Distributed quantum phase sensing for arbitrary positive and negative weights

Estimation of a global parameter defined as a weighted linear combination of unknown multiple parameters can be enhanced by using quantum resources. Advantageous quantum strategies may vary depending on the weight distribution, but have rarely been studied. Here, we propose a distributed quantum phase sensing scheme using Gaussian states in order to take into account an arbitrary distribution of the weights with positive and negative signs. The optimal estimation precision of the proposed scheme is derived, and shown to be achievable by using squeezed states injected into linear beam-splitter networks and performing homodyne detection on them. It is shown that entanglement of Gaussian states is advantageous only among the modes assigned with equal signs of the weights, whereas is not useful between the modes with opposite weight signs. We also provide a deeper understanding of our finding by focusing on the two-mode case, in comparison with the cases using non-Gaussian probe states. We expect this work to motivate further studies on quantum-enhanced distributed sensing schemes considering various types of physical parameters with an arbitrary weight distribution.