= Product decomposition of a Fourier phase state
{title2=$N^{-1/2}\sum_l e^{i\phi l}|l\rangle=\bigotimes_{j=1}^n\frac{|0\rangle+e^{i\phi2^{n-j}}|1\rangle}{\sqrt2}$}
Write the integer label in most-significant-bit-first order as $l=\sum_jb_j2^{n-j}$. The phase factors as $e^{i\phi l}=\prod_je^{i\phi b_j2^{n-j}}$, giving the displayed <product state>. For dyadic Fourier phases this explains why the Fourier basis states are unentangled even though a general <quantum Fourier transform> can create <entanglement>.
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