  Metric adjusted skew information, induced from quantum Fisher information, is
a well-known family of resource measures in the resource theory of asymmetry.
However, its asymptotic rates are not valid asymmetry monotone since it has an
asymptotic discontinuity. We here introduce a new class of asymmetry measures
with the smoothing technique, which we term smooth metric adjusted skew
information. We prove that its asymptotic sup- and inf-rates are valid
asymptotic measures in the resource theory of asymmetry. Furthermore, it is
proven that the smooth metric adjusted skew information rates provide a lower
bound for the coherence cost and an upper bound for the distillable coherence.
