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 mechanically conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for thermodynamically conjugate variables - specifically, a classical intensive parameter θ 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 θ is quantified by its quantum Fisher information mathcal{F}_θ. The framework is based on an exact integral representation that relates mathcal{F}_θ 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: Δθ overline{ΔO} ge k_BT with overline{ΔO}equivpartial_θlanglehat{O}rangle Δθ 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.