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

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.