  Different ensembles of quantum states can have the same average nonpure
state. Distinguishing between such constructions, via different mixing
procedures of the same nonpure quantum state, is known to entail signaling. In
parallel, different superpositions of pure quantum states can lead to the same
pure state. We show that the possibility of distinguishing between such
preparations, via different interferometric setups leading to the same pure
quantum state, also implies signaling. The implication holds irrespective of
whether the distinguishing procedure is deterministic or probabilistic.
