Contributions of control theory to fundamental quantum mechanics and its applications

In this paper we survey some results in Control Theory that have had a significant impact on quantum mechanics and its applications. We shall consider three specific examples: 1) The concept of controllability, in its various forms, has provided a practical test to verify the feasibility of a number of putative experiments in quantum physics. The verification of controllability involves the study of the Lie algebra structure underlying the system. This study itself has proved a valuable tool to design experiments. 2) Results on the uniform generation of compact Lie groups have been applied to show that almost every set of quantum gates is universal for quantum computation. This result can be further refined by showing that the number of gates necessary to implement quantum logic operations is uniformly bounded by a value which only depends on the number of quantum bits processed. 3) Techniques of Optimal Control Theory have been employed to show that the pulses used to drive particles with spin 1/2 in Nuclear Magnetic Resonance are the optimal pulses which minimize the time of state transfer (subject to an energy constraint) and the energy (subject to a time constraint).