Proposed optical realisation of a two photon, four-qubit entangled $\chi$ state

The four-qubit states $\lvert\chi^{ij}\rangle$, exhibiting genuinely multi-partite entanglement have been shown to have many interesting properties and have been suggested for novel applications in quantum information processing. In this work we propose a simple quantum circuit and its corresponding optical embodiment with which to prepare photon pairs in the $\lvert\chi^{ij}\rangle$ states. Our approach uses hyper-entangled photon pairs, produced by the type-I spontaneous parametric down-conversion (SPDC) process in two contiguous nonlinear crystals, together with a set of simple linear-optical transformations. Our photon pairs are maximally hyper-entangled in both their polarisation and orbital angular momentum (OAM). After one of these daughter photons passes through our optical setup, we obtain photon pairs in the hyper-entangled state $\lvert\chi^{00}\rangle$, and the $\lvert\chi^{ij}\rangle$ states can be achieved by further simple transformations.