Universal Operational Privacy in Distributed Quantum Sensing

TL;DR

This work introduces a universal operational privacy framework for distributed quantum sensing, formulated in terms of the experimentally accessible classical Fisher information matrix and applicable to arbitrary protocols characterized by singular information structures.

Abstract

Privacy is a fundamental requirement in distributed quantum sensing 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 classical 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 sensing networks and provide a foundation for practical, privacy-preserving quantum sensing beyond full-rank regimes.

Figures (4)

• Figure 1: Conceptual schematic of a private distributed quantum sensing network. A shared quantum probe state is encoded with spatially distributed parameters $\bm{\phi}$ such that no information about individual parameters is accessible to untrusted servers. The servers perform local measurements on the encoded state, and the resulting outcomes $\{x_1,\ldots,x_m\}$ are communicated to the clients, who estimate a global function $f(\bm{\phi})=\bm{w}^{\mathrm{T}}\bm{\phi}$.
• Figure 2: Experimental implementation of a private distributed quantum sensing network. A polarization-entangled Bell state is generated in a Sagnac interferometer and distributed through a beam splitter network (BSN) to prepare the two-photon state $|\Psi_4^2\rangle$. The state is sent to spatially separated servers via 3-km optical fiber links, where local phase encoding $\hat{U}(\bm{\phi})$ is applied. Each server performs projective measurements in the $\hat{\sigma}_x$, and two-photon coincidence events are recorded and transmitted to the clients for estimating global functions of the distributed phases. SNSPD: superconducting nanowire single-photon detector; QWP: quarter waveplate; HWP: half waveplate; PBS: polarizing beam splitter; DM: dichroic mirror; DWP: dual wavelength PBS; DWM: dual wavelength mirror; DWH: dual wavelength HWP; PPKTP: periodically-poled KTiOPO$_4$; BSN: beam splitter network; FBS: $50/50$ fiber beam splitter.
• Figure 3: Experimentally measured two-photon outcome probabilities for the distributed quantum sensing protocol. The sixteen probability surfaces are fitted to the experimental data points using the probability functions $P_{\mu\nu}^{\pm\pm}$ and $P_{\mu\nu}^{\pm\mp}$ in Eq. (\ref{['eq:P']}), with $\{\mu \nu\}=\{12,23,34,41\}$, obtained by scanning $\phi_1$, $\phi_2$, $\phi_3$, and $\phi_4$ within the interval $[0,\pi]$. Error bars are smaller than the marker size.
• Figure 4: Experimental verification of privacy in distributed quantum sensing via the singular CFIM. Eigenvalues and eigenvectors of the CFIM, where the horizontal and vertical axes denote eigenvalues and the corresponding vector components, respectively. (a) Theoretical CFIM. (b) Experimentally reconstructed CFIM at $\phi = 0.25\pi$. (c) Experimentally reconstructed CFIM at $\phi = 0.75\pi$.