Engineering quantum correlations for m x n spatially encoded two-photons states

Discretizing transverse linear momentum of photon pairs, generated by spontaneous parametric down conversion (SPDC), is one of the simplest methods for producing bipartite entangled states in high dimensions. So far, it has been employed only to prepare states in dimensions m x m. In this work, we study the generalization for engineering entangled states in dimensions m x n. Our approach relies on the manipulation of the pump beam transverse profile and the phase-matching function of the SPDC process to prepare, behind an m- and n-slit aperture, different m x n spatial entangled states. We demonstrate the technique experimentally for some 2 x 3 states. Compared to previous approaches in producing 2 x n photonic entanglement, which require either more than two photons or hybrid entanglement, our scheme is less demanding and simpler since we employ only two photons and a single degree of freedom to encode the states.