Measurement incompatibility in Bayesian multiparameter quantum estimation

Francesco Albarelli; Dominic Branford; Jesús Rubio

doi:10.22331/q-2026-08-13-2192

Measurement incompatibility in Bayesian multiparameter quantum estimation

Francesco Albarelli^1,2, Dominic Branford³, and Jesús Rubio^4,5

¹Università di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy
²INFN—Sezione di Milano-Bicocca, gruppo collegato di Parma, I-43124 Parma, Italy
³Dipartimento di Fisica e Astronomia, Università di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
⁴School of Mathematics and Physics, University of Surrey, Guildford GU2 7XH, United Kingdom
⁵Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom

Published:	2026-08-13, volume 10, page 2192
Editor:	Christos Gagatsos
Eprint:	arXiv:2511.16645v4
Doi:	https://doi.org/10.22331/q-2026-08-13-2192
Citation:	Quantum 10, 2192 (2026).

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Abstract

We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation. Within this framework, we analyse the role of measurement incompatibility and establish its quantitative effect on attainable precision. We achieve this by deriving upper bounds based on the pretty good measurement – a notion from hypothesis testing – combined with the evaluation of the Nagaoka-Hayashi lower bound. In general, we prove that, as in the many-copy regime of local estimation theory, incompatibility can at most double the minimum loss relative to the idealised scenario in which individually optimal measurements are assumed jointly implementable. Therefore, in practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through applications of discrete phase imaging, phase and dephasing estimation, and qubit sensing, provide analytical and numerical tools for assessing ultimate precision limits and the role of measurement incompatibility in Bayesian multiparameter quantum metrology, including an open-source package for all the bounds discussed here.

Popular summary

Quantum technologies can enhance the precision with which physical parameters are measured. However, when several parameters are estimated simultaneously, the measurements that are individually optimal for different parameters may be incompatible—impossible to implement simultaneously. Understanding the resulting loss of precision is a central problem in quantum metrology.

Most previous results address this question in a local regime, where their validity is often contingent on the parameters already being approximately known. Here, we instead study measurement incompatibility within Bayesian quantum estimation, a global framework that explicitly incorporates prior knowledge and is particularly relevant when experimental data are limited.

We show that the effect of measurement incompatibility is fundamentally limited: the minimum estimation loss can be at most twice the idealised benchmark obtained by assuming that all individually optimal measurements can be simultaneously implemented. We also demonstrate that prior information can conceal the practical effect of incompatibility.

To practically narrow down the range of this type of incompatibility, we explore complementary upper and lower bounds on the achievable estimation precision. We illustrate the framework with quantum phase imaging, simultaneous phase and dephasing estimation, and qubit sensing (for which we also derive exact attainable bounds), and provide open-source numerical tools for evaluating the bounds.

Overall, our results lay the groundwork for determining when measurement incompatibility is a genuine concern in Bayesian quantum sensing.

► BibTeX data

@article{Albarelli2026measurement,
  doi = {10.22331/q-2026-08-13-2192},
  url = {https://doi.org/10.22331/q-2026-08-13-2192},
  title = {Measurement incompatibility in {B}ayesian multiparameter quantum estimation},
  author = {Albarelli, Francesco and Branford, Dominic and Rubio, Jes{\'{u}}s},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2192},
  month = aug,
  year = {2026}
}

► References

[1] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011), arXiv:1102.2318.
https://doi.org/10.1038/nphoton.2011.35
arXiv:1102.2318

[2] E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, Photonic quantum metrology, AVS Quantum Sci. 2, 024703 (2020), arXiv:2003.05821.
https://doi.org/10.1116/5.0007577
arXiv:2003.05821

[3] J. Liu, M. Zhang, H. Chen, L. Wang, and H. Yuan, Optimal Scheme for Quantum Metrology, Adv. Quantum Technol. 5, 2100080 (2022), arXiv:2111.12279.
https://doi.org/10.1002/qute.202100080
arXiv:2111.12279

[4] L. Jiao, W. Wu, S.-Y. Bai, and J.-H. An, Quantum Metrology in the Noisy Intermediate-Scale Quantum Era, Adv. Quantum Techol. 8, 2300218 (2025), arXiv:2307.07701.
https://doi.org/10.1002/qute.202300218
arXiv:2307.07701

[5] V. Montenegro, C. Mukhopadhyay, R. Yousefjani, S. Sarkar, U. Mishra, M. G. A. Paris, and A. Bayat, Review: Quantum metrology and sensing with many-body systems, Phys. Rep. 1134, 1 (2025), arXiv:2408.15323.
https://doi.org/10.1016/j.physrep.2025.05.005
arXiv:2408.15323

[6] R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, Quantum Limits in Optical Interferometry, in Progress in Optics, Volume 60, edited by E. Wolf (Elsevier, Amsterdam, 2015) pp. 345–435, arXiv:1405.7703.
https://doi.org/10.1016/bs.po.2015.02.003
arXiv:1405.7703

[7] J. Sahota and N. Quesada, Quantum correlations in optical metrology: Heisenberg-limited phase estimation without mode entanglement, Phys. Rev. A 91, 013808 (2015), arXiv:1404.7110.
https://doi.org/10.1103/PhysRevA.91.013808
arXiv:1404.7110

[8] N. Friis, D. Orsucci, M. Skotiniotis, P. Sekatski, V. Dunjko, H. J. Briegel, and W. Dür, Flexible resources for quantum metrology, New J. Phys. 19, 063044 (2017), arXiv:1610.09999.
https://doi.org/10.1088/1367-2630/aa7144
arXiv:1610.09999

[9] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

[10] R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, Multi-parameter estimation beyond quantum Fisher information, J. Phys. A 53, 363001 (2020), arXiv:2001.11742.
https://doi.org/10.1088/1751-8121/ab8ef3
arXiv:2001.11742

[11] W. Górecki and R. Demkowicz-Dobrzański, Multiparameter quantum metrology in the Heisenberg limit regime: Many-repetition scenario versus full optimization, Phys. Rev. A 106, 012424 (2022), arXiv:2203.09541.
https://doi.org/10.1103/PhysRevA.106.012424
arXiv:2203.09541

[12] M. Tsang, Ziv-Zakai Error Bounds for Quantum Parameter Estimation, Phys. Rev. Lett. 108, 230401 (2012), arXiv:1111.3568.
https://doi.org/10.1103/PhysRevLett.108.230401
arXiv:1111.3568

