Statistical theory of controlled quantum dynamics

我们目前正在进入一个新的技术时代，利用量子力学效应是从计算到密码学和计量学等应用的核心。量子开放系统的建模、估计和识别是一系列量子工程任务的关键支持工具，如状态准备和断层扫描、量子反馈控制、误差校正和高精度计量。随着量子设备和测量技术变得越来越复杂，实验者必须解释越来越复杂的测量数据集，以获得有关系统状态和动态的准确信息。因此，有必要扩展量子统计的数学基础，超越传统的状态估计的i.i.d（独立、同分布）框架，以解决涉及相互作用系统相关状态的新推理问题，并具有现实的噪声建模和最佳实验设计。本文的目标是在量子输入-输出（I-O）动力学的框架下建立量子随机过程的统计理论。I-O形式主义将感兴趣的系统（例如量子设备或传感器）描述为通过输入和输出通道与外部世界交互的黑盒。这些可以模拟不需要的影响，如减相和泄漏，或者可以用来监测和控制系统使用量子反馈。该建议的中心前提是，有关动态的信息被连续编码到输出状态，并且该资源可以被利用来执行诸如系统识别，估计和量子增强计量等任务。三个中心目标是：(1)发展与中心极限、量子过程集中和大偏差有关的新的数学理论，与有限相关态和隐马尔可夫过程的理论密切相关；(2)建立了以局部渐近正态性为中心的I-O系统量子估计的一般统计框架；(3)将这些理论结果用于设计新的量子计量装置，这些装置利用动态相变等特征，包括使用新方法进行时间相关参数的统计学习。

We are currently entering a new technological era where harnessing quantum mechanical effects is at the core of applications ranging from computation to cryptography and metrology. The modelling, estimation and identification of quantum open systems are key enabling tools for a broad array of quantum engineering tasks such as state preparation and tomography, quantum feedback control, error correction, and high precision metrology. As quantum devices and measurement techniques become more sophisticated, experimenters have to interpret increasingly complex measurement datasets to obtain accurate information about the system's state and dynamics. Therefore, there is a need for expanding the mathematical foundations of quantum statistics beyond the traditional i.i.d. (independent, identically distributed) framework of state estimation, in order to tackle new inference problems involving correlated states of interacting systems, with realistic modelling of noise, and optimal experimental design. The goal of this proposal is to build a statistical theory of quantum stochastic processes in the framework of quantum input-output (I-O) dynamics. The I-O formalism describes the system of interest (e.g. quantum device, or sensor) as a black-box interacting with the outside world via input and output channels. These can model unwanted effects such as dephasing and leakage, or can be used to monitor and control the system using quantum feedback. The central premise of the proposal is that information about the dynamics is continuously encoded into the output state, and this resource can be exploited to perform tasks such as system identification, estimation and quantum enhanced metrology. The three central objectives are: (1) to develop new mathematical theory pertaining to central limit, concentration for quantum processes, and large deviations, in close relation to the theories of finitely correlated states and hidden Markov processes; (2) to build a general statistical framework for quantum estimation with I-O systems centred around the concept of local asymptotic normality; and (3) to employ these theoretical results for the design of new quantum metrology setups which exploit features such as dynamical phase transitions, including the use of novel methods for statistical learning of time-dependent parameters.