Heisenberg scaling precision in multi-mode distributed quantum metrology

We consider the estimation of an arbitrary parameter φ , such as the temperature or a magnetic field, affecting in a distributed manner the components of an arbitrary linear optical passive network, such as an integrated chip. We demonstrate that Heisenberg scaling precision (i.e. of the order of 1/ N , where N is the number of probe photons) can be achieved without any iterative adaptation of the interferometer hardware and by using only a...