Landauer's erasure principle states that the irreversible erasure of a one-bit memory, embedded in a thermal environment, is accompanied with a work input of at least $k_{\text{B}}T\ln2$. Fundamental to that principle is the assumption that the physical states representing the two possible logical states are close to thermal equilibrium. Here, we propose and theoretically analyze a minimalist mechanical model of a one-bit memory operating with squeezed thermal states. It is shown that the Landauer energy bound is exponentially lowered with increasing squeezing factor. Squeezed thermal states, which may naturally arise in digital electronic circuits operating in a pulse-driven fashion, thus can be exploited to reduce the fundamental energy costs of an erasure operation.
A known aspect of the Clausius inequality is that an equilibrium system subjected to a squeezing $\d S<0$ of its entropy must release at least an amount $|\dbarrm Q|=T|\d S|$ of heat. This serves as a basis for the Landauer principle, which puts a lower bound $T\ln 2$ for the heat generated by erasure of one bit of information. Here we show that in the world of quantum entanglement this law is broken, suggesting that quantum carriers of information can be more efficient than assumed so far.
We present a concise derivation of Landauer's erasure principle from the postulates of statistical mechanics, along with a small number of additional but uncontroversial axioms.
Landauer principle states that erasure of $N$ bit information requires an entropic cost of $Nk_B \ln 2$. This fact can easily be demonstrated by relaxation of an ideal gas consisting of $N$ particles inside a fixed volume. In this paper we discuss the modification of Landauer bound when we use non-ideal gas with inter-particle interactions for erasure procedure. We have found that the bound on the entropy production can be lowered when the interaction between particles is square-well potential.
It is widely accepted that information erasure entails heat dissipation. Here we analyze asymmetric memory states to show that this energy cost can be shuffled around to any step in a write-erase cycle and need not accompany the logically irreversible step in a computation. We discuss the resulting symmetry between logically irreversible and reversible operations, along with the implications for resolutions of Maxwell's demon paradox.
We present an experiment in which a one-bit memory is constructed, using a system of a single colloidal particle trapped in a modulated double-well potential. We measure the amount of heat dissipated to erase a bit and we establish that in the limit of long erasure cycles the mean dissipated heat saturates at the Landauer bound, i.e. the minimal quantity of heat necessarily produced to delete a classical bit of information. This result demonstrates the intimate link between information theory and thermodynamics. To stress this connection we also show that a detailed Jarzynski equality is verif…
According to Landauer's principle, erasure of information is the only part of a computation process that unavoidably involves energy dissipation. If done reversibly, such an erasure generates the minimal heat of $k_BT\ln 2$ per erased bit of information. The goal of this work is to discuss the actual reversal of the optimal erasure which can serve as the basis for the Maxwell's demon operating with ultimate thermodynamic efficiency as dictated by the second law of thermodynamics. The demon extracts $k_BT\ln 2$ of heat from an equilibrium reservoir at temperature $T$ per one bit of information …
Landauer's principle states that the erasure of information must be a dissipative process. In this paper, we carefully analyze the recording and erasure of information on a physical memory. On the one hand, we show that in order to record some information, the memory has to be driven out of equilibrium. On the other hand, we derive a differential version of Landauer's principle: We link the rate at which entropy is produced at every time of the erasure process to the rate at which information is erased.
A known aspect of the Clausius inequality is that an equilibrium system subjected to a squeezing $\d S$ of its entropy must release at least an amount $|\dbarrm Q|=T|\d S|$ of heat. This serves as a basis for the Landauer principle, which puts a lower bound $T\ln 2$ for the heat generated by erasure of one bit of information. Here we show that in the world of quantum entanglement this law is broken. A quantum Brownian particle interacting with its thermal bath can either generate less heat or even {\it adsorb} heat during an analogous squeezing process, due to entanglement with the bath. The e…
We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauer's Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauer's bound.
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 time during quasi-static protocols, but for fast ones, the…
New concepts from nonequilibrium thermodynamics are used to show that Landauer's principle can be understood in terms of time asymmetry in the dynamical randomness generated by the physical process of the erasure of digital information. In this way, Landauer's principle is generalized, showing that the dissipation associated with the erasure of a sequence of bits produces entropy at the rate $k_{\rm B}I$ per erased bit, where $I$ is Shannon's information per bit.
