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Universal Laws of Thermodynamics

Manabendra N. Bera,1, ∗Arnau Riera,1Maciej Lewenstein,1, 2 and Andreas Winter2, 3

1ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain

2ICREA – Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain

3Departament de Física: Grup d’Informació Quàntica,

Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain

Thermodynamics is one of the most successful physical theories ever formulated. Though it was initially

developed to deal with steam engines and, in particular, the problem of conversion of heat into mechanical

work, it has prevailed even after the scientiﬁc revolutions of relativity and quantum mechanics. Despite its wide

range of applicability, it is known that the laws of thermodynamics break down when systems are correlated

with their environments. In the presence of correlations, anomalous heat ﬂows from cold to hot baths become

possible, as well as memory erasure accompanied by work extraction instead of heat dissipation.

Here, we generalize thermodynamics to physical scenarios which allow presence of correlations, including

those where strong correlations are present. We exploit the connection between information and physics, and

introduce a consistent redeﬁnition of heat dissipation by systematically accounting for the information ﬂow

from system to bath in terms of the conditional entropy. As a consequence, the formula for the Helmholtz free

energy is accordingly modiﬁed. Such a remedy not only ﬁxes the apparent violations of Landauer’s erasure

principle and the second law due to anomalous heat ﬂows, but it also leads to a reformulation of the laws of

thermodynamics that are universally respected. In this information-theoretic approach, correlations between

system and environment store work potential. Thus, in this view, the apparent anomalous heat ﬂows are the

refrigeration processes driven by such potentials.

I. INTRODUCTION

Thermodynamics lays one of the basic foundations of our

current understanding of the physical world. It prevails de-

spite the revolutions that took place in science through rela-

tivity and quantum mechanics. Beyond its own domain, so

far, it has been successfully applied to black holes [1,2], tiny

quantum engines comprised of only a few qubits [3–6]. It was

initially developed to deal with macroscopic heat engines, in

particular, to investigate conversion of energy into mechanical

work and heat, long before quantum mechanics had been de-

veloped. It is therefore plausible that thermodynamics in the

microscopic regime, where the quantum properties dominate,

departs signiﬁcantly from its macroscopic counterpart. This

is indeed the case. Inspired by resource theories [7–13], re-

cently developed in the domain of quantum information [14–

16], a renewed eﬀort has been made to understand the founda-

tions of thermodynamics in the quantum domain, giving rise

to many interesting results [17–27], including its connections

to statistical mechanics [28–31] and information theory [32–

42]. Studying extractable work from a thermal machine in-

volving individual quantum system not only requires modi-

ﬁed forms of free energies [18,19], but also relies on many

second laws to dictate state transformations [22,24,25]. No-

tably, all the results converge to their macroscopic analogues

for large ensemble of identical systems [17]. Heat, in the mi-

croscopic regime, was initially studied in connection with a

system’s information by Landauer [39] to validate the second

law by exorcising Maxwell’s demon [34,37,38], which led

to Landauer’s erasure principle. It was then further extended

to information theory in the context of information erasure

[32–36,41]. Heat dissipation due to erasing information from

∗mnbera@gmail.com

individual quantum systems has recently been studied system-

atically [32].

An important feature in the microscopic regime is that the

quantum particles can exhibit non-trivial correlations, such as

entanglement [9] and other quantum correlations [43]. Ther-

modynamics in the presence of correlations has been consid-

ered only in limited physical situations. It is assumed, in

nearly all cases of thermodynamical processes, that system

and bath are initially uncorrelated, although correlations may

appear in the course of the process. In fact, it has been noted

that in the presence of such correlations, Landauer’s erasure

principle could be violated [32]. Even more strikingly, with

strong quantum correlation between two thermal baths of dif-

ferent temperatures, heat could ﬂow from the colder bath to

the hotter one [44–47]. In other words, the fundamental laws

of thermodynamics are violated in the presence of system-

bath correlations. There is an eﬀort to resolve this issue using

the concept of relative thermalization [48], but it is far from

being a complete remedy to the violations of the fundamental

laws.

The aim of the present work is to re-establish the laws of

thermodynamics in all possible physical scenarios, including

initial inter-system correlations. Our ﬁrst step employs a con-

sistent redeﬁnition of heat dissipation based on a systematic

account of information ﬂow. Thereby we establish a con-

crete inter-relation between thermodynamics and information.

This information-theoretic approach towards thermodynamics

also results in a redeﬁnition of the Helmholtz free energy, and

therefore a consistent quantiﬁcation of work. Our approach

not only ﬁxes the apparent violations of Landauer’s erasure

principle and the second law due to anomalous heat ﬂows, but

also leads to reformulations of the laws of thermodynamics

that are universally respected.

