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Since $U$ is a normal matrix, the spectral theorem applies, i.e. we can write $$ U=\sum_n\lambda_n|\lambda_n\rangle\langle\lambda_n|, $$ where $\lambda_n$ are the eigenvalues, and $|\lambda_n\rangle$ are the eigenvectors. Moreover, since $UU^\dagger=I$, we know that $|\lambda_n|^2=1$, and thus we can write $\lambda_n=e^{-i\theta_n}$ for $\theta_n$ in the ...


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