Dynamical Decomposition of Bilinear Control Systems subject to Symmetries

TL;DR

This paper introduces a method to analyze and decompose the dynamics of control systems on Lie groups with symmetries, using generalized Young symmetrizers, with applications to quantum multipartite systems.

Contribution

It develops a novel decomposition technique for bilinear control systems with symmetries, leveraging representation theory to handle tensor product spaces.

Findings

Applicable to quantum spin networks and multipartite systems

Demonstrated through multiple example applications

Provides a framework for future research directions

Abstract

We describe a method to analyze and decompose the dynamics of a control system on a Lie group subject to symmetries. The method is based on the concept of generalized Young symmetrizers of representation theory. It naturally applies to the situation where the system evolves on a tensor product space and there exists a finite group of symmetries for the dynamics which interchanges the various factors. This is the case for quantum mechanical multipartite systems, such as spin networks, where each factor of the tensor product represents the state of one of the component systems. We present several examples of applications and indicate directions for future research.