Examining the validity of Schatten-$p$-norm-based functionals as coherence measures

It has been asked by different authors whether the two classes of Schatten-$p$-norm-based functionals $C_p(\rho)=\min_{\sigma\in\mathcal{I}}||\rho-\sigma||_p$ and $\tilde{C}_p(\rho)= \|\rho-\Delta\rho\|_{p}$ with $p\geq 1$ 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 $\Delta\rho$ 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.