Thermodynamic tradeoff relations quantify the fundamental concept of "no free lunch" in the physical world, suggesting that faster and more precise physical processes come at a higher thermodynamic cost. The key elements in these trade-off relations are the thermodynamic uncertainty relation and speed limit, which are closely tied to information inequalities from which other trade-off relations are derived. Concentration inequalities are relations that complement information inequalities in statistical analyses and have been widely used in various fields. However, their role in thermodynamic trade-off relations remains unclear. This Letter develops thermodynamic concentration inequalities that provide bounds for the distribution of observables in quantum and classical Markov processes. We derive a set of trade-off relations that generalize speed limits and thermodynamic uncertainty relations from the developed thermodynamic concentration inequalities. The derived trade-off relations hold under minimal assumptions of the underlying physical processes. This Letter clarifies the role of concentration inequalities in thermodynamics, paving the way for deriving new trade-off relations.
Thermodynamic tradeoff relations quantify the fundamental concept of "no free lunch" in the physical world, suggesting that faster and more precise physical processes come at a higher thermodynamic cost. The key elements in these trade-off relations are the thermodynamic uncertainty relation and speed limit, which are closely tied to information inequalities from which other trade-off relations are derived. Concentration inequalities are relations that complement information inequalities in statistical analyses and have been widely used in various fields. However, their role in thermodynamic trade-off relations remains unclear. This Letter develops thermodynamic concentration inequalities that provide bounds for the distribution of observables in quantum and classical Markov processes. We derive a set of trade-off relations that generalize speed limits and thermodynamic uncertainty relations from the developed thermodynamic concentration inequalities. The derived trade-off relations hold under minimal assumptions of the underlying physical processes. This Letter clarifies the role of concentration inequalities in thermodynamics, paving the way for deriving new trade-off relations.
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 importance of Fisher information is increasing in nonequilibrium thermodynamics, as it has played a fundamental role in trade-off relations such as thermodynamic uncertainty relations and speed limits. In this work, we investigate temporal Fisher information, which measures the temporal information content encoded in probability distributions, for both classical and quantum systems. We establish that temporal Fisher information is bounded from above by physical costs, such as entropy production in classical Langevin and Markov processes, and the quantum entropy production and the expectation of the effective interaction Hamiltonian in open quantum systems. Conversely, temporal Fisher information is bounded from below by statistical distances (e.g., the Bhattacharyya arccos distance), leading to classical and quantum speed limits that constrain the minimal time required for state transformations. We perform numerical simulations on two quantum dot models to validate the obtained bounds. Our work provides a unified perspective on speed limits from the point of view of temporal Fisher information in both classical and quantum dynamics.
Abstract We derive new concentration bounds for time averages of measurement outcomes in quantum Markov processes. This generalizes well-known bounds for classical Markov chains, which provide constraints on finite-time fluctuations of time-additive quantities around their averages. We employ spectral, perturbation and martingale techniques, together with non-commutative L2 L 2 theory, to derive: (i) a Bernstein-type concentration bound for time averages of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type concentration bound for the same process, (iii) a generalization of the Bernstein-type concentration bound for counting processes of continuous-time quantum Markov processes, (iv) new concentration bounds for empirical fluxes of classical Markov chains which broaden the range of applicability of the corresponding classical bounds beyond empirical averages. We also suggest potential application of our results to parameter estimation and consider extensions to reducible quantum channels, multi-time statistics and time-dependent measurements, and comment on the connection to so-called thermodynamic uncertainty relations.
Traditional thermodynamic trade-off relations usually apply to quantities that depend linearly on probability distributions. In contrast, many important information-theoretic measures, such as entropies, are nonlinear and therefore difficult to analyze with existing frameworks. Motivated by replica methods in quantum information and spin-glass theory, we introduce replica Markov processes, i.e., Markovian dynamics of K independent, identical copies, and derive trade-off relations that bound relative moments of replica observables in terms of the dynamical activity. By choosing appropriate replica observables, these inequalities translate into bounds on nonlinear quantities of the original single process. Focusing on entropic measures of uncertainty, we obtain upper bounds on both the time derivative and the value of the Tsallis entropy for general trajectory-observable distributions. Moreover, we derive upper bounds on the Rényi and Tsallis entropies, where the bounds involve the initial dynamical activity. Analogous bounds can be derived for the entropies of the state distribution. In particular, we provide upper bounds on both the time derivative and the value of the Tsallis entropy of state distributions. Moreover, we show that the Rényi and Tsallis entropies of the state distribution are bounded from above by terms involving the local escape rate from the initial node. We also illustrate how replicas can be used to study other nonlinear functionals such as extreme-value observables. Finally, we extend the construction to continuously monitored open quantum systems, where the bounds are expressed in terms of the quantum dynamical activity. These results provide entropic counterparts to activity-based uncertainty relations and establish a general method for constraining nonlinear information-theoretic quantities in stochastic and quantum thermodynamics.
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 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.