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arXiv:2602.10189 · Quantum Science and Technology

Entanglement percolation in random quantum networks

Alessandro Romancino, Jordi Romero-Pallejà, G. Massimo Palma, Anna Sanpera

quant-phSubmitted Feb 10, 2026• Announced Feb 12, 2026• 1 citation
v1 · Feb 10, 2026v2 · Apr 16, 2026
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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.

Journal reference
Quantum Sci. Technol. (2026) 11 025044
DOI
10.1088/2058-9565/ae5fca

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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.
Quantum communication demands efficient distribution of quantum entanglement across a network of connected partners. The search for efficient strategies for the entanglement distribution may be based on percolation theory, which describes evolution of network connectivity with respect to some network parameters. In this framework, the probability to establish perfect entanglement between two remote partners decays exponentially with the distance between them before the percolation transition point, which unambiguously defines percolation properties of any classical network or lattice. Here we introduce quantum networks created with local operations and classical communication, which exhibit non-classical percolation transition points leading to striking communication advantages over those offered by the corresponding classical networks. We show, in particular, how to establish perfect entanglement between any two nodes in the simplest possible network-the 1D chain-using imperfectly entangled pairs of qubits.
Abstract We study entanglement percolation in qubit-based planar quantum network models of arbitrary topology, where neighboring nodes are initially connected by pure states with quenched disorder in their entanglement. To address this, we develop a physics-informed heuristic algorithm designed to find a sequence of entanglement swapping and distillation operations to connect any pair of distant nodes. The algorithm combines locally optimal percolation strategies between nodes at a maximum distance of one swapping operation. If this fails to produce a maximally entangled state, it looks for alternative paths surrounding intermediate states within the process. We analytically find and numerically verify thresholds in quantum percolation, which depend on the initial network configuration and entanglement, and are associated with specific percolation strategies. We classify these strategies based on the connectivity, a quantity that relates the entanglement in the final state and the level of integrity of the network at the end of the process. We find distinct regimes of quantum percolation, which are clearly separated by the percolation thresholds of the employed strategies and vastly vary according to the network topology.
Abstract Quantum communication demands efficient distribution of quantum entanglement across a network of connected partners. The search for efficient strategies for the entanglement distribution may be based on percolation theory, which describes evolution of network connectivity with respect to some network parameters. In this framework, the probability to establish perfect entanglement between two remote partners decays exponentially with the distance between them before the percolation transition point, which unambiguously defines percolation properties of any classical network or lattice. Here we introduce quantum networks created with local operations and classical communication, which exhibit non-classical percolation transition points leading to striking communication advantages over those offered by the corresponding classical networks. We show, in particular, how to establish perfect entanglement between any two nodes in the simplest possible network—the 1D chain—using imperfectly entangled pairs of qubits.
Quantum networks have good prospects for applications in the future. Compared with classical networks, small-world quantum networks have some interesting properties. The topology of the network can be changed through entanglement exchange operations, and different network topologies will result in different percolation thresholds when performing entanglement percolation. A lower percolation threshold means that quantum networks require fewer minimum resources for communication. Since a shared singlet between two nodes can still be a limitation, concurrency percolation theory (ConPT) can be used to relax the condition. In this paper, we investigate how entanglement distribution is performed in small-world quantum networks to ensure that nodes in the network can communicate with each other by establishing communication links through entanglement swapping. Any node can perform entanglement swapping on only part of the connected edges, which can reduce the influence of each node in the network during entanglement swapping. In addition, the ConPT method is used to reduce the percolation threshold even further, thus obtaining a better network structure and reducing the resources required.
Quantum networks are interconnected by nodes, between singlets which are formed to ensure the successful transmission of information with a probability of 1. However, in real quantum networks, nodes often share a partially entangled state instead of a singlet due to factors such as environmental noise. Therefore, it is necessary to convert the partially entangled state into a singlet for efficient communication. Percolation happens during the conversion of connected edges in the whole network. As a result, when the singlet conversion probability (SCP) is greater than the percolation threshold, a giant interconnected cluster that meets the basic requirements of communication will appear in the network. The percolation threshold of the network reveals the minimum resources required to carry out large scale quantum communication. In this paper, we investigate the effect of different q-swaps on the percolation threshold in quantum entanglement percolation of small world networks. We show that Quantum Entanglement Percolation (QEP) has a better percolation performance than Classical Entanglement Percolation (CEP). By using different q swaps in Watts Strogatz (WS) small world networks and Kleinberg networks for simulation, we also show that the percolation threshold is minimized when SCP is equal to the average degree of the network. Furthermore, we introduce quantum walk as a new scheme to have an extra reduction in the percolation threshold.
Quantum entanglement as a non-local correlation between particles is critical to the transmission of quantum information in quantum networks (QNs); the key challenge lies in establishing long-distance entanglement transmission between distant targets. This issue aligns with percolation theory, and as a result, an entanglement distribution scheme called “Classical Entanglement Percolation” (CEP) has been proposed. While this scheme provides an effective framework, “Quantum Entanglement Percolation” (QEP) indicates a lower percolation threshold through quantum preprocessing strategies, which will modify the network topology. Meanwhile, an emerging statistical theory known as “Concurrence Percolation” reveals the unique advantages of quantum networks, enabling entanglement transmission under lower conditions. It fundamentally belongs to a different universality class from classical percolation. Although these studies have made significant theoretical advancements, most are based on an idealized pure state network model. In practical applications, quantum states are often affected by thermal noise, resulting in mixed states. When these mixed states meet specific conditions, they can be transformed into pure states through quantum operations and further converted into singlets with a certain probability, thereby facilitating entanglement percolation in mixed state networks. This finding greatly broadens the application prospects of quantum networks. This review offers a comprehensive overview of the fundamental theories of quantum percolation and the latest cutting-edge research developments.