Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems
Abstract
Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap and the peripheral spectrum is described in a precise manner. Next two perturbations of this operator are studied. This allows to derive limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation Principle) for the empirical mean of functions of the Markov chain as well as the Lyapounov exponent of the underlying random dynamical system.
Cite
@article{arxiv.2402.03879,
title = {Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems},
author = {Tristan Benoist and Arnaud Hautecoeur and Clément Pellegrini},
journal= {arXiv preprint arXiv:2402.03879},
year = {2025}
}
Comments
with respect to v1, shorter proof of former Theorem 3.3 (on the advice of anonymous referee) and minor corrections. Former Appendix B has been deleted accordingly. To be published in JFA