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A note on the Landauer principle in quantum statistical mechanics

Claude‐Alain Pillet

Published Aug 26, 2015• 14 citations
OpenAlex

Abstract

Abstract. The Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than kBT log 2. We discuss Landauer’s principle for quantum statistical models describing a finite level quantum system S coupled to an infinitely extended thermal reservoirR. Using Araki’s perturbation theory of KMS states and the Avron-Elgart adiabatic theorem we prove, under a natural ergodicity assumption on the joint system S +R, that Landauer’s bound saturates for adiabatically switched interactions. The recent work [ReWo] on the subject is discussed and compared. 1

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Landauer's erasure principle, which is one of the defining elements of modern information theory, states the minimum work required to erase 1 bit of information is kT ln(2), where T is the temperature of a thermal reservoir and k is Boltzmann's constant [1]. This principle arises from the maximisation of the entropy of the whole system (memory bit plus thermal reservoir) subject to energy conservation. In this talk we explore a different kind of statistical mechanics based on more general constraints other than energy conservation. Such systems have the potential to provide new mechanisms for erasing information, initialising the state of quantum systems and providing novel "heat" engines. We consider a gas of spin-1/2 systems (e.g. energy degenerate two-state atoms) which comes to equilibrium via angular momentum conserving interactions (e.g. elastic collisions). We model the gas as a canonical ensemble of spin particles constrained by the conservation of angular momentum. The gas acts as a spin reservoir whose macrostate is characterised by the total (fixed) average angular momentum of the gas. Our 1 bit memory device is represented by one spin-1/2 particle in an arbitrary mixed state. The state of the memory is erased by performing controlled operations on the memory particle and ancillary spin-1/2 particles and letting the memory and ancilla particles come to equilibrium with the spin reservoir. The cost of erasing 1 bit of memory is shown to be the angular momentum expended in the controlled operations. In principle there is no energy cost. We then apply this erasure mechanism to the Maxwell's demon problem. Imagine the demon is using the hot molecules of a thermal gas to produce mechanical work and we want to erase the demon's memory. If another thermal reservoir is used for the erasure, the associated energy cost will be greater than the mechanical work extracted by the demon [2]. However, we can avoid this energy cost by using the spin-reservoir erasure mechanism. In principle, all the heat can be extracted from the thermal reservoir as mechanical work for zero energy cost. The second law of thermodynamics and energy conservation are not violated by this process. The cost is in terms of angular momentum expended and an increase in the entropy of the spin reservoir. Moreover, the entropy of the whole system is increased in the process. [1] R. Landauer, IBM J. Res. Lett., 67 661 (1961). [2] C.H. Bennett, Int. J. Theor. Phys. 21, 905 (1982).

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Landauer's principle provides a fundamental lower bound for energy dissipation occurring 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 off-diagonal (coherence) elements of the density matrix are decoupled. In this paper, we aimed to broaden the comparative study to include the influence of quantum coherence induced by the tilted system-reservoir interaction direction. By examining their dependence on the initial state of the information-bearing system, we find that the following properties of the bounds are generically held regardless of whether the influence of the coherence is present or not: the entropic bound serves as the tighter bound for a sufficiently mixed initial state, while the thermodynamic bound is tighter when the purity of the initial state is sufficiently high. The exception is the case where the system dynamics involve only phase relaxation; in this case, the two bounds coincide when the initial coherence is zero; otherwise, the thermodynamic bound serves the tighter bound. We also find the quantum information erasure inevitably accompanies constant energy dissipation caused by the creation of system-reservoir correlation, which may cause an additional source of energetic cost for the erasure.

arXiv:2203.16724•Entropy•Apr 13, 2022•3 citations
Abstract. The Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than kBT log 2. We discuss Landauer’s principle for quantum statistical models describing a finite level quantum system S coupled to an infinitely extended thermal reservoirR. Using Araki’s perturbation theory of KMS states and the Avron-Elgart adiabatic theorem we prove, under a natural ergodicity assumption on the joint system S +R, that Landauer’s bound saturates for adiabatically switched interactions. The recent work [ReWo] on the subject is discussed and compared. 1
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.
A known aspect of the Clausius inequality is that an equilibrium system subjected to a squeezing dS < 0 of its entropy must release at least an amount the absolute value of dQ = T times the absolute value of dS 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 absorb heat during an analogous squeezing process, due to entanglement with the bath. The effect exists even for weak but fixed coupling with the bath, provided that temperature is low enough. This invalidates the Landauer bound in the quantum regime, and suggests that quantum carriers of information can be more efficient than assumed so far.
Landauer's principle gives a fundamental limit to the thermodynamic cost of erasing information. Its saturation requires a reversible isothermal process, and hence infinite time. We develop a finite-time version of Landauer's principle for a bit encoded in the occupation of a single fermionic mode, which can be strongly coupled to a reservoir. By solving the exact non-equilibrium dynamics, we optimize erasure processes (taking both the fermion's energy and system-bath coupling as control parameters) in the slow driving regime through a geometric approach to thermodynamics. We find analytic expressions for the thermodynamic metric and geodesic equations, which can be solved numerically. Their solution yields optimal processes that allow us to characterize a finite-time correction to Landauer's bound, fully taking into account non-markovian and strong coupling effects.
Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.
Landauer's erasure principle, which is one of the defining elements of modern information theory, states the minimum work required to erase 1 bit of information is kT ln(2), where T is the temperature of a thermal reservoir and k is Boltzmann's constant [1]. This principle arises from the maximisation of the entropy of the whole system (memory bit plus thermal reservoir) subject to energy conservation. In this talk we explore a different kind of statistical mechanics based on more general constraints other than energy conservation. Such systems have the potential to provide new mechanisms for erasing information, initialising the state of quantum systems and providing novel "heat" engines. We consider a gas of spin-1/2 systems (e.g. energy degenerate two-state atoms) which comes to equilibrium via angular momentum conserving interactions (e.g. elastic collisions). We model the gas as a canonical ensemble of spin particles constrained by the conservation of angular momentum. The gas acts as a spin reservoir whose macrostate is characterised by the total (fixed) average angular momentum of the gas. Our 1 bit memory device is represented by one spin-1/2 particle in an arbitrary mixed state. The state of the memory is erased by performing controlled operations on the memory particle and ancillary spin-1/2 particles and letting the memory and ancilla particles come to equilibrium with the spin reservoir. The cost of erasing 1 bit of memory is shown to be the angular momentum expended in the controlled operations. In principle there is no energy cost. We then apply this erasure mechanism to the Maxwell's demon problem. Imagine the demon is using the hot molecules of a thermal gas to produce mechanical work and we want to erase the demon's memory. If another thermal reservoir is used for the erasure, the associated energy cost will be greater than the mechanical work extracted by the demon [2]. However, we can avoid this energy cost by using the spin-reservoir erasure mechanism. In principle, all the heat can be extracted from the thermal reservoir as mechanical work for zero energy cost. The second law of thermodynamics and energy conservation are not violated by this process. The cost is in terms of angular momentum expended and an increase in the entropy of the spin reservoir. Moreover, the entropy of the whole system is increased in the process. [1] R. Landauer, IBM J. Res. Lett., 67 661 (1961). [2] C.H. Bennett, Int. J. Theor. Phys. 21, 905 (1982).
Landauer's principle provides a fundamental lower bound for energy dissipation occurring 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 off-diagonal (coherence) elements of the density matrix are decoupled. In this paper, we aimed to broaden the comparative study to include the influence of quantum coherence induced by the tilted system-reservoir interaction direction. By examining their dependence on the initial state of the information-bearing system, we find that the following properties of the bounds are generically held regardless of whether the influence of the coherence is present or not: the entropic bound serves as the tighter bound for a sufficiently mixed initial state, while the thermodynamic bound is tighter when the purity of the initial state is sufficiently high. The exception is the case where the system dynamics involve only phase relaxation; in this case, the two bounds coincide when the initial coherence is zero; otherwise, the thermodynamic bound serves the tighter bound. We also find the quantum information erasure inevitably accompanies constant energy dissipation caused by the creation of system-reservoir correlation, which may cause an additional source of energetic cost for the erasure.