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arXiv:2509.22103 · npj Quantum Information

Privacy in distributed quantum sensing with Gaussian quantum networks

Uesli Alushi, Roberto Di Candia

Published May 21, 2026• 2 citations
arXivHTMLPDF

Abstract

We study the privacy properties of distributed quantum sensing protocols in a Gaussian quantum network, where each node encodes a parameter via a local phase shift. We first show that perfect privacy and optimal precision are jointly achievable using specifically tailored multimode photon-number correlated states. We then consider Gaussian states, which are experimentally less demanding as they can be implemented using only linear optics and two-photon parametric processes. Focusing on fully symmetric Gaussian states, we show that for networks with more than two nodes, perfect privacy can be achieved only asymptotically, in the limit of large photon numbers. However, we show that optimized fully-symmetric Gaussian states enable improved privacy levels while maintaining near-optimal sensing performance. We also show that local homodyne detection is essentially optimal, achieving quadratic scaling of precision with the total number of photons. We further analyze the impact of thermal noise in the preparation stage on both privacy and estimation precision. Our results pave the way for the development of practical, private distributed quantum sensing protocols in continuous-variable quantum networks.

DOI
10.1038/s41534-026-01266-3

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We study the privacy properties of distributed quantum sensing protocols in a Gaussian quantum network, where each node encodes a parameter via a local phase shift. We first show that perfect privacy and optimal precision are jointly achievable using specifically tailored multimode photon-number correlated states. We then consider Gaussian states, which are experimentally less demanding as they can be implemented using only linear optics and two-photon parametric processes. Focusing on fully symmetric Gaussian states, we show that for networks with more than two nodes, perfect privacy can be achieved only asymptotically, in the limit of large photon numbers. However, we show that optimized fully-symmetric Gaussian states enable improved privacy levels while maintaining near-optimal sensing performance. We also show that local homodyne detection is essentially optimal, achieving quadratic scaling of precision with the total number of photons. We further analyze the impact of thermal noise in the preparation stage on both privacy and estimation precision. Our results pave the way for the development of practical, private distributed quantum sensing protocols in continuous-variable quantum networks.
Private distributed quantum sensing aims to estimate an authorized collective parameter while preventing independent estimation of individual local parameters. Here, we analytically characterize Gaussian quantum networks satisfying this perfect local privacy condition. Using a graph representation of the Gaussian pairing matrix, we show that two-mode squeezed vacuum states constitute the essential building blocks of privacy-preserving Gaussian states. We then analytically derive the optimal sensitivity under perfect local privacy and determine a Gaussian probe that achieves Heisenberg scaling with respect to both the photon number and the number of sensing modes. Using local homodyne measurements and maximum-likelihood estimation, we numerically verify quantum-enhanced sensitivity beyond the shot-noise limit while preserving perfect local privacy. We further show that the local-privacy condition is preserved under arbitrary phase-independent quantum channels, including optical loss. Our results provide a general framework for constructing and optimizing privacy-preserving continuous-variable quantum sensing networks.
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.
In the discrete variable setting, entangled resource states allow a set of parties to learn a global function of a set of spatially separated systems, whilst keeping the local parameters of those systems completely private. In the continuous variable setting, distributed sensing has been carried out using Gaussian resource states, but without the same guarantees about privacy. Here, we show that perfect privacy is impossible to achieve for any distributed sensing protocol that uses Gaussian states as a resource. We also introduce a measure of relative privacy, bounding the degree to which any Gaussian distributed sensing protocol can keep local parameters hidden.
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.
Privacy is a fundamental requirement in distributed quantum sensor networks, where multiple clients estimate spatially distributed parameters using shared quantum resources while interacting with potentially untrusted servers. Despite its importance, existing privacy conditions rely on idealized quantum bounds and do not fully capture the operational constraints imposed by realistic measurements. Here, we introduce a universal operational privacy framework for distributed quantum sensing, formulated in terms of the experimentally accessible Fisher information matrix and applicable to arbitrary protocols characterized by singular information structures. The proposed condition provides a protocol-independent criterion, ensuring that no information about individual parameters is accessible to untrusted parties. We further experimentally demonstrate that a distributed quantum sensing protocol employing fewer photons than the number of estimated parameters simultaneously satisfies the universal privacy condition and achieves Heisenberg-limited precision. Our results establish universal operational constraints governing privacy in distributed quantum sensor networks and provide a foundation for practical, privacy-preserving quantum sensing beyond full-rank regimes.
Quantum sensing is a promising technology capable of demonstrating clear advantage over comparable classical techniques for precise measurement. One application of quantum sensing is in function estimation, which can be done using a network of entangled quantum sensors, allowing for measurements with greater optimal sensitivity than unentangled sensing protocols. In cases where quantum sensor networks will be used to measure data that should remain private (e.g., biomedical data), it is imperative that these protocols include a privacy mechanism to hide sensitive information. In this work, we show that entangled sensor networks are vulnerable to certain privacy-violating attacks. To mitigate these attacks, we introduce secure sensing protocols endowed with differential privacy. We reconcile differential privacy with retaining Heisenberg-limited scaling, and introduce several protocols achieving varying balances between the two. We show that our main protocol, an nnn-node network sensing protocol that injects noise directly into the sensing Hamiltonian, exhibits a tradeoff between the desirable O(1/n2)O(1/n^2)O(1/n2) Heisenberg scaling of the mean-squared error of the function estimate and the level of privacy attainable. Under assumptions on the network (a common source of randomness and a constant fraction of honest parties), we show that this protocol is locally implementable and achieves (O(1),δ)(O(1), δ)(O(1),δ)-differential privacy for arbitrarily small δδδ while retaining Heisenberg scaling of the mean-squared error. We prove that our protocols are resilient to attacks by broad classes of classical and quantum adversaries, and find advantages in the privacy-utility tradeoff when using quantum techniques.