Let $ \mathcal{C} $ be the quantum channel defined by $$ \mathcal{C}(\rho)=\operatorname{Tr}_{2}\left[U(\rho \otimes|0\rangle\langle 0|) U^{\dagger}\right], $$ where $ U $ is the two-qubit unitary gate $$ U=I \otimes I+\left(e^{i t}-1\right)\left|\Psi^{-}\right\rangle\left\langle\Psi^{-}\right|, \quad t \in \mathbb{R} $$ $ \left|\Psi^{-}\right\rangle=(|0\rangle \otimes|1\rangle-|1\rangle \otimes|0\rangle) / \sqrt{2} $, and $ \operatorname{Tr}_{2} $ denotes the partial trace on the second qubit. Show that $ \mathcal{C} $ can be corrected if and only if $ t=0 $.

I wonder if there are any other ways to do this without expanding all the terms.

  • $\begingroup$ What are the Kraus operators of the channel? $\endgroup$
    – DaftWullie
    Dec 8 '21 at 7:51

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