Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 6 / a / Solution

One implementation uses type-II phase matching for spontaneous parametric down-conversion in a nonlinear birefringent crystal pumped by a coherent laser. A pump photon produces two lower-frequency photons of orthogonal polarizations, subject to energy and momentum phase matching. Collect two spatial modes in which the alternatives “horizontal photon in Alice's mode and vertical photon in Bob's mode” and the reversed assignment are indistinguishable. Compensation crystals, filters and matched optical paths erase polarization-dependent timing and spectral information. The amplitudes then add coherently to give a state proportional to $|HV⟩+e^{i\phi }|VH⟩$; a birefringent phase element tunes $\phi$.

The essential requirement is coherent indistinguishability of the two assignments. Producing two orthogonally polarized photons with information revealing which assignment occurred would instead give a mixture without the required polarization entanglement.