Response uncertainty relations in stochastic thermodynamics extend precision bounds to the sensitivity of observables under external perturbations. Here we derive a quantum response kinetic uncertainty relation for continuously monitored Markovian open quantum systems in the steady state of the Lindblad master equation. The response precision of a measured trajectory observable is bounded by two contributions: the conventional quantum dynamical activity and a perturbation-induced intersubspace transition term. The latter is absent in the classical limit and captures a genuinely quantum part of the response cost. We identify simple conditions under which either contribution vanishes, and we further clarify the structure of the intersubspace term through a symmetry-resolved decomposition and exact sector-selection rules. The bound and its structure are illustrated in a driven two-level atom.
Response uncertainty relations in stochastic thermodynamics extend precision bounds to the sensitivity of observables under external perturbations. Here we derive a quantum response kinetic uncertainty relation for continuously monitored Markovian open quantum systems in the steady state of the Lindblad master equation. The response precision of a measured trajectory observable is bounded by two contributions: the conventional quantum dynamical activity and a perturbation-induced intersubspace transition term. The latter is absent in the classical limit and captures a genuinely quantum part of the response cost. We identify simple conditions under which either contribution vanishes, and we further clarify the structure of the intersubspace term through a symmetry-resolved decomposition and exact sector-selection rules. The bound and its structure are illustrated in a driven two-level atom.
The precision and response of trajectory observables offer valuable insights into the behavior of nonequilibrium systems. For classical systems, trade-offs between these characteristics and thermodynamic costs, such as entropy production and dynamical activity, have been established through uncertainty relations. Quantum systems, however, present unique challenges, where quantum coherence can enhance precision and violate classical uncertainty relations. In this study, we derive trade-off relations for stochastic observables in Markovian open quantum systems. Specifically, we present three key results: (i) a quantum generalization of the thermokinetic uncertainty relation, which bounds the relative fluctuations of currents in terms of entropy production and dynamical activity; (ii) a quantum inverse uncertainty relation, which constrains the relative fluctuations of arbitrary counting observables based on their instantaneous fluctuations and the spectral gap of the symmetrized Liouvillian; and (iii) a quantum response kinetic uncertainty relation, which bounds the response of general observables to kinetic perturbations in terms of dynamical activity. These fundamental bounds, validated numerically using a three-level maser and a boundary-driven X X Z spin chain, provide a comprehensive framework for understanding the interplay between precision, response, and thermodynamic costs in quantum systems.
Kinetic uncertainty relations bound current precision of stochastic processes by dynamical activity. The extension of these bounds to quantum systems has been impeded by coherence, strong system-reservoir coupling, and the subtlety of defining dynamical activity in the quantum regime. Here, we introduce a partial dynamical activity through the quantum Fisher information associated with the rescaling of the system-reservoir coupling and show that it bounds current precision via a universal susceptibility-kinetic uncertainty relation. The general validity of this relation for any open quantum system is guaranteed by the natural contribution of a susceptibility term, which is experimentally accessible by tuning the system-reservoir coupling strength. We show how the partial dynamical activity encompasses previous definitions of activity in the weak-coupling Markovian limit and that it provides an information-geometric interpretation of correlator-based activities. We illustrate the tight constraint on precision that our bound provides with the example of steady-state transport through a double quantum dot, where quantum effects invalidate previously developed kinetic uncertainty relations. We expect our bound to provide a powerful tool for optimizing precision in arbitrary quantum systems.
The thermodynamic and kinetic uncertainty relations indicate trade-offs between the relative fluctuation of observables and thermodynamic quantities such as dissipation and dynamical activity. Although these relations have been well studied for classical systems, they remain largely unexplored in the quantum regime. In this Letter, we investigate such trade-off relations for Markovian open quantum systems whose underlying dynamics are quantum jumps, such as thermal processes and quantum measurement processes. Specifically, we derive finite-time lower bounds on the relative fluctuation of both dynamical observables and their first passage times for arbitrary initial states. The bounds imply that the precision of observables is constrained not only by thermodynamic quantities but also by quantum coherence. We find that the product of the relative fluctuation and entropy production or dynamical activity is enhanced by quantum coherence in a generic class of dissipative processes of systems with nondegenerate energy levels. Our findings provide insights into the survival of the classical uncertainty relations in quantum cases.
Thermodynamic uncertainty relations (TURs) and kinetic uncertainty relations (KURs) provide tradeoff relations between measurement fluctuation and thermodynamic cost, such as entropy production and activity. Conventionally, these relations are derived using the Cramér-Rao inequality, which involves an auxiliary perturbation in deterministic differential equations governing the time evolution of the system’s probability distribution. In this study, without relying on the previous formulation based on a deterministic evolving equation, we demonstrate that the main previously discovered uncertainty relations can be derived solely through the stochastic representation of the same dynamics. For this purpose, we propose a unified method based on stochastic representations for general Markovian dynamics. Extending beyond classical systems, we apply this method to Markovian open quantum systems by unraveling their dynamics, deriving quantum uncertainty relations that are tighter than existing ones in regimes where quantum effects play a significant role. This fully establishes uncertainty relations for both classical and quantum systems as intrinsic properties of their stochastic nature. Thermodynamic and kinetic uncertainty relations describe trade-offs between measurement fluctuations and thermodynamic costs, such as entropy production and activity. In this work, the authors present a general framework for deriving uncertainty relations in classical and quantum systems, leveraging the intrinsic stochastic properties of these systems.
The intuition that the precision of observables is constrained by thermodynamic costs has recently been formalized through thermodynamic and kinetic uncertainty relations. While such trade-offs have been extensively studied in Markovian systems, corresponding constraints in the non-Markovian regime remain largely unexplored. In this Letter, we derive universal bounds on the precision of generic observables in open quantum systems that interact with their environments at arbitrary coupling strengths and are subjected to two-point measurements. By introducing an asymmetry term that quantifies the disparity between forward and backward processes, we show that the relative fluctuation of any time-antisymmetric current is constrained not only by entropy production but also by this asymmetry. For general observables, we further prove that their relative fluctuation is always bounded from below by a generalized activity term that characterizes environmental changes. These results establish a comprehensive framework for understanding the fundamental limits of precision in a broad class of open quantum systems, beyond the traditional Markovian setting.
We derive a thermodynamic uncertainty relation for work in an open quantum system driven by a time-dependent protocol and subject to repeated projective energy measurements. The protocol is represented by a sequence of protocol quenches separated by finite-time Lindblad evolution, so that work is accumulated at the quenches while dissipation occurs between them. The resulting work statistics obey the Gallavotti-Cohen symmetry to give rise to the Crooks and Jarzynski fluctuation relations. By perturbing the dissipative dynamics and combining the Cramér--Rao inequality with the quantum Fisher information, we obtain VarW/[τ∂τ⟨W⟩]2≥1/F≥max(1/A,2/E), where ⟨W⟩ is the work expectation value, VarW is the variance of work, F is the Fisher information, A is the dynamical activity, E is the dimensionless entropy production, and τ is the switching time interval between quenches. The two thermodynamic bounds follow from two independent perturbations that generate the same response of the work statistics. We illustrate the relation for a dissipative two-level system under square-wave and sinusoidal driving and show that the tighter thermodynamic bound can depend on the driving protocol and switching time scale.