Finite-Time Landauer Principle

Unified hierarchical relationship between thermodynamic tradeoff relations

Recent years have witnessed a surge of discoveries in the studies of thermodynamic inequalities: the thermodynamic uncertainty relation (TUR) and the entropic bound (EB) provide a lower bound on the entropy production (EP) in terms of nonequilibrium currents; the classical speed limit (CSL) expresses the lower bound on the EP using the geometry of probability distributions; the power-efficiency (PE) tradeoff dictates the maximum power achievable for a heat engine given the level of its thermal efficiency. In this study, we show that there exists a unified hierarchical structure encompassing all of these bounds, with the fundamental inequality given by an extension of the TUR (XTUR) that incorporates the most general range of currentlike and state-dependent observables. By selecting more specific observables, the TUR and the EB follow from the XTUR, and the CSL and the PE tradeoff follow from the EB. Our derivations cover both Langevin and Markov jump systems, with the first proof of the EB for the Markov jump systems and a more generalized form of the CSL. We also present concrete examples of the EB for the Markov jump systems and the generalized CSL.

Optimal control theory for maximum power of Brownian heat engines

The pursuit of achieving the maximum output power in microscopic heat engines has gained increasing attention in the field of stochastic thermodynamics. We employ the optimal control theory to study Brownian heat engines and determine the optimal heat-engine cycles in a generic damped situation, which were previously known only in the overdamped and the underdamped limits. These optimal cycles include two isothermal processes, two adiabatic processes, and an extra isochoric relaxation process at the high stiffness constraint. Our results determine the maximum output power under realistic control constraints, and also bridge the gap of the optimal cycles between the overdamped and the underdamped limits. Hence, we solve an outstanding problem in the studies of heat engines by employing the optimal control theory to stochastic thermodynamics. These findings bring valuable insights for the design of high-performance Brownian heat engines in experimental setups.

Optimal Control of Underdamped Systems: An Analytic Approach

Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback-Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.

Shortcut to finite-time memory erasure

To achieve fast computation, it is crucial to reset the memory to a desired state within a limited time. However, the inherent delay in the system's response often prevents reaching the desired state once the control process is completed in finite time. To address this challenge, we propose a shortcut strategy that incorporates an auxiliary control to guide the system towards an equilibrium state that corresponds to the intended control, thus enabling memory reset to desired accuracy regardless of the erasure speed. Through the application of thermodynamic geometry, we derive an optimal shortcut protocol for erasure processes that minimizes the energy cost. This research provides an effective design principle for realizing the finite-time erasure process to desired accuracy while simultaneously reducing the energy cost, thereby alleviating the burden of heat dissipation.

Efficiency bounds for bipartite information-thermodynamic systems

In this paper, we introduce an approach to derive a lower bound for the entropy production rate of a subsystem by utilizing the Cauchy-Schwarz inequality. It extends to establishing comprehensive upper and lower bounds for the efficiency of two subsystems. These bounds are applicable to a wide range of Markovian stochastic processes, which enhances the accuracy in depicting the range of energy conversion efficiency between subsystems. Empirical validation is conducted using a two-quantum-dot system model, which serves to confirm the effectiveness of our inequality in refining the boundaries of efficiency.

Beyond Linear Response: Equivalence between Thermodynamic Geometry and Optimal Transport

A fundamental result of thermodynamic geometry is that the optimal, minimal-work protocol that drives a nonequilibrium system between two thermodynamic states in the slow-driving limit is given by a geodesic of the friction tensor, a Riemannian metric defined on control space. For overdamped dynamics in arbitrary dimensions, we demonstrate that thermodynamic geometry is equivalent to L^{2} optimal transport geometry defined on the space of equilibrium distributions corresponding to the control parameters. We show that obtaining optimal protocols past the slow-driving or linear response regime is computationally tractable as the sum of a friction tensor geodesic and a counterdiabatic term related to the Fisher information metric. These geodesic-counterdiabatic optimal protocols are exact for parametric harmonic potentials, reproduce the surprising nonmonotonic behavior recently discovered in linearly biased double well optimal protocols, and explain the ubiquitous discontinuous jumps observed at the beginning and end times.

Qubit reset with a shortcut-to-isothermal scheme

Landauer's principle shows that the minimum energy cost to reset a classical bit in a bath with temperature T is k_{B}Tln2 in the infinite time. However, the task to reset the bit in finite time has posted a new challenge, especially for quantum bit (qubit) where both the operation time and controllability are limited. We design a shortcut-to-isothermal scheme to reset a qubit in finite time τ with limited controllability. The energy cost is minimized with the optimal control scheme with and without bound. This optimal control scheme can provide a reference to realize qubit reset with minimum energy cost for the limited time.

Thermodynamic Geometry of Nonequilibrium Fluctuations in Cyclically Driven Transport

Nonequilibrium thermal machines under cyclic driving generally outperform steady-state counterparts. However, there is still lack of coherent understanding of versatile transport and fluctuation features under time modulations. Here, we formulate a theoretical framework of thermodynamic geometry in terms of full counting statistics of nonequilibrium driven transports. We find that, besides the conventional dynamic and adiabatic geometric curvature contributions, the generating function is also divided into an additional nonadiabatic contribution, manifested as the metric term of full counting statistics. This nonadiabatic metric generalizes recent results of thermodynamic geometry in near-equilibrium entropy production to far-from-equilibrium fluctuations of general currents. Furthermore, the framework proves geometric thermodynamic uncertainty relations of near-adiabatic thermal devices, constraining fluctuations in terms of statistical metric quantities and thermodynamic length. We exemplify the theory in experimentally accessible driving-induced quantum chiral transport and Brownian heat pump.

Estimating entanglement entropy via variational quantum circuits with classical neural networks

Entropy plays a crucial role in both physics and information science, encompassing classical and quantum domains. In this paper, we present the quantum neural entropy estimator (QNEE), an approach that combines classical neural network (NN) with variational quantum circuits to estimate the von Neumann and Rényi entropies of a quantum state. QNEE provides accurate estimates of entropy while also yielding the eigenvalues and eigenstates of the input density matrix. Leveraging the capabilities of classical NN, QNEE can classify different phases of quantum systems that accompany the changes of entanglement entropy. Our numerical simulation demonstrates the effectiveness of QNEE by applying it to the 1D XXZ Heisenberg model. In particular, QNEE exhibits high sensitivity in estimating entanglement entropy near the phase transition point. We expect that QNEE will serve as a valuable tool for quantum entropy estimation and phase classification.

Generalized Landauer bound from absolute irreversibility

In this work, we introduce a generalization of the Landauer bound for erasure processes that stems from absolutely irreversible dynamics. Assuming that the erasure process is carried out in an absolutely irreversible way so that the probability of observing some trajectories is zero in the forward process but finite in the reverse process, we derive a generalized form of the bound for the average erasure work, which is valid also for imperfect erasure and asymmetric bits. The generalized bound obtained is tighter than or, at worst, as tight as existing ones. Our theoretical predictions are supported by numerical experiments and the comparison with data from previous works.

Thermodynamic trade-off relation for first passage time in resetting processes

Resetting is a strategy for boosting the speed of a target-searching process. Since its introduction over a decade ago, most studies have been carried out under the assumption that resetting takes place instantaneously. However, due to its irreversible nature, resetting processes incur a thermodynamic cost, which becomes infinite in the case of instantaneous resetting. Here, we take into consideration both the cost and the first passage time (FPT) required for a resetting process, in which the reset or return to the initial location is implemented using a trapping potential over a finite but random time period. An iterative generating function and a counting functional method à la Feynman and Kac are employed to calculate the FPT and the average work for this process. From these results, we obtain an explicit form of the time-cost trade-off relation, which provides the lower bound of the mean FPT for a given work input when the trapping potential is linear. This trade-off relation clearly shows that instantaneous resetting is achievable only when an infinite amount of work is provided. More surprisingly, the trade-off relation derived from the linear potential seems to be valid for a wide range of trapping potentials. In addition, we have also shown that the fixed-time or sharp resetting can further enhance the trade-off relation compared to that of the stochastic resetting.

