They have built a QKD network prototype, a test bed for QKD-based technologies where research on quantum protocols and its integration within standard optical communication networks is being carried out. The first QKD network was unique in that it departed with previous thinking about QKD networks in not relying in intermediate trusted nodes connected through non-shared optical fibres. This was tantamount to have a separate, ad hoc network for the quantum signals alone.

The network sought a full classical and quantum network integration by using the usual core and access architecture of a metropolitan area network. Standard, off-the-shelf, components were used to research the limits of quantum-classical coexistence in a CWDM metro area core network and using a GPON access network. The objective was to study the limits in stablishing a quantum link crossing a full metropolitan area network using standard components and sharing the already deployed optical infrastructure [ Lancho et al. The second QKD network prototype was less concerned with the coexistence of classical and quantum signals and sought to maximize the number of users in a quantum network.

In this network, the classical signals were, at first, limited to those needed to keep the quantum channel stable and to distill the keys in the QKD link. Finally, this limitation was also raised. The result was a demonstrator were up to 32 QKD users could use simultaneously a fully addressable metropolitan area network i.

The network would, in an almost fully passive way, automatically direct the quantum and corresponding classical channels from the emitter to the receiver by using a mixture of CWDM and DWDM channels using standard components [ Ciurana et al. This network was also capable to distribute entanglement [ Ciurana et al. The third network prototype present , now in development, is researching the integration of QKD in novel network architectures, in particular using the Software Defined Networking and associated Network Function Virtualization paradigms.

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These new paradigms are designed for maximum flexibility by stepping away from proprietary appliances that work mainly as separated entities. Relying on as much common hardware as possible that is accessed through common interfaces, a logically centralized SW entity —the SDN controller— can manage the SDN network taking into account the peculiarities of all new hardware installed, as long as it exports the needed functionalities and the corresponding interfaces. The number of incoming, outgoing, and total edges is known as incoming, outgoing, and total degree of a node, respectively.

The sum of the corresponding weights defines the incoming, outgoing, and total strength of that node, respectively. Networks are often characterized by how node degree and strength are distributed and correlated. Systems modeled by uncorrelated networks with homogeneous degree distribution are known as Erdos—Renyi networks, whereas systems with power-law degree distribution are known as scale-free networks.

We refer to refs. The use of various aspects of graph and network theory can be found in all aspects of quantum theory, yet not all networks are complex. Although the idea of a complex network is not defined in a strict sense, the definition is typically that of a network that exhibits an emergent property, such as a non-trivial distribution in node degree. This is in contrast to graph theory, which applies graph theory or tensor network reasoning to deduce and determine properties of quantum systems.

Here, we will focus on topics in quantum systems that are known to be connected with the same sort of complexity considered in complex networks. To define quantum networks based on entangled states, let us start from the state of each i th qubit, written without loss of generality, as. Let us consider a quantum system with two qubits, i. If the two qubits are not entangled, i. Instead of distributing entanglement on regular graphs, such as uniform lattices typically studied in condensed matter physics, it has been shown that it is possible to tune the amount of entanglement between two nodes in such a way that it equals the probability to have a link in classical Erdos—Renyi graphs 6.

Such random graphs can be defined by the family of networks G N , p , where N is the number of nodes and p the probability to find a link between any two nodes. However, in this case, although the connections are identical and pure they encode non-maximally entangled pairs. Entanglement connecting distant network nodes.

A pair of distant cavities in a quantum communication network are driven by a shared squeezed light source S. In the systems steady state the two atoms A and B entangle, forming a network edge If each link, i. It follows that the fraction of existing entangled states converted to maximally entangled ones by LOCC corresponds to the probability of having a link between nodes in the corresponding classical random network 6.

## Physics Course Information

By varying the value of the parameter z , i. This is useful to create special multipartite states, such as the Greenberger—Horne—Zeilinger state, which exhibits non-classical correlations This bridge between complex network theory and quantum theory provides a powerful tool to investigate the critical properties of a quantum system.

For instance, in the case of regular lattices, it has been shown that the probability p opt to establish a perfect quantum channel between the nodes can be mapped to the probability of distributing links among each pair of nodes in the lattice 6 , a scenario that can be studied using the well-established bond-percolation theory from statistical physics. This result allows one to calculate the critical probability above which the system will exhibit an infinite connected cluster and, in the case of qubits, it has been shown that the probability of having an entangled path with infinite length—i.

However, for product states this probability is zero, denoting the existence of a sharp transition between these two scenarios.

### Description

However, local measurements based on this approach, called classical entanglement percolation CEP , are not optimal, in general, to generate maximally entangled states: CEP is not even asymptotically optimal for two-dimensional lattices and new quantum protocols based on quantum entanglement percolation have to be used instead 6. A novel critical phenomenon, defining an entanglement phase transition, emerges from this new strategy, where the critical parameter is the degree of entanglement required to be distributed in order to establish a quantum channel with probability that does not decay exponentially with the size of the system, at variance with CEP.

