Generalized uncertainty relation between thermodynamic variables in quantum thermodynamics

The description of thermodynamic states turns possible by using time-independent intensive and extensive variables in the weak coupling regime. However, neither classical nor quantum cases make this an obvious task when one is dealing with finite systems whose statistical fluctuations have a significant impact, i.e., not in the thermodynamic limit, or when the coupling between a system and its environment is of their energy magnitude. On the other hand, in recent years, metrology has been extended to the quantum regime in such a way that the fluctuation of physical quantities plays a crucial role in the precise estimation of parameters. Using metrology tools, it is possible to derive an uncertainty relation between the internal energy and temperature of systems for arbitrary linear coupling scales, showing an important connection between the two research fields. Our work is dedicated to the generalization of the thermodynamic uncertainty relations between an intensive and an extensive quantity for all coupling regimes in the quantum scenario through the generalized Gibbs ensemble (GGE). First, we demonstrate a fundamental limit between the intensive and extensive quantity for a total GGE state, which makes it possible to take the trace of the degrees of freedom of one of the systems and evaluate the uncertainty relation in the system of interest. After that, we performed a series of examples to corroborate the results already existing in the literature, thus showing the versatility of our method.