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arXiv:2511.16645 · Quantum

Measurement incompatibility in Bayesian multiparameter quantum estimation

Francesco Albarelli, Dominic Branford, Jesús Rubio

quant-phSubmitted Nov 20, 2025• Announced Nov 21, 2025• 0 citations• Revised Aug 3, 2026 (v4)

v1 · Nov 20, 2025 v2 · Dec 4, 2025 v3 · May 27, 2026 v4 · Aug 3, 2026

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.

Journal reference: Quantum 10, 2192 (2026)
DOI: 10.22331/q-2026-08-13-2192
Comments: 33 pages, 3 figures. v4: minor corrections

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In multiparameter quantum estimation, the optimal measurements for different parameters encoded in a quantum state are in general incompatible, giving rise to nontrivial tradeoffs between their attainable precisions. Understanding and characterizing such tradeoffs is essential for determining the ultimate precision limits in multiparameter quantum estimation and is therefore a central topic in quantum metrology. In this article, we present an approach that precisely quantifies the tradeoff resulting from incompatible optimal measurements in multiparameter estimation. We derive a tight analytical tradeoff relation that determines the ultimate precision limits for estimating an arbitrary number of parameters encoded in pure quantum states. Additionally, we provide a systematic methodology for constructing optimal measurements that saturate this tight bound in an analytical and structured manner. To demonstrate the power of our findings, we apply our methodology to quantum radar, resulting in a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity.

Bayesian quantum estimation offers a finite-data framework for quantum sensing and metrology, yet a unified geometric formulation for multiparameter Bayes risk has been lacking. We introduce Bayesian monotone metrics by evaluating Petz monotone metrics on the prior-averaged state, providing a Bayesian extension of the full class of statistically meaningful (CPTP) quantum metrics. This framework yields Bayesian quantities, including quantum posterior-mean operators and a quantum Bayesian dual Fisher-information matrix, and it leads to a systematic family of computable lower bounds on the Bayes risk. The resulting bounds naturally incorporate multiparameter measurement incompatibility and, for every monotone metric in the family, we prove a universal dominance over the corresponding quantum van Trees (Bayesian Cramér--Rao) bound. Moreover, we show that optimizing over all operator monotone functions collapses to a one-parameter subfamily, turning the tightest bound into a tractable optimization with a clear geometric interpretation. In representative examples, the optimized bounds are strictly tighter than the Bayesian SLD and RLD bounds. Our results establish Bayesian monotone metrics as a unifying information-geometric perspective on Bayesian quantum estimation, enabling systematic and computable performance limits in multiparameter settings.

The incompatibility of the measurements constrains the achievable precisions in multiparameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general

p

-local measurements, which are the measurements that can be performed collectively on at most

p

copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precisions of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cram\'er-Rao bound under

p

-local measurements, which recovers the partial commutative condition at

p=1

and the weak commutative condition at $p=\ensuremath{\infty}$. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators.