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Problem with an operator

  1. May 2, 2013 #1

    ShayanJ

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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:
    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
     
  2. jcsd
  3. May 2, 2013 #2

    Simon Bridge

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    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.
     
  4. May 2, 2013 #3

    ShayanJ

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    Gold Member

    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.
     
    Last edited: May 2, 2013
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