Privacy in distributed quantum sensing with Gaussian quantum networks

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.