Purification of a monitored qubit: Exact path-integral solution
Phys. Rev. E 114, 044131 – Published 19 October, 2026
DOI: https://doi.org/10.1103/cjfz-qn1b
Abstract
We investigate the purification dynamics of a single qubit under continuous-in-time monitoring. By employing a collisional model framework where the system interacts sequentially with ancillary qubits, we describe the conditioned evolution of the system's density matrix through a stochastic master equation. We show that for initial mixed states, the dynamics reduce to a multiplicative Langevin equation for a single scalar parameter representing the state's purity. This stochastic process is solved exactly using the Onsager-Machlup path-integral formalism, allowing us to derive the full probability distribution for the qubit's trajectories. Our analytical results reveal that purification is characterized by a dynamical crossover from a diffusion-dominated regime to a measurement-dominated regime, visible in the emergence of a bimodal state distribution. The analytical solutions are in strong agreement with numerical simulations, providing a robust theoretical benchmark for the study of information extraction in monitored quantum systems.
