  The indistinguishability of non-orthogonal pure states lies at the heart of
quantum information processing. Although the indistinguishability reflects the
impossibility of measuring complementary physical quantities by a single
measurement, we demonstrate that the distinguishability can be perfectly
retrieved simply with the help of posterior classical partial information. We
demonstrate this by showing an ensemble of non-orthogonal pure states such that
a state randomly sampled from the ensemble can be perfectly identified by a
single measurement with help of the post-processing of the measurement outcomes
and additional partial information about the sampled state, i.e., the label of
subensemble from which the state is sampled. When an ensemble consists of two
subensembles, we show that the perfect distinguishability of the ensemble with
the help of the post-processing can be restated as a matrix-decomposition
problem. Furthermore, we give the analytical solution for the problem when both
subensembles consist of two states.