This type of enhancement with respect to the classical case has been reported for different network topologies, such as Erdos—Renyi, scale-free, and small-world networks Unexpected quantum effects emerging from network effects have been reported.

Cardillo et al. More recently, Carvacho et al.

### Introduction

The static entangled states providing the network connectivity described here, will be replaced in the next section by dynamical processes on networked quantum systems. Another wide area where network concepts have found applicability consists of quantum systems physically interconnected, such as atoms or superconducting quantum electronics 35 , 36 , 37 , 38 , 39 , These types of systems provide fertile ground where quantum algorithms are tested 58 , 59 , 60 , 61 , 62 and quantum information transport systems are studied 63 , 64 , Stochastic non-quantum walks are also a central model in complex network theory—see the review Any quantum process can be viewed as a single-particle walk on a graph.

Single-particle quantum walks represent a universal model of quantum computation—meaning that any algorithm for a quantum computer can be translated into a quantum walk on a graph—and, in addition, quantum walks have been widely studied in the realm of quantum search on graphs, in both continuous and discrete time via coined walks see e.

The computational advantages of quantum versus stochastic random walk based algorithms has attracted wide interest with typical focus being on general graphs, which consequently do not exhibit complex features. However, many works have compared properties of stochastic 68 , 69 and quantum random walks 63 , 64 , 65 on complex networks Recently, Faccin et al.

These differences push forward a general understanding which can lead to a theory explaining novel complex features in quantum systems. Quantum walks on complex networks represent both a practical model of transport 70 as well as an interesting stage of comparison between the quantum and stochastic cases. As a closed quantum system exhibits fluctuations in the probabilities in time, typically a long time average is considered. Physically, this is the best approximation one can hope for, provided that there is no knowledge of when the walk started. In this case, the probability to find a quantum walker in the i th node is given by.

The non-symmetric adjacency matrix representing the directed connectivity of the World Wide Web, a. The eigenvector p corresponds to the limiting distribution of occupation probabilities of a random web surfer—it represents a unique attractor for the dynamics independently of the initial state. The vector p is known as the Page-Rank.

A dumping or teleportation factor is often included in the computation in order to assure the Perron—Frobenius theorem satisfactibility. Garnerone et al. This fact leads to a definition of a Hermitian operator which can play the role of a Hamiltonian, defined as:. Using a quantum computer to accelerate the calculation of various network properties has been considered widely 58 , 59 , 60 , As Page-Rank relies on finding the vector corresponding to the lowest eigenvalue of the Google matrix, the adiabatic algorithm opens the door up to accelerate network calculations using quantum computers.

These complex phases in many cases do not affect transfer probabilities: the theory explaining this finding was developed in refs. The case of open systems has been investigated as well in ref. In the scenarios where the addition of complex phases affects transfer probabilities, the underlying system breaks time-reversal symmetry and, consequently, the probability flow into the quantum system is biased. This fact enables directed state transfer without requiring a biased or non-local distribution in the initial states, or coupling to an environment.

When the underlying graph is bipartite e. Transport suppression is indeed possible however Bipartite graphs include trees, linear chains and generally, graphs with only even cycles. These results point to a subtle interplay between the topology of the underlying graph, giving rise to a new challenge for dynamical control of probability transfer when considering walks on complex networks 12 , 65 , 71 , The area of open quantum systems 75 studies noise and its effects in quantum systems.

The adiabatic version of Page-Rank 9 uses a quantum stochastic quantum walk as proposed by 64 see also refs. Quantum stochastic walks are defined by a quantum walk undergoing dissipative dynamics. The latter follows the quantum master equation in the Lindbladian form:. Paparo et al. In the case of quantum Page-Rank, this is set to the instantaneous probability P i , t of finding the walker at node i at the time-step t.

To obtain a fixed value for the quantum Page-Rank a time average is calculated as long as with its variance as a measure for quantum fluctuations.

Another approach 11 involves defining a Markov quantum evolution similar to Eq. The interaction between network science and quantum information science has led to the development of theoretical and computational tools that benefitted from both fields. On the one hand, quantum-inspired tools, such as information entropies and quantum distance measures, have been successfully applied to practical problems concerning classical complex networks On the other hand, classical network descriptors have been ported to the quantum realm to gain better insights about the structure and the dynamics of networked quantum systems The cross-pollination between the two fields—including, among others, quantum statistics for modeling the dynamics of classical networks and their geometry 79 , 80 see also ref.

Here, we briefly review the advances concerning quantum-inspired entropic measures for networks and network-inspired measures for quantum systems. Historically, the concept of entropy has been successfully used to quantify the complexity of many systems 82 , Recently, the possibility of using quantum entropy and other quantum information theoretical measures has been explored by the community of network scientists.

The combinatorial Laplacian matrix L , obtained from the adjacency matrix representing the network, is rescaled by the number of edges in the network. Network entropy is defined according to von Neumann quantum entropy as. This entropy has been generalized to the case of multilayer systems 86 , composite networks where units exhibit different types of relationships that are generally modeled as different layers see ref.