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：我们研究了由具有非确定性纠缠交换能力的设备组成的量子网络中，向一对节点进行多路径纠缠分发的问题。多路径纠缠分发允许网络利用任意数量的可用路径（预先建立链路级纠缠）建立端到端的纠缠链路。另一方面，概率性纠缠交换限制了节点之间共享的纠缠量；当由于架构和其他实际约束，交换必须在时间上近似同步执行时，这种情况尤为明显。我们将关注点限于网络中仅生成二分纠缠态的情形，将问题形式化为两个希望通信的量子端节点之间的广义流最大化实例。我们提出了一种混合整数二次约束规划（MIQCP）来解决任意拓扑网络中的这一流问题。随后，通过求解所有可能由概率性纠缠链路存在与否生成的网络状态下的流问题，并进一步对所有网络状态容量进行平均，我们计算了整体网络容量，定义为单位时间内分发给用户的EPR态的最大数量。该MIQCP也可应用于具有复用链路的网络。尽管我们计算整体网络容量的方法存在一个不良特性——状态总数随链路复用能力呈指数增长——但它能提供精确解，可作为易于实现但非最优的纠缠路由算法吞吐量性能的上界比较基准。我们将此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。