How to train your demon to do fast information erasure without heat production

Time-dependent protocols that perform irreversible logical operations, such as memory erasure, cost work and produce heat, placing bounds on the efficiency of computers. Here we use a prototypical computer model of a physical memory to show that it is possible to learn feedback-control protocols to do fast memory erasure without input of work or production of heat. These protocols, which are enacted by a neural-network "demon," do not violate the second law of thermodynamics because the demon generates more heat than the memory absorbs. The result is a form of nonlocal heat exchange in which one computation is rendered energetically favorable while a compensating one produces heat elsewhere, a tactic that could be used to rationally design the flow of energy within a computer.

Adiabatic computing for optimal thermodynamic efficiency of information processing

Landauer's principle makes a strong connection between information theory and thermodynamics by stating that erasing a one-bit memory at temperature [Formula: see text] requires an average energy larger than [Formula: see text], with [Formula: see text] Boltzmann's constant. This tiny limit has been saturated in model experiments using quasistatic processes. For faster operations, an overhead proportional to the processing speed and to the memory damping appears. In this article, we show that underdamped systems are a winning strategy to reduce this extra energetic cost. We prove both experimentally and theoretically that, in the limit of vanishing dissipation mechanisms in the memory, the physical system is thermally insulated from its environment during fast erasures, i.e., fast protocols are adiabatic as no heat is exchanged with the bath. Using a fast optimal erasure protocol, we also show that these adiabatic processes produce a maximum adiabatic temperature [Formula: see text], and that Landauer's bound for fast erasures in underdamped systems becomes the adiabatic bound: [Formula: see text].

Logical and thermodynamical reversibility: Optimized experimental implementation of the not operation

The not operation is a reversible transformation acting on a 1-bit logical state and should be achievable in a physically reversible manner at no energetic cost. We experimentally demonstrate a bit-flip protocol based on the momentum of an underdamped oscillator confined in a double-well potential. The protocol is designed to be reversible in the ideal dissipationless case, and the thermodynamic work required is inversely proportional to the quality factor of the system. Our implementation demonstrates an energy dissipation significantly lower than the minimal cost of information processing in logically irreversible operations. It is, moreover, performed at high speed: A fully equilibrated final state is reached in only half a period of the oscillator. The results are supported by an analytical model that takes into account the presence of irreversibility. This Research Letter concludes with a discussion of optimization strategies.

Landauer Bound and Continuous Phase Transitions

In this review, we establish a relation between information erasure and continuous phase transitions. The order parameter, which characterizes these transitions, measures the order of the systems. It varies between 0, when the system is completely disordered, and 1, when the system is completely ordered. This ordering process can be seen as information erasure by resetting a certain number of bits to a standard value. The thermodynamic entropy in the partially ordered phase is given by the information-theoretic expression for the generalized Landauer bound in terms of error probability. We will demonstrate this for the Hopfield neural network model of associative memory, where the Landauer bound sets a lower limit for the work associated with 'remembering' rather than 'forgetting'. Using the relation between the Landauer bound and continuous phase transition, we will be able to extend the bound to analog computing systems. In the case of the erasure of an analog variable, the entropy production per degree of freedom is given by the logarithm of the configurational volume measured in units of its minimal quantum.

Dissipation during the Gating Cycle of the Bacterial Mechanosensitive Ion Channel Approaches the Landauer Limit

The Landauer principle sets a thermodynamic bound of kBT ln 2 on the energetic cost of erasing each bit of information. It holds for any memory device, regardless of its physical implementation. It was recently shown that carefully built artificial devices can attain this bound. In contrast, biological computation-like processes, e.g., DNA replication, transcription and translation use an order of magnitude more than their Landauer minimum. Here, we show that reaching the Landauer bound is nevertheless possible with biological devices. This is achieved using a mechanosensitive channel of small conductance (MscS) from E. coli as a memory bit. MscS is a fast-acting osmolyte release valve adjusting turgor pressure inside the cell. Our patch-clamp experiments and data analysis demonstrate that under a slow switching regime, the heat dissipation in the course of tension-driven gating transitions in MscS closely approaches its Landauer limit. We discuss the biological implications of this physical trait.

