The Jordan-Wigner transformation from pair operators to Pauli matrices is $$ \begin{align} A_p &= \frac{X_p+iY_p}{2} \tag{25}\\ A_p^{\dagger} &= \frac{X_p-iY_p}{2}, \tag{26} \end{align} $$ where \( P_i\equiv \left(\bigotimes_{n=1}^{i-1}I\right)\otimes P\otimes\left(\bigotimes_{n=i+1}^NI\right) \) where \( P \in \{X,Y,Z\} \) and \( N \) is the total number of particles.