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There's no need to increase the dimension of $\newcommand{\calH}{{\mathcal H}}\calH_E$ to $D+1$. All you care about is that $U$ must send orthonormal vectors to orthonormal vectors. What the state $|0\rangle_E$ actually represents is purely conventional. You could put an arbitrary (pure) state there, and get the same channel (as long as you also change the ...

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