Truly work-like work extraction via a single-shot analysis

Abstract

The work content of non-equilibrium systems in relation to a heat bath is often analysed in terms of expectation values of an underlying random work variable. However, when optimizing the expectation value of the extracted work, the resulting extraction process is subject to intrinsic fluctuations, uniquely determined by the Hamiltonian and the initial distribution of the system. These fluctuations can be of the same order as the expected work content per se, in which case the extracted energy is unpredictable, thus intuitively more heat-like than work-like. This raises the question of the 'truly' work-like energy that can be extracted. Here we consider an alternative that corresponds to an essentially fluctuation-free extraction. We show that this quantity can be expressed in terms of a one-shot relative entropy measure introduced in information theory. This suggests that the relations between information theory and statistical mechanics, as illustrated by concepts like Maxwell's demon, Szilard engines and Landauer's principle, extends to the single-shot regime.

ARTICLE

Received 12 Dec 2012 | Accepted 7 Mar 2013 | Published 26 Jun 2013

The work content of non-equilibrium systems in relation to a heat bath is often analysed in terms of expectation values of an underlying random work variable. However, when optimizing the expectation value of the extracted work, the resulting extraction process is subject to intrinsic uctuations, uniquely determined by the Hamiltonian and the initial distribution of the system. These uctuations can be of the same order as the expected work content per se, in which case the extracted energy is unpredictable, thus intuitively more heat-like than work-like. This raises the question of the truly work-like energy that can be extracted. Here we consider an alternative that corresponds to an essentially uctuation-free extraction. We show that this quantity can be expressed in terms of a one-shot relative entropy measure introduced in information theory. This suggests that the relations between information theory and statistical mechanics, as illustrated by concepts like Maxwells demon, Szilard engines and Landauers principle, extends to the single-shot regime.

DOI: 10.1038/ncomms2712

Truly work-like work extraction via a single-shot analysis

Johan berg1,2

1 Institute for Physics, University of Freiburg, Hermann-Herder-Strasse 3, Freiburg D-79104, Germany. 2 Institute for Theoretical Physics, ETH Zurich, Zurich 8093, Switzerland. Correspondence and requests for materials should be addressed to J.A. (email: mailto:[email protected]

Web End [email protected] ).

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The amount of useful energy that can be harvested from non-equilibrium systems not only characterizes practical energy extraction and storage, but is also a fundamental

thermodynamic quantity. Intuitively, we wish to extract ordered and predictable energy, that is, work, as opposed to disordered random energy in the form of heat. The catch is that, in statistical systems, the work cost or yield of a given transformation is typically a random variable1. As an example, one can think of the friction that an object experiences when forced through a viscous medium. On a microscopic level, this force resolves into chaotic molecular collisions and thus results in a random work cost each time we perform this transformation. This is further illustrated by experimental tests24 of uctuation theorems, which characterize the randomness in the work cost (or entropy production) of non-equilibrium processes1. These observations raise the question of a quantitative notion of work content that truly reects the idea of work as ordered energy.

Here, we show that standard expressions for the work content58 can correspond to a very noisy and thus heat-like energy, but we also introduce an alternative that quanties the amount of ordered energy that can be extracted. The latter can be expressed in terms of a non-equilibrium generalization of the free energy, or equivalently in terms of a one-shot information-theoretic relative entropy, which quanties how far the given non-equilibrium system is from thermal equilibrium.

Standard information theory typically quanties the resources needed to perform a given information-theoretic task averaged over many repetitions, for example, the average number of bits needed to send many independent messages9. In contrast, one-shot (or single-shot) information theory rather focuses on single instances of such tasks (see, for example, refs 10, 11). Given the strong historical links between standard information theory and the work extraction problem, via concepts like Szilard engines, Landauers principle and Maxwells demon12,13, it is reasonable to ask whether also one-shot information theory has a counterpart in statistical mechanics. Together with refs 1419, the results of this investigation suggest that this is indeed the case. A direct consequence of the present investigation is that the results of refs 14, 15, 19 is brought into a more physical setting, allowing, for example, systems with non-trivial Hamiltonians, proof of near-optimality, as well as a connection to uctuation theorems1. The latter suggests that the effects we consider become relevant in the typical regimes of uctuation theorems. Similar results as in this study have been obtained independently in ref. 16. See also recent results in ref. 17 based on ideas in ref. 18. (For further discussions on the relations to the existing literature, see Supplementary Note 1.)

