  We develop multipolar theory of nonlinear generation of entangled photons
from subwavelength dielectric particles due to the spontaneous parametric
downconversion. We demonstrate that optical excitation in resonance with the
high-quality supercavity mode of the aluminum gallium arsenide (AlGaAs)
nanodisk leads to a strong enhancement of generation of entangled photon pairs
associated with electric and magnetic dipole modes. Our rigorous numerical
results are corroborated by an analytical model, universally describing
formation of high-$Q$ resonant states and dark states due to the interference
and interplay of the parent multipoles, namely, magnetic dipoles and magnetic
octupole. Our findings and description can be instructive for quantum and
nonlinear nanophotonics applications.
