  Statistical estimation and test of unknown channels have attracted interest
of many researchers. In optimizing the process of inference, an important step
is optimization of the input state, which in general do depend on the kind of
inference (estimation or test, etc.), on the error measure, and so on. But
sometimes, there is a universally optimal input state, or an input state best
for all the statistical inferences and for all the risk functions. In the
paper, the existence of a universally optimal state is shown for group
covariant/contravariant channels, unital qubit channels and some measurement
families. To prove these results, theory of "comparison of state families" are
used. We also discuss about effectiveness of entanglement and adaptation of
input states.