ResultsWork extraction. The amount of work that a system can perform while it equilibrates with respect to an environment of temperature T is often58 expressed as

Aq; h kT ln2DqjjGh: 1 Here q is the state of the system, G(h) its equilibrium state, h the system Hamiltonian and k Boltzmanns constant. For the simple model we employ here, q is a probability distribution over a nite set of energy levels, and D(q||p) P

nqnlog2qn P

a b

q

h

h

h

G(h)

Wn = hnhn

nqnlog2pn is

the relative Shannon entropy (KullbackLeibler divergence)9, and log2 denotes the base 2 logarithm.

The quantity A(q, h), and the closely related cost of

information erasure (Landauers principle), is often understood as an expectation value of an underlying random work yield (see, for example, refs 5,7,20,21). However, this tells us very little about the uctuations, and thus the quality of the extracted energy. Here, we show that optimizing the expected gain leads to intrinsic

uctuations. These can be of the same order as the expected work content A(q, h) per se, in which case the work extraction does not

act as a truly ordered energy source. As an alternative, we introduce the E-deterministic work content, which quanties the maximal amount of energy that can be extracted if we demand to always get precisely this energy each single time we run the extraction process, apart from a small probability of failure E.

Hence, in contrast to the expected work extraction, where we do not put any restrictions on how broadly distributed the random energy gain is, we do in the E-deterministic work extraction demand that the probability distribution should be very peaked, that is, very predictable. In other words, the E-deterministic work content formalizes the idea of an almost perfectly ordered energy source.

The model. Our analysis is based on a very simple model of a system interacting with a heat bath of xed temperature T (see Fig. 1). Akin to, for example, refs 15, 21, 22, we model the Hamiltonian of the system as nite set of energy levels h (h1,y, hN). The state of the system we regard as a random

variable N , with a probability distribution q (q1,y, qN). On

this system, we have two elementary operations.

The rst of these two operations changes the energy levels h to a new set of energy levels h0, but leaves the state, and thus the probability distribution q, intact. We refer to this as level transformations (LTs). (For a quantum system, this would essentially correspond to adiabatic evolution with respect to some external control parameters, that is, in the limit of innitely slow changes of the control we alter the energy levels, but not how they are occupied.) Via the LTs we dene what work is in our model. If we perform an LT that changes h to h0, and if the system is in state n, then this results in a work gain hn h0n (or work cost

h0n hn). As the work gain depends on the state of the system, a

random state implies a random work gain.

The second elementary operation corresponds to thermalization, where one can imagine that we connect the system to the heat bath, let it thermalize and slowly de-connect it again. We model this by putting the system into the random state

N described by the Gibbs distribution, P(N n) Gn(h), where

Gnh e bhn=Zh, b 1/(kT) and Zh Pn e bhn is the

partition function. It is furthermore assumed that the state (regarded as a random variable) after a thermalization is independent of the state before.

We construct processes by combining these two types of elementary operations into any sequence of our choice. The

h

Figure 1 | The model. The system can be in any of the states 1,y,N, each assigned an energy level (horizontal lines) h (h1,y,hN). Two elementary

operations modify the system: (a) LT. This lifts or lowers the energy levels h to a new conguration h0, but leaves the state (circle) intact. If the system is in state n, the work cost of the LT is dened as Wn h0n hn. (b)

Thermalization with respect to a heat bath of temperature T. This changes the initial probability distribution (bars) q (q1,y,qN) over the states, into

the Gibbs distribution G(h), but leaves the energy levels h intact, and has no work cost. We build up processes by combining LTs and thermalizations.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2712 ARTICLE

hf

a

b

q

hi

h = kT ln(q)

Figure 2 | ITR processes. In the space of energy level congurations, we connect an initial conguration hiA RN with the nal hfARN by a smooth path (grey line). Given an L-step discretization of this path, we construct a sequence of LTs (arrows) sandwiched by thermalizations (circles).

