Distributed quantum sensing with multi-mode $N00N$ states

Distributed quantum sensing, which estimates a global parameter across distant nodes, has attracted significant interest for applications such as quantum imaging, sensor networks, and global-scale clock synchronization. $N00N$ states are regarded as one of the optimal quantum resources for quantum metrology, enabling the Heisenberg scaling. Recently, the concept of $N00N$ states has been extended to multi-mode $N00N$ states for quantum-enhanced multiple-parameter estimation. However, the application of multi-mode $N00N$ states in distributed quantum sensing remains unexplored. Here, we propose a distributed quantum sensing scheme that achieves the Heisenberg scaling using multi-mode $N00N$ states. We theoretically show that multi-mode $N00N$ states can reach the Heisenberg scaling by examining both the Cram\'er-Rao bound and the quantum Cram\'er-Rao bound. For experimental demonstration, we employ a four-mode $2002$ state to estimate the average of two spatially distributed phases, achieving a 2.74 dB sensitivity enhancement over the standard quantum limit. We believe that utilizing multi-mode $N00N$ states for distributed quantum sensing offers a promising approach for developing entanglement-enhanced sensor networks.