We study the problem of the energetic cost of information erasure by looking at it through the lens of the Jarzynski equality. We observe that the Landauer bound, $\langle W \rangle \geq kT \ln 2$, on average dissipated work $\langle W \rangle$ associated to an erasure process, literally emerges from the underlying second law bound as formulated by Kelvin, $\langle W \rangle \geq 0$, as consequence of a spontaneous breaking of the Crooks-Tasaki fluctuation-symmetry, that accompanies logical irreversibility. We illustrate and corroborate this insight with numerical simulations of the process of…
Information erasure at the molecular scale during the depolymerization of copolymers is shown to require a minimum entropy production in accordance with Landauer's principle and as a consequence of the second law of thermodynamics. This general result is illustrated with an exactly solvable model of copolymerization, which also shows that the minimum entropy production that is possible for a specific molecular mechanism of depolymerization may be larger than the minimum required by Landauer's principle.
In two recent papers, Maroney and Turgut separately and independently show generalisations of Landauer's erasure principle to indeterministic logical operations, as well as to logical states with variable energies and entropies. Here we show that, although Turgut's generalisation seems more powerful, in that it implies but is not implied by Maroney's and that it does not rely upon initial probability distributions over logical states, it does not hold for non-equilibrium states, while Maroney's generalisation holds even in non-equilibrium. While a generalisation of Turgut's inequality to non-e…
A Brownian particle in a symmetric double well potential is used as a representation for a single bit memory, where, the location of the particle in either well denotes one of the two states of a single bit memory. This article analyzes the effect of modifications to a symmetric double well potential on the minimum heat dissipation associated with erasure of the information stored in a single bit memory. Two types of modifications are considered, viz., overlap between the two wells and the asymmetry between the two wells of a bit of memory. Moreover, the analysis presented here, takes into acc…
We consider an overdamped nanoparticle in a driven double-well potential as a generic model of an erasable one-bit memory. We study in detail the statistics of the heat dissipated during an erasure process and show that full erasure may be achieved by dissipating less heat than the Landauer bound. We quantify the occurrence of such events and propose a single-particle experiment to verify our predictions. Our results show that Landauer's principle has to be generalized at the nanoscale to accommodate heat fluctuations.
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 coh…
Landauer's principle states that information erasure requires heat dissipation. Landauer's original result focused on equilibrium memories. We here investigate the reset of information stored in a nonequilibrium state of a symmetric two-state memory. We derive a nonequilibrium generalization of the erasure principle and demonstrate that the corresponding bounds on work and heat may be reduced to zero. We further introduce reset protocols that harness the initial preparation energy and entropy and so allow to reach these nonequilibrium bounds. We finally provide numerical simulations with reali…
Fooling the Landauer bound with a demon biased thermal bath
Institut de science et d’ingénierie supramoléculaires (ISIS) - UMR 7006, université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心超分子科学与工程研究所）；CNRS, ENS de Lyon, Laboratoire de Physique（法国国家科学研究中心里昂高等师范学校物理实验室）
The Landauer principle states that any logically irreversible information processing must be accompanied by dissipation into the environment. In this study, 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 …
Second law of thermodynamics can be apparently violated for systems whose dynamics depends on acquired information by measurement. However, when one consider measurement and erasure process together along with the system it saves the second law. We consider a simple example of information machine where information is used as a resource to increase its performance. The system is connected to two baths, a work source and a moving tape which is used as an information reservoir. The performance of the device is autonomous. The system acts as an engine, erasure or refrigerator. Even combination of …
Information erasure inevitably leads to heat dissipation. Minimizing this dissipation will be crucial for developing small-scale information processing systems, but little is known about the optimal procedures required. We have obtained closed-form expressions for maximally efficient erasure cycles for deletion of a classical bit of information stored by the position of a particle diffusing in a double-well potential. We find that the extra dissipation beyond the Landauer bound is proportional to the square of the Hellinger distance between the initial and final states divided by the cycle dur…
