Bayesian multiparameter metrology in the International Year of Quantum Science and Technology
Torrejón de Ardoz, 31 December 2025
On this last day of the International Year of Quantum Science and Technology, it seems fitting to highlight a recent manuscript, arXiv.2511.16645, addressing a fundamental but unresolved question in quantum estimation theory: how measurement incompatibility limits achievable precision in Bayesian multiparameter settings.
In this work, written with Francesco Albarelli and Dominic Branford—always a pleasure to collaborate with—we develop a comprehensive and self-contained formulation that sets a solid foundation for organising research in this field, and use it to quantify in operational terms the impact of measurement incompatibility.
Our main result shows that, even at the single-shot level, measurement incompatibility can increase the minimum quadratic loss by at most a factor of two relative to the idealised benchmark where individually optimal measurements are assumed compatible. This benchmark is defined by what we term the symmetric posterior mean (SPM) bound, which was explicitly identified in arXiv:1906.04123, with subsequent discussions in arXiv:1907.06628 and arXiv:2001.11742.
Furthermore, we derive explicit optimality conditions and provide a hierarchy of upper and lower bounds on the minimum quadratic loss based on monotone metrics and ideas from hypothesis testing, together with practical tools for evaluating precision limits in concrete models. These include the open-source multipar_bayes Python package, available on GitHub. The framework is illustrated through detailed examples covering quantum imaging, phase and diffusion estimation, and planar tomography, with accompanying code archived on Zenodo.
The work opens several promising avenues, including pushing the boundaries of what adaptive sensing schemes can achieve and providing a genuinely universal framework for performing measurements relevant to fundamental physics. Perhaps more importantly, it sits within a long-standing effort to understand quantum measurement and estimation theory, building on foundations laid by Heisenberg, Schrödinger, Born, Dirac, and later by Jaynes, Helstrom, Personick, and Holevo, as well as many others across the last century of developments in quantum and information sciences.
One can only hope that the next hundred years will continue to bring both conceptual clarity and new discoveries in this fascinating field.