  We propose defect extremal surface as the holographic counterpart of boundary
quantum extremal surface. The defect extremal surface is defined by minimizing
the Ryu-Takayanagi surface corrected by the defect theory. This is particularly
interesting when the RT surface crosses or terminates on the defect. In a
simple set up of AdS/BCFT, we find that the defect extremal surface formula
gives precisely the same results of the boundary quantum extremal surface. We
provide a decomposition procedure of an AdS bulk with a defect brane to see
clearly how Island formula emerges from a brane world system with gravity glued
to a flat space quantum field theory.
