Generalized Laws of Thermodynamics in the Presence of Correlations
Manabendra Nath Bera, Arnau Riera, Maciej Lewenstein, Andreas Winter
GGeneralized Laws of Thermodynamics in the Presence of Correlations
Manabendra N. Bera, ∗ Arnau Riera,
1, 2
Maciej Lewenstein,
1, 3 and Andreas Winter
3, 4 ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany ICREA – Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain Departament de Física: Grup d’Informació Quàntica,Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
Abstract
The laws of thermodynamics, despite their wide range ofapplicability, are known to break down when systems are cor-related with their environments. Here, we generalize thermo-dynamics to physical scenarios which allow presence of corre-lations, including those where strong correlations are present.We exploit the connection between information and physics,and introduce a consistent redefinition of heat dissipation bysystematically accounting for the information flow from sys-tem to bath in terms of the conditional entropy. As a con-sequence, the formula for the Helmholtz free energy is ac-cordingly modified. Such a remedy not only fixes the appar-ent violations of Landauer’s erasure principle and the secondlaw due to anomalous heat flows, but also leads to a generallyvalid reformulation of the laws of thermodynamics. In thisinformation-theoretic approach, correlations between systemand environment store work potential. Thus, in this view, theapparent anomalous heat flows are the refrigeration processesdriven by such potentials.
Introduction
Thermodynamics is one of the most successful physicaltheories ever formulated. Though it was initially developedto deal with steam engines and, in particular, the problem ofconversion of heat into mechanical work, it has survived evenafter the scientific revolutions of relativity and quantum me-chanics. Inspired by resource theories, recently developed inquantum information, a renewed e ff ort has been made to un-derstand the foundations of thermodynamics in the quantumdomain [1–11], including its connections to statistical me-chanics [12–14] and information theory [15–25]. However,all these approaches assume that the system is initially uncor-related from the bath. In fact, in the presence of correlations,the laws of thermodynamics can be violated. In particular,when there are inter-system correlations, phenomena such asanomalous heat flows from cold to hot baths [26], and mem-ory erasure accompanied by work extraction instead of heatdissipation [24] become possible. These two examples indi-cate a violation of the second law in its Clausius formulation,and the Landauer’s principle of information erasure [15] re-spectively. Due to the interrelation between the di ff erent lawsof thermodynamics, the zeroth law and the first law can alsobe violated (see Supplementary Note 4 for simple and explicitexamples of these violations). The theory of thermodynamics can be summarized in itsthree main laws. The zeroth law introduces the notion of ther-mal equilibrium as an equivalence relation of states, wheretemperature is the parameter that labels the di ff erent equiv-alence classes. In particular, the transitive property of theequivalence relation implies that if a body A is in equilibriumwith a body B, and B is with a third body C, then A and Care also in equilibrium. The first law assures energy conser-vation. It states that in a thermodynamic process not all of en-ergy changes are of the same nature and distinguishes betweenwork, the type of energy that allows for “useful” operations asraising a weight, and its complement heat, any energy changewhich is not work. Finally, the second law establishes an ar-row of time. It has several formulations and perhaps the mostcommon one is the Clausius statement, which reads: No pro-cess is possible whose sole result is the transfer of heat from acooler to a hotter body. Such a restriction not only introducesthe fundamental limit on how and to what extent various formsof energy can be converted to accessible mechanical work, butalso implies the existence of an additional state function, theentropy, which has to increase. There is also the third law ofthermodynamics; we shall, however, leave it out of the discus-sion, as it is beyond immediate context of the physical scenar-ios considered here.Although the laws of thermodynamics were developed phe-nomenologically, they have profound implications in informa-tion theory. The paradigmatic example is the Landauer era-sure principle, which states: “Any logically irreversible ma-nipulation of information, such as the erasure of a bit or themerging of two computation paths, must be accompanied bya corresponding entropy increase in non-information-bearingdegrees of freedom of the information-processing apparatusor its environment” [17]. Therefore, an erasing operation isbound to be associated with a heat flow to the environment.An important feature in the microscopic regime is that thequantum particles can exhibit non-trivial correlations, such asentanglement [27] and other quantum correlations [28]. Ther-modynamics in the presence of correlations has been consid-ered only in limited physical situations. It is assumed, innearly all cases of thermodynamical processes, that systemand bath are initially uncorrelated, although correlations mayappear in the course of the process. In fact, it has been notedthat in the presence of such correlations, Landauer’s erasureprinciple could be violated [15]. Even more strikingly, withstrong quantum correlation between two thermal baths of dif-ferent temperatures, heat could flow from the colder bath to a r X i v : . [ qu a n t - ph ] D ec the hotter one [26, 29, 30].The impact of inter-system correlations resulting from astrong system-bath coupling and its role in thermodynamicshas been studied for some specific solvable models [31–33],and for general classical systems [34, 35]. It has been notedthat presence of correlations requires certain adjustments ofwork and heat to fulfil the second law and the Landauer prin-ciple. Also, from an information theoretic perspective, bothextractable work from correlations and work cost to createcorrelations have been studied [25, 36–38]. However, in allthese works, there is no explanation of how to deal with gen-eral correlated scenarios irrespective of where the correlationscome from and in systems away from thermal equilibrium.Here we show that the violations of the laws of thermo-dynamics (see Supplementary Note 4) indicate that correla-tions between two systems, irrespective of the correspond-ing marginals being thermal states or not, manifest out-of-equilibrium phenomena. In order to re-establish the lawsof thermodynamics, one not only has to look at the localmarginal systems, but also the correlations between them. Inparticular, we start by redefining the notions of heat and work,then establish a generalized Landauer’s principle and intro-duce the generalized Helmholtz free-energy. The resultinglaws are general in the sense that they rely on the least set ofassumptions to formulate thermodynamics: a system, a con-siderably large thermal bath at well defined temperature, andseparable initial and final Hamiltonians. The first two assump-tions are obvious. The third assumption is basically requiredfor system’s and bath’s energies to be well defined (see Sup-plementary Note 2 for details). ResultsDefinition of heat
