2状態系
σ ^ 1 σ ^ 2 = − σ ^ 2 σ ^ 1 = i σ ^ 3 , σ ^ 2 σ ^ 3 = − σ ^ 3 σ ^ 2 = i σ ^ 1 , σ ^ 3 σ ^ 1 = − σ ^ 1 σ ^ 3 = i σ ^ 2 {\displaystyle {\hat {\sigma }}_{1}{\hat {\sigma }}_{2}=-{\hat {\sigma }}_{2}{\hat {\sigma }}_{1}=i{\hat {\sigma }}_{3},\quad {\hat {\sigma }}_{2}{\hat {\sigma }}_{3}=-{\hat {\sigma }}_{3}{\hat {\sigma }}_{2}=i{\hat {\sigma }}_{1},\quad {\hat {\sigma }}_{3}{\hat {\sigma }}_{1}=-{\hat {\sigma }}_{1}{\hat {\sigma }}_{3}=i{\hat {\sigma }}_{2}}
を満たす。
特に状態ベクトル |1⟩, |2⟩ を特定の基底
| 1 ⟩ = ( 1 0 ) , | 2 ⟩ = ( 0 1 ) {\displaystyle |1\rangle ={\begin{pmatrix}1\\0\end{pmatrix}},\,|2\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}}
と対応させたときに、ˆσk は
σ ^ 1 = ( 0 1 1 0 ) , σ ^ 2 = ( 0 − i i 0 ) , σ ^ 3 = ( 1 0 0 − 1 ) {\displaystyle {\hat {\sigma }}_{1}={\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}{\mbox{, }}\quad {\hat {\sigma }}_{2}={\begin{pmatrix}0&-i\\i&0\\\end{pmatrix}}{\mbox{, }}\quad {\hat {\sigma }}_{3}={\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}}}