The remainder of the article is organized as follows. We

start with brieﬂy reviewing the laws of thermodynamics in

arXiv:1612.04779v1 [quant-ph] 14 Dec 2016

2

section II, and deﬁning the notion of heat in section III, where

we also justify why it should be the correct notion compared

to the others considered in the literature. Then in section IV,

we sketch the violations of the laws of thermodynamics and

Landauer’s erasure principle in the presence of system-bath

correlations. To resolve these contradictions, we propose uni-

versal laws of thermodynamic, under which we can restore

Landauer’s erasure principle and the laws of thermodynamics

by systematically accounting for the information stored in the

correlations, in section V. We conclude in section VI.

II. LAWS OF THERMODYNAMICS

The theory of thermodynamics can be summarized in its

three main laws. The zeroth law introduces the notion of ther-

mal equilibrium as an equivalence relation of states, where

temperature is the parameter that labels the diﬀerent equiv-

alence classes. In particular, the transitive property of the

equivalence relation implies that if a body Ais in equilibrium

with a body B, and Bis with a third body C, then Aand C

are also in equilibrium. The ﬁrst law assures energy conser-

vation. It states that in a thermodynamic process not all of

energy changes are of the same nature and distinguishes be-

tween work, the type of energy that allows for “useful” opera-

tions as raising a weight, and its complement heat, any energy

change which is not work. Finally, the second law establishes

an arrow of time. It has several formulations and perhaps the

most common one is the Clausius statement, which reads: No

process is possible whose sole result is the transfer of heat

from a cooler to a hotter body. Such a restriction not only

introduces the fundamental limit on how and to what extent

various forms of energy can be converted to accessible me-

chanical work, but also implies the existence of an additional

state function, the entropy, which has to increase. There is

also third law of thermodynamics; we shall, however, leave it

out of the discussion, as it is beyond immediate context of the

physical scenarios 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 the

merging of two computation paths, must be accompanied by

a corresponding entropy increase in non-information-bearing

degrees of freedom of the information-processing apparatus

or its environment” [34]. Therefore, an erasing operation is

bound to be associated with a heat ﬂow to the environment.

III. HEAT

In thermodynamics, heat is deﬁned as the ﬂow of energy

from a system to its environment, normally considered as a

thermal bath at certain temperature, in some way diﬀerent

from work. Work, on the other hand, is quantiﬁed as the

ﬂow of energy, say to a bath or to an external agent, that

could be extractable (or accessible). Operationally, it is en-

ergy that can be used to raise a weight or similar battery sys-

tem. Therefore, the diﬀerence between change in internal en-

ergy and extractable work quantiﬁes the heat ﬂow to the bath.

More explicitly, let us consider a thermal bath with Hamilto-

nian HBand at temperature Trepresented by the Gibbs state

ρB=τB=1

ZBexp( −HB

kT ), where kthe Boltzmann constant and

ZB=Tr hexp(−HB

kT )iis the partition function. Then for a pro-

cess that transforms the thermal bath ρB→ρ0

Bwith the ﬁxed

Hamiltonian HB, the heat transfer to the bath is quantiﬁed (see

section A) as

∆Q=kT ∆SB,(1)

where ∆SB=S(ρ0

B)− S(ρB) is the change in bath’s von Neu-

mann entropy, S(ρB)=−Tr ρBlog2ρB. The work stored

in the bath is ∆FB, where F(ρB)=E(ρB)−kT S(ρB) is the

Helmholtz free energy, with E(ρB)=Tr (HBρB). Heat ex-

pressed in Eq. (1) is the correct quantiﬁcation of heat, which

can be justiﬁed in two ways. First, it has a clear information-

theoretic interpretation, which accounts for the information

ﬂow to the bath. Second, it is the ﬂow of energy to the bath

other than work and, with the condition of entropy preserva-

tion, any other form of energy ﬂow to the bath will be stored

as extractable work, and thus not converted into heat (see also

section V C). Note that there is another deﬁnition of heat that

can be found in the literature, e.g. [32,46], where heat is de-

ﬁned as the change in the internal energy of the bath. In ap-

pendix A, we extensively compare the two deﬁnitions; both

coincide in the scenario of a very large bath that always re-

mains in equilibrium.

IV. VIOLATIONS OF LAWS OF THERMODYNAMICS

In order to highlight how the laws of thermodynamics break

down in the presence of correlations, let us discuss the follow-

ing two examples. In the ﬁrst, the system Sis purely classi-

cally correlated with the bath Bat temperature T, while in

the other they are jointly in a pure state and share quantum

entanglement. In both the examples the Hamiltonians of the

system and bath (HSand HB) remain unchanged throughout

the processes.

Example 1 – Classical correlations.

