Response Kinetic Uncertainty Relation for Markovian Open Quantum System

Abstract: The thermodynamics and kinetics of a nonequilibrium classical system fundamentally constrain the precision of an observable regarding the celebrated thermodynamic uncertainty relation (TUR) and the kinetic uncertainty relation (KUR). They have been extended to open quantum systems obeying the Lindblad master equation where unique quantum effects are identified. Recently, a new set of principles that further incorporate the response of a classical system to an external perturbation have been discovered, named the response TUR and the response KUR (R-KUR). In this work, we generalize the classical R-KUR using the quantum Cram\'er-Rao bound to the steady state of the Lindblad master equation. The precision and the sensitivity of a measured observable are bounded from above by the conventional quantum dynamical activity and a perturbation-induced inter-subspace transition term. We provide sufficient conditions such that either of the two contributions vanishes, implying their equal importance. We verify our result and illustrate the conditions in an exactly solvable two-level atom.