An area law for the maximally-mixed ground state in arbitrarily degenerate systems with good AGSP

I will show how an area law in the mutual information for the maximally-mixed state in a highly-degenerate ground-space of a many-body Hamiltonian follows from the existence of a `good' approximation to the ground state projector (a good AGSP). Good AGSPs have been a pivotal ingredient in former area-law proofs, but so far, they been used only for proving area-law when the ground-space degeneracy is at most polynomial. Our proof, however, uses new tools from quantum information theory to show that the maximally-mixed state in the ground space satisfies an area-law, *regardless* of the underlying degeneracy.

As a corollary, we use existing constructions of good AGSPs to prove such area-law for the case of highly-degenerate gapped 1D systems and frustration-fee, locally-gapped, 2D systems.

Finally, I will also show how in 1D our results imply the existence of an efficient MPO approximation for the maximally-mixed state, with an inverse polynomially small error in the trace distance.

Joint work with Raz Firanko Rahul Jain, based on arXiv:2310.19028