Multipartite separability of Laplacian matrices of graphs

TL;DR

This paper investigates the conditions under which Laplacian matrices of graphs are multipartite entangled or separable, extending previous work to various graph products and addressing open questions in quantum graph theory.

Contribution

It identifies specific bipartite graphs with entangled Laplacian matrices and provides degree-based conditions for entanglement, extending prior results to multipartite cases and graph products.

Findings

Complete bipartite graphs can have Laplacian matrices that are always entangled.

Vertex degree conditions influence the entanglement of Laplacian matrices.

Laplacian matrices of graph products are shown to be multipartite separable.

Abstract

Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in [1]. Finally, we extend some of the results in [1,2] to the multipartite case and show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, categorical, etc.) is multipartite separable.