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In an article I'm reading, the author defines an operator as below:
[itex]
\hat{U}_{CNOT}(\theta)=\exp{(-i \theta \hat{U}_{CNOT})}=\hat{1} \cos{\theta}-i \hat{U}_{CNOT} \sin{\theta}
[/itex]
Where [itex] \hat{U}_{CNOT} [/itex] is the controlled not gate(http://en.wikipedia.org/wiki/Controlled_NOT_gate).
Then the operator is applied to a state of the form [itex] (\alpha |0\rangle+\beta|1\rangle) \otimes |\psi\rangle [/itex] and the resulting state is:
[itex]
(\alpha e^{-i \theta} |0\rangle+\beta \cos{\theta} |1\rangle)\otimes |\psi\rangle-i \beta \sin{\theta} |1\rangle \otimes (\hat{\sigma}_x |\psi\rangle)
[/itex]
where [itex]\hat{\sigma}_x=\begin{pmatrix}
0&1\\
1&0
\end{pmatrix} [/itex]
Then the author says:
Thanks
[itex]
\hat{U}_{CNOT}(\theta)=\exp{(-i \theta \hat{U}_{CNOT})}=\hat{1} \cos{\theta}-i \hat{U}_{CNOT} \sin{\theta}
[/itex]
Where [itex] \hat{U}_{CNOT} [/itex] is the controlled not gate(http://en.wikipedia.org/wiki/Controlled_NOT_gate).
Then the operator is applied to a state of the form [itex] (\alpha |0\rangle+\beta|1\rangle) \otimes |\psi\rangle [/itex] and the resulting state is:
[itex]
(\alpha e^{-i \theta} |0\rangle+\beta \cos{\theta} |1\rangle)\otimes |\psi\rangle-i \beta \sin{\theta} |1\rangle \otimes (\hat{\sigma}_x |\psi\rangle)
[/itex]
where [itex]\hat{\sigma}_x=\begin{pmatrix}
0&1\\
1&0
\end{pmatrix} [/itex]
Then the author says:
Now my problem is that I can't understand how that happens. I don't know how to work with [itex] \hat{Z}_A(\theta)\hat{U}_{CNOT}(\theta) [/itex].I'll appreciate any suggestion.The relative phase [itex] e^{i \theta} [/itex] between [itex]|0\rangle[/itex] and [itex] |1\rangle [/itex] is a complication in the calculations that will follow.To avoid this problem, instead of [itex] \hat{U}_{CNOT}(\theta) [/itex],we will use interactions of the form [itex] \hat{Z}_A(\theta) \hat{U}_{CNOT}(\theta)[/itex],where[itex] \hat{Z}_A(\theta)=(\exp{(i \hat{\sigma}_z \theta/2)}\otimes \hat{1})[/itex]...The single-bit rotation exactly undoes the extra relative phase produced by [itex] \hat{U}_{CNOT} [/itex],while changing nothing else.
Thanks