complete boundedness ($cb{\hspace{1pt}}$-norm in what follows) of a linear map between the algebras $\B(\H_B)$ and $\B(\H_A)$ (cf. [@Paul]) induces (by duality) the norm $$\label{d-n-def}
\|\Phi\|_{\rm cb}\doteq\sup_{\rho\in\T(\H_{AR}),\|\rho\|_1\leq 1}\|\Phi\otimes {\mathrm{Id}}_R(\rho)\|_1$$ on the set of all linear maps between Banach spaces $\T(\H_A)$ and $\T(\H_B)$, where $\H_R$ is a separable Hilbert space and $\H_{AR}=\H_{A}\otimes \H_{R}$. If $\Phi$ is a Hermitian preserving map then the supremum in (\[d-n-def\]) can be taken over the set $\S(\H_{AR})$ [@Watrous Ch.3].

The Kretschmann-Schlingemann-Werner theorem (the KSW-theorem in what follows) obtained in states that $$\frac{\|\Phi-\Psi\|_{\rm cb}}{\sqrt{\|\Phi\|_{\rm cb}}+\sqrt{\|\Psi\|_{\rm cb}}}\leq\inf_{V_{\Phi},V_{\Psi}}\|V_{\Phi}-V_{\Psi}\|\leq\sqrt{\|\Phi-\Psi\|_{\rm cb}},$$ where the infimum is over all common Stinespring representations $$\label{c-S-r}
\Phi(\rho)={\mathrm{Tr}}_E V_{\Phi}\rho V^*_{\Phi}\quad\textrm{and}\quad\Psi(\rho)={\mathrm{Tr}}_E V_{\Psi}\rho V^*_{\Psi}.$$ In the proof of the KSW theorem it is shown that the quantity $\inf_{V_{\Phi},V_{\Psi}}\|V_{\Phi}-V_{\Psi}\|$ coincides with the *Bures distance* between the maps $\Phi$ and $\Psi$ defined by the expression $$\label{b-dist+}
\beta(\Phi,\Psi)=\sup_{\rho\in\S(\H_{AR})} \beta\!\left(\Phi\otimes {\mathrm{Id}}_R(\rho),\Psi\otimes {\mathrm{Id}}_R(\rho)\right),$$ in which $\H_R$ is a separable Hilbert space and $\beta(\cdot,\cdot)$ in the r.h.s. is the Bures distance between operators in $\T_+(\H_{BR})$ defined as $$\label{B-d-s}
  \beta(\rho,\sigma)=\sqrt{\|\rho\|_1+\|\sigma\|_1-2\sqrt{F(\rho,\sigma)}},$$ where $$\label{fidelity}
  F(\rho,\sigma)=\|\sqrt{\rho}\sqrt{\sigma}\|^2_1$$ is the fidelity of the operators $\rho$ and $\sigma$ [@H-SCI; @Watrous; @Wilde]. The Bures distance between CP linear maps $\Phi$ and $\Psi$ is connected to the *operational fidelity* of these maps introduced in .

