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}≲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 ψ_{0}. We provide a partial solution, showing that under mild assumptions t_{exit}(ε)≈ε/sqrt[Δ(H^{2})], with Δ(H^{2}) the Hamiltonian variance in ψ_{0}. 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.
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}≲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 ψ_{0}. We provide a partial solution, showing that under mild assumptions t_{exit}(ε)≈ε/sqrt[Δ(H^{2})], with Δ(H^{2}) the Hamiltonian variance in ψ_{0}. 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.
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, trec≲texit(ε)(1/ε)d, where d is the Hilbert-space dimension, ε the neighborhood size, and texit(ε) the escape time from this neighborhood. For pure states evolving under a Hamiltonian H, estimating texit is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state ψt needs to depart from the ε-vicinity of the initial state ψ0. We provide a partial solution, showing that under mild assumptions texit(ε)≈ε/Δ(H2), with Δ(H2) the Hamiltonian variance in ψ0. We show that our upper bound on trec 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.
We study the time it takes for all states of a finite quantum system to return simultaneously to their original configuration. In particular, we define the recurrence time for a quantum system to be the time at which all time-evolved states are close to their initial configuration, and at least one state has deviated significantly during this interval. Considering finite-dimensional quantum systems evolving unitarily, we find bounds on this notion of recurrence time, for continuous time and discrete time, by using Dirichlet's approximation theorem. We show how the problem of finding a bound on recurrence time can be related to approximating the difference of real numbers by rationals. We present a mathematical result on the latter, which we then use to obtain tighter bounds on recurrence time.
The Poincaré recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most studies treat recurrence in an approximate, distance-based sense, here we address the problem of {exact}, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary's spectrum. This computationally tractable approach yields both positive results, enumerating all candidate recurrence times, and definitive negative results, rigorously ruling out candidate recurrence times for a given set of Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrences, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.
The expected return time to the original state is a key concept characterizing systems obeying both classical or quantum dynamics. We consider iterated open quantum dynamical systems in finite-dimensional Hilbert spaces, a broad class of systems that includes classical Markov chains and unitary discrete-time quantum walks on networks. Starting from a pure state, the time evolution is induced by repeated applications of a superoperator (quantum channel) in each time step followed by a measurement to detect whether the system has returned to the original state. We prove that if the superoperator is unital in the part of the Hilbert space explored by the system, then the expectation value of the return time is an integer, equal to the dimension of this relevant Hilbert space. We illustrate our results on partially coherent quantum walks on finite graphs. Our work shows that the expected return time is a quantitative measure of the size of the part of the Hilbert space available to the system when the dynamics is started from a certain state.
The traditional quantum speed limits are not attainable for many physical processes, as they tend to be loose and fail to determine the exact time taken by quantum systems to evolve. To address this we derive exact quantum speed limits for the unitary dynamics of pure-state quantum systems that outperform the existing quantum speed limits. Using these exact quantum speed limits, we can precisely estimate the evolution time for two- and higher-dimensional quantum systems. Additionally, for both finite- and infinite-dimensional quantum systems, we derive an improved Mandelstam-Tamm bound for pure states and show that this bound always saturates for any unitary generated by self-inverse Hamiltonians. Furthermore, we show that our speed limits establish an upper bound on the quantum computational circuit complexity and yield a tighter quantum Cramér-Rao bound. These results will have a significant impact on our understanding of quantum physics as well as rapidly developing quantum technologies, such as quantum computing, quantum control, and quantum batteries.
Recurrence time quantifies the duration required for a physical system to return to its initial state, playing a pivotal role in understanding the predictability of complex systems. In quantum systems with subspace measurements, recurrence times are governed by Anandan-Aharonov phases, yielding fractionally quantized recurrence times. However, the fractional quantization phenomenon in interacting quantum systems remains unexplored. Here, we address this gap by establishing universal lower and upper bounds for recurrence times in interacting many-body spin systems. Notably, we investigate scenarios where these bounds are approached, shedding light on the speed of quantum processes under monitoring. In specific cases, our findings reveal that the complex many-body system can be effectively mapped onto a dynamical system with a single quasi-particle, leading to integer-quantized recurrence times. Our work demonstrates a valuable link between recurrence times and the number of dark states in the system, thus providing a deeper understanding of the intricate interplay between Hilbert-space fragmentation, ergodicity breaking, measurements, and interaction effects. Finally, our findings have been implemented on an IBM quantum computer, revealing resonances and fractional quantization in agreement with theoretical predictions. This demonstrates the resilience of nonequilibrium topological fractional quantization to noise and highlights its potential use for benchmarking quantum devices and probing dark states.