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：
Show that the transpose, complex conjugation, and adjoint operations distribute over the tensor product,$ (A \otimes B)^{*}=A^{*} \otimes B^{*} ;(A \otimes B)^{T}=A^{T} \otimes B^{T} ;(A \otimes B)^{\dagger}=A^{\dagger} \otimes B^{\dagger} $；
解答
$ (A \otimes B)^{*}=\left[\begin{array}{ccc}a_{11} B & a_{12} B & \ldots \\ a_{21} B & a_{22} B & \ldots \\ \vdots & & \ddots\end{array}\right]^{*} =\left[\begin{array}{ccc}a_{11}^{*} B^{*} & a_{21}^{*} B^{*} & \ldots \\ a_{12}^{*} B^{*} & a_{22}^{*} B^{*} & \ldots \\ \vdots & & \ddots\end{array}\right]=A^{*} \otimes B^{*} $;
$ (A \otimes B)^{T}=\left[\begin{array}{ccc}a_{11} B & a_{12} B & \ldots \\ a_{21} B & a_{22} B & \ldots \\ \vdots & & \ddots\end{array}\right]^{T}=\left[\begin{array}{ccc}a_{11}^{T} B^{T} & a_{12}^{T} B^{T} & \ldots \\ a_{21}^{T} B^{T} & a_{22}^{T} B^{T} & \ldots \\ \vdots & & \ddots\end{array}\right]=A^{T} \otimes B^T $;
$(A \otimes B)^{\dagger}=((A \otimes B)^{T})^*=(A^{T} \otimes B^{T})^*=(A^{T})^* \otimes (A^{T})^*=A^{\dagger} \otimes B^{\dagger}$;
参考
[1]www.qtumist.com
参与者
作者：HKL, W65
贡献者:Dingyan, Wjw,Wxw
