Limit theorems for quantum trajectories and analyticity of the pressure by quasi-compactness of Markov operators

Arnaud Hautecoeur

Published Oct 9, 2025• 0 citations

Abstract

This thesis investigates limit theorems for quantum trajectories and the analyticity of the pressure function for random matrix products.Quantum trajectories are Markov chains that model the state of a quantum system undergoing repeated indirect measurements. The long-term behavior of these Markov chains is studied through the spectral properties of the associated Markov operator. More precisely, we show that this operator is quasi-compact, which implies the existence of a spectral gap. By analytically perturbing this operator, the quasi-compactness property is preserved, allowing us to derive a central limit theorem, Berry–Esseen bounds, and a large deviation principle.We also show in this thesis that the pressure function for matrix products is analytic under suitable contractivity and irreducibility conditions. This analysis also relies on the quasi-compactness of certain Markov operators. In addition, we establish a variational principle for the pressure.The thesis contains three introductory chapters. The first discusses quasi-compactness; the second introduces the formalism of quantum trajectories and their connection to matrix products; and the third provides elements of statistical mechanics to introduce the notion of pressure. Chapter 4 presents the limit theorems obtained for quantum trajectories, and Chapter 5 contains the proof of the analyticity of the pressure.

DOI: 10.70675/e24c367bz427ez4daez8d71zdb51a97cda67

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