This process has the random work cost W PL 1l0hl1N

hlN

q

G(h)

, where

N l is the state at the l-th step, which is Gibbs distributed G(hl). In

the limit of an innitely ne discretization, the expected work cost is limL-N/WS F(hf) F(hi). The independence of the work costs of the

subsequent LTs yields limL-N(/W2S /WS2) 0, that is, the work cost

is essentially deterministic.

h

h

G(h)

c

h

Figure 3 | Expected work extraction. For an initial state N with

distribution q (bars) and energy levels h (horizontal lines), the expected work content A(q, h) is obtained by a cyclic three-step process. The

idea is to avoid unnecessary dissipation when the system is put in contact with the heat bath. To this end, we make an LT to a set of energy levels h0 for which G(h0) q. The total process is: (a) LT that transforms

hn into h0n kT ln qn. (b) Thermalization, resulting in the Gibbs

distribution G(h0) q. (c) ITR process from h0 back to h. The resulting

random work yield is Wyield kT lnqN kT lnGN(h), with expectation

value /WyieldS kT ln(2)D(q||G(h)).

resulting work yield of the process is dened as the sum of the work yields of all the LTs. (For a more detailed description of the model, see Supplementary Note 2.) An example is given in Fig. 2, where we construct the analogue of isothermal reversible (ITR) processes, which serve as a building block in our analysis (see Supplementary Note 3). As opposed to other processes we will consider, the ITRs have essentially uctuation-free work costs.

Expected work extraction. Given an initial state N with dis

tribution q, we can reproduce equation (1) within our model. A cyclic three-step process, as described in Fig. 3, gives the random work yield

Wyield kT ln qN kT ln GN h: 2

By taking the expectation value, we obtain equation (1). The positivity of relative entropy, D(q||p)Z0, can be used to show that no process can give a better expected work yield (Supplementary Note 4). One can in a similar fashion determine the minimal expected work cost for information erasure (see Supplementary Note 5).

Fluctuations in expected work extraction. How large are the uctuations for a process that maximizes the expected work extraction, and thus achieves A(q, h)? Equation (2) determines

the noise of the specic process in Fig. 3, but it turns out that it actually species the uctuations for all processes that optimize the expected work extraction. (For the exact statement, see Methods, or Supplementary Note 6.) We can conclude that to analyse the noise in the optimal expected work extraction, it is enough to consider equation (2). As we will conrm later, these uctuations can be of the same order as

A(q, h) itself.

e-deterministic work extraction. As the optimal expected work extraction suffers from uctuations, a natural question is how much (essentially) noise-free energy can be extracted. We say that a random variable X has the (E, d)-deterministic value x, if the probability to nd X in the interval [x d, x d] is larger than

1 E. Hence, d is the precision by which the value x is taken, and

E the largest probability by which this fails. (See Supplementary Note 7 for further properties.) We dene AEdq; h as the highest

possible (E, d)-deterministic work yield among all processes that operate on the initial distribution q with initial and nal energy levels h. Next, we dene the E-deterministic work content as

AEq; h limd!0AEdq; h, that is, we take the limit of innite

precision, thus formalizing the idea of an energy extraction that is essentially free from uctuations.

AE(q, h) can be expressed in terms of the E-free energy,

which is dened via restrictions to sufciently likely subsets of energy levels. Given a subset L, we dene ZLh Pn2L e bhn.

We minimize ZL(h) among all subsets L such that q(L) PnALqn41 E. If L* is such a minimizing set, then the

E-free energy is dened as F E(q, h) kTlnZL*(h). (See

Supplementary Note 8 for further explanations.) The concept of one-shot free energy has been introduced independently in ref. 16.

The distribution of uctuations is clearly important for determining the value of AE(q, h). It is thus maybe not

surprising that a variation (see Methods) of Crooks uctua

tion theorem23 can be used to prove

AEq; h FEq; h Fh; 3 where E signies that the equality is true up to a small error (see Methods, or Supplementary Note 9 and Supplementary Note 10). The error is small in the sense that it can be regarded as the energy of a sufciently likely equilibrium uctuation (see Methods and Supplementary Note 11). An example of a process that gives the right-hand side of equation (3) is described in Fig. 4. In the case of completely degenerate energy levels h (r,y, r), equation (3) reduces to the result

in ref. 14. (See also Supplementary Note 12 and Supplementary Note 13 for the E-deterministic cost of information erasure).