Quantum Landauer's principle provides a fundamental lower bound for energy dissipation occurred with information erasure in the quantum regime. While most studies have related the entropy reduction incorporated with the erasure to the lower bound~(entropic bound), recent efforts have also provided another lower bound associated with the thermal fluctuation of the dissipated energy~(thermodynamic bound). The coexistence of the two bounds has stimulated comparative studies of their properties; however, these studies were performed for systems where the time-evolution of diagonal (population) and…
We study the thermodynamics of a Brownian particle under the influence of a time multiplexed harmonic potential of finite width. The memory storage mechanism and the erasure protocol realized by time multiplexed potentials are utilized to experimentally realize erasure with work close to the Landauer's bound. We quantify the work done on the system with respect to the duty-ratio of time multiplexing, which also provides a handle to approach reversible erasures. A Langevin dynamics based simulation model is developed for the proposed memory bit and the erasure protocol, which guides the experim…
Landauer's erasure principle is generalized to nondeterministic processes on systems having an arbitrary number of non-symmetrical logical states. The condition that the process is applied in the same way, irrespective of the initial logical state, imposes some restrictions on the individual heat exchanges associated with each possible transition. The complete set of such restrictions are derived by a statistical analysis of the phase-space flow induced by the process. Landauer's erasure principle can be derived from and is a special case of these.
Recent analyses have calculated the minimal thermodynamic work required to perform a computation pi when two conditions hold: the output of pi is independent of its input (e.g., as in bit erasure); we use a physical computer C to implement pi that is specially tailored to the environment of C, i.e., to the precise distribution over C's inputs, P_0. First I extend these analyses to calculate the work required even if the output of pi depends on its input, and even if C is not used with the distribution P_0 it was tailored for. Next I show that if C will be re-used, then the minimal work to run …
The fundamental lower bounds of the thermodynamic energy cost (work) needed for the measurement and the erasure of information are found. The lower bound for the erasure vindicates the "Landauer's principle" for a special case, but otherwise implies its breakdown, indicating that no unique relationship exists between logical reversibility and physical one. Our results constitute the second law of "information thermodynamics", in which the information content and thermodynamic variables are treated on an equal footing.
We confirm Landauer's 1961 hypothesis that reducing the number of possible macroscopic states in a system by a factor of two requires work of at least kT ln 2. Our experiment uses a colloidal particle in a time-dependent, virtual potential created by a feedback trap to implement Landauer's erasure operation. In a control experiment, similar manipulations that do not reduce the number of system states can be done reversibly. Erasing information thus requires work. In individual cycles, the work to erase can be below the Landauer limit, consistent with the Jarzynski equality.
Using the operational framework of completely positive, trace preserving operations and thermodynamic fluctuation relations, we derive a lower bound for the heat exchange in a Landauer erasure process on a quantum system. Our bound comes from a non-phenomenological derivation of the Landauer principle which holds for generic non-equilibrium dynamics. Furthermore the bound depends on the non-unitality of dynamics, giving it a physical significance that differs from other derivations. We apply our framework to the model of a spin-1/2 system coupled to an interacting spin chain at finite temperat…
The Landauer principle states that at least $k_B T \ln 2$ 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 micro-mechanical 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 b…
In this article, we focus on erasure of a bit of information in finite time. Landauer's principle states that the average heat dissipation due to erasure of information is k_B T ln 2, which is achievable only in an asymptotic manner. Recent theoretical developments in non-equilibrium thermodynamics and stochastic control, predict a more general bound for finite time erasure dependent on the Wasserstein distances between the initial and final configurations. These predictions suggest improvements to experimental protocol with regards to minimizing average heat dissipation for bit erasure in fin…
A single bit memory system is made with a brownian particle held by an optical tweezer in a double-well potential and the work necessary to erase the memory is measured. We show that the minimum of this work is close to the Landauer's bound only for very slow erasure procedure. Instead a detailed Jarzynski equality allows us to retrieve the Landauer's bound independently on the speed of this erasure procedure. For the two separated subprocesses, i.e. the transition from state 1 to state 0 and the transition from state 0 to state 0, the Jarzynski equality does not hold but the generalized versi…
