# 问题：

The Pauli matrices (Figure 2.2 on page 65) can be considered as operators with respect to an orthonormal basis$\left|0\right\rangle$, $\left|1\right\rangle$ for a two-dimensional Hilbert space. Express each of the Pauli operators in the outer product notation.

## 解答

1, $\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$，外积表示为$1|0\rangle\langle 0|+0|0\rangle\langle 1|+0|1\rangle\langle 0|+1|1\rangle\langle 1|$.

2, $\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right)$，外积表示为$0|0\rangle\langle 0|+1|0\rangle\langle 1|+1|1\rangle\langle 0|+0|1\rangle\langle 1|$.

3, $\left(\begin{array}{ll} 0 & -i \\ i & 0 \end{array}\right)$，外积表示为$0|0\rangle\langle 0|+(-i)|0\rangle\langle 1|+i|1\rangle\langle 0|+0|1\rangle\langle 1|$.

4, $\left(\begin{array}{ll} 1 & 0 \\ 0 & -1 \end{array}\right)$，外积表示为$1|0\rangle\langle 0|+0|0\rangle\langle 1|+0|1\rangle\langle 0|+(-1)|1\rangle\langle 1|$.

#### 参考

[1]www.qtumist.com