Quantifying the quantumness of ensembles

We quantify the quantumness of quantum ensembles in terms of noncommutativity between pairs of the square root of the constituent states. We prove that this measure satisfies desirable and intuitive properties required naturally for a measure of quantumness, such as positivity, unitary invariance, subadditivity, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Some applications and implications are indicated.