Product decomposition of a Fourier phase state ($N^{-1/2}{\sum }_l{e}^{i\varphi l}|l⟩={⨂}_{j=1}^n\frac{|0⟩+e^{i\varphi 2^{n-j}}|1⟩}{\sqrt{2}}$)

Write the integer label in most-significant-bit-first order as $l={\sum }_j{b}_j{2}^{n-j}$. The phase factors as $e^{i\varphi l}={\prod }_j{e}^{i\varphi b_j{2}^{n-j}}$, 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.