Mismatched dimensions in a tensor product with CNOT

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SUMMARY

The discussion centers on the application of the CNOT gate to a quantum state represented as α|0>|00> + β|1>|00> in the context of Nielson and Chuang's "Quantum Computation." The author encounters a dimensional mismatch when applying the CNOT matrix, which is a 4x4 matrix, to an 8x1 vector. The resolution involves recognizing that the CNOT operation must be applied using the tensor product with the identity matrix to accommodate the dimensions correctly, specifically using CNOT ⊗ I or I ⊗ CNOT for the respective particles.

PREREQUISITES
  • Understanding of quantum states and notation, specifically |0>, |1>, and |00>
  • Familiarity with the CNOT gate and its matrix representation
  • Knowledge of tensor products in quantum mechanics
  • Proficiency in matrix multiplication and dimensional analysis
NEXT STEPS
  • Study the tensor product of matrices, particularly in quantum mechanics contexts
  • Learn about the CNOT gate's role in quantum circuits and its matrix representation
  • Explore the implications of dimensionality in quantum state transformations
  • Review the "drudge method" for converting quantum states to matrix form
USEFUL FOR

Quantum computing students, researchers in quantum mechanics, and anyone working with quantum gates and state transformations will benefit from this discussion.

nomadreid
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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.
 
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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