The rate of purification of quantum trajectories

TL;DR

This paper proves that quantum trajectories conditioned on measurements tend to purify exponentially fast, and demonstrates that quantum state estimation converges rapidly, quantifying information gain during measurement.

Contribution

It provides a simplified proof of purification and establishes exponential convergence rates for quantum trajectories and state estimation using Lyapunov methods.

Findings

Quantum trajectories almost surely purify over time.

Purification occurs at an exponential rate in expectation.

Estimated states converge exponentially fast to true states.

Abstract

We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, K\"{u}mmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process.