  For a large class of density matrices in semiclassical gravity, it is shown
that the reduced density matrix which corresponds to tracing over the degrees
of freedom in a spatial subregion is dominated by states for which the area of
the boundary of the subregion is minimised. In the semiclassical limit, the
entropy of the reduced density matrix is found to have a leading order
contribution equal to one quarter of the minimal area in natural units. This is
consistent with the Ryu-Takayanagi conjecture. An extension to higher
derivative theories of gravity is established, for which the area is replaced
by a dynamical generalisation of the Wald entropy.
