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 $\mathrm{poly}(|L|/\varepsilon)$.

翻译：我们证明了一维有隙局域哈密顿量 $H$ 的（简并）基空间中最大混合态 $\Omega$ 服从具有对数修正的面积律，该结果与底层基空间简并度无关。形式化地，对于 $\varepsilon>0$ 和一维格点上的二分划 $L\cup L^c$，我们证明： $$\mathrm{I}^{\varepsilon}_{\max}(L:L^c)_{\Omega} \leq O(\log(|L|)+\log(1/\varepsilon)),$$ 其中 $|L|$ 表示 $L$ 中的量子比特数，$\mathrm{I}^{\epsilon}_{\max}(L:L^c)_{\Omega}$ 表示 $\Omega$ 中关于 $L:L^c$ 分划的 $\varepsilon$-“平滑最大互信息”。作为推论，我们得到形如 $\mathrm{I}(L:R)_\Omega \leq O(\log |L|)$ 的互信息面积律。此外，我们证明 $\Omega$ 可以在迹范数下被一个 Schmidt 秩至多为 $\mathrm{poly}(|L|/\varepsilon)$ 的态近似到 $\varepsilon$ 精度内。