Fundamental precision limits in finite-dimensional quantum thermal machines

Abstract

Enhancing the precision of a thermodynamic process inevitably necessitates a thermodynamic cost. This notion was recently formulated as the thermodynamic uncertainty relation, which states that the lower bound on the relative variance of thermodynamic currents decreases as entropy production increases. From another viewpoint, the thermodynamic uncertainty relation implies that if entropy production were allowed to become infinitely large, the lower bound on the relative variance could approach zero. However, it is evident that realizing infinitely large entropy production is infeasible in reality. This indicates that physical constraints impose precision limits on the system, independent of its dynamics. In this study, we derive fundamental precision limits, dynamics-independent bounds on the relative variance and the expectations of observables for open quantum thermal machines operating within a finite-dimensional system and environment. These bounds are set by quantities such as dimensions and energy bandwidth, which depend only on the initial configuration and are independent of the dynamics. Using a quantum battery model, the fundamental precision limits show that there is a trade-off between the amount of energy storage and the charging precision. Additionally, we investigate how quantum coherence affects these fundamental limits, demonstrating that the presence of coherence can improve the precision limits. Our findings provide insights into fundamental limits on the precision of quantum thermal machines.

Figures (2)

• Figure 1: Conceptual illustration of thermodynamic bounds. Consider the initial state $\rho_{SE} = \rho_S \otimes \rho_E$, where the system undergoes a unitary transformation. The large circles represent the set of all states accessible from $\rho_{SE}$ by any possible unitary transformation. (a) Thermodynamic uncertainty relations focus on a single time evolution under a specific unitary $U$. The precision is evaluated for the state $U \rho_{SE} U^\dagger$, and its lower bound is determined by the entropy production $\Sigma$, which depends on the chosen unitary $U$. As a result, this bound applies only to the particular time evolution generated by $U$. (b) Fundamental precision limits, on the other hand, consider all possible time evolutions starting from $\rho_{SE}$, that is, any state within the circle. Here, the lower bound of the precision is determined by quantities that are independent of the specific unitary transformation. Therefore, these bounds apply universally to all states within the circle.
• Figure 2: Open quantum models. (a) The basic model comprising the system $S$ and the environment $E$. The joint unitary $U$ acts on $S$ and $E$; after the interaction, $E$ is measured with a Hermitian observable $G$. Here, we assume that the initial environmental state is the Gibbs state $\gamma_E$. (b) In the quantum battery model, the battery is charged through its interaction with a charger. If we regard the charger as the system and the battery as the environment, this charging process falls within the framework of (a). (c) In the collision model, the system interacts sequentially with ancillae, and each ancilla is measured after its interaction. If we treat the ancillae as the environment, this model also falls within the framework of (a).