The above result can be reformulated in terms of an E-smoothed relative Rnyi 0-entropy, dened as DE0qjjp

log2minqL4 1 E Pj2L pj: This relative entropy was (up to

some technical differences) introduced in ref. 24 in the context of one-shot information theory. (See refs 25, 26 for quantum versions.) One can see that

FEq; h Fh kT ln2DE0qjjGh: 4

Comparisons. An immediate question is how A(q, h) compares

with AE(q, h), and with the uctuations in the optimal expected

work extraction. The latter we measure by the s.d. of Wyield in

equation (2), sW

yield

i21=2. We compare

how these three quantities scale with increasing system size (for example, in number of spins, or other units).

hW2yieldi hW

yield

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Our rst example is a collection of m systems whose state distributions are independent and identical, qm(n1,y, nm)

q(n1)yq(nm), and which have non-interacting identical Hamiltonians, corresponding to energy levels hm(n1,y,nm)

h(n1) y h(nm). In this case A(qm, hm) mA(q, h), and

sWmyield

m

p kT ln2sq k Gh, where s(q||r)2 P

nqn[log2

(qn/rn)]2 D(q||r)2. By BerryEsseens theorem27,28 (see Methods

and Supplementary Note 14), one can show that AE(qm, hm) is

equal to mA(q, h) to the leading order in m (see ref. 29 for a

similar result in a resource theory framework). The difference only appears at the next to leading order. Hence, in these systems the uctuations are comparably small, and the dominant contribution to AE(qm, hm) is A(qm, hm). It appears reasonable

to expect similar results for non-equilibrium systems with sufciently fast spatial decay of both correlations and interactions, which may explain why issues concerning A as a

measure of work content appear to have gone largely unnoticed.

A simple modication of the state distribution in the previous example results in a system with large uctuations. With probability 1 E (independent of m), the system is prepared

in the joint ground state 0,y, 0, and with probability E in the Gibbs distribution. This results in qml1;... ; lm 1 Edl1;0 dlm;0 EGl1h Glmh; and yields A(qm, hm)B mkT ln(2)(1 E)

log2G0(h), sWmyield mkT ln2

E1 E

p log2G0h and

AE(qm, hm)B mkT ln(2)log2G0(h). Hence, all three quantities

grow proportionally to m.

For a second case of large uctuations, we choose the distribution qml1;... ; lm d m, for a collection of d-level

systems. For large m, we assume that the energy levels are dense enough that they can be replaced by a spectral density.

One example is Wigners semicircle law, where f mx

2

Rm2 x2

q =pRm2 for |x|rR(m). With Rm 2

p dm=2,

this is the asymptotic energy level distribution of large random matrices from the Gaussian unitary ensemble30. For the semicircle distribution A(qm, hm)BR(m), sWmyield Rm=2, and AE(qm,hm)Bc(E)R(m), where c(E) is independent of m.

DiscussionWe have here employed what one could refer to as a discrete classical model. Relevant extensions include a classical phase-space picture, as well as a quantum setting that allows superpositions between different energy eigenstates (for example, in the spirit of refs 16, 29, 31) and where the work-extractor can possess quantum information about the system15. An operational approach, based on what work is supposed to achieve, rather than ad hoc denitions, may yield deeper insights to the question of the truly work-like energy content.

It is certainly justied to ask for the relevance of the effects we have considered here. The evident role of uctuations suggests that the noise in the expected work extraction should become noticeable in the same nano-regimes as where uctuation theorems are relevant. The considerable experimental progress on the latter (see, for example, refs 24) should reasonably be applicable also to the former. Also the theoretical aspects of the link to uctuation theorems merits further investigations.

We have seen that the E-deterministic work content to the leading order becomes equal to the expected work content for systems with identical non-interacting Hamiltonians and identical uncorrelated state distributions. However, we have also demonstrated by simple examples that the expected work extraction can become very noisy when we deviate from this simple setup. In these cases, the expected work extraction thus fails to capture our intuitive notion of work as ordered energy,

while the E-deterministic work extraction is predictable by construction. One might object that many realistic systems are approximately non-interacting and approximately uncorrelated, and thus presumably show no signicant difference between the E-deterministic and the expected work extraction. However, as we here consider a general non-equilibrium setting, there is no particular reason to assume, for example, weak correlations. It is maybe also worth to point out that the uctuations in the expected work extraction can be large also outside the microscopic regime, as this only requires a sufciently violent relation between the non-equilibrium state and the Hamiltonian of the system. As opposed to the expected work content, the E-deterministic work content retains its interpretation as the ordered energy. It is no coincidence that this is much analogous to how single-shot information theory generalizes standard information theory10,11. In this spirit, the present study, along with refs 1419, can be viewed as the rst glimpse of a single-shot statistical mechanics.

