  We report a refinement of Robertson-Schroedinger uncertainty relation via
Wigner-Yanase skew information. Besides the well known quantum uncertainty
arising from the noncommutativity of observables, there is classical
uncertainty arising from the mixedness of the states that is quantified by the
difference between the variance and the skew information. Our refined
uncertainty relation for canonical observables is saturated by all the Gaussian
states, pure or mixed, and thus provides an alternative measure for the
non-Gaussianity of quantum states. Generalizations to the case of metric
adjusted skew information are presented, unifying and refining most of previous
results.