To reformulate thermodynamics, we start with redefining heatby properly accounting for the information flow and therebyrestoring Landauer’s erasure principle. In general, heat is de-fined as the flow of energy from the environment, normallyconsidered as a thermal bath at certain temperature, to a sys-tem, in some way di ff erent from work. Work, on the otherhand, is quantified as the flow of energy, say to a bath or to anexternal agent, that could be extractable (or accessible). Con-sider a thermal bath with Hamiltonian H B and at temperature T represented by the Gibbs state ρ B = τ B = Z B exp( − H B kT ),where k is the Boltzmann constant, and Z B = Tr (cid:104) exp( − H B kT ) (cid:105) isthe partition function. The degrees of freedom in B are con-sidered to be a part of a large thermal super-bath, at temper-ature T . Then, for a process that transforms the thermal bath ρ B → ρ (cid:48) B with the fixed Hamiltonian H B , the heat transfer tothe bath is quantified (see Supplementary Note 1) as ∆ Q = − kT ∆ S B , (1)where ∆ S B = S ( ρ (cid:48) B ) − S ( ρ B ) is the change in bath’s von Neu-mann entropy, S ( ρ B ) = − Tr (cid:2) ρ B log ρ B (cid:3) . Note that ρ (cid:48) B is notin general thermal. In fact, the work stored in the bath is ∆ F B , where F ( ρ B ) = E ( ρ B ) − kT S ( ρ B ) is the Helmholtz free en-ergy, with E ( ρ B ) = Tr ( H B ρ B ). Heat expressed in Eq. (1) isthe correct quantification of heat (for further discussion seeSupplementary Note 1), which can be justified in two ways.First, it has a clear information-theoretic interpretation, whichaccounts for the information flow to the bath. Second, it isthe flow of energy to the bath other than work and, with thecondition of entropy preservation, any other form of energyflow to the bath will be stored as extractable work, and thuswill not converted into heat. The process-dependent characterof heat as defined here can be seen from the fact that it can-not be written as a di ff erence of state functions of the system.In the Supplementary Note 1, this issue is discussed and thesources of irreversibility, i.e. the reasons for not saturating theClausius inequality, are re-examined.The transformations considered in our framework are en-tropy preserving operations. More explicitly, given a system-bath setting initially in a state ρ SB , in which the reduced stateof the system ρ S is arbitrary while ρ B is thermal, we considertransformations ρ (cid:48) SB = Λ ( ρ SB ) such that the von Neumann en-tropy is unchanged i. e. S ( ρ (cid:48) SB ) = S ( ρ SB ). The Hamiltoniansof the system and the bath are the same before and after thetransformation Λ ( · ). Note that we do not demand energy con-servation, rather assuming that a suitable battery takes care ofthat. In fact, the work cost of such an operation Λ ( · ) is quan-tified by the global internal energy change ∆ W = ∆ E S + ∆ E B .Another comment to make is that we implicitly assume a bathof unbounded size; namely, it consists of the part ρ B of whichwe explicitly track the correlations with S, but also of arbitrar-ily many independent degrees of freedom. Also, we are im-plicitly considering always the asymptotic scenario of n → ∞ copies of the state in question (“thermodynamic limit”). Theseoperations are general and include any process and situationin standard thermodynamics involving a single bath. It is theresult of abstracting the essential elements of thermodynamicprocesses: existence of a thermal bath and global entropypreservation operations.In extending thermodynamics in correlated scenarios andlinking thermodynamics with information, we consider thequantum conditional entropy as the natural quantity to rep-resent information content in the system as well as in the cor-relations. For a joint system-bath state ρ SB , the informationcontent in the system S, given all the information availablein the bath B at temperature T , is quantified by the condi-tional entropy S (S | B) = S ( ρ SB ) − S ( ρ B ). It vanishes when thejoint system-environment state is perfectly classically corre-lated and can even become negative in the presence of entan-glement. Generalized second law of information
With quantum conditional entropy, the generalized second lawof information can be stated as follows. For an entropy pre-serving operation ρ (cid:48) SB = Λ SB ( ρ SB ), with the reduced statesbefore (after) the evolution denoted ρ S ( ρ (cid:48) S ) and ρ B ( ρ (cid:48) B ), re-spectively, we have ∆ S B = − ∆ S (S | B) , (2)where ∆ S B = S ( ρ (cid:48) B ) − S ( ρ B ) is the change in (von Neumann)entropy of the bath, and ∆ S (S | B) = S ( S (cid:48) | B (cid:48) ) − S (S | B) is thechange in conditional entropy of the system. Note that in thepresence of initial correlations, the informational second lawcould be violated if one considers only system entropy (seeSupplementary Note 3).Let us point out that the conditional entropy of the systemfor a given bath is also used in [24] in the context of eras-ing. There, it is shown that the conditional entropy quanti-fies the amount of work necessary to erase quantum informa-tion. The formalism in [24] considers energy preserving butnon-entropy preserving operations and that perfectly enablesto quantify work. In contrast, in our formalism, as we at-tempt to quantify heat in connection with information flow,it is absolutely necessary to guarantee information conserva-tion, thereby restrict ourselves to entropy preserving opera-tions. This leads us to quantify heat in terms of conditionalentropy. Both approaches are di ff erent and complement eachother. In one, the conditional entropy quantifies work and onthe other, it quantifies heat. Generalized Landauer’s principle
The Landauer principle is required to be expressed in termsof conditional entropy of the system, rather than its local en-tropy. Therefore, the dissipated heat associated to informationerasure of a system S connected to a bath B at temperature T by an entropy preserving operation ρ (cid:48) SB = Λ SB ( ρ SB ), is equalto ∆ Q = kT ∆ S (S | B) . (3)Note that, in complete information erasure, the final condi-tional entropy vanishes, then ∆ Q = − kT S (S | B).
