Distributed quantum phase sensing using an SU(1,1) Perelomov coherent state in a Mach-Zehnder interferometer network

We present a theoretical study of distributed quantum sensing using a reconfigurable network of N Mach--Zehnder interferometers to simultaneously estimate weighted linear combinations of relative phase shifts. The proposed scheme relies on injecting a Perelomov coherent state, generated from the SU(1,1) group representation, into one port of a passive quantum circuit while the remaining modes are initialized in the vacuum state, thereby establishing non-trivial quantum correlations among the sensors. Based on multi-parameter quantum estimation theory, our analysis shows that this entangled strategy outperforms the standard shot-noise limit and asymptotically achieves the Heisenberg limit. Furthermore, we highlight the crucial role of the first beam splitter's asymmetry; specifically, the optimal transmission coefficient that maximizes sensitivity corresponds to its maximum imbalance. Compared to a separable approach utilizing the same total average photon number, the entangled strategy yields a more efficient use of quantum resources, guaranteeing an accuracy gain between 1 and N. This demonstrates that the entangled strategy is systematically superior to its separable counterpart. Ultimately, these results illustrate the potential of entangled distributed quantum sensor networks for high-precision measurements and provide clear guidelines for optimizing interferometric architectures in quantum metrology.