Quantum state engineering of maximally entangled photon pairs by path identity

Cutting-edge quantum technologies lean on sources of high-dimensional entangled states (HDES) that reliably prepare high-fidelity target states. The idea to overlap photon paths from distinct but indistinguishable sources was recently introduced for the creation of HDES, known as entanglement by path identity. In this regard, the use of orbital angular momentum (OAM) modes is promising, as they offer a high-dimensional and discrete Hilbert space to encode information. While entanglement by path identity with OAM has been verified experimentally, a detailed investigation of how the OAM distribution of photon pairs can be engineered to maximize the entanglement is lacking. We address this gap and identify an optimal dimensionality for maximally entangled states (MESs) when the spatial engineering of pump beam and the path identity approach are combined. Our theoretical study reveals notable limitations for the fidelity of high-dimensional target states and also establishes their equivalence to entangled biphoton states pumped by a spatially engineered beam. These findings constitute a valuable step toward the optimized preparation of MESs in high dimensions.