Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements

TL;DR

This paper proves a strong law of large numbers for measurement outcomes in disordered quantum trajectories modeled as Markov chains in random environments, extending previous noise-free results to disordered settings.

Contribution

It establishes ergodic theorems and a law of large numbers for disordered quantum trajectories, covering various types of disorder including i.i.d., Markovian, periodic, quasiperiodic, and constant.

Findings

Proved a strong law of large numbers for measurement outcomes.

Extended ergodic theorems to disordered quantum systems.

Unified treatment of different disorder types in quantum trajectories.

Abstract

We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCRE) with dynamically-defined transition probabilities; we call these disordered quantum trajectories. Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of K\"ummerer and Maassen to the present disordered framework.