Quantum random networks

Abstract

In recent years, new algorithms and cryptographic protocols based on the laws of quantum physics have been designed to outperform classical communication and computation. We show that the quantum world also opens up new perspectives in the field of complex networks. Already the simplest model of a classical random network changes dramatically when extended to the quantum case, as we obtain a completely distinct behavior of the critical probabilities at which different subgraphs appear. In particular, in a network of N nodes, any quantum subgraph can be generated by local operations and classical communication if the entanglement between pairs of nodes scales as N −2 .On the one hand, complex networks describe a wide variety of systems in nature and society, as chemical reactions in a cell, the spreading of diseases in populations or communications using the Internet [1]. Their study has traditionally been the territory of graph theory, which initially focused on regular graphs, and was extended to random graphs by the mathematicians Paul Erdős and Alfréd Rényi in a series of seminal papers [2,3,4] in the 1950s and 1960s. With the improvement of computing power and the emergence of large databases, these theoretical models have become increasingly important, and in the past few years new properties which seem universal in real networks have been described, as a small-world [5] or a scale-free [6] behavior.On the other hand, quantum networks are expected to be developed in a near future in order to achieve, for instance, perfectly secure communications [7,8]. These networks are based on the laws of quantum physics and will offer us new opportunities and phenomena as compared to their classical counterpart. Recently it has been shown that quantum phase transitions may occur in the entanglement properties of quantum networks defined on regular lattices, and that the use of joint strategies may be beneficial, for example, for quantum teleportation between nodes [9,10]. In this work we introduce a simple model of complex quantum networks, a new class of systems that exhibit some totally unexpected properties. In fact we obtain a completely different classification of their behavior as compared to what one would expect from their classical counterpart.A classical network is mathematically represented by a graph, which is a pair of sets G = (V, E) where V is a set of N nodes (or vertices) and E is a set of L edges (or links) connecting two nodes. The theory of random graphs, aiming to tackle networks with a complex topology, considers graphs in which each pair of nodes i and j are joined by a link with probability p i,j . The simplest and most studied model is the one where this probability is independent of the nodes, with p i,j = p, and the resulting graph is denoted G N,p . The construction of these graphs can be considered as an evolution process:Evolution process of a classical random graph with N = 10 nodes: starting from isolated nodes, we randomly add edges with increasing probability p, to eventually get the co...

Control principles of complex systems

“…The notion of quantum networks has been recently proposed by the quantum information community (Acín et al, 2007;Calsamiglia, 2009, 2012;Czekaj et al, 2012;Hahn et al, 2012;Lapeyre et al, 2009;Perseguers, 2010;Perseguers et al, 2008Perseguers et al, , 2013Perseguers et al, , 2010, offering fresh perspectives in the field of complex networks. In a quantum network, each node possesses exactly one qubit for each of its neighbors.…”

“…As z approaches −1, trees and cycles of all orders appear (Albert and . Surprisingly, in quantum networks any subgraph can be generated by local operations and classical communication, provided that the entangle-ment between pairs of nodes scales with the graph size as p ∼ N −2 (Perseguers et al, 2010). In other words, thanks to the superposition principle and the ability to coherently manipulate the qubits at the stations, even for the lowest non-trivial connection probability that is just sufficient to get simple connections in a classical graph, we obtain quantum subgraphs of any complexity.…”

Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

“…Complex networks model systems as diverse as the brain and the internet; however, up till now they have been obtained in quantum systems by explicitly enforcing complex network structure in their quantum connections [2][3][4][5][6][7], e.g. entanglement percolation on a complex network [4].…”

“…However, at the most basic level we can first ask, are quantum systems inherently complex? Must we impose complexity on quantum systems to obtain it [2][3][4][5][6][7], or is there a regime in which complexity naturally emerges, even in ground states of regular lattice models? In this Letter we show that emergent complexity can be well quantified in the simplest of 1D lattice quantum simulator models in terms of complexity measures around QCPs in direct analogy to similar measurements on the brain; moreover we establish a much-needed new set of tools for quantifying the complexity of far-from-equilibrium quantum dynamics.…”

A Single-Photon Router in Quantum Fluctuation of Field

“…In recent years, much research has been conducted in the field of quantum information science [1,2]. Photons can carry quantum information like electrons [3][4][5].…”