The second law of thermodynamics is probabilistic in nature. Its formulation requires that the state of a system be described by a probability distribution. A natural question, thereby, arises as to whether a prior knowledge about the state of the system affects the second law. This question has now been nurtured over a century and its inception was done by C. Maxwell through his famous thought experiment wherein comes the idea of Maxwell's demon. The next important step in this direction was provided by L. Szilard who demonstrated a theoretical model for an information engine incorporating Ma…
We study the thermodynamic cost associated with the erasure of one bit of information over a finite amount of time. We present a general framework for minimizing the average work required when full control of a system's microstates is possible. In addition to exact numerical results, we find simple bounds proportional to the variance of the microscopic distribution associated with the state of the bit. In the short-time limit, we get a closed expression for the minimum average amount of work needed to erase a bit. The average work associated with the optimal protocol can be up to a factor of f…
Maxwell's Demon, 'a being whose faculties are so sharpened that he can follow every molecule in its course', has been the centre of much debate about its abilities to violate the second law of thermodynamics. Landauer's hypothesis, that the Demon must erase its memory and incur a thermodynamic cost, has become the standard response to Maxwell's dilemma, and its implications for the thermodynamics of computation reach into many areas of quantum and classical computing. It remains, however, still a hypothesis. Debate has often centred around simple toy models of a single particle in a box. Despi…
According to Landauer's principle, erasing a memory requires an average work of at least $kT\ln2$ per bit. Recent experiments have confirmed this prediction for a one-bit memory represented by a symmetric double-well potential. Here, we present an experimental study of erasure for a memory encoded in an asymmetric double-well potential. Using a feedback trap, we find that the average work to erase can be less than $kT\ln2$. Surprisingly, erasure protocols that differ subtly give measurably different values for the asymptotic work, a result we explain by showing that one protocol is symmetric w…
Landauer's erasure principle puts a fundamental constraint on the amount of work required to erase information using thermal reservoirs. Recently this bound was improved to include corrections for finite-sized thermal reservoirs. In conventional information-erasure schemes, conservation of energy plays a key role with the cost of erasure. However, it has been shown that erasure can be achieved through the manipulation of spin angular momentum rather than energy, using a reservoir composed of energy-degenerate spin particles under the constraint of the conservation of spin angular momentum, in …
Landauer's principle, often regarded as the foundation of the thermodynamics of information processing, holds that any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information bearing degrees of freedom of the information processing apparatus or its environment. Conversely, it is generally accepted that any logically reversible transformation of information can in principle be accomplished by an appropriate physical mechanism operating in a thermodynamical…
We observe that the yield strength of a variety of materials, including highly structured and densely packed metals, alloys and semi-crystalline polymers is reasonably approximated by the thermal energy density of the material. This suggests that it is related to the entropic cost of the irreversible work done during plastic deformation rather than the enthalpic cost that depends on the elastic modulus of the material. Here we propose that the entropic cost of material rearrangement in crystalline solids arises from the difference in the uncertainty in building block positions before and after…
Quantum Szilard engine constitutes an adequate interplay of thermodynamics, information theory and quantum mechanics. Szilard engines are in general operated by a Maxwell's Demon where Landauer's principle resolves the apparent paradoxes. Here we propose a Szilard engine setup without featuring an explicit Maxwell's demon. In a demonless Szilard engine, the acquisition of which-side information is not required, but erasure and the related heat dissipation still take place implicitly by the very nature of the work extraction process. We see that the insertion of the partition in a quantum Szila…
Computations implemented on a physical system are fundamentally limited by the laws of physics. A prominent example for a physical law that bounds computations is the Landauer principle. According to this principle, erasing a bit of information requires a concentration of probability in phase space, which by Liouville's theorem is impossible in pure Hamiltonian dynamics. It therefore requires dissipative dynamics with heat dissipation of at least $k_BT\log 2$ per erasure of one bit. Using a concrete example, we show that when the dynamic is confined to a single energy shell it is possible to c…
