Interpreting the von Neumann entropy of graph Laplacians, and coentropic graphs

TL;DR

This paper links the von Neumann entropy of graph Laplacians to quantum entanglement, introduces coentropic graphs, and explores their properties and classifications in a quantum information framework.

Contribution

It provides a novel interpretation of graph Laplacian entropy through quantum entanglement and introduces the concept of coentropic graphs with new classification insights.

Findings

Partial traces of a rank-1 operator relate to Laplacian and edge-Laplacian.

Cospectral graphs correspond to locally unitarily equivalent quantum states.

Constructed smallest coentropic graphs with 8 vertices.

Abstract

For any graph, we define a rank-1 operator on a bipartite tensor product space, with components associated to the set of vertices and edges respectively. We show that the partial traces of the operator are the Laplacian and the edge-Laplacian. This provides an interpretation of the von Neumann entropy of the (normalized)\ Laplacian as the amount of quantum entanglement between two systems corresponding to vertices and edges. In this framework, cospectral graphs correspond exactly to local unitarily equivalent pure states. Finally, we introduce the notion of coentropic graphs, that is, graphs with equal von Neumann entropy. The smallest coentropic (but not cospectral) graphs that we are able to construct have 8 vertices. The number of equivalence classes of coentropic graphs with n vertices and m edges is a lower bound to the number of (pure) bipartite entanglement classes with subsystems of corresponding dimension.