@mastersthesis {391,
title = {Entanglement distribution in quantum complex networks},
volume = {Doctorat en F{\'\i}sica},
year = {2012},
month = {11/2012},
school = {Universitat Aut{\`o}noma de Barcelona},
type = {PhD},
address = {Barcelona},
abstract = {This thesis deals with the study of quantum networks with a complex structure, the implications this structure has in the distribution of entanglement and how their functioning can be enhanced by operating in the quantum regime. We first consider a complex network of bipartite states, both pure and mixed, and study the distribution of long-distance entanglement. Then, we move to a network with noisy channels and study the creation and distribution of large, multipartite states.
The work contained in this thesis is primarily motivated by the idea that the interplay between quantum information and complex networks may give rise to a new understanding and characterization of natural systems. Complex networks are of particular importance in communication infrastructures, as most present telecommunication networks have a complex structure. In the case of quantum networks, which are the necessary framework for distributed quantum processing and for quantum communication, it is very plausible that in the future they acquire a complex topology resembling that of existing networks, or even that methods will be developed to use current infrastructures in the quantum regime.
A central task in quantum networks is to devise strategies to distribute entanglement among its nodes. In the first part of this thesis, we consider the distribution of bipartite entanglement as an entanglement percolation process in a complex network. Within this approach, perfect entanglement is established probabilistically between two arbitrary nodes. We see that for large networks, the probability of doing so is a constant strictly greater than zero (and independent of the size of the network) if the initial amount of entanglement is above a certain critical value. Quantum mechanics offer here the possibility to change the structure of the network without need to establish new, "physical" channels. By a proper local transformation of the network, the critical entanglement can be decreased and the probability increased. We apply this transformation to complex network models with arbitrary degree distribution.
In the case of a noisy network of mixed states, we see that for some classes of states, the same approach of entanglement percolation can be used. For general mixed states, we consider a limited-path-length entanglement percolation constrained by the amount of noise in the connections. We see how complex networks still offer a great advantage in the probability of connecting two nodes.
In the second part, we move to the multipartite scenario. We study the creation and distribution of graph states with a structure that mimic the underlying communication network. In this case, we use an arbitrary complex network of noisy channels, and consider that operations and measurements are also noisy. We propose an efficient scheme to distribute and purify small subgraphs, which are then merged to reproduce the desired state. We compare this approach with two bipartite protocols that rely on a central station and full knowledge of the network structure. We show that the fidelity of the generated graphs can be written as the partition function of a classical disordered spin system (a spin glass), and its decay rate is the analog of the free energy. Applying the three protocols to a one-dimensional network and to complex networks, we see that they are all comparable, and in some cases the proposed subgraph protocol, which needs only local information of the network, performs even better.
},
author = {Cuquet, Mart{\'\i}}
}
@article {cuquet_growth_2012,
title = {Growth of graph states in quantum networks},
journal = {Physical Review A},
volume = {86},
number = {4},
year = {2012},
pages = {042304},
abstract = {We propose a scheme to distribute graph states over quantum networks in the presence of noise in the channels and in the operations. The protocol can be implemented efficiently for large graph sates of arbitrary (complex) topology. We benchmark our scheme with two protocols where each connected component is prepared in a node belonging to the component and subsequently distributed via quantum repeaters to the remaining connected nodes. We show that the fidelity of the generated graphs can be written as the partition function of a classical Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear cluster and results for its decay rate in random graphs with arbitrary (uncorrelated) degree distributions.},
doi = {10.1103/PhysRevA.86.042304},
url = {http://link.aps.org/doi/10.1103/PhysRevA.86.042304},
author = {Cuquet, Mart{\'\i} and John Calsamiglia}
}
@article {303,
title = {Limited-path-length entanglement percolation in quantum complex networks},
journal = {Physical Review A},
volume = {83},
number = {3},
year = {2011},
month = {03/2011},
pages = {032319-14},
abstract = {We study entanglement distribution in quantum complex networks where nodes are connected by bipartite entangled states. These networks are characterized by a complex structure, which dramatically affects how information is transmitted through them. For pure quantum state links, quantum networks exhibit a remarkable feature absent in classical networks: it is possible to effectively rewire the network by performing local operations on the nodes. We propose a family of such quantum operations that decrease the entanglement percolation threshold of the network and increase the size of the giant connected component. We provide analytic results for complex networks with an arbitrary (uncorrelated) degree distribution. These results are in good agreement with numerical simulations, which also show enhancement in correlated and real-world networks. The proposed quantum preprocessing strategies are not robust in the presence of noise. However, even when the links consist of (noisy) mixed-state links, one can send quantum information through a connecting path with a fidelity that decreases with the path length. In this noisy scenario, complex networks offer a clear advantage over regular lattices, namely, the fact that two arbitrary nodes can be connected through a relatively small number of steps, known as the small-world effect. We calculate the probability that two arbitrary nodes in the network can successfully communicate with a fidelity above a given threshold. This amounts to working out the classical problem of percolation with a limited path length. We find that this probability can be significant even for paths limited to few connections and that the results for standard (unlimited) percolation are soon recovered if the path length exceeds by a finite amount the average path length, which in complex networks generally scales logarithmically with the size of the network.},
doi = {10.1103/PhysRevA.83.032319},
url = {http://pra.aps.org/abstract/PRA/v83/i3/e032319},
author = {Cuquet, Mart{\'\i} and John Calsamiglia}
}
@article {Cuquet2009,
title = {Entanglement Percolation in Quantum Complex Networks},
journal = {Physical Review Letters},
volume = {103},
number = {24},
year = {2009},
month = {12/2009},
pages = {240503{\textendash}4},
publisher = {APS},
abstract = {Quantum networks are essential to quantum information distributed applications, and communicatingover them is a key challenge. Complex networks have richand intriguing properties, which are as yet unexplored in thequantum setting. Here, we study the effect of entanglement percolationas a means to establish long-distance entanglement between arbitrary nodesof quantum complex networks. We develop a theory to analyticallystudy random graphs with arbitrary degree distribution and give exactresults for some models. Our findings are in good agreementwith numerical simulations and show that the proposed quantum strategiesenhance the percolation threshold substantially. Simulations also show a clearenhancement in small-world and other real-world networks.},
keywords = {complex networks, entanglement distribution, entanglement percolation, erdos-renyi, generating function, quantum complex networks, quantum networks, scale-free, small-world},
doi = {10.1103/PhysRevLett.103.240503},
url = {http://link.aps.org/abstract/PRL/v103/e240503},
author = {Cuquet, Mart{\'\i} and John Calsamiglia}
}