Burkhard Kuemmerer, Hans Maassen
If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states.
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Asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement
本文研究了不完美测量下量子轨迹的渐近稳定性和遍历性质，推导了估计轨迹收敛到真实轨迹的条件，并证明了不变测度的唯一性。
In 1989 Kusuoka started the study of probability measures on the shift space that are defined with the help of products of matrices. In particular, he derived a sufficient condition for the ergodicity of such measures, which have since been referred to as Kusuoka measures. We observe that repeated measurements of a unitarily evolving quantum system generate a Kusuoka measure on the space of sequences of measurement outcomes. We show that if the measurement consists of scaled projections, then Kusuoka's sufficient ergodicity condition can be significantly simplified. We then prove that this con…
Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements
研究无序广义测量下的量子轨迹，建立测量结果的强大数定律，扩展了Kümmerer和Maassen的噪声自由结果。
We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the `nodal point-X-point complex' mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is ma…
This is a short review in the theory of chaos in Bohmian Quantum Mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally we explore the effect of quantum entanglement on the evolution of the Bohmian traj…
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, quantum ergodic regime and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates and per gate dissipation rate induced b…
Ergodic dynamics in iterated quantum protocols
本文研究了迭代量子协议中测量诱导的非线性动力学，发现特定协议在纯态下呈现遍历性，同时混合态动力学表现出统计混合与纯化的共存。
Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments
本文提出了一种在量子计算机上模拟线性伊藤随机微分方程的通用框架，通过量子轨迹和李纳布兰特方程实现高效模拟。
Unlike classical system, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entan…
Limit Theorems for Disordered Quantum Trajectories
本文针对无序环境中的离散时间量子轨迹，在淬火Doeblin条件下建立了随机后验律等变族的唯一存在性，并证明遍历环境下时间平均的收敛性、方差渐近性、中心极限定理及Berry-Esseen界。
Exact Hilbert-space ergodicity from continuous monitoring
通过连续监测构造量子系统，证明其量子轨迹的晚期平衡分布精确对应Scrooge系综，仅需跳变算子形成变形酉1-设计，数值验证热态平衡。
Invariant measures of randomized quantum trajectories
本文研究随机化探针可观测量的量子轨迹性质，证明非奇异随机化使轨迹纯化并具有唯一不变测度，引入乘法原生性概念并计算典型量子通道的不变测度。
Dark Subspaces and Invariant Measures of Quantum Trajectories
本文研究了量子轨迹的不变测度，通过分析暗子空间上的马尔可夫过程并利用等距族最小性概念，建立了量子轨迹不变测度与遍历测度的完整分类。
Limit theorems for Quantum Trajectories
本文研究量子轨迹马尔可夫过程，在纯化与不可约条件下证明 LLN、泛函中心极限定理、重对数律及中偏差原理，并分别以遍历定理、调和函数分析和鞅逼近完成证明。
Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called "Stochastic Schrödinger Equations", which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a "purification" condition on the generator of the evolution…
An introduction to monitored quantum systems and quantum trajectories: spectrum, typicality, and phases
本文介绍受监控量子系统的理论框架及最新发展，探讨量子轨迹的谱特性及其与相位转变的关系。
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