ρS B =X

i

pi|iihi|S⊗ |iihi|B

Uc

S B

−−−→ ρ0

S B =|φihφ|S⊗X

i

pi|iihi|B,

Example 2 – Entanglement.

|ΨiS B =X

i

√pi|iiS|iiB

Ue

S B

−−−→ |Ψi0

S B =|φiS⊗ |φiB,

where in both examples |φiX=Pi√pi|iiXwith X∈ {S,B}

and 1 >pi≥0 for all i. Note that the unitaries, Uc

S B and Ue

S B,

leave the local energies of system and bath unchanged, and

Uc

S B does not change the bath state.

3

1. Violations of ﬁrst law

In Example 1, the Helmholtz free energy of the system

increases F(|φiS)>F(ρS) and therefore a work −∆WS=

∆FS>0 is performed on the system. To assure the energy

conservation of the system, an equal amount of heat is re-

quired to be transferred to the bath. Surprisingly, however, no

heat is transferred to the bath as it remains unchanged. Thus

∆ES,−∆WS−∆Q, i.e. the energy conservation is violated

and so the ﬁrst law.

A further violation can also be seen in Example 2 involving

system-bath quantum entanglement. In this case, a non-zero

work −∆WS= ∆FS>0 has been performed on the system,

and a heat ﬂow to the bath is expected. In contrast, there is a

negative heat ﬂow to the bath! Therefore, it violates the ﬁrst

law, i.e. ∆ES,−∆WS−∆Q.

2. Violations of second law and anomalous heat ﬂows

We now show how correlation could result in a viola-

tion of the Kelvin-Planck statement of the second law, which

states: No process is possible whose sole result is the ab-

sorption of heat from a reservoir and the conversion of this

heat into work. In Example 1, no change in the local bath

state indicates that there is no transfer of heat. However, the

change in the Helmholtz free energy of the local system is

−∆WS= ∆FS>0. Thus, a non-zero amount of work is per-

formed on the system without even absorbing heat from the

bath (∆Q=0).

The situation becomes more striking in Example 2, with ini-

tial system-bath entanglement. In this case, −∆WS= ∆FS>0

amount of work is performed on the system. However, not

only is there no heat ﬂow from the bath to the system, but

there is a negative heat ﬂow to the bath! Thus, the second law

is violated.

We next see how the presence of correlations can lead to

anomalous heat ﬂows and thereby a violation of the Clau-

sius statement based second law (see Fig. 1). Such violations

were known for the other deﬁnition of heat ∆Q= ∆EB(see

[46] and references therein). Here we show that such viola-

tions are also there with new heat deﬁnition ∆Q=kT ∆SB.

Let ρAB ∈ HA⊗ HBbe an initial bipartite ﬁnite dimensional

state whose marginals ρA=Tr BρAB =1

ZAexp[−HA

kTA] and

ρB=1

ZBexp[−HB

kTB] are thermal states at diﬀerent temperatures

TAand TBand with Hamiltonians HAand HB. In absence of

initial correlations between the baths Aand B, any energy pre-

serving unitary will respect Clausius’ statement of the second

law. However, if initial correlations are present, this will not

be necessarily the case.

Consider a state transformation ρ0

S B =UA BρAB U†

AB where

UAB is a energy preserving unitary acting on ρAB. As the ther-

mal state minimizes the free energy, the ﬁnal reduced states

ρ0

Sand ρ0

Bhave increased their free energy,

∆EA−kTA∆SA≥0 (2)

∆EB−kTB∆SB≥0,(3)

HC

CCCCC

HHHHHHH

Figure 1. In presence of correlations, spontaneous heat ﬂows from

cold to hot baths are possible [46]. This is a clear violation of second

law.

where TA/Bis the initial temperature of the baths, and ∆EA/B

and ∆SA/Bare the change in internal energy and entropy re-

spectively.

By adding Eqs. (2) and (3), and considering energy conser-

vation, we get

TA∆SA+TB∆SB≤0.(4)

Due to the conservation of total entropy, the change in mutual

information is simply ∆I(A:B)= ∆SA+ ∆SB, with I(A:

B)=SA+SB−SAB. This allows us to rewrite Eq. (4) in terms

of only the entropy change in Aas

(TA−TB)∆SA≤ −TB∆I(A:B).(5)

If the initial state ρAB =ρA⊗ρBis uncorrelated, then the

change in mutual information is necessarily positive ∆I(A:

B)≥0, and

k(TA−TB)∆SA= ∆QA

TA−TB

TA≤0.(6)

To see that this equation is precisely the Clausius statement,

consider without loss of generality that Ais the hot bath and

TA−TB>0. Then, inequality (6) implies an entropy reduction

of the hot bath ∆SA≤0 i. e. a heat ﬂow from the hot bath to

the cold one.

However, if the the system is initially correlated, the pro-

cess can reduce the mutual information, ∆I(A:B)<0, and

Eq. (5) allows a heat ﬂow from the cold bath to the hot one.

3. Violations of zeroth law

The zeroth law establishes the notion of thermal equilib-

rium as an equivalence relation, in which temperature labels

the diﬀerent equivalent classes. To see that the presence of

correlations also invalidates the zeroth law, we show that the

transitive property of the equivalence relation is not fulﬁlled.

