The Lie algebra structure of spin systems and their controllability properties

In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.