  We formulate an algorithm to lower bound the fidelity between quantum
many-body states only from partial information, such as the one accessible by
few-body observables. Our method is especially tailored to permutationally
invariant states, but it gives nontrivial results in all situations where this
symmetry is even partial. This property makes it particularly useful for
experiments with atomic ensembles, where relevant many-body states can be
certified from collective measurements. As an example, we show that a
$\xi^2\approx-6\;\text{dB}$ spin squeezed state of $N=100$ particles can be
certified with a fidelity up to $F=0.999$, only from the measurement of its
polarization and of its squeezed quadrature. Moreover, we show how to
quantitatively account for both measurement noise and partial symmetry in the
states, which makes our method useful in realistic experimental situations.
