  In Phys. Rev. Lett. 128, 200501 (2022) the authors consider the thermodynamic
cost of quantum metrology. One of the main results is $\mathcal{S} \geq \log(2)
\| h_\lambda \|^{-2} F_Q [\psi_\lambda]$, which purports to relate the Shannon
entropy $\mathcal{S}$ of an optimal measurement (i.e., in the basis of the
symmetric logarithmic derivative) to the quantum Fisher information $F_Q$ of
the pure state $|\psi_\lambda\rangle$. However, we show that in the setting
considered by the authors we have $\mathcal{S} = \log(2)$ and $\| h_\lambda
\|^{2} = \max_{\psi_\lambda} F_Q[\psi_\lambda]$, so that their inequality
reduces to the trivial inequality $\max_{\psi_\lambda} F_Q[\psi_\lambda] \geq
F_Q[\psi_\lambda]$, and does not in fact relate the entropy $\mathcal{S}$ to
the quantum Fisher information. Moreover, for pure state quantum metrology,
there exist optimal measurements (though not in the basis of the symmetric
logarithmic derivative) for which $0 \leq \mathcal{S} \leq \log(2)$, leading to
violations of the inequality for some states $|\psi_\lambda\rangle$.
