Non-compression of quantum phase information

We raise a general question of quantum information theory: whether quantum phase information can be compressed and retrieved. A general qubit contains both amplitude and phase information, while an equatorial qubit contains only phase information. We study whether it is possible to compress the phase information of n equatorial qubits into m general qubits, with m being less than n, and still perfectly retrieve that information. We prove that this process is not allowed by quantum mechanics.