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Journal of Physics A Mathematical and Theoretical

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

M. A. Jafarizadeh, F. Eghbalifam, S. Nami

Published Jan 19, 2018• 5 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/aaa1da

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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.
We investigate the quantum networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information one part of a network shares with the other part of the system. The networks which we studied in this paper, are called strongly regular graphs (SRG). These kinds of graphs have some special properties like they have three strata in the stratification basis. The Schur complement method is used to calculate the Schmidt numbers and entanglement entropy between two parts of graph. We could obtain analytically, all blocks of adjacency matrix in several scalable sets of strongly regular graphs. Also the entanglement entropy in the large coupling limit is considered in these graphs and the relationship between entanglement entropy and the ratio of size of boundary to size of the system is found. After that, area-law is studied to show that there are no entanglement entropy for the highest size of system. Then, the graph isomorphism problem is considered in SRGs by using the entanglement entropies in some partitioning of graph. Two SRGs with the same parameters: are isomorphic if they can be made identical by relabeling their vertices. So the adjacency matrices of two isomorphic SRGs become identical by replacing of rows and columns. The non-isomorph SRGs could be distinguished by using the elements of blocks of adjacency matrices in the stratification basis, numerically.
Abstract We investigate the entanglement entropy in the ground state of the supersymmetric fermion lattice model. The Hamiltonian of this model is block diagonal in the basis of a fixed number of particles, so we compute the entanglement entropy for the blocks of the Hamiltonian separately. We explain the way of calculating the correlation matrix and the entanglement entropy for the ground state of supersymmetric Hamiltonian on diverse graphs. Then we show that the entanglement entropy exhibits the volume law scaling in various choices of the number of fermions in regular graphs. Also, we give the analytical formula for the entanglement entropy in one scalable set of strongly regular graphs in terms of their parameters, and we prove that the entanglement entropy satisfies a volume law.
We investigate the graph isomorphism (GI) in some cospectral networks. Two graph are isomorphic when they are related to each other by a relabeling of the graph vertices. We want to investigate the GI in two scalable (n + 2)-regular graphs G4(n; n + 2) and G5(n; n + 2), analytically by using the multiparticle quantum walk. These two graphs are a pair of non-isomorphic connected cospectral regular graphs for any positive integer n. In order to investigation GI in these two graphs, we rewrite the adjacency matrices of graphs in the antisymmetric fermionic basis and show that they are different for thesepairs of graphs. So the multiparticle quantum walk is able to distinguish pairs of non- isomorph graphs. Also we construct two new graphs T4(n; n + 2) and T5(n; n + 2) and repeat the same process of G4 and G5 to study the GI problem by using multiparticle quantum walk. Then we study GI by using the entanglement entropy. To this aim, we calculate entanglement entropy between two parts of network. In our model the nodes are considered as identical quantum harmonic oscillators. The entanglement entropy between two special parts of G4(n; n+2) and G5(n; n+2) are calculated analytically. It is shown that the entanglement entropy can distinguish pairs of non-isomorphic cospectral graphs too.
We study the ground-state entanglement entropy of free fermions on random graphs. We first establish a general criterion for the volume law on random graphs, based on the spectral characteristics of the Hamiltonian. We then apply this criterion to the Erdős--Rényi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we show that the ground-state entanglement entropy obeys an exact volume law in Erdős--Rényi graphs in the thermodynamic limit, with a universal coefficient that is independent of the edge probability of the graph. This coefficient is confirmed numerically to take the value approximately 0.38630.38630.3863 nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.
We investigate multipartite information and entanglement measures in the ground state of a free-fermion model defined on a Hamming graph. Using the known diagonalization of the adjacency matrix, we solve the model and construct the ground-state correlation matrix. Moreover, we find all the eigenvalues of the chopped correlation matrix when the subsystem consists of n disjoint Hamming subgraphs embedded in a larger one. These results allow us to find an exact formula for the entanglement entropy of disjoint graphs, as well as for the mutual and tripartite information. We use the exact formulas for these measures to extract their asymptotic behavior in two distinct thermodynamic limits, and find excellent match with the numerical calculations. In particular, we find that the entanglement entropy admits a logarithmic violation of the area law which decreases the amount of entanglement compared to the area law scaling.
Free Fermions on vertices of distance-regular graphs are considered. Bipartitions are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in detail within that framework and the existence of the commuting matrix is shown to allow for an analytic diagonalization of the restricted two-point correlation matrix and hence for an explicit determination of the entanglement entropy.