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arXiv:2606.25920

Finite-Shot Sensitivity for Moment Estimation in Quantum Metrology

Shaowei Du, Shuheng Liu, Weidong Li, Luca Pezzè, Augusto Smerzi, Qiongyi He

quant-phSubmitted Jun 24, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

The quantum Cramér-Rao bound can be saturated only asymptotically and does not specify how many measurements are needed for a concrete estimator to approach it. We develop a finite-measurement theory for method-of-moments estimation, where the parameter is inferred from the sample mean of a calibrating observable rather than from the full likelihood. For general quantum statistical models, the expansion is written in terms of the calibration curve and the central moments of the measured observable. Nonlinear calibration curves make the usual moment estimator biased at finite measurement number; we construct a bias-corrected estimator with bias $O(ν^{-3})$. This gives sensitivity corrections beyond the leading error-propagation term of the chosen moment protocol. We identify a general density-matrix condition under which the full $1/ν^2$ correction vanishes. In unitary examples, the leading residual correction appears at order $1/ν^3$, is governed by calibration curvature, and can be reduced or cancelled by higher-rank components of the same measured observable. The resulting thresholds quantify how many measurements are needed before the asymptotic sensitivity of a moment-estimation protocol is operationally visible.

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16 pages, 5 figures

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The quantum Cramér-Rao bound can be saturated only asymptotically and does not specify how many measurements are needed for a concrete estimator to approach it. We develop a finite-measurement theory for method-of-moments estimation, where the parameter is inferred from the sample mean of a calibrating observable rather than from the full likelihood. For general quantum statistical models, the expansion is written in terms of the calibration curve and the central moments of the measured observable. Nonlinear calibration curves make the usual moment estimator biased at finite measurement number; we construct a bias-corrected estimator with bias O(ν−3)O(ν^{-3})O(ν−3). This gives sensitivity corrections beyond the leading error-propagation term of the chosen moment protocol. We identify a general density-matrix condition under which the full 1/ν21/ν^21/ν2 correction vanishes. In unitary examples, the leading residual correction appears at order 1/ν31/ν^31/ν3, is governed by calibration curvature, and can be reduced or cancelled by higher-rank components of the same measured observable. The resulting thresholds quantify how many measurements are needed before the asymptotic sensitivity of a moment-estimation protocol is operationally visible.
In quantum metrology, one of the major applications of quantum technologies, the ultimate precision of estimating an unknown parameter is often stated in terms of the Cramér-Rao bound. Yet, the latter is no longer guaranteed to carry an operational meaning in the regime where few measurement samples are obtained, which we illustrate through a simple example. We instead propose to quantify the quality of a metrology protocol by the probability of obtaining an estimate with a given accuracy. This approach, which we refer to as probably approximately correct (PAC) metrology, ensures operational significance in the finite-sample regime. The accuracy guarantees hold for any value of the unknown parameter, unlike the Cramér-Rao bound which assumes it is approximately known. We establish a strong connection to multi-hypothesis testing with quantum states, which allows us to derive an analogue of the Cramér-Rao bound which contains explicit corrections relevant to the finite-sample regime. We further study the asymptotic behavior of the success probability of the estimation procedure for many copies of the state and apply our framework to the example task of phase estimation with an ensemble of spin-1/2 particles. Overall, our operational approach allows the study of quantum metrology in the finite-sample regime and opens up a plethora of new avenues for research at the interface of quantum information theory and quantum metrology.
Many results in the quantum metrology literature use the Cramér–Rao bound and the Fisher information to compare different quantum estimation strategies. However, there are several assumptions that go into the construction of these tools, and these limitations are sometimes not taken into account. While a strategy that utilizes this method can considerably simplify the problem and is valid asymptotically, to have a rigorous and fair comparison we need to adopt a more general approach. In this work we use a methodology based on Bayesian inference to understand what happens when the Cramér–Rao bound is not valid. In particular we quantify the impact of these restrictions on the overall performance of a wide range of schemes including those commonly employed for the estimation of optical phases. We calculate the number of observations and the minimum prior knowledge that are needed such that the Cramér–Rao bound is a valid approximation. Since these requirements are state-dependent, the usual conclusions that can be drawn from the standard methods do not always hold when the analysis is more carefully performed. These results have important implications for the analysis of theory and experiments in quantum metrology.
