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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

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