Quantifying Complexity in Networks: The von Neumann Entropy

The Von Neumann Entropy

The state of a quantum mechanical system with a Hilbert space of finite dimension n is described by a density matrix. Each density matrix ρ is a positive semidefinite matrix with Tr(ρ) = 1. Here we consider a matrix representation based on the combinatorial Laplacian to associate graphs to specific density matrices.

Let G = (V, E) be a simple undirected graph with set of vertices V(G) = {1, 2, …,n} and set of edges . The adjacency matrix of G is denoted by A(G) and defined by [A(G)]_u,v = 1 if {u, v} ∈ E(G) and [A(G)]_u,v = 0, otherwise.