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arXiv:2404.02203

James-Stein Estimation in Quantum Gaussian Sensing

Wilfred Salmon, Sergii Strelchuk, David R. M. Arvidsson-Shukur

Published Apr 2, 2024• 0 citations
arXivHTMLPDF

Abstract

The James-Stein estimator is a biased estimator -- for a finite number of samples its expected value is not the true mean. The maximum-likelihood estimator (MLE), is unbiased and asymptotically optimal. Yet, when estimating the mean of $3$ or more normally-distributed random variables, the James-Stein estimator has a smaller total (expected) error than the MLE. We introduce the James-Stein estimator to the field of quantum metrology, from both the frequentist and Bayesian perspectives. We characterise the effect of quantum phenomena on the James-Stein estimator through the lens of quantum Gaussian sensing, the task of estimating the mean of an unknown multivariate quantum Gaussian state. We find that noiseless entanglement or coherence improves performance of the James-Stein estimator, but diminishes its advantage over the MLE. In the presence of noise, the James-Stein advantage is restored. Quantum effects can also boost the James-Stein advantage. We demonstrate this by investigating multivariate postselective metrology (generalised weak-value amplification), a strategy that uses quantum effects to measure parameters with imperfect detectors. Simply by post-processing measured data differently, our techniques reduce errors in quantum experiments.

