  Controlled generation and manipulation of photon states encoded in their
spatial degrees of freedom is a crucial ingredient in many quantum information
tasks exploiting higher-than-two dimensional encoding. Here, we prove the
impossibility to arbitrarily modify $d$-level state superpositions (qu$d$its)
for $d>2$, encoded in the transverse modes of light, with optical components
associated to the group of symplectic transforms (Gaussian operations).
Surprisingly, we also provide an explicit construction of how non-Gaussian
operations acting on mode subspaces do enable to overcome the limit $d=2$. In
addition, this set of operations realizes the full SU(3) algebra.
