Bounds on quantum Fisher information and uncertainty relations for thermodynamically conjugate variables

Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter $\theta$ and its corresponding extensive quantum operator $\hat{O}$ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter $\theta$ is quantified by its quantum Fisher information $\mathcal{F}_\theta$. The framework is based on an exact integral representation that relates $\mathcal{F}_{\theta}$ to the autocorrelation function of operator $\hat{O}$. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: $\Delta\theta\,\overline{\Delta O} \ge k_\text{B}T$ with $\overline{\Delta O}\equiv\partial_\theta\langle\hat{O}\rangle\,\Delta\theta$ and $T$ is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.