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arXiv:2303.06108 · Physical Review Letters

Hierarchies of Frequentist Bounds for Quantum Metrology: From Cramér-Rao to Barankin

Manuel Gessner, Augusto Smerzi

Published Jun 29, 2023• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.130.260801

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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.
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Quantum metrology establishes fundamental precision limits for parameter estimation, yet standard Quantum Cramér‐Rao Bound (QCRB) becomes insufficient in the finite‐data regime. In this article, the QCRB is systematically tighten using a hybrid classical‐quantum framework by incorporating generalized unbiasedness constraints through test observables. These frequentist bounds form a hierarchy converging asymptotically to the QCRB as more measurements are performed. These hierarchical frequentist bounds are validated for single‐qubit phase estimation on a photonic platform.
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.
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