We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by the propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is “less chaotic.” In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.

我们在量子环面的设置下，研究了克劳斯算子重复间接测量下的半经典量子粒子的轨迹。在测量之间，系统通过哈密顿传播器或元波算子演化。我们在这两种情况下都展示了量子轨迹的总变分与其相应的经典轨迹的收敛性，如半经典缺陷测量的传播所定义的。这种收敛符合经典系统的埃伦菲斯特时间，当系统“不太混乱”时，埃伦菲斯特时间就更大。此外，我们还对这些效应进行了数值模拟。在证明这个结果时，我们提供了一种半经典缺陷度量的表征，我们称之为统一缺陷度量。我们还证明了由环面上的流组成的函数的导数估计。