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:

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.
Thanks

Meaningless without citations.
Which article - where - when - who is the author?

But I think this is just a handy notation to remove a nasty bit of math.
Z does what it is defined to do. The author is only using it by defining it.
See what happens next.

Yeah,I wasn't clear enough,sorry.
This is the link to the article:http://arxiv.org/abs/quant-ph/0108132
But the part that I mentioned,doesn't need much knowledge about the other parts of the article.
My problem is that when I apply [itex] \hat{Z}_A(\theta) [/itex] to [itex] \hat{U}_{CNOT}(\theta)(\alpha |0 \rangle +\beta|1\rangle)\otimes |\psi \rangle [/itex],there is still a relative phase between [itex] |0\rangle [/itex] and [itex] |1\rangle [/itex] which leads me to think I'm missing sth.
Also I can't get help from the article itself,because it doesn't contain calculations that I want.