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Outer products of 2 qubits

  1. Apr 4, 2014 #1
    Forming the matrix representation of say 1><1 is no problem but how does one calculate the matrix representation of 11><11 ? Is it
    0 0 0 0
    0 1 0 1
    0 0 0 0
    0 1 0 1
    Any help? thanks jack
     
    Last edited: Apr 4, 2014
  2. jcsd
  3. Apr 7, 2014 #2
    i suppose 1> is (1 0)t then 11 is (1 0 0 0)t and the matrix is 1 in the upper left corner and all the rest 0.
     
  4. Apr 11, 2014 #3
    Clarification

    I take that 11>. Is (0 1 0 1 ) and < 10. Refers to ( 0 1 1 0 ) . The outer product as a matrix has more than 1 non zero entry.,so I'm still stuck. Any clarification on the correct way to do this.? Thanks
     
  5. Apr 11, 2014 #4
    Unless 11< refers to ( 0 0 0 1) and 10< refers to (0 1 0 0 ) ,01< to (0 0 1 0) Is this it??
     
  6. Apr 12, 2014 #5

    Fredrik

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    The matrix elements depend on the order of the basis vectors, so you need to choose a way to order them, e.g. (|00>,|01>,|10>,|11>). The matrix of |11><11| with respect to this ordered basis is
    \begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} This could be anticipated from the fact that |11><11| is a projection operator for a 1-dimensional subspace of the vector space, which is 4-dimensional.

    The matrix of any operator T with respect to this ordered basis is
    \begin{pmatrix}\langle 00|T|00\rangle & \langle 00|T|01\rangle &\dots & &\\ \langle 01|T|00\rangle & \langle 01|T|01\rangle & \dots \\ \vdots & \vdots & \ddots \\ \end{pmatrix}
    For more information, see the https://www.physicsforums.com/showthread.php?t=694922 [Broken] about the relationship between linear operators and matrices.
     
    Last edited by a moderator: May 6, 2017
  7. Apr 14, 2014 #6
    If my previous reply didn't register, thanks,my problem was that I didn't order the basis vectors correctly. Now I can proceed , although something bugs me about ordering of the basis, nature doesn't seem to care about humans need to order anything. Thanks
     
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