Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 58 / 1 / b / Solution

Use most-significant-bit-first ordering, so $l={\sum }_{j=1}^n{b}_j{2}^{n-j}$ and $|l⟩=|b_1⟩\otimes \cdots \otimes |b_n⟩$. Then the coefficient factors as $e^{i{\varphi }_{m}l}={\prod }_j{e}^{i{\varphi }_m{b}_j{2}^{n-j}}$. Expanding a tensor product over all bit strings proves the product decomposition of a Fourier phase state:Each factor is a normalized one-qubit state. The whole register is therefore a product state, hence a separable state; no claim that arbitrary outputs of the Fourier transform are separable is needed. Reversing the bit convention reverses the order of these factors.