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：
The set $L_V$ of linear operators on a Hilbert space $V$ is obviously a vector space —- the sum of two linear operators is a linear operator, $zA$ is a linear operator if $A$ is a linear operator and $z$ is a complex number, and there is a zero element $0$. An important additional result is that the vector space $L_V$ can be given a natural inner product structure, turning it into a Hilbert space.
$(1)$ Show that the function $(·, ·)$ on $L_V × L_V$ defined by $(A, B) ≡ tr(A^† B)$
is an inner product function. This inner product is known as the Hilbert–Schmidt or trace inner product.
$(2)$ If $V$ has d dimensions show that $L_V$ has dimension $d^2$ .
$(3)$ Find an orthonormal basis of Hermitian matrices for the Hilbert space $L_V$ .
算子的$Hilbert–Schmidt$內积函数空间（将线性算子看做空间的元素）.
解答
$(1)$说明定义的函数满足內积的三条性质：
这里只说明$i=2$的情况，$(A, B_{1}+B_{2}) = tr(A^† (B_{1}+B_{2}))=tr(A^† B_{1})+tr(A^† B_{2})$；
$ (A, z B)=\operatorname{tr}\left(A^{\dagger} z B\right)=\sum_{i k} A_{i k}^{\dagger} z B_{k i}=z \sum_{i k} A_{i k}^{\dagger} B_{k i}=z(A, B) $；
$ (A, B)=\sum_{i k} A_{i k}^{\dagger} B_{k i}=\sum_{i k} A_{k i}^{} B_{k i}=\sum_{i k} B_{i k}^{ \dagger} A_{i i}^{}=(B, A)^{} $.
$(2)$说明定义的函数空间的维度是$d^2$；
由向量空间$V$的维度为$d$,则对一组标准正交基而言，一个算子$A$将一个向量可以映射到$n$个标准正交基上而具有$n$种情况，故共有$n^2$种情况，即有$n^2$个算子就可以组合出向量空间$V$上的任意一个算子，而算子个数不能少于$n^2$；
$(3)$给出一组算子函数空间的标准正交基：
在第$(2)$小节的基础上，先给出一组算子基为$A_{ij}=|v_{i}\rangle\langle v_{j}|$,其中$|v_{i}\rangle$是一组标准正交基，而只考虑Hermitian算子的集合的话，考虑算子函数元素的Hermitian性质，构造$ \tilde{A}{ij}=\left(A{ij}+A_{ij}^{\dagger}\right) / 2 $,则任意一个Hermitian算子都能由这些算子构造出来.
参考
[1]www.qtumist.com
参与者
作者：HKL, W65
贡献者:Dingyan, Wjw,Wxw
