Measurement incompatibility in Bayesian multiparameter quantum estimation
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AbstractWe 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 summaryQuantum 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. 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This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe 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 summaryQuantum 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. 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Suzuki, Bayesian Monotone Metrics for Multiparameter Quantum Estimation (2026), arXiv:2607.01685. https://doi.org/10.48550/arXiv.2607.01685 arXiv:2607.01685Cited by[1] El Mustapha Mansouri and Keigo Arai, "Fisher Glasses: Tail-Certified Quantum Metrology in Quenched Environments", arXiv:2607.01085, (2026). [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). The above citations are from SAO/NASA ADS (last updated successfully 2026-08-13 13:58:15). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-08-13 13:58:08: Could not fetch cited-by data for 10.22331/q-2026-08-13-2192 from Crossref. 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