  We introduce a semidefinite programming algorithm to find the minimal quantum
Fisher information compatible with an arbitrary dataset of mean values. This
certification task allows one to quantify the resource content of a quantum
system for metrology applications without complete knowledge of the quantum
state. We implement the algorithm to study quantum spin ensembles. We first
focus on Dicke states, where our findings challenge and complement previous
results in the literature. We then investigate states generated during the
one-axis twisting dynamics, where in particular we find that the metrological
power of the so-called multi-headed cat states can be certified using simple
collective spin observables, such as fourth-order moments for small systems,
and parity measurements for arbitrary system sizes.