Geometric Brownian information engine: Essentials for the best performance

We investigate a geometric Brownian information engine (GBIE) in the presence of an error-free feedback controller that transforms the information gathered on the state of Brownian particles entrapped in monolobal geometric confinement into extractable work. Outcomes of the information engine depend on the reference measurement distance x_{m}, the feedback site x_{f}, and the transverse force G. We determine the benchmarks for utilizing the available information in an output work and the optimum operating requisites for best achievable work. Transverse bias force (G) tunes the entropic contribution in the effective potential and hence the standard deviation (σ) of the equilibrium marginal probability distribution. We recognize that the amount of extractable work reaches a global maximum when x_{f}=2x_{m} with x_{m}∼0.6σ, irrespective of the extent of the entropic limitation. Because of the higher loss of information during the relaxation process, the best achievable work of a GBIE is lower in an entropic system. The feedback regulation also bears the unidirectional passage of particles. The average displacement increases with growing entropic control and is maximum when x_{m}∼0.81σ. Finally, we explore the efficacy of the information engine, a quantity that regulates the efficiency in utilizing the information acquired. With x_{f}=2x_{m}, the maximum efficacy reduces with increasing entropic control and shows a crossover from 2 to 11/9. We discover that the condition for the best efficacy depends only on the confinement lengthscale along the feedback direction. The broader marginal probability distribution accredits the increased average displacement in a cycle and the lower efficacy in an entropy-dominated system.

Fundamental energy cost of finite-time parallelizable computing

The fundamental energy cost of irreversible computing is given by the Landauer bound of [Formula: see text]/bit, where k is the Boltzmann constant and T is the temperature in Kelvin. However, this limit is only achievable for infinite-time processes. We here determine the fundamental energy cost of finite-time parallelizable computing within the framework of nonequilibrium thermodynamics. We apply these results to quantify the energetic advantage of parallel computing over serial computing. We find that the energy cost per operation of a parallel computer can be kept close to the Landauer limit even for large problem sizes, whereas that of a serial computer fundamentally diverges. We analyze, in particular, the effects of different degrees of parallelization and amounts of overhead, as well as the influence of non-ideal electronic hardware. We further discuss their implications in the context of current technology. Our findings provide a physical basis for the design of energy-efficient computers.

Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics

Stochastic thermodynamics lays down a broad framework to revisit the venerable concepts of heat, work and entropy production for individual stochastic trajectories of mesoscopic systems. Remarkably, this approach, relying on stochastic equations of motion, introduces time into the description of thermodynamic processes-which opens the way to fine control them. As a result, the field of finite-time thermodynamics of mesoscopic systems has blossomed. In this article, after introducing a few concepts of control for isolated mechanical systems evolving according to deterministic equations of motion, we review the different strategies that have been developed to realize finite-time state-to-state transformations in both over and underdamped regimes, by the proper design of time-dependent control parameters/driving. The systems under study are stochastic, epitomized by a Brownian object immersed in a fluid; they are thus strongly coupled to their environment playing the role of a reservoir. Interestingly, a few of those methods (inverse engineering, counterdiabatic driving, fast-forward) are directly inspired by their counterpart in quantum control. The review also analyzes the control through reservoir engineering. Besides the reachability of a given target state from a known initial state, the question of the optimal path is discussed. Optimality is here defined with respect to a cost function, a subject intimately related to the field of information thermodynamics and the question of speed limit. Another natural extension discussed deals with the connection between arbitrary states or non-equilibrium steady states. This field of control in stochastic thermodynamics enjoys a wealth of applications, ranging from optimal mesoscopic heat engines to population control in biological systems.

