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Acta Sinica Quantum Optica

Entanglement Percolation in Small-World Quantum Networks

Zhu Shi-Qun

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Long-distance entanglement establishment is investigated on small-world complex network topologies.The quantum entanglement percolation(QEP) protocol is proposed to optimize networked quantum communications.If the additional local target-swap quantum operations are performed in the classical entanglement percolation(CEP) protocol,the QEP can be obtained.The QEP has great advantage over CEP with respect to the efficiency of long-distance entanglement establishment.The advantage may be due to the changes of the degree distributions in the QEP.

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arXiv:0906.2977•Physical Review Letters•Dec 10, 2009•90 citations
Long-distance entanglement establishment is investigated on small-world complex network topologies.The quantum entanglement percolation(QEP) protocol is proposed to optimize networked quantum communications.If the additional local target-swap quantum operations are performed in the classical entanglement percolation(CEP) protocol,the QEP can be obtained.The QEP has great advantage over CEP with respect to the efficiency of long-distance entanglement establishment.The advantage may be due to the changes of the degree distributions in the QEP.
In the paper, the analysis of percolation of quantum entanglement in complex networks was given, including the preparation of long-range quantum entangled states in two-dimensional complex networks of small word. A scheme of quantum entanglement percolation optimization of quantum communication network is proposed. By introducing the localized “target-swap” quantum operation into the classical entanglement percolation protocol (CEP), the quantum entanglement percolation protocol is formed. It also shows that the quantum entanglement percolation protocol (QEP) is significantly better than CEP in the performance of long-range entanglement preparation in two-dimensional complex networks of small world.
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 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.
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.
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 networks are essential to quantum information distributed applications, and communicating over them is a key challenge. Complex networks have rich and intriguing properties, which are as yet unexplored in the quantum setting. Here, we study the effect of entanglement percolation as a means to establish long-distance entanglement between arbitrary nodes of quantum complex networks. We develop a theory to analytically study random graphs with arbitrary degree distribution and give exact results for some models. Our findings are in good agreement with numerical simulations and show that the proposed quantum strategies enhance the percolation threshold substantially. Simulations also show a clear enhancement in small-world and other real-world networks.