The Future of Quantum Computing

Measurement incompatibility in Bayesian multiparameter quantum estimation

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 measurem