An Area Law for the Maximally-Mixed Ground State in Arbitrarily Degenerate Systems with Good AGSP

We show an area law in the mutual information for the maximally-mixed state Ω in the ground space of general Hamiltonians, which is independent of the underlying ground space degeneracy. Our result assumes the existence of a ‘good’ approximation to the ground state projector (a good AGSP), a crucial ingredient in former area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped local Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any є>0 and any bi-partition L∪ Lc of the system,