Topological Speed Limit

Any physical system evolves at a finite speed that is constrained not only by the energetic cost but also by the topological structure of the underlying dynamics. In this Letter, by considering such structural information, we derive a unified topological speed limit for the evolution of physical states using an optimal transport approach. We prove that the minimum time required for changing states is lower bounded by the discrete Wasserstein distance, which encodes the topological information of the system, and the time-averaged velocity. The bound obtained is tight and applicable to a wide range of dynamics, from deterministic to stochastic, and classical to quantum systems. In addition, the bound provides insight into the design principles of the optimal process that attains the maximum speed. We demonstrate the application of our results to chemical reaction networks and interacting many-body quantum systems.

Minimally Dissipative Information Erasure in a Quantum Dot via Thermodynamic Length

In this Letter, we explore the use of thermodynamic length to improve the performance of experimental protocols. In particular, we implement Landauer erasure on a driven electron level in a semiconductor quantum dot, and compare the standard protocol in which the energy is increased linearly in time with the one coming from geometric optimization. The latter is obtained by choosing a suitable metric structure, whose geodesics correspond to optimal finite-time thermodynamic protocols in the slow driving regime. We show experimentally that geodesic drivings minimize dissipation for slow protocols, with a bigger improvement as one approaches perfect erasure. Moreover, the geometric approach also leads to smaller dissipation even when the time of the protocol becomes comparable with the equilibration timescale of the system, i.e., away from the slow driving regime. Our results also illustrate, in a single-electron device, a fundamental principle of thermodynamic geometry: optimal finite-time thermodynamic protocols are those with constant dissipation rate along the process.

Speed Limit for a Highly Irreversible Process and Tight Finite-Time Landauer's Bound

Landauer's bound is the minimum thermodynamic cost for erasing one bit of information. As this bound is achievable only for quasistatic processes, finite-time operation incurs additional energetic costs. We find a tight finite-time Landauer's bound by establishing a general form of the classical speed limit. This tight bound well captures the divergent behavior associated with the additional cost of a highly irreversible process, which scales differently from a nearly irreversible process. We also find an optimal dynamics which saturates the equality of the bound. We demonstrate the validity of this bound via discrete one-bit and coarse-grained bit systems. Our Letter implies that more heat dissipation than expected occurs during high-speed irreversible computation.

Minimal energy cost to initialize a bit with tolerable error

Landauer's principle imposes a fundamental limit on the energy cost to perfectly initialize a classical bit, which is only reached under the ideal operation with infinitely long time. The question on the cost in the practical operation for a bit has been raised under the constraint by the finiteness of operation time. We discover a raise-up of energy cost by L^{2}(ε)/τ from the Landaeur's limit (k_{B}Tln2) for a finite-time τ initialization of a bit with an error probability ε. The thermodynamic length L(ε) between the states before and after initializing in the parametric space increases monotonously as the error decreases. For example, in the constant dissipation coefficient (γ_{0}) case, the minimal additional cost is 0.997k_{B}T/(γ_{0}τ) for ε=1% and 1.288k_{B}T/(γ_{0}τ) for ε=0.1%. Furthermore, the optimal protocol to reach the bound of minimal energy cost is proposed for the bit initialization realized via a finite-time isothermal process.

Optimal control with a strong harmonic trap

Quadratic trapping potentials are widely used to experimentally probe biopolymers and molecular machines and drive transitions in steered molecular-dynamics simulations. Approximating energy landscapes as locally quadratic, we design multidimensional trapping protocols that minimize dissipation. The designed protocols are easily solvable and applicable to a wide range of systems. The approximation does not rely on either fast or slow limits and is valid for any duration provided the trapping potential is sufficiently strong. We demonstrate the utility of the designed protocols with a simple model of a periodically driven rotary motor. Our results elucidate principles of effective single-molecule manipulation and efficient nonequilibrium free-energy estimation.

