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LETTER doi:10.1038/nature10123

The thermodynamic meaning of negative entropy Ldia del Rio1* , Johan Aberg 1* , Renato Renner 1, Oscar Dahlsten 1,2,3 & Vlatko Vedral 2,3,4

The heat generated by computations is not only an obstacle tocircuit miniaturization but also a fundamental aspect of the rela-tionship between information theory and thermodynamics. Inprinciple, reversible operations may be performed at no energy cost; given that irreversible computations can always be de-composed into reversible operations followed by the erasure of data 1,2 , the problem of calculating their energy cost is reduced tothe study of erasure. Landauers principle states that the erasure of data stored in a system has an inherent work cost and thereforedissipates heat 38 . However, this consideration assumes that theinformation about the system to be erased is classical, and doesnotextendto thegeneral case where an observer mayhave quantum

informationabout thesystemtobe erased,for instance bymeans ofa quantum memory entangled with the system. Herewe show that thestandard formulation and implications of Landauers principle areno longer valid in the presence of quantum information. Our mainresult is that theworkcost of erasure is determined by the entropy of thesystem, conditionedon thequantuminformationanobserver hasabout it. In other words, the more an observer knows about thesystem, the less it costs to erase it. This result gives a direct ther-modynamic significance to conditional entropies, originally intro-duced in information theory. Furthermore, it provides new boundson the heat generation of computations: because conditional entro-pies can become negative in the quantum case, an observer who isstrongly correlated with a system may gain work while erasing it,thereby cooling the environment.

Erasure of a system is defined as taking it to a pre-defined purestate, j0 (a familiar example is disk formatting, where a sequence of zero bits is written onto the disk). Landauers principle asserts that theenergy dissipated to erase a system, S, using an optimal process in anenvironment of temperature T is given by

W (S)~ H (S)kT ln(2) 1

where k is the Boltzmann constant 3,813 . Here the von Neumannentropy, H (S) 5 2 Tr[r log 2 (r )], quantifies the uncertainty about sys-tem S, whose state is described by r .

Different observers may have different knowledge about the samesystem. For instance, an observer, Alice, can prepare n quantum bits(qubits; for example n spin-1/2 particles) in a pure state of her choice,keeping a record of that state. However, from the point of view of another observer, Bob, who does not have access to Alices record,thestate of thesystemis completely unknown: he describes it as a fully mixed state, of maximal entropy. Hence, rather than W (S) being defined as the cost of erasure, it may be described as the cost of erasure for observer C , which we denote by W (SjC ). This leads tothe following reformulation of equation (1), where H (SjC ) denotesthe uncertainty that observer C has about system S:

W (SjC )~ H (SjC )kT ln(2) 2

Typically, the observer C is assumed to be classical. More explicitly,classical observers represent their information about S in a memory that consists of classical bits (as in the case of Alice and Bob above).

Our contribution is to go beyond this classical picture and considerobservers who may have access to information that is represented asthestate of a quantum system: a quantum memory. As an example, wecould imagine a third observer, Quasimodo, who has a memory thatincludes n qubits, each maximally entangled with a particle of S.

Our main result characterizes the work, W (SjQ), that an observerwith accessto a quantum memory, Q, needs to performto erase systemS. Forsimplicity,we formulate it here fora special case, which could bereferred to as a thermodynamic limit of erasure, where the observererases many identical copies of S jointly. In this case, we show thatthere exists an erasure process whose work cost does not exceed

W (SjQ)~ H (SjQ)kT ln(2) 3

per copy of S. Here H (SjQ) is the conditional von Neumann entropy,H (SjQ) 5 H (SQ) 2 H (Q). We show that this work cost is optimal,under the assumption that Landauers principle holds for a classicalobserver (Methods Summary). In its general form, our main result(Theorem 1 in Methods Summary) includes the more natural casewhere a single set of data, rather than a collection of identical copies,is to be erased.

Crucially, we require that the information stored in Q be preservedby the erasure processa non-trivial condition, given that accessing aquantum memory can in principle change it. This information-preservation property is particularly important if we consider theerasure process to be part of a larger procedure (see, for instance,Fig. 1). For example, suppose that we erase system S and later wouldlike to erase anothersystem, Z . Iftheerasureof Sremoved the informa-tion about Z , erasing Z could become unnecessarily costly. Moregenerally, we can consider a reference system,R, that models all othersystems about which the memory can haveinformation (technically, Ris a purification of Q and S). The information-preservation conditioncan then be formulated as the requirement that the joint state of thememory and the reference system, r QR, be preserved by the erasureprocess (see also Fig. 2).

