Entanglement entropy in the spinless free fermion model and its application to the graph isomorphism problem

Abstract

In this paper, the entanglement entropy is investigated in the ground state of a spinless free fermion Hamiltonian where its hopping matrix is given by the adjacency matrix of the graph. The bipartite entanglement entropy is calculated from the eigenvalues of the correlation matrix. The entanglement entropy and mutual information are calculated for some scalable graphs including a complete graph and three sets of strongly regular graphs (SRGs). In addition, the volume law scaling has been shown for these graphs. Then, we use the entanglement entropy as a tool for studying the graph isomorphism problem in some non-isomorphic pairs of graphs in order to distinguish them from each other. Two graphs are isomorphic when they are related to each other by relabeling the graph vertices. We study the graph isomorphism in non-isomorphic SRGs, distance-regular graphs and pseudo distance-regular graphs. Most pairs of non-isomorphic SRGs can be distinguished from each other by using fermionic entanglement entropy.

Entanglement in Fermionic Chains and Bispectrality

“…It would obviously be of relevance to extend the approach to other space limiting. Studies of entanglement of Fermions (and Bosons) on different graphs have been undertaken [26,27]. We plan on examining how the considerations developed in this paper could extend in that context.…”

Entanglement of Free Fermions on Hadamard Graphs

“…We here extend this study to systems of free Fermions on graphs and in particular on those of Hadamard. While the entanglement of such Fermionic configurations has been looked at in [19,20], our main interest will be to identify the commuting matrices in the case of distance-regular graphs. This will prove possible when the limitings will respect the distance structure of the graph.…”