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arXiv:2610.26319

Squeezing-type lower bounds on generalized metrological sensitivity

Irene García-Martínez, Manuel Gessner

quant-phSubmitted Oct 19, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

We introduce squeezing-type lower bounds for a hierarchy of variance bounds under specific unbiasedness constraints, which generalizes the quantum Cramér-Rao bound. Our construction directly extends standard squeezing coefficients to the corresponding generalized metrological sensitivities while retaining their experimentally accessible structure. We show how observables maximizing these bounds can be identified analytically within families of accessible observables. If arbitrary observables are accessible, the optimized squeezing-type bound recovers the exact generalized metrological sensitivity. These bounds provide efficient and experimentally viable characterizations of generalized metrological sensitivities that are especially relevant in finite-shot scenarios and in the transition from local to global parameter estimation.

Comments
16 pages, 4 figures

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We introduce squeezing-type lower bounds for a hierarchy of variance bounds under specific unbiasedness constraints, which generalizes the quantum Cramér-Rao bound. Our construction directly extends standard squeezing coefficients to the corresponding generalized metrological sensitivities while retaining their experimentally accessible structure. We show how observables maximizing these bounds can be identified analytically within families of accessible observables. If arbitrary observables are accessible, the optimized squeezing-type bound recovers the exact generalized metrological sensitivity. These bounds provide efficient and experimentally viable characterizations of generalized metrological sensitivities that are especially relevant in finite-shot scenarios and in the transition from local to global parameter estimation.
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We derive lower bounds on the variance of estimators in quantum metrology by choosing test observables that define constraints on the unbiasedness of the estimator. The quantum bounds are obtained by analytical optimization over all possible quantum measurements and estimators that satisfy the given constraints. We obtain hierarchies of increasingly tight bounds that include the quantum Cramér-Rao bound at the lowest order. In the opposite limit, the quantum Barankin bound is the variance of the locally best unbiased estimator in quantum metrology. Our results reveal generalizations of the quantum Fisher information that are able to avoid regularity conditions and identify threshold behavior in quantum measurements with mixed states, caused by finite data.
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.
The quantum Cramér-Rao bound (QCRB) stands as a cornerstone of quantum metrology. Yet, akin to its classical counterpart, it provides only local information and overlooks higher-order details. We leverage the theory of higher-order asymptotics to circumvent these issues, providing corrections to the performance of estimators beyond the QCRB. While the QCRB often yields a whole family of optimal states and several optimal measurements, our approach allows us to identify optimal states and measurements within the family that are otherwise equivalent according to the QCRB alone; these are requisite for optimal metrology before reaching the asymptotic limit. These results are especially pertinent when dealing with unitary processes.
The well-known metrological linear squeezing parameters (such as quadrature or spin squeezing) efficiently quantify the sensitivity of Gaussian states. Yet, these parameters are insufficient to characterize the much wider class of highly sensitive non-Gaussian states. Here, we introduce a class of metrological nonlinear squeezing parameters obtained by analytical optimization of measurement observables among a given set of accessible (possibly nonlinear) operators. This allows for the metrological characterization of non-Gaussian quantum states of discrete and continuous variables. Our results lead to optimized and experimentally feasible recipes for a high-precision moment-based estimation of a phase parameter and can be used to systematically construct multipartite entanglement and nonclassicality witnesses for complex quantum states.
We propose a quantum version of the Barankin bound as an alternative to the quantum Cramér-Rao bound for quantum parameter estimation. The quantum Barankin bound provides a lower bound on the mean squared error of estimators satisfying arbitrarily chosen bias constraints at arbitrarily chosen parameter points. In particular, unbiasedness over the entire parameter space can be imposed, yielding precision limits for globally unbiased quantum parameter estimation. In contrast to the recently derived quantum Barankin bound, which is based on a symmetric division superoperator, our bound is based on a nonsymmetric right-division superoperator. This formulation enables us to find analytic expressions for the Barankin matrix for Gaussian states, which allows for an efficient calculation of the bound. We demonstrate the usefulness of our results by applying them to various examples that exhibit the threshold effect in the few-shot regime, which is invisible to the standard quantum Cramér-Rao bound approach.