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arXiv:1211.2580 · Physical Review A

Information sharing in quantum complex networks

Alessio Cardillo, Fernando Galve, David Zueco, Jesús Gómez‐Gardeñes

Published May 15, 2013• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.87.052312

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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 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 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 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 introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of random graphs known as complex networks. In the case of classical random graphs, we derive an analytic expression for the averaged entanglement entropy while for general complex networks we rely on numerics. For a large number of nodes n we find a scaling where both the prefactor c and the sub-leading O (1) term g e are characteristic of the different classes of complex networks. In particular, g e encodes topological features of the graphs and is named network topological entropy. Our results suggest that quantum entanglement may provide a powerful tool for the analysis of large complex networks with non-trivial topological properties.
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.
To utilize a scalable quantum network and perform a quantum state transfer within distant arbitrary nodes, coherence and control of the dynamics of couplings between the information units must be achieved as a prerequisite ingredient for quantum information processing within a hierarchical structure. Graph theoretic approach provides a powerful tool for the characterization of quantum networks with non-trivial clustering properties. By encoding the topological features of the underlying quantum graphs, relations between the quantum complexity measures are presented revealing the intricate links between a quantum and a classical networks dynamics.