We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, { \it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group $S_n$. Because of the symmetry, the underlying Hilbert space, ${ \cal H}=(\mathbb{ C}^d)^{ \otimes n}$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(d^n)$, denoted here by $u^{ S_n}(d^n)$. The dynamical Lie algebra ${ \cal L}$, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra $u^{ S_n}(d^n)$. If ${ \cal L}$ acts as $su\left( \dim(V) \right)$ on each of the invariant subspaces $V$, the system is called { \it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space ${ \cal H}$ coincides with the { \it Clebsch-Gordan} splitting of $(\mathbb{ C}^d)^{ \otimes n}$ into { \it irreducible representations} of $su(d)$. In this view, $u^{ S_n}(d^n)$, is the direct sum of certain $su(n_j)$ for some $n_j$'s we shall specify, and its { \it center} which is the Abelian (Lie) algebra generated by the { \it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general $n$ and $d$ but we recast previous results on $n$ qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of $n=3$, $d=3$, that is, { \it three qutrits}.
Categories: quant-ph