We consider the problem of multi-path entanglement distribution to a pair of nodes in a quantum network consisting of devices with non-deterministic entanglement swapping capabilities. Multi-path entanglement distribution enables a network to establish end-to-end entangled links across any number of available paths with pre-established link-level entanglement. Probabilistic entanglement swapping, on the other hand, limits the amount of entanglement that is shared between the nodes; this is especially the case when, due to architectural and other practical constraints, swaps must be performed in temporal proximity to each other. Limiting our focus to the case where only bipartite entangled states are generated across the network, we cast the problem as an instance of generalized flow maximization between two quantum end nodes wishing to communicate. We propose a mixed-integer quadratically constrained program (MIQCP) to solve this flow problem for networks with arbitrary topology. We then compute the overall network capacity, defined as the maximum number of EPR states distributed to users per time unit, by solving the flow problem for all possible network states generated by probabilistic entangled link presence and absence, and subsequently by averaging over all network state capacities. The MIQCP can also be applied to networks with multiplexed links. While our approach for computing the overall network capacity has the undesirable property that the total number of states grows exponentially with link multiplexing capability, it nevertheless yields an exact solution that serves as an upper bound comparison basis for the throughput performance of easily-implementable yet non-optimal entanglement routing algorithms. We apply our capacity computation method to several networks, including a topology based on SURFnet -- a backbone network used for research purposes in the Netherlands.

翻译：我们研究量子网络中一对节点间的多路径纠缠分发问题，该网络由具备非确定性纠缠交换能力的设备构成。多路径纠缠分发使网络能够利用预先建立的链路级纠缠，沿任意数量的可用路径建立端到端纠缠链路。然而，概率性纠缠交换限制了节点间共享的纠缠量——特别是当交换操作需在时间上邻近执行（受架构及其他实际约束）时更为显著。本文将研究范围限定在仅生成双粒子纠缠态的网络场景，将此问题转化为两个拟通信量子端节点间的广义流最大化实例。针对任意拓扑网络，我们提出混合整数二次约束规划（MIQCP）求解该流问题。通过遍历所有由概率性纠缠链路存在/缺失产生的网络状态，分别求解各状态的流问题，再对所有网络状态容量取平均，最终计算出整体网络容量——定义为每时间单位可向用户分发的EPR态最大数量。该MIQCP方法同样适用于复用链路网络。尽管整体网络容量计算存在状态总数随链路复用能力指数增长的不足，但这种方法能获得精确解，为易于实现但非最优的纠缠路由算法的吞吐量性能提供上界比较基准。我们将该容量计算方法应用于多个网络，包括基于SURFnet（荷兰用于研究目的的主干网络）的拓扑结构。