Raz Firanko

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 and locally-gapped local Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any eps>0 and any bi-partition Lcup L^c of the system, beginalign* I^eps_max(L:L^c)_Ømega łe bigO łog (|L|) + łog(1/eps), endalign* where |L| represents the number of sites in L and I^eps_max(L:L^c)_Ømega represents the eps-emphsmoothed maximum mutual information with respect to the L:L^c partition in Ω. From this bound we then conclude I(L:L^c)_Ømega łe bigOłog(|L|) – an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that Ω can be approximated up to an eps in trace norm with a state of Schmidt rank of at most poly(|L|/eps). Similar corollaries are derived for the mutual information of 2D frustration-free locally-gapped local Hamiltonians.