Global controllability with a single local actuator

We show that we can achieve global density-operator controllability for most $N$-dimensional bilinear Hamiltonian control systems with general fixed couplings using a single, locally acting actuator that modulates one energy-level transition. Controllability depends upon the position of the actuator and relies on the absence of either decompositions into noninteracting subgroups or symmetries restricting the dynamics to a subgroup of $\mathrm{SU}(N)$. These results are applied to spin-chain systems and used to explicitly construct control sequences for a single binary-valued switch actuator.