Ergodic Theorems for Continuous-Time Quantum Walks on Crystal Lattices and the Torus

We give several quantum dynamical analogs of the classical Kronecker–Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk \(\exp (-\textrm{i}t \Delta ) \psi \) $exp(-\text{i}t\mathrm{\Delta })\psi$ starting from a localized initial state \(\psi \) $\psi$ . Then, the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.