We show an area law in the mutual information for the maximally-mixed state \ (\) 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 previous area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any \ (>0\) and any bipartition \ (L Lᶜ\) of the system, where | L | represents the number of sites in L, d is the dimension of a site and \ (I^ \, \! \! (L: Lᶜ) \) represents the \ (\) - smoothed maximum mutual information with respect to the \ (L: Lᶜ\) partition in \ (\). From this bound we then conclude \ (I \, \! \! (L: Lᶜ) _ O ( (|L| (d) ) ) \) – an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that \ (\) can be approximated in trace norm up to \ (\) with a state of Schmidt rank of at most \ (poly (|L| (d) /) \), leading to a good MPO approximation for \ (\) with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.
Arad et al. (Thu,) studied this question.