We consider a quantum sensor network of qubit sensors coupled to a field $f(\mathbit{x};\mathbit{\ensuremath{\theta}})$ analytically parameterized by the vector of parameters $\mathbit{\ensuremath{\theta}}$. The qubit sensors are fixed at positions ${\mathbit{x}}_{1},\ensuremath{\cdots},{\mathbit{x}}_{d}$. While the functional form of $f(\mathbit{x};\mathbit{\ensuremath{\theta}})$ is known, the parameters $\mathbit{\ensuremath{\theta}}$ are not. We derive saturable bounds on the precision of measuring an arbitrary analytic function $q(\mathbit{\ensuremath{\theta}})$ of these parameters and construct the optimal protocols that achieve these bounds. Our results are obtained from a combination of techniques from quantum information theory and duality theorems for linear programming. They can be applied to many problems, including optimal placement of quantum sensors, field interpolation, and the measurement of functionals of parametrized fields.
We consider a quantum sensor network of qubit sensors coupled to a field f(\mathbitx;\mathbit\ensuremathθ) analytically parameterized by the vector of parameters \mathbit\ensuremathθ. The qubit sensors are fixed at positions \mathbitx1,\ensuremath⋯,\mathbitxd. While the functional form of f(\mathbitx;\mathbit\ensuremathθ) is known, the parameters \mathbit\ensuremathθ are not. We derive saturable bounds on the precision of measuring an arbitrary analytic function q(\mathbit\ensuremathθ) of these parameters and construct the optimal protocols that achieve these bounds. Our results are obtained from a combination of techniques from quantum information theory and duality theorems for linear programming. They can be applied to many problems, including optimal placement of quantum sensors, field interpolation, and the measurement of functionals of parametrized fields.
Quantum sensor networks promise precision advantages over classical and single-sensor strategies, in particular when the estimator is non-local. We address the problem of finding such estimators through a framework we connote spatial quantum sensing: given an underlying field interrogated by a network of quantum sensors at fixed positions, construct an estimator for a property of the field, for example, distinguishing a source of signal, or evaluating the field or its derivatives at an arbitrary point. We first treat polynomial fields, casting the task as an interpolation problem, and then generalize to fields modeled by analytic functions, which yields general least-squares estimators. A central and largely unaddressed question is under what conditions on sensor placement these estimators are well-defined and error-free. For m-dimensional arrays we give explicit constructions and proofs in the interpolation setting using algebraic geometry, and establish necessary and sufficient conditions in the general case. Comparing a non-local entangled protocol with the best local strategy, we show that entanglement yields maximal precision in distributed sensing under global resource constraints. Finally, we introduce error-free subspaces, a technique that translates prior knowledge of the field into a reduction in the number of required sensors. We expect these techniques to be broadly useful in sensing problems across scales, ranging from earth-scale experiments to local applications such as biological imaging.
We consider the problem of estimating multiple analytic functions of a set of local parameters via qubit sensors in a quantum sensor network. To address this problem, we highlight a generalization of the sensor symmetric performance bounds of Rubio et al., [J. Phys. A 53, 344001 (2020)] and develop an optimized sequential protocol for measuring such functions. We compare the performance of both approaches to one another and to local protocols that do not utilize quantum entanglement, emphasizing the geometric significance of the coefficient vectors of the measured functions in determining the best choice of measurement protocol. We show that, in many cases, especially for a large number of sensors, the optimized sequential protocol results in more accurate measurements than the other strategies. In addition, in contrast to the sensor symmetric approach, the sequential protocol is known to always be explicitly implementable. The sequential protocol is very general and has a wide range of metrological applications.
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
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.
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.