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In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation: $$\varepsilon(\rho) = tr_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$

They show an example where

$$\rho_{env} = |0\rangle \langle0|$$ $$U = CNOT$$

And they claim the final solution is: $P_0\rho P_0 + P_1\rho P_1$ where $P_0$ is $|0\rangle \langle0|$ and P1 is $|1\rangle \langle 1|$.

These are my steps so far to get this, but I don't know how to trace out environment after this:

Let $\rho$ be $|\psi \rangle \langle \psi |$

So, $\rho \otimes \rho_{env} = |\psi 0\rangle \langle \psi 0|$

After applying the unitary,

$ |00 \rangle \langle 00| \psi 0 \rangle \langle \psi 0 | 00 \rangle \langle 00 |$

$ + |00 \rangle \langle 00| \psi 0 \rangle \langle \psi 0 | 10 \rangle \langle 11 | $

$+ |11 \rangle \langle 10| \psi 0 \rangle \langle \psi 0 | 00 \rangle \langle 00 |$

$ + |11 \rangle \langle 10| \psi 0 \rangle \langle \psi 0 | 10 \rangle \langle 11 |$

I don't know how to trace out environment for the above state.

Also, I realize that I have considered only a pure state, if anyone can show it for a general state that would be great.

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Let's start with a general state $$ \rho\otimes\rho_0=\sum_{x,y\in\{0,1\}}\langle x|\rho|y\rangle|x\rangle\langle y|\otimes |0\rangle\langle 0|. $$ If we apply the controlled-not, we have $$ \rightarrow\rho_{\text{final}}=\sum_{x,y\in\{0,1\}}\langle x|\rho|y\rangle|x\rangle\langle y|\otimes |x\rangle\langle y|. $$

Now we want to take the partial trace over the second subsystem. This means calculating $$ \sum_k(\mathbb{I}\otimes\langle k|)\rho_{\text{final}}(\mathbb{I}\otimes|k\rangle)=\sum_k\sum_{x,y\in\{0,1\}}\langle x|\rho|y\rangle|x\rangle\langle y|\times \langle k|x\rangle\langle y|k\rangle. $$ If we perform the sums over $x$ and $y$, we find that $x=y=k$, so $$ =\sum_k\langle k|\rho|k\rangle|k\rangle\langle k|, $$ which is entirely equivalent to removing all the off-diagonal elements of $\rho$.

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