Distribution of Graph States via Quantum Routers with Network Coding

Quantum repeaters are devices that subdivide a long quantum channel into smaller segments [1, 2]. They are designed to tackle errors like noise and losses which would spoil the transmitted quantum state. Long-distance quantum communication via quantum repeaters allows distant parties to perform various quantum information protocols. A prominent application is quantum key distribution [3–5]. Several proposals for quantum repeaters are known. They may be classified by the direction of classical communication: two-way communication is used for repeat-until-success strategies for transmission between neighbouring repeaters [6–8] and for entanglement distillation protocols, while one-way communication suffices for schemes based on quantum error correction [9–12]. Our results can be obtained for both types of repeaters, but we focus on repeaters of the latter kind in this talk [13]. The previous descriptions apply to a single channel connecting two parties A and B. It is natural to consider the multipartite generalization of this scenario: several sites A, B, C, ... are connected by quantum channels. Such a quantum network corresponds to a mathematical graph, where the sites are represented by nodes and the quantum channels are the edges of the graph. This way the mathematical graph models the physical infrastructure of the quantum network. In practice the capacity of each link is constrained. For simplicity we assume that each channel allows to transmit a single qudit of fixed dimension D per time step. Some sites are special in the sense that they will share a qudit of the finally distributed entangled quantum state. We continue to call these sites parties, while we call the other nodes quantum repeaters or quantum routers if they have vertex degree two or larger than two, respectively.