[13] J. Rubio and J. Dunningham, Quantum metrology in the presence of limited data, New J. Phys. 21, 043037 (2019), arXiv:1810.12857.
https://doi.org/10.1088/1367-2630/ab098b
arXiv:1810.12857

[14] M. Valeri, E. Polino, D. Poderini, I. Gianani, G. Corrielli, A. Crespi, T. Osellame, N. Spagnolo, and F. Sciarrino, Experimental adaptive Bayesian estimation of multiple phases with limited data, npj Quantum Inf. 6, 92 (2020), arXiv:2002.01232.
https://doi.org/10.1038/s41534-020-00326-6
arXiv:2002.01232

[15] M. Gessner and A. Smerzi, Hierarchies of Frequentist Bounds for Quantum Metrology: From Cramér-Rao to Barankin, Phys. Rev. Lett. 130, 260801 (2023), arXiv:2303.06108.
https://doi.org/10.1103/PhysRevLett.130.260801
arXiv:2303.06108

[16] M. Hassani, C. Macchiavello, and L. Maccone, Digital Quantum Estimation, Phys. Rev. Lett. 119, 200502 (2017), arXiv:1705.06666.
https://doi.org/10.1103/PhysRevLett.119.200502
arXiv:1705.06666

[17] V. Gebhart, M. Gessner, and A. Smerzi, Fundamental bounds for parameter estimation with few measurements, Phys. Rev. Res. 6, 043261 (2024), arXiv:2402.14495.
https://doi.org/10.1103/PhysRevResearch.6.043261
arXiv:2402.14495

[18] J. R. Hervas, A. Z. Goldberg, A. S. Sanz, Z. Hradil, J. Řeháček, and L. L. Sánchez-Soto, Beyond the Quantum Cramér-Rao Bound, Phys. Rev. Lett. 134, 010804 (2025a).
https://doi.org/10.1103/PhysRevLett.134.010804

[19] J. R. Hervas, A. Z. Goldberg, A. S. Sanz, Z. Hradil, J. Řeháček, and L. L. Sánchez-Soto, Higher-order corrections to the quantum Cramér-Rao bound, Phys. Rev. A 112, 022426 (2025b).
https://doi.org/10.1103/ltz3-163t

[20] J. J. Meyer, S. Khatri, D. Stilck França, J. Eisert, and P. Faist, Quantum Metrology in the Finite-Sample Regime, PRX Quantum 6, 030336 (2025), arXiv:2307.06370.
https://doi.org/10.1103/qbn1-p6bq
arXiv:2307.06370

[21] C. Mukhopadhyay, V. Montenegro, and A. Bayat, Current trends in global quantum metrology, J. Phys. A 58, 063001 (2025), arXiv:2411.03850.
https://doi.org/10.1088/1751-8121/adb112
arXiv:2411.03850

[22] A. M. Polloreno, J. L. Beckey, J. Levin, A. Shlosberg, J. K. Thompson, M. Foss-Feig, D. Hayes, and G. Smith, Opportunities and Limitations in Broadband Sensing, Phys. Rev. Appl. 19, 014029 (2023), arXiv:2203.05520.
https://doi.org/10.1103/PhysRevApplied.19.014029
arXiv:2203.05520

[23] R. R. Allen, F. Machado, I. L. Chuang, H.-Y. Huang, and S. Choi, Quantum Computing Enhanced Sensing (2025), arXiv:2501.07625.
https://doi.org/10.48550/arXiv.2501.07625
arXiv:2501.07625

[24] S. Personick, Efficient analog communication over quantum channels, Ph.D. thesis, Dep. Elec. Eng., M.I.T., Cambridge, Mass. (1969).

[25] S. Personick, Application of quantum estimation theory to analog communication over quantum channels, IEEE Trans. Inf. Theory 17, 240 (1971).
https://doi.org/10.1109/TIT.1971.1054643

[26] A. S. Holevo, Statistical decision theory for quantum systems, J. Multivar. Anal. 3, 337 (1973a).
https://doi.org/10.1016/0047-259X(73)90028-6

[27] A. S. Holevo, Statistical problems in quantum physics, in Proc. Second Jpn.-USSR Symp. Probab. Theory, edited by G. Maruyama and Yu. V. Prokhorov (Springer, Berlin, Heidelberg, 1973) pp. 104–119.
https://doi.org/10.1007/BFb0061483

[28] H. Yuen and M. Lax, Multiple-parameter quantum estimation and measurement of nonselfadjoint observables, IEEE Trans. Inf. Theory 19, 740 (1973).
https://doi.org/10.1109/TIT.1973.1055103

[29] C. Helstrom and R. Kennedy, Noncommuting observables in quantum detection and estimation theory, IEEE Trans. Inf. Theory 20, 16 (1974).
https://doi.org/10.1109/TIT.1974.1055173

[30] S. E. D'Aurelio, M. Valeri, E. Polino, V. Cimini, I. Gianani, M. Barbieri, G. Corrielli, A. Crespi, R. Osellame, F. Sciarrino, and N. Spagnolo, Experimental investigation of Bayesian bounds in multiparameter estimation, Quantum Sci. Technol. 7, 025011 (2022), arXiv:2304.06405.
https://doi.org/10.1088/2058-9565/ac5124
arXiv:2304.06405

[31] M. Valeri, V. Cimini, S. Piacentini, F. Ceccarelli, E. Polino, F. Hoch, G. Bizzarri, G. Corrielli, N. Spagnolo, R. Osellame, and F. Sciarrino, Experimental multiparameter quantum metrology in adaptive regime, Phys. Rev. Res. 5, 013138 (2023), arXiv:2208.14473.
https://doi.org/10.1103/PhysRevResearch.5.013138
arXiv:2208.14473

[32] V. Cimini, E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, Benchmarking Bayesian quantum estimation, Quantum Sci. Technol. 9, 035035 (2024a), arXiv:2401.14900.
https://doi.org/10.1088/2058-9565/ad48b3
arXiv:2401.14900

[33] V. Cimini, M. Valeri, S. Piacentini, F. Ceccarelli, G. Corrielli, R. Osellame, N. Spagnolo, and F. Sciarrino, Variational quantum algorithm for experimental photonic multiparameter estimation, npj Quantum Inf. 10, 26 (2024b), arXiv:2308.02643.
https://doi.org/10.1038/s41534-024-00821-0
arXiv:2308.02643

