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arXiv:2608.06888

Correlation Geometry of Quantum Sensor Networks: Local-Global Information Flow and Local Privacy

Gong-Chu Li, Lei Chen, Xu-Song Hong, Hua-Qing Xu, Yuancheng Liu, Si-Qi Zhang, Jia-Hao Zhao, Geng Chen, Chuan-Feng Li, Guang-Can Guo

quant-phSubmitted Aug 7, 2026• Announced Aug 10, 2026• 0 citations• Revised Aug 13, 2026 (v2)
v1 · Aug 7, 2026v2 · Aug 13, 2026
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Abstract

Quantum sensor networks (QSN) typically encode N unknown parameters while targeting a single linear combination, rendering the N-1 remaining parameters as nuisance directions. To rigorously quantify estimation precision under such nuisances, we use the effective quantum Fisher information (EQFI) and establish a ``barrel-effect'' bottleneck: the global EQFI cannot exceed the weakest weighted local sensing capacity. To elucidate the information allocation mechanism underlying this bottleneck, we derive an exact local--global phase map that delineates how the trade-off between local and global EQFI depends dynamically on quantum correlations, and accordingly we identify concrete conditions for saturating the bottleneck bound. Notably, this geometric map uncovers a counterintuitive ``overcorrelated'' regime where excessive correlations actively degrade both local and global performance. Finally, we apply the phase map to intrinsic local privacy and identify the condition under which every local parameter is inaccessible while the desired global combination remains estimable. Overall, our work provides a principled methodology for engineering optimal network states in quantum sensing architectures.

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5 pages, 3 figures

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Quantum sensor networks (QSN) typically encode N unknown parameters while targeting a single linear combination, rendering the N-1 remaining parameters as nuisance directions. To rigorously quantify estimation precision under such nuisances, we use the effective quantum Fisher information (EQFI) and establish a barrel-effect'' bottleneck: the global EQFI cannot exceed the weakest weighted local sensing capacity. To elucidate the information allocation mechanism underlying this bottleneck, we derive an exact local--global phase map that delineates how the trade-off between local and global EQFI depends dynamically on quantum correlations, and accordingly we identify concrete conditions for saturating the bottleneck bound. Notably, this geometric map uncovers a counterintuitive overcorrelated'' regime where excessive correlations actively degrade both local and global performance. Finally, we apply the phase map to intrinsic local privacy and identify the condition under which every local parameter is inaccessible while the desired global combination remains estimable. Overall, our work provides a principled methodology for engineering optimal network states in quantum sensing architectures.
We establish a quantum Fisher information (QFI) duality for distributed quantum sensor networks with local phase encoding. For any NNN-qubit probe state, where NNN denotes the number of sensors, FQ(w⊤θ)+FQ(v⊤θ)≤NF_Q(\boldsymbol{w}^\top \boldsymbolθ) + F_Q(\boldsymbol{v}^\top \boldsymbolθ) \leq NFQ​(w⊤θ)+FQ​(v⊤θ)≤N for all unit orthogonal sensing directions w\boldsymbol{w}w and v\boldsymbol{v}v, with equality for all equatorial states when N=2N=2N=2 and for Greenberger--Horne--Zeilinger (GHZ) states when N≥2N\geq 2N≥2. Heisenberg-limited precision for direction w\boldsymbol{w}w, FQ(w⊤θ)=NF_Q(\boldsymbol{w}^\top \boldsymbolθ)=NFQ​(w⊤θ)=N, saturates the bound and simultaneously forces zero QFI for all other independent directions. This can be interpreted as the condition for parameter privacy in distributed quantum sensing: attaining Heisenberg-limited precision for the sensing target renders all alternative privacy-intrusive estimations impossible.
We treat privacy in a network of quantum sensors where accessible information is limited to specific functions of the network parameters, and all other information remains private. We develop an analysis of privacy in terms of a manipulation of the quantum Fisher information matrix, and find the optimal state achieving maximum privacy in the estimation of linear combination of the unknown parameters in a network of quantum sensors. We also discuss the effect of uncorrelated noise on the privacy of the network. Moreover, we illustrate our results with an example where the goal is to estimate the average value of the unknown parameters in the network. In this example, we also introduce the notion of quasiprivacy (ε privacy), quantifying how close the state is to being private.
Can a distributed network of quantum sensors estimate a global parameter while protecting every locally encoded value? We answer this question affirmatively by introducing and analysing a protocol for distributed quantum sensing in the continuous-variable regime. We consider a multipartite network in which an unknown local phase is imprinted at each node on a shared entangled Gaussian state. We show that the average phase can be estimated with high precision, exhibiting Heisenberg scaling in the total photon number, while individual phases are inaccessible. We further prove a no-go theorem showing that, for three or more parties, no finite-energy Gaussian probe can provide complete privacy of the average phase, meaning that all phase combinations orthogonal to the average remain entirely hidden. This identifies the two-party case as an exceptional Gaussian setting that can achieve complete privacy. We further investigate the impact of displacements and optical losses, revealing trade-offs between estimation accuracy and privacy. Finally, we benchmark the protocol against other continuous-variable resource states.
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.
Privacy of a quantum metrological protocol concerns the extent to which single parameters can be kept inaccessible to an observer or to other users of the network. In this work, an information geometric framework is developed to quantify privacy and accessibility of functions of parameters effectively, that is, up to a finite accuracy in state discrimination. Both quantities are defined by measuring volumes in the parameter space induced by the underlying quantum states. This construction subsumes previous definitions of privacy based on the degeneracy of quantum Fisher information, naturally encompassing imperfect implementations. Using extended-GHZ states as a representative example of a quantum network scenario, privacy and accessibility are characterized by quantum correlations and accuracy, providing scaling laws depending on imperfect measurements and entanglement.
Can a distributed network of quantum sensors estimate a global parameter while protecting every locally encoded value? We answer this question affirmatively by introducing and analyzing a protocol for distributed quantum sensing in the continuous-variable regime. We consider a multipartite network in which an unknown local phase is imprinted at each node on a shared entangled Gaussian state. We show that the average phase can be estimated with high precision, exhibiting Heisenberg scaling in the total photon number, while individual phases are inaccessible. We further prove a no-go theorem showing that, for three or more parties, no finite-energy Gaussian probe can provide complete privacy of the average phase, meaning that all phase combinations orthogonal to the average remain entirely hidden. This identifies the two-party case as an exceptional Gaussian setting that can achieve complete privacy. We further investigate the impact of displacements and optical losses, revealing trade-offs between estimation accuracy and privacy. Finally, we benchmark the protocol against other continuous-variable resource states.