Heisenberg scaling precision in multi-mode distributed quantum metrology

We propose an $N$-photon Gaussian measurement scheme which allows the estimation of a parameter $\varphi$ encoded into a multi-port interferometer with a Heisenberg scaling precision (i.e. of order $1/N$). In this protocol, no restrictions on the structure of the interferometer are imposed other than linearity and passivity, allowing the parameter $\varphi$ to be distributed over several components. In all previous proposals Heisenberg scaling has been obtained provided that both the input state and the measurement at the output are suitably adapted to the unknown parameter $\varphi$. This is a serious drawback which would require in practice the use of iterative procedures with a sequence of trial input states and measurements, which involve an unquantified use of additional resources. Remarkably, we find that only one stage has to be adapted, which leaves the choice of the other stage completely arbitrary. We also show that our scheme is robust against imperfections in the optimized stage. Moreover, we show that the adaptive procedure only requires a preliminary classical knowledge (i.e to a precision $1/\sqrt{N}$) on the parameter, and no further additional resources. As a consequence, the same adapted stage can be employed to monitor with Heisenberg-limited precision any variation of the parameter of the order of $1/\sqrt{N}$ without any further adaptation.