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The starting point is:
<math>{\color{Blue}\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)</math> which can also be represented as the vector <math>\frac{1}{2}\bigl( \begin{smallmatrix}1\\ -1 \\ 1 \\ -1 \end{smallmatrix} \bigr)</math> where each element corresponds to states <math>\bigl( \begin{smallmatrix}|00\rangle\\|01\rangle\\|10\rangle\\|11\rangle\end{smallmatrix} \bigr)</math>.
Applying <math>f(x)</math> gives
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