  We prove that majorization relations hold step by step in the Quantum Fourier
Transformation (QFT) for phase-estimation algorithms considered in the
canonical decomposition. Our result relies on the fact that states which are
mixed by Hadamard operators at any stage of the computation only differ by a
phase. This property is a consequence of the structure of the initial state and
of the QFT, based on controlled-phase operators and a single action of a
Hadamard gate per qubit. As a consequence, Hadamard gates order the probability
distribution associated to the quantum state, whereas controlled-phase
operators carry all the entanglement but are immaterial to majorization. We
also prove that majorization in phase-estimation algorithms follows in a most
natural way from unitary evolution, unlike its counterpart in Grover's
algorithm.
