Purification of a monitored qubit: exact path-integral solution

TL;DR

This paper provides an exact analytical solution for the purification process of a monitored qubit using path-integral methods, revealing a crossover in the dynamics and offering a benchmark for quantum measurement studies.

Contribution

It introduces an exact path-integral solution for qubit purification dynamics under continuous monitoring, connecting stochastic equations to full trajectory distributions.

Findings

Purification dynamics exhibit a crossover from diffusion to measurement dominance.

The full probability distribution of qubit trajectories is derived analytically.

Results agree well with numerical simulations, validating the theoretical approach.

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.