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, requiring the study of an optimal scheme achieving a maximal quantum advantage for a given sensing scenario. In this work, we propose a Heisenberg-limited distributed quantum phase sensing scheme using Gaussian states for an arbitrary distribution of the weights with positive and negative signs. The proposed scheme exploits entanglement of Gaussian states only among the modes assigned with equal signs of the weights, but separates the modes with opposite weight signs. We show that the estimation precision of the scheme exhibits the Heisenberg scaling in the mean photon number and it can be achieved by injecting two single-mode squeezed states into the respective linear beam-splitter networks and performing homodyne detection on them in the absence of loss. Interestingly, the proposed scheme is proven to be optimal for Gaussian probe states with zero displacement. We also provide an intuitive understanding of our results 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.