  The quantification of the quantumness of a quantum ensemble has theoretical
and practical significance in quantum information theory. We propose herein a
class of measures of the quantumness of quantum ensembles using the unitary
similarity invariant norms of the commutators of the constituent density
operators of an ensemble. Rigorous proof shows that they share desirable
properties for a measure of quantumness, such as positivity, unitary
invariance, concavity under probabilistic union, convexity under state
decomposition, decreasing under coarse graining, and increasing under fine
graining. Several specific examples illustrate the applications of these
measures of quantumness in studying quantum information.
