Notes on the Graph-Encoded Manifolds for the (Genuine) Multi-Entropy

Abstract

Genuine multi-entropy is the part of the $\mathtt{q}$-partite multi-entropy that captures entanglement genuinely shared among all $\mathtt{q}$ parties, rather than entanglement already present among fewer subsystems. It was recently proposed that this genuine part -- so-called a multipartite entanglement signal -- can be studied via the graph-encoded manifold (GEM) built from the underlying $\mathtt{q}$-partite multi-entropy permutation family: reading its $\mathtt{q}$-colored contraction graph ${\Gamma }_{\mathtt{q},n}$ as a triangulation by $(\mathtt{q}-1)$-simplices. Focusing on $\mathtt{q}=4$, we classify the topology of every vertex link of ${\Gamma }_{4,n}$ and find ${\chi }_{\mathrm{link}}(n)=n(3-n)$: the graph satisfies the GEM condition, namely every vertex link is a two-sphere, if and only if $n=2$, while for every $n\ge 3$ the vertex links are closed surfaces of strictly positive genus $g_n=\frac{1}{2}(n-1)(n-2)$, so that the graph instead defines a simplicial complex with a conical singularity at every vertex.

34 pages, 5 figures