Suppose $U$ is a unitary matrix of dimensions $n\times n$; then we know ($U^{\dagger}$ is conjugate transpose):
\begin{eqnarray} \sum_k U_{nk}U_{km}^{\dagger}=\delta_{nm}. \end{eqnarray} How does this change when we modify the LHS as: \begin{eqnarray} \sum_k U_{nk}e^{i\theta_k}U_{km}^{\dagger}, \end{eqnarray} where $\theta_k\in [0,2\pi]$ ? That is, when the columns get each multiplied by one complex factor. It is not supposed to be orthogonal anymore I assume, but is there a way to derive a closed expression for the RHS in that case? Namely, how far away are we from the $\delta_{nm}$ depending on the factors $\theta_k$, and any good references to look this up?
So to reframe the question, if we're given that $UU^\dagger = I$ (where $I$ is the identity matrix), then what can be said about $UDU^\dagger$, where $D$ is diagonal with diagonal entries $e^{i \theta_1},\dots,e^{i\theta_n}$?
The answer is that $UDU^\dagger$ will be unitary, have eigenvectors equal to the columns of $U$, and eigenvalues equal to $e^{i \theta_1},\dots,e^{i\theta_n}$. The "distance" of this matrix from $I$ (going by any typical metric for matrices, i.e. any "unitarily invariant" metric) is equal to the distance between $D$ and $I$.