  We investigate the coherence measures induced by fidelity and trace norm,
based on the recent proposed coherence quantification in [Phys. Rev. Lett. 113,
140401, 2014]. We show that the fidelity of coherence does not in general
satisfy the monotonicity requirement as a measure of coherence under the
subselection of measurements condition. We find that the trace norm of
coherence can act as a measure of coherence for qubit case and some special
class of qutrits.
