Quantum Control: Approach based on Scattering Theory for Non-commutative Markov Chains and Multivariate Operator Theory

量子控制：基于非交换马尔可夫链散射理论和多元算子理论的方法

基本信息

批准号：
EP/G039275/1
负责人：
Rolf Gohm
金额：
$ 31.17万
依托单位：
Aberystwyth University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG039275%2F1
关键词：
Quantum Control Approach based Scattering

项目摘要

Modern control theory has frequently used concepts and results from abstract mathematics. The aim of this proposal is to explore genuinely non-commutativeversions with a view toward direct applications to the emergent discipline of quantum control. Some elements of open-loop and closed-loop control have already been developed, and recently there has been much interest in fully quantum (or, coherent) control models. Experimental advances mean that physicists have anunprecedented ability to manipulate quantum mechanical systems, and from the technological point of view there is currently much interest in deriving atheory of quantum engineering as the foundation for a much anticipated quantum technological revolution.The proposed research is on the theoretical side and aims to enlarge the toolkit of mathematical methods available to control quantum systems. The classical theory of control has many deep and productive connections with disciplines of mathematical analysis. Complex analysis is indispensable in the use of Laplace transforms and transfer functions for controlled systems. Based on that the theory of state space models and robust control profited from ideas in operator theory. But there are difficulties to adapt these methods to the world of noncommutative mathematics needed for quantum control. There is partial success in specific models but an integrated, first-principles discipline of quantum control is still missing.A basic idea of this proposal is to make use of recent developments in multivariate operator theory. While in classical operator theory a single operator is analysed, in multivariate operator theory the joint action of a family of operators is studied. These operators may not commute with each other. Nevertheless there are analogues to classical results in complex analysis such as the idea of multi-analytic operators. In fact, many of the operator results which are relevant for classical control theory can be extended to this setting. We propose to develop these tools with applications to quantum control. Scattering theory for non-commutative Markov chains is a theory about open quantum systems with many connections to operator theory. Recently the wave operator occurring in this theory has been rewritten as a multi-analytic operator. On the other hand it is possible to interpret this theory as a version of open-loop control, for example it has been successfully applied to the preparation of states in a micromaser interacting with a stream of atoms. Hence it is very natural to start here to develop the methods of multivariate operator theory as applied to the problems in quantum control.Once the bridge between quantum control and multivariate operator theory is understood in the specific directions described above we speculate that a considerable amount of related and deep mathematics becomes available for engineering applications. In the later parts of the project this also includes the more advanced methods of robust control.The investigators Dr Gohm and Professor Gough are the members of the Quantum Control Research Group at Aberystwyth University, set up in 2007. Both have a strong background in mathematical physics and they share the conviction that building bridges between the demands of an applied discipline such as quantum control and ideas in pure mathematics such as multivariate operator theory is as interesting as it is useful.

现代控制理论经常使用抽象数学的概念和结果。该提案的目的是探索真正的非交换版本，以期直接应用于新兴的量子控制学科。开环和闭环控制的一些要素已经被开发出来，最近人们对全量子（或相干）控制模型产生了很大的兴趣。实验的进步意味着物理学家拥有前所未有的操纵量子力学系统的能力，从技术的角度来看，目前人们对导出量子工程理论作为备受期待的量子技术革命的基础非常感兴趣。一方面，旨在扩大可用于控制量子系统的数学方法工具包。经典控制理论与数学分析学科有着许多深刻而富有成效的联系。在受控系统中使用拉普拉斯变换和传递函数时，复杂分析是必不可少的。在此基础上，状态空间模型和鲁棒控制理论借鉴了算子理论的思想。但要使这些方法适应量子控制所需的非交换数学领域存在困难。在特定模型中取得了部分成功，但仍然缺乏完整的量子控制第一原理学科。该提案的基本思想是利用多元算子理论的最新发展。在经典算子理论中分析单个算子，而在多元算子理论中研究一系列算子的联合作用。这些运营商可能不会相互通勤。尽管如此，复杂分析中还是有一些与经典结果类似的东西，例如多分析算子的思想。事实上，许多与经典控制理论相关的算子结果都可以扩展到这种设置。我们建议开发这些工具并应用于量子控制。非交换马尔可夫链的散射理论是一种关于开放量子系统的理论，与算子理论有很多联系。最近，该理论中出现的波算子已被重写为多解析算子。另一方面，可以将该理论解释为开环控制的一种版本，例如，它已成功应用于与原子流相互作用的微脉泽中的状态准备。因此，从这里开始开发应用于量子控制问题的多元算子理论方法是很自然的。一旦在上述特定方向上理解了量子控制和多元算子理论之间的桥梁，我们推测大量的相关且深入的数学可用于工程应用。在该项目的后期部分，还包括更先进的鲁棒控制方法。研究人员Gohm博士和Gough教授是阿伯里斯特威斯大学量子控制研究小组的成员，该研究小组成立于2007年。两人都拥有深厚的数学背景他们都坚信，在量子控制等应用学科的需求与多元算子理论等纯数学思想之间架起桥梁既有趣又有用。