Inferring potential landscapes from noisy trajectories of particles within an optical feedback trap

While particle trajectories encode information on their governing potentials, potentials can be challenging to robustly extract from trajectories. Measurement errors may corrupt a particle’s position, and sparse sampling of the potential limits data in higher energy regions such as barriers. We develop a Bayesian method to infer potentials from trajectories corrupted by Markovian measurement noise without assuming prior functional form on the potentials. As an alternative to Gaussian process priors over potentials, we introduce structured kernel interpolation to the Natural Sciences which allows us to extend our analysis to large datasets. Structured-Kernel-Interpolation Priors for Potential Energy Reconstruction (SKIPPER) is validated on 1D and 2D experimental trajectories for particles in a feedback trap.

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A feedback trap was used to generate noisy Langevin microbead trajectories

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The potential energy surface is recovered using a Bayesian formulation

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The formulation uses a structured-kernel-interpolation Gaussian process (SKI-GP) to tractably approximate Gaussian process regression for larger datasets

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Thanks to our adaptation of SKI-GP, we have broadened the use of Gaussian processes for natural science applications

Verification of Information Thermodynamics in a Trapped Ion System

Information thermodynamics has developed rapidly over past years, and the trapped ions, as a controllable quantum system, have demonstrated feasibility to experimentally verify the theoretical predictions in the information thermodynamics. Here, we address some representative theories of information thermodynamics, such as the quantum Landauer principle, information equality based on the two-point measurement, information-theoretical bound of irreversibility, and speed limit restrained by the entropy production of system, and review their experimental demonstration in the trapped ion system. In these schemes, the typical physical processes, such as the entropy flow, energy transfer, and information flow, build the connection between thermodynamic processes and information variation. We then elucidate the concrete quantum control strategies to simulate these processes by using quantum operators and the decay paths in the trapped-ion system. Based on them, some significantly dynamical processes in the trapped ion system to realize the newly proposed information-thermodynamic models is reviewed. Although only some latest experimental results of information thermodynamics with a single trapped-ion quantum system are reviewed here, we expect to find more exploration in the future with more ions involved in the experimental systems.

Thermodynamic Principle for Quantum Metrology

The heat dissipation in quantum metrology represents not only an unavoidable problem towards practical applications of quantum sensing devices but also a fundamental relationship between thermodynamics and quantum metrology. However, a general thermodynamic principle which governs the rule of energy consumption in quantum metrology, similar to Landauer's principle for heat dissipation in computations, has remained elusive. Here, we establish such a physical principle for energy consumption in order to achieve a certain level of measurement precision in quantum metrology, and show that it is intrinsically determined by the erasure of quantum Fisher information. The principle provides a powerful tool to investigate the advantage of quantum resources, not only in measurement precision but also in energy efficiency. It also serves as a bridge between thermodynamics and various fundamental physical concepts related in quantum physics and quantum information theory.

Inverse linear versus exponential scaling of work penalty in finite-time bit reset

A bit reset is a basic operation in irreversible computing. This costs work and dissipates energy in the computer, creating a limit on speeds and energy efficiency of future irreversible computers. It was recently shown by Zhen et al. [Phys. Rev. Lett. 127, 190602 (2021)0031-900710.1103/PhysRevLett.127.190602] that for a finite-time reset protocol, the additional work on top of the quasistatic protocol can always be minimized by considering a two-level system, and then be lower bounded through a thermodynamical speed limit. An important question is to understand under what protocol parameters, including a bit reset error and maximum energy shift, this penalty decreases exponentially vs inverse linearly in the protocol time. Here we provide several analytical results to address this question, as well as numerical simulations of specific examples of protocols.

