  Quantum entanglement is commonly assumed to be a central resource for quantum
computing and quantum simulation. Nonetheless, the capability to detect it in
many-body systems is severely limited by the absence of sufficiently scalable
and flexible certification tools. This issue is particularly critical in
situations where the structure of entanglement is a priori unknown, and where
one cannot rely on existing entanglement witnesses. Here, we implement a scheme
in which the knowledge of the mean value of arbitrary observables can be used
to probe multipartite entanglement in a scalable, certified and systematic
manner. Specifically, we rely on positive semidefinite conditions, independent
of partial-transposition-based criteria, necessarily obeyed if the data can be
reproduced by a separable state. The violation of any of these conditions
yields a specific entanglement witness, tailored to the data of interest,
revealing the salient features of the data which are impossible to reproduce
without entanglement. We validate this approach by probing theoretical
many-body states of several hundreds of qubits relevant to existing
experiments: a single-particle quench in a one-dimensional $XX$ chain; a
many-body quench in a two-dimensional $XX$ model with $1/r^3$ interactions; and
thermal equilibrium states of Heisenberg and transverse-field Ising chains. In
all cases, these investigations have lead us to discover new entanglement
witnesses, some of which could be characterized analytically, generalizing
existing results in the literature. In summary, our paper introduces a flexible
data-driven entanglement detection technique for uncharacterized quantum
many-body states, of immediate relevance to experiments in a quantum advantage
regime.
