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Statistical estimation and test of unknown channels have attracted interestof many researchers. In optimizing the process of inference, an important stepis optimization of the input state, which in general do depend on the kind ofinference (estimation or test, etc.), on the error measure, and so on. Butsometimes, there is a universally optimal input state, or an input state bestfor all the statistical inferences and for all the risk functions. In thepaper, the existence of a universally optimal state is shown for groupcovariant/contravariant channels, unital qubit channels and some measurementfamilies. To prove these results, theory of "comparison of state families" areused. We also discuss about effectiveness of entanglement and adaptation ofinput states.

When is an input state always better than the others ?: universally optimal input states for statistical inference of quantum channels