弱測定
| i ⟩ = 2 3 | 1 ⟩ + i 1 3 | − 1 ⟩ {\displaystyle |i\rangle ={\sqrt {\frac {2}{3}}}|1\rangle +i{\sqrt {\frac {1}{3}}}|-1\rangle }
| f ⟩ = i 2 3 e i φ | 1 ⟩ + 1 3 e − i φ | − 1 ⟩ {\displaystyle |f\rangle =i{\sqrt {\frac {2}{3}}}e^{i\phi }|1\rangle +{\sqrt {\frac {1}{3}}}e^{-i\phi }|-1\rangle }
と選ぶとき
A w = − i 2 3 e − i φ − i 1 3 e i φ − i 2 3 e − i φ + i 1 3 e i φ = 2 e − i φ + e i φ 2 e − i φ − e i φ = 3 cos  φ − i sin  φ cos  φ − i 3 sin  φ = 3 + i 8 sin  φ 1 + 8 sin 2  φ {\displaystyle A_{\mathrm {w} }={\frac {-i{\frac {2}{3}}e^{-i\phi }-i{\frac {1}{3}}e^{i\phi }}{-i{\frac {2}{3}}e^{-i\phi }+i{\frac {1}{3}}e^{i\phi }}}={\frac {2e^{-i\phi }+e^{i\phi }}{2e^{-i\phi }-e^{i\phi }}}={\frac {3\cos {\phi }-i\sin {\phi }}{\cos {\phi }-i3\sin {\phi }}}={\frac {3+i8\sin {\phi }}{1+8\sin ^{2}{\phi }}}}