Symmetric Laplacians, Quantum Density Matrices and their Von-Neumann Entropy

TL;DR

This paper links symmetric graph Laplacians to quantum states, interpreting their Von Neumann entropy as bipartite entanglement, and analyzes entropy extremities across different graph classes.

Contribution

It introduces a quantum interpretation of symmetric Laplacians and studies entropy bounds, revealing new extremal properties of specific graph types.

Findings

Complete graph has maximum entropy.

2-regular graph has minimum Rfeyn-2 entropy among k-regular graphs.

Star graph's entropy behavior contrasts with combinatorial Laplacian results.

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 R\'enyi-$p$ entropy. Specifically, we show that (1) the complete graph achieves maximum entropy, (2) the $2$-regular graph: a) achieves minimum R\'enyi-$2$ entropy among all $k$-regular graphs, b) is within $\log 4/3$ of the minimum R\'enyi-$2$ entropy and $\log4\sqrt{2}/3$ 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. Keywords: Symmetric; Laplacian; Quantum; Entropy; Bounds; R\'enyi.