Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state

Phase precision in optimal 2-channel quantum interferometry is studied in the limit of large photon number $N\gg 1$, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the losses or the number of photons increase. We further show that fixing the loss paramater determines a scale for quantum metrology -- a crossover value of the photon number $N_c$ beyond which the supra-classical precision is progressively lost. For large losses the optimal state also has a different structure from those considered previously.