Asymptotic Purification of Quantum Trajectories under Random Generalized Measurements

TL;DR

The paper addresses asymptotic purification of quantum trajectories under repeated random generalized measurements with stationary noise in disordered environments. It generalizes the Kümmerer–Maassen framework by introducing a measurable random-subspace darkness concept and modeling trajectories as a time-inhomogeneous Markov process in a random environment. The main contribution is a necessary-and-sufficient condition: asymptotic purification occurs if and only if the collection of random dark subspaces is almost surely empty. Overall, the work provides a structured method to diagnose purification via darkness in disorder, supported by illustrative examples with implications for quantum control under randomized measurements.

Abstract

We develop a general framework to study quantum trajectories resulting from repeated random measurements subject to stationary noise, and generalize results of Kümmerer and Maassen to this setting. The resulting trajectory of quantum states is a time-inhomogeneous Markov chain in a random environment. Kümmerer and Maassen introduced the concept of dark subspaces for noise-free processes, establishing that their absence is equivalent to asymptotic purification of the system state. We clarify the notion of dark subspaces in the disordered setting by defining a measurable correspondence consisting of a collection of random subspaces satisfying a darkness condition. We further prove that asymptotic purification occurs if and only if this collection of random dark subspaces is empty. Several examples of these phenomena are provided.