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arXiv:2503.05483 · Quantum Science and Technology

Metrological symmetries in singular quantum multi-parameter estimation

George Mihailescu, Saubhik Sarkar, Abolfazl Bayat, Steve Campbell, Andrew K. Mitchell

Published Oct 24, 2025• 9 citations
arXivHTMLPDF

Abstract

Abstract The theoretical foundation of quantum sensing is rooted in the Cramér–Rao formalism, which establishes quantitative precision bounds for a given quantum probe. In many practical scenarios, where more than one parameter is unknown, the multi-parameter Cramér–Rao bound (CRB) applies. Since this is a matrix inequality involving the inverse of the quantum Fisher information matrix (QFIM), the formalism breaks down when the QFIM is singular. In this paper, we examine the physical origins of such singularities, showing that they result from an over-parameterization on the metrological level. This is itself caused by emergent metrological symmetries , whereby the same set of measurement outcomes are obtained for different combinations of system parameters. Although the number of effective parameters is equal to the number of non-zero QFIM eigenvalues, the Cramér–Rao formalism typically does not provide information about the effective parameter encoding. Instead, we demonstrate through a series of concrete examples that Bayesian estimation can provide deep insights. In particular, the metrological symmetries appear in the Bayesian posterior distribution as lines of persistent likelihood running through the space of unknown parameters. These lines are contour lines of the effective parameters which, through suitable parameter transformations, can be estimated and follow their own effective CRBs.

DOI
10.1088/2058-9565/ae1757

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Abstract The theoretical foundation of quantum sensing is rooted in the Cramér–Rao formalism, which establishes quantitative precision bounds for a given quantum probe. In many practical scenarios, where more than one parameter is unknown, the multi-parameter Cramér–Rao bound (CRB) applies. Since this is a matrix inequality involving the inverse of the quantum Fisher information matrix (QFIM), the formalism breaks down when the QFIM is singular. In this paper, we examine the physical origins of such singularities, showing that they result from an over-parameterization on the metrological level. This is itself caused by emergent metrological symmetries , whereby the same set of measurement outcomes are obtained for different combinations of system parameters. Although the number of effective parameters is equal to the number of non-zero QFIM eigenvalues, the Cramér–Rao formalism typically does not provide information about the effective parameter encoding. Instead, we demonstrate through a series of concrete examples that Bayesian estimation can provide deep insights. In particular, the metrological symmetries appear in the Bayesian posterior distribution as lines of persistent likelihood running through the space of unknown parameters. These lines are contour lines of the effective parameters which, through suitable parameter transformations, can be estimated and follow their own effective CRBs.
In quantum multi-parameter estimation, the precision of estimating unknown parameters is bounded by the Cramér-Rao bound (CRB), defined via the inverse of the Fisher information matrix (FIM). However, in certain scenarios—such as distributed quantum sensing—the FIM becomes non-invertible due to parameter redundancy, which depends on the probe state and measurement. This issue is often handled using a weaker form of the CRB, potentially overestimating the uncertainty and underrepresenting achievable precision. Here, we propose an alternative approach by introducing equality constraints to remove redundancy and define the CRB via the Moore-Penrose pseudoinverse of the FIM. This framework enables systematic treatment of both simultaneous estimation and distributed sensing cases. We demonstrate its utility by reanalyzing several known examples within this unified perspective, highlighting improved interpretability and practical relevance. Our results offer a concrete guideline for addressing non-invertible FIMs and enhancing the precision of quantum multi-parameter estimation in realistic scenarios.
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.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.
Abstract Chapter 24 provides a compact introduction to the topic of quantum metrology, focusing on Hamiltonian parameter estimation in the frequentist and in the Bayesian paradigms. We first discuss how estimates of non-directly measurable quantities such as phases are obtained from measurement statistics in the frequentist approach. We then discuss the Cramér-Rao bound and the Fisher information and study single-qubit phase estimation in the light of this result. We then turn to the multi-qubit setting and discuss the quantum Cramér-Rao bound, for which the quantum Fisher information is the central quantity of interest. In this context we discuss the Uhlmann fidelity in detail. We further contrast Heisenberg scaling with the standard quantum limit in the phase-estimation scenario. Finally, we analyse phase estimation in the Bayesian-estimation approach and derive the van Trees inequality as a Bayesian version of the Cramér-Rao bound
Abstract This review aims at gathering the most relevant quantum multi-parameter estimation methods that go beyond the direct use of the quantum Fisher information concept. We discuss in detail the Holevo Cramér–Rao bound, the quantum local asymptotic normality approach as well as Bayesian methods. Even though the fundamental concepts in the field have been laid out more than forty years ago, a number of important results have appeared much more recently. Moreover, the field drew increased attention recently thanks to advances in practical quantum metrology proposals and implementations that often involve estimation of multiple parameters simultaneously. Since the topics covered in these review are spread in the literature and often served in a very formal mathematical language, one of the main goals of this review is to provide a largely self-contained work that allows the reader to follow most of the derivations and get an intuitive understanding of the interrelations between different concepts using a set of simple yet representative examples involving qubit and Gaussian shift models.
As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.