[34] F. Belliardo, V. Cimini, E. Polino, F. Hoch, B. Piccirillo, N. Spagnolo, V. Giovannetti, and F. Sciarrino, Optimizing quantum-enhanced Bayesian multiparameter estimation of phase and noise in practical sensors, Phys. Rev. Res. 6, 023201 (2024), arXiv:2211.04747.
https://doi.org/10.1103/PhysRevResearch.6.023201
arXiv:2211.04747

[35] G. Minati, E. Urbani, N. Spagnolo, V. Cimini, and F. Sciarrino, Multiparameter quantum-enhanced adaptive metrology with squeezed light, Nat. Commun. 17, 6418 (2026), arXiv:2510.14739.
https://doi.org/10.1038/s41467-026-73069-1
arXiv:2510.14739

[36] J. Glatthard, J. Rubio, R. Sawant, T. Hewitt, G. Barontini, and L. A. Correa, Optimal cold atom thermometry using adaptive bayesian strategies, PRX Quantum 3, 040330 (2022), arXiv:2204.11816.
https://doi.org/10.1103/PRXQuantum.3.040330
arXiv:2204.11816

[37] M. Overton, J. Rubio, N. Cooper, D. Baldolini, D. Johnson, J. Anders, and L. Hackermüller, Adaptive, symmetry-informed bayesian metrology for precise quantum technology measurements, Phys. Rev. Lett. 136, 140801 (2026), arXiv:2410.10615.
https://doi.org/10.1103/rq95-zq9r
arXiv:2410.10615

[38] H. Espinós, O. Dhungel, A. Wickenbrock, D. Budker, R. Puebla, and E. Torrontegui, Microwave-free vector magnetometry and crystal orientation determination with Nitrogen-Vacancy centers using Bayesian inference (2025), arXiv:2512.13835.
https://doi.org/10.48550/arXiv.2512.13835
arXiv:2512.13835

[39] A. Rivas and A. Luis, Sub-Heisenberg estimation of non-random phase shifts, New J. Phys. 14, 093052 (2012), arXiv:1105.6310.
https://doi.org/10.1088/1367-2630/14/9/093052
arXiv:1105.6310

[40] L. Pezzè, Sub-Heisenberg phase uncertainties, Phys. Rev. A 88, 060101 (2013).
https://doi.org/10.1103/PhysRevA.88.060101

[41] P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Quantum Enhanced Multiple Phase Estimation, Phys. Rev. Lett. 111, 070403 (2013), arXiv:1307.7653.
https://doi.org/10.1103/PhysRevLett.111.070403
arXiv:1307.7653

[42] P. A. Knott, T. J. Proctor, A. J. Hayes, J. F. Ralph, P. Kok, and J. A. Dunningham, Local versus global strategies in multiparameter estimation, Phys. Rev. A 94, 062312 (2016), arXiv:1601.05912.
https://doi.org/10.1103/PhysRevA.94.062312
arXiv:1601.05912

[43] J. Rubio and J. Dunningham, Bayesian multiparameter quantum metrology with limited data, Phys. Rev. A 101, 032114 (2020), arXiv:1906.04123.
https://doi.org/10.1103/PhysRevA.101.032114
arXiv:1906.04123

[44] W. Górecki and R. Demkowicz-Dobrzański, Multiple-Phase Quantum Interferometry: Real and Apparent Gains of Measuring All the Phases Simultaneously, Phys. Rev. Lett. 128, 040504 (2022), arXiv:2107.10863.
https://doi.org/10.1103/PhysRevLett.128.040504
arXiv:2107.10863

[45] W. Górecki, R. Demkowicz-Dobrzański, H. M. Wiseman, and D. W. Berry, ${\pi}$-Corrected Heisenberg Limit, Phys. Rev. Lett. 124, 030501 (2020), arXiv:1907.05428.
https://doi.org/10.1103/PhysRevLett.124.030501
arXiv:1907.05428

[46] R. Ravell Rodríguez and S. Morelli, Knowledge-dependent optimal Gaussian strategies for phase estimation, SciPost Phys. 18, 167 (2025), arXiv:2412.16023.
https://doi.org/10.21468/SciPostPhys.18.5.167
arXiv:2412.16023

[47] E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, 2003).
https://doi.org/10.1017/CBO9780511790423

[48] V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezzè, and C. Bonato, Learning quantum systems, Nat. Rev. Phys. 5, 141 (2023), arXiv:2207.00298.
https://doi.org/10.1038/s42254-022-00552-1
arXiv:2207.00298

[49] Y. Berquin, Bayesian statistical learning using density operators (2023), arXiv:2212.14715.
https://doi.org/10.48550/arXiv.2212.14715
arXiv:2212.14715

[50] L. Banchi, J. L. Pereira, S. T. Jose, and O. Simeone, Statistical Complexity of Quantum Learning, Adv. Quantum Technol. 8, 2300311 (2024), arXiv:2309.11617.
https://doi.org/10.1002/qute.202300311
arXiv:2309.11617

[51] S. Chen, C. Oh, S. Zhou, H.-Y. Huang, and L. Jiang, Tight Bounds on Pauli Channel Learning without Entanglement, Phys. Rev. Lett. 132, 180805 (2024a), arXiv:2309.13461.
https://doi.org/10.1103/PhysRevLett.132.180805
arXiv:2309.13461

[52] C. Oh, S. Chen, Y. Wong, S. Zhou, H.-Y. Huang, J. A. H. Nielsen, Z.-H. Liu, J. S. Neergaard-Nielsen, U. L. Andersen, L. Jiang, and J. Preskill, Entanglement-Enabled Advantage for Learning a Bosonic Random Displacement Channel, Phys. Rev. Lett. 133, 230604 (2024), arXiv:2402.18809.
https://doi.org/10.1103/PhysRevLett.133.230604
arXiv:2402.18809

[53] A. Dutt, Accelerating Learning of Quantum Systems Using Prior Information, Ph.D thesis, Massachusetts Institute of Technology (2024).
https://dspace.mit.edu/handle/1721.1/153668

[54] Y. Wang, C. Oh, J. Liu, L. Jiang, and S. Zhou, Advancing quantum imaging through learning theory, Nat Commun 17, 1124 (2025), arXiv:2501.15685.
https://doi.org/10.1038/s41467-025-67884-1
arXiv:2501.15685

