Large Time Behaviour and Convergence Rate for Non Demolition Quantum Trajectories

TL;DR

This paper analyzes the long-term behavior of quantum trajectories under continuous measurement, showing convergence to a pure state and quantifying the rate of this convergence, with implications for quantum state estimation.

Contribution

It provides a rigorous analysis of the asymptotic behavior and convergence rate of quantum trajectories in the context of Quantum Non Demolition measurements.

Findings

Quantum trajectories converge to a random pure state over time.

The exponential rate of convergence is explicitly determined.

Implications for quantum state estimation are discussed.

Abstract

A quantum system S undergoing continuous time measurement is usually described by a jump-diffusion stochastic differential equation. Such an equation is called a stochastic master equation and its solution is called a quantum trajectory. This solution describes actually the evolution of the state of S. In the context of Quantum Non Demolition measurement, we investigate the large time behavior of this solution. It is rigorously shown that, for large time, this solution behaves as if a direct Von Neumann measurement has been performed at time 0. In particular the solution converges to a random pure state which is related to the wave packet reduction postulate. Using theory of Girsanov transformation, we determine precisely the exponential rate of convergence towards this random state. The important problem of state estimation (used in experiment) is also investigated.