Weighted permutation unitarity criterion ($U|j⟩=w_j|\pi (j)⟩\text{ is unitary}⟺|w_j|=1\forall j$)

For a permutation $\pi$ of an orthonormal basis, the weighted image columns are mutually orthogonal and have squared norms $|w_j{|}^2$. Hence $U^{†}U=\mathrm{diag}(|w_1{|}^2,\dots ,|w_d{|}^2)$ 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.