Quantum Control and Representation Theory

TL;DR

This paper introduces von Neumann controllability for quantum systems, a new concept leveraging superposition, characterized via Lie group representation theory, offering a fresh perspective on quantum control.

Contribution

It defines von Neumann controllability, demonstrates its relation to existing notions, and provides a characterization using Lie group unitary representations.

Findings

Von Neumann controllability is strictly weaker than traditional controllability.

Characterization of controllability using Lie group representation theory.

Application examples involving compact and nilpotent Lie groups.

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.