[55] H.-Y. Hu, M. Ma, W. Gong, Q. Ye, Y. Tong, S. T. Flammia, and S. F. Yelin, Ansatz-Free Hamiltonian Learning with Heisenberg-Limited Scaling, PRX Quantum 6, 040315 (2025), arXiv:2502.11900.
https://doi.org/10.1103/j7b8-pb77
arXiv:2502.11900

[56] F. Belliardo, E. M. Gauger, M. H. Abobeih, T. H. Taminiau, Y. Altmann, and C. Bonato, Multidimensional Quantum Estimation and Model Learning Framework Based on Variational Bayesian Inference, PRX Quantum 7, 020360 (2026), arXiv:2507.23130.
https://doi.org/10.1103/6yhz-9vr6
arXiv:2507.23130

[57] T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, J. Phys. A 49, 123001 (2016), arXiv:1511.07548.
https://doi.org/10.1088/1751-8113/49/12/123001
arXiv:1511.07548

[58] O. Gühne, E. Haapasalo, T. Kraft, J.-P. Pellonpää, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys. 95, 011003 (2023), arXiv:2112.06784.
https://doi.org/10.1103/RevModPhys.95.011003
arXiv:2112.06784

[59] M. Szczykulska, T. Baumgratz, and A. Datta, Multi-parameter quantum metrology, Adv. Phys. X 1, 621 (2016), arXiv:1604.02615.
https://doi.org/10.1080/23746149.2016.1230476
arXiv:1604.02615

[60] G. Li and X.-M. Lu, Ultimate tradeoff relation of quantum precision limits in multiparameter linear measurement, Quantum Sci. Technol. 11, 035058 (2026), arXiv:2412.15031.
https://doi.org/10.1088/2058-9565/ae8df4
arXiv:2412.15031

[61] B.-S. Hu and X.-M. Lu, Quantum Parameter Estimation Uncertainty Relation (2025), arXiv:2506.15352.
https://doi.org/10.48550/arXiv.2506.15352
arXiv:2506.15352

[62] I. Kull, P. A. Guérin, and F. Verstraete, Uncertainty and trade-offs in quantum multiparameter estimation, J. Phys. Math. Theor. 53, 244001 (2020), arXiv:2002.05961.
https://doi.org/10.1088/1751-8121/ab7f67
arXiv:2002.05961

[63] S. Razavian, M. G. A. Paris, and M. G. Genoni, On the Quantumness of Multiparameter Estimation Problems for Qubit Systems, Entropy 22, 1197 (2020), arXiv:2010.12630.
https://doi.org/10.3390/e22111197
arXiv:2010.12630

[64] F. Belliardo and V. Giovannetti, Incompatibility in quantum parameter estimation, New J. Phys. 23, 063055 (2021), arXiv:2102.13417.
https://doi.org/10.1088/1367-2630/ac04ca
arXiv:2102.13417

[65] A. Candeloro, M. G. A. Paris, and M. G. Genoni, On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation, J. Phys. A 54, 485301 (2021), arXiv:2107.13426.
https://doi.org/10.1088/1751-8121/ac331e
arXiv:2107.13426

[66] A. Candeloro, Z. Pazhotan, and M. G. A. Paris, Dimension matters: Precision and incompatibility in multi-parameter quantum estimation models, Quantum Sci. Technol. 9, 045045 (2024), arXiv:2403.07106.
https://doi.org/10.1088/2058-9565/ad7498
arXiv:2403.07106

[67] F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging, Phys. Lett. A 384, 126311 (2020), arXiv:1911.12067.
https://doi.org/10.1016/j.physleta.2020.126311
arXiv:1911.12067

[68] J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A 53, 023001 (2020), arXiv:1907.08037.
https://doi.org/10.1088/1751-8121/ab5d4d
arXiv:1907.08037

[69] L. Pezzè and A. Smerzi, Advances in multiparameter quantum sensing and metrology (2025), arXiv:2502.17396.
https://doi.org/10.48550/arXiv.2502.17396
arXiv:2502.17396

[70] C. W. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. A 25, 101 (1967).
https://doi.org/10.1016/0375-9601(67)90366-0

[71] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439

[72] O. E. Barndorff-Nielsen and R. D. Gill, Fisher information in quantum statistics, J. Phys. A 33, 4481 (2000), arXiv:quant-ph/9808009.
https://doi.org/10.1088/0305-4470/33/24/306
arXiv:quant-ph/9808009

[73] M. Hayashi and K. Matsumoto, Asymptotic performance of optimal state estimation in qubit system, J. Math. Phys. 49, 102101 (2008), arXiv:quant-ph/0411073.
https://doi.org/10.1063/1.2988130
arXiv:quant-ph/0411073

[74] J. Kahn and M. Guţă, Local Asymptotic Normality for Finite Dimensional Quantum Systems, Commun. Math. Phys. 289, 597 (2009), arXiv:0804.3876.
https://doi.org/10.1007/s00220-009-0787-3
arXiv:0804.3876

[75] K. Yamagata, A. Fujiwara, and R. D. Gill, Quantum local asymptotic normality based on a new quantum likelihood ratio, Ann. Stat. 41, 2197 (2013), arXiv:1210.3749.
https://doi.org/10.1214/13-AOS1147
arXiv:1210.3749

[76] Y. Yang, G. Chiribella, and M. Hayashi, Attaining the Ultimate Precision Limit in Quantum State Estimation, Commun. Math. Phys. 368, 223 (2019a), arXiv:1802.07587.
https://doi.org/10.1007/s00220-019-03433-4
arXiv:1802.07587

[77] A. Fujiwara and K. Yamagata, Efficiency of estimators for locally asymptotically normal quantum statistical models, Ann. Stat. 51, 1159 (2023), arXiv:2209.00832.
https://doi.org/10.1214/23-AOS2285
arXiv:2209.00832

[78] M. Tsang, F. Albarelli, and A. Datta, Quantum Semiparametric Estimation, Phys. Rev. X 10, 031023 (2020), arXiv:1906.09871.
https://doi.org/10.1103/PhysRevX.10.031023
arXiv:1906.09871

[79] A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, On quantumness in multi-parameter quantum estimation, J. Stat. Mech. Theory Exp. 2019, 094010 (2019), arXiv:1911.11621.
https://doi.org/10.1088/1742-5468/ab3ccb
arXiv:1911.11621

