Uncertainty lower bounds for parameter estimations associated with a unitary family of mixed-state density matrices are obtained by embedding the space of density matrices in the Hilbert space of square-root density matrices. In the Hilbert-space setup the measure of uncertainty is given by the skew information of the second kind, while the uncertainty lower bound is given by the Wigner-Yanase skew information associated with the conjugate observable. Higher-order corrections to the uncertainty lower bound are determined by higher-order quantum skew moments; expressions for these moments are worked out in closed form.
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