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 $\mathit{\text{mechanically}}$ conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for $\mathit{\text{thermodynamically}}$ conjugate variables – specifically, a classical intensive parameter $\theta$ and its corresponding extensive quantum operator $\widehat{O}$ – 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 $\widehat{O}$. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: $\mathrm{\Delta }\theta \overline{\mathrm{\Delta }O}\ge k_{\text{B}}T$ with $\overline{\mathrm{\Delta }O}\equiv {\mathrm{\partial }}_{\theta }⟨\widehat{O}⟩\mathrm{\Delta }\theta$ 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.