  Multipartite entanglement, characterized by the quantum Fisher information
(QFI), plays a central role in quantum-enhanced metrology and understanding
quantum many-body physics. With a dynamical generalization of the Mazur-Suzuki
relations, we provide a rigorous lower bound on the QFI for the thermal Gibbs
states in terms of dynamical symmetries, i.e., operators with periodic time
dependence. We demonstrate that this bound can be saturated when considering a
complete set of dynamical symmetries. Moreover, this lower bound with dynamical
symmetries can be generalized to the QFI matrix and to the QFI for the thermal
pure states, predicted by the eigenstate thermalization hypothesis. Our results
reveal a new perspective to detect multipartite entanglement and other
generalized variances in an equilibrium system, from its nonstationary
dynamical properties, and is promising for studying emergent nonequilibrium
many-body physics.
