Entanglement percolation in random quantum networks

TL;DR

This work investigates entanglement percolation in quantum networks under realistic randomness: edge entanglement is not identical but drawn from a distribution. It shows that the classical entanglement percolation (CEP) protocol's success depends only on the average entanglement and percolates if this average exceeds the network's threshold $p_c$. In contrast, the quantum entanglement percolation (QEP) protocol, which reshapes network topology via $q$-swaps, becomes less effective as the entanglement distribution broadens, with a width threshold $w^*\approx 0.067$ beyond which CEP outperforms QEP. These results suggest CEP may be the more robust strategy for entanglement distribution in noisy, randomly entangled quantum networks and provide a baseline for future multipartite generalizations and optimization frameworks across complex network topologies.

Abstract

Entanglement percolation aims at generating maximal entanglement between any two nodes of a quantum network by utilizing strategies based solely on local operations and classical communication between the nodes. As it happens in classical percolation theory, the topology of the network is crucial, but also the entanglement shared between the nodes of the network. In a network of identically partially entangled states, the network topology determines the minimum entanglement needed for percolation. In this work, we generalize the protocol to scenarios where the initial entanglement shared between each two nodes of the network is not the same but has some randomness. In such cases, we find that for classical entanglement percolation, only the average initial entanglement is relevant. In contrast, the quantum entanglement percolation protocol generally performs worse under these more realistic conditions.

Figures (12)

• Figure 1: Example of a quantum network consisting of identical copies of state $\ket{\omega}$ with Singlet Conversion Probability (SCP) $p_{\omega} = 2\lambda^\omega_2$. Nodes $i$ and $j$ are highlighted.
• Figure 2: Representation of an entanglement swapping procedure, where a joined bell measurement is performed on systems $R_1$ and $R_2$.
• Figure 3: A single (successful) run of a classical entanglement percolation protocol. The final singlet is created between node 0 and node 3 after applying entanglement swapping on nodes 2 and 5.
• Figure 4: An example of a $5$-swap operation. (a) Initial star graph $S_5$ with a central node $0$ connected with two identical states $\ket{\omega}$ with each of the other nodes. (b) End result after performing the $5$-swap between the $0$ node and all other nodes. This creates new entangled states between the external nodes and erases node $0$ from the network.
• Figure 5: Representation of the initial and final networks after aplying the QEP protocol to a double-bond honeycomb lattice.