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Mismatched dimensions in a tensor product with CNOT

  1. Jan 3, 2015 #1
    I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am ignoring the constant factor here.) The result is α|0>|00> + β|1>|10>.
    But I am working this through via the drudge method, that is, converting everything to old-fashioned matrix formulation, and I run into a problem: the CNOT matrix is a 4x4 matrix, and α|0>|00> + β|1>|00> is a 8 x 1 vector. Using ordinary matrix multiplication, this is a mismatch in dimensions. It appears to me that they are saying that
    CNOT (α|0>|00>)= α|0>[CNOT (|00>)]. I am confused.
  2. jcsd
  3. Jan 8, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
  4. Jan 11, 2015 #3
    I believe I have the missing link now. Instead of purely the CNOT matrix, I need the tensor product of CNOT with the identity (4 x 4) matrix for one of the particles, and the identity tensor product with CNOT for the other one.
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