  Recent results on the stability of the spectral gap under general
perturbations for frustration-free Hamiltonians, have motivated the following
question: Does the entanglement entropy of quantum states that are connected to
states satisfying an area law along gapped Hamiltonian paths, also satisfy an
area law? We answer this question in the affirmative, combining recent advances
in quasi-adiabatic evolution and Lieb-Robinson bounds with ideas from the proof
of the 1D area law.
