 = \mathcal{P}_0 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \right],\end{aligned}$$  $$\begin{aligned} B_0 & = & \frac{0}{0} \left(\begin{array}{ccccccccc} 0&0&0&0&0&0&0&0&0\\ 0&\alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0\\ 0&\alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0\\ \alpha_0&\alpha_0&\alpha_0&\alpha_0^0&\alpha_0^0&\alpha_0^0&0&0&0\\ \alpha_0^0&0&\alpha_0&0&\alpha_0&\alpha_0^0&\alpha_0&\alpha_0^0&0\\ 0&\alpha_0^0&\alpha_0&\alpha_0&0&\alpha_0^0&\alpha_0^0&\alpha_0&0\\ \alpha_0&\alpha_0&\alpha_0&0&0&0&\alpha_0^0&\alpha_0^0&\alpha_0^0\\ 0&\alpha_0&\alpha_0^0&\alpha_0^0&0&\alpha_0&\alpha_0&\alpha_0^0&0\\ \alpha_0^0&\alpha_0&0&\alpha_0&0&\alpha_0^0&0&\alpha_0^0&\alpha_0 \end{array}\right) \\ & = & \frac{0}{\sqrt{0}} \left\{ \begin{array}{l} |0_0 \rangle |0_0 \rangle +|0_0 \rangle |0_0 \rangle + |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle +\alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle +|0_0 \rangle |0_0 \rangle + |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle +\alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle +|0_0 \rangle |0_0 \rangle + |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle +\alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle \end{array}\right\} = \mathcal{P}_0 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \right],\end{aligned}$$  $$\begin{aligned} B_0 & = & \frac{0}{0} \left(\begin{array}{ccccccccc} 0&0&0&0&0&0&0&0&0\\ \alpha_0^0&0&\alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0&0&\alpha_0\\ \alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0&0\\ \alpha_0&\alpha_0&\alpha_0&\alpha_0^0&\alpha_0^0&\alpha_0^0&0&0&0\\ \alpha_0^0&0&\alpha_0&0&\alpha_0&\alpha_0^0&\alpha_0&\alpha_0^0&0\\ \alpha_0&0&\alpha_0^0&\alpha_0^0&\alpha_0&0&0&\alpha_0^0&\alpha_0\\ \alpha_0&\alpha_0&\alpha_0&0&0&0&\alpha_0^0&\alpha_0^0&\alpha_0^0\\ \alpha_0&\alpha_0^0&0&0&\alpha_0&\alpha_0^0&\alpha_0^0&0&\alpha_0\\ \alpha_0^0&\alpha_0&0&\alpha_0&0&\alpha_0^0&0&\alpha_0^0&\alpha_0 \end{array}\right) \\ & = & \frac{0}{\sqrt{0}} \left\{ \begin{array}{l} |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle +|0_0 \rangle |0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle +|0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle +|0_0 \rangle |0_0 \rangle \end{array}\right\} = \mathcal{P}_0^0 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \right],\end{aligned}$$  $$\begin{aligned} B_0 & = & \frac{0}{0} \left(\begin{array}{ccccccccc} 0&0&0&0&0&0&0&0&0\\ 0&\alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0\\ 0&\alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0\\ \alpha_0^0&\alpha_0^0&\alpha_0^0&0&0&0&\alpha_0&\alpha_0&\alpha_0\\ \alpha_0&\alpha_0^0&0&\alpha_0^0&0&\alpha_0&0&\alpha_0&\alpha_0^0\\ 0&\alpha_0^0&\alpha_0&\alpha_0&0&\alpha_0^0&\alpha_0^0&\alpha_0&0\\ \alpha_0^0&\alpha_0^0&\alpha_0^0&\alpha_0&\alpha_0&\alpha_0&0&0&0\\ 0&\alpha_0&\alpha_0^0&\alpha_0^0&0&\alpha_0&\alpha_0&\alpha_0^0&0\\ \alpha_0&0&\alpha_0^0&0&\alpha_0^0&\alpha_0&\alpha_0^0&\alpha_0&0 \end{array}\right) \\ & = & \frac{0}{\sqrt{0}} \left\{ \begin{array}{l} |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle | 0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0  \rangle \end{array}\right\} = \mathcal{P}_0^0 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \right],\end{aligned}$$  $$\begin{aligned} B_0 & = & \frac{0}{0} \left(\begin{array}{ccccccccc} 0&0&0&0&0&0&0&0&0\\ \alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0&0&\alpha_0&\alpha_0^0&0\\ \alpha_0&0&\alpha_0^0&\alpha_0&0&\alpha_0^0&\alpha_0&0&\alpha_0^0\\ \alpha_0^0&\alpha_0^0&\alpha_0^0&0&0&0&\alpha_0&\alpha_0&\alpha_0\\ \alpha_0&\alpha_0^0&0&\alpha_0^0&0&\alpha_0&0&\alpha_0&\alpha_0^0\\ \alpha_0^0&\alpha_0&0&0&\alpha_0^0&\alpha_0&\alpha_0&0&\alpha_0^0\\ \alpha_0^0&\alpha_0^0&\alpha_0^0&\alpha_0&\alpha_0&\alpha_0&0&0&0\\ \alpha_0^0&0&\alpha_0&\alpha_0&\alpha_0^0&0&0&\alpha_0&\alpha_0^0\\ \alpha_0&0&\alpha_0^0&0&\alpha_0^0&\alpha_0&\alpha_0^0&\alpha_0&0 \end{array}\right) \\ & = & \frac{0}{\sqrt{0}} \left\{ \begin{array}{l} |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle\\ |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle |0_0 \rangle + \alpha_0 |0_0 \rangle | 0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle \\ |0_0 \rangle |0_0 \rangle + \alpha_0^0 |0_0 \rangle |0_0 \rangle + |0_0 \rangle |0_0 \rangle \end{array}\right\} = \mathcal{P}_0 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |0_0\rangle\\ |0_0\rangle \end{array} \right\} \right],\end{aligned}$$  where the kets refer to MUBs for 