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arXiv:2602.12193

A Framework for Spatial Quantum Sensing

Luís Bugalho, Yasser Omar, Damian Markham

quant-phSubmitted Feb 12, 2026• Announced Feb 13, 2026• 0 citations
v1 · Feb 12, 2026v2 · May 14, 2026
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Abstract

Analytical and algebraic geometry are valuable tools for dealing with problems involving analytical functions and polynomials. In what we connote as spatial quantum sensing the goal is, given an underlying field and a set of quantum sensors interrogating the field in a set of positions, to find an estimator for some property the field. This property can have multiple forms, be it distinguishing the source of a target signal, or evaluating the field (or a derivative thereof) in an arbitrary position. In this work we also link this problem to networks of quantum sensors, and the role and usefulness of entangling these sensors. We find that the estimators that come out as a solution to the problem are such that a non-local entangled strategy provides maximum precision. We start by working under the assumption of polynomial fields, which relates to the interpolation problem, and then generalize for any signal that is modeled via analytical functions, giving rise to any general least-squares estimator. We discuss the effects of the placement of the sensors in the estimation, namely, how to find well defined, construction error-free placements for the sensors. In the case of interpolation we provide concrete examples and proofs in a $m$-dimensional array of sensors, and discuss necessary and sufficient conditions for the more general cases. We provide clear examples of the possible use-cases and statements, and compare a non-local entangled strategy with the best local strategy for an interpolation problem, showing the benefit in terms of precision in a distributed sensing scenario. This is a key tool for a wide-range of problem in sensing problems, ranging from large-scale such as earth-sized experiments, to local-scale, such has biological experiments.

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Keywords: Quantum Distributed Sensing, Quantum Information

