# Percolation Distribution in Small-World Quantum Networks
## 4 Concurrence Percolation in Small-World Quantum Networks  
Whether using CEP or optimized QEP, a link composed of a singlet is required to accomplish communication between any two nodes in the network, and concurrence  
Figure 8: Distribution of network clustering coefficients obtained with different swaps. Figure 7: The relationship between GCC and conversion probability in the network. “WS” represents the percolation result in original WS network. “\(WS_{cn}\)” represents the result of CEP after adding the central node. “\(WS_{4 swap}\)” and “\(WS_{4 pswap}\)” represent the percolation result in networks with the central node after 4-swap and 4-pswap, respectively. percolation [13] can reduce the percolation threshold of the network while relaxing this condition. Concurrency is a metric of pure state entanglement, denoted as \(c:=\sin 2\theta\) (\(c\) is defined as \(c(\psi)=\left|\sum_{i}a_{i}^{2}\right|\) for a pure state \(\left|\psi\right\rangle=\sum_{i}a_{i}|e_{i}\rangle\), where \(|e_{i}\rangle\) is the basis state) [7]. Compared with traditional percolation analysis, concurrence percolation theory (ConPT) discards the measurement of cluster size and uses the \"sponge-crossing\" [14] probability \(P_{SC}\), which is the probability that two distant nodes are interconnected by an open path. \(P_{SC}\) can be calculated by combining the occupation probability of each path that connects two distant nodes using connectivity rules [13]. The concurrency of \"sponge-crossing\" is denoted as \(C_{SC}\). According to the thermodynamic limit, both \(P_{SC}\) and \(C_{SC}\) should jump from 0 to 1 once approaching the percolation threshold \(P_{th}\) and \(C_{th}\) respectively [13]. In the analysis of the network, the original network can be transformed by the following three rules to obtain the \(P_{SC}\) and \(C_{SC}\) of the network between two nodes. 1. Serial rule: for any 3 nodes A, B, J in the quantum network, A and J are connected by a link of concurrency \(c_{AJ}\), while B and J are connected by a link of concurrency \(c_{BJ}\). By performing a projection measurement on J (XZ is chosen as the measurement base), the final average concurrency obtained by the measurement is \[C=c_{AJ}c_{BJ}.\] (2)
2. Parallel rule: Nodes A and B are connected by two parallel links, which consist of product state, specified as: \[|\varphi_{AB}(\theta_{1})\rangle\bigotimes|\varphi_{AB}(\theta_{1})\rangle=( \cos\theta_{1}|00\rangle+\sin\theta_{1}|11\rangle)((\cos\theta_{2}|00\rangle+ \sin\theta_{2}|11\rangle)).\] (3) when \(\cos\theta_{1}\cos\theta_{2}\leq\frac{1}{\sqrt{2}}\), according to [PERSON]'s theorem, the maximally entangled state can be obtained by conversion. The average concurrency obtained at this point \(C=\sum_{k}\omega_{k}C_{k}\) is optimal, where \(\omega_{k}\) is the probability that concurrency \(C_{k}\) obtained after measurement is in pure state. 3. Star-mesh (SM) transform: The star graph of size \(n\) is transformed into a complete graph of size \(n-1\), and then one node in the complete graph of \(n-1\) is selected as the root node to obtain a star graph of \(n-1\). After several iterations, any two nodes in the connected cluster of the network share a connection. The series-parallel rule can be represented by Table 1[13]. The above three methods are applied to the WS quantum network to analyze the percolation threshold of the network. The number of nodes in the WS quantum network are selected as \(N=10\) and \(N=20\), the probability of reconnecting the network with disconnected edges \(p_{re}=0.5\), the average degree of the network \(\bar{k}=6\), both ConPT and classical methods are simulated. As shown in Figure 9, the percolation threshold obtained by the ConPT method is smaller: \(C_{th}=0.253\) and \(p_{th}=0.406\). Unlike the structure of regular networks (e.g., cellular networks, Bethe lattice networks, etc. ), WS quantum networks can have different topologies when different \(p_{re}\) are selected. Even if the same \(p_{re}\) is selected, the topologies obtained may be different as well.