043034

[63] A.S. Holevo. ``Covariant measurements and uncertainty relations''. Reports on Mathematical Physics 16, 385–400 (1979).
https://doi.org/10.1016/0034-4877(79)90072-7

[64] Masahito Hayashi. ``Group representation for quantum theory''. Springer. (2017).
https://doi.org/10.1007/978-3-319-44906-7

[65] Yingkai Ouyang and Rui Chao. ``Permutation-invariant constant-excitation quantum codes for amplitude damping''. IEEE Transactions on Information Theory 66, 2921–2933 (2019).
https://doi.org/10.1109/TIT.2019.2956142

[66] Manuel Erhard, Mario Krenn, and Anton Zeilinger. ``Advances in high-dimensional quantum entanglement''. Nature Reviews Physics 2, 365–381 (2020).
https://doi.org/10.1038/s42254-020-0193-5

[67] Alfredo Luis. ``Quantum mechanics as a geometric phase: phase-space interferometers''. Journal of Physics A: Mathematical and General 34, 7677 (2001).
https://doi.org/10.1088/0305-4470/34/37/317

[68] Xiaoguang Wang and Paolo Zanardi. ``Simulation of many-body interactions by conditional geometric phases''. Phys. Rev. A 65, 032327 (2002).
https://doi.org/10.1103/PhysRevA.65.032327

[69] Masahito Hayashi. ``A linear programming approach to attainable cramér-rao type bounds and randomness condition'' (1997). arXiv:quant-ph/9704044.
arXiv:quant-ph/9704044

[70] Hiroshi Nagaoka. ``A new approach to Cramér-Rao bounds for quantum state estimation''. In Masahito Hayashi, editor, Asymptotic Theory Of Quantum Statistical Inference: Selected Papers. Pages 100–112. World Scientific (2005).
https://doi.org/10.1142/9789812563071_0009

[71] Shun'ichi Amari and Hiroshi Nagaoka. ``Methods of information geometry''. Volume 191. American Mathematical Soc. (2000).
https://doi.org/10.1007/978-4-431-55978-8

[72] Masahito Hayashi. ``Fourier analytic approach to quantum estimation of group action''. Communications in Mathematical Physics 347, 3–82 (2016).
https://doi.org/10.1007/s00220-016-2738-0

[73] Akio Fujiwara. ``Estimation of SU(2) operation and dense coding: An information geometric approach''. Phys. Rev. A 65, 012316 (2001).
https://doi.org/10.1103/PhysRevA.65.012316

[74] Hiroshi Imai and Akio Fujiwara. ``Geometry of optimal estimation scheme for SU(D) channels''. Journal of Physics A: Mathematical and Theoretical 40, 4391 (2007).
https://doi.org/10.1088/1751-8113/40/16/009

[75] Masahito Hayashi. ``Phase estimation with photon number constraint''. Progress in InformaticsPages 81–87 (2011).
https://doi.org/10.2201/NiiPi.2011.8.9

[76] Jonas Kahn and Mădălin Guţă. ``Local asymptotic normality for finite dimensional quantum systems''. Communications in Mathematical Physics 289, 597–652 (2009).
https://doi.org/10.1007/s00220-009-0787-3

[77] Jonas Kahn. ``Quantum local asymptotic normality and other questions of quantum statistics''. Leiden University. (2008). url: https://hdl.handle.net/1887/12956.
https://hdl.handle.net/1887/12956

[78] Jae-Gyun Baak and Uwe R. Fischer. ``Self-consistent many-body metrology''. Phys. Rev. Lett. 132, 240803 (2024).
https://doi.org/10.1103/PhysRevLett.132.240803

[79] G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter. ``Permutationally invariant quantum tomography''. Phys. Rev. Lett. 105, 250403 (2010).
https://doi.org/10.1103/PhysRevLett.105.250403

[80] Tobias Moroder, Philipp Hyllus, Géza Tóth, Christian Schwemmer, Alexander Niggebaum, Stefanie Gaile, Otfried Gühne, and Harald Weinfurter. ``Permutationally invariant state reconstruction''. New Journal of Physics 14, 105001 (2012).
https://doi.org/10.1088/1367-2630/14/10/105001

[81] Christian Schwemmer, Géza Tóth, Alexander Niggebaum, Tobias Moroder, David Gross, Otfried Gühne, and Harald Weinfurter. ``Experimental comparison of efficient tomography schemes for a six-qubit state''. Phys. Rev. Lett. 113, 040503 (2014).
https://doi.org/10.1103/PhysRevLett.113.040503

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