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arXiv · 2605.12783

Purification of a monitored qubit: exact path-integral solution

Matheus M. R. Poltronieri Martins·Henrique Santos Lima
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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 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.

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Matheus M. R. Poltronieri Martins, Henrique Santos Lima. 2026-05-12. Purification of a monitored qubit: exact path-integral solution. https://arxiv.org/abs/2605.12783

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