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

Bounding the fidelity of quantum many-body states from partial information