Repeated quantum non-demolition measurements: convergence and continuous-time limit

TL;DR

This paper studies repeated quantum non-demolition measurements, proving convergence of the system's state and density matrix, and deriving continuous-time limits modeled by stochastic differential equations.

Contribution

It establishes convergence results for repeated indirect measurements and derives continuous-time quantum measurement models from discrete processes.

Findings

System state probabilities converge after many measurements.

Convergence rate is exponential, related to mean relative entropies.

Continuous-time models are derived and shown to converge over large times.

Abstract

We analyze general enough models of repeated indirect measurements in which a quantum system interacts repeatedly with randomly chosen probes on which Von Neumann direct measurements are performed. We prove, under suitable hypotheses, that the system state probability distribution converges after a large number of repeated indirect measurements, in a way compatible with quantum wave function collapse. Similarly a modified version of the system density matrix converges. We show that the convergence is exponential with a rate given by some relevant mean relative entropies. We also prove that, under appropriate rescaling of the system and probe interactions, the state probability distribution and the system density matrix are solutions of stochastic differential equations modeling continuous-time quantum measurements. We analyze the large time convergence of these continuous-time processes and prove convergence.