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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
| <math>\frac{1}{2}(|0\rangle(|1\rangle - |0\rangle) + |1\rangle(|1\rangle - |0\rangle))</math>
| <math>{\color{Blue}\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)}\frac{1}{\sqrt{2}}(|1\rangle - |0\rangle)</math>
|}
In vector notation this is:
{| class="wikitable"
|-
! Function
! Type
! Output vector
! Disregard last bit
! After hadamard
|-
| <math>f(0)=0, f(1)=0</math>
| Constant
| <math>\frac{1}{2}\bigl( \begin{smallmatrix}1\\ -1 \\ 1 \\ -1 \end{smallmatrix} \bigr)</math>
| <math>\frac{1}{\sqrt{2}}\bigl( \begin{smallmatrix}1\\ 1\end{smallmatrix} \bigr)</math>
| <math>\bigl( \begin{smallmatrix}1\\ 0\end{smallmatrix} \bigr)</math>
|-
| <math>f(0)=0, f(1)=1</math>
| Balanced
| <math>\frac{1}{2}\bigl( \begin{smallmatrix}1\\ -1 \\ -1 \\ 1 \end{smallmatrix} \bigr)</math>
| <math>\frac{1}{\sqrt{2}}\bigl( \begin{smallmatrix}1\\ -1\end{smallmatrix} \bigr)</math>
| <math>\bigl( \begin{smallmatrix}0\\ 1\end{smallmatrix} \bigr)</math>
|-
| <math>f(0)=1, f(1)=0</math>
| Balanced
| <math>\frac{1}{2}\bigl( \begin{smallmatrix}-1\\ 1 \\ 1 \\ -1 \end{smallmatrix} \bigr)</math>
| <math>\frac{1}{\sqrt{2}}\bigl( \begin{smallmatrix}1\\ -1\end{smallmatrix} \bigr)</math>
| <math>\bigl( \begin{smallmatrix}0\\ 1\end{smallmatrix} \bigr)</math>
|-
| <math>f(0)=1, f(1)=1</math>
| Constant
| <math>\frac{1}{2}\bigl( \begin{smallmatrix}-1\\ 1 \\ -1 \\ 1 \end{smallmatrix} \bigr)</math>
| <math>\frac{1}{\sqrt{2}}\bigl( \begin{smallmatrix}1\\ 1\end{smallmatrix} \bigr)</math>
| <math>\bigl( \begin{smallmatrix}1\\ 0\end{smallmatrix} \bigr)</math>
|}
| 