Cours doctoral Arnaud Hautecoeur

Dr. Arnaud Hautecoeur de l'Institut de Mathématiques de Toulouse dispense des cours doctoraux au sein de l'Institut CY Advanced Studies du lundi 8 juin au vendredi 12 juin 2026 de 9h00 à 11h30 sur le thème : "Introduction to Quantum Trajectories"

Quantum trajectories are Markov chains representing the evolution of the state of a quantum system subjected to repeated indirect measurements. In this course, after introducing the physical model and the mathematical formalism of quantum physics, we study the mathematical properties of the associated Markov chain. In particular, we will study the spectral properties of the Markov operator in order to derive limit theorems for the Markov chain.

Plan of the course

1. What are quantum trajectories?
• Introduction to the quantum trajectory model
• Mathematical formalism of quantum mechanics
• Modeling of quantum trajectories
• Examples

2. Existence and uniqueness of the invariant measure
• Definition of the Markov operator
• Why is this Markov chain singular?
• Exponential convergence toward the invariant measure.

3. Limit theorems for Markov chains
• Outline of the proof strategy for the results
I. Introduction to quasi-compactness
II. Introduction to the study of random matrix products.
III. Proof of the limit theorems

4. Further directions?
• Can the purification assumption be removed?