Time-Multiplexed Distributed Quantum Sensing

Abstract

Quantum metrology enables parameter estimation beyond classical limits by exploiting nonclassical resources such as squeezing and entanglement. In distributed quantum sensing, Heisenberg scaling has been extended from $1/N^2$ to $1/(NM)^2$ through entanglement across both particles and spatial modes, where $N$ denotes the photon number and $M$ the number of spatially distributed modes. However, the overall sensitivity has remained limited to linear scaling with the number of measurement repetitions $R$. Here, we show that exploiting entanglement across temporal modes via time-domain multiplexing enables a scaling advantage with respect to $R$. As a result, the sensitivity can asymptotically approach simultaneous Heisenberg scaling in photons, spatial modes, and repetitions, yielding an overall sensitivity approaching $Δ^2 φ\propto 1/(NMR)^2$. Using the Bogoliubov transformation formalism, we prove the optimality of the protocol within the class of Gaussian states and show that the scaling is realizable via homodyne detection and maximum-likelihood estimation. We further show that the advantage persists under optical loss and propose an experimentally feasible loop-based photonic sensing scheme. Our results open a route to incorporating time-multiplexing techniques into quantum metrology.

Figures (2)

• Figure 1: Distributed quantum sensing with (a) time-separable (TS) and (b) time-multiplexed (TM) protocols. The probe state $\hat{\rho}$ acquires spatial phase shifts $\{\phi_j\}$ and the average phase is inferred from correlated measurements. The shaded region indicates entanglement between spatiotemporal modes. (c) Variance of $\phi_{\mathrm{avg}}=\sum_{i=1}^{M}\phi_i/M$ versus the number of measurement repetitions $R$. The TM scheme exhibits improved scaling compared with the TS protocol and the standard quantum limit (SQL), with $\bar{n}=2$, $M=2$, and $\eta=1$ fixed.
• Figure 2: (a) Time-multiplexed distributed sensing protocol. A squeezed probe is distributed across spatial modes via the interferometer $\hat{U}_S$ and across temporal modes via $\hat{U}_T$. The protocol is repeated $R_r$ times with $R_t$ temporal modes per run, giving a total number of measurements $R=R_rR_t$. (b) Sensitivity in the presence of optical loss $\eta$. The TM protocol retains a clear advantage over the TS scheme and the standard quantum limit (SQL) over a broad range of loss. Here, $\bar{n}=2$, $M=2$, and $R=10^{3}$, with $\eta=1$ corresponding to the lossless case.