Methods

Randomness in optimal expected work extraction. In the main text, we briey mentioned the fact that processes that optimize the expected work extraction converge to the random variable in equation (2). We can phrase this result more precisely as follows. For a process P that operates on an initial state N

with distribution q, we let Wyield(P, N ) denote the corresponding random work

yield. We here consider cyclic processes that starts and ends in the energy levels h. If Pm1m1 is a family of processes such that limm-N/W

yield(Pm, N )S A(q, h), then Wyield(Pm, N )-kT lnqN kT lnGN(h) in probability. (For a

proof see Supplementary Note 6).

Bounds on the e-deterministic work content. The exact statement of equation (3) is

0 AEq; h FEq; h Fh kT ln1 E: 5

In Supplementary Note 11, it is shown that kT ln(1 E) is an upper bound to

the E-deterministic work content of equilibrium systems. Equation (5) thus determines the value of AE(q, h) up to an error with the size of a sufciently

probable equilibrium uctuation. We obtain the lower bound in equation (5) by the process described in Fig. 4. The upper bound is obtained by a combination of a variation (Supplementary Note 10, Supplementary Equation (S73)) on Crooks uctuation theorem23 and a work bound for LTs (Supplementary Note 10). For a discussion on an alternative single-shot work extraction quantity, and its relation to

AE, see Supplementary Note 15.

a b

q

h

q

G(h)

h

h

q() > 1

G(h)

c

h

Figure 4 | e-deterministic work extraction. For a state distribution q and energy levels h, let L* be a subset of the energy levels such that

FE(q, h) kTlnZL*(h). (a) LT that lifts all energy levels not in L* to a

very high value, that is, h0n hn if nAL*, while h0n hn E if neL*.

(b) Thermalization, resulting in the Gibbs distribution G(h0). (c) ITR process from h0 back to h, which gives the essentially deterministic work yield

F(h0) F(h). In the limit E- N, this process gives the work yield

FE(q, h) F(h) with a probability larger than 1 E. This is a lower

bound to AE(q, h), but is also close to it for small E.

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Expansion of Ae in the multi-copy case. In the case of a state distribution qm(n1,y,nm) q(n1)yq(nm), and energy levels hm(n1,y,nm) h(n1) y

h(nm), the E-deterministic work content has the expansion

AEqm; hm mAq; h

m

p kT ln2F 1EsqjjGh

o m

p ;

where o

m

p , and F 1 is the inverse of the cumulative distribution function of the standard normal distribution.

The smaller our error tolerance E, the more the correction term lowers the value of

AE(qm, hm) as compared with A(qm, hm). This expansion is proved via Berry

Esseens theorem27,28, which determines the convergence rate in the central limit theorem (see Supplementary Note 14).

References

1. Jarzynski, C. Equalities and inequalities: irreversibility and the second law of thermodynamics at the nanoscale. Annu. Rev. Condens. Matter Phys. 2, 329351 (2011).

2. Liphardt, J., Dumont, S., Smith, S. B., Tinoco, Jr I. & Bustamante, C. Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynskis equality. Science 296, 18321835 (2002).

3. Collin, D., Ritort, F., Jarzynski, C., Smith, S. B., Tinoco, Jr I. & Bustamante, C. Verication of the Crooks uctuation theorem and recovery of RNA folding free energies. Nature 437, 231234 (2005).

4. Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E. & Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys. 6, 988992 (2010).

5. Procaccia, I. & Levine, R. D. Potential work: a statistical-mechanical approach for systems in disequilibrium. J. Chem. Phys. 65, 33573364 (1967).

6. Lindblad, G. Non-Equilibrium Entropy and Irreversibility (Reidel, Lancaster, 1983).

7. Takara, K., Hasegawa, H.-H. & Driebe, D. J. Generalization of the second law for a transition between nonequilibrium states. Phys. Lett. A 375, 8892 (2010).

