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

Stability of the Area Law for the Entropy of Entanglement