Quantum network is the key to enable distributed quantum information processing. As the single-link communication rate decays exponentially with the distance, to enable reliable end-to-end quantum communication, the number of nodes needs to grow with the network scale. For highly connected networks, we identify a threshold transition in the capacity as the density of network nodes increases---below a critical density, the rate is almost zero, while above the threshold the rate increases linearly with the density. Surprisingly, above the threshold the typical communication capacity between two nodes is independent of the distance between them, due to multi-path routing enabled by the quantum network. In contrast, for less connected networks such as scale-free networks, the end-to-end capacity saturates to constants as the number of nodes increases, and always decays with the distance. Our results are based on capacity evaluations, therefore the minimum density requirement for an appreciable capacity applies to any general protocols of quantum networks.

翻译：量子网络是允许分布量子信息处理的关键。 奇怪的是, 单链通信率随着距离而指数衰减, 从而能够实现可靠的端到端量子通信, 节点的数量需要随着网络规模的增加而增长。 对于高度连接的网络, 我们确定一个阈值转换, 即网络节点的密度会增加- 低于临界密度, 比率几乎为零, 而比阈值高出线性速度随密度线性增长。 奇怪的是, 两个节点之间的典型通信能力由于量子网络的多路径路径连接而与它们之间的距离无关。 相反, 对于连接较少的网络, 如无比例网络, 端到端能力的饱和度会随着节点数量的增加而恒定, 并且总是随距离而衰减。 我们的结果以能力评估为基础, 因此, 两个节点之间的典型通信能力的最低密度要求 适用于量子网络的一般协议 。