# How do we restrict to a limited number of dimensions, say 3 for qutrits, while using OAM states of light?

When a azimuthal phase $$\mathrm{e}^{il\phi}$$ is applied to gaussian beams having plane wavefront, they develop a corkscrew sort of structure and therefore possess an orbital angular momentum in addition to the spin angular momentum. Due to the continuity of the function, it should have the same value after a $$2\pi$$ rotation hence $$l$$ can only take integer values. One can create these states by passing Gaussian beams into spiral wave plates. I want to know if we want to perform say a QKD protocol using let's say qutrits how do we restrict ourselves to just 3 dimensions?

As for any platform, one has to choose a suitable $$d$$-dimensional "computational" subspace. Suitability depends on your application, but generally it means that one should be able to perform operations on that subspace and couple it to other qudits. In practice, these operations will couple the qudit to degrees of freedom outside of the subspace which will effectively lead to noise. Additionally, one often want the subspace to be "robust" against e.g. thermal noise. In practise, this could mean that one chooses the lowest energy eigenstates of a system which are hopefully separated by a large energy gap from the rest.
As far as I know from conversations with friends working on OAM & entanglement, the qubit case is often realised in a $$\pm l$$ subspace since it is particularly easy to generate a maximally entangled state using e.g. parametric down conversion. As far as I know, higher-dimensional implementations are less common, but there are a few groups working on this. Here's a recent paper (open access) by Vienna, Tampere and Ottawa groups showing how to it for $$d \leq 5$$: https://www.osapublishing.org/optica/abstract.cfm?uri=optica-7-2-98 and this paper from the Viennise groups demonstrate higher-dimensional entanglement using OAM https://www.nature.com/articles/nphoton.2016.12