Consider a bipartite system AC in an initial correlated state

ρAC, like in Examples 1 and 2, and a third party Bwhich is

in a thermal state at the same temperature of the marginals

ρAand ρC(see Fig. 2). Then, while the subsystems AB and

BC are mutually in equilibrium, the subsystems AC are not,

clearly violating transitivity. There are several ways to real-

ize that the parties AC are not in equilibrium. One way is to

see that any energy preserving unitary, except for the iden-

tity, decreases the amount of correlations between the parties,

4

A

B

C

AC

AC

FAC

Figure 2. In presence of correlations, the notion of equilibrium is not

an equivalence relation. While Ais in equilibrium with Band Bis

in equilibrium with C,Aand Care not in equilibrium. The transitive

property is violated. This is justiﬁed, on the right, as F(ρAC)>F(ρA⊗

ρC).

∆I(A:C)<0, which implies that the initial state is not sta-

ble. This can be shown from Eq. (4) for the particular case of

equal temperatures and the deﬁnition of mutual information.

Another way is to see that the Helmholtz free energy follows

F(ρAC)>F(ρA⊗ρC).

4. Violations of Landauer’s erasure principle

Another thermodynamic principle that breaks down when

correlations are present is Landauer’s erasure principle. Lan-

dauer postulated that in order to erase one bit of information

in presence of a bath at temperature T, an amount of heat

needed to be dissipated is kT log 2. As noted in [32], when

the system is classically correlated, there exists erasing pro-

cess which does not increase entropy of the bath (see Exam-

ple 1). The situation becomes more striking when the system

shares quantum entanglement with the bath. This is the case

of Example 2 with initial entanglement, where instead of in-

creasing, an erasing process reduces the entropy of the bath

and the bath is cooled down.

V. UNIVERSAL QUANTUM THERMODYNAMICS

To summarize the previous discussion, the violations of the

laws of thermodynamics indicate that correlations between

two systems, irrespective of the corresponding marginals be-

ing thermal states or not, manifest out-of-equilibrium phe-

nomena. In order to re-establish the laws of thermodynamics,

one not only has to look at the local marginal systems, but

also the correlations between them. To do so, we start with re-

deﬁning heat by properly accounting for the information ﬂow

and thereby restoring Landauer’s erasure principle. Then we

proceed to restore the fundamental laws.

The transformations considered in the rest of the paper

are entropy preserving operations. More explicitly, given a

system-bath setting in a state ρS B, we consider transforma-

tions ρ0

S B = Λ(ρS B) such that the von Neumann entropy is

unchanged i. e. S(ρ0

S B)=S(ρS B ). The Hamiltonians of the

system and the bath are the same before and after the trans-

formation Λ(·). Note that we do not demand energy conserva-

tion, rather assuming that a suitable battery takes care of that.

In fact, the work cost of such an operation Λ(·) is quantiﬁed

by the global internal energy change ∆W= ∆ES+ ∆EB. An-

other comment to make is that we implicitly assume a bath of

unbounded size; namely, it consists of the part ρBof which we

explicitly track the correlations with S, but also of arbitrarily

many independent degrees of freedom.

An important remark is that this set of entropy preserv-

ing operations is strictly larger than the unitary transforma-

tions considered in many texts, and in particular in the so-

called thermal operations [17]. Although, both coincide in

the asymptotic limit of many copies or runs. In this work

we are implicitly considering always the asymptotic scenario

of n→ ∞ copies of the state in question (“thermodynamic

limit”).

A. Universal Landauer’s erasure principle and heat

In extending thermodynamics in correlated scenarios and

linking thermodynamics with information, we consider the

quantum conditional entropy as the natural quantity to rep-

resent information content in the system. For a joint system-

bath state ρS B, the information content in the system S, given

all the information available in the bath Bat temperature T,

is quantiﬁed by the conditional entropy S(S|B)=S(ρS B)−

S(ρB). Note that this quantity not only counts the informa-

tion content in the local system, but also takes into account

the information stored in the correlation. It vanishes when the

joint system-environment state is perfectly classically corre-

lated. However, it can even become negative in the presence

of entanglement. With quantum conditional entropy as the

correct quantiﬁer of system information, the second law of

information can be restored as follows.

Lemma 1 (Universal informational second law).Let ρS B ∈

HS⊗HBbe an initial bipartite system-bath state and ΛS B an

entropy preserving operation acting on ρS B, and resulting in

ρ0

S B = ΛS B (ρS B). Then, with the reduced states before (after)

the evolution denoted ρS(ρ0

S) and ρB(ρ0

B), respectively, we

have

∆SB=−∆S(S|B),(7)

where ∆SB=S(ρ0

B)− S(ρB)is the change in (von Neumann)

entropy of the bath, and ∆S(S|B)=S(S0|B0)− S(S|B)is the

change in conditional entropy of the system.