[80] K. Matsumoto, A new approach to the Cramér-Rao-type bound of the pure-state model, J. Phys. A 35, 3111 (2002), arXiv:quant-ph/9711008.
https://doi.org/10.1088/0305-4470/35/13/307
arXiv:quant-ph/9711008

[81] R. D. Gill and S. Massar, State estimation for large ensembles, Phys. Rev. A 61, 042312 (2000), arXiv:quant-ph/9902063.
https://doi.org/10.1103/PhysRevA.61.042312
arXiv:quant-ph/9902063

[82] X.-M. Lu and X. Wang, Incorporating Heisenberg's Uncertainty Principle into Quantum Multiparameter Estimation, Phys. Rev. Lett. 126, 120503 (2021), arXiv:2008.08888.
https://doi.org/10.1103/PhysRevLett.126.120503
arXiv:2008.08888

[83] H. Chen, Y. Chen, and H. Yuan, Incompatibility measures in multiparameter quantum estimation under hierarchical quantum measurements, Phys. Rev. A 105, 062442 (2022a), arXiv:2109.05807.
https://doi.org/10.1103/PhysRevA.105.062442
arXiv:2109.05807

[84] H. Chen, Y. Chen, and H. Yuan, Information Geometry under Hierarchical Quantum Measurement, Phys. Rev. Lett. 128, 250502 (2022b), arXiv:2206.13095.
https://doi.org/10.1103/PhysRevLett.128.250502
arXiv:2206.13095

[85] H. Chen, L. Wang, and H. Yuan, Simultaneous measurement of multiple incompatible observables and tradeoff in multiparameter quantum estimation, npj Quantum Inf. 10, 98 (2024b), arXiv:2310.11925.
https://doi.org/10.1038/s41534-024-00894-x
arXiv:2310.11925

[86] L. Wang, H. Chen, and H. Yuan, Tight tradeoff relation and optimal measurement for multiparameter quantum estimation, Phys. Rev. A 114, 012609 (2026), arXiv:2504.09490.
https://doi.org/10.1103/m1z7-f7b8
arXiv:2504.09490

[87] H. Nagaoka, A New Approach to Cramér-Rao Bounds for Quantum State Estimation, IEICE Tech. Rep. IT 89-42, 9 (1989).
https://doi.org/10.1142/9789812563071_0009

[88] H. Nagaoka, A Generalization of the Simultaneous Diagonalization of Hermitian Matrices and its Relation to Quantum Estimation Theory, Trans. Jpn. Soc. Ind. Appl. Math. 1, 43 (1991).
https://doi.org/10.1142/9789812563071_0012

[89] M. Hayashi, On simultaneous measurement of noncommutative observables, Dev. Infin.-Dimens. Non-Commut. Anaysis , 96 (1999).

[90] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Efficient computation of the Nagaoka–Hayashi bound for multiparameter estimation with separable measurements, npj Quantum Inf. 7, 110 (2021), arXiv:2008.02612.
https://doi.org/10.1038/s41534-021-00414-1
arXiv:2008.02612

[91] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, The gap persistence theorem for quantum multiparameter estimation (2024), arXiv:2208.07386.
https://doi.org/10.48550/arXiv.2208.07386
arXiv:2208.07386

[92] S. K. Yung, L. O. Conlon, J. Zhao, P. K. Lam, and S. M. Assad, Comparison of estimation limits for quantum two-parameter estimation, Phys. Rev. Res. 6, 033315 (2024), arXiv:2407.12466.
https://doi.org/10.1103/PhysRevResearch.6.033315
arXiv:2407.12466

[93] A. Das, L. O. Conlon, J. Suzuki, S. K. Yung, P. K. Lam, and S. M. Assad, Holevo Cramér-Rao bound: How close can we get without entangling measurements?, Quantum 9, 1867 (2025), arXiv:2405.09622.
https://doi.org/10.22331/q-2025-09-30-1867
arXiv:2405.09622

[94] L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Role of the extended Hilbert space in the attainability of the quantum Cramér–Rao bound for multiparameter estimation, Phys. Lett. A 542, 130445 (2025), arXiv:2404.01520.
https://doi.org/10.1016/j.physleta.2025.130445
arXiv:2404.01520

[95] M. Hayashi and Y. Ouyang, Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming, Quantum 7, 1094 (2023), arXiv:2209.05218.
https://doi.org/10.22331/q-2023-08-29-1094
arXiv:2209.05218

[96] M. Hayashi and Y. Ouyang, Finding the optimal probe state for multiparameter quantum metrology using conic programming, npj Quantum Inf. 10, 111 (2024), arXiv:2401.05886.
https://doi.org/10.1038/s41534-024-00905-x
arXiv:2401.05886

[97] H. Yuen, R. Kennedy, and M. Lax, On optimal quantum receivers for digital signal detection, Proc. IEEE 58, 1770 (1970).
https://doi.org/10.1109/PROC.1970.8004

[98] H. Yuen, RS. Kennedy, and M. Lax, Optimum testing of multiple hypotheses in quantum detection theory, IEEE Trans. Inf. Theory 21, 125 (1975).
https://doi.org/10.1109/TIT.1975.1055351

[99] B. Coutts, M. Girard, and J. Watrous, Certifying optimality for convex quantum channel optimization problems, Quantum 5, 448 (2021), arXiv:1810.13295.
https://doi.org/10.22331/q-2021-05-01-448
arXiv:1810.13295

[100] D. C. Brody and B. Meister, Bayesian inference in quantum systems, Physica A 223, 348 (1996).
https://doi.org/10.1016/0378-4371(95)00342-8

[101] J. Rubio, First-principles construction of symmetry-informed quantum metrologies, Phys. Rev. A 110, L030401 (2024), arXiv:2402.16410.
https://doi.org/10.1103/PhysRevA.110.L030401
arXiv:2402.16410

[102] V. P. Belavkin and B. A. Grishanin, Optimum Estimation in Quantum Channels by the Generalized Heisenberg Inequality Method, Probl. Peredachi Inf. 9, 44 (1973).

