Measurement compatibility in multiparameter quantum interferometry

While the performance of single-parameter quantum sensors is immediate to assess by means of the Cram\'er-Rao bound, extension to multiple parameters demands more caution. Different aspects need to be captured at once, including, critically, compatibility. In this article we consider compatibility in quantum interferometry for an important class of probe states and for double homodyne measurement, standard benchmarks for these applications. We include the presence of loss and phase diffusion in the estimation of a phase. Our results illustrate how different weighting of the precision on individual parameters affects their compatibility, adding to the list of considerations for quantum multiparameter estimation.