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arXiv:2607.12651

James-Stein estimation for quantum sensing schemes

Luke Alexander Rhodes, Sean William Moore, Jacob A. Dunningham

quant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations
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Abstract

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.

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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.
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.
A longstanding problem in quantum metrology is how to extract as much information as possible in realistic scenarios with not only multiple unknown parameters, but also limited measurement data and some degree of prior information. Here we present a practical solution to this: We derive a Bayesian multi-parameter quantum bound, construct the optimal measurement when our bound can be saturated for a single shot, and consider experiments involving a repeated sequence of these measurements. Our method properly accounts for the number of measurements and the degree of prior information, and we illustrate our ideas with a qubit sensing network and a model for phase imaging, clarifying the nonasymptotic role of local and global schemes. Crucially, our technique is a powerful way of implementing quantum protocols in a wide range of practical scenarios that tools such as the Helstrom and Holevo Cram\'er-Rao bounds cannot normally access.
Science relies on our practical ability to extract information from reality, since processing this information is essential for developing theories that explain our world. This thesis is precisely the study of how to extract and process information using quantum systems when a constrained amount of resources means that the available data is limited. The natural framework for this task is quantum metrology, a set of tools to model and design quantum measurement strategies. Equipped with this theory, we advocate a Bayesian approach as the appropriate formalism to study systems with a finite amount of resources, which is a non-asymptotic problem, and we propose a methodology for non-asymptotic quantum metrology. To start with, we show the consistency of taking those solutions that are optimal in the asymptotic regime of many trials as a guide to calculate a generalised measure of uncertainty in the Bayesian framework. This provides an approximate but useful way of studying the non-asymptotic regime whenever a direct Bayesian optimisation is intractable, and it avoids non-physical results that can arise when only the asymptotic theory is employed. Secondly, we construct a new non-asymptotic Bayesian bound without relying on the previous approximation by first selecting the optimal quantum strategy for a single shot, and then simulating a sequence of repetitions of this scheme, which is suitable for experiments where we do not wish or cannot correlate different trials. These methods are applied to a Mach-Zehnder interferometer, which is a single-parameter problem, and to quantum sensing networks where the nodes are either qubits or optical modes, which are multi-parameter protocols. Our results provide a detailed characterisation of how the interplay between prior information, entanglement and a limited amount of data affects the performance of quantum metrology protocols, which has important implications for the analysis of theory and experiments in this field.
The number of times that we can access a system to extract information via quantum metrology is always finite, and possibly small, and realistic amounts of prior knowledge tend to be moderate. Thus theoretical consistency demands a methodology that departs from asymptotic approximations and restricted parameter locations, while practical convenience requires that it is also flexible and easy to use in applications with limited data. We submit that this methodology can and should be built on a Bayesian framework, and in this thesis we propose, construct, explore and exploit a new non-asymptotic quantum metrology. First we show the consistency of taking those solutions that are optimal in the asymptotic regime of many trials as a guide to calculate a Bayesian measure of uncertainty. This provides an approximate but useful way of studying the non-asymptotic regime whenever an exact optimisation is intractable, and it avoids the non-physical results that can arise when only the asymptotic theory is used. Secondly, we construct a new non-asymptotic Bayesian bound without relying on the previous approximation by first selecting a single-shot optimal quantum strategy, and then simulating a sequence of repetitions of this scheme. These methods are then applied to a single-parameter Mach-Zehnder interferometer, and to multi-parameter qubit and optical sensing networks. Our results provide a detailed characterisation of how the interplay between prior information, correlations and a limited amount of data affects the performance of quantum metrology protocols, which opens the door to a vast set of unexplored possibilities to enhance non-asymptotic schemes. Finally, we provide practical researchers with a numerical toolbox for Bayesian metrology, while theoretical workers will benefit from the broader and more fundamental perspective that arises from the unified character of our methodology.
The simultaneous estimation of multiple unknown parameters is the most general scenario in quantum sensing. Quantum multiparameter estimation theory provides fundamental bounds on the achievable precision of simultaneous estimation. However, these bounds can become singular (no finite bound exists) in multiparameter sensing due to parameter interdependencies, limited probe accessibility, and insufficient measurement outcomes. Here, we address the singularity issue in quantum sensing through a simple mechanism based on a sequential measurement strategy. This sensing scheme overcomes the singularity constraint and enables the simultaneous estimation of multiple parameters with a local and fixed measurement throughout the sensing protocol. This is because sequential measurements, involving consecutive steps of local measurements followed by probe evolution, inherently produce correlated measurement data that grows exponentially with the number of sequential measurements. Finally, through two different examples, namely a strongly correlated probe and a light-matter system, we demonstrate how such singularities are reflected when inferring the unknown parameters through Bayesian estimation.
Abstract Quantum parameter estimation offers solid conceptual grounds for the design of sensors enjoying quantum advantage. This is realised not only by means of hardware supporting and exploiting quantum properties, but data analysis has its impact and relevance, too. In this respect, Bayesian methods have emerged as an effective and elegant solution, with the perk of incorporating naturally the availability of a priori information. In this article we present an evaluation of Bayesian methods for multiple phase estimation, assessed based on bounds that work beyond the usual limit of large samples assumed in parameter estimation. Importantly, such methods are applied to experimental data generated from the output statistics of a three-arm interferometer seeded by single photons. Our studies provide a blueprint for a more comprehensive data analysis in quantum metrology.