Dynamics-independent bounds on state transformations and precision in open quantum systems

We derive dynamics-independent upper bounds on achievable quantum state transformations. Modeling the evolution as a joint unitary on the system and its environment, we show that the R\'enyi divergence between the initial system state and any state reachable via the dynamics is bounded from above by a quantity determined solely by the eigenvalues of the initial system and environment density operators. As a consequence, we establish dynamics-independent lower bounds on the relative variance for arbitrary measurements, which parallel thermodynamic uncertainty relations. Moreover, we obtain dynamics- and measurement-independent lower bounds on the variance of parameter estimators. These results depend only on the initial eigenvalues of the system and environment and hold for any joint unitary, providing computable bounds for open quantum systems.