Dynamics of Information Erasure and Extension of Landauer's Bound to Fast Processes

Using a double-well potential as a physical memory, we study with experiments and numerical simulations the energy exchanges during erasure processes, and model quantitatively the cost of fast operation. Within the stochastic thermodynamics framework we find the origins of the overhead to Landauer's bound required for fast operations: in the overdamped regime this term mainly comes from the dissipation, while in the underdamped regime it stems from the heating of the memory. Indeed, the system is thermalized with its environment at all times during quasistatic protocols, but for fast ones, the inefficient heat transfer to the thermostat is delayed with respect to the work influx, resulting in a transient temperature rise. The warming, quantitatively described by a comprehensive statistical physics description of the erasure process, is noticeable on both the kinetic and potential energy: they no longer comply with equipartition. The mean work and heat to erase the information therefore increase accordingly. They are both bounded by an effective Landauer's limit k_{B}T_{eff}ln2, where T_{eff} is a weighted average of the actual temperature of the memory during the process.

Experimental Verification of the Work Fluctuation-Dissipation Relation for Information-to-Work Conversion

We study experimentally work fluctuations in a Szilard engine that extracts work from information encoded as the occupancy of an electron level in a semiconductor quantum dot. We show that as the average work extracted per bit of information increases toward the Landauer limit k_{B}Tln2, the work fluctuations decrease in accordance with the work fluctuation-dissipation relation. We compare the results to a protocol without measurement and feedback and show that when no information is used, the work output and fluctuations vanish simultaneously, contrasting the information-to-energy conversion case where increasing amount of work is produced with decreasing fluctuations. Our study highlights the importance of fluctuations in the design of information-to-work conversion processes.

Geometric Brownian information engine: Upper bound of the achievable work under feedback control

We design a geometric Brownian information engine by considering overdamped Brownian particles inside a two-dimensional monolobal confinement with irregular width along the transport direction. Under such detention, particles experience an effective entropic potential which has a logarithmic form. We employ a feedback control protocol as an outcome of error-free position measurement. The protocol comprises three stages: measurement, feedback, and relaxation. We reposition the center of the confinement to the measurement distance (x_p) instantaneously when the position of the trapped particle crosses x_p for the first time. Then, the particle is allowed for thermal relaxation. We calculate the extractable work, total information, and unavailable information associated with the feedback control using this equilibrium probability distribution function. We find the exact analytical value of the upper bound of extractable work as (53-2ln2)k_BT. We introduce a constant force G downward to the transverse coordinate (y). A change in G alters the effective potential of the system and tunes the relative dominance of entropic and energetic contributions in it. The upper bound of the achievable work shows a crossover from (53-2ln2)k_BT to 12k_BT when the system changes from an entropy-dominated regime to an energy-dominated one. Compared to an energetic analog, the loss of information during the relaxation process is higher in the entropy-dominated region, which accredits the less value in achievable work. Theoretical predictions are in good agreement with the Langevin dynamics simulation studies.

Finite-Time Quantum Landauer Principle and Quantum Coherence

The Landauer principle states that any logically irreversible information processing must be accompanied by dissipation into the environment. In this Letter, we investigate the heat dissipation associated with finite-time information erasure and the effect of quantum coherence in such processes. By considering a scenario wherein information is encoded in an open quantum system whose dynamics are described by the Markovian Lindblad equation, we show that the dissipated heat is lower bounded by the conventional Landauer cost, as well as a correction term inversely proportional to the operational time. To clarify the relation between quantum coherence and dissipation, we derive a lower bound for heat dissipation in terms of quantum coherence. This bound quantitatively implies that the creation of quantum coherence in the energy eigenbasis during the erasure process inevitably leads to additional heat costs. The obtained bounds hold for arbitrary operational time and control protocol. By following an optimal control theory, we numerically present an optimal protocol and illustrate our findings by using a single-qubit system.

Universal Bound on Energy Cost of Bit Reset in Finite Time

We consider how the energy cost of bit reset scales with the time duration of the protocol. Bit reset necessarily takes place in finite time, where there is an extra penalty on top of the quasistatic work cost derived by Landauer. This extra energy is dissipated as heat in the computer, inducing a fundamental limit on the speed of irreversible computers. We formulate a hardware-independent expression for this limit in the framework of stochastic processes. We derive a closed-form lower bound on the work penalty as a function of the time taken for the protocol and bit reset error. It holds for discrete as well as continuous systems, assuming only that the master equation respects detailed balance.