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Analytical and algebraic geometry are valuable tools for dealing with problems involving analytical functions and polynomials. In what we connote as spatial quantum sensing the goal is, given an underlying field and a set of quantum sensors interrogating the field in a set of positions, to find an estimator for some property the field. This property can have multiple forms, be it distinguishing the source of a target signal, or evaluating the field (or a derivative thereof) in an arbitrary position. In this work we also link this problem to networks of quantum sensors, and the role and usefulness of entangling these sensors. We find that the estimators that come out as a solution to the problem are such that a non-local entangled strategy provides maximum precision. We start by working under the assumption of polynomial fields, which relates to the interpolation problem, and then generalize for any signal that is modeled via analytical functions, giving rise to any general least-squares estimator. We discuss the effects of the placement of the sensors in the estimation, namely, how to find well defined, construction error-free placements for the sensors. In the case of interpolation we provide concrete examples and proofs in a mmm-dimensional array of sensors, and discuss necessary and sufficient conditions for the more general cases. We provide clear examples of the possible use-cases and statements, and compare a non-local entangled strategy with the best local strategy for an interpolation problem, showing the benefit in terms of precision in a distributed sensing scenario. This is a key tool for a wide-range of problem in sensing problems, ranging from large-scale such as earth-sized experiments, to local-scale, such has biological experiments.
The theoretical framework for networked quantum sensing has been developed to a great extent in the past few years, but there are still a number of open questions. Among these, a problem of great significance, both fundamentally and for constructing efficient sensing networks, is that of the role of inter-sensor correlations in the simultaneous estimation of multiple linear functions, where the latter are taken over a collection local parameters and can thus be seen as global properties. In this work we provide a solution to this when each node is a qubit and the state of the network is sensor-symmetric. First we derive a general expression linking the amount of inter-sensor correlations and the geometry of the vectors associated with the functions, such that the asymptotic error is optimal. Using this we show that if the vectors are clustered around two special subspaces, then the optimum is achieved when the correlation strength approaches its extreme values, while there is a monotonic transition between such extremes for any other geometry. Furthermore, we demonstrate that entanglement can be detrimental for estimating non-trivial global properties, and that sometimes it is in fact irrelevant. Finally, we perform a non-asymptotic analysis of these results using a Bayesian approach, finding that the amount of correlations needed to enhance the precision crucially depends on the number of measurement data. Our results will serve as a basis to investigate how to harness correlations in networks of quantum sensors operating both in and out of the asymptotic regime.
Abstract The theoretical framework for networked quantum sensing has been developed to a great extent in the past few years, but there are still a number of open questions. Among these, a problem of great significance, both fundamentally and for constructing efficient sensing networks, is that of the role of inter-sensor correlations in the simultaneous estimation of multiple linear functions, where the latter are taken over a collection local parameters and can thus be seen as global properties. In this work we provide a solution to this when each node is a qubit and the state of the network is sensor-symmetric. First we derive a general expression linking the amount of inter-sensor correlations and the geometry of the vectors associated with the functions, such that the asymptotic error is optimal. Using this we show that if the vectors are clustered around two special subspaces, then the optimum is achieved when the correlation strength approaches its extreme values, while there is a monotonic transition between such extremes for any other geometry. Furthermore, we demonstrate that entanglement can be detrimental for estimating non-trivial global properties, and that sometimes it is in fact irrelevant. Finally, we perform a non-asymptotic analysis of these results using a Bayesian approach, finding that the amount of correlations needed to enhance the precision crucially depends on the number of measurement data. Our results will serve as a basis to investigate how to harness correlations in networks of quantum sensors operating both in and out of the asymptotic regime.
We introduce a general model for a network of quantum sensors, and we use this model to consider the question: when do correlations (quantum or classical) between quantum sensors enhance the precision with which the network can measure an unknown set of parameters? We rigorously answer this question for a range of practically important problems. When each sensor in the network measures a single parameter, we show that correlations between sensors cannot increase the estimation precision beyond what can be achieved with an uncorrelated scheme, regardless of the particular details of the estimation problem in question. We also consider the more general setting whereby each sensor may be used to measure multiple parameters, e.g., the three spatial components of a magnetic field. In this case, we show that correlations between sensors can only provide, at best, a small constant precision enhancement, over uncorrelated estimation techniques. Finally, we consider optimizing the network for measuring a single linear function of the unknown parameters, e.g., the average of all of the parameters. Here quantum correlations between the sensors can provide a significant precision enhancement over uncorrelated techniques, and this enhancement factor scales with the number of sensors. To illustrate the broad implications of this work, we apply our results to a wide range of estimation problems of practical interest, including multi-mode optical interferometry, networks of atomic sensors, and networks of clocks. Our findings shed light on a number of results in the literature, provide a rigorous general framework for future research on networked quantum sensors, and have implications for both quantum multi-parameter estimation theory, and quantum sensing technologies.
We introduce a general model for a network of quantum sensors, and we use this model to consider the following question: When can entanglement between the sensors, and/or global measurements, enhance the precision with which the network can measure a set of unknown parameters? We rigorously answer this question by presenting precise theorems proving that for a broad class of problems there is, at most, a very limited intrinsic advantage to using entangled states or global measurements. Moreover, for many estimation problems separable states and local measurements are optimal, and can achieve the ultimate quantum limit on the estimation uncertainty. This immediately implies that there are broad conditions under which simultaneous estimation of multiple parameters cannot outperform individual, independent estimations. Our results apply to any situation in which spatially localized sensors are unitarily encoded with independent parameters, such as when estimating multiple linear or nonlinear optical phase shifts in quantum imaging, or when mapping out the spatial profile of an unknown magnetic field. We conclude by showing that entangling the sensors can enhance the estimation precision when the parameters of interest are global properties of the entire network.
The problem of optimally measuring an analytic function of unknown local parameters each linearly coupled to a qubit sensor is well understood, with applications ranging from field interpolation to noise characterization. Here we resolve a number of open questions that arise when extending this framework to Mach-Zehnder interferometers and quadrature displacement sensing. In particular, we derive lower bounds on the achievable mean square error in estimating a linear function of either local phase shifts or quadrature displacements. In the case of local phase shifts, these results prove, and somewhat generalize, a conjecture by Proctor []. For quadrature displacements, we extend proofs of lower bounds to the case of arbitrary linear functions. We provide optimal protocols achieving these bounds up to small (multiplicative) constants and describe an algebraic approach to deriving new optimal protocols, possibly subject to additional constraints. Using this approach, we prove necessary conditions for the amount of entanglement needed for any optimal protocol for both local phase and displacement sensing. Published by the American Physical Society 2024
One of the most promising applications of quantum networks is entanglement-assisted sensing. The field of quantum metrology exploits quantum correlations to improve the precision bound for applications such as precision timekeeping, field sensing, and biological imaging. When measuring multiple spatially distributed parameters, current literature focuses on quantum entanglement in the discrete-variable case and quantum squeezing in the continuous-variable case, distributed amongst all of the sensors in a given network. However, it can be difficult to ensure that all sensors preshare entanglement of sufficiently high fidelity. This work probes the space between fully entangled and fully classical sensing networks by modeling a star network with probabilistic entanglement generation that is attempting to estimate the average of local parameters. The quantum Fisher information is used to determine which protocols best utilize entanglement as a resource for different network conditions. It is shown that without entanglement distillation there is a threshold fidelity below which classical sensing is preferable. For a network with a given number of sensors and links characterized by a certain initial fidelity and probability of success, this work outlines when and how to use entanglement, when to store it, and when it needs to be distilled.