An increase (decrease) in bath entropy is compensated by

a decrease (increase) in conditional entropy of the system.

Therefore information content is preserved under entropy pre-

serving (EP) operations. Note that in the presence of initial

correlations, the informational second law could be violated

if one considers only system entropy (see appendix B). Thus,

auniversal Landauer’s principle is required to be expressed

in terms of conditional entropy of the system, rather than its

local entropy, and is stated in Theorem 2.

5

Theorem 2 (Universal Landauer’s erasure principle).The dis-

sipated heat associated to information erasure of a system S

connected to a bath B at temperature T by an entropy preserv-

ing operation, is equal to

∆Q=kT S(S|B),(8)

where ∆S(S|B)is the change in conditional entropy of the sys-

tem.

Proof. It follows immediately from the deﬁnition of heat in

Eq. (1), and Lemma 1, noting that the ﬁnal conditional entropy

vanishes, S(S0|B0)=0.

Now one can easily see that Landauer’s erasure principle is

respected for arbitrary erasing process, cf. Examples 1 and 2.

B. Work extraction

Next, we address extraction of work from a system Scou-

pled to a bath Bat temperature T. The system may share cor-

relations, be they classical or quantum, with the bath. Without

loss of generality, we assume that the system Hamiltonian HS

is unchanged in the process. Note that the extractable work

has two contributions: one comes from system-bath correla-

tions and the other from the local system alone, irrespective

of its correlations with the bath. Here we consider these two

contribution separately.

Here we put forward the following lemma which will be

useful for our work extraction protocols later.

Lemma 3. For an arbitrary system-bath state ρS B , there ex-

ists an additional systems A with degenerate Hamiltonian

HA=0, and a state ρA, such that there is an energy- and

entropy-preserving transformation ρA⊗ρS B

ΛAS B

−−−−→ ρ0

A⊗ρS⊗ρB

with

S(AS |B)ρA⊗ρS B =S(A0S|B)ρ0

A⊗ρS⊗ρB.(9)

Proof. It is evident that if Ais large enough and we start it in

a thermal state, the condition for entropy preservation can be

satisﬁed.

As the local system and bath states remain unchanged under

the transformation, and the system Ahas trivial Hamiltonian,

also the (expected) energy is preserved.

The change in entropy of the additional system is exactly

equal, and opposite in sign, to the change in system-bath

correlation (which is nothing but the mutual information):

SA− SA0=SS− S(S|B)=I(S:B).

1. Extractable work from correlation

The extractable work only from the correlation (see Fig. 3)

is given by the following result. Here, by extracting work from

the correlation, we mean any process that returns the system

and the bath in the original reduced states, ρSand ρB=τB,

respectively.

H

C

CCCCS

HHHHHHB

W

+

H

C

CCCCS

HHHHHHB

W

+

Figure 3. Correlations can be understood as a work potential. See

section V B for details.

Theorem 4. For a system-bath state ρSB , with the bath at

temperature T, the maximum extractable work solely from the

correlation, using entropy preserving operations, is given by

∆WC=kT I(S:B),(10)

where I(S:B)=SS+SB− SS B is the mutual information.

Proof. Consider the following work extraction protocol in

three steps:

Step 1. We attach to ρS B an ancillary system Awith trivial

Hamiltonian HA=0, consisting of I(B:S) qubits in the

maximally mixed state τA=I2

2⊗I(S:B)(which is thermal!).

Step 2. By using a global entropy preserving operation, ac-

cording to Lemma 3, we convert τA⊗ρS B into |φihφ|A⊗ρS⊗ρB,

i.e. turning the additional state into a pure state ρ0

A=|φihφ|

of Awhile leaving the marginal system and bath states un-

changed. Clearly, the extractable work stored in the correla-

tion is now transferred to the new additional system state ρ0

A.

Step 3. Work is extracted from ρ0

Aat temperature T, equal to

∆WC=I(S:B)ρSB kT .

2. Universal Helmholtz free energy

The maximum extractable work from a state ρS, disregard-

ing the correlations with a bath at temperature T, is given

by ∆WL=F(ρS)−F(τS), where τS=1

ZSexp[−HS

kT ] is the

corresponding thermal state of the system in equilibrium with

the bath. Now, in addition to this “local work”, we have the

work due to correlations, and so the total extractable work

∆WS= ∆WC+ ∆WL, comprising both the system and the cor-

relation, becomes

∆WS= ∆WL+kT I(S:B)ρS B .(11)

Note that, for the system alone, the Helmholtz free energy

F(ρS)=ES−kT SS. However, in the presence of correlations,

the free energy is modiﬁed to

F(ρS)=ES−kT S(S|B).(12)

The above is simply achieved by adding kT I(S:B)ρSB to

F(ρS). Therefore, for a system-bath state ρS B, maximum

extractable work from the system can be given as ∆WS=

6

F(ρS)− F(τS), where F(τS)=F(τS). Then for a trans-

formation, where initial and ﬁnal states are ρS B and σS B re-

spectively, the maximum extractable work from the system, is

∆WS=−∆F=F(ρS)− F (σS).

We observe that all this is of course consistent with what

we know from situations with an uncorrelated bath. Indeed,

we can simply make the conceptual step of calling S B “the

system”, allowing for arbitrary correlations between Sand B,

with a suitable inﬁnite bath B0that is uncorrelated from S B.

Then, the free energy as we know it is

F(ρS B)=ES+EB−kT S(ρS B )

=ES−kT S(S|B)+EB−kT S(τB),(13)

where the ﬁrst term is the modiﬁed free energy in Eq. (12),

and the second term is the free energy of the bath in its thermal

state. As the latter cannot become smaller in any entropy-

preserving operation, the maximum extractable work is −∆F.

C. Universal ﬁrst law

As we have shown, restoring the (information based) sec-

ond law requires proper accounting of information to correctly

quantify the heat ﬂow. The lessons learned from the universal

Landauer erasure principle, now lead to the following univer-

sal ﬁrst law of thermodynamics for closed systems.

Theorem 5 (Universal ﬁrst law).Given an entropy preserving

operation ρS B →ρ0

S B, where the local system state follow the

transformation ρS→ρ0

S, the distribution of the change in the

system’s internal energy into work and heat satisﬁes

∆ES=−(∆WS−∆WB)−(∆WB+ ∆Q),(14)

where the heat dissipated to the bath is given by

∆Q=−kT ∆S(S|B),(15)

and the maximum extractable work from the system is

∆WS=−(∆ES−kT ∆S(S|B)),(16)

and the work performed on the bath is ∆WB= ∆EB−kT ∆SB.

Proof. The proof relies on adding Eqs. (15) and (16). The

quantity ∆WS=−(∆ES−kT ∆S(S|B))was shown to be the

maximum extractable work in the preceding Section V B.

The maximum work ∆WSis extracted by thermodynam-

ically reversible processes. Any irreversible process leads

to ∆WB>0, which means that some work has been per-

formed on the bath. Finally, in an equilibration process, which

will happen due to spontaneous relaxation of the bath, such

amount of work will be transformed into heat and hence can-

not be accessed any more. Note that such an equilibration

process is not entropy preserving [28].

In the light of the above universal ﬁrst law, we reconsider

the examples from before. In Example 1, it indicates that an

initial classically correlated state ρS B is thermodynamically

equivalent to the ﬁnal product state ρ0

S B. It is also clear from

the fact that, in both the states, local thermal states, local av-

erage energy and moreover the system’s conditional informa-

tion are same. The case in Example 2, which considers an

initial entangled state that is unitarily transformed to a ﬁnal

pure product state, is more intriguing. It says that quantum

entanglement can be exploited to perform work (due to nega-

tive heat ﬂow) on the bath by leaving the system in pure state.

The resulting pure state can further be used to extract work.

Therefore a system having entanglement with the bath has

more work potential than the uncorrelated one with the same

marginals.

D. Universal second law

Now taking into account the information content in the

marginal system as well as in the correlation with the bath,

in terms of conditional entropy of the system, the second law

is also modiﬁed, as follows:

Theorem 6 (Kelvin-Planck statement of the universal second

law).No process is possible whose sole result is the absorp-

tion of heat from a reservoir and the conversion of this heat

into work, where heat and work are deﬁned as in Theorem 5.

Clearly the previously mentioned second law violations,

based on Examples 1 and 2, are ﬁxed. In Example 1, there is

no work is done on the system as both initial and ﬁnal system-

bath states are thermodynamically equivalent and the system

contains same work potentials. In Example 2, one part of the

work stored in the correlation is used to refrigerate the bath,

while other part is transferred to the system by lowering its

entropy.

Theorem 7 (Clausius statement of the universal second law).

No process is possible whose sole result is the transfer of

heat from a cooler to a hotter body, where the work poten-

tial stored in the correlations, as deﬁned in Theorem 4, does

not decrease.

Proof. Equation (5) in the derivation of the anomalous heat

ﬂow of the previous section together with the work potential

deﬁned in Theorem 4lead to

∆QA

TA−TB

TA≤ −∆WC,(17)

which implies the statement of Theorem 7.

Then, we can interpret the anomalous heat ﬂow as a re-

frigerator driven by the work potential stored in correlations.

From the refrigerator perspective, it is interesting to determine

its eﬃciency, that from Eq. (17) leads to

∆WC

∆QA

≤TA−TB

TA

(18)

which is nothing else than the Carnot eﬃciency. This is a nice

reconciliation with traditional thermodynamics. The Carnot

eﬃciency is a consequence of the fact that reversible processes

7

are optimal, otherwise the perpetual mobile could be build by

concatenating a "better" process and a reversed reversible one.

