Limits on noisy nonlinear quantum metrology

In quantum metrology the precision to which one can estimate a parameter is limited by the amount of resources available in the probe. In an optical interferometer, for example, the parameter is a small phase shift and the resource is the number of photons in the probe state. Metrology puts limits on parameter estimation such as the standard quantum limit (SQL) or the lower Heisenberg limit which requires using more exotic probe states. A nonlinear interferometer encodes the phase parameter onto the probe in a nonlinear manner. This can drastically improve photon number scaling of the parameter inference. The use of nonlinearity is not without drawbacks though. We show that in the same way that the nonlinearity can boost parameter encoding, it also boosts noise, not just making them large in size, but also making them potentially uncorrectable. Furthermore we provide a go/no-go result that can determine the largest nonlinearity given the amount of noise present, and provide several ways to still achieve improved nonlinear scaling even in the presence of noise.