that $\M(\Phi,\Psi)$ does not depend on this representation.

To show that $\M(\Phi,\Psi)\subseteq\N(\Phi,\Psi)$ it suffices to find for any contraction $C\in\B(\H_E)$ a common Stinespring representation for $\Phi$ and $\Psi$ with the operators $\tilde{V}_{\Phi}$ and $\tilde{V}_{\Psi}$ from $\H_A$ to $\H_B\otimes\H_{\tilde{E}}$ such that $\tilde{V}^*_{\Phi}\tilde{V}_{\Psi}=V_{\Phi} ^*(I_B\otimes C)V_{\Psi}$.

Let $\H_{\tilde{E}}=\H^1_E\oplus\H^2_E$, where $\H^1_E$ and $\H^2_E$ are copies of $\H_E$. For given $C$ define the operators $\tilde{V}_{\Phi}$ and $\tilde{V}^C_{\Psi}$ from $\H_A$ into $\H_B\otimes(\H_{E_1}\oplus\H_{E_2})=\H_B\otimes\H_{E_1}\oplus\H_B\otimes\H_{E_2}$ by setting $$\label{bar-v}
\tilde{V}_{\Phi}|\varphi\rangle=V_{\Phi}|\varphi\rangle\oplus|0\rangle,\quad
\tilde{V}^C_{\Psi}|\varphi\rangle=(I_B\otimes C)V_{\Psi}|\varphi\rangle\oplus \left(I_B\otimes\sqrt{I_{E}-C^*C}\right)V_{\Psi}|\varphi\rangle$$ for any $\varphi\in\H_A$, where we assume that the operators $V_{\Phi}$ and $V_{\Psi}$ act from $\H_A$ to $\H_B\otimes \H^1_E$ and $\H_B\otimes \H^2_E$ correspondingly, while the contraction $C$ acts from $\H^2_E$ to $\H^1_E$. It is easy to see that the operators $\tilde{V}_{\Phi}$ and $\tilde{V}^C_{\Psi}$ form a common Stinespring representation for the maps $\Phi$ and $\Psi$ with the required property.

To prove that $\N(\Phi,\Psi)\subseteq\M(\Phi,\Psi)$ take any common Stinespring representation for the maps $\Phi$ and $\Psi$ with the operators $\tilde{V}_{\Phi}$ and $\tilde{V}_{\Psi}$ from $\H_A$ to $\H_B\otimes\H_{\tilde{E}}$. By Theorem 6.2.2 in [@H-SCI] there exist partial isometries $W_{\Phi}$ and $W_{\Psi}$ from $\H_E$ to $\H_{\tilde{E}}$ such that $\tilde{V}_{\Phi}=(I_B\otimes W_{\Phi})V_{\Phi}$ and $\tilde{V}_{\Psi}=(I_B\otimes W_{\Psi})V_{\Psi}$. So, $\tilde{V}^*_{\Phi}\tilde{V}_{\Psi}=V^*_{\Phi}(I_B\otimes W^*_{\Phi}W_{\Psi})V_{\Psi}\in \M(\Phi,\Psi)$, since $\|W^*_{\Phi}W_{\Psi}\|\leq 1$.

Since $\N(\Phi,\Psi)=\M(\Phi,\Psi)$, the infimum in (\[beta-e\]) can be taken over the set $\M(\Phi,\Psi)$. This implies 