Hence, it is natural that the refrigeration process driven by the

work stored in the correlations preserves Carnot statement of

second law.

E. Universal zeroth law

Our reconstruction of the zeroth law is a redeﬁnition of

equilibrium which, as we have previously seen, cannot be an

equivalence relation when correlations between systems are

present. Therefore the universally respected version of the

law becomes:

Deﬁnition 8 (Universal zeroth law).A collection {ρX}Xof

states is said to be in mutual thermal equilibrium with each

other, at a certain temperature T , if and only if no work can be

extracted from any of their combinations under entropy pre-

serving operations.

Note that the notion of thermal equilibrium is extended be-

yond the transitivity relation used in the traditional zeroth law.

The thermal equilibria ABand BCnot necessarily as-

sure AC. Rather, the systems {A,B,C}are guaranteed to

be in mutual equilibria if and only if no work is can be ex-

tracted from individual systems as well as from their arbitrary

collections, using entropy preserving operations.

VI. CONCLUSIONS

We have seen that the laws of thermodynamics break down

in presence of correlations e. g., anomalous heat ﬂows from

cold to a hot baths become possible [46], as well as violations

of Landauer’s principle [32] or, even, erasing memory leads to

work extraction instead of heat dissipation [42]. In this work

we have generalized thermodynamics to the physical scenar-

ios that involve correlations, including those where strong cor-

relations are present. To do so, we have exploited the connec-

tion between information and physics, and introduced a con-

sistent redeﬁnition of heat dissipation by systematically ac-

counting the information ﬂow from system to bath in terms of

the conditional entropy. As a consequence, the Helmholtz free

energy has been accordingly modiﬁed. Such a remedy, not

only ﬁxes the violations of Landauer’s erasure principle [39]

and the second law due to anomalous heat ﬂows, but also leads

to the reformulation of the laws of thermodynamics that are

universally respected. In this information-theoretic approach,

correlations between system and environment store work po-

tential and apparent anomalous heat ﬂows are a refrigeration

process driven by such a potentials.

An important remark is that all our derivations, so far, have

been made in the asymptotic limit of many copies. A relevant

question is how the laws of thermodynamics can be formu-

lated for a single system. In our forthcoming paper [49], we

will address these questions by discussing consistent notions

of one-shot heat, one-shot Landauer erasure, and of one-shot

work extraction from correlations.

VII. ACKNOWLEDGEMENT

We thank V. Pellegrini and R. B. Harvey for useful discus-

sions.

We acknowledge ﬁnancial support from the European

Commission (FETPRO QUIC, STREP EQuaM and STREP

RAQUEL), the European Research Council (AdG OSYRIS

and AdG IRQUAT), the Spanish MINECO (grants no.

FIS2013-46768-P FOQUS and FIS2008-01236, and Severo

Ochoa Excellence Grant SEV-2015-0522) with the support of

FEDER funds, the Generalitat de Catalunya (grants no. SGR

874, 875 and 966), and Fundació Privada Cellex.

Appendix A: Deﬁnitions of heat

We have mathematically deﬁned heat in Eq. (1) according

to the common description: “ﬂow of energy to a bath some

way other than through work”. Note, however, that this is not

the most extended deﬁnition of heat that one ﬁnds in many

works, e.g., [32,46], where heat is deﬁned as the change in

the internal energy of the bath, i. e.

∆˜

Q= ∆EB,(A1)

and no diﬀerent 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 less

extended, seems the most appropriate.

The ambiguity in deﬁning heat comes from the diﬀerent

ways in which the change in the internal energy of the sys-

tem EScan be decomposed. More explicitly, let us con-

sider a unitary process US B acting on a system-bath state ρS B

with ρB=Tr SρS B =τB∝e−HB/kT and global Hamiltonian

H=HS⊗I+I⊗HB. The change in the total internal en-

ergy ∆ES B is the sum of system and bath internal energies

∆ES B = ∆ES+ ∆EB, or equivalently

∆ES= ∆ES B −∆EB.(A2)

Many text-books identify in this decomposition ∆ES B := ∆W

as work and ∆EB:= ∆ ˜

Qas heat. Nevertheless, note that it

also assigns to heat increases of the internal energy that are

not irreversibly lost and can be recovered when having a bath

at our disposal.

To highlight the incompleteness of the above deﬁnition, let

us consider a reversible process US B =I⊗UBthat acts trivially

on the system. Then, even though the state of the system is

untouched in such a process, the amount of heat dissipated is

∆˜

Q= ∆EB=Tr [HB(ρB−UBρBU†

B)].

In order to avoid this kind of paradoxes and in the spirit of

the deﬁnition given above, we subtract from ∆˜

Qits compo-

nent of energy that can still be extracted (accessed). Then for

a transformation ρB→ρ0

B, the heat transferred is given as

∆Q= ∆EB−∆FB,(A3)

=kT ∆SB,(A4)

where ∆FB=F(ρ0

B)−F(ρB) is the work stored on the

bath and can be extracted. Here, F(ρX)=EX−kT S(ρX)

8

is the Helmholtz free energy, EXis the internal energy and

∆SB=S(ρ0

B)−S(ρB) is the change in the bath’s von Neumann

entropy, S(ρB)=−Tr ρBlog2ρB. Throughout this work, we

consider log2as the unit of entropy.

Let us ﬁnally remark that in practical situations, with large

baths whose internal dynamics rapidly make them indistin-

guishable from thermal, both deﬁnitions coincide. However,

when studying thermodynamics at the quantum regime with

small machines approaching the nanoscale such conceptual

diﬀerences are crucial to extend, for instance, the domain of

standard thermodynamics to situations where the correlations

become relevant.

Appendix B: Landauer’s erasure principle (LEP):

connecting heat and information

The information theory and statistical mechanics have long-

standing and intricate relation. In particular, to exorcise

Maxwell’s demon in the context of statistical thermodynam-

ics, Landauer ﬁrst indicated that information is physical and

any manipulation of that has thermodynamic cost. As put for-

ward by Bennett [34], the Landauer information erasure prin-

ciple implies that “[a]ny logically irreversible manipulation of

information, such as the erasure of a bit or the merging of two

computation paths, must be accompanied by a corresponding

entropy increase in non-information-bearing degrees of free-

dom of the information-processing apparatus or its environ-

ment.”

Following the deﬁnition of heat, it indicates that, in such

processes, entropy increase in non-information-bearing de-

grees of freedom of a bath is essentially associated with ﬂow

of heat to the bath. The major contribution of this work is to

exclusively 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 process

where an event, denoted with i, happens with the probability

pi. Then storing (classical) information memorizing the pro-

cess means constructing a d-dimensional system (a memory-

dit) in a state ρS=Pipi|iihi|, where {|ii} are the orthonor-

mal basis correspond to the event i. In other words, memoriz-

ing the physical process is nothing but constructing a memory

state ρS=Pipi|iihi|from a memoryless state |iihi|where i

could assume any values 1 6i6d. On the contrary, pro-

cess of erasing requires the transformation of a memory state

ρS=Pipi|iihi|to a memoryless state |iihi|for any i. Lan-

dauer’s erasure principle (LEP) implies that erasing informa-

tion, a process involving a global evolution of the memory-dit

system and its environment, is inevitably associated with an

increase in entropy in the environment.

In establishing the connection between information eras-

ing and heat dissipation, we make two assumptions to start

with. First, the memory-system (S) and bath (B) are both de-

scribed by the Hilbert space HS⊗ HB. Secondly, the eras-

ing process involves entropy preserving operation ΛS B, i.e.,

ρ0

S B = ΛS B (ρS B). The latter assumption is most natural

and important, as it preserves information content in the joint

memory-environment system. Without loss of generality, one

can further assume that the system and bath Hamiltonians re-

main unchanged throughout the erasing process, to ease the

derivations.

Now we consider the simplest information erasing scenario,

which leads to LEP in its traditional form. In this scenario, a

system ρSis brought in contact with a bath ρBand the system

is transformed to a information erased state, say |0ih0|S, by

performing a global entropy-preserving operation ΛS B, i.e.,

ρS⊗ρB

ΛS B

−−−→ |0ih0|S⊗ρ0

B,(B1)

where initial and ﬁnal joint system-bath states are uncorre-

lated. The joint operation guarantees that the decrease in sys-

tem’s entropy is exactly equal to the increase in bath entropy

and heat dissipated to the bath is ∆Q=kT ∆SB. It clearly

indicates that an erasure process is expected to heat up the

bath. This in turn also says that ∆Q=−kT ∆SS, where

∆SS=S(ρ0

S)− S(ρS). In the case where the d-dimensional

system memorizes maximum information, or in other words

it is maximally mixed and contains log2dbits of information,

the process dissipates an amount kT log2dof heat to com-

pletely erase the information. In other words, to erase one bit

of information system requires the dissipation of kT of heat

and we denote it as one heat-bit or `-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 he system, i.e.,

∆Q>−kT ∆SS.(B2)

This is what is generally known as the Landauer’s erasure

principle (LEP), in terms of heat.

The above formulation of LEP crucially relies on the fact

that any change in system entropy leads to a larger change

in the bath entropy, which is also traditionally known as the

second law for the change in the information, i.e.,

∆SB>−∆SS.(B3)

However, it is limited by the assumptions made above and can

be violated with initial correlations. Consider the examples

proposed in the main text, see section IV. In both the exam-

ples, ∆SB−∆SS. Therefore, one has to replace it with

universal informational second law (see Lemma 1).

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