Comment on “Quantum coherence measures based on Fisher information with applications”

In their recent paper [Li et al., Phys. Rev. A 103, 012401 (2021)], the authors proposed two quantities $C_{\mathrm{F}}^D\left(\rho \right|\mathcal{K})$ in Eq. (11) and $C_{\mathrm{F}}^C\left(\rho \right|\mathcal{K})$ in Eq. (12). The authors proved that $C_{\mathrm{F}}^C\left(\rho \right|\mathcal{K})$ is a valid coherence measure in Theorem 5 and proved that $C_{\mathrm{F}}^D\left(\rho \right|\mathcal{K})=C_{\mathrm{F}}^C\left(\rho \right|\mathcal{K})$ in Proposition 4. In this Comment, we show that the proof of Proposition 4 is not justified, that is, $C_{\mathrm{F}}^D\left(\rho \right|\mathcal{K})=C_{\mathrm{F}}^C\left(\rho \right|\mathcal{K})$ is not true, in general. Consequently, whether $C_{\mathrm{F}}^D\left(\rho \right|\mathcal{K})$ is a valid coherence measure is still an open problem.