= Weighted permutation unitarity criterion
{title2=$U|j\rangle=w_j|\pi(j)\rangle\text{ is unitary}\iff |w_j|=1\ \forall j$}
For a <permutation> $\pi$ of an <orthonormal basis>, the weighted image columns are mutually <orthogonal> and have squared <norms> $|w_j|^2$. Hence $U^\dagger U=\operatorname{diag}(|w_1|^2,\ldots,|w_d|^2)$ and the displayed criterion follows. All nonzero weights suffice for invertibility, but <unitarity> requires unit-modulus weights. This describes <basis> permutations decorated by phases without introducing unwanted changes of amplitude.
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