Quantum-private distributed sensing

TL;DR

The paper addresses private distributed sensing by embedding privacy into quantum metrology via private parameter estimation (PPE) using $n$-qubit GHZ states. It combines a verification protocol based on stabilizer measurements with a remote-sensing protocol that encodes local phases and estimates a global phase with Heisenberg-limited precision, while bounding information leakage about local parameters through $\varepsilon_p$-privacy. The experimental demonstration with $n=3$ sensors achieves high global-signal precision and shows substantial suppression of local information, supported by both state tomography and direct QFI analysis; it also outlines practical challenges and potential improvements toward scalable, quantum-secure sensing networks. This work integrates privacy into distributed quantum sensing, signaling a crucial step toward secure, private quantum networks and metrological applications.

Abstract

Quantum networks can enhance both security and privacy conditions for multi-user communication, delegated computation, and distributed sensing tasks. An example quantum protocol is private parameter estimation (PPE) where only the aggregate information is accessible while individual sensor data remain confidential. Specifically, the protocol enables the estimation of a global function of remote sensor parameters without revealing local parameters to any entity. We implement the PPE protocol by distributing a three-photon Greenberger-Horne-Zeilinger (GHZ) state, among three sensors, which is verified using stabilizer measurements to establish privacy and precision bounds for the sensing task. We demonstrate Heisenberg-limited precision scaling of the global parameter while suppressing the metrological information of the local parameters by up to three orders of magnitude. This work, which integrates privacy in distributed quantum sensing, marks a crucial step towards developing advanced quantum-secure-and-private protocols in complex quantum networks.

Figures (8)

• Figure 1: Illustration of private parameter estimation. Sensors in a network monitor a global function of parameters while local values $\{\theta_i\}$ remain secret. An honest verifier establishes private pairwise channels with each sensor (not shown). An untrusted server sends $N_{\textrm{t}}$ copies of the resource state to all sensors, who store them locally in a quantum memory. They perform the verification protocol to ensure they share a state close to a GHZ state. If verification is successful, each sensor encodes their local parameter on one copy of the shared state to perform parameter estimation.
• Figure 2: Experimental setup preparing a three-qubit GHZ state and implementing PPE. A Ti:sapph laser pumps two entangled photon-pair sources (EPS). One photon from each source is sent to a polarising beamsplitter gate (PBS-G). Each sensor has a local phase encoding stage ($\theta_i$). See Methods for detail. HWP: half-wave plate, QWP: quarter-wave plate, PBS: polarising beamsplitter, DM: dichroic mirror, FPC: fibre polarisation controller, $\Delta$: delay stage, aKTP: aperiodically-poled KTP crystal, SNSPD: superconducting nanowire single photon detector.
• Figure 3: Verification protocol results. (a) Using stabilizer measurements the failure rate $f$ is measured (dark blue dots) for each delay, $\Delta$ then we evaluate the lower bound fidelity (dark grey dots). We perform quantum state tomography, reconstruct the density matrices, then estimate the GHZ state fidelity (light blue dots). Solid lines are a visual guide for datasets and not fits. (b) Using the density matrices we directly calculate QFI for estimating $\phi$ for each $\Delta$ which has a maximum QFI $=9$ (orange line). We directly calculate privacy parameters $\varepsilon(\theta_i)$ and protocol privacy $\varepsilon_p = \max_{i}{[\varepsilon (\theta_i)]}$ from the density matrices. From the verification procedure we obtain an upper-bound on $\varepsilon_p$ (blue line) for our optimal state. We evaluate the upper-bound for $|+++\rangle$ (red line) using the verification process. For comparison, the green line is the directly evaluated $\varepsilon_p$ from the density matrices. All error bars derived from Monte Carlo sampling with 200 samples, assuming Poissonian statistics. Red star denotes the optimal setting, $\Delta=0$.
• Figure 4: Private parameter estimation spanning the parameter space, $\phi \in [-1.05, 1.05]$ for a sample size $\nu=50$ using $n=3$ sensors. To collect statistics and evaluate the mean squared error, each estimation scheme is repeated 100 times. Solid circle data points represent evaluations of the mean value of the global phase $\phi$, while the other markers represent the evaluation of individual sensor phase values $\theta_i$. The red solid line is the theoretical Cramer-Rao bound obtained from estimating the Fisher information of the estimator with non-unit visibility. The dashed lines represent the theoretical lower-bounds based on the standard quantum limit precision scaling and Heisenberg limit scaling respectively. Error bars for all data points are evaluated using standard error, with three standard deviations shown.
• Figure SM1: The correction term, $2\sqrt{c}/n$, is evaluated for a range of total copies $N_t$ used in the verification stage for $n=3$. This is shown for three values of failure probabilities $p_{fail}$, which corresponds to the liklihood that the verification procedure fails to correctly lower bound the fidelity and upper bound $\varepsilon_p$.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3