  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.
