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IEEE International Symposium on Information Theory (ISIT)

On Two-Stage Quantum Estimation and Asymptotics of Quantum-Enhanced Transmittance Sensing

Zihao Gong, Boulat A. Bash

Published Jul 7, 2024• 1 citation
Publisher

Abstract

Quantum Cramér-Rao bound is the ultimate limit of the mean squared error for unbiased estimation of an unknown parameter embedded in a quantum state. While it can be achieved asymptotically for large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. This paradox was addressed by Hayashi and Matsumoto using a two-stage approach in 2005. Unfortunately, their analysis imposes conditions that severely restrict the class of classical estimators applied to the quantum measurement outcomes, hindering applications of this method. We relax these conditions to substantially broaden the class of usable estimators at the cost of slightly weakening the asymptotic properties of the two-stage method. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.

DOI
10.1109/isit57864.2024.10619296

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Quantum Cramér-Rao bound is the ultimate limit of the mean squared error for unbiased estimation of an unknown parameter embedded in a quantum state. While it can be achieved asymptotically for large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. This paradox was addressed by Hayashi and Matsumoto using a two-stage approach in 2005. Unfortunately, their analysis imposes conditions that severely restrict the class of classical estimators applied to the quantum measurement outcomes, hindering applications of this method. We relax these conditions to substantially broaden the class of usable estimators at the cost of slightly weakening the asymptotic properties of the two-stage method. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.
We consider estimation of a single unknown parameter embedded in a quantum state. Quantum Cramér-Rao bound (QCRB) is the ultimate limit of the mean squared error for any unbiased estimator. While it can be achieved asymptotically for a large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. Prior work addresses this paradox using a two-stage approach: in the first stage, a preliminary estimate is obtained by applying, on a vanishing fraction of quantum state copies, a sub-optimal measurement that does not depend on the parameter of interest. In the second stage, the preliminary estimate is used to construct the QCRB-achieving measurement that is applied to the remaining quantum state copies. This is akin to two-step estimators for classical problems with nuisance parameters. Unfortunately, the original analysis imposes conditions that severely restrict the class of classical estimators applied to the quantum measurement outcomes, hindering applications of this method. We relax these conditions to substantially broaden the class of usable estimators for single-parameter problems at the cost of slightly weakening the asymptotic properties of the two-stage method. We also account for nuisance parameters. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.
Quantum-enhanced (i.e., higher performance by quantum effects than any classical methods) mean value estimation of observables is a fundamental task in various quantum technologies; in particular, it is an essential subroutine in quantum computing algorithms. Notably, the quantum estimation theory identifies the ultimate precision of such an estimator, which is referred to as the quantum Cramér-Rao (QCR) lower bound or equivalently the inverse of the quantum Fisher information. Because the estimation precision directly determines the performance of those quantum technological systems, it is highly demanded to develop a generic and practically implementable estimation method that achieves the QCR bound. Under imperfect conditions, however, such an ultimate and implementable estimator for quantum mean values has not been developed. In this paper, we propose a quantum-enhanced mean value estimation method in a depolarizing noisy environment that asymptotically achieves the QCR bound in the limit of a large number of qubits. To approach the QCR bound in a practical setting, the method adaptively optimizes the amplitude amplification and a specific measurement that can be implemented without any knowledge of state preparation. We provide a rigorous analysis for the statistical properties of the proposed adaptive estimator such as consistency and asymptotic normality. Furthermore, several numerical simulations are provided to demonstrate the effectiveness of the method, particularly showing that the estimator needs only a modest number of measurements to almost saturate the QCR bound.
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 field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.
We consider the problem of estimating unknown transmittanceθ\thetaθof a target bathed in thermal background light. As quantum estimation theory yields the fundamental limits, we employ the lossy thermal-noise bosonic channel model, which describes sensor-target interaction quantum mechanically in many practical active-illumination systems (e.g., using emissions at optical, microwave, or radio frequencies). We prove that quantum illumination using two-mode squeezed vacuum (TMSV) states asymptotically achieves minimal quantum Cramér-Rao bound (CRB) over all quantum states (not necessarily Gaussian) in the limit of low transmitted power. We characterize the optimal receiver structure for TMSV input, and show its advantage over other receivers using both analysis and Monte Carlo simulation.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.