Distributed Quantum Systems - from Graph States to Quantum Networks

Quantum networks promise to provide secure communication and better computational power than classical computer networks. An important resource for such networks is multipartite entanglement. Challenges in real-world applications include imperfections and noise. This thesis contributes to the characterisation of multipartite entanglement and the design of quantum networks, taking the effects of noise into account.
Currently, one open problem is the characterisation of multipartite entangled states. The first part of this thesis focuses on a family of such multipartite entangled states which can be described by the graphical formalism of graph and hypergraph states. We use the graphical properties to develop easy-to-compute criteria for assessing the equivalence of entangled states under local unitary transformations. We also extend our analysis to the hypergraph state formalism for continuous variable systems and develop graphical transformation rules characterising multipartite continuous variable entanglement. Additionally, we address how to deal with noise by identifying robust graph state structures that yield high-fidelity post-measurement states. Subsequently, we develop methods for reducing the number of input states in purification protocols of hypergraph states, another option to deal with noise.
The second part of this thesis focuses on quantum network architectures. We analyse the probabilistic nature of entanglement distribution across network segments, deriving expressions for the average waiting time required to establish entangled links in simple repeater chains. We also show how to achieve maximal key rates in networks with imperfect quantum channels. Finally, we show that the performance of a simple repeater chain can be improved by adding memory qubits to the repeater stations.