1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Conditional rotation in the bloch sphere with a 2-qubit system

  1. Feb 24, 2014 #1
    1. The problem statement, all variables and given/known data
    The problem is as follows. I have two spins, [itex]m_S[/itex] and [itex]m_I[/itex]. The first spin can either be [itex]\uparrow[/itex] or [itex]\downarrow[/itex] , and the second spin can be -1, 0 or 1.
    Now, I envision the situation as the first spin being on the bloch sphere, with up up to and down at the bottom.
    What I want to do is as follows:
    Given an initial situation [itex]\left|\psi\right> = \left|\psi_1\right> \otimes \left|\psi_2\right> = \left|\uparrow\right> \otimes \frac{1}{\sqrt{3}}\left( \left|-1\right> + \left|0\right>+ \left|1\right> \right)[/itex]

    I want to rotate m_S around the x-axis by pi/2, followed by a waiting time t. In this waiting time t, I want m_S to rotate around the z-axis, conditional on the state of m_I. If m_I is -1, m_S should rotate clockwise, if it is 0, m_S should not rotate, and if it is 1, m_S should rotate anticlockwise.

    After this has happened, I want to perform another rotation, this time around the y-axis. This way, the state of m_S becomes entangled with the state m_I.

    3. The attempt at a solution

    Now, the rotation part I know how to do, as that can simply be written as

    [itex]\left|\psi\right> = R_x (\frac{\pi}{2}) \left|\psi_1\right> \otimes \left|\psi_2\right> = \frac{1}{\sqrt{2}} \left( \left|\uparrow\right> -i \left|\downarrow\right> \right) \otimes \frac{1}{\sqrt{3}}\left( \left|-1\right> + \left|0\right>+ \left|1\right> \right)[/itex]

    But here I get to the point of the conditional rotation, and I don't know how to proceed. Could anyone help me start with this?
    Last edited: Feb 24, 2014
  2. jcsd
  3. Mar 12, 2014 #2
    Distribute out the tensor product. [itex]\sum_i \alpha_i \left|i \right> \otimes \sum_j \beta_j \left|j \right>=\sum_{i,j} \alpha_i \beta_j \left|i\right> \otimes \left|j \right>[/itex]. Then apply the controlled operation to every basis state.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted