  It is known that the variance and entropy of quantum observables decompose
into intrinsically quantum and classical contributions. Here a general method
of constructing quantum-classical decompositions of resources such as
uncertainty is discussed, with the quantum contribution specified by a measure
of the noncommutativity of a given set of operators relative to the quantum
state, and the classical contribution generated by the mixedness of the state.
Suitable measures of noncommutativity or 'quantumness' include quantum Fisher
information, and the asymmetry of a given set, group or algebra of operators,
and are generalised to nonprojective observables and quantum channels. Strong
entropic uncertainty relations and lower bounds for R\'enyi entropies are
obtained, valid for arbitrary discrete observables, that take the mixedness of
the state into account via a classical contribution to the lower bound. These
relations can also be interpreted without reference to quantum-classical
decompositions, as tradeoff relations that bound the asymmetry of one
observable in terms of the entropy of another.
