Generalized uncertainty relation between thermodynamic variables in quantum thermodynamics

Thermodynamic equilibrium properties of a macroscopic system emerge from an interaction with a thermal bath. In the weak coupling regime, the description of thermodynamic states turns possible to describe the system in terms of its time-independent intensive and extensive variables. However, this is not an obvious task in both the classical and quantum cases when either dealing with finite systems described by a few degrees of freedom when the statistical fluctuations play an important role, i.e., not in the thermodynamic limit, or the coupling between the system and the environment is of the same order of magnitude as their energies. 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.