Many results in the quantum metrology literature use the Cramér-Rao bound and the Fisher information to compare different quantum estimation strategies. However, there are several assumptions that go into the construction of these tools, and these limitations are sometimes not taken into account. While a strategy that utilises this method can considerably simplify the problem and is valid asymptotically, to have a rigorous and fair comparison we need to adopt a more general approach. In this work we use a methodology based on Bayesian inference to understand what happens when the Cramér-Rao bound is not valid. In particular we quantify the impact of these restrictions on the overall performance of a wide range of schemes including those commonly employed for the estimation of optical phases. We calculate the number of observations and the minimum prior knowledge that are needed such that the Cramér-Rao bound is a valid approximation. Since these requirements are state-dependent, the usual conclusions that can be drawn from the standard methods do not always hold when the analysis is more carefully performed. These results have important implications for the analysis of theory and experiments in quantum metrology.
In quantum metrology, a major application of quantum technologies, the ultimate precision of estimating an unknown parameter is often stated in terms of the Cramér-Rao bound. Yet, the latter fails to completely characterize the distribution of estimates in the nonasymptotic regime. Optimizing metrology protocols with respect to the Cramér-Rao bound can therefore lead to surprisingly poor finite-sample performance. This can be avoided by quantifying the quality of a metrology protocol by the probability of obtaining an estimate with a given accuracy. Using this intrinsically single-shot quantity naturally accommodates the finite-sample regime. We show that the fundamental limits of this figure of merit can be quantified through a multihypothesis testing problem between quantum states. Building on this connection, we derive an analogue of the Cramér-Rao bound that contains explicit corrections relevant to the finite-sample regime. We apply the finite-sample metrology framework to the example task of phase estimation with an ensemble of spin-1/2 particles, giving unambiguous evidence that large quantum Fisher information does not guarantee that a metrology protocol has a good finite-sample performance. Overall, analyzing the probability of success allows the reliable study of quantum metrology in the finite-sample regime and opens up a plethora of new avenues for research at the interface of quantum information theory and quantum metrology.
Quantum Fisher information and the associated quantum Cramér-Rao bound (QCRB) are fundamental tools in frequentist quantum metrology, offering both analytical simplicity and practical precision limits for parameter estimation. The QCRB sets a lower bound on the mean square error (MSE) in the idealized limit of infinite measurement trials ( ν → ∞ ). Here we perform a systematic expansion in powers of 1 / ν to refine MSE estimates in realistic, finite-resource scenarios. These corrections reveal differences between measurements that appear equally optimal under the QCRB. They also help to distinguish among multiple optimal state families for estimating an unknown unitary transformation. Additionally, we explore the Bhattacharyya bound and its quantum counterpart, which constrain these corrections. Our results are relevant for preasymptotic metrology, enabling optimized protocols with limited resources without reliance on numerical simulations.
Quantum Cramér--Rao theory is intrinsically local: it bounds precision near a specified parameter value, and its saturating measurement generally depends on that value. Barankin-type bounds use finite parameter displacements, but remain anchored to a chosen reference value. This leaves open a basic global-estimation problem: when the parameter is known only within a broad domain, what precision can be guaranteed by a single estimator and a single measurement strategy fixed before the true value is localized? We answer this question by introducing global score functions tied to a weighted variance over the whole parameter domain. Their correlations generate a hierarchy of precision bounds: global Cramér--Rao and Barankin-type bounds arise as restricted levels, whereas unrestricted score correlations yield a fully global bound for the prescribed weighted variance. The hierarchy recovers local Cramér--Rao theory in the many-repetition limit and reveals genuinely global precision limits for finite data over broad domains. In the quantum setting, the construction identifies when this fully global bound can be realized by a single parameter-independent measurement. The same framework extends to Bayesian estimation, recovering the Van Trees bound in the local limit while yielding stronger finite-width lower bounds on the Bayesian mean-square error beyond this limit.