Geometric uncertainty relations on Wigner-Yanase skew information

We formulate uncertainty relations based on Wigner-Yanase skew information. By using the Kirillov-Kostant-Souriau Kahler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kahler structure plays a vital role in the proof.