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arXiv:1609.01618 · New Journal of Physics

Valid lower bound for all estimators in quantum parameter estimation

Jing Liu, Haidong Yuan

Published Sep 6, 2016• 25 citations
arXivHTMLPDF

Abstract

The widely used quantum Cramér–Rao bound (QCRB) sets a lower bound for the mean square error of unbiased estimators in quantum parameter estimation, however, in general QCRB is only tight in the asymptotical limit. With a limited number of measurements biased estimators can have a far better performance for which QCRB cannot calibrate. Here we introduce a valid lower bound for all estimators, either biased or unbiased, which can serve as standard of merit for all quantum parameter estimations.

DOI
10.1088/1367-2630/18/9/093009

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The widely used quantum Cramér–Rao bound (QCRB) sets a lower bound for the mean square error of unbiased estimators in quantum parameter estimation, however, in general QCRB is only tight in the asymptotical limit. With a limited number of measurements biased estimators can have a far better performance for which QCRB cannot calibrate. Here we introduce a valid lower bound for all estimators, either biased or unbiased, which can serve as standard of merit for all quantum parameter estimations.
Seeking the available precision limit of unknown parameters is a significant task in quantum parameter estimation. One often resorts to the widely utilized quantum Cramér-Rao bound (QCRB) based on unbiased estimators to finish this task. Nevertheless, most actual estimators are usually biased in the limited number of trials. For this reason, we introduce two effective error bounds for biased estimators based on a unitary evolution process in the framework of the quantum optimal biased bound. Furthermore, we show their estimation performance by two specific examples of the unitary evolution process, including the phase encoding and the SU(2) interferometer process. Our findings will provide an useful guidance for finding the precision limit of unknown parameters.
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.
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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.
To approach the fundamental limits on the estimation precision for random parameters in quantum systems, we propose a quantum version of the Weiss-Weinstein family of lower bounds on estimation errors. The quantum Weiss-Weinstein bounds (QWWB) include the popular quantum Cram\'er-Rao bound (QCRB) as a special case, and do not require the differentiability of prior distributions and conditional quantum states as the QCRB does; thus, the QWWB is a superior alternative for the QCRB. We show that the QWWB well captures the insurmountable error caused by the ambiguity of the phase in quantum states, which cannot be revealed by the QCRB. Furthermore, we use the QWWB to expose the possible shortcomings of the QCRB when the number of independent and identically distributed systems is not sufficiently large.
The Cramér-Rao bound serves as a crucial lower limit for the mean square error of an estimator in frequentist parameter estimation. Paradoxically, it requires highly accurate prior knowledge of the estimated parameter for constructing the optimal unbiased estimator. In contrast, Bhattacharyya bounds offer a more robust estimation framework with respect to prior accuracy by introducing additional constraints on the estimator. In this work, we examine divergences that arise in the computation of these bounds and establish the conditions under which they remain valid. Notably, we show that when the number of constraints exceeds the number of measurement outcomes, an estimator with finite variance typically does not exist. Furthermore, we systematically investigate the properties of these bounds using paradigmatic examples, comparing them to the Cramér-Rao and Bayesian approaches.