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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.
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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.
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 [36]. 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 [713], re-
cently developed in the domain of quantum information [14
16], a renewed eort has been made to understand the founda-
tions of thermodynamics in the quantum domain, giving rise
to many interesting results [1727], including its connections
to statistical mechanics [2831] 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
[3236,41]. Heat dissipation due to erasing information from
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 [4447]. In other words, the fundamental laws
of thermodynamics are violated in the presence of system-
bath correlations. There is an eort 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
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
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.
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 dierent 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.
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 dierent
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 dierence 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
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 ρBρ0
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.
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
pi|iihi|S⊗ |iihi|B
S B =|φihφ|SX
Example 2 – Entanglement.
|ΨiS B =X
→ |Ψi0
S B =|φiS⊗ |φiB,
where in both examples |φiX=Pipi|iiXwith X∈ {S,B}
and 1 >pi0 for all i. Note that the unitaries, Uc
S B and Ue
S B,
leave the local energies of system and bath unchanged, and
S B does not change the bath state.
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,WSQ, 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,WSQ.
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
kTA] and
kTB] are thermal states at dierent 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
AB where
UAB is a energy preserving unitary acting on ρAB. As the ther-
mal state minimizes the free energy, the final reduced states
Sand ρ0
Bhave increased their free energy,
EAkTASA0 (2)
Figure 1. In presence of correlations, spontaneous heat flows from
cold to hot baths are possible [46]. This is a clear violation of second
where TA/Bis the initial temperature of the baths, and EA/B
and SA/Bare the change in internal energy and entropy re-
By adding Eqs. (2) and (3), and considering energy conser-
vation, we get
Due to the conservation of total entropy, the change in mutual
information is simply I(A:B)= ∆SA+ ∆SB, with I(A:
B)=SA+SBSAB. This allows us to rewrite Eq. (4) in terms
of only the entropy change in Aas
(TATB)SA≤ −TBI(A:B).(5)
If the initial state ρAB =ρAρBis uncorrelated, then the
change in mutual information is necessarily positive I(A:
B)0, and
To see that this equation is precisely the Clausius statement,
consider without loss of generality that Ais the hot bath and
TATB>0. Then, inequality (6) implies an entropy reduction
of the hot bath SA0 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 dierent 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,
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
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
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.
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
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
HSHBbe an initial bipartite system-bath state and ΛS B an
entropy preserving operation acting on ρS B, and resulting in
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
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.
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-
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 ρAρS B
S(AS |B)ρAρS B =S(A0S|B)ρ0
Proof. It is evident that if Ais large enough and we start it in
a thermal state, the condition for entropy preservation can be
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,
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 τAρS B into |φihφ|AρSρB,
i.e. turning the additional state into a pure state ρ0
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
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
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)=ESkT SS. However, in the presence of correlations,
the free energy is modified to
F(ρS)=ESkT 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=
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+EBkT S(ρS B )
=ESkT S(S|B)+EBkT 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 ρSρ0
S, the distribution of the change in the
system’s internal energy into work and heat satisfies
ES=(WSWB)(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=(ESkT S(S|B)),(16)
and the work performed on the bath is WB= ∆EBkT SB.
Proof. The proof relies on adding Eqs. (15) and (16). The
quantity WS=(ESkT 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
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
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
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 eciency, that from Eq. (17) leads to
which is nothing else than the Carnot eciency. This is a nice
reconciliation with traditional thermodynamics. The Carnot
eciency is a consequence of the fact that reversible processes
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.
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.
We thank V. Pellegrini and R. B. Harvey for useful discus-
We acknowledge financial support from the European
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 dierent 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 dierent
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 =τBeHB/kT and global Hamiltonian
H=HSI+IHB. 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 =IUBthat 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
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 ρBρ0
B, the heat transferred is given as
Q= ∆EBFB,(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)=EXkT S(ρX)
is the Helmholtz free energy, EXis the internal energy and
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
dierences 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-
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.,
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
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.,
→ |0ih0|Sρ0
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
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-
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.,
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, SBSS. Therefore, one has to replace it with
universal informational second law (see Lemma 1).
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We establish an operational theory of coherence (or of superposition) in quantum systems. Namely, we introduce the two basic concepts --- "coherence distillation" and "coherence cost" in the processing quantum states under so-called \emph{incoherent operations} [Baumgratz \emph{et al.}, Phys. Rev. Lett. 113:140401 (2014)]. We then show that in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state but no coherence could be distilled from it. Further we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all the reversible coherent states.
Thermodynamics is traditionally concerned with systems comprised of a large number of particles. Here we present a framework for extending thermodynamics to individual quantum systems, including explicitly a thermal bath and work-storage device (essentially a 'weight' that can be raised or lowered). We prove that the second law of thermodynamics holds in our framework, and gives a simple protocol to extract the optimal amount of work from the system, equal to its change in free energy. Our results apply to any quantum system in an arbitrary initial state, in particular including non-equilibrium situations. The optimal protocol is essentially reversible, similar to classical Carnot cycles, and indeed, we show that it can be used to construct a quantum Carnot engine.
When studying thermalization of quantum systems, it is typical to ask whether a system interacting with an environment will evolve towards a local thermal state. Here, we show that a more general and relevant question is "when does a system thermalize relative to a particular reference?" By relative thermalization we mean that, as