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 (qudits) 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.
Cite
@article{arxiv.0712.1197,
title = {Manipulation of single-photon states encoded in transverse spatial modes: possible and impossible tasks},
author = {Gabriel F. Calvo and Antonio Picón},
journal= {arXiv preprint arXiv:0712.1197},
year = {2011}
}