= Fourier transform of a basis-copying isometry
{c}
{title2=$C|\widehat j\rangle=N^{-1/2}\sum_{m+n=j}|\widehat m\rangle|\widehat n\rangle$}
For the normalized <discrete Fourier transform> <basis> over $\mathbb Z_N$, products of coefficient <functions> satisfy $f_mf_n=N^{-1/2}f_{m+n}$. 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 $N\geq2$. This explains concretely why copying one <basis> is incompatible with copying its Fourier <basis>.
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