8. Esposito, M. & Van den Broeck, C. Second law and Landauer principle far from equilibrium. Euro. Phys. Lett. 95, 40004 (2011).

9. Cover, T. M. & Thomas, J. A. Elements of Information Theory 2nd edn (Wiley, Hoboken, 2006).

10. Holenstein, T. & Renner, R. On the randomness of independent experiments. IEEE Trans. Inf. Theor. 57, 18651871 (2011).

11. Tomamichel, M., Colbeck, R. & Renner, R. A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theor. 55, 58405847 (2009).

12. Leff, H. S. & Rex, A. F. Maxwells Demon: Entropy, Information, Computing (Taylor and Francis, 1990).

13. Leff, H. S. & Rex, A. F. Maxwells Demon 2: Entropy, Classical and Quantum Information, Computing (Taylor and Francis, 2002).

14. Dahlsten, O., Renner, R., Rieper, E. & Vedral, V. Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys. 13, 053015 (2011).

15. del Rio, L., berg, J., Renner, R., Dahlsten, O. & Vedral, V. The thermodynamic meaning of negative entropy. Nature 474, 6163 (2011).

16. Horodecki, M. & Oppenheim, J. Fundamental limitations for quantum and nano thermodynamics. Nat. Commun. doi:10.1038/ncomms3059 (2013).

17. Egloff, D., Dahlsten, O., Renner, R. & Vedral, V. Laws of thermodynamics beyond the von Neumann regime. Preprint at http://arxiv.org/abs/1207.0434

Web End =http://arxiv.org/abs/1207.0434 (2012).

18. Egloff, D. Work Value of Mixed States in Single Instance Work Extraction Games. Masters thesis, ETH Zurich (2010).

19. Faist, P., Dupuis, F., Oppenheim, J. & Renner, R. A quantitative Landauers principle. Preprint at http://arxiv.org/abs/1211.1037

Web End =http://arxiv.org/abs/1211.1037 (2012).

20. Shizume, K. Heat generation required by information erasure. Phys. Rev. E 52,

34953499 (1995).

21. Piechocinska, B. Information erasure. Phys. Rev. A 61, 062314 (2000).22. Crooks, G. E. Nonequilibrium measurements of free energy differences for microscopically reversible markovian systems. J. Stat. Phys. 90, 14811487 (1998).

23. Crooks, G. E. Entropy production uctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 27212726 (1999).

24. Wang, L., Colbeck, R. & Renner, R. Simple channel coding bounds. IEEE Int Symp Inf Theory 2009, 18041808 (2009).

25. Datta, N. Min- and max-relative entropies and a new entanglement monotone. IEEE Trans. Inf. Theor. 55, 28162826 (2009).

26. Wang, L. & Renner, R. One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108, 200501 (2012).

27. Berry, A. C. The accuracy of the gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49, 122136 (1941).

28. Esseen, C.-G. On the Liapunoff limit of error in the theory of probability. Ark. Mat. Astr. o. Fys. 28A, 119, (1942).

29. Brandao, F. G. S. L., Horodecki, M., Oppenheim, J., Renes, J. M. & Spekkens, R.W. The resource theory of quantum states out of thermal equilibrium. Preprint at http://arxiv.org/abs/1111.3882

Web End =http://arxiv.org/abs/1111.3882 (2011).30. Mehta, M. L. Random Matrices (Elsevier, 2004).31. Janzing, D., Wocjan, P., Zeier, R., Geiss, R. & Beth, Th. Thermodynamic cost of reliability and low temperatures: tightening Landauers principle and the second law. Int. J. Theor. Phys. 39, 27172753 (2000).

Acknowledgements

I thank Ldia del Rio, Renato Renner and Paul Skrzypczyk for useful comments. This research was supported by the Swiss National Science Foundation through the National Centre of Competence in Research Quantum Science and Technology, by the European Research Council, grant no. 258932, and the Excellence Initiative of the German Federal and State Governments (grant ZUK 43).

Author contributions

J.. is responsible for the whole content of this paper.

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How to cite this article: berg J. Truly work-like work extraction. Nat. Commun. 4:1925 doi: 10.1038/ncomms2712 (2013).

p is a correction term that grows slower than

m

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