We consider the problem of multipath entanglement distribution to a pair of nodes in a quantum network consisting of devices with nondeterministic entanglement swapping capabilities. Multipath entanglement distribution enables a network to establish end-to-end entangled links across any number of available paths with preestablished 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 practical constraints, swaps must be performed in temporal proximity to each other. Limiting our focus to the case where only bipartite entanglement is 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 Einstein-Podolsky-Rosen (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 more easily implementable yet nonoptimal entanglement routing algorithms.

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