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\begin{document}
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%Traitement quantique de l'information II   \hfill
%EPFL Semestre de printemps 2010/2011\\
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\homeworksheet{S\'{e}rie 11\\}{}{24 Mai 2013}
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\begin{exercise}

\end{exercise}

Consid\'{e}rez les canaux bit-flip, phase-flip, bit \& phase-flip et
d\'{e}polarisant vus au cours et donner leur repr\'{e}sentation en terme de la
sph\`{e}re de Bloch.

\begin{exercise}

\end{exercise}

On consid\`{e}re le cannal d\'{e}polarisant
\[
\mathcal{E}\left(  \rho\right)  =p\frac{I}{2}+\left(  1-p\right)  \rho.
\]
Avec probabilit\'{e} $p$ le qubit est d\'{e}polaris\'{e}, c'est-\`{a}-dire
transform\'{e} dans l'\'{e}tat compl\'{e}tement m\'{e}lang\'{e} $\frac{I}{2}$,
et avec probabilit\'{e} $\left(  1-p\right)  $ le qubit reste inchang\'{e}. Le
but de cet exercice est de montrer que ce cannal est identique au cannal
suivant%
\[
\mathcal{E}\left(  \rho\right)  =\left(  1-p^{\prime}\right)  \rho
+\frac{p^{\prime}}{3}\left(  X\rho X+Y\rho Y+Z\rho Z\right)
\]
o\`{u} $p=\frac{4}{3}p^{\prime}$.

\begin{description}
\item[a)] Montrez la relation suivante, valable pour toute matrice densit\'{e}
$\rho:$%
\[
\frac{I}{2}=\frac{\rho+X\rho X+Y\rho Y+Z\rho Z}{4}.%
\]
Astuce: utilisez la repr\'{e}sentation $\rho=\frac{I}{2}+\vec{N}\cdot
\vec{\sigma}$.

\item[b)] Dans la d\'{e}finition du cannal d\'{e}polarisant, remplacez le
terme $\frac{I}{2}$ par la formule trouv\'{e}e en a) et effectuez le
changement de parametrisation $p=\frac{4}{3}p^{\prime}$.
\end{description}


\end{document}