[103] X.-B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, Quantum information with Gaussian states, Phys. Rep. 448, 1 (2007), arXiv:0801.4604.
https://doi.org/10.1016/j.physrep.2007.04.005
arXiv:0801.4604

[104] J. Suzuki, Bayesian Nagaoka-Hayashi Bound for Multiparameter Quantum-State Estimation Problem, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E107.A, 510 (2024), arXiv:2302.14223.
https://doi.org/10.1587/transfun.2023TAP0014
arXiv:2302.14223

[105] J. Zhang and J. Suzuki, Bayesian Logarithmic Derivative Type Lower Bounds for Quantum Estimation (2024), arXiv:2405.10525.
https://doi.org/10.48550/arXiv.2405.10525
arXiv:2405.10525

[106] R. Kaubruegger, A. Shankar, D. V. Vasilyev, and P. Zoller, Optimal and Variational Multiparameter Quantum Metrology and Vector-Field Sensing, PRX Quantum 4, 020333 (2023), arXiv:2302.07785.
https://doi.org/10.1103/PRXQuantum.4.020333
arXiv:2302.07785

[107] J. Bavaresco, P. Lipka-Bartosik, P. Sekatski, and M. Mehboudi, Designing optimal protocols in bayesian quantum parameter estimation with higher-order operations, Phys. Rev. Res. 6, 023305 (2024), arXiv:2311.01513.
https://doi.org/10.1103/PhysRevResearch.6.023305
arXiv:2311.01513

[108] V. P. Belavkin, Optimum distinction of non-orthogonal quantum signals, Radiotekhnika Elektron. 20, 1177 (1975a).
https://ui.adsabs.harvard.edu/abs/1975RaEl...20.1177B

[109] V. P. Belavkin, Optimal multiple quantum statistical hypothesis testing, Stochastics 1, 315 (1975b).
https://doi.org/10.1080/17442507508833114

[110] A. S. Holevo, On Asymptotically Optimal Hypothesis Testing in Quantum Statistics, Theory Probab. Its Appl. 23, 411 (1979).
https://doi.org/10.1137/1123048

[111] P. Hausladen and W. K. Wootters, A `Pretty Good' Measurement for Distinguishing Quantum States, J. Mod. Opt. 41, 2385 (1994).
https://doi.org/10.1080/09500349414552221

[112] M. Hayashi, Quantum Information Theory (Springer, Berlin, Heidelberg, 2017).
https://doi.org/10.1007/978-3-662-49725-8

[113] multipar_bayes available on github: dbranford/multipar_bayes (2026).
https://github.com/dbranford/multipar_bayes

[114] G. Chiribella, G. M. D'Ariano, and D. Schlingemann, How continuous quantum measurements in finite dimensions are actually discrete, Phys. Rev. Lett. 98, 190403 (2007), arXiv:quant-ph/0702068.
https://doi.org/10.1103/PhysRevLett.98.190403
arXiv:quant-ph/0702068

[115] W. von derLinden, V. Dose, and U. von Toussaint, Bayesian Probability Theory: Applications in the Physical Sciences (Cambridge University Press, 2014).

[116] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).

[117] W. Górecki, Heisenberg Limit beyond Quantum Fisher Information, Ph.D. thesis, Faculty of Physics, Univ. of Warsaw (2023), arXiv:2304.14370.
arXiv:2304.14370

[118] E. T. Jaynes and O. Kempthorne, Confidence intervals vs bayesian intervals, in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science: Proceedings of an International Research Colloquium held at the University of Western Ontario, London, Canada, 10–13 May 1973 Volume II Foundations and Philosophy of Statistical Inference, edited by W. L. Harper and C. A. Hooker (Springer Netherlands, Dordrecht, 1976) pp. 175–257.
https://doi.org/10.1007/978-94-010-1436-6_6

[119] J. Boeyens, S. Seah, and S. Nimmrichter, Uninformed Bayesian quantum thermometry, Phys. Rev. A 104, 052214 (2021), arXiv:2108.07025.
https://doi.org/10.1103/PhysRevA.104.052214
arXiv:2108.07025

[120] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, Springer Basel, 2011).

[121] J. Rubio, J. Anders, and L. A. Correa, Global Quantum Thermometry, Phys. Rev. Lett. 127, 190402 (2021), arXiv:2011.13018.
https://doi.org/10.1103/PhysRevLett.127.190402
arXiv:2011.13018

[122] J. Boeyens, J. Glatthard, E. Gandar, S. Nimmrichter, L. A. Correa, and J. Rubio, On the role of symmetry and geometry in global quantum sensing, Quantum Sci. Technol. 10, 045053 (2025), arXiv:2502.14817.
https://doi.org/10.1088/2058-9565/ae08e1
arXiv:2502.14817

[123] M. Tsang, Generalized conditional expectations for quantum retrodiction and smoothing, Phys. Rev. A 105, 042213 (2022), arXiv:1912.02711.
https://doi.org/10.1103/PhysRevA.105.042213
arXiv:1912.02711

[124] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016), arXiv:1608.02634.
https://doi.org/10.1103/PhysRevA.94.052108
arXiv:1608.02634

[125] F. Albarelli and R. Demkowicz-Dobrzański, Probe Incompatibility in Multiparameter Noisy Quantum Metrology, Phys. Rev. X 12, 011039 (2022), arXiv:2104.11264.
https://doi.org/10.1103/PhysRevX.12.011039
arXiv:2104.11264

[126] Z. Hu, S. Wang, L. Qiao, T. Isogawa, C. Li, Y. Yang, G. Wang, H. Yuan, and P. Cappellaro, Control incompatibility in multiparameter quantum metrology (2024), arXiv:2411.18896.
https://doi.org/10.48550/arXiv.2411.18896
arXiv:2411.18896

[127] A. S. Holevo, Investigations in the General Theory of Statistical Decisions, Proc. Steklov Institute of Mathematics No. 124 (American Mathematical Soc., 1978 [1976]).
https://www.mathnet.ru/eng/tm3122

[128] J. S. Sidhu and P. Kok, Geometric perspective on quantum parameter estimation, AVS Quantum Sci. 2, 014701 (2020), arXiv:1907.06628.
https://doi.org/10.1116/1.5119961
arXiv:1907.06628

[129] J. Yang, S. Pang, Y. Zhou, and A. N. Jordan, Optimal measurements for quantum multiparameter estimation with general states, Phys. Rev. A 100, 032104 (2019b), arXiv:1806.07337.
https://doi.org/10.1103/PhysRevA.100.032104
arXiv:1806.07337

