Contents:

I always end up looking up or recalculating these things, so it’s easier to have them in one place.

Pauli matrices

\[\begin{equation} \sigma_x = \left( \begin{array}{cc} 0 && 1 \\ 1 && 0 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \sigma_y = \left( \begin{array}{cc} 0 && -i \\ i && 0 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \sigma_z = \left( \begin{array}{cc} 1 && 0 \\ 0 && -1 \\ \end{array} \right) \end{equation}\]

Eigenvectors

\[\begin{equation} \vert x_+ \rangle = \vert\leftarrow\rangle= \frac{1}{\sqrt{2}}\left( \begin{array}{c} 1 \\ 1 \\ \end{array} \right), \vert x_- \rangle = \vert\rightarrow\rangle = \frac{1}{\sqrt{2}}\left( \begin{array}{c} 1 \\ -1 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert y_+ \rangle = \frac{1}{\sqrt{2}}\left( \begin{array}{c} 1 \\ i \\ \end{array} \right), \vert y_- \rangle = \frac{1}{\sqrt{2}}\left( \begin{array}{c} 1 \\ -i \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert z_+ \rangle = \vert\uparrow\rangle = \left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right), \vert z_- \rangle = \vert\downarrow\rangle = \left( \begin{array}{c} 0 \\ 1 \\ \end{array} \right) \end{equation}\]

Density matrices for each eigenvector

\[\begin{equation} \vert x_+ \rangle\langle x_+\vert = \frac{1}{2}\left( \begin{array}{cc} 1 && 1 \\ 1 && 1 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert x_-\rangle\langle x_-\vert = \frac{1}{2}\left( \begin{array}{cc} 1 && -1 \\ -1 && 1 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert y_+\rangle\langle y_+ \vert = \frac{1}{2}\left( \begin{array}{cc} 1 && -i \\ i && 1 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert y_-\rangle\langle y_-\vert = \frac{1}{2}\left( \begin{array}{cc} 1 && i \\ -i && 1 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert z_+ \rangle\langle z_+ \vert = \left( \begin{array}{cc} 1 && 0 \\ 0 && 0 \\ \end{array} \right) \end{equation}\] \[\begin{equation} \vert z_-\rangle\langle z_-\vert = \left( \begin{array}{cc} 0 && 0 \\ 0 && 1 \\ \end{array} \right) \end{equation}\]

Two point discrete Fourier transform

This is just

\[\begin{equation} F = \frac{1}{\sqrt{2}}\left( \begin{array}{cc} 1 && 1 \\ 1 && -1 \\ \end{array} \right) \end{equation}\]