## Appendix A Proofs of Equations (22) and (23)  
Combining (20) and (21), (22) can be rewritten as  
\[\big{(}Q_{Exp}W_{1}\big{)}(y_{1})=\int_{0}^{\infty}\biggl{(}\frac{1}{\varepsilon} e^{-\frac{x}{\varepsilon}}\biggr{)}.\frac{1}{\sqrt{2\pi Hx\sigma^{2}\zeta^{2}}}e^{- \frac{(y_{1}-Hx)^{2}}{2\pi\zeta^{2}\zeta^{2}}}dx\]  
\[\xRightarrow{\frac{Hx=x^{\prime}}{\sqrt{2\pi\left(\varepsilon^{\prime}\right) ^{2}\sigma^{2}\zeta^{2}}}}\int_{0}^{\infty}\frac{1}{\varepsilon H}e^{-\frac{x ^{\prime}}{\sqrt{2\pi x^{\prime}\sigma^{2}\zeta^{2}}}}e^{-\frac{(y_{1}-x^{ \prime})^{2}}{2\pi\zeta^{2}\zeta^{2}}}dx^{\prime}\]  
\[\xRightarrow{\frac{\varepsilon H=x^{\prime}}{\sqrt{2\pi\left(\varepsilon^{ \prime}\right)^{2}\sigma^{2}\zeta^{2}}}}\int_{0}^{\infty}\left(x^{\prime} \right)^{-\frac{1}{2}}.e^{-\frac{(2x^{2}\zeta^{2}+\varepsilon^{\prime})}{2 \pi\left(\varepsilon^{\prime}\right)^{2}\sigma^{2}\zeta^{2}}}dx^{\prime}\]  
\[=\frac{1}{\sqrt{2\pi\left(\varepsilon^{\prime}\right)^{2}\sigma^{2}\zeta^{2} }}.e^{\frac{y_{1}}{\sqrt{2\pi\left(\varepsilon^{\prime}\right)^{2}\sigma^{2} \zeta^{2}}}}\int_{0}^{\infty}\left(x^{\prime}\right)^{-\frac{1}{2}}.e^{-(\frac {2x^{2}\zeta^{2}+\varepsilon^{\prime}}{2\pi\left(\varepsilon^{\prime} \right)^{2}\sigma^{2}\zeta^{2}})\cdot x^{\prime}}.e^{-\frac{y_{1}^{2}}{2 \sigma^{2}\zeta^{2}}\frac{1}{\sqrt{2\pi\left(\varepsilon^{\prime}\right)^{2} \sigma^{2}\zeta^{2}}}}dx^{\prime}. \tag{10}\]  
According to the elementary function integral formula (3.471.15) in [24], that is,  
\[\int_{0}^{\infty}\left(x\right)^{-\frac{1}{2}}.e^{-ax}.e^{-\frac{b}{\pi}}dx= \frac{\sqrt{\pi}e^{-2\sqrt{ab}}}{\sqrt{a}}(a>0,b>0), \tag{11}\]  
and after some necessary manipulation and simplification, we achieve the final expression of (22).  
The proof of (23) is based on the data processing inequality for relative entropies [23], i.e.,  
\[D(Q||Q_{Exp})\geq D(QW_{1}||Q_{Exp}W_{1}), \tag{12}\]  
where \((QW_{1})(\cdot)\) denotes the \(Y_{1}\) channel output distribution with an input distribution \(Q(\cdot)\). By the definition of the relative entropy [23], the left-hand side of (12) could be evaluated by  
\[D(Q||Q_{Exp})\triangleq\int_{-\infty}^{\infty}Q(x)log\biggl{(}\frac{Q(x)}{Q_ {Exp}(x)}\biggr{)}dx. \tag{13}\]  
By (20), (13) can be denoted as  
\[D(Q||Q_{Exp}) =-h_{Q}(x)-E_{\text{Q}}[log(\frac{1}{\varepsilon}e^{-\frac{x}{ \varepsilon}})]\] \[=-h(x)+log(\varepsilon)+1. \tag{14}\]With (22), the right-hand side of (A3) could be rewritten as  
\[D(QW_{1}\|Q_{Exp}W_{1})\triangleq\int_{-\infty}^{\infty}QW_{1}(y)log(\frac{QW_{1} (y)}{Q_{Exp}W_{1}(y)})dx\]  
\[=-h(Y_{1})+\frac{1}{2}log(\epsilon H)+\frac{1}{2}log\Big{(}\epsilon H+2\sigma^{2 }\xi^{2}\Big{)}-\frac{\epsilon H}{\sigma^{2}\xi^{2}}+\]  
\[\frac{1}{\sigma^{2}\xi^{2}}\sqrt{\frac{\epsilon H+2\sigma^{2}\xi^{2}}{\epsilon H }}E_{(QW_{1})}[|Y_{1}|]\]  
\[\geq-h(Y_{1})+\frac{1}{2}log(\epsilon H)+\frac{1}{2}log\Big{(}\epsilon H+2 \sigma^{2}\xi^{2}\Big{)}-\frac{\epsilon H}{\sigma^{2}\xi^{2}}+\frac{\sqrt{ \epsilon H(\epsilon H+2\sigma^{2}\xi^{2})}}{\sigma^{2}\xi^{2}}.\] (A6)  
Here we have used the Jensen inequality with the convex function \(|\cdot|\) to get  
\[E_{(QW_{1})}[|Y_{1}|]\geq\Big{|}E_{(QW_{1})}[Y_{1}]\Big{|}=\epsilon H.\] (A7)  
Combining (A3), (A5) with (A6) yields (23) and (24).