  Various forms of optimality for quantum observables described as normalized
positive operator valued measures (POVMs) are studied in this paper. We give
characterizations for observables that determine the values of the measured
quantity with probabilistic certainty or a state of the system before or after
the measurement. We investigate observables which are free from noise caused by
classical post-processing, mixing, or pre-processing of quantum nature.
Especially, a complete characterization of pre-processing and post-processing
clean observables is given, and necessary and sufficient conditions are imposed
on informationally complete POVMs within the set of pure states. We also
discuss joint and sequential measurements of optimal quantum observables.
