  We investigate an original family of quantum distinguishability problems,
where the goal is to perfectly distinguish between $M$ quantum states that
become identical under a completely decohering map. Similarly, we study
distinguishability of $M$ quantum channels that cannot be distinguished when
one is restricted to decohered input and output states. The studied problems
arise naturally in the presence of a superselection rule, allow one to quantify
the amount of information that can be encoded in phase degrees of freedom
(coherences), and are related to time-energy uncertainty relation. We present a
collection of results on both necessary and sufficient conditions for the
existence of $M$ perfectly distinguishable states (channels) that are
classically indistinguishable.
