Feedback can convert continuously monitored quantum jumps into a thermodynamic resource. We formulate full counting statistics for open quantum systems under unital jump feedback by incorporating the feedback maps into the tilted generator. The resulting trajectory ensemble determines both current fluctuations and the Fisher information of the measurement record. We show that feedback can enhance reservoir-parameter estimation and clock precision without necessarily changing average thermodynamic currents. This enhanced precision is not bounded by reservoir entropy production alone. By embedding the reduced dynamics in an enlarged measurement-feedback process, we derive a feedback-modified thermodynamic uncertainty relation in which the information entropy production of the feedback apparatus supplies the missing cost. A charge-monitored double quantum dot illustrates the framework: jump-conditioned feedback improves thermometry and chemical-potential sensing, and stabilizes a quantum clock defined by output-current ticks.
Feedback can convert continuously monitored quantum jumps into a thermodynamic resource. We formulate full counting statistics for open quantum systems under unital jump feedback by incorporating the feedback maps into the tilted generator. The resulting trajectory ensemble determines both current fluctuations and the Fisher information of the measurement record. We show that feedback can enhance reservoir-parameter estimation and clock precision without necessarily changing average thermodynamic currents. This enhanced precision is not bounded by reservoir entropy production alone. By embedding the reduced dynamics in an enlarged measurement-feedback process, we derive a feedback-modified thermodynamic uncertainty relation in which the information entropy production of the feedback apparatus supplies the missing cost. A charge-monitored double quantum dot illustrates the framework: jump-conditioned feedback improves thermometry and chemical-potential sensing, and stabilizes a quantum clock defined by output-current ticks.
Fluctuations are intrinsic to microscopic systems and impose fundamental limits on nonequilibrium precision, as captured by the thermodynamic uncertainty relation (TUR), which links current fluctuations to entropy production. While feedback control is expected to further suppress fluctuations, its role within the TUR framework has remained unclear, particularly in quantum systems where control is inherently information-driven. In this Letter, we consider open quantum systems weakly coupled to a thermal environment, in which quantum jumps are continuously monitored, and Markovian feedback is applied. Using quantum mutual information to quantify the information contribution induced by feedback, we derive a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information. Our results uncover how feedback control suppresses fluctuations together with thermodynamic cost and establish a fundamental precision bound imposed by information-based control. As an application, we analyze a quantum clock model and demonstrate that the clock precision can be enhanced by feedback control in the presence of a single thermal reservoir.
We derive a thermodynamic uncertainty relation (TUR) under quantum continuous measurement and feedback control. By incorporating the quantum-classical-transfer entropy, which quantifies the information gained by continuous measurement, we show that the precision of currents is constrained by information-thermodynamic costs such as the entropy production and information gain. Our result shows that information gain has the potential to enhance the precision of currents beyond the bounds set by the conventional TUR. We illustrate the bound with a driven two-level system under continuous measurement and feedback, demonstrating that feedback achieves higher precision of currents while suppressing the entropy production.
In this work, we consider a general feedback protocol based on quantum-jump detections, where the last detected jump channel is stored in a memory and subsequently used to implement a feedback action, such as modifying the system Hamiltonian conditioned on the last jump. We show that the time evolution of this general protocol can be described by a Lindblad master equation defined in a hybrid classical-quantum space, where the classical part encodes the stored measurement record (memory) and the quantum part represents the monitored system. Moreover, we show that this new representation can be used to fully characterize the counting statistics of a system subject to a general jump-based feedback protocol. We apply the formalism to a three-level system coupled to two thermal baths operating as a thermal machine, and we show that jump-based feedback can be used to convert the information obtained from the jump detections into work. Our framework provides analytical tools that enable the characterization of key statistical properties of any counting observable under jump-based feedback, such as the average current, noise, correlation functions, and power spectrum.
Kinetic uncertainty relations impose a fundamental constraint on counting precision by requiring large dynamical activity to suppress fluctuations. For N independent constituents, the activity is additive and scales as O(N), so the corresponding precision bound improves only as O(1/N). A central question is whether many-body quantum systems can instead achieve Heisenberg scaling, with relative variance suppressed as O(1/N2). Such scaling could enable substantially enhanced precision in quantum devices, including quantum clocks. Superradiance appears to provide a natural route because its collective emission activity can scale as O(N2). We show, however, that superradiance alone fails to achieve Heisenberg scaling because this enhanced activity is transient. We then demonstrate that jump-conditioned quantum feedback stabilizes the superradiant high-activity regime and converts this collective enhancement into Heisenberg scaling. We establish this mechanism analytically using a many-body kinetic uncertainty relation and mean-field analysis, and demonstrate through quantum-trajectory simulations that the actual relative counting fluctuation scales as O(1/N2).
The thermodynamic uncertainty relation posits that higher thermodynamic costs are essential for a system to function with greater precision. Recent discussions have expanded thermodynamic uncertainty relations beyond classical non-equilibrium systems, investigating how quantum characteristics can be utilized to improve precision. In this Letter, we explore how quantum feedback, a control technique used to manipulate quantum systems, can enhance the precision. Specifically, we derive a quantum thermodynamic uncertainty relation for feedback control under jump measurement, which provides the lower bound to the scaled variance of the number of jumps. We find that the presence of feedback control can increase the accuracy of continuous measured systems, which is verified with numerical simulations. Moreover, we derive a quantum thermodynamic uncertainty relation for feedback control under homodyne detection.
We introduce the coherent-incoherent correspondence as a framework for deriving quantum thermodynamic uncertainty relations under continuous measurement in Lindblad dynamics. The coherent-incoherent correspondence establishes a mapping between the original quantum system that undergoes \textit{ coherent} evolution and its corresponding \textit{incoherent} system without coherent dynamics. The coherent-incoherent correspondence relates quantities across these two systems, including jump statistics, dynamical activity, and entropy production. Since the classical-like properties of the incoherent system allow us to derive thermodynamic uncertainty relations within it, these relations can be transferred to the coherent system via the coherent-incoherent correspondence. This enables us to derive quantum thermodynamic uncertainty relations for the original coherent system. Unlike existing quantum uncertainty relations, which typically require explicit quantum correction terms, our approach avoids these additional terms. This means that we can establish a lower bound for quantum entropy production using only current statistics. This approach opens up new possibilities for inferring entropy production in quantum systems. Through numerical calculations for a model with coherent jump operators, we show that steady-state coherence lowers the bounds on precision (i.e., allows higher precision).