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Weiss-Weinstein Family of Error Bounds for Quantum Parameter Estimation

Xiao-Ming Lu, Mankei Tsang

Published Nov 29, 2015• 0 citations
OpenAlex

Abstract

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.

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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.
Abstract Sensing and imaging are among the most important applications of quantum information science. To investigate their fundamental limits and the possibility of quantum enhancements, for decades researchers have relied on the quantum Cramér-Rao lower error bounds pioneered by Helstrom. Recent work, however, has called into question the tightness of those bounds for highly nonclassical states in the non-asymptotic regime, and better methods are now needed to assess the attainable quantum limits in reality. Here we propose a new class of quantum bounds called quantum Weiss-Weinstein bounds, which include Cramér-Rao-type inequalities as special cases but can also be significantly tighter to the attainable error. We demonstrate the superiority of our bounds through the derivation of a Heisenberg limit and phase estimation examples.
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.
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.
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.
I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cramér-Rao bounds for quantum parameter estimation. From a simple form of the proposed bounds, I derive both a Heisenberg error limit that scales with the average energy and a limit similar to the quantum Cramér-Rao bound that scales with the energy variance. These results are further illustrated by applying the bound to a few examples of optical phase estimation, which show that a quantum Ziv-Zakai bound can be much higher and thus tighter than a quantum Cramér-Rao bound for states with highly non-gaussian photon-number statistics in certain regimes and also stay close to the latter where the latter is expected to be tight.
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.