在阅读该页内容之前,我们向量子计算的开创者费曼和Deutsch致敬,同时向三位量子信息学的奠基人Charles H. Bennett, David Deutsch, Peter Shor表示敬意.
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?
计算 $Pauli$矩阵的张量积的矩阵表示：
它们可对易嘛？
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] $；
令$X \otimes Z=A,I \otimes Z=B,Z \otimes I=C，AB=\left[\begin{array}{rrrr}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right],BA=\left[\begin{array}{rrrr}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0\end{array}\right]$,$\{A,B\}=0$,它们具有反对易关系；
$BC=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]=CB$,$[A, B]=0$,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
参与者
作者：HKL, W65
贡献者:Dingyan, Wjw,Wxw
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