The generalizationoferasure to thequantumcaseexposesfeaturesnotpresent in a classical setting. In particular, equation (3) implies that thework required for erasure may be negative for an observer with aquantum memory: theprocessresults ina netgainofwork.Thishappensbecause theuncertainty H (SjQ) canbecome negative forquantumobser- vers. For instance, Quasimodos conditional von Neumann entropy isH (SjQ)

5 0 2 n (because the joint state of S and Q is pure and thereduced state of the memory, Q, is fully mixed). Our result provides athermodynamicoperational meaning fornegative conditionalentropies,which until now only had information-theoretical interpretations; forinstance, they measure the entanglement needed to send a state to areceiver withsideinformation 14 (state merging),andquantify violationsof Heisenbergs uncertainty principle15 .

Theproofof equation (3)uses a probabilistic methodto find appro-priate erasure procedures. In the simple examples of our three obser- vers, Alice, Bob and Quasimodo,we can describe them explicitly.Alice,who holds a classical description of the pure state of S, has no uncer-tainty about the system: H (Sj A) 5 0. As expected, shedoes not need to

* These authors contributed equally to this work.

1 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland. 2 Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK. 3 Centre for Quantum Technologies, National University ofSingapore, 117543 Singapore. 4 Department of Physics, National University of Singapore, 117542 Singapore.

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do any workto erase S: she may consult her record to check the state of S and change it to j0 with a reversible transformation (Fig. 3).

Bob, however, has no access to Alices record and, so, has maximaluncertainty: H (SjB) 5 n. He can perform a simple erasure process10,11that lets the system interact with a heat bath at temperature T (Fig. 4).The work cost of this process is nkT ln(2).

Turning now to Quasimodo, recall that his memory contains nqubits maximally entangled with S. We call this part of his memory Q1 and denote the entangled state jSQ1. In addition, the rest of hismemory, Q2 , is correlated with a referencesystem, R, in state jQ2 R. Toerase S, Quasimodo combines two elementary procedures: the erasureprocess used by Bob, and its reverse, a work-extraction process,whereby he transforms an initially pure state into a maximally mixedstate, gaining energy kT ln(2) perqubit (Fig. 4).He startsby performing the elementary work extraction on the 2n-qubit state jSQ1 to gain

energy 2nkT ln(2), leaving S and Q1 maximally mixed. Then he per-forms the elementary erasure process on S, which costs him nkT ln(2)in energy. In total,his energygain is nkT ln(2), matching the predictionof equation (3). The final state of Quasimodos memory and the ref-erence is r QR~ 2

{ n1Q1 6 jQ2RihQ2Rj, where 22 n1Q 1 is the fully mixed

state on Q1 . This is precisely the initial state of QR, because theoriginalstate of Q1 S had a fully mixed marginal in Q1 the information-preservation condition is therefore satisfied.

In general, correlations between Sand the memoryare not asneatasforQuasimodo. Nevertheless,this specialexamplecontainstheessenceof the general case. Using decoupling results16 , we show that, indepen-

dently of the exact form of the correlations between S and Q, it ispossible to find a subsystem of S and Q that is (approximately) in apure state. This pure subsystem furthermore has a special structurethat allowsus toextract work fromit, thus replacingit with a maximally mixed state, without changing the state of the memory and the ref-erence(Fig. 5). Thesize of the pure state foundand, therefore, thework gained depends on the entropy of S conditioned on the informationheld by the quantum memory (see Supplementary Information,section I, for a formal proof). In its general single-shot form, theerasure procedure we introduce has a failure probability that can bemade arbitrarily small, at thecost of increasing thework performed by an additive term(Theorem1 in MethodsSummary; non-deterministicwork extraction has been discussed before for classical observers17 ).

The erasure processes contemplated in this work require con-siderable control over the quantum systems involved, and it may notbe clear why such idealizations are interesting. As an analogy we canthink of the Carnot cycle. Although the ideal performance of a Carnotengine may be an unattainable limit in practice, it neverthelessprovides the theoretical foundation for the behaviour of real heatengines. In a similar spirit, our investigation bounds the ideal mini-mum work cost for the implementation of irreversible processes.

Such arguments are particularly relevant to the study of heatgeneration in computation, oneof the majorobstacles to theminiatur-ization of circuitry. Computation can in principle be made reversible,

Heat bath

T S

Memory, Q

Battery

Reference

a

b

c

Figure 2 | Erasure setting. An observer,here represented by a machinewith aquantummemory, Q, erasesa system, S. Allmemorycontentsabouta referencesystem must be preserved. We assume that theinitial Hamiltonian of Sand Q isfully degenerate (forexamplelike that of paramagnets in a zero magnetic field).We use a simple model for erasure, with the following options. a , The observermay couple S to a heat bath at temperature T ; the bath thermalizes S, leaving itin a Gibbs state. b, The observer may manipulate S and Q, by (1) applying unitary operations to those systems, and (2) raising or lowering any energy levels of their Hamiltonian (for example by tuning a magnetic field). By raising or lowering an occupied level by D E , the observer uses or, respectively, gainsenergy D E ; empty levels can be changed at no energy cost. c, The observer may store energy in and withdraw it from a battery.

0

E 0 E 0

E E

a b

U

0

Figure 3 | Erasure of a pure state. The circles represent the energy eigenstatesof system S, and a filled circle means that the system can be found in that statewith certainty. a , Alice knows that thesystem is in a particular pure state. b, Sheperforms a unitary transformation, U , that swaps that state with |0. If theHamiltonian of S is degenerate, this operation has no energy cost.

0 0 0

Heat bath Heat bath Heat bath

nkT ln(2)

E 0 E 0E 0

a b c

T T T

Figure 4 | Erasure of a fully mixed state and work extraction. a , Initially, S isin a fully mixed state. b, Keeping one energy level untouched, Bob raises allotherlevels in smallsteps, letting Sthermalize after each step. As these levelsareraised it is more likely to findSin the untouched level. c, Bob decouplesS fromthe bath and lowers all states back to the original level, in one single step. Thework cost of the whole process in the quasistatic limit is kT ln(2) per qubit. By running this process in reverse, we obtain a work-extraction procedure, whichgiven an initial pure state yields an energy gain of kT ln(2) per qubit, at the costof leaving the state maximally mixed.

X F

| x |0 x

| x |f ( x ) x

(2a)

|0 |0

|0

U f

Erase FMeasure X

Erase F

Measure X U X 1 F

U X 1

(1)

(2b)

(3b)(3a)

X nal X

nal

X nal

Figure 1 | Erasure in quantum computation. Asan example, we consider theperiod-finding algorithm for a function f , used in the quantum part of Shorsfactoring algorithm. (1) Initially the algorithm evaluates f on a superposition of all inputs, x , creating an entangled state between two registers, X and F (this isdone with a unitary transformation, U f , on the two registers). Given statisticalknowledge about the properties of f , it is possible to find highly correlatedsubsystems of X and F (in blue). (2a) The second part of the algorithm consistsonly of local operations on X (a unitary transformation, U X , followed by ameasurement; the final stateof X is r final X ). (3a) Usually F is erased at the end of the algorithm, when correlations between X and F have been destroyed. (2b)Instead, it is possible to erase F while it is still partlyentangled with X , using thecorrelations(in blue)to decrease thework cost of theerasure.If the erasure canbe performed in a computationally efficient manner, it may be incorporated inthe algorithm. (3b) The information-preservation condition ensures that therest of thealgorithm is notaffected by the early erasure.For a concrete exampleand further discussion, see Supplementary Information, section V.

RESEARCH LETTER

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but at the expense of keeping extra data in a memory 1,2 . A part of thatmemory may then be erased, keeping the rest intact. Naively, the work cost of that operation, and the corresponding heat generation, wouldbe given solely by the entropy of the part to be erased. However, our

analysis shows that erasure can be optimized if information stored inother parts of the memory is used (Fig. 1).The correspondence between conditional entropy and work that we

found can also be used to quantify quantum correlations 18 . More pre-cisely,because an energygainin erasure relieson entanglement betweenthe system and the memory, an erasure processwith negativeworkcostcanserveas an entanglement witness 19 . Similarly,our work is related todiscord 2023 . Ourresults suggest thatdiscord canquantifythedifferencebetween the respective work costs of erasure using quantum and clas-sical memories24 . Because our relation is valid for a single instance of erasure, it may be used to obtaina single-shot generalizationof discord.

In this work, we have used information-theoretical tools, such asdecoupling 14,16,25 , to prove a physical result. It is also possible to translatethermodynamic statements into information-theoretical ones. For

instance, the work required to erase a system S cannot be reduced by locally processinginformationabout S(seeSupplementaryInformation,Lemma I.6). Using our results, we can infer that the conditional entropy H (SjQ) cannot decrease under local operations on Q, which is a fun-damental result in information theory known as the data processing inequality. In general, we expect our results to strengthen the link between information theory and statistical physics.

METHODS SUMMARYHere we characterize a single instance of erasure. The work cost of erasure,W (SjQ), is a random variable and may fluctuate. Our main result is an upperbound for W (SjQ) that is violated only with a very small probability. This we stateas Theorem 1: there existsa process to erase a system S, conditioned on memory Q,at temperature T , whose work cost satisfies

W (SjQ) H emax

(SjQ)z D kT ln(2)

except with probability less than d~ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2{ D=2z 12ep for all D $ 0 and all e $ 0.The quantity H emax (SjQ) denotes the smooth max-entropy of system S condi-

tioned on the quantum memory, Q, and is a single-shot generalization of the vonNeumann entropy 26 (Supplementary Information, section II).

We can fix d and then adjust e to minimize the work cost. For instance, if weallow a probability of failure of 3%, we pay an extra price of D < 20 in the totalwork cost, independently of the size of S.

Theproof ofTheorem 1 issketchedin Fig. 5 andcan be foundin Supplementary Information, section I.

To understand the exact meaning of equation (3), we consider a thermodyn-amic limit of erasure, wherea largecollection of independent and identicalcopiesof the system is erased. We define the work cost rate as the averagew (SjQ)~ (1=n)W (S6 n jQ6 n) for an optimal erasure process, in the limit of largen (Supplementary Information, Definition I.3). A result from information theory,

the quantum asymptotic equipartition property 27

, asserts that the smooth

max-entropy converges to the von Neumann entropy for many identical copiesof the state (Supplementary Information, section II.B). Combining Theorem 1with the asymptotic equipartition property, we find that there exists an erasureprocess such that w (SjQ)# H (SjQ)kT ln(2); hence, the average work cost neverexceeds W (SjQ) 5 H (SjQ)kT ln(2). Furthermore, if we assume that Landauersprinciple holds for a classical observer (equation (2)), we can show that thequantum bound is tight in this limit (Supplementary Information, Lemma I.5).

Received 24 November 2010; accepted 18 April 2011.

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Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements Wethank R. Colbeck for discussions. We acknowledge supportfrom the Swiss National Science Foundation (L.d.R., J.A., R.R. and O.D.; grantno.200021-119868 and the NCCR QSIT), the Portuguese Fundaa o para a Cie ncia eTecnologia(L.d.R.; grant no. SFRH/BD/43263/2008), the EuropeanResearch Council(R.R.; grant no. 258932) and Singapores National Research Foundation and Ministryof Education (V.V.).

Author Contributions The main ideas were developed by all authors. L.d.R., J.A. andR.R. formulated and proved the main technical claims. L.d.R. and J.A. wrote themanuscript.

Author Information Reprints and permissions information is available atwww.nature.com/reprints . The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should be

addressed to L.d.R. (delrio@phys.ethz.ch) .

Reference

S PS 1 Q

Figure 5 | General erasure procedure. The erasure proceeds in three steps.First we find a subsystem,S1 , that is decoupled from the reference; the reducedstate of S1 is approximately fully mixed16 . The size of S1 is limited by thecorrelationsbetween Sandthe reference. These areweakif Sis highly correlatedwith Q (because the global state is pure28 ), such thatlog 2 ( |S1 |) < [log 2 ( |S|) 2 H (S|Q)]/2. Because S1 is decoupled from thereference, it is purified by a subsystem, P , of S6 Q. The state of S16 P ismaximally entangled. Then we extract work from the pure state of S16 P ,gaining energy 2log 2 ( |S1 |)kT ln(2). The reduced state of P , originally fully mixed, is preserved by this process. Finally we erase S, performing work log 2 ( |S|)kT ln(2). The total work cost of the process is approximately H (S|Q)kT ln(2). See Supplementary Information for details.

LETTER RESEARCH

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CORRECTIONS & AMENDMENTSADDENDUMdoi:10.1038/nature10395

The thermodynamic meaning of negative entropy Ldia del Rio, Johan A berg, Renato Renner, Oscar Dahlsten& Vlatko Vedral

Nature 474, 6163 (2011)

To clarify the implications of our result, we note that, although theerasure processes we considered in our Letter can have negative work cost (that is, they can yield work), they do notviolate the second law of thermodynamics, because they are not cyclic processes. A negativework cost is associated with the consumption of entanglement, whichcanonlybe restoredby doingwork.Ourresults arealso consistent withthe original unconditional form of Landauers principle, which saysthat if thereis no informationavailableaboutthedata beingerased,thecost of erasure is always positive. Similarly, because in a computationwith deterministic classical output the joint entropy of all registersconditioned on the output cannot be negative, the overall work costof such a computation is always positive or zero (even though tem-porary quantum correlations may be created and exploited during thecourse of the computation). In fact, standard techniques of reversibleinformation processing allow any deterministic classical algorithm tobe performed, on a classical or quantum computer, in a thermodyna-mically reversible fashion, with work cost arbitrarily close to zero1,2 .These clarificationsare developed in moredetail in theSupplementary Information to this Addendum. We thank Charles H. Bennett forremarks on reversible computation.Supplementary Information is linked to the online version of the Addendum atwww.nature.com/nature .

1. Bennett, C. H. The thermodynamics of computationa review. Int. J. Theor. Phys.

21, 905940 (1982).2. Watrous, J. Quantum computational complexity. In Encyclopaedia of Complexity and Systems Science (ed. Meyers, R. A.) Part 17, 71747201 (Springer, 2009).

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