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I'm trying to understand an example from Nielsen and Chuang, on p.359. The setup is that we have some state $\rho$, an environment state $\rho_{env} = |0 \rangle \langle 0 |$, and $U$ is the CNOT gate applied with $\rho$ as the control bit and $\rho_{env}$ as the target bit.

They use the formula $$\epsilon(\rho) = tr_{env}\left[U(\rho \otimes \rho_{env})U^+ \right]$$ to conclude that in the above example we get $$\epsilon(\rho) = P_0 \rho P_0 + P_1 \rho P_1,$$ where $P_i = |i\rangle \langle i|$ for $i=0,1$. I'm having trouble seeing why this is the case. I think the idea is to "pull out" $U \rho U^+$ and "trace out" the other part, but I'm having trouble seeing how this is done since the CNOT operator requires $\rho$ as a sort of input.

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  • $\begingroup$ Wht kind of operator is $\times$ (is it supposed to be the tensor product $\otimes$, perchance?) and what do you know about it's relationship with the trace? $\endgroup$ – ACuriousMind Nov 27 '16 at 16:05
  • $\begingroup$ Yes, that is a tensor product. I edited my post. I know that $tr_B(\rho \otimes \sigma) = \rho tr(\sigma)$ where $\sigma$ is the density operator of system $B$. $\endgroup$ – theQman Nov 27 '16 at 16:13
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    $\begingroup$ Possible duplicate of Trouble with operator-sum representation of a quantum operation $\endgroup$ – Norbert Schuch Nov 30 '16 at 7:34