Generalized Helmholtz free-energy
We address extraction of work from a system S possibly corre-lated to a bath B at temperature T . Without loss of generality,we assume that the system Hamiltonian H S is unchanged inthe process. Note that the extractable work has two contribu-tions: one comes from system-bath correlations (cf. [25]) andthe other from the local system alone, irrespective of its corre-lations with the bath. Here we consider these two contributionseparately.By extracting work from the correlation, we mean any pro-cess that returns the system and the bath in the original re-duced states, ρ S and ρ B = τ B . The maximum extractable worksolely from the correlation, using entropy preserving opera-tions, is given by W C = kT I (S : B) , (4)where I (S : B) = S S + S B − S SB is the mutual information.This is illustrated in Fig. 1. The proof is given by the protocoldescribed in Box 1 . HCCCCCSHHHHHHB W + HCCCCCSHHHHHHB W + Figure 1.
Correlations as a work potential.
Correlations can beunderstood as a work potential, as quantitatively expressed in Eq. (4).
Box 1 | Work extraction from correlations.
Addition of an ancillary system : We attach to ρ SB an ancillarysystem A with trivial Hamiltonian H A =
0, consisting of I (S :B) qubits in the maximally mixed state τ A = (cid:16) I (cid:17) ⊗I (S:B) (whichis thermal!). Removing the correlations between S and B : By using aglobal entropy preserving operation, we make a transformation τ A ⊗ ρ SB → τ (cid:48) A ⊗ ρ S ⊗ ρ B , where S (AS | B) τ A ⊗ ρ SB = S (A (cid:48) S | B) τ (cid:48) A ⊗ ρ S ⊗ ρ B , (5)and thereby turning the additional state into a pure state τ (cid:48) A = | φ (cid:105)(cid:104) φ | of A, while leaving the marginal system and bath statesunchanged. Clearly, the extractable work stored in the correla-tion is now transferred to the new additional system state τ (cid:48) A . Work extraction : Work is extracted from τ (cid:48) A at temperature T ,equal to W C = I (S : B) ρ SB kT . Disregarding the correlations with a bath at temperature T ,the maximum extractable work from a state ρ S is given by ∆ W L = F ( ρ S ) − F ( τ S ), where τ S = Z S exp[ − H S kT ] is the cor-responding thermal state of the system in equilibrium withthe bath. Now, in addition to this “local work”, we have thework due to correlations, and so the total extractable work ∆ W S = ∆ W L + kT I (S : B) ρ SB . Note that, for the system alone,the Helmholtz free energy F ( ρ S ) = E S − kT S S . However,in the presence of correlations, it is modified to generalizedHelmholtz free energy, by adding kT I (S : B) ρ SB to F ( ρ S ), as F ( ρ SB ) = E S − kT S (S | B) . (6)Unlike the traditional free energy, the generalized free energyis not only a state function of the system S, but also of thosedegrees of freedom of the bath correlated with it. This is anunavoidable feature of the generalised formalism. Therefore,for a system-bath state ρ SB , maximum extractable work fromthe system can be given as ∆ W S = F ( ρ SB ) − F ( τ S ⊗ τ B ),where F ( τ S ⊗ τ B ) = F ( τ S ). Then, for a transformation,for which initial and final states are ρ SB and σ SB , respec-tively, the maximum extractable work from the system, is ∆ W S = − ∆ F = F ( ρ SB ) − F ( σ SB ). We observe that all thisis of course consistent with what we know from situationswith an uncorrelated bath. Indeed, we can simply make theconceptual step of calling SB “the system”, allowing for ar-bitrary correlations between S and B, with a suitable infinitebath B’ that is uncorrelated from SB. Then, the free energy aswe know it is F ( ρ SB ) = E S − kT S (S | B) + E B − kT S ( τ B ), wherethe first term is the modified free energy in Eq. (6), and thesecond term is the free energy of the bath in its thermal state.As the latter cannot become smaller in any entropy-preservingoperation, the maximum extractable work is − ∆ F . Generalized laws of thermodynamics
Now, equipped with the proper definition of heat (as inEq. (3)) and work (based on generalized free energy in Eq. (6))in the presence of correlations, we put forward the generalizedlaws of thermodynamics.We start with generalized first law, which states: given anentropy preserving operation ρ SB → ρ (cid:48) SB , the distribution ofthe change in the system’s internal energy into work and heatsatisfies ∆ E S = − ( ∆ W S + ∆ F B ) + ( ∆ Q + ∆ F B ) , (7)where the heat dissipated to the bath is given by ∆ Q = − kT ∆ S (S | B), the maximum extractable work from the systemis ∆ W S = − ( ∆ E S − kT ∆ S (S | B)), and the work performed onthe bath is ∆ F B = ∆ E B − kT ∆ S B (cid:62) ∆ W S = − ( ∆ E S − kT ∆ S (S | B)) was shown tobe the maximum extractable work, as it is equal to − ∆ F S . Themaximum work ∆ W S is extracted by thermodynamically re-versible processes. Irreversible processes require that somework is performed on the bath ∆ F B > ∆ F B / T , and in that case heat flow from thebath is exactly equal to the decrease of its internal energy. H CCCCCCHHHHHHH
Figure 2.
Anomalous heat flows.
In the presence of correlations,spontaneous heat flows from cold to hot baths are possible [26]. Thisis an apparent violation of second law, if one ignores the work poten-tial stored in correlation. Otherwise, it is a refrigeration process.
In this new approach, the second law is also modified. TheClausius statement of the generalized second law states thatno process is possible whose sole result is the transfer of heatfrom a cooler to a hotter body, where the work potential storedin the correlations, as defined in Eq. (4), does not decrease.To prove it, consider a state transformation ρ (cid:48) AB = Λ AB ( ρ AB )where Λ AB is an entropy preserving and and energy non-increasing operation. As the thermal state minimizes the freeenergy, the final reduced states ρ (cid:48) S and ρ (cid:48) B have increased theirfree energy, i.e., ∆ E A − kT A ∆ S A (cid:62) ∆ E B − kT B ∆ S B (cid:62) T A / B , ∆ E A / B and ∆ S A / B are the initial temperatures, changes in internal energy and entropy of the baths respec-tively. By adding the former inequalities and considering en-ergy non-increasing, we get T A ∆ S A + T B ∆ S B (cid:54)
0. Due tothe conservation of total entropy, the change in mutual infor-mation is simply ∆ I (A : B) = ∆ S A + ∆ S B , with I (A : B) = S A + S B − S AB . This allows us to conclude − ∆ Q A ( T B − T A ) (cid:62) kT A T B ∆ I (A : B) , (8)which implies Clausius statement of the generalized secondlaw.Note that if the initial state ρ AB is correlated, then thechange in mutual information could be negative, ∆ I (A : B) (cid:54)
0, and − ∆ Q A ( T B − T A ) (cid:54)
0. Note that for T A (cid:54) T B and ∆ I (A : B) (cid:54)
0, there could be a heat flow from the cold tothe hot bath ∆ Q A (cid:62)
0, i.e., an apparent anomalous heat flow.From our new perspective, we interpret the anomalous heatflow as a refrigeration driven by the work potential stored incorrelations. In this case, it is interesting to determine its co-e ffi cient of performance η cop , that from Eq. (8) leads, with thework performed on the hot bath ∆ W C ( T B ) = − kT B ∆ I (A : B),to η cop (cid:66) ∆ Q A ∆ W C ( T B ) (cid:54) T A T B − T A (9)which is nothing else than the Carnot coe ffi cient of perfor-mance (see Fig. 2). Note that we have taken the work valueof the correlations W C with respect to the hot bath T B . This isdue to the fact that for this refrigeration process the hot bathis the one acting as a reservoir.Equation (9) is a nice reconciliation with traditional ther-modynamics. The Carnot coe ffi cient of performance is a con-sequence of the fact that reversible processes are optimal, oth-erwise the perpetual mobile could be build by concatenatinga "better" process and a reversed reversible one. Hence, it isnatural that the refrigeration process driven by the work storedin the correlations preserves Carnot statement of second law. A B C
A CA C F A C Figure 3.
Violation of the zeroth law.
In the presence of correla-tions, the notion of equilibrium is not an equivalence relation. Con-sider 3-party state ρ B ⊗ ρ AC with all marginals thermal states. Thethermal equilibria A (cid:11) B and B (cid:11)
C implies that A, B and C sharethe same temperature. But, in the presence of correlations betweenA and C, that does not assure the equilibrium A (cid:11)
C. Therefore, thetransitive property of the equivalence relation is violated. This is jus-tified, on the right, as F ( ρ AC ) > F ( ρ A ⊗ ρ C ). Thus, the generalizedzeroth law has to overcome these limitations. Now, we reconstruct the zeroth law which can be violatedin the presence of correlations as shown in Fig. 3. To do this,we redefine the notion of equilibrium beyond an equivalencerelation when correlations between systems are present. Thus,the Universal zeroth law states that, a collection { ρ X } X of statesis said to be in mutual thermal equilibrium with each other ifand only if no work can be extracted from any of their combi-nations under entropy preserving operations. This is the caseif and only if all the parties X are uncorrelated and each ofthem is in a thermal state with the same temperature. Discussion
Landauer exorcised Maxwell’s demon and saved the sec-ond law of thermodynamics by taking into account the workpotential of information. In this work, we extend this idea toinclude also the information about the system that is storedin its correlations with the environment. With this approach,we easily resolve the apparent violations of thermodynamicsin correlated scenarios, and generalize it by reformulating itszero-th, first, and second laws.An important remark is that, our generalized thermodynam-ics is formulated in the asymptotic limit of many copies. Arelevant question is how the laws of thermodynamics are ex-pressed for a single system. In our forthcoming paper, wewill address these questions by discussing consistent notionsof one-shot heat, one-shot Landauer erasure, and of one-shotwork extraction from correlations.
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We thank R. Alicki, R. B. Harvey, K. Gawedzki, P. Grang-ier, J. Kimble, V. Pellegrini, R. Quidant, D. Reeb, F. Schmidt-Kaler, P. Walther and H. Weinfurter for useful discussionsand comments in both theoretical and experimental aspectsof our work. We acknowledge financial support from the Eu-ropean Commission (FETPRO QUIC (H2020-FETPROACT-2014 No. 641122), STREP EQuaM and STREP RAQUEL),the European Research Council (AdG OSYRIS and AdGIRQUAT), the Spanish MINECO (grants no. FIS2008-01236, FIS2013-46768-P FOQUS, FISICATEAMO FIS2016-79508-P and Severo Ochoa Excellence Grant SEV-2015-0522) with the support of FEDER funds, the Generalitat deCatalunya (grants no. SGR 874, 875 and 966), CERCA Pro-gram / Generalitat de Catalunya and Fundació Privada Cellex.AR also thanks support from the CELLEX-ICFO-MPQ fel-lowship.
Author contributions
M. N. B. and A. R. have equally contributed to this work. M. L. and A. W. have supervised the project. All authorsdiscussed the results and contributed to the final manuscript.
SUPPLEMENTARY INFORMATIONSUPPLEMENTARY NOTE 1:DEFINITIONS OF HEAT
In the main text, the heat dissipated in a process involving asystem and a bath B has been defined as ∆ Q = kT ∆ S B , suchas the common description: “flow of energy to a bath someway other than through work” suggests. Note, however, thatthis is not the most extended definition of heat that one finds inmany works, e.g., [ ? ? ], where heat is defined as the changein the internal energy of the bath, i. e. ∆ ˜ Q (cid:66) − ∆ E B , (10)and no di ff erent types of energy are distinguished in this in-crease of energy. In this section, we compare these two def-initions and argue why the approach taken here, though lessextended, seems the most appropriate.The ambiguity in defining heat comes from the di ff erentways in which the change in the internal energy of the sys-tem E S can be decomposed. More explicitly, let us con-sider a unitary process U SB acting on a system-bath state ρ SB with ρ B = Tr S ρ SB = τ B ∝ e − H B / kT and global Hamiltonian H = H S ⊗ I + I ⊗ H B . The change in the total internal en-ergy ∆ E SB is the sum of system and bath internal energies ∆ E SB = ∆ E S + ∆ E B , or equivalently ∆ E S = ∆ E SB − ∆ E B . (11)Many text-books identify in this decomposition ∆ W (cid:66) − ∆ E SB as work and ∆ ˜ Q = − ∆ E B as heat. Nevertheless, notethat it also assigns to heat increases of the internal energy thatare not irreversibly lost and can be recovered when having abath at our disposal.To highlight the incompleteness of the above definition, letus consider a reversible process U SB = I ⊗ U B that acts triviallyon the system. Then, even though the state of the system isuntouched in such a process, the amount of heat dissipated is ∆ ˜ Q = − ∆ E B = Tr [ H B ( ρ B − U B ρ B U † B )].In order to avoid this kind of paradoxes and in the spirit ofthe definition given above, we subtract from ∆ E B its compo-nent of energy that can still be extracted (accessed). Then fora transformation ρ B → ρ (cid:48) B , the heat transferred is given as ∆ Q = − ( ∆ E B − ∆ F B ) , = − kT ∆ S B , (12)where ∆ F B = F ( ρ (cid:48) B ) − F ( ρ B ) is the work stored on thebath and can be extracted. Here, F ( ρ X ) = E X − kT S ( ρ X )is the Helmholtz free energy, E X is the internal energy and ∆ S B = S ( ρ (cid:48) B ) −S ( ρ B ) is the change in the bath’s von Neumannentropy, S ( ρ B ) = − Tr (cid:2) ρ B log ρ B (cid:3) . Throughout this work, weconsider log as the unit of entropy.Let us remark that in practical situations, in the limit oflarge baths, both definitions coincide. To see it, take Supple-mentary Eq. (12) and note that both definitions only di ff er in the free energy di ff erence term, which together with the factthat the free energy is minimized by the thermal state, impliesthat the di ff erence is very small when the bath is slightly per-turbed. However, when studying thermodynamics at the quan-tum regime with small machines approaching the nanoscalesuch conceptual di ff erences are crucial to extend, for instance,the domain of standard thermodynamics to situations wherethe correlations become relevant.Note finally that both definitions express a path dependentquantity of the system like heat in terms of a di ff erence of statefunctions of the bath. The path dependence character comesfrom the fact that there are several processes that leave thesystem in the same state but the bath in a di ff erent one. Thisconnects with Clausius inequality, which is usually stated as (cid:73) d QT (cid:54) ∆ S S = − ∆ S B + ∆ I (S : B) = ∆ QT + ∆ I (S : B) (14)where I (S : B) = S S + S B −S SB is the mutual information. Foran initially uncorrelated system-bath, the mutual informationcan only increase ∆ I (S : B) ≥
0, and ∆ QT (cid:54) ∆ S S . (15)For the definition of heat as an increase of the internal en-ergy, we have ∆ ˜ Q = ∆ Q − ∆ F B (cid:54) ∆ Q , (16)where we have used Supplementary Eq. (12) and the positivityof the free energy change. In sum, for the case of initiallyuncorrelated states, we recover the Clausius inequality, ∆ ˜ Q (cid:54) ∆ Q (cid:54) T ∆ S . (17)The deficit for the first inequality to be saturated is ∆ F B , thatis, the energy that can still extracted from the bath. If one has alimited access to the bath, an apparent relaxation process willfollow and the bath will thermalize keeping its energy con-stant. This will imply an entropy increase of the bath ∆ F B / T which will make ∆ ˜ Q and ∆ Q coincide.The deficit to saturate the second inequality in Supplemen-tary Eq. (17) is ∆ I (S : B), that is, the amount of enabled corre-lations during the process. One of the main ideas of this workis to show that these correlations capture a free energy that canbe extracted. SUPPLEMENTARY NOTE 2:SET OF OPERATIONS
The set of operations that we consider in this manuscript isthe so called entropy preserving operations. Given a systeminitially in a state ρ , the set of entropy preserving operationsare all the operations that change arbitrarily the state but keepits entropy constant ρ → σ : S ( ρ ) = S ( σ ) , (18)where S ( ρ ) (cid:66) − Tr ( ρ log ρ ) is the Von Neumann entropy. Itis important to note that an operation that acting on ρ pro-duces a state with the same entropy does not mean that willalso preserve entropy when acting on other states. In otherwords, such entropy preserving operations are in general notlinear, since they have to be constraint to some input state. Infact, in [39], it is shown that a quantum channel Λ ( · ) that pre-serves entropy and respects linearity, i. e. Λ ( p ρ + (1 − p ) ρ ) = p Λ ( ρ ) + (1 − p ) Λ ( ρ ), has to be necessarily unitary.One could think then that the extension of the unitaries to aset of entropy preserving operations is rather unphysical sincethey are not linear. However, they can be microscopically de-scribed by global unitaries in the limit of many copies [40].That is, given any two states ρ and σ with equal entropies S ( ρ ) = S ( σ ), then there exists a unitary U and an additionalsystem of O ( (cid:112) n log n ) ancillary qubits such thatlim n →∞ (cid:107) Tr anc (cid:16) U ρ ⊗ n ⊗ η U † (cid:17) − σ ⊗ n (cid:107) = , (19)where (cid:107) · (cid:107) is the one-norm and the partial trace is performedon the ancillary qubits. The reverse statement is also true, i. e.if two states can be related as in Supplementary Eq. (19) thenthey have equal entropies. This is proven in Theorem 4 ofRef. [40].Sometimes it can be interesting to restrict entropy preserv-ing operations to also be energy preserving. The set of en-ergy and entropy preserving channels can also be describedas a global energy preserving unitary in the many copy limit.More explicitly, in Theorem 1 of Ref. [40], it is proventhat two states ρ and σ having equal entropies and energies( S ( ρ ) = S ( σ ) and E ( ρ ) = E ( σ )) is equivalent to the existenceof some U and an additional system A with O ( (cid:112) n log n ) an-cillary qubits with Hamiltonian (cid:107) H A (cid:107) (cid:54) O ( n / ) in some state η for which Supplementary Eq. (19) is fulfilled. Note that theamount of energy and entropy of the ancillary system per copyvanishes in the large n limit.In sum, considering the set of entropy preserving operationsmeans implicitly taking the limit of many copies and globalunitaries. In addition, as that the set of entropy preservingoperations contains the set of unitaries, any constraint that ap-pears as a consequence of entropy preservation will be alsorespected by individual quantum systems.The Hamiltonians of the system and the bath are the samebefore and after the transformation Λ ( · ). This can be donewithout loss of generality since, when this is not the case and the final Hamiltonian is di ff erent from the initial one, thetwo situations are related by a simple quench (instantaneouschange of the Hamiltonian). More explicitly, let us considera process (a) with equal initial and final Hamiltonian, and anidentical process (b) with di ff erent ,(a) ( H , ρ i ) → ( H , ρ f ) (20)(b) ( H , ρ i ) → ( H (cid:48) , ρ f ) (21)where ρ i / f is the inital / final state, H the inital Hamiltonian and H (cid:48) the final Hamiltonian of the process with di ff erent Hamil-tonians. Then, it is trivial to relate the work and heat involvedin both processes W (cid:48) = W + Tr (cid:0) ( H − H (cid:48) ) ρ f (cid:1) (22) Q (cid:48) = Q , (23)where W (cid:48) and Q (cid:48) are the work and heat associated to the pro-cess (b) and we have only used that the process (b) is the com-position of the process (a) followed by a quantum quench.Let us finally point out that initially and finally the Hamilto-nians of system and bath are not interacting, or in other words,the system is decoupled from the bath H = H S ⊗ I + I ⊗ H B , (24)with H S / B the Hamiltonian of the system / bath. This is a nec-essary condition to be able to consider system and bath as in-dependent systems each with a well defined notion of energy.Otherwise, assigning an energy to the system and to the bathwould not be possible beyond the weak coupling limit. Notethat the system and the bath interact (arbitrarily strongly) dur-ing the process, in which for instance a non-product unitarycould be performed. SUPPLEMENTARY NOTE 3:THE LANDAUER PRINCIPLE
The information theory and statistical mechanics have long-standing and intricate relation. In particular, to exorciseMaxwell’s demon in the context of statistical thermodynam-ics, Landauer first indicated that information is physical andany manipulation of that has thermodynamic cost. As putforward by Bennett [ ? ], the Landauer information erasureprinciple (LEP) implies that “any logically irreversible ma-nipulation of information, such as the erasure of a bit or themerging of two computation paths, must be accompanied bya corresponding entropy increase in non-information-bearingdegrees of freedom of the information-processing apparatusor its environment.”Following the definition of heat, it indicates that, in suchprocesses, entropy increase in non-information-bearing de-grees of freedom of a bath is essentially associated with flowof heat to the bath. The major contribution of this work is toexclusively quantify heat in terms of flow of information, in-stead of counting it with the flow of non-extractable energy,the work. To establish this we start with the case of infor-mation erasure of a memory. Consider a physical processwhere an event, denoted with i , happens with the probability p i . Then storing (classical) information memorizing the pro-cess means constructing a d -dimensional system (a memory-dit) in a state ρ S = (cid:80) i p i | i (cid:105)(cid:104) i | , where {| i (cid:105)} are the orthonor-mal basis correspond to the event i . In other words, memoriz-ing the physical process is nothing but constructing a memorystate ρ S = (cid:80) i p i | i (cid:105)(cid:104) i | from a memoryless state | i (cid:105)(cid:104) i | where i could assume any values 1 (cid:54) i (cid:54) d . On the contrary, pro-cess of erasing requires the transformation of a memory state ρ S = (cid:80) i p i | i (cid:105)(cid:104) i | to a memoryless state | i (cid:105)(cid:104) i | for any i . Lan-dauer’s erasure principle (LEP) implies that erasing informa-tion, a process involving a global evolution of the memory-ditsystem and its environment, is inevitably associated with anincrease in entropy in the environment.In establishing the connection between information eras-ing and heat dissipation, we make two assumptions to startwith. First, the memory-system (S) and bath (B) are both de-scribed by the Hilbert space H S ⊗ H B . Secondly, the eras-ing process involves entropy preserving operation Λ SB , i.e., ρ (cid:48) SB = Λ SB ( ρ SB ). The latter assumption is most natural andimportant, as it preserves information content in the jointmemory-environment system. Without loss of generality, onecan further assume that the system and bath Hamiltonians re-main unchanged throughout the erasing process, to ease thederivations.Now we consider the simplest information erasing scenario,which leads to LEP in its traditional form. In this scenario, asystem ρ S is brought in contact with a bath ρ B and the systemis transformed to a information erased state, say | (cid:105)(cid:104) | S , byperforming a global entropy-preserving operation Λ SB , i.e., ρ S ⊗ ρ B Λ SB −−−→ | (cid:105)(cid:104) | S ⊗ ρ (cid:48) B , (25)where initial and final joint system-bath states are uncorre-lated. The joint operation guarantees that the decrease insystem’s entropy is exactly equal to the increase in bath en-tropy and heat dissipated to the bath is ∆ Q = − kT ∆ S B . Itclearly indicates that an erasure process is expected to heat upthe bath. This in turn also says that ∆ Q = kT ∆ S S , where ∆ S S = S ( ρ (cid:48) S ) − S ( ρ S ). In the case where the d -dimensionalsystem memorizes maximum information, or in other wordsit is maximally mixed and contains log d bits of information,the process dissipates an amount kT log d of heat to com-pletely erase the information. In other words, to erase one bitof information system requires the dissipation of kT of heatand we denote it as one heat-bit or (cid:96) -bit (in honour of Lan-dauer).In the case where the final state may be correlated, the dis-sipated heat in general is lower bounded by the entropy reduc-tion in the system, i.e., | ∆ Q | (cid:62) kT | ∆ S S | . (26)This is what is generally known as the Landauer’s erasureprinciple (LEP), in terms of heat. The above formulation of LEP crucially relies on the factthat any change in system entropy leads to a larger changein the bath entropy, which is also traditionally known as thesecond law for the change in the information, i.e., ∆ S B (cid:62) − ∆ S S . (27)However, it is limited by the assumptions made above and canbe violated with initial correlations. Consider the examplesin section of the Supplementary Information. In both the ex-amples, ∆ S B (cid:3) − ∆ S S . Therefore, one has to replace it withuniversal informational second law. SUPPLEMENTARY NOTE 4:VIOLATIONS OF LAWS OF THERMODYNAMICS
In order to highlight how the laws of thermodynamics breakdown in the presence of correlations, let us discuss the follow-ing two examples. In the first, the system S is purely classi-cally correlated with the bath B at temperature T , while inthe other they are jointly in a pure state and share quantumentanglement. In both the examples the Hamiltonians of thesystem and bath ( H S and H B ) remain unchanged throughoutthe processes. Example 1 – Classical correlations. ρ SB = (cid:88) i p i | i (cid:105)(cid:104) i | S ⊗ | i (cid:105)(cid:104) i | B U c SB −−−→ ρ (cid:48) SB = | φ (cid:105)(cid:104) φ | S ⊗ (cid:88) i p i | i (cid:105)(cid:104) i | B , Example 2 – Entanglement. | Ψ (cid:105) SB = (cid:88) i √ p i | i (cid:105) S | i (cid:105) B U e SB −−−→ | Ψ (cid:105) (cid:48) SB = | φ (cid:105) S ⊗ | φ (cid:105) B , where in both examples | φ (cid:105) X = (cid:80) i √ p i | i (cid:105) X with X ∈ { S , B } and 1 > p i ≥ i . Note that the unitaries, U c SB and U e SB ,leave the local energies of system and bath unchanged, and U c SB does not change the bath state. Violations of first law
In Example 1, the Helmholtz free energy of the systemincreases F ( | φ (cid:105) S ) > F ( ρ S ) and therefore a work − ∆ W S =∆ F S > ∆ E S (cid:44) − ∆ W S + ∆ Q , i.e. the energy conservation is violatedand so the first law.A further violation can also be seen in Example 2 involvingsystem-bath quantum entanglement. In this case, a non-zerowork − ∆ W S = ∆ F S > ∆ E S (cid:44) − ∆ W S + ∆ Q .0 Violations of second law and anomalous heat flows
We now show how correlation could result in a violationof the Kelvin-Planck statement of the second law, whichstates: No process is possible whose sole result is the ab-sorption of heat from a reservoir and the conversion of thisheat into work. In Example 1, no change in the local bathstate indicates that there is no transfer of heat. However, thechange in the Helmholtz free energy of the local system is − ∆ W S = ∆ F S >
0. Thus, a non-zero amount of work is per-formed on the system without even absorbing heat from thebath ( ∆ Q = − ∆ W S = ∆ F S > ∆ Q = − ∆ E B (see [ ? ] andreferences therein). Here we show that such violations arealso there with new heat definition ∆ Q = − kT ∆ S B . Let ρ AB ∈ H A ⊗ H B be an initial bipartite finite dimensionalstate whose marginals ρ A = Tr B ρ AB = Z A exp[ − H A kT A ] and ρ B = Z B exp[ − H B kT B ] are thermal states at di ff erent temperatures T A and T B and with Hamiltonians H A and H B . In absence ofinitial correlations between the baths A and B, any energy pre-serving unitary will respect Clausius’ statement of the secondlaw. However, if initial correlations are present, this will notbe necessarily the case.Consider a state transformation ρ (cid:48) SB = U AB ρ AB U † AB where U AB is a energy preserving unitary acting on ρ AB . As the ther-mal state minimizes the free energy, the final reduced states ρ (cid:48) S and ρ (cid:48) B have increased their free energy, ∆ E A − kT A ∆ S A ≥ ∆ E B − kT B ∆ S B ≥ , (29)where T A / B is the initial temperature of the baths, and ∆ E A / B and ∆ S A / B are the change in internal energy and entropy re-spectively.By adding Supplementary Eqs. (28) and (29), and consid-ering energy conservation, we get T A ∆ S A + T B ∆ S B ≤ . (30)Due to the conservation of total entropy, the change in mutualinformation is simply ∆ I (A : B) = ∆ S A + ∆ S B , with I (A :B) = S A + S B − S AB . This allows us to rewrite SupplementaryEq. (30) in terms of only the entropy change in A as( T A − T B ) ∆ S A ≤ − T B ∆ I (A : B) . (31)If the initial state ρ AB = ρ A ⊗ ρ B is uncorrelated, then thechange in mutual information is necessarily positive ∆ I (A : B) ≥
0, and k ( T A − T B ) ∆ S A = − ∆ Q A T A − T B T A ≤ . (32)To see that this equation is precisely the Clausius statement,consider without loss of generality that A is the hot bath and T A − T B >
0. Then, Supplementary inequality (32) impliesan entropy reduction of the hot bath ∆ S A ≤ ∆ I (A : B) <
0, andSupplementary Eq. (31) allows a heat flow from the cold bathto the hot one.
Violations of zeroth law
The zeroth law establishes the notion of thermal equilib-rium as an equivalence relation, in which temperature labelsthe di ff erent equivalent classes. To see that the presence ofcorrelations also invalidates the zeroth law, we show that thetransitive property of the equivalence relation is not fulfilled.Consider a bipartite system AC in an initial correlated state ρ AC , like in Examples 1 and 2, and a third party B which isin a thermal state at the same temperature of the marginals ρ A and ρ C . Then, while the subsystems AB and BC are mutuallyin equilibrium, the subsystems AC are not, clearly violatingtransitivity. There are several ways to realize that the partiesAC are not in equilibrium. One way is to see that any en-ergy preserving unitary, except for the identity, decreases theamount of correlations between the parties, ∆ I (A : C) < F ( ρ AC ) > F ( ρ A ⊗ ρ C ). Violations of Landauer’s erasure principle
Another thermodynamic principle that breaks down whencorrelations are present is Landauer’s erasure principle. Lan-dauer postulated that in order to erase one bit of informationin the presence of a bath at temperature T , an amount of heatneeded to be dissipated is kT log 2. As noted in [ ??