Extrapolating the thermodynamic length with finite-time measurements

The thermodynamic length, though providing a lower bound for the excess work required in a finite-time thermodynamic process, is determined by the properties of the equilibrium states reached by the quasistatic process and is thus beyond the direct experimental measurement. We propose an experimental strategy to measure the thermodynamic length of an open classical or quantum system by extrapolating finite-time measurements. The current proposal enables the measurement of the thermodynamic length for a single control parameter without requiring extra effort to find the optimal control scheme, and is illustrated with examples of the quantum harmonic oscillator with tuning frequency and the classical ideal gas with changing volume. Such a strategy shall shed light on the experimental design of the lacking platforms to measure the thermodynamic length.

Steps minimize dissipation in rapidly driven stochastic systems

Micro- and nanoscale systems driven by rapid changes in control parameters (control protocols) dissipate significant energy. In the fast-protocol limit, we find that protocols that minimize dissipation at fixed duration are universally given by a two-step process, jumping to and from a point that balances jump size with fast relaxation. Jump protocols could be exploited by molecular machines or thermodynamic computing to improve energetic efficiency, and implemented in nonequilibrium free-energy estimation to improve accuracy.

Optimality of nonconservative driving for finite-time processes with discrete states

An optimal finite-time process drives a given initial distribution to a given final one in a given time at the lowest cost as quantified by total entropy production. We prove that for a system with discrete states this optimal process involves nonconservative driving, i.e., a genuine driving affinity, in contrast to the case of a system with continuous states. In a multicyclic network, the optimal driving affinity is bounded by the number of states within each cycle. If the driving affects forward and backwards rates nonsymmetrically, the bound additionally depends on a structural parameter characterizing this asymmetry.

Information and Thermodynamics: Fast and Precise Approach to Landauer's Bound in an Underdamped Micromechanical Oscillator

The Landauer principle states that at least k_{B}Tln2 of energy is required to erase a 1-bit memory, with k_{B}T the thermal energy of the system. We study the effects of inertia on this bound using as one-bit memory an underdamped micromechanical oscillator confined in a double-well potential created by a feedback loop. The potential barrier is precisely tunable in the few k_{B}T range. We measure, within the stochastic thermodynamic framework, the work and the heat of the erasure protocol. We demonstrate experimentally and theoretically that, in this underdamped system, the Landauer bound is reached with a 1% uncertainty, with protocols as short as 100 ms.

Quantum Fluctuations Hinder Finite-Time Information Erasure near the Landauer Limit

Information is physical but information is also processed in finite time. Where computing protocols are concerned, finite-time processing in the quantum regime can dynamically generate coherence. Here we show that this can have significant thermodynamic implications. We demonstrate that quantum coherence generated in the energy eigenbasis of a system undergoing a finite-time information erasure protocol yields rare events with extreme dissipation. These fluctuations are of purely quantum origin. By studying the full statistics of the dissipated heat in the slow-driving limit, we prove that coherence provides a non-negative contribution to all statistical cumulants. Using the simple and paradigmatic example of single bit erasure, we show that these extreme dissipation events yield distinct, experimentally distinguishable signatures.

Optimal finite-time bit erasure under full control

We study the finite-time erasure of a one-bit memory consisting of a one-dimensional double-well potential, with each well encoding a memory macrostate. We focus on setups that provide full control over the form of the potential-energy landscape and derive protocols that minimize the average work needed to erase the bit over a fixed amount of time. We allow for cases where only some of the information encoded in the bit is erased. For systems required to end up in a local-equilibrium state, we calculate the minimum amount of work needed to erase a bit explicitly, in terms of the equilibrium Boltzmann distribution corresponding to the system's initial potential. The minimum work is inversely proportional to the duration of the protocol. The erasure cost may be further reduced by relaxing the requirement for a local-equilibrium final state and allowing for any final distribution compatible with constraints on the probability to be in each memory macrostate. We also derive upper and lower bounds on the erasure cost.