Simple rules for two-photon state preparation with linear optics

Entangling photons is a critical challenge for photonic quantum information processing: entanglement is a crucial resource for quantum communication and computation but can only be performed in a probabilistic manner when using linear optics. In this work, we leverage a two-photon state matrix representation to derive necessary and sufficient conditions on two-photon entangling operations with linear optics. We give a characterization of the input photonic states that can be used to prepare arbitrary two-qudit states in d-rail encoding with post-selection. We determine how many auxiliary photons are required to prepare any two-photon state with heralding. In addition, we present a construction for generalized post-selected n-qubit control-rotation gates.