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 $L_V$ .

解答

$(1)$说明定义的函数满足內积的三条性质：

$(A, z B)=\operatorname{tr}\left(A^{\dagger} z B\right)=\sum_{i k} A_{i k}^{\dagger} z B_{k i}=z \sum_{i k} A_{i k}^{\dagger} B_{k i}=z(A, B)$；

$(A, B)=\sum_{i k} A_{i k}^{\dagger} B_{k i}=\sum_{i k} A_{k i}^{} B_{k i}=\sum_{i k} B_{i k}^{ \dagger} A_{i i}^{}=(B, A)^{}$.

$(2)$说明定义的函数空间的维度是$d^2$；

$(3)$给出一组算子函数空间的标准正交基：

参考

[1]www.qtumist.com