  Entanglement plays a central role in our understanding of quantum many body
physics, and is fundamental in characterising quantum phases and quantum phase
transitions. Developing protocols to detect and quantify entanglement of
many-particle quantum states is thus a key challenge for present experiments.
Here, we show that the quantum Fisher information, representing a witness for
genuinely multipartite entanglement, becomes measurable for thermal ensembles
via the dynamic susceptibility, i.e., with resources readily available in
present cold atomic gas and condensed-matter experiments. This moreover
establishes a fundamental connection between multipartite entanglement and
many-body correlations contained in response functions, with profound
implications close to quantum phase transitions. There, the quantum Fisher
information becomes universal, allowing us to identify strongly entangled phase
transitions with a divergent multipartiteness of entanglement. We illustrate
our framework using paradigmatic quantum Ising models, and point out potential
signatures in optical-lattice experiments.
