Quantum trajectories. Spectral gap, quasi-compactness & limit theorems

IF 1.7 2区 数学 Q1 MATHEMATICS
Tristan Benoist, Arnaud Hautecœur, Clément Pellegrini
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引用次数: 0

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 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.
量子轨迹。谱间隙、拟紧性和极限定理
量子轨迹是马尔可夫过程,模拟了量子系统在重复独立测量下的演化。在矩阵随机积理论的启发下,证明了这些马尔可夫过程在净化和不可约假设下具有唯一不变测度。本文主要研究底层马尔可夫算子的谱问题。利用拟紧性,证明了该算子允许谱隙,并能精确地描述外围谱。然后研究了该算子的两个摄动。这允许导出极限定理(中心极限定理,Berry-Esseen界和大偏差原理)的马尔可夫链的经验均值函数以及底层随机动力系统的李亚普诺夫指数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
3.20
自引率
5.90%
发文量
271
审稿时长
7.5 months
期刊介绍: The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published. Research Areas Include: • Significant applications of functional analysis, including those to other areas of mathematics • New developments in functional analysis • Contributions to important problems in and challenges to functional analysis
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