Joint estimation of position and momentum with arbitrarily high precision using non-Gaussian states

We address the joint estimation of changes in the position and linear momentum of a quantum particle or, equivalently, changes in the complex field of a bosonic mode. Although these changes are generated by non-commuting operators, we show that leveraging non-Gaussianity enables their simultaneous estimation with arbitrarily high precision and arbitrarily low quantum incompatibility. Specifically, we demonstrate that any pure non-Gaussian state provides an advantage over all Gaussian states, whether pure or mixed. Moreover, properly tuned non-Gaussian mixtures of Gaussian states can also serve as a resource.