On the role of symmetry and geometry in global quantum sensing

TL;DR

The paper addresses global quantum sensing under finite data by unifying two prior-ignorance paradigms (invariance-based and geometry-based) through a location-isomorphic framework built on a symmetry function $f$. It derives exact multi-shot estimators $\tilde{\vartheta}_{\boldsymbol{y}}(\boldsymbol{x})$ and single-shot POVMs via a Lyapunov equation, and provides a general prescription to construct $f$ from maximum-ignorance priors, via either problem symmetries or information geometry. Through case studies on exponential-rate estimation, coherence under depolarising noise, and atomic lifetimes, the work demonstrates when symmetry-based or geometry-based priors yield faster convergence and how to implement adaptive protocols in practice. The framework yields a practical blueprint for designing high-precision quantum sensors that operate robustly with finite data and varying prior information, bridging foundational concepts with actionable sensing strategies.

Abstract

Global quantum sensing enables parameter estimation across arbitrary ranges with a finite number of measurements. Among the various existing formulations, the Bayesian paradigm stands as a flexible approach for optimal protocol design under minimal assumptions. Within this paradigm, however, there are two fundamentally different ways to capture prior ignorance and uninformed estimation; namely, requiring invariance of the prior distribution under specific parameter transformations, or adhering to the geometry of a state space. In this paper we carefully examine the practical consequences of both the invariance-based and the geometry-based approaches, and show how to apply them in relevant examples of rate and coherence estimation in noisy settings. We find that, while the invariance-based approach often leads to simpler priors and estimators and is more broadly applicable in adaptive scenarios, the geometry-based one can lead to faster posterior convergence in a well-defined measurement setting. Crucially, by employing the notion of location-isomorphic parameters, we are able to unify the two formulations into a single practical and versatile framework for optimal global quantum sensing, detailing when and how each set of assumptions should be employed to tackle any given estimation task. We thus provide a blueprint for the design of novel high-precision quantum sensors.

Figures (7)

• Figure 1: Diagram illustrating the reasoning behind adaptive Bayesian parameter estimation. One begins with a prior distribution $p_0(\theta)$ and a probe state $\rho_{y_0}(\theta)$ that encodes information about the parameter of interest $\theta$. Here, $y_k$ is the control parameter used in shot $k$. A POVM $M_{y_k}(x_k)$ is applied, yielding outcome $x_k$ with likelihood $p(x_k | \theta, y_k) = \Tr\left[ M_{y_k}(x_k) \rho_{y_k}(\theta) \right]$. The prior is updated via Bayes's rule and can be used to optimise the probe state and measurement settings for subsequent shots. When desired, the full set of outcomes $x_1, \ldots, x_n$ can be post-processed to produce an estimate for the unknown parameter. Further details are provided in Sec. \ref{['sec:simmetry-informed_estimation']}.
• Figure 2: Diagram illustrating the construction of loss functions based on specific symmetries or the geometry of the state space. Given the nature of the unknown parameter $\theta$, in the first approach, the ignorance prior satisfies the functional equation \ref{['eq:prior-transformation']}, which requires identifying a symmetry $\tau_\gamma (\theta)$ of the problem. In the information geometry approach, the geodesic length is given by Jeffreys's rule and is directly proportional to the cumulative probability under the ignorance prior. In both frameworks, the symmetry function $f(\theta)$, calculated via Eq. \ref{['eq:symmetry-equation']}, can be computed to map the parameter to a location variable with an associated quadratic loss function (see Sec. \ref{['sec:simmetry-informed_estimation']}). Further details about the scope and limitations of these approaches are given in Sec. \ref{['sec:symmetry-section']}
• Figure 3: Intrinsic precision gain in Eq. \ref{['eq:intrinsic-gain']} for the estimation of a coherence parameter in a depolarising channel with noise rate $\lambda = 0.1$, using geometry- (solid) and transformation-informed (dashed) frameworks. A measurement within the geometry-informed approach provides less information---i.e., it renders a lower $\varepsilon$---because, being more informed initially, it leaves less room for precision improvement. The inset shows the prior for both approaches with $a=0.95$
• Figure 4: Variability of coherence estimates simulated in a depolarising channel with noise rate $\lambda=1/10$ using the transformation- (green circles) and geometry-based (blue triangles) approaches for $\mu=120$ (top panel) and $\mu = 20$ (bottom panel) repetitions of the single-shot optimal measurement rendered by Eq. \ref{['eq:lyapunov']}. The prior width in all cases is $a=1-10^{-5}$. The results obtained from $m = 10$ different realisations are shown, on which both estimation strategies are applied. The underlying true coherence is set to $\zeta = 0.72$ (dashed grey). Insets (a), (b) show the typical convergence of estimates with the number of measurements. We see that the different strategies can give different results for a small dataset, but become similar for a large number of data.
• Figure 5: Intrinsic precision gain in Eq. \ref{['eq:intrinsic-gain']} in the estimation of the lifetime of an atomic state, shown for scale estimation (dashed) and information geometry (solid), as a function of the prior width $b$. The inset shows the ignorance prior of each approach (here, we have set $t = 1\,$s for representation purposes). We see that the prior of scale estimation is concentrated at low values of $\theta$, growing unbounded as $\theta$ is reduced, while the prior of information geometry is concentrated around an intermediate value. This results, for wide prior ranges, in a more uninformed state of knowledge for scale estimation and, consequently, a higher precision gain due to the increased potential for information update. For small prior ranges, the intrinsic precision gain becomes the same for both approaches, reflecting that the prior densities approach a Dirac delta centred at $b=1$.