Fundamental Precision Limits in Finite-Dimensional Quantum Thermal Machines

Enhancing the precision of a thermodynamic process inevitably comes with a thermodynamic cost. This idea was recently formalized as the thermodynamic uncertainty relation, which states that the lower bound on the relative variance of thermodynamic currents decreases as entropy production increases. Equivalently, the relation suggests that if entropy production could become arbitrarily large, the lower bound on the relative variance could approach zero. In practice, however, achieving infinitely large entropy production is impossible. This implies that physical constraints impose limits on precision that are independent of the system's dynamics. In this work, we derive fundamental precision limits, dynamics-independent bounds on the relative variance and expectation values of observables, for open quantum thermal machines with finite-dimensional systems and environments. These bounds are determined by quantities such as the Hilbert-space dimensions, energy bandwidths, and the smallest initial eigenvalue. They depend only on the initial configuration and not on the dynamics. Using a quantum battery as an example, we show that these fundamental limits reveal a trade-off between the amount of stored energy and the charging precision. We also analyze the role of quantum coherence and demonstrate that coherence can improve achievable precision. Our results thus clarify the fundamental limits on the precision of quantum thermal machines.