$E>0$ the distance $\beta_E^G(\Phi,\Psi)$ generates the same topology on bounded subsets of $\F(A,B)$ as any of the energy-constrained $cb{\hspace{1pt}}$-norms (\[E-sn\]). The results of calculation of $\beta_E^G(\Phi,\Psi)$ for real quantum channels can be found in [@Nair].

Now we can formulate the $E$-version of KSW-theorem.

\[KSW-E\] *Let ${G}$ be a positive semidefinite densely defined operator on $\H_A$ satisfying condition (\[H-cond\]) and $E>0$. Let $\|\cdot\|^G_{\mathrm{cb},E}$ and $\|\cdot\|_E^G$ be, respectively, the energy-constrained $cb{\hspace{1pt}}$-norm and the operator $E\textrm{-}$norm induced by ${G}$. For any CP linear maps $\,\Phi$ and $\,\Psi$ from $\,\T(\H_A)$ to $\,\T(\H_B)$ the following inequalities hold $$\label{KSW-rel}
 \frac{\|\Phi-\Psi\|^G_{\mathrm{cb},E}}{\sqrt{\|\Phi\|^G_{\mathrm{cb},E}}+\sqrt{\|\Psi\|^G_{\mathrm{cb},E}}}\leq\inf_{V_{\Phi},V_{\Psi}}
 \|V_{\Phi}-V_{\Psi}\|_E^G\leq\sqrt{\|\Phi-\Psi\|^G_{\mathrm{cb},E}},$$ where the infimum is over all common Stinespring representation (\[c-S-r\]). The quantity $\,\inf_{V_{\Phi},V_{\Psi}}\|V_{\Phi}-V_{\Psi}\|_E^G$ coincides with the energy-constrained Bures distance $\beta_E^G(\Phi,\Psi)$ defined in (\[ec-b-dist\]). The infimum in (\[KSW-rel\]) is attainable.*

*Proof.* We will follow the proof of the KSW theorem (given in ) with necessary modifications concerning the use of the energy-constrained $cb{\hspace{1pt}}$-norms and the operator $E$-norms (instead of the ordinary $cb{\hspace{1pt}}$-norm and the operator norm).

To prove the first inequality in (\[KSW-rel\]) assume that $\rho$ is a state in $\S(\H_{AR})$ such that ${\mathrm{Tr}}{G}\rho_A\leq E$. For a given common Stinespring representation (\[c-S-r\]) we have $$\begin{array}{c}
\|(\Phi-\Psi)\otimes {\mathrm{Id}}_R(\rho)\|_1\leq \|V_{\Phi}\otimes I_R \cdot\rho\cdot V^*_{\Phi}\otimes I_R-V_{\Psi}\otimes I_R \cdot\rho\cdot V^*_{\Psi}\otimes I_R\|_1\\\\
