On the Bipartite Entanglement Capacity of Quantum Networks

TL;DR

This work introduces a MIQCP-based framework to compute the bipartite entanglement capacity (BEC) of quantum networks under probabilistic entanglement swapping and multi-path routing. By modeling each network state as a directed, leaky s-t flow with unit-edge capacity and enforcing flow integrity, disjointness, and valid s-t paths, it yields an exact capacity per state; averaging over all states provides the overall network capacity. The approach extends to multiplexed links and is validated on several realistic topologies, offering exact upper-bounds for throughput and a benchmark for heuristic routing. The results demonstrate the potential throughput gains from multi-path routing and multiplexing, while also highlighting scenarios (e.g., long-hop SURFNet) where improvements or alternative strategies are essential for practical quantum networks.

Abstract

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.

Figures (13)

• Figure 1: Entanglement swapping. Quantum network nodes $A$, $B$, and $C$ begin by generating elementary link-level entangled states $\ket{\Psi^+}_{AB}$ and $\ket{\Psi^+}_{BC}$. A successful swap/BSM (Bell state measurement) at node $B$ results in an entangled state $\ket{\Psi^+}_{AC}$ shared between $A$ and $C$.
• Figure 2: Example of a quantum network where two nodes, $s$ and $t$, wish to share entanglement. Link $(i,j)$ generates entanglement with probability $p_{ij}$, and node $n_i$ performs entanglement swapping with success probability $q_i$.
• Figure 3: Two possible network snapshots after link-level entanglement generation attempts.
• Figure 4: Different scenarios relevant to Lemma \ref{['lemma:noxijknoflow']} proof. Directed arrows indicate presence of flow in that direction.
• Figure 5: A valid flow scenario described in Section \ref{['sec:miqcp:path_val']}.

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof