It has been asked by different authors whether the two classes of Schatten-$p$-norm-based functionals $C_p(rho)=min_{sigmainmathcal{I}}||rho-sigma||_p$ and $ tilde{C}_p(rho)= |rho-Deltarho|_{p}$ with $pgeq 1$ are valid coherence measures under incoherent operations, strictly incoherent operations, and genuinely incoherent operations, respectively, where $mathcal{I}$ is the set of incoherent states and $Deltarho$ is the diagonal part of density operator $rho$. Of these questions, all we know is that $C_p(rho)$ is not a valid coherence measure under incoherent operations and strictly incoherent operations, but all other aspects remain open. In this paper, we prove that (1) $tilde{C}_1(rho)$ is a valid coherence measure under both strictly incoherent operations and genuinely incoherent operations but not a valid coherence measure under incoherent operations, (2) $C_1(rho)$ is not a valid coherence measure even under genuinely incoherent operations, and (3) neither ${C}_{p>1}(rho)$ nor $tilde{C}_{p>1}(rho)$ is a valid coherence measure under any of the three sets of operations. This paper not only provides a thorough examination on the validity of taking $C_p(rho)$ and $tilde{C}_p(rho)$ as coherence measures, but also finds an example that fulfills the monotonicity under strictly incoherent operations but violates it under incoherent operations.
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