Area laws and tensor networks for maximally mixed ground states

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 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 $\varepsilon>0$ and any bi-partition $L\cup L^c$ of the system, \begin{align*} \mathrm I_{\max}^\varepsilon (L:L^c)_Ω \le \mathrm O \big( \log (|L|\log(d))+\log(1/\varepsilon)\big), \end{align*} where $|L|$ represents the number of sites in $L$, $d$ is the dimension of a site and $ \mathrm I_{\max}^\varepsilon (L:L^c)_Ω $ represents the $\varepsilon$-\emph{smoothed maximum mutual information} with respect to the $L:L^c$ partition in $Ω$. From this bound we then conclude $\mathrm I (L:L^c)_Ω \le \mathrm O\big(\log(|L|\log(d))\big)$ -- an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that $Ω$ can be approximated in trace norm up to $\varepsilon$ with a state of Schmidt rank of at most $\mathrm{poly}(|L|/\varepsilon)$, leading to a good MPO approximation for $Ω$ with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.