Pauli Matrix

Info

I tend to refer to this note using the plurised form Pauli Matrixes. I apologise for the absurd grammar: plurasing notes is not as easy as I’d like it to be.

Pauli matrices operate on Qubits. Pauli matrices are represented as:

${\sigma }_1={\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ ${\sigma }_2={\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$ ${\sigma }_3={\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ Their relations can be summarised by a Cayley table:

$$\begin{array}{cccc}\times & {\sigma }_x& {\sigma }_y& {\sigma }_z\\ {\sigma }_x& 1_2& i{\sigma }_{}& -i{\sigma }_y\\ {\sigma }_y& i{\sigma }_{}& 1_2& i{\sigma }_x\\ {\sigma }_z& i{\sigma }_y& -i{\sigma }_z& 1_2\end{array}$$

Which has a shorter description, with indices taken module $3$, as

• ${\sigma }_x^2={\sigma }_y^2={\sigma }_z^2=I$
• ${\sigma }_i{\sigma }_{i+1}=i{\sigma }_{i+2}$
• ${\sigma }_i{\sigma }_{i-1}=-i{\sigma }_{i-2}$