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Hint: Start by representing $\psi$ and $\rho$ in the basis $\{ \vert x\rangle,\vert y\rangle\}$. Shouldn't be too difficult to calculate the action of $U$ once you've done that. If you don't know how to take the partial trace, post the intermediate result and ask back. These steps will produce an equation like $$ A_{xy}(\psi,\rho)\vert x\rangle\langle ...



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