[130] P. Horodecki, Ł. Rudnicki, and K. Życzkowski, Five Open Problems in Quantum Information Theory, PRX Quantum 3, 010101 (2022), arXiv:2002.03233.
https://doi.org/10.1103/PRXQuantum.3.010101
arXiv:2002.03233

[131] H. I. Nurdin, Corrections to ``Saturability of the Quantum Cramér-Rao Bound in Multiparameter Quantum Estimation at the Single-Copy Level'', IEEE Control Syst. Lett. 8, 2111 (2024).
https://doi.org/10.1109/LCSYS.2024.3451468

[132] L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases, Phys. Rev. Lett. 119, 130504 (2017), arXiv:1705.03687.
https://doi.org/10.1103/PhysRevLett.119.130504
arXiv:1705.03687

[133] J. Rubio, Quantum scale estimation, Quantum Sci. Technol. 8, 015009 (2022), arXiv:2111.11921.
https://doi.org/10.1088/2058-9565/aca04b
arXiv:2111.11921

[134] H. Barnum and E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity, J. Math. Phys. 43, 2097 (2002), arXiv:quant-ph/0004088.
https://doi.org/10.1063/1.1459754
arXiv:quant-ph/0004088

[135] H. K. Mishra, L. Lami, and M. M. Wilde, Near-optimal performance of square-root measurement for general score functions and quantum ensembles (2025), arXiv:2505.20447.
https://doi.org/10.48550/arXiv.2505.20447
arXiv:2505.20447

[136] D. Petz, Monotone metrics on matrix spaces, Linear Algebra Its Appl. 244, 81 (1996).
https://doi.org/10.1016/0024-3795(94)00211-8

[137] M. Scandi, P. Abiuso, J. Surace, and D. De Santis, Quantum Fisher information and its dynamical nature, Rep. Prog. Phys. 88, 076001 (2025), arXiv:2304.14984.
https://doi.org/10.1088/1361-6633/ade453
arXiv:2304.14984

[138] K. Hunter, Measurement does not always aid state discrimination, Phys. Rev. A 68, 012306 (2003), arXiv:quant-ph/0211148.
https://doi.org/10.1103/PhysRevA.68.012306
arXiv:quant-ph/0211148

[139] S. M. Barnett and S. Croke, Quantum state discrimination, Adv. Opt. Photonics 1, 238 (2009), arXiv:0810.1970.
https://doi.org/10.1364/AOP.1.000238
arXiv:0810.1970

[140] M. Quadeer, Haar-random and pretty good measurements for Bayesian state estimation (2024), arXiv:2310.20565.
https://doi.org/10.48550/arXiv.2310.20565
arXiv:2310.20565

[141] R. D. Gill, Conciliation of Bayes and Pointwise Quantum State Estimation: Asymptotic information bounds in quantum statistics, in Quantum Stoch. Inf. Stat. Filter. Control, edited by V. P. Belavkin and M. Guţă (World Scientific, 2008) pp. 239–261, arXiv:math/0512443.
https://doi.org/10.1142/9789812832962_0011
arXiv:math/0512443

[142] M. Jarzyna and R. Demkowicz-Dobrzański, Quantum interferometry with and without an external phase reference, Phys. Rev. A 85, 011801 (2012), arXiv:1108.3844.
https://doi.org/10.1103/PhysRevA.85.011801
arXiv:1108.3844

[143] C. Macchiavello, Optimal estimation of multiple phases, Phys. Rev. A 67, 062302 (2003), arXiv:quant-ph/0304126.
https://doi.org/10.1103/PhysRevA.67.062302
arXiv:quant-ph/0304126

[144] V. Gebhart, A. Smerzi, and L. Pezzè, Bayesian quantum multiphase estimation algorithm, Phys. Rev. Appl. 16, 014035 (2021), arXiv:2010.09075.
https://doi.org/10.1103/PhysRevApplied.16.014035
arXiv:2010.09075

[145] M. J. W. Hall and H. M. Wiseman, Does Nonlinear Metrology Offer Improved Resolution? Answers from Quantum Information Theory, Phys. Rev. X 2, 041006 (2012), arXiv:1205.2405.
https://doi.org/10.1103/PhysRevX.2.041006
arXiv:1205.2405

[146] V. Giovannetti and L. Maccone, Sub-Heisenberg Estimation Strategies Are Ineffective, Phys. Rev. Lett. 108, 210404 (2012), arXiv:1201.1878.
https://doi.org/10.1103/PhysRevLett.108.210404
arXiv:1201.1878

[147] J. Rubio, P. A. Knott, and J. A. Dunningham, Non-asymptotic analysis of quantum metrology protocols beyond the Cramér-Rao bound, J. Phys. Commun. 2, 015027 (2018), arXiv:1707.05022.
https://doi.org/10.1088/2399-6528/aaa234
arXiv:1707.05022

[148] D. Branford and J. Rubio, Average number is an insufficient metric for interferometry, New J. Phys. 23, 123041 (2021), arXiv:2107.06698.
https://doi.org/10.1088/1367-2630/ac3571
arXiv:2107.06698

[149] F. Albarelli, D. Branford, and J. Rubio, Code and data for Measurement incompatibility in Bayesian multiparameter quantum estimation , 10.5281/zenodo.17814129 (2026).
https://doi.org/10.5281/zenodo.17814129

[150] M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, Joint estimation of phase and phase diffusion for quantum metrology, Nat. Commun. 5, 3532 (2014), arXiv:1410.5353.
https://doi.org/10.1038/ncomms4532
arXiv:1410.5353

[151] E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M. G. Genoni, and M. Barbieri, Entangling measurements for multiparameter estimation with two qubits, Quantum Sci. Technol. 3, 01LT01 (2018), arXiv:1704.03327.
https://doi.org/10.1088/2058-9565/aa9212
arXiv:1704.03327

[152] A. Kobus and R. Demkowicz-Dobrzański, Asymptotically optimal joint phase and dephasing strength estimation using spin-squeezed states (2025), arXiv:2507.22997.
https://doi.org/10.48550/arXiv.2507.22997
arXiv:2507.22997

[153] S. K. Yung, A. Das, J. Suzuki, P. K. Lam, J. Zhao, L. O. Conlon, and S. M. Assad, Saturating the Quantum Cramér–Rao Bound in Prioritised Parameter Estimation (2025), arXiv:2511.06704.
https://doi.org/10.48550/arXiv.2511.06704
arXiv:2511.06704

