discrete unbounded operator (see Def.\[D-H\]) then the topology generated by any of the norms (\[E-sn\]) on bounded subsets of $\F(A,B)$ coincides with the strong convergence topology generated by the family of seminorms $\Phi\mapsto\|\Phi(\rho)\|_1$, $\rho\in\T(\H_A)$ [@SCT Proposition 3].[^14]

Following [@CID] introduce the *energy-constrained Bures distance* $$\label{ec-b-dist}
\beta_E^G(\Phi,\Psi)=\sup_{\rho\in\S(\H_{AR}):{\mathrm{Tr}}{G}\rho_A\leq E} \beta(\Phi\otimes {\mathrm{Id}}_R(\rho),\Psi\otimes {\mathrm{Id}}_R(\rho)), \quad E>0,$$ between CP linear maps $\Phi$ and $\Psi$ from $\T(\H_A)$ to $\T(\H_B)$, where $\beta(\cdot,\cdot)$ in the r.h.s. is the Bures distance between operators in $\T_+(\H_{BR})$ defined in (\[B-d-s\]) and $\H_R$ is a separable Hilbert space.

\[ec-b-dist-r\] The infimum in (\[ec-b-dist\]) can be taken only over pure states $\rho\in\S(\H_{AR})$. This follows from the freedom of choice of $R$, which implies possibility to purify any mixed state in $\S(\H_{AR})$ by extending system $R$. We have only to note that the Bures distance between operators in $\T_{+}(\H_{XY})$ defined in (\[B-d-s\]) does not increase under partial trace: $\beta(\rho,\sigma)\geq \beta(\rho_X,\sigma_X)$ for any $\rho$ and $\sigma$ in $\T_{+}(\H_{XY})$ [@H-SCI; @Watrous; @Wilde].

The distance $\beta_E^G(\Phi,\Psi)$ turns out to be extremely useful in quantitative continuity analysis of capacities of energy-constrained infinite-dimensional quantum channels [@CID Theorem 2]. By using the well known relations between the trace norm and the Bures distance (\[B-d-s\]) one can show that for any 