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Recurrence Time for Finite Quantum Systems
Phys. Rev. Lett. 137, 140201 – Published 30 September, 2026
DOI: https://doi.org/10.1103/392k-bgpt
Abstract
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.
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Note that we do not require all states to recur non-trivially, as some states might be stationary (e.g., energy eigestates for Hamiltonian evolution).
Note that this is a strict inequality, and hence, a slightly stronger condition than Eq. (c2).
If you have items places in pigeonholes, one of the pigeonholes must contain at least two items.
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/392k-bgpt for further details on rational approximation for difference of reals using tiling.
