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, $t_{\mathrm{rec}}\lesssim t_{\mathrm{exit}}(\epsilon )(1/\epsilon {)}^d$, where $d$ is the Hilbert-space dimension, $\epsilon$ the neighborhood size, and $t_{\mathrm{exit}}(\epsilon )$ the escape time from this neighborhood. For pure states evolving under a Hamiltonian $H$, estimating $t_{\mathrm{exit}}$ is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state ${\psi }_t$ needs to depart from the $\epsilon$ vicinity of the initial state ${\psi }_0$. We provide a partial solution, showing that under mild assumptions $t_{\mathrm{exit}}(\epsilon )\approx \epsilon /\sqrt{\mathrm{\Delta }(H^2)}$, with $\mathrm{\Delta }(H^2)$ the Hamiltonian variance in ${\psi }_0$. We show that our upper bound on $t_{\mathrm{rec}}$ 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.

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• Quantum information theory