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arXiv:2512.10541

Encoding parameters by measurement: Forgetting can be better in quantum metrology

Shuva Mondal, Priya Ghosh, Ujjwal Sen

quant-phSubmitted Dec 11, 2025• Announced Dec 12, 2025• 0 citations• Revised Jun 14, 2026 (v2)
v1 · Dec 11, 2025v2 · Jun 14, 2026
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Abstract

We introduce quantum parameter estimation with the encoding being via a quantum measurement. We quantify the precision for estimating parameters characterizing a general two-outcome qubit measurement, considering two cases: when the outcomes of the encoding measurement are recorded and when the same are ignored. We find that in a large variety of such estimation scenarios, forgetting the outcomes yields higher precision. We derive a necessary criterion under which remembering the measurement outcomes provides better precision in comparison to the outcome-forgotten strategy. Furthermore, we establish a necessary and sufficient criterion for the simultaneous estimation of multiple parameters encoded by an arbitrary quantum process, including those involving measurements, using qubit probes, and find when the quantum Cramér$-$Rao bound is valid and achievable. For simultaneous estimation of two parameters characterizing the measurement, we find that the achievable quantum Cramér$-$Rao bound can be a valid precision bound only when the measurement direction depends on the parameters of interest.

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11 pages, 2 figures

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We introduce quantum parameter estimation with the encoding being via a quantum measurement. We quantify the precision for estimating parameters characterizing a general two-outcome qubit measurement, considering two cases: when the outcomes of the encoding measurement are recorded and when the same are ignored. We find that in a large variety of such estimation scenarios, forgetting the outcomes yields higher precision. We derive a necessary criterion under which remembering the measurement outcomes provides better precision in comparison to the outcome-forgotten strategy. Furthermore, we establish a necessary and sufficient criterion for the simultaneous estimation of multiple parameters encoded by an arbitrary quantum process, including those involving measurements, using qubit probes, and find when the quantum Cramér−-−Rao bound is valid and achievable. For simultaneous estimation of two parameters characterizing the measurement, we find that the achievable quantum Cramér−-−Rao bound can be a valid precision bound only when the measurement direction depends on the parameters of interest.
Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
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.
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