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arXiv:1203.2380 · Journal of Physics A Mathematical and Theoretical

Quantum control and representation theory

Alberto Ibort, Juan Manuel Pérez-Pardo

Published Apr 28, 2009• 8 citations
arXivHTMLPDF

Abstract

A new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced. We call such notion von Neumann controllabilty and it is shown that it is strictly weaker than the usual notion of pure state and operator controlability. We provide a simple and effective characterization of it by using tools from the theory of unitary representations of Lie groups. In this sense we are able to approach the problem of control of quantum states from a new perspective, that of the theory of unitary representations of Lie groups. A few examples of physical interest and the particular instances of compact and nilpotent dynamical Lie groups are discussed.

DOI
10.1088/1751-8113/42/20/205301

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A new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced. We call such notion von Neumann controllabilty and it is shown that it is strictly weaker than the usual notion of pure state and operator controlability. We provide a simple and effective characterization of it by using tools from the theory of unitary representations of Lie groups. In this sense we are able to approach the problem of control of quantum states from a new perspective, that of the theory of unitary representations of Lie groups. A few examples of physical interest and the particular instances of compact and nilpotent dynamical Lie groups are discussed.
Abstract One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie‐theoretical developments in finite‐dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control‐an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in quantum technology. The purpose of this paper is to analyze such controllability aspects for finite dimensional bilinear quantum control systems. We use a unified approach based on Lie-algebraic methods from nonlinear control theory to revisit known and to establish new results for closed and open quantum systems. In particular, we provide a simplified characterization of different notions of controllability for closed quantum systems described by the Liouville-von Neumann equation. We derive new necessary and sufficient conditions for accessibility of open quantum systems modelled by the Lindblad-Kossakowski master equation. To this end, we exploit a well-studied topic of differential geometry, namely the classification of all matrix Lie-groups which act transitively on the Grassmann manifold or the punctured Euclidean space. For the special case of coupled spin-1/2 systems, we obtain a remarkably simple characterization of accessibility. These accessibility results correct and refine previous statements in the quantum control literature.
In this paper, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability. The paper also contains results on the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove that, for the system of two interacting spin 1/2 particles, not every state transfer can be obtained in arbitrary small time.
The introduction of control theory in quantum mechanics has created a rich, new interdisciplinary scientific field, which is producing novel insight into important theoretical questions at the heart of quantum physics. Exploring this emerging subject, Introduction to Quantum Control and Dynamics presents the mathematical concepts and fundamental physics behind the analysis and control of quantum dynamics, emphasizing the application of Lie algebra and Lie group theory. To advantage students, instructors and practitioners, and since the field is highly interdisciplinary, this book presents an introduction with all the basic notions in the same place. The field has seen a large development in parallel with the neighboring fields of quantum information, computation and communication. The author has maintained an introductory level to encourage course use. After introducing the basics of quantum mechanics, the book derives a class of models for quantum control systems from fundamental physics. It examines the controllability and observability of quantum systems and the related problem of quantum state determination and measurement. The author also uses Lie group decompositions as tools to analyze dynamics and to design control algorithms. In addition, he describes various other control methods and discusses topics in quantum information theory that include entanglement and entanglement dynamics. Changes to the New Edition: New Chapter 4: Uncontrollable Systems and Dynamical Decomposition New section on quantum control landscapes A brief discussion of the experiments that earned the 2012 Nobel Prize in Physics Corrections and revised concepts are made to improve accuracy Armed with the basics of quantum control and dynamics, readers will invariably use this interdisciplinary knowledge in their mathematics, physics and engineering work.
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.