Entangled graphs: A classification of four-qubit entanglement

TL;DR

This paper introduces a novel classification of four-qubit entanglement using entangled graphs with weighted edges, based on generalized Schmidt decomposition and bipartite concurrences, providing a visual and structural understanding of entanglement.

Contribution

It presents a new classification scheme for four-qubit entanglement that does not rely on LOCC or SLOCC, utilizing entangled graphs and generalized Schmidt decomposition.

Findings

Every entangled graph corresponds to a pure state with matching bipartite concurrences.

Global entanglement is indicated by a circle around the graph.

The classification captures both bipartite and multipartite entanglement structures.

Abstract

We use the concept of \textit{entangled graphs} with weighted edges to present a classification for four-qubit entanglement which is based neither on the LOCC nor the SLOCC. Entangled graphs, first introduced by Plesch et al. [Phys. Rev. A 67, (2003) 012322], are structures such that each qubit of a multi-qubit system is represented as a vertex and an edge between two vertices denotes bipartite entanglement between the corresponding qubits. Our classification is based on the use of generalized Schmidt decomposition of pure states of multi-qubit systems. We show that for every possible entangled graph one can find a pure state such that the reduced entanglement of each pair, measured by concurrence, represents the weight of the corresponding edge in the graph. We also use the concept of tripartite and quadripartite concurrences as a proper measure of global entanglement of the states. In this case a circle including the graph indicates the presence of global entanglement.