Laplacian matrices of weighted digraphs represented as quantum states

Abstract

Graph representation of quantum states is becoming an increasingly important area of research to investigate combinatorial properties of quantum states which are nontrivial to comprehend in standard linear algebraic density matrix based approach of quantum states. 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 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 represent 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. It is important to observe that all these entangled pure states share a common combinatorial structure.

Quantum discord of states arising from graphs

“…In this section we provide a brief review on simple graphs and describe the quantum states arising from them [5,6]. A simple graph G = (V (G), E(G)) consists of a vertex set V (G) and an edge set…”

“…We denote ρ l (G) and ρ q (G) together by ρ(G) when no confusion arises. It is important to note that L(G) and Q(G) depend on the vertex labellings, and hence different labellings on the vertex set of a graph generate different quantum states [5,6]. Given a graph G on N = mn vertices the vertex set V (G) can be partitioned into m classes, say C 1 , C 2 , .…”

“…It has made significant contributions to quantum physics [2] and information theory [3,4,5]. Graphs provide a pictorial representation of quantum states [6]. They have been used to interpret separability property of quantum states [7], and to model useful unitary operations [8].…”

Bipartite separability and nonlocal quantum operations on graphs

“…Representing a quantum state by a graph is beneficial for research in both quantum information theory as well as complex networks. Graphs provide a platform to visualize quantum states pictorially [18], such that different states have different pictographic representations [17] and some important unitary evolutions can also be represented by changes in their representations [19]. In this way, graphs form an intuitively appealing framework for quantum information and communication.…”

“…In this work, we are concerned with the density matrices corresponding to L(G) and Q(G) only. They are defined as [17] …”