  We consider the problem of discriminating qubit states that are sent over a
quantum channel and derive a necessary and sufficient condition for an optimal
measurement to be preserved by the channel. We apply the result to the
characterization of optimal measurement preserving (OMP) channels for a given
qubit ensemble, e.g., a set of two states or a set of multiple qubit states
with equal a priori probabilities. Conversely, we also characterize qubit
ensembles for which a given channel is OMP, such as unitary and depolarization
channels. Finally, we show how the sets of OMP channels for a given ensemble
can be constructed.