DOI
10.48550/arxiv.2404.02203

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The James-Stein estimator is a biased estimator -- for a finite number of samples its expected value is not the true mean. The maximum-likelihood estimator (MLE), is unbiased and asymptotically optimal. Yet, when estimating the mean of 333 or more normally-distributed random variables, the James-Stein estimator has a smaller total (expected) error than the MLE. We introduce the James-Stein estimator to the field of quantum metrology, from both the frequentist and Bayesian perspectives. We characterise the effect of quantum phenomena on the James-Stein estimator through the lens of quantum Gaussian sensing, the task of estimating the mean of an unknown multivariate quantum Gaussian state. We find that noiseless entanglement or coherence improves performance of the James-Stein estimator, but diminishes its advantage over the MLE. In the presence of noise, the James-Stein advantage is restored. Quantum effects can also boost the James-Stein advantage. We demonstrate this by investigating multivariate postselective metrology (generalised weak-value amplification), a strategy that uses quantum effects to measure parameters with imperfect detectors. Simply by post-processing measured data differently, our techniques reduce errors in quantum experiments.
Quantum metrology protocols typically consist of four steps: state preparation, evolution, measurement, and data processing. Often, the first three steps are prioritised when designing a scheme as they contain all the quantum elements. The data analysis is generally considered an add-on with an implicit assumption that this step is well behaved and so standard data techniques can be applied. However, the situation can be more nuanced, such as when the available data are limited. In limited-data quantum metrology the choice of data analysis technique and cost function of the estimator is of great importance, and a reliable prior distribution of the unknown parameters is required for Bayesian analysis. An interesting question is what we should do when no such prior is available. In this work, we consider how the James-Stein estimator can give significant advantages when measuring multiple unknown parameters with limited data and, importantly, does not require any prior distribution. We demonstrate the advantage by applying this methodology to simple quantum metrology schemes.
We propose the first near-optimal quantum algorithm for estimating in Euclidean norm the mean of a vector-valued random variable with finite mean and covariance. Our result aims at extending the theory of multivariate sub-Gaussian estimators [Lugosi and Mendelson, 2019] to the quantum setting. Unlike classically, where any univariate estimator can be turned into a multivariate estimator with at most a logarithmic overhead in the dimension, no similar result can be proved in the quantum setting. Indeed, [Heinrich, 2004] ruled out the existence of a quantum advantage for the mean estimation problem when the sample complexity is smaller than the dimension. Our main result is to show that, outside this low-precision regime, there does exist a quantum estimator that outperforms any classical estimator. More precisely, we prove that the approximation error can be decreased by a factor of about the square root of the ratio between the dimension and the sample complexity. Our approach is substantially more involved than in the univariate setting, where most quantum estimators rely only on phase estimation. We exploit a variety of additional algorithmic techniques such as linear amplitude amplification, the Bernstein-Vazirani algorithm, and quantum singular value transformation. Our analysis is also deeply rooted in proving concentration inequalities for multivariate truncated statistics. Finally, we describe several applications of our algorithms, notably in measuring the expectation values of commuting observables and in the field of machine learning.
We undertake a detailed study of the performance of maximum likelihood (ML) estimators of the density matrix of finite-dimensional quantum systems, in order to interrogate generic properties of frequentist quantum state estimation. Existing literature on frequentist quantum estimation has not rigorously examined the finite sample performance of the estimators and associated methods of hypothesis testing. While ML is usually preferred on the basis of its asymptotic properties - it achieves the Cramer-Rao (CR) lower bound - the finite sample properties are often less than optimal. We compare the asymptotic and finite-sample properties of the ML estimators and test statistics for two different choices of measurement bases: the average case optimal or mutually unbiased bases (MUB) and a representative set of suboptimal bases, for spin-1/2 and spin-1 systems. We show that, in both cases, the standard errors of the ML estimators sometimes do not contain the true value of the parameter, which can render inference based on the asymptotic properties of the ML unreliable for experimentally realistic sample sizes. The results indicate that in order to fully exploit the information geometry of quantum states and achieve smaller reconstruction errors, the use of Bayesian state reconstruction methods - which, unlike frequentist methods, do not rely on asymptotic properties - is desirable, since the estimation error is typically lower due to the incorporation of prior knowledge.
Probabilistic metrology attempts to improve parameter estimation by occasionally reporting an excellent estimate and the rest of the time either guessing or doing nothing at all. Here we show that probabilistic metrology can never improve quantum limits on estimation of a single parameter, both on average and asymptotically in number of trials, if performance is judged relative to mean-square estimation error. We extend the result by showing that for a finite number of trials, the probability of obtaining better estimates using probabilistic metrology, as measured by mean-square error, decreases exponentially with the number of trials. To be tight, the performance bounds we derive require that likelihood functions be approximately normal, which in turn depends on how rapidly specific distributions converge to a normal distribution with number of trials.
Recently, there has been a great deal of interest surrounding the calibration of quantum sensors using machine learning techniques. This work explores the use of regression to infer a machine-learned point estimate of an unknown parameter. Although the analysis is necessarily frequentist—relying on repeated estimates to build up statistics—the authors clarify that this machine-learned estimator converges to the Bayesian maximum a posteriori estimator (subject to some regularity conditions). When the number of training measurements is large, this is identical to the well-known maximum-likelihood estimator (MLE), and using this fact, the authors argue that the Cramér–Rao sensitivity bound applies to the mean-square error cost function and can therefore be used to select optimal model and training parameters. The machine-learned estimator inherits the desirable asymptotic properties of the MLE, up to a limit imposed by the resolution of the training grid. Furthermore, the authors investigate the role of quantum noise in the training process and show that this noise imposes a fundamental limit on the number of grid points. This manuscript paves the way for machine-learning to assist the calibration of quantum sensors, thereby allowing maximum-likelihood inference to play a more prominent role in the design and operation of the next generation of ultra-precise sensors.
Accurately inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware. The predominant state estimation procedure, maximum likelihood estimation (MLE), generally reports an estimate with zero eigenvalues. These cannot be justified. Furthermore, the MLE estimate is incompatible with error bars, so conclusions drawn from it are suspect. I propose an alternative procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues, its eigenvalues provide a bound on their own uncertainties, and under certain circumstances it is provably the most accurate procedure possible. I show how to implement BME numerically, and how to obtain natural error bars that are compatible with the estimate. Finally, I briefly discuss the differences between Bayesian and frequentist estimation techniques.