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

We follow recent works analyzing precision bounds to the estimation of multiple parameters generated by a unitary evolution with non-commuting Hamiltonians that form a closed algebra. We consider the 3-parameter estimation of SU(2) and SU(1,1) unitaries and investigate the scaling of the precision in a pragmatic framework where the estimation is performed via the so-called method of moments, consisting in the estimation of phases via expectation values of time-evolved observables, which we restrict to be the first two moments of the Hamiltonian generators. We consider optimal or close-to-optimal initial states, that can be obtained by maximizing the quantum Fisher information matrix, and analyze the maximal precision achievable from measuring only the first two moments of the generators. As a result, we find that in our context with limited resources accessible, the twin-Fock state emerges as the best probe state, that allows the estimation of two out of the three parameters with Heisenberg precision scaling. Moreover, in other practical scenarios with less limitations we identify useful probe states that can in principle achieve Heisenberg scaling for the three parameters considered.