Controlled NOT Gate - Proof of Operation

Proof of Operation

Let \left\{ |0\rangle =\! \begin{bmatrix} 1 \\ 0 \end{bmatrix}, |1\rangle =\! \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} be the orthonormal basis (using Bra-ket notation).

Let .

Let be the flip qubit of .

Recall that |\alpha\rangle\!\otimes\!|\beta\rangle = |\alpha\rangle |\beta\rangle = |\alpha,\beta\rangle .

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