  We prove that the time required for sustained information scrambling in any
Hamiltonian quantum system is universally at least logarithmic in the
entanglement entropy of scrambled states. This addresses two foundational
problems in nonequilibrium quantum dynamics. (1) It sets the earliest possible
time for the applicability of equilibrium statistical mechanics in a quantum
system coupled to a bath at a finite temperature. (2) It proves a version of
the fast scrambling conjecture, originally motivated in models associated with
black holes, as a fundamental property of quantum mechanics itself. Our result
builds on a refinement of the energy-time uncertainty principle in terms of the
infinite temperature spectral form factor in quantum chaos. We generalize this
formulation to arbitrary initial states of the bath, including finite
temperature states, by mapping Hamiltonian dynamics with any initial state to
nonunitary dynamics at infinite temperature. A regularized spectral form factor
emerges naturally from this procedure, whose decay is universally constrained
by analyticity in complex time. This establishes an exact speed limit on
information scrambling by the most general quantum mechanical Hamiltonian,
without any restrictions on locality or the nature of interactions.
