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arXiv:1411.0310

Entanglement entropy in the Hypercube networks

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

Published Nov 2, 2014• 2 citations
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Abstract

We investigate the hypercube 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 each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hypercube network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis which is the matrix representation of the angular momentum. Then the entanglement entropy and Schmidt number for special partitions are calculated an- alytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary equal subsets (two equal subsets have the same number of vertices) in H(3, 2) and H(4, 2) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hypercube network. Then we can conjecture the minimum and maximum values of entanglement entropy for equal subsets in H(d, 2).

DOI
10.48550/arxiv.1411.0310

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We investigate the hypercube 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 each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hypercube network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis which is the matrix representation of the angular momentum. Then the entanglement entropy and Schmidt number for special partitions are calculated an- alytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary equal subsets (two equal subsets have the same number of vertices) in H(3, 2) and H(4, 2) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hypercube network. Then we can conjecture the minimum and maximum values of entanglement entropy for equal subsets in H(d, 2).
We investigate the Hamming 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 each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hamming network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis. Then the entanglement entropy and Schmidt number for special partitions are calculated analytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary subsets (two equal subsets have the same number of vertices) in H(2; 3) and H(2; 4) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hamming network.
We investigate the entanglement of the ground state in the quantum networks that their nodes are considered as quantum harmonic oscillators. To this aim, the Schmidt numbers and entanglement entropy between two arbitrary partitions of a network, are calculated. In partitioning an arbitrary graph into two parts there are some nodes in each parts which are not connected to the nodes of the other part. So these nodes of each part, can be in distinct subsets. Therefore the graph separates into four subsets. The nodes of the first and last subsets are those which are not connected to the nodes of other part. In theorem I, by using generalized Schur complement method in these four subsets, we prove that all graphs which their connections between all two alternative subsets are complete, have the same entropy. A large number of graphs satisfy this theorem. Then the entanglement entropy in the limit of large coupling and large size of system, is investigated in these graphs. One of important quantities about partitioning, is conductance of graph. The con- ductance of graph is considered in some various graphs. In these graphs we compare the conductance of graph and the entanglement entropy.
We investigate the entanglement of the ground state in the quantum networks that their nodes are considered as quantum harmonic oscillators. To this aim, the Schmidt numbers and entanglement entropy between two arbitrary partitions of a network are calculated. In partitioning an arbitrary graph into two parts there are some nodes in each part which are not connected to the nodes of the other part. So, these nodes of each part can be in distinct subsets. Therefore, the graph is separated into four subsets. The nodes of the first and last subsets are those which are not connected to the nodes of the other part. In theorem 1 , by using the generalized Schur complement method in these four subsets, we prove that all the graphs whose connections between the two alternative subsets are complete, have the same entropy. A large number of graphs satisfy this theorem. Then the entanglement entropy in the limit of the large coupling and large size of the system is investigated in these graphs. Also, the asymptotic behaviors of the Schmidt numbers and entanglement entropy in the limit of infinite coupling are shown. One important quantity about partitioning is the conductance of the graph. The conductance of the graph is considered in various graphs. In these graphs we compare the conductance of the graph and the 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.
We introduce the use of entanglement entropy as a tool for studying the amount of information shared between the nodes of quantum complex networks. By considering the ground state of a network of coupled quantum harmonic oscillators, we compute the information that each node has on the rest of the system. We show that the nodes storing the largest amount of information are not the ones with the highest connectivity, but those with intermediate connectivity, thus breaking down the usual hierarchical picture of classical networks. We show both numerically and analytically that the mutual information characterizes the network topology. As a by-product, our results point out that the amount of information available for an external node connecting to a quantum network allows one to determine the network topology.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H\mathcal{H}H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H\mathcal{H}H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.