测量单量子位的状态

泡利矩阵对应基上的期望值

⟨X⟩=⟨q∣X∣q⟩=⟨q∣0⟩⟨1∣q⟩+⟨q∣1⟩⟨0∣q⟩=2⟨q∣0⟩⟨1∣q⟩⟨Y⟩=⟨q∣Y∣q⟩=−i⟨q∣0⟩⟨1∣q⟩+i⟨q∣1⟩⟨0∣q⟩=0⟨Z⟩=⟨q∣Z∣q⟩=⟨q∣0⟩⟨0∣q⟩−⟨q∣1⟩⟨1∣q⟩=∣⟨0∣q⟩∣2−∣⟨1∣q⟩∣2\begin{aligned} \langle X \rangle &=\langle q | X | q\rangle =\langle q|0\rangle\langle 1|q\rangle + \langle q|1\rangle\langle 0|q\rangle =2\langle q |0\rangle\langle 1 | q\rangle\\ \langle Y \rangle &=\langle q | Y | q\rangle =-i\langle q|0\rangle\langle 1|q\rangle + i\langle q|1\rangle\langle 0|q\rangle =0\\ \langle Z \rangle &=\langle q | Z | q\rangle =\langle q|0\rangle\langle 0|q\rangle - \langle q|1\rangle\langle 1|q\rangle =|\langle 0 |q\rangle|^2 - |\langle 1 | q\rangle|^2 \end{aligned} \\