# 问题：

Calculate the matrix representation of the tensor products of the Pauli operators

(a) $X$ and $Z;$

(b)$I$ and $X;$

(c) $X$ and $I.$

Is the tensor product commutative?

## 解答

$X \otimes Z=\left[\begin{array}{rrrr}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0\end{array}\right]$；

$I \otimes Z=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\end{array}\right]$；

$Z \otimes I=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1\end{array}\right]$；

$BC=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]=CB$,$[A, B]=0$,它们具有对易关系；

$AC=\left[\begin{array}{rrrr}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]=CA$,$[A, C]=0$,它们具有对易关系.

#### 参考

[1]www.qtumist.com