Jesús Rubio

We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation, providing explicit conditions for achieving minimum quadratic losses. Within this framework, we analyse the role of measurement incompatibility and establish its quantitative effect on attainable precision. We achieve this by deriving upper bounds based on the pretty good measurement—a notion originally developed for hypothesis testing—combined with the evaluation of the Nagaoka–Hayashi lower bound. In general, we prove that, as in the many-copy regime of local estimation theory, incompatibility can at most double the minimum loss relative to the idealised scenario in which individually optimal measurements are assumed jointly implementable. This result implies that, in many practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through a range of applications, including discrete quantum phase imaging, phase and dephasing estimation, and qubit sensing, provide analytical and numerical tools for assessing ultimate precision limits and the role of measurement incompatibility in Bayesian multiparameter quantum metrology. We also provide an open-source package that implements all bounds discussed here, enabling practical evaluation and comparison across quantum metrological models.

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.

Global sensing enables parameter estimation across arbitrary parameter ranges with a finite number of shots. While various formulations exist, the Bayesian paradigm offers a flexible approach to optimal protocol design under minimal assumptions. However, there are several sets of assumptions capturing the notions of prior ignorance and uninformed estimation, leading to two main approaches: invariance of the prior distribution under specific parameter transformations, and adherence to the geometry of a state space. While the first approach often leads to simpler priors and estimators and is more broadly applicable in adaptive settings, the second can lead to faster posterior convergence in a well-defined measurement setting. We examine the practical consequences of both approaches and show how to apply them in examples of rate and coherence estimation in noisy scenarios. More importantly, by employing the notion of location-isomorphic parameters, we unify the two approaches into a practical and versatile framework for optimal global quantum sensing, detailing when and how each set of assumptions should be employed—a blueprint for the design of quantum sensors.

Bayesian methods promise enhanced device performance and accelerated data collection. We demonstrate an adaptive Bayesian measurement strategy for atom number estimation in a quantum technology experiment, utilising a symmetry-informed loss function. Compared to a standard unoptimised strategy, our method yields a five-fold reduction in the fractional variance of the atom number estimate. Equivalently, it achieves the target precision with 40% fewer data points. We provide general expressions for the optimal estimator and error for any quantity amenable to symmetry-informed strategies, facilitating the application of these strategies in quantum computing, communication, metrology, and the wider quantum technology sector.

Combining quantum and Bayesian principles leads to optimality in metrology, but the optimization equations involved are often hard to solve. This Letter mitigates this problem with a class of measurement strategies for quantities isomorphic to location parameters, which are shown to admit a closed-form optimization. The resulting framework admits any parameter range, prior information, or state, and the associated estimators apply to finite samples. As an example, the metrology of relative weights is formulated from first principles and shown to require hyperbolic errors. The primary advantage of this approach lies in its simplifying power: it reduces the search for good strategies to identifying which symmetry leaves a state of maximum ignorance invariant. This will facilitate the application of quantum metrology to fundamental physics, where symmetries play a key role. Published by the American Physical Society 2024

The tunable design of protein redox potentials promises to open a range of applications in biotechnology and catalysis. Here we introduce a method to calculate redox potential changes by combining fluctuation relations with molecular dynamics simulations. It involves the simulation of reduced and oxidized states, followed by the instantaneous conversion between them. Energy differences introduced by the perturbations are obtained using the Kubo-Onsager approach. Using a detailed fluctuation relation coupled with Bayesian inference, these are post-processed into estimates for the redox potentials in an efficient manner. This new method, denoted MD+CB, is tested on a de novo four-helix bundle heme protein (the m4D2 ‘maquette’) and five designed mutants, including some mutants characterized experimentally in this work. The MD+CB approach is found to perform reliably, giving redox potential shifts with reasonably good correlation (0.85) to the experimental values for the mutants. The MD+CB approach also compares well with redox potential shift predictions using a continuum electrostatic method. The estimation method employed within the MD+CB approach is straightforwardly transferable to standard equilibrium MD simulations, and holds promise for redox protein engineering and design applications.

Precise temperature measurements on systems of few ultracold atoms is of paramount importance in quantum technologies, but can be very resource intensive. Here, we put forward an adaptive Bayesian framework that substantially boosts the performance of cold atom temperature estimation. Specifically, we process data from real and simulated release-recapture thermometry experiments on few potassium atoms cooled down to the microkelvin range in an optical tweezer. From simulations, we demonstrate that adaptively choosing the release-recapture times to maximize information gain does substantially reduce the number of measurements needed for the estimate to converge to a final reading. Unlike conventional methods, our proposal systematically avoids capturing and processing uninformative data. We also find that a simpler nonadaptive method exploiting all the a priori information can yield competitive results, and we put it to the test on real experimental data. Furthermore, we are able to produce much more reliable estimates, especially when the measured data are scarce and noisy, and they converge faster to the real temperature in the asymptotic limit. Importantly, the underlying Bayesian framework is not platform specific and can be adapted to enhance precision in other setups, thus opening new avenues in quantum thermometry.

Quantum scale estimation, as introduced and explored here, establishes the most precise framework for the estimation of scale parameters that is allowed by the laws of quantum mechanics. This addresses an important gap in quantum metrology, since current practice focuses almost exclusively on the estimation of phase and location parameters. For given prior probability and quantum state, and using Bayesian principles, a rule to construct the optimal probability-operator measurement is provided. Furthermore, the corresponding minimum mean logarithmic error is identified. This is then generalised as to accommodate the simultaneous estimation of multiple scale parameters, and a procedure to classify practical measurements into optimal, almost-optimal or sub-optimal is highlighted. As a means of illustration, the new framework is exploited to generalise scale-invariant global thermometry, as well as to address the estimation of the lifetime of an atomic state. On a more conceptual note, the optimal strategy is employed to construct an observable for scale parameters, an approach which may serve as a template for a more systematic search of quantum observables. Quantum scale estimation thus opens a new line of enquire-the precise measurement of scale parameters such as temperatures and rates-within the quantum information sciences.

We argue that analysing schemes for metrology solely in terms of the average particle number can obscure the number of particles effectively used in informative events. For a number of states we demonstrate that, in both frequentist and Bayesian frameworks, the average number of a state can essentially be decoupled from the aspects of the total number distribution associated with any metrological advantage.

A paradigm shift in quantum thermometry is proposed. To date, thermometry has relied on local estimation, which is useful to reduce statistical fluctuations once the temperature is very well known. In order to estimate temperatures in cases where few measurement data or no substantial prior knowledge are available, we build instead a method for global quantum thermometry. Based on scaling arguments, a mean logarithmic error is shown here to be the correct figure of merit for thermometry. Its full minimization provides an operational and optimal rule to postprocess measurements into a temperature reading, and it establishes a global precision limit. We apply these results to the simulated outcomes of measurements on a spin gas, finding that the local approach can lead to biased temperature estimates in cases where the global estimator converges to the true temperature. The global framework thus enables a reliable approach to data analysis in thermometry experiments.