Maximal-value condition of coherence measures holds for mixed states if and only if it does for pure states

While various coherence measures based on the framework for quantifying coherence [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)] have been proposed, it is often asked whether a coherence measure fulfills the maximal value condition that only maximally coherent states should achieve the maximal value of a coherence measure [Y. Peng, Y. Jiang, and H. Fan, Phys. Rev. A 93, 032326 (2016)]. Although it may be easy to rule out the measures that violate the condition by giving a counterexample, it is usually complicated to confirm the validity of the condition for a general coherence measure due to the deal with all mixed states. In this paper, we prove that the maximal value condition of coherence measures holds for mixed states if and only if it holds for a special subset of pure states. Our finding can greatly reduce the examination of whether a coherence measure fulfills the maximal value condition since one only needs to consider a subset of pure states, avoiding the work of considering mixed states.