Fundamental trade-off relations, such as quantum speed limit and quantum thermodynamic uncertainty relation, describe the performance limits of quantum systems by imposing that improvements in speed or precision necessitate a substantial thermodynamic cost. Quantum feedback control, which is a pivotal technique for manipulating quantum dynamics based on measurement outcomes, is widely employed to enhance system performance. Nevertheless, its impact on these fundamental bounds remains an open question. This work elucidates this influence by establishing a theoretical framework for quantum speed limit and quantum thermodynamic uncertainty relation under a paradigmatic Markovian feedback protocol. We derive general inequalities incorporating the effects of feedback control on speed and precision. Through numerical simulations on a simple two-level system and quantum error correction, a key application of quantum feedback control, we validate our derived bounds and demonstrate that feedback control can indeed improve both speed and precision beyond those achievable limits in uncontrolled systems. Next, to elucidate the mechanism behind these improvements and the qualitative difference from uncontrolled dynamics, we analyze the governing thermodynamic costs, which are the fundamental quantities that constrain speed and precision, within a simple model. Our analysis reveals that feedback can improve the time scaling order of these costs. This modification of the dynamical scaling is the origin of the qualitative performance gain, signifying that the feedback-induced improvements of performance are not merely quantitative but represent a fundamental shift. Consequently, our work offers a comprehensive understanding of how feedback control impacts the fundamental limits on the speed and precision of quantum systems, providing crucial insights for designing high-performance quantum technologies.
Fundamental trade-off relations, such as quantum speed limit and quantum thermodynamic uncertainty relation, describe the performance limits of quantum systems by imposing that improvements in speed or precision necessitate a substantial thermodynamic cost. Quantum feedback control, which is a pivotal technique for manipulating quantum dynamics based on measurement outcomes, is widely employed to enhance system performance. Nevertheless, its impact on these fundamental bounds remains an open question. This work elucidates this influence by establishing a theoretical framework for quantum speed limit and quantum thermodynamic uncertainty relation under a paradigmatic Markovian feedback protocol. We derive general inequalities incorporating the effects of feedback control on speed and precision. Through numerical simulations on a simple two-level system and quantum error correction, a key application of quantum feedback control, we validate our derived bounds and demonstrate that feedback control can indeed improve both speed and precision beyond those achievable limits in uncontrolled systems. Next, to elucidate the mechanism behind these improvements and the qualitative difference from uncontrolled dynamics, we analyze the governing thermodynamic costs, which are the fundamental quantities that constrain speed and precision, within a simple model. Our analysis reveals that feedback can improve the time scaling order of these costs. This modification of the dynamical scaling is the origin of the qualitative performance gain, signifying that the feedback-induced improvements of performance are not merely quantitative but represent a fundamental shift. Consequently, our work offers a comprehensive understanding of how feedback control impacts the fundamental limits on the speed and precision of quantum systems, providing crucial insights for designing high-performance quantum technologies.
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.
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.
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.
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, energy bandwidth, and initial minimum eigenvalue, 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.
Abstract One of the most widely known building blocks of modern physics is Heisenberg’s indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit , and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach , where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.