Sharp implications of AGSPs for degenerate ground spaces

We generalize the `off-the-rack' AGSP$\Rightarrow$entanglement bound implication of [Arad, Landau, and Vazirani '12] from unique ground states to degenerate ground spaces. Our condition $R\Delta\le1/2$ on a $(\Delta,R)$-AGSP matches the non-degenerate case, whereas existing tools in the literature of spin chains would only be adequate to prove a less natural implication which assumes $R^{\text{Const}}\Delta\le c$. To show that $R\Delta\le1/2$ still suffices in the degenerate case we prove an optimal error reduction bound which improves on the literature by a factor $\delta\mu$ where $\delta=1-\mu$ is the viability. The generalized off-the-rack bound implies the generalization of a recent 2D subvolume law of [Anshu, Arad, and Gosset '19] from the non-degenerate case to the sub-exponentially degenerate case.