Laplacian matrices of weighted digraphs represented as quantum states

TL;DR

This paper introduces a comprehensive framework using Laplacian matrices of weighted directed graphs to represent and analyze quantum states, including criteria for purity, mixedness, and entanglement.

Contribution

It generalizes graph-based density matrix constructions to complex weighted directed graphs and introduces a new signed Laplacian matrix for quantum state analysis.

Findings

Criteria for pure and mixed states derived from graph properties

Identification of graphs representing entangled pure states

Shared combinatorial structure among entangled states

Abstract

Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.