## 4 Methodology  
Entanglement is a type of the quantum correlations [6; 43; 44; 45; 46; 47; 48] that distinguishes the behavior of quantum mechanics. A two-qubit system becomes an entangled states if the state of one qubit cannot be separated from the state of the other. Entanglement cannot be detected directly, so entanglement measures [49; 50; 51] were proposed to determine the degree of entanglement in a quantum system. Various measurements are proposed to measure the strength of the entanglement between two-qubit systems such as concurrence, witness, and negativity among others [52; 53; 50]. However, the strength of entanglement in a two-qubit system is often measured using concurrence measure [50; 51]. An arbitrary, pure two-qubit state has the form  
\[|\psi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle, \tag{2}\]  
its concurrence measure is defined theoretically as follows [54]:  
\[C=2|\alpha\delta-\gamma\beta|, \tag{3}\]  
where \(0\leq C\leq 1\). It is obvious that if the state of the two-qubit system is defined by the state \(\alpha|00\rangle+\beta|11\rangle\), the concurrence value becomes \(2|\alpha\beta|\)[55]. Concurrence measure operator, denoted \(M_{z}\), was proposed to solve some quantum computation problems [56]. For example, some researchers used this operator to increase the computational speed for testing junta variables [57] and measuring the Hamming distance between boolean functions [58] quantum mechanically. The circuit model of \(M_{z}\) operator was proposed as shown in Figure 2 when applied on arbitrary two qubits \(|u\rangle\) and \(|v\rangle\). The state of the qubit \(|v\rangle\) is initialized in the vacuum state \(|0\rangle\) and the state of the qubit \(|u\rangle\) is arbitrary. The \(M_{z}\) is a unitary operator that applies the \(CNOT\) gate between the qubits \(|u\rangle\) and \(|v\rangle\) followed by measuring the degree of entanglement in between through the operator \(D\). The \(M_{z}\) operator creates the entangled state \(|uv\rangle=\alpha|00\rangle+\delta|11\rangle\), then it measures the concurrence between the qubits \(|u\rangle\) and \(|v\rangle\) as \(C>0\) through the operator \(D\) only if the input qubit \(|u\rangle\) is in the state \(|u\rangle=\alpha|0\rangle+\delta|1\rangle\), while it leaves them disentangled in the state \(|uv\rangle=|00\rangle\), then it measures the concurrence value in between as \(C=0\) through the operator \(D\) only if the input qubit \(|u\rangle\) is in the state \(|u\rangle=|0\rangle\). Some suggestions to implement the operator \(D\) using experimental setups are depicted in [51; 55; 59]. However, concretely in this paper, we propose to implement the operator \(D\) in the operator \(M_{z}\) using [PERSON] et al. protocol [54]. This protocol implements Equation (3) quantum mechanically, and the circuit model of this protocol is depicted in Figure 3. This protocol requires two copies of the two-qubit pure state given by Equation (2). This protocol is summarized in the following steps:  
1. Prepare two copies of the two-qubit state given by Equation (2) as follows: \[|\eta_{0}\rangle=|\psi\rangle(\sigma_{y}\otimes\sigma_{y}|\psi\rangle).\]
2. \(CNOT\) gate is applied between the second and the forth qubits, respectively, followed by the rotation \(R\) gate as follows: \[|\eta_{1}\rangle=R\ CNOT_{\eta_{0_{2}}\eta_{0_{4}}}|\eta_{0}\rangle,\] where the subscripts in the \(CNOT_{\eta_{0_{2}}\eta_{0_{4}}}\) gate represent the control and the target gate, respectively. However, the unitary \(R\) gate rotate the state of the qubit as follows: \[R|0\rangle=\frac{|0\rangle-|1\rangle}{\sqrt{2}},\quad R|1\rangle=\frac{|0 \rangle+|1\rangle}{\sqrt{2}}.\]  
Figure 2: Quantum circuit of \(M_{z}\) operator when applied on two qubits \(u\) and \(v\), where \(u\) is the control qubit and \(v\) is the target qubit. Therefore, the state of the system is as follows:  
\[