We use a min-max procedure on the Allen–Cahn energy functional to construct geodesics closed, $2$‑dimensional Riemannian manifolds, as motivated by work of Guaraco [Gua18]. Borrowing classical blowup and curvature estimates from geometric analysis, well novel due Wang–Wei [WW19], we manage study fine structure potential singular points at diffuse level, show that problem reduces understanding “...

The von Neumann entropy of pure quantum states and the min-cut function weighted hypergraphs are both symmetric submodular functions. In this article, we explain coincidence by proving that any hypergraph can be approximated (up to an overall rescaling) entropies known as stabilizer states. We do so constructing a novel ensemble random states, built from tensor networks, whose entanglement stru...