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：
Verify the commutation relations $ [X, Y]=2 i Z ; \quad[Y, Z]=2 i X ; \quad[Z, X]=2 i Y $.
There is an elegant way of writing this using $ \epsilon_{j k l} $, the antisymmetric tensor on three indices, for which $ \epsilon_{j k l} =0$ except for $ \epsilon_{123}=\epsilon_{231}=\epsilon_{312}=1 $ and $ \epsilon_{321}=\epsilon_{213}=\epsilon_{132}=-1 $:$ \left[\sigma_{j}, \sigma_{k}\right]=2 i \sum_{l=1}^{3} \epsilon_{j k l} \sigma_{l} $.
解答
$ X Y=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{cc}0 & -i \\ i & 0\end{array}\right]=\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right]=i Z $;
$ Y X=\left[\begin{array}{rr}0 & -i \\ i & 0\end{array}\right]\left[\begin{array}{rr}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{rr}-i & 0 \\ 0 & i\end{array}\right]=-i Z $;
$ Y Z=\left[\begin{array}{rr}0 & -i \\ i & 0\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}0 & i \\ i & 0\end{array}\right]=i X $;
$ Z Y=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{rr}0 & -i \\ i & 0\end{array}\right]=\left[\begin{array}{rr}0 & -i \\ -i & 0\end{array}\right]=-i X $;
$ Z X=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]=i Y $;
$ X Z=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]=-i Y $;
令$X=\sigma_{1},Y=\sigma_{2},Z=\sigma_{3}$即可.
参考
[1]www.qtumist.com
参与者
作者：HKL, W65
贡献者:Dingyan, Wjw,Wxw
