  We find and investigate the optimal scheme of quantum distributed Gaussian
sensing for estimation of the average of independent phase shifts. We show that
the ultimate sensitivity is achievable by using an entangled symmetric Gaussian
state, which can be generated using a single-mode squeezed vacuum state, a
beam-splitter network, and homodyne detection on each output mode in the
absence of photon loss. Interestingly, the maximal entanglement of a symmetric
Gaussian state is not optimal although the presence of entanglement is
advantageous as compared to the case using a product symmetric Gaussian state.
It is also demonstrated that when loss occurs, homodyne detection and other
types of Gaussian measurements compete for better sensitivity, depending on the
amount of loss and properties of a probe state. None of them provide the
ultimate sensitivity, indicating that non-Gaussian measurements are required
for optimality in lossy cases. Our general results obtained through a
full-analytical investigation will offer important perspectives to the future
theoretical and experimental study for quantum distributed Gaussian sensing.
