## 4 Methodology
\begin{array}{ll}|\eta_{1}\rangle=\frac{1}{\sqrt{2}}&\{\ \ \ (\beta\gamma- \alpha\delta)|0000\rangle+(\beta\gamma+\alpha\delta)|0100\rangle+(\alpha\gamma- \beta\delta)|0001\rangle-(\alpha\gamma+\beta\delta)|0101\rangle+2\gamma\delta| 1100\rangle\\ &-2\alpha\beta|0110\rangle+(\beta^{2}-\alpha^{2})|0011\rangle+(\beta^{2}+ \alpha^{2})|0111\rangle+(\gamma^{2}-\delta^{2})|1001\rangle-(\gamma^{2}+ \delta^{2})|1101\rangle\\ &+(\beta\gamma-\alpha\delta)|1010\rangle-(\beta\gamma+\alpha\delta)|1110\rangle +(\alpha\gamma+\beta\delta)|1111\rangle-(\alpha\gamma-\beta\delta)|1011\rangle \}.\end{array}  \tag{4}\]  
Comparing Equation (3) and Equation (4), we obtain  
\[C=2\sqrt{2P_{0000}}\ \ \ or\ \ \ C=2\sqrt{2P_{1010}}\, \tag{5}\]  
where \(P_{0000}\) and \(P_{1010}\) are the success probability for obtaining the state \(|0000\rangle\) and \(|1010\rangle\), respectively. end. So, taking this into account, our proposal to implement the operator \(D\) using [PERSON] et al. circuit (2016), transforms the circuit model of the \(M_{z}\) operator from Figures 2-4. It is obvious that \(M_{z}\) operator applies two main operations. In the first operation, the \(CNOT\) gate is applied on each replica of the two-qubit systems \(|u\rangle\) and \(|v\rangle\) as the control qubit and as a target qubit, respectively. Accordingly, there are two cases:  
(i) The state of each replica of the two-qubit \(|uv\rangle\) system will be entangled, \(|uv\rangle=\alpha|00\rangle+\delta|11\rangle\), only if \(|u\rangle=\alpha|0\rangle+\delta|1\rangle\). Consequently, when the second operation of the operator \(M_{z}\) operator is applying the operator \(D\) on the state \(|uv\rangle\otimes|uv\rangle\) as shown in Figure 4. Therefore, the state of the system is as follows:  
\[|\eta_{1}\rangle=\frac{1}{\sqrt{2}}\{-\alpha\delta|0000\rangle+\alpha\delta|010 0\rangle-\alpha^{2}|0011\rangle-\alpha\delta|1010\rangle+\alpha^{2}|0111 \rangle-\delta^{2}|1001\rangle-\delta^{2}|1101\rangle-\alpha\delta|1110\rangle\}. \tag{6}\]  
Comparing Equation (3) and Equation (6), we obtain  
\[C=2\sqrt{2P_{0000}},\ C=2\sqrt{2P_{0100}},\ C=2\sqrt{2P_{1010}}\ \ \ or\ \ \ C=2\sqrt{2P_{1110}}. \tag{7}\]  
Then the concurrence can be estimated using one of the formulas shown in Equation (7). (ii) On the other hand, the state of the system will be separable in the state \(|uv\rangle=|00\rangle\) if the state of the qubit \(|u\rangle\) is \(|0\rangle\). Again, when the second operation of the operator \(M_{z}\) operator is applying the operator \(D\) on the state \(|uv\rangle\otimes|uv\rangle\) as shown in Figure 4. Therefore, the state of the system is as follows:  
\[|\eta_{1}\rangle=\frac{1}{\sqrt{2}}\{-\alpha^{2}|0011\rangle+\alpha^{2}|0111 \rangle\}. \tag{8}\]  
So, in this case the concurrence value is \(C=0\). Eventually, the operator \(D\) in \(M_{z}\) operator measures the concurrence value between the two qubits \(|u\rangle\) and \(|v\rangle\), by estimating the probability of obtaining the probability of the state \(|0000\rangle\), \(|0100\rangle\), \(|1010\rangle\), or \(|1110\rangle\), therefore the concurrence value is given by Equation (7). \(M_{z}\) will be used in the last step int the proposed algorithm (see Section 5). The proposed Quantum Classification Algorithm Based on Competitive Learning and Entanglement Measure: Case Study  
Quantum competitive neural networks consists of two layers namely quantum-storing layer and quantum-competitive layer [21; 38]. Quantum-storing layer stores \(M\) defined prototype patterns in advance with equal probabilities via \(n\) neurons using the [PERSON]'s storage model (for more details see Section 5.1). Here, we propose the quantum classification algorithm based on competitive neural network. As for the quantum-competitive layer, the QCPNN algorithm classifies the input pattern into one of two classes \"0\" or \"1\". In this layer, the steps of the QCPNN algorithm conduct a competition between the prototype patterns to mark, by entanglement, the class of the closest prototype pattern to the input pattern.