Asymmetry and tighter uncertainty relations for R\'enyi entropies via quantum-classical decompositions of resource measures

Abstract

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.