[154] M. G. Genoni, S. Olivares, and M. G. A. Paris, Optical Phase Estimation in the Presence of Phase Diffusion, Phys. Rev. Lett. 106, 153603 (2011), arXiv:1012.1123.
https://doi.org/10.1103/PhysRevLett.106.153603
arXiv:1012.1123

[155] M. Szczykulska, T. Baumgratz, and A. Datta, Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion, Quantum Sci. Technol. 2, 044004 (2017), arXiv:1701.07520.
https://doi.org/10.1088/2058-9565/aa7fa9
arXiv:1701.07520

[156] Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Deterministic realization of collective measurements via photonic quantum walks, Nat. Commun. 9, 1414 (2018), arXiv:1710.10045.
https://doi.org/10.1038/s41467-018-03849-x
arXiv:1710.10045

[157] M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Beating the Rayleigh Limit Using Two-Photon Interference, Phys. Rev. Lett. 121, 250503 (2018), arXiv:1803.07096.
https://doi.org/10.1103/PhysRevLett.121.250503
arXiv:1803.07096

[158] Y. Yuan, Z. Hou, J.-F. Tang, A. Streltsov, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Direct estimation of quantum coherence by collective measurements, npj Quantum Inf. 6, 1 (2020), arXiv:2001.01384.
https://doi.org/10.1038/s41534-020-0280-6
arXiv:2001.01384

[159] L. O. Conlon, T. Vogl, C. D. Marciniak, I. Pogorelov, S. K. Yung, F. Eilenberger, D. W. Berry, F. S. Santana, R. Blatt, T. Monz, P. K. Lam, and S. M. Assad, Approaching optimal entangling collective measurements on quantum computing platforms, Nat. Phys. 19, 351 (2023), arXiv:2205.15358.
https://doi.org/10.1038/s41567-022-01875-7
arXiv:2205.15358

[160] M. Mi, B. Wang, and L. Zhang, Approaching the Multiparameter Quantum Cramér-Rao Bound via Classical Correlation and Entangling Measurements, Phys. Rev. Lett. 135, 110804 (2025), arXiv:2509.10196.
https://doi.org/10.1103/37kb-qq6k
arXiv:2509.10196

[161] D. V. Vasilyev, A. Shankar, R. Kaubruegger, and P. Zoller, Optimal Multiparameter Metrology: The Quantum Compass Solution (2024), arXiv:2404.14194.
https://doi.org/10.48550/arXiv.2404.14194
arXiv:2404.14194

[162] W. Górecki, S. Zhou, L. Jiang, and R. Demkowicz-Dobrzański, Optimal probes and error-correction schemes in multi-parameter quantum metrology, Quantum 4, 288 (2020), arXiv:1901.00896.
https://doi.org/10.22331/q-2020-07-02-288
arXiv:1901.00896

[163] E. L. André, J. Bavaresco, and M. Mehboudi, Strategy optimization for Bayesian quantum parameter estimation with finite copies: Adaptive greedy, parallel, sequential, and general strategies, Quantum Sci. Technol. 11, 035040 (2026), arXiv:arXiv:2602.09655.
https://doi.org/10.1088/2058-9565/ae846c
arXiv:arXiv:2602.09655

[164] R. Demkowicz-Dobrzański, Optimal phase estimation with arbitrary a priori knowledge, Phys. Rev. A 83, 061802 (2011), arXiv:1102.0786.
https://doi.org/10.1103/PhysRevA.83.061802
arXiv:1102.0786

[165] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photonics 12, 724 (2018), arXiv:1811.01969.
https://doi.org/10.1038/s41566-018-0301-6
arXiv:1811.01969

[166] M. Barbieri, Optical Quantum Metrology, PRX Quantum 3, 010202 (2022), arXiv:2507.22680.
https://doi.org/10.1103/PRXQuantum.3.010202
arXiv:2507.22680

[167] Y.-D. Wu, Y. Zhu, G. Chiribella, and N. Liu, Efficient learning of continuous-variable quantum states, Phys. Rev. Res. 6, 033280 (2024), arXiv:2303.05097.
https://doi.org/10.1103/PhysRevResearch.6.033280
arXiv:2303.05097

[168] F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V. Giovannetti, L. Lami, L. Leone, and S. F. E. Oliviero, Learning quantum states of continuous-variable systems, Nature Physics 21, 2002 (2025), arXiv:2405.01431.
https://doi.org/10.1038/s41567-025-03086-2
arXiv:2405.01431

[169] L. Bittel, F. A. Mele, A. A. Mele, S. Tirone, and L. Lami, Optimal estimates of trace distance between bosonic Gaussian states and applications to learning, Quantum 9, 1769 (2025), arXiv:2411.02368.
https://doi.org/10.22331/q-2025-06-12-1769
arXiv:2411.02368

[170] M. Fanizza, C. Rouzé, and D. S. França, Efficient Hamiltonian, structure and trace distance learning of Gaussian states (2025), arXiv:2411.03163.
https://doi.org/10.48550/arXiv.2411.03163
arXiv:2411.03163

[171] S. Morelli, A. Usui, E. Agudelo, and N. Friis, Bayesian parameter estimation using Gaussian states and measurements, Quantum Sci. Technol. 6, 025018 (2021), arXiv:2009.03709.
https://doi.org/10.1088/2058-9565/abd83d
arXiv:2009.03709

[172] S. Chang, M. G. Genoni, and F. Albarelli, Efficiently evaluating Holevo, RLD and SLD Cramér-Rao bounds for multiparameter quantum estimation with Gaussian states, Commun. Phys. 9, 126 (2026), arXiv:2504.17873.
https://doi.org/10.1038/s42005-026-02550-6
arXiv:2504.17873

[173] J. Zhang, K. Yamagata, and J. Suzuki, Bayesian Monotone Metrics for Multiparameter Quantum Estimation (2026), arXiv:2607.01685.
https://doi.org/10.48550/arXiv.2607.01685
arXiv:2607.01685

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[2] Jianchao Zhang, Koichi Yamagata, and Jun Suzuki, "Bayesian Monotone Metrics for Multiparameter Quantum Estimation", arXiv:2607.01685, (2026).

[3] Leo Bia and Christos N. Gagatsos, "Saturating the Bayesian Nagaoka-Hayashi bound within numerical precision for the depolarization SU(2) rotation channel", arXiv:2607.15398, (2026).

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