Characterizing resources for multiparameter estimation of SU(2) and SU(1,1) unitaries

We analyze the task of estimating a multi-parameter unitary belonging to the $SU(2)$ or $SU(1,1)$ groups, in a two-bosonic-mode scenario and investigate the scaling of the precision in terms of the total particle number. For the $SU(2)$ case, the total particle number is conserved by the evolution and we discuss optimal states in fixed-$n$ subspaces, identifying eigenstates of $J_z^2$ as useful resources, even allowing simultaneous Heisenberg precision scaling for all three parameters. In the $SU(1,1)$ case instead, the conserved quantity is the particle number difference between the two modes, and we identify useful probe states in the sector with an equal number of particles in the two modes. These states are analogous to the $SU(2)$ case and would also allow simultaneous Heisenberg precision scaling for all three parameters. We then consider the more pragmatic scenario of an estimation via expectation values of time-evolved observables, which we restrict to be the first two moments of the generators. We analyze the maximal precision achievable in this setting and we find that the twin-Fock state emerges in both the $SU(2)$ and the $SU(1,1)$ cases as the only one potentially allowing Heisenberg scaling for the estimation of two out of the three parameters. As a complement, we also consider other probe states with fluctuating number of particles, with measurements restricted to quadratic expressions in the mode operators. In this scenario, simultaneous Heisenberg scaling in multiple parameters seems mostly forbidden, with the only exception being an input two-mode squeezed state for the estimation of a two-parameter $SU(2)$. This extends to the multiparameter scenario the well-established intuition that the performance of a $SU(2)$ interferometer can be enhanced by a prior $SU(1,1)$ operation.