Measurement incompatibility in Bayesian multiparameter quantum estimation

TL;DR

This work develops a comprehensive Bayesian framework for multiparameter quantum estimation, clarifying how prior information and measurement incompatibility constrain the minimum mean squared loss. It derives explicit optimality conditions for the POVMs under the MSL, introduces a robust incompatibility figure of merit, and establishes tight upper and lower bounds (notably via the Pretty Good Measurement and the Nagaoka–Hayashi bound) that quantify how incompatibility can impact precision. The authors demonstrate these concepts through applications in discrete quantum phase imaging, phase and dephasing estimation, and qubit planar tomography, and provide open-source tools for practical evaluation. By connecting Bayesian and local quantum estimation perspectives and highlighting the role of priors, this work offers both theoretical insight and practical benchmarks for designing information-optimal quantum sensors.

Abstract

We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation, providing explicit conditions for achieving minimum quadratic losses. 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 originally developed for 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. This result implies that, in many practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through a range of applications, including discrete quantum 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. We also provide an open-source package that implements all bounds discussed here, enabling practical evaluation and comparison across quantum metrological models.

Figures (3)

• Figure 1: Evaluation of measurement incompatibility for a discrete quantum phase imaging protocol based on generalised N00N states. The shaded region represents the range of incompatibility defined in Eq. \ref{['eq:incomp-range']} as a function of the number of parameters $d$, choosing an initial state with $\alpha=\sqrt{d}$. The upper boundary of this region corresponds to the incompatibility quantifier associated with the PGM bound enhanced by the PM estimator (dash–dot), while the lower boundary corresponds to that associated with the NH bound (solid). The first inset displays the reference MSL, $\mathcal{L}_{\mathrm{SPM}}$, and the dotted line indicates the incompatibility quantifier associated with the a priori MSL. As observed, the latter decreases with the number of parameters, thereby fundamentally limiting the extent to which incompatibility can influence the overall precision; a zoomed view for larger parameter values is shown in the second inset. Finally, for completeness, we also include the incompatibility quantifier associated with the standard PGM bound, which, as seen, is considerably larger than its a priori MSL counterpart and thus trivial for this system.
• Figure 2: Evaluation of measurement incompatibility over $n$ copies of the qubit state \ref{['eq:state-phase-diffusion']}, for prior widths $W_1 = \pi/2$ and $W_2 = 5$. The shaded region represents the range of incompatibility defined in Eq. \ref{['eq:incomp-range']} as a function of $n$. Unlike in Sec. \ref{['sec:imaging']}, the upper boundary (inverted triangles) now corresponds to the incompatibility quantifier associated with the MSL obtained from a tomographic measurement. The lower boundary (triangles) is as before, i.e., given by the incompatibility quantifier associated with the NH bound. The inset shows the reference MSL, $\mathcal{L}_{\mathrm{SPM}}$. The line marked with crosses indicates the incompatibility quantifier associated with the a priori MSL (the corresponding loss is constant, but the decreasing $\mathcal{L}_{\mathrm{SPM}}$ means the incompatibility is increasing). We also include the incompatibility quantifier associated with the standard PGM bound (circles), which, as in Sec. \ref{['sec:imaging']}, is substantially larger than its a priori MSL counterpart and therefore again trivial. Finally, the upper and lower dotted lines correspond to the respective bounds on measurement incompatibility in Eq. \ref{['eq:I-between-0-and-1']}.
• Figure 3: Top panel: Measurement incompatibility bounds as a function of the shape parameter $\beta$ of the beta distribution used in constructing the two-parameter prior, for fixed prior widths $W_1 = 0.85$ and $W_2 = 0.51$. Bottom panel: Measurement incompatibility bounds as a function of the prior width $W_2$ for the parameter $\theta_2$, for fixed shape parameter $\beta = 0.07$ and prior width $W_1 = 0.83$ for the parameter $\theta_1$. The two insets display the MSL for the corresponding parameter values in each panel, provided as a reference. Graphical conventions distinguishing curves as in Sec. \ref{['sec:imaging']}.