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arXiv:2511.06704

Saturating the Quantum Cramér--Rao Bound in Prioritised Parameter Estimation

Simon K. Yung, Aritra Das, Jun Suzuki, Ping Koy Lam, Jie Zhao, Lorcán O. Conlon, Syed M. Assad

quant-phSubmitted Nov 10, 2025• Announced Nov 11, 2025• 0 citations
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Abstract

Measurement incompatibility is a cornerstone of quantum mechanics. In the context of estimating multiple parameters of a quantum system, this manifests as a fundamental trade-off between the precisions with which different parameters can be estimated. Often, a parameter can be optimally measured, but at the cost of gaining no information about incompatible parameters. Here, we report that there are systems where one parameter's information can be maximised while not completely losing information about the other parameters. In doing so, we find attainable trade-off relations for quantum parameter estimation with a structure that is different to typical Heisenberg-type trade-offs. We demonstrate our findings by implementing an optimal entangling measurement on a Quantinuum trapped-ion quantum computer.

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Measurement incompatibility is a cornerstone of quantum mechanics. In the context of estimating multiple parameters of a quantum system, this manifests as a fundamental trade-off between the precisions with which different parameters can be estimated. Often, a parameter can be optimally measured, but at the cost of gaining no information about incompatible parameters. Here, we report that there are systems where one parameter's information can be maximised while not completely losing information about the other parameters. In doing so, we find attainable trade-off relations for quantum parameter estimation with a structure that is different to typical Heisenberg-type trade-offs. We demonstrate our findings by implementing an optimal entangling measurement on a Quantinuum trapped-ion quantum computer.
A fundamental challenge in multiparameter quantum estimation arises from the incompatibility of optimal measurements for different parameters, leading to intricate precision trade-offs that obscure the understanding of ultimate quantum limits. Here, we present an approach that precisely quantifies these trade-offs for an arbitrary number of parameters encoded in pure quantum states. Our approach not only derives tight analytical bounds for the trade-offs induced by measurement incompatibility but also provides a systematic methodology to design optimal measurement strategies that saturate these limits. To demonstrate the practical significance of our findings, we apply our framework to quantum radar and obtain a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity with any given amount of entanglement. This showcases the transformative potential of our findings for a wide range of applications in quantum metrology, sensing, and beyond.
Abstract Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cramér–Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state-independent uncertainty relation between the parameters of a qubit system.
The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be understood as a result of extended regimes of optimization. Two experimentally feasible examples are presented to illustrate the theory, saturation of these bounds and estimation precisions beyond the Heisenberg limit are predicted and discussed. A setup to measure non-Hermitian observable is also proposed.
The quantum multiparameter estimation is very different from the classical multiparameter estimation due to Heisenberg's uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inaccuracy of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenberg's uncertainty principle into quantum multiparameter estimation by giving a trade-off relation between the measurement inaccuracies for estimating different parameters. For pure quantum states, this trade-off relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the trade-off between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the trade-off relation. We also show that our approach can be readily used to derive the trade-off between the errors of jointly estimating the phase shift and phase diffusion without explicitly parametrizing quantum measurements.
Abstract Quantum metrology aims at delivering new quantum-mechanical improvement to technologies of parameter estimations with precision bounded by the quantum Cramér-Rao bound. The currently used quantum Cramér-Rao bound was established with measurements of observables restricted to be Hermitian. This constrains the bound and limits the precision of parameter estimation. In this paper, we lift the constraint and derive a previously unknown quantum Cramér-Rao bound. We find that the new bound can reach arbitrary small value with mixed states and it breaks the Heisenberg limit in some cases. We construct a setup to measure non-Hermitian operators and discuss the saturation of the present bound. Two examples—the phase estimation with Greenberger-Horne-Zeilinger states of trapped ions and the adiabatic quantum parameter estimation with the nuclear magnetic resonance—are employed to demonstrate the theory. The present study might open a new research direction—non-Hermitian quantum metrology.
Measurement estimation bounds for local quantum multiparameter estimation, which provide lower bounds on possible measurement uncertainties, have so far been formulated in two ways: by extending the classical Cramér-Rao bound (e.g., the quantum Cramér-Rao bound and the Nagaoka Cramér-Rao bound) and by incorporating the parameter estimation framework with the uncertainty principle, as in the Lu-Wang uncertainty relation. In this work, we present a general framework that allows a direct comparison between these different types of estimation limits. Specifically, we compare the attainability of the Nagaoka Cramér-Rao bound and the Lu-Wang uncertainty relation, using analytical and numerical techniques. We show that these two limits can provide different information about the physically attainable precision. We present an example where both limits provide the same attainable precision and an example where the Lu-Wang uncertainty relation is not attainable even for pure states. We further demonstrate that the unattainability in the latter case arises because the figure of merit underpinning the Lu-Wang uncertainty relation (the difference between the quantum and classical Fisher information matrices) does not necessarily agree with the conventionally used figure of merit (mean-squared error). The results offer insights into the general attainability and applicability of the Lu-Wang uncertainty relation. Furthermore, our proposed framework for comparing bounds of different types may prove useful in other settings. Published by the American Physical Society 2024