Quantum trajectories are Markov processes that describe the time evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called Stochastic Schrödinger Equations, which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a “purification” condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast toward this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.

量子轨迹是马尔可夫过程，描述经历连续间接测量的量子系统的时间演化。在数学上，它们被定义为所谓的随机Schrödinger方程的解，该方程是由Poisson和Wiener过程驱动的非线性随机微分方程。本文致力于量子轨迹的不变度量的研究。特别是，我们证明了在平均时间演化过程中的遍历性条件下以及演化生成者的“纯化”条件下不变度量是唯一的。我们进一步证明，量子轨道在律上以指数形式快速收敛于该不变量度。我们用示例说明我们的结果，在这些示例中我们可以得出不变测度的显式表达式。