Multiparameter quantum metrology at Heisenberg scaling for an arbitrary two-channel linear interferometer with squeezed light

We present a framework for simultaneously estimating all four real parameters of a general two-channel unitary $\mathrm{U}(2)$ with Heisenberg-scaling precision. We derive analytical expressions for the quantum Fisher information matrix and show that all parameters attain the $1/N$ scaling in the precision by using experimentally feasible Gaussian probes, such as two-mode squeezed states or two single-mode squeezed states. Our results extend multiparameter metrology to its most general two-mode setting and establish concrete design principles for experimental implementations of Heisenberg-scaling, multiparameter optical interferometry with experimentally feasible resources. It not only sheds light on the fundamental interface between quantum interference of squeezed light and quantum metrological advantage in multiparameter estimation, but it also provides an important stepping stone towards the development of a wide range of quantum technologies based on distributed quantum metrology in arbitrary optical networks.