Fourier transform of a basis-copying isometry ($C|\hat{j}⟩=N^{-1/2}{\sum }_{m+n=j}|\hat{m}⟩|\hat{n}⟩$)

For the normalized discrete Fourier transform basis over ${\mathbb{Z}}_N$, products of coefficient functions satisfy $f_m{f}_n=N^{-1/2}{f}_{m+n}$. Taking adjoints of the multiplication rule for the basis-copying isometry gives the displayed identity, with addition modulo $N$. Its $N$ output terms are orthonormal, so the prefactor is required by norm preservation. Each reduced quantum state is $I/N$, making the output a maximally entangled state for $N\ge 2$. This explains concretely why copying one basis is incompatible with copying its Fourier basis.