Move the 3rd qbit to the 1st qbit remotely:
Φ 00=[12 0 0 12],Φ 01=[12 0 0 −12],Φ 10=[0 12 12 0],Φ 11=[0 12 −12 0] [12 0 0 12]⊗[v 1 v 2]+[12 0 0 −12]⊗[v 1 −v 2]+[0 12 12 0]⊗[v 2 v 1]+[0 12 −12 0]⊗[−v 2 v 1] =[12v 1 12v 2 0 0 0 0 12v 1 12v 2]+[12v 1 −12v 2 0 0 0 0 −12v 1 12v 2]+[0 0 12v 2 12v 1 12v 2 12v 1 0 0]+[0 0 −12v 2 12v 1 12v 2 −12v 1 0 0] =2[12v 1 0 0 12v 1 12v 2 0 0 12v 2] =2[v 1 v 2]⊗[12 0 0 12]\begin{aligned}&\Phi_{00} = \begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\0\\\frac{1}{\sqrt{2}}\end{bmatrix},\Phi_{01} = \begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\0\\\frac{-1}{\sqrt{2}}\end{bmatrix},\Phi_{10} = \begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\end{bmatrix},\Phi_{11} = \begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{-1}{\sqrt{2}}\\0\end{bmatrix}\\&\begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\0\\\frac{1}{\sqrt{2}}\end{bmatrix} \otimes \begin{bmatrix}v_1\\v_2\end{bmatrix} + \begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\0\\\frac{-1}{\sqrt{2}}\end{bmatrix} \otimes \begin{bmatrix}v_1\\-v_2\end{bmatrix} + \begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\end{bmatrix} \otimes \begin{bmatrix}v_2\\v_1\end{bmatrix} + \begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{-1}{\sqrt{2}}\\0\end{bmatrix} \otimes \begin{bmatrix}-v_2\\v_1\end{bmatrix}\\&= \begin{bmatrix}\frac{1}{\sqrt{2}} v_1\\\frac{1}{\sqrt{2}} v_2\\0\\0\\0\\0\\\frac{1}{\sqrt{2}} v_1\\\frac{1}{\sqrt{2}} v_2\end{bmatrix} + \begin{bmatrix}\frac{1}{\sqrt{2}} v_1\\\frac{-1}{\sqrt{2}} v_2\\0\\0\\0\\0\\\frac{-1}{\sqrt{2}} v_1\\\frac{1}{\sqrt{2}} v_2\end{bmatrix} + \begin{bmatrix}0\\0\\\frac{1}{\sqrt{2}} v_2\\\frac{1}{\sqrt{2}} v_1\\\frac{1}{\sqrt{2}} v_2\\\frac{1}{\sqrt{2}} v_1\\0\\0\end{bmatrix} + \begin{bmatrix}0\\0\\\frac{-1}{\sqrt{2}} v_2\\\frac{1}{\sqrt{2}} v_1\\\frac{1}{\sqrt{2}} v_2\\\frac{-1}{\sqrt{2}} v_1\\0\\0\end{bmatrix}\\&= 2 \begin{bmatrix}\frac{1}{\sqrt{2}} v_1\\0\\0\\\frac{1}{\sqrt{2}} v_1\\\frac{1}{\sqrt{2}} v_2\\0\\0\\\frac{1}{\sqrt{2}} v_2\end{bmatrix}\\&= 2 \begin{bmatrix}v_1\\v_2\end{bmatrix} \otimes \begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\0\\\frac{1}{\sqrt{2}}\end{bmatrix}\end{aligned}
