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, ${\overline{n}}_T$, we find analytically a sensitivity scaling $1/{\overline{n}}_T^2$ for the entangled case that provides a gain factor $d$ with respect to the separable case where the sensitivity scales as $d/{\overline{n}}_T^2$. 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.

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• Quantum measurements
• Quantum metrology
• Quantum parameter estimation