  We examine the entanglement properties of the spin-half Heisenberg model on
the two-dimensional square-lattice bilayer based on quantum Monte Carlo
calculations of the second R\'enyi entanglement entropy. In particular, we
extract the dominant area-law contribution to the bipartite entanglement
entropy that shows a non-monotonous behavior upon increasing the inter-layer
exchange interaction: a local maximum in the area-law coefficient is located at
the quantum critical point separating the antiferromagnetically ordered region
from the disordered dimer-singlet regime. Furthermore, we consider subleading
logarithmic corrections to the R\'enyi entanglement entropy scaling. Employing
different subregion shapes, we isolate the logarithmic corner term from the
logarithmic contribution due to Goldstone modes that is found to be enhanced in
the limit of decoupled layers. At the quantum critical point, we estimate a
contribution of $0.016(1)$ due to each $90^{\circ}$ corner. This corner term at
the SU(2) quantum critical point deviates from the Gaussian theory value, while
it compares well with recent numerical linked cluster calculations on the
bilayer model.
