  How much of the uncertainty in predicting measurement outcomes for
non-commuting quantum observables is genuinely quantum mechanical? We provide a
natural decomposition of the total entropic uncertainty of two non-commuting
observables into a classical component, and an intrinsically quantum mechanical
component. We show that the total quantum component in a state is never lower
or upper bounded by any state-independent quantities, but instead admits
"purity-based" lower bounds that generalize entropic formulations such as the
Maassen-Uffink relation. These relations reveal a non-trivial interplay between
quantum and classical randomness in any finite-dimensional state.
