ماتریس‌های پائولی

${\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\qquad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

${\displaystyle [\sigma _{i},\,\sigma _{j}]=2i\,\epsilon _{ijk}\;\sigma _{k}}$

${\displaystyle \{\sigma _{i},\,\sigma _{j}\}=2\delta _{ij}}$

${\displaystyle \sigma _{i}^{\dagger }=\sigma _{i},\qquad (\sigma _{i})^{2}=I_{2\times 2},\qquad \det \sigma _{i}=-1,\qquad \operatorname {Tr} \sigma _{i}=0}$