Symmetric Laplacians, quantum density matrices and their Von-Neumann entropy

Abstract

Abstract. We show that the (normalized) symmetric Laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite Hilbert space (one part corresponding to the vertices, the other corresponding to the edges). This suggests an interpretation of the symmetric Laplacian's Von Neumann entropy as a measure of bipartite entanglement present between the two parts of the state. We then study extreme values for a connected graph's generalized Rényi-p entropy. Specifically, we show that(1) the complete graph achieves maximum entropy, (2) the 2-regular graph: (a) achieves minimum Rényi-2 entropy among all k-regular graphs, (b) is within log 4/3 of the minimum Rényi-2 entropy and log 4 √ 2/3 of the minimum Von Neumann entropy among all connected graphs, (c) achieves a Von Neumann entropy less than the star graph. Point (2) contrasts sharply with similar work applied to (normalized) combinatorial Laplacians, where it has been shown that the star graph almost always achieves minimum Von Neumann entropy. In this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound.

The von Neumann Theil index: characterizing graph centralization using the von Neumann index

“…Let G be a graph on n vertices. Let f be the functional (15), which assigns an arbitrary positive number to each vertex v i . Then the entropy of the graph with respect to f is defined to be…”

“…For example, in [1] an information function was defined on each of the vertices of a graph in order to associate an entropic quantity with the network; in [2], the entropic properties of graph ensembles have been studied; while in [3,4] Shannon entropy was used to study the topological uncertainty for networks embedded within a spatial domain. There have also been studies of the von Neumann entropy 1 [7] (from herein referred to as the von Neumann index) of graphs [2,8,9,10,11,6,12,13,14,15,16]. This work focuses on such a study.…”

On the Distribution of Random Geometric Graphs

“…This information would enable us to study not only connectivity, but also important features such as topological structure and complexity through the lens of graph entropy [8]. Applications of entropy-based methods to the study of networked systems are abundant and include problems related to molecular struc-ture classification [9], social networks [10], [11], data compression [12], and quantum entanglement [13], [14]. Graph entropy has also been invoked in the study of communication networks to quantify node and route stability [15] with the aim of improving link prediction [16] and routing protocols [17], [18].…”

On the conditional entropy of wireless networks

“…This formalism is deeply rooted in statistical physics and information theory, and it allows one to quantitatively characterize the uncertainty or inherent information content of systems that can be described by a graphical model [4]- [9]. Applications of entropy-based methods to the study of networked systems are abundant and include problems related to molecular structure classification [10], social networks [11], [12], data compression [13], and quantum entanglement [14], [15]. With regard to communication networks, graph entropy has been used to quantify node and route stability [16] with the aim of improving link prediction [17] and routing protocols [18], [19].…”