Distributed quantum sensing with squeezed-vacuum light in a configurable array of Mach-Zehnder interferometers

We study an entangled distributed quantum sensing scheme based on an array of Mach-Zehnder interferometers (MZIs) for the estimation of relative phase shifts. The scheme uses coherent states and a single squeezed-vacuum state that is distributed among the MZIs by a quantum circuit (QC). The protocol can be optimized analytically: it overcomes the shot-noise limit and reaches the Heisenberg limit with respect to the average total number of probe particles, , for the estimation of arbitrary linear combinations of the phases. We compare the entangled strategy with a separable one that uses coherent and squeezed-vacuum states and the same . The entangled strategy benefits for a substantial reduction of resource overhead and can achieve a maximum gain equal to when using the same total squeezed-light intensity as the separable strategy. Interestingly, the entangled strategy using a single squeezed-vacuum state can reach the same sensitivity as the separable strategy that uses copies of the same state. Finally, given a random choices of the QC, we identify the optimal linear combination of the phases that can be estimated with maximum sensitivity. Our scheme paves the ways for a variety of applications in distributed quantum sensing with photonic and atomic interferometers.