  The antiferromagnetic coupling and entanglement between skyrmion lattices are
treated in magnetic bilayer systems. We first formulate the problem of large
bilayer skyrmions using $\mathbb{CP}^{1}\otimes \mathbb{CP}^{1}$ theory. We
have considered bilayer skyrmions under the presence of Dzyaloshinskii-Moriya
(DMI) and Zeeman interactions confined in a two-dimensional chiral magnet such
as Fe$_{0.5}$Co$_{0.5}$Si. We parametrize bilayer skyrmions using $SU(4)$
representation, and represent each skyrmion and antiskyrmion using Schmidt
decomposition. The reduced density matrices for skyrmion and antiskyrmion has
been calculated. The conditions for maximal, partial entanglement and separable
bilayer skyrmions are presented. Our results can be used for generating
entanglement in systems with large number of spins.
