Area law for the maximally mixed ground state in degenerate 1D gapped systems

We show an area law with logarithmic correction for the maximally mixed state $\Omega$ in the (degenerate) ground space of a 1D gapped local Hamiltonian $H$, which is independent of the underlying ground space degeneracy. Formally, for $\varepsilon>0$ and a bi-partition $L\cup L^c$ of the 1D lattice, we show that $$\mathrm{I}^{\varepsilon}_{\max}(L:L^c)_{\Omega} \leq O(\log(|L|)+\log(1/\varepsilon)),$$ where $|L|$ represents the number of qudits in $L$ and $\mathrm{I}^{\epsilon}_{\max}(L:L^c)_{\Omega}$ represents the $\varepsilon$- 'smoothed maximum mutual information' with respect to the $L:L^c$ partition in $\Omega$. As a corollary, we get an area law for the mutual information of the form $\mathrm{I}(L:R)_\Omega \leq O(\log |L|)$. In addition, we show that $\Omega$ can be approximated up to an $\varepsilon$ in trace norm with a state of Schmidt rank of at most $\mathrm{poly}(|L|/\varepsilon)$.