  Two-photon state with spatial entanglement is an essential resource for
testing fundamental laws of quantum mechanics and various quantum applications.
Its creation typically relies on spontaneous parametric down-conversion in
bulky nonlinear crystals where the tunability of spatial entanglement is
limited. Here, we predict that ultrathin nonlinear lithium niobate metasurfaces
can generate and diversely tune spatially entangled photon pairs. The spatial
properties of photons including the emission pattern, rate, and degree of
spatial entanglement are analysed theoretically with the coupled mode theory
and Schmidt decomposition method. We show that by leveraging the strong angular
dispersion of the metasurface, the degree of spatial entanglement quantified by
the Schmidt number can be decreased or increased by changing the pump laser
wavelength and a Gaussian beam size. This flexibility can facilitate diverse
quantum applications of entangled photon states generated from nonlinear
metasurfaces.
