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Stochastic Processes and their Applications

Limit theorems for quantum trajectories

Tristan Benoist, Jan-Luka Fatras, Clément Pellegrini

Published Jul 20, 2023• 11 citations
Publisher
DOI
10.1016/j.spa.2023.07.014

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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.

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Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure - see Benoist et al. Probab. Theory Relat. Fields 2019. In this article we prove, finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff's ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums.

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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.
Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure - see Benoist et al. Probab. Theory Relat. Fields 2019. In this article we prove, finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff's ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums.