此图片的alt属性为空;文件名为1625798919-6446d860dbbfe54.png

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

发表评论

后才能评论