Entanglement of multi-qubit states representing directed networks and its detection with quantum computing

TL;DR

This paper explores the entanglement properties of multi-qubit states derived from directed weighted graphs, establishing relationships with graph properties and proposing a quantum protocol for entanglement quantification.

Contribution

It introduces a method to relate the geometric measure of entanglement to graph properties and develops a quantum protocol for entanglement detection in quantum graph states.

Findings

Entanglement relates to in-degree and out-degree of graph vertices.

The geometric measure of entanglement can be calculated for arbitrary graphs.

A quantum protocol for entanglement quantification is proposed.

Abstract

We consider quantum graph states that can be mapped to directed weighted graphs, also known as directed networks. The geometric measure of entanglement of the states is calculated for the quantum graph states corresponding to arbitrary graphs. We find relationships between the entanglement and the properties of the corresponding graphs. Namely, we obtain that the geometric measure of entanglement of a qubit with other qubits in the graph state is related to the weights of ingoing and outgoing arcs with respect to the vertex representing the qubit, outdegree and indegree of the corresponding vertex in the graph. For unweighted and undirected graphs, the entanglement depends on the degree of the corresponding vertex. Quantum protocol for quantifying of the entanglement of the quantum graph states is constructed. As an example, a quantum graph state corresponding to a chain is examined, and the entanglement of the state is calculated on AerSimulator.