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.
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.
For classic systems, the thermodynamic uncertainty relation (TUR) states that the fluctuations of a current have a lower bound in terms of the entropy production. Some TURs are rooted in information theory, particularly derived from relations between observations (mean and variance) and dissimilarities, such as the Kullback-Leibler divergence, which plays the role of entropy production in stochastic thermodynamics. We generalize this idea for quantum systems, where we find a lower bound for the uncertainty of quantum observables given in terms of the quantum relative entropy. We apply the result to obtain a quantum thermodynamic uncertainty relation in terms of the quantum entropy production, valid for arbitrary dynamics and nonthermal environments.
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.
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.
In quantum thermodynamics, entropy production is usually defined in terms of the quantum relative entropy between two states. We derive a lower bound for the quantum entropy production in terms of the mean and variance of quantum observables, which we refer to as a thermodynamic uncertainty relation (TUR) for the entropy production. In the absence of coherence between the states, our result reproduces classic TURs in stochastic thermodynamics. For the derivation of the TUR, we introduce a lower bound for a quantum generalization of the χ^{2} divergence between two states and discuss its implications for stochastic and quantum thermodynamics, as well as the limiting case where it reproduces the quantum Cramér-Rao inequality.
The thermodynamic uncertainty relation (TUR) is a fundamental principle in nonequilibrium thermodynamics that relates entropy production to fluctuations in a system, establishing a trade-off between the precision of an observable and the thermodynamic cost. Investigating TUR violations challenges classical thermodynamic limits, offering the potential for improved precision-entropy trade-offs, which is crucial for enhancing performance and optimization in quantum technologies. In this work, we investigate the thermodynamic uncertainty relation within a quantum collisional model, which offers the advantage of discretizing interactions into successive collisions with auxiliaries, allowing for precise tracking of dynamics and the incorporation of memory effects and non-Markovian behavior. We consider three types of dynamics in the collisional model: one is Markovian evolution, achieved by taking the continuous-time limit and imposing the stability condition, while the other two are non-Markovian dynamics—one arising from increasing the collision time between the system and the auxiliaries, and the other from incorporating interactions between the auxiliaries. For the Markovian dynamics, we examine the classical and quantum TUR bounds in the nonequilibrium steady-state regime, and also the finite-time TUR bound. We identify two distinct regimes of classical TUR violation: in some cases, the maximum violation occurs in the steady state, while in others, it is necessarily transient—appearing only at early times and vanishing with further evolution. For the two non-Markovian approaches, we find that both the degree and type of non-Markovianity crucially affect TUR violations. The second approach shows more pronounced violations during transient times, while the first approach has much stronger violations in the steady-state regime for a certain parameter window.
Thermodynamic (TUR) and kinetic (KUR) uncertainty relations are fundamental bounds constraining the fluctuations of current observables in classical, nonequilibrium systems. Several works have verified, however, violations of these classical bounds in open quantum systems, motivating the derivation of new quantum TURs and KURs that account for the role of quantum coherence. Here, we go one step further by deriving multidimensional KUR and TUR for multiple observables in open quantum systems undergoing Markovian dynamics. Our derivation exploits a multiparameter metrology approach, in which the Fisher information matrix plays a central role. Crucially, our bounds are tighter than previously derived quantum TURs and KURs for single observables, precisely because they incorporate correlations between multiple observables. We also find an intriguing quantum signature of correlations that is captured by the off-diagonal element of the Fisher information matrix, which vanishes for classical stochastic dynamics. By considering two examples, namely a coherently driven qubit system and the three-level maser, we demonstrate that the multidimensional quantum KUR bound can even be saturated when the observables are perfectly correlated.