Ultimate precision limit of quantum thermal machines

Abstract: Enhancing the precision of a thermodynamic process inevitably necessitates a thermodynamic cost, a principle formulated as the thermodynamic uncertainty relation. The thermodynamic uncertainty relation states that the relative variance of thermodynamic currents, calculated as the variance divided by the squared mean, decreases as entropy production increases. This means that if entropy production were allowed to become infinitely large, the relative variance could approach zero. However, it is evident that realizing infinitely large entropy production is infeasible in reality. In this Letter, we establish the ultimate limits of precision for open quantum thermal machines operating within a finite-dimensional system and environment. We derive bounds on the relative variance and the expectation of observables, applicable to any unitary evolution of the composite system. These bounds are governed by the \textit{minimum eigenvalue factor}, which serves as a maximal entropy production attainable by the quantum dynamics and parallels the concept of dynamical activity in classical stochastic thermodynamic. Additionally, we investigate how quantum coherence in the initial environmental state affects these fundamental bounds, showing that the presence of coherence can actually improve the ultimate precision limits. Our findings provide insights into fundamental limits on the precision of quantum thermal machines and the role of quantum effects in thermodynamic processes.