  Photon pairs generated from spontaneous parametric down-conversion are a
well-established method to realize entangled bipartite photonic systems.
Laguerre-Gaussian modes, which carry orbital angular momentum (OAM), are
commonly exploited to engineer high-dimensional entangled quantum states
experimentally. For Hilbert spaces with dimension d>2, maximally entangled
states (MES) help to improve the capacity and security of quantum communication
protocols, among several other promising features. However, the direct
generation of MES in well-defined high-dimensional subspaces of the infinite
OAM basis has remained a challenge. Here, we formalize how the spatial
distribution of the pump beam and the nonlinear profile of the crystal can be
utilized to generate MES without additional spatial filtering of OAM modes
within a subspace. We illustrate our approach with maximally entangled qutrits
(d=3) and ququints (d=5).
