Heisenberg-limited continuous-variable distributed quantum metrology with arbitrary weights

Distributed quantum metrology (DQM) enables the estimation of global functions of d distributed parameters beyond the capability of separable sensors. Continuous-variable DQM involves using a linear network with at least one nonclassical input. Here we fully elucidate the structure of linear networks with two non-vacuum inputs which allows us to prove a number of fundamental properties of continuous-variable DQM. While measuring the sum of d parameters at the Heisenberg limit can be achieved with a single non-vacuum input, we show that two inputs, one of which can be classical, is required to measure an arbitrary linear combination of d parameters and an arbitrary global function of the parameters. We obtain a universal and tight upper bound on the sensitivity of DQM networks with two inputs, and completely characterize the properties of the nonclassical input required to obtain a quantum advantage. This reveals that a wide range of nonclassical states make this possible, including a squeezed vacuum. We also show that for a class of nonclassical inputs local photon number detection will achieve the maximum sensitivity. Finally we show that a general DQM network has two distinct regimes. The first achieves Heisenberg scaling. In the second the nonclassical input is much weaker than the coherent input, nevertheless providing a multiplicative enhancement to the otherwise classical sensitivity.