# Problem with an operator

1. May 2, 2013

### ShayanJ

In an article I'm reading, the author defines an operator as below:
$\hat{U}_{CNOT}(\theta)=\exp{(-i \theta \hat{U}_{CNOT})}=\hat{1} \cos{\theta}-i \hat{U}_{CNOT} \sin{\theta}$
Where $\hat{U}_{CNOT}$ is the controlled not gate(http://en.wikipedia.org/wiki/Controlled_NOT_gate).
Then the operator is applied to a state of the form $(\alpha |0\rangle+\beta|1\rangle) \otimes |\psi\rangle$ and the resulting state is:
$(\alpha e^{-i \theta} |0\rangle+\beta \cos{\theta} |1\rangle)\otimes |\psi\rangle-i \beta \sin{\theta} |1\rangle \otimes (\hat{\sigma}_x |\psi\rangle)$
where $\hat{\sigma}_x=\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$
Then the author says:
Now my problem is that I can't understand how that happens. I don't know how to work with $\hat{Z}_A(\theta)\hat{U}_{CNOT}(\theta)$.I'll appreciate any suggestion.
Thanks

2. May 2, 2013

### Simon Bridge

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.

3. May 2, 2013

### ShayanJ

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 $\hat{Z}_A(\theta)$ to $\hat{U}_{CNOT}(\theta)(\alpha |0 \rangle +\beta|1\rangle)\otimes |\psi \rangle$,there is still a relative phase between $|0\rangle$ and $|1\rangle$ 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.

Last edited: May 2, 2013