Distributed Quantum Multiparameter Estimation with Optimal Local Measurements

We study the multiparameter sensitivity bounds of a network of d spatially distributed Mach-Zehnder interferometers (MZIs). A generic single nonclassical state is mixed with d-1 vacuums to create a d-mode entangled state, each mode entering one input port of a MZI, while a coherent state enters its second port. We show that local measurements, independently performed on each MZI, are sufficient to provide a sensitivity saturating the quantum Cramér-Rao bound. The sensor network can overcome the shot noise limit for the estimation of arbitrary linear combinations of the d phase shifts, provided that the nonclassical probe state has an antisqueezed quadrature variance. We compare the sensitivity bounds of this sensor with that achievable with d independent MZIs, each probed with a nonclassical state and a coherent state. We find that the d independent interferometers can achieve the same sensitivity of the entangled protocol, but at the cost of using d nonclassical states rather than a single one. Considering the same average number of particles per shot in the two protocols, n[over ¯]_{T}, we find analytically a sensitivity scaling 1/n[over ¯]_{T}^{2} for the entangled case that provides a gain factor d with respect to the separable case where the sensitivity scales as d/n[over ¯]_{T}^{2}. We have numerical evidence that the gain factor d is also found when fixing the total average number of particles, namely when optimizing with respect to the number of repeated measurements.