Why is the subscript like this in the equation $sum_i |psi_irangle langlepsi_i| = sum_{ijk} u_{ij} u_{ik}^{*}|phi_jrangle langlephi_k|$?

Question

In Nielsen's book when proving "Unitary freedom in the ensemble for density matrices"(Theorem 2.6):
$$text{Suppose }|widetilde{psi_i}rangle = sumlimits_{j}u_{ij} |widetilde{phi_j}rangle$$
Then in Equation 2.168:
$$ sum_i |widetilde{psi_i}rangle langlewidetilde{psi_i}| = sum_{ijk} u_{ij} u_{ik}^{*}|widetilde{phi_j}rangle langlewidetilde{phi_k}|$$
In equation 2.168 adjoint of the tilded psi has now the element in the unitary matrix u being ik conjugated($u_{ik}^*$). Now I understand that the column index after the adjoint will not be the same due to the transpose(hence k instead of j), what I don't understand is why the row index (i) is unchanged. I know it's probably something simple that I am missing, but I would appreciate your help.

Rammus · Accepted Answer

The proof begins with let $|psi_irangle = sum_j u_{ij} |varphi_jrangle$ where $U = (u_{ij})_{ij}$ is some unitary matrix. But now,
$$
begin{aligned}
|psi_irangle langle psi_i| &= left(sum_j u_{ij} |varphi_jrangle right)left(sum_k u_{ik} |varphi_krangleright)^{dagger} 
&= left(sum_j u_{ij} |varphi_jrangle right)sum_k u_{ik}^* langlevarphi_k| 
&= sum_{jk} u_{ij} u_{ik}^* |varphi_jrangle langlevarphi_k|.
end{aligned}
$$
On the first line $dagger$ denotes the hermitian conjugate (adjoint operator); on the second line we used that $dagger$ is conjugate-linear (here $u_{ik}^*$ is the complex conjugate of the complex number $u_{ik}$) and on the last line we just rearranged the sums and moved the complex numbers to the front.

Why is the subscript like this in the equation $sum_i |psi_irangle langlepsi_i| = sum_{ijk} u_{ij} u_{ik}^{*}|phi_jrangle langlephi_k|$?

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