Tight bounds on recurrence time in closed quantum systems

TL;DR

This work provides deterministic, rigorous upper bounds on recurrence times t_rec(ε) for closed quantum systems in finite dimensions by linking recurrence to an exit time t_exit(ε) and a packing-number argument. It derives an inverse quantum speed limit that bounds the exit time in terms of Hamiltonian fluctuations Δ(H^{2}) and Δ(H^{4}), with an explicit ε_* controlling the bound's validity. The results show the first recurrence bound t_rec(ε) ≤ t_exit(ε) (4π/ε)^{d−1}, extendable to subsequent recurrences and to unitary-channel dynamics, and demonstrate that random Hamiltonians saturate these bounds. The analysis further reveals how initial-state coherence and effective support influence recurrence, and provides sharper bounds for non-interacting (symmetric) dynamics, while highlighting limitations of effective dimension as a predictor for recurrence time. Overall, the findings offer a rigorous framework for understanding recurrence in quantum dynamics with broad implications for quantum statistical mechanics and chaotic dynamics.

Abstract

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, $t_{\mathrm{rec}} \lesssim t_{\mathrm{exit}}(ε)(1/ε)^d$, where $d$ is the Hilbert-space dimension, $ε$ the neighborhood size, and $t_{\mathrm{exit}}(ε)$ 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 $ψ_t$ needs to depart from the $ε$-vicinity of the initial state $ψ_0$. We provide a partial solution, showing that under mild assumptions $t_{\mathrm{exit}}(ε) \approx ε/\sqrt{ Δ(H^2)}$, with $Δ(H^2)$ the Hamiltonian variance in $ψ_0$. 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.

Figures (2)

• Figure 1: The Hamiltonian evolution of a closed quantum system started at an initial state $\psi_0$. After time $t_{\mathrm{exit}}$ the system leaves the $\varepsilon$-ball around the initial point and subsequently after time $t_{\mathrm{rec}} > t_{\mathrm{exit}}$ the system has undergone recurrence.
• Figure 2: Main part of the proof of \ref{['prop:t+N']}. If the number of points $x_k= \varphi^k(x_0)$ is greater than the packing number ${\cal N}_{\rm pack}(X,\varepsilon)$, there exists a point $x_{j-i}$ within distance $\varepsilon$ of $x_0$. The ball of radius $\varepsilon$ around $x_0$ is shown in red.

Theorems & Definitions (35)

• Theorem 1: Recurrence for states
• Theorem 2: Inverse quantum speed limit
• Theorem 3
• Proposition 1
• proof
• Lemma 1
• proof
• Remark 1
• Remark 2
• Lemma 2