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Comment on "Thermodynamic Principle for Quantum Metrology"

In Phys. Rev. Lett. 128, 200501 (2022) the authors consider the thermodynamic cost of quantum metrology. One of the main results is $\mathcal{S} \geq \log(2) \| h_λ\|^{-2} F_Q [ψ_λ]$, 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 $|ψ_λ\rangle$. However, we show that in the setting considered by the authors we have $\mathcal{S} = \log(2)$ and $\| h_λ\|^{2} = \max_{ψ_λ} F_Q[ψ_λ]$, so that their inequality reduces to the trivial inequality $\max_{ψ_λ} F_Q[ψ_λ] \geq F_Q[ψ_λ]$, 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 $|ψ_λ\rangle$.