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The cost of information erasure in atomic and spin systems

Joan A. Vaccaro, Stephen Mark Barnett

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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 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 the limit of an infinite number of particles. In this case the erasure cost is in terms of dissipation of spin angular momentum. Here we analyze the erasure of memory using a finite-sized spin reservoir. We compute the erasure cost to compare it with its infinite counterpart and determine what size of finite reservoir gives similar erasure cost statistics using the Jensen-Shannon Divergence as the measure of difference. Our findings show that erasure with finite-sized reservoirs results in the erasure of less information compared to the infinite reservoir counterpart when compared on this basis. In addition we discuss the cost of resetting the state of the ancillary spin particles used in the erasure process, and we investigate the degradation in erasure performance when a finite reservoir is repeatedly reused to erase a sequence of memories.

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The Cost of Erasing Information

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Landauer has argued that the process of erasing 1 bit of information requires a minimum amount of work equal to kTln(2) where T is the temperature of a thermal reservoir that consumes the entropy of the bit. We show that, in principle, the thermal reservoir can be replaced by a reservoir of N spin‐1/2 particles which is subject to the conservation of spin angular momentum. The cost of erasing 1 bit of information in this case is found to be at least β−1ln(2) where e−ℏβ = (Nℏ+2〈Jz〉)/(Nℏ−2〈Jz〉) and 〈Jz〉 is the z component of the total spin angular momentum of the spin reservoir.

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According to Landauer's principle, erasing one bit of information incurs a minimum energy cost. Recently, Vaccaro and Barnett (VB) explored information erasure within the context of generalized Gibbs ensembles and demonstrated that for energy-degenerate spin reservoirs the cost of erasure can be solely in terms of a minimum amount of spin angular momentum and no energy. As opposed to the Landauer case, the cost of erasure in this case is associated with an intrinsically discrete degree of freedom. Here we study the discrete fluctuations in this cost and the probability of violation of the VB bound. We also obtain a Jarzynski-like equality for the VB erasure protocol. We find that the fluctuations below the VB bound are exponentially suppressed at a far greater rate and more tightly than for an equivalent Jarzynski expression for VB erasure. We expose a trade-off between the size of the fluctuations and the cost of erasure. We find that the discrete nature of the fluctuations is pronounced in the regime where reservoir spins are maximally polarized. We also state the first laws of thermodynamics corresponding to the conservation of spin angular momentum for this particular erasure protocol. Our work will be important for novel heat engines based on information erasure schemes that do not incur an energy cost.

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Landauer's Principle states that the work cost of erasure of one bit of information has a fundamental lower bound of kT ln(2). Here we prove a quantitative Landauer's principle for arbitrary processes, providing a general lower bound on their work cost. This bound is given by the minimum amount of (information theoretical) entropy that has to be dumped into the environment, as measured by the conditional max-entropy. The bound is tight up to a logarithmic term in the failure probability. Our result shows that the minimum amount of work required to carry out a given process depends on how much correlation we wish to retain between the input and the output systems, and that this dependence disappears only if we average the cost over many independent copies of the input state. Our proof is valid in a general framework that specifies the set of possible physical operations compatible with the second law of thermodynamics. We employ the technical toolbox of matrix majorization, which we extend and generalize to a new kind of majorization, called lambda-majorization. This allows us to formulate the problem as a semidefinite program and provide an optimal solution.

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Landauer's erasure principle is a cornerstone of thermodynamics and information theory [R. Landauer, IBM J. Res. Dev. 5, 183 (1961)]. According to this principle, erasing information incurs a minimum energy cost. Recently, Vaccaro and Barnett [J. A. Vaccaro and S. M. Barnett, Proc. R. Soc. A 467, 1770 (2011)] explored information erasure in the context of multiple conserved quantities and showed that the erasure cost can be solely in terms of spin angular momentum. As Landauer's erasure principle plays a fundamental role in heat engines, their result considerably widens the possible configurations that heat engines can have. Motivated by this, we propose here an optical heat engine that operates under a single thermal reservoir and a spin angular momentum reservoir coupled to a three-level system with two energy degenerate ground states. The proposed heat engine operates without producing waste heat and goes beyond the traditional Carnot engine where the working fluid is subjected to two thermal baths at different temperatures.

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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.

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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).
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 the limit of an infinite number of particles. In this case the erasure cost is in terms of dissipation of spin angular momentum. Here we analyze the erasure of memory using a finite-sized spin reservoir. We compute the erasure cost to compare it with its infinite counterpart and determine what size of finite reservoir gives similar erasure cost statistics using the Jensen-Shannon Divergence as the measure of difference. Our findings show that erasure with finite-sized reservoirs results in the erasure of less information compared to the infinite reservoir counterpart when compared on this basis. In addition we discuss the cost of resetting the state of the ancillary spin particles used in the erasure process, and we investigate the degradation in erasure performance when a finite reservoir is repeatedly reused to erase a sequence of memories.
Landauer has argued that the process of erasing 1 bit of information requires a minimum amount of work equal to kTln(2) where T is the temperature of a thermal reservoir that consumes the entropy of the bit. We show that, in principle, the thermal reservoir can be replaced by a reservoir of N spin‐1/2 particles which is subject to the conservation of spin angular momentum. The cost of erasing 1 bit of information in this case is found to be at least β−1ln(2) where e−ℏβ = (Nℏ+2〈Jz〉)/(Nℏ−2〈Jz〉) and 〈Jz〉 is the z component of the total spin angular momentum of the spin reservoir.
According to Landauer's principle, erasing one bit of information incurs a minimum energy cost. Recently, Vaccaro and Barnett (VB) explored information erasure within the context of generalized Gibbs ensembles and demonstrated that for energy-degenerate spin reservoirs the cost of erasure can be solely in terms of a minimum amount of spin angular momentum and no energy. As opposed to the Landauer case, the cost of erasure in this case is associated with an intrinsically discrete degree of freedom. Here we study the discrete fluctuations in this cost and the probability of violation of the VB bound. We also obtain a Jarzynski-like equality for the VB erasure protocol. We find that the fluctuations below the VB bound are exponentially suppressed at a far greater rate and more tightly than for an equivalent Jarzynski expression for VB erasure. We expose a trade-off between the size of the fluctuations and the cost of erasure. We find that the discrete nature of the fluctuations is pronounced in the regime where reservoir spins are maximally polarized. We also state the first laws of thermodynamics corresponding to the conservation of spin angular momentum for this particular erasure protocol. Our work will be important for novel heat engines based on information erasure schemes that do not incur an energy cost.
Landauer's Principle states that the work cost of erasure of one bit of information has a fundamental lower bound of kT ln(2). Here we prove a quantitative Landauer's principle for arbitrary processes, providing a general lower bound on their work cost. This bound is given by the minimum amount of (information theoretical) entropy that has to be dumped into the environment, as measured by the conditional max-entropy. The bound is tight up to a logarithmic term in the failure probability. Our result shows that the minimum amount of work required to carry out a given process depends on how much correlation we wish to retain between the input and the output systems, and that this dependence disappears only if we average the cost over many independent copies of the input state. Our proof is valid in a general framework that specifies the set of possible physical operations compatible with the second law of thermodynamics. We employ the technical toolbox of matrix majorization, which we extend and generalize to a new kind of majorization, called lambda-majorization. This allows us to formulate the problem as a semidefinite program and provide an optimal solution.
Landauer's erasure principle is a cornerstone of thermodynamics and information theory [R. Landauer, IBM J. Res. Dev. 5, 183 (1961)]. According to this principle, erasing information incurs a minimum energy cost. Recently, Vaccaro and Barnett [J. A. Vaccaro and S. M. Barnett, Proc. R. Soc. A 467, 1770 (2011)] explored information erasure in the context of multiple conserved quantities and showed that the erasure cost can be solely in terms of spin angular momentum. As Landauer's erasure principle plays a fundamental role in heat engines, their result considerably widens the possible configurations that heat engines can have. Motivated by this, we propose here an optical heat engine that operates under a single thermal reservoir and a spin angular momentum reservoir coupled to a three-level system with two energy degenerate ground states. The proposed heat engine operates without producing waste heat and goes beyond the traditional Carnot engine where the working fluid is subjected to two thermal baths at different temperatures.
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.