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Estimation of a global parameter defined as a weighted linear combination ofunknown multiple parameters can be enhanced by using quantum resources.Advantageous quantum strategies may vary depending on the weight distribution,requiring the study of optimal schemes achieving a maximal quantum advantagefor a given sensing scenarios. In this work, we propose an optimal distributedquantum phase sensing scheme using Gaussian states with zero displacement foran arbitrary distribution of the weights with positive and negative signs. Theestimation precision of the optimal scheme is derived, and shown to beachievable by using squeezed states injected into linear beam-splitter networksand performing homodyne detection on them in the absence of loss.Interestingly, the optimal scheme exploits entanglement of Gaussian states onlyamong the modes assigned with equal signs of the weights, but separates themodes with opposite weight signs. We also provide a deeper understanding of ourfinding by focusing on the two-mode case, in comparison with the cases usingnon-Gaussian probe states. We expect this work to motivate further studies onquantum-enhanced distributed sensing schemes considering various types ofphysical parameters with an arbitrary weight distribution.

Distributed quantum phase sensing for arbitrary positive and negative weights