Tight bounds on recurrence time in closed quantum systems

The evolution of an isolated quantum system inevitably exhibits recurrence: the state returns to the vicinity of its initial condition after finite time. Despite its fundamental nature, a rigorous quantitative understanding of recurrence has been lacking. We establish upper bounds on the recurrence time, t_rec lesssim t_exit(ε)(1/ε)^d, where d is the Hilbert-space dimension, ε the neighborhood size, and t_exit(ε) the escape time from this neighborhood. For pure states evolving under a Hamiltonian H, estimating t_exit is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state ψ_t needs to depart from the ε-vicinity of the initial state ψ₀. We provide a partial solution, showing that under mild assumptions t_exit(ε) approx ε/sqrt{ Δ\(H²\)}, with Δ\(H²\) the Hamiltonian variance in ψ₀. We show that our upper bound on t_rec is generically saturated for random Hamiltonians. Finally, we analyze the impact of coherence of the initial state in the eigenbasis of H on recurrence behavior.