Privacy in Distributed Quantum Sensing with Gaussian Quantum Networks

TL;DR

It is shown that optimized fully symmetric Gaussian states enable improved privacy levels while maintaining near-optimal sensing performance, and local homodyne detection achieves a quadratic scaling of precision with the total number of photons.

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. For networks with more than two nodes, achieving perfect privacy is possible 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 show that local homodyne detection achieves a 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.

Figures (4)

• Figure 1: Scheme of a DQS protocol with a Gaussian quantum network. An entangled probe state $\rho$ is prepared by applying a beam splitter array (BSA) to an initial product state $S\rho_1S^\dag \otimes \left(\bigotimes_{i=2}^M \rho_i\right)$, where $S$ denotes a squeezing operation and each $\rho_i$ is a thermal state at the same temperature. This procedure produces a fully symmetric (permutation-invariant) state. Each node then locally encodes a phase parameter $\theta_i$ onto its share of the entangled state. Finally, each node performs a measurement and transmits the outcome to a central server, which processes the data to estimate a linear function of the local parameters.
• Figure 2: Isothermal FSG states maximizing estimation precision.Top: Estimation precision $\xi$ achievable for the estimation of the mean function, depending on the total number of photons $N_{\rm tot}$, for a)$n_{\rm th}=0$ (pure states), b)$n_{\rm th}=1$, and c)$n_{\rm th}=5$, and for different numbers of nodes $M$. Pure states achieve the ultimate precision $\xi_{\rm opt}$ for this task; however, for ${n_{\rm th}>0}$, the quadratic scaling with $N_{\rm tot}$ is still observed. Bottom: Privacy deficit $1-\mathcal{P}$ in semi-logarithmic scale for the states maximizing precision. For pure states, as shown in Eq. \ref{['privacyottimo']}, privacy can reach 1, but only for $N_{\rm tot} \gg M$. d) illustrates how rapidly this convergence occurs. e) and f) show the same results for $n_{\rm th}=1$ and $n_{\rm th}=5$ respectively, where privacy still approaches 1 but at a much slower rate.
• Figure 3: Isothermal FSG states maximizing privacy.Top: Ratio $\mathcal{R}$ between the estimation precision of FSG states maximizing privacy and that of FSG states maximizing precision, as a function of the total number of photons $N_{\rm tot}$, for a)$n_{\rm th}=0$ (pure states), b)$n_{\rm th}=1$, and c)$n_{\rm th}=5$, and for different numbers of nodes $M$. When maximizing privacy, only a constant factor is lost in precision, which is close to $2$ for $M=2$ and close to $1$ for $M>2$. This implies that the quadratic scaling with $N_{\rm tot}$ is preserved if an optimal measurement is performed. Bottom: Privacy deficit $1-\mathcal{P}$ in semi-logarithmic scale optimized over FSG states, for d)$n_{\rm th}=0$ (pure states), e)$n_{\rm th}=1$, and f)$n_{\rm th}=5$, and for different numbers of nodes $M$. While estimation precision is almost untouched, privacy deficit can improve considerably with respect to the results in Fig. \ref{['Fig2']}. For instance, for $n_{\rm th}=0$, $M=4$, and $N_{\rm tot}=100$, we have $1-\mathcal{P}\simeq 10^{-2.42}$ in contrast with $10^{-1.83}$ that one gets in Eq. \ref{['privacyottimo']}.
• Figure 4: Precision achievable with homodyne detection. Ratio $\mathcal{R}_{\rm HD}$ between the estimation precision achievable with homodyne detection on the privacy-optimized FSG states, and the precision optimized over generic collective measurements (as shown in Fig. \ref{['Fig3']}), plotted as a function of the total photon number $N_{\rm tot}$. Results are shown for a)$n_{\rm th}=0$ (pure states), b)$n_{\rm th}=1$, and c)$n_{\rm th}=5$, each for different numbers of nodes $M$. For pure states, homodyne detection is optimal for $M>2$, while for $M=2$ it loses a factor of 2 in precision. For ${n_{\rm th}>0}$, the loss in precision is larger; however, the loss factor approaches a constant for large $N_{\rm tot}$, so the quadratic scaling with $N_{\rm tot}$ is preserved.