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 any 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 (within the q-swap framework) degrades under these more realistic conditions as the width of the distribution increases, suggesting that Random CEP may become the optimal LOCC strategy in sufficiently heterogeneous quantum networks.
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 any 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 (within the q-swap framework) degrades under these more realistic conditions as the width of the distribution increases, suggesting that Random CEP may become the optimal LOCC strategy in sufficiently heterogeneous quantum networks.
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.
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.
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.
In a quantum communication network, links represent entanglement between qubits located at different nodes. Even if two nodes are not directly linked by shared entanglement, they can still communicate via routing protocols. However, in contrast to classical communication, each quantum communication event removes all participating links along the routed path, disrupting the quantum communication network. Here, we propose a simple model, where randomly selected pairs of nodes communicate through the shortest paths. Each time such a path is used, all participating links are eliminated, leading to a correlated percolation process that we call "path percolation." We study path percolation both numerically and analytically and present the phase diagram of the steady states as a function of the rate at which new links are being added to the network. As a key result, the steady state is found to be independent of the initial network topologies when new links are added randomly between disconnected components. We close by discussing extensions of path percolation and link replenishment, along with their potential applications.
Quantum networks (QNs) are expected to play a key role in next-generation networks. A QN is comprised of quantum nodes and links (or bonds) representing entangled qubit pairs (EQPs) shared between two nodes. The probability of establishing a pure maximally entangled pair, known as Einstein-Podolsky-Rosen (EPR) pair, between two neighboring nodes represents the linking probability (or bond occupation probability). If such probability exceeds the so-called percolation threshold, the QN exhibits a percolation phenomenon characterized by the emergence of a giant cluster spanning the entire network. In the presence of such a giant cluster, any two arbitrary nodes in the system are connected by a chain of links with a non-zero probability, thus establishing EPR pairs at remote nodes. This work introduces a procedure for obtaining QNs from initially isolated quantum nodes. In the proposed procedure, two copies of EPR pairs are prepared at a source node and two qubits, one from each pair, are sent to a neighboring node through a noisy quantum channel. Then, a pair of mixed quantum states is established between the source node and the neighboring node. By performing local operations and classical communication, an EPR pair can be established between these two nodes with non-zero probability. Repeating this procedure for every pair of source node and neighboring node, a QN exhibiting a percolation phenomenon can be obtained.