$\cal H$ and the drift Schrödinger operator $H_0$ (the *internal Hamiltonian*) has discrete spectrum. The control functions $u_1(\cdot),\ldots,u_p(\cdot)$, representing external fields, are real-valued and $\psi(\cdot)$ takes values in the unit sphere of ${\cal H}$.

The controllability of system (\[12\]) is a well-established topic when the state space $\mathcal{H}$ is finite-dimensional (see for instance [@dalessandro-book] and reference therein), thanks to general controllability methods for left-invariant control systems on compact Lie groups ([@brock; @jur; @JK81; @GB82; @AGK96]).

When $\mathcal{H}$ is infinite-dimensional, it is known that the bilinear Schrödinger equation is not exactly controllable (see [@BMS; @turinici]). Hence, one has to look for weaker controllability properties as, for instance, approximate controllability or controllability between eigenstates of the internal Hamiltonian $H_{0}$ (which are the most relevant physical states). In certain cases, when the space variable is one-dimensional, a description of reachable sets has been provided (see [@beauchard-coron; @camillo]). In dimension larger than one or for more general situations, the exact description of the reachable set appears to be more difficult and at the moment only approximate controllability results are available. Most of them are for the single-input case (see, in particular, [@noi; @mirrahimi-aihp; @Nersy; @nersesyan; @beauchard-nersesyan; @ancoranoi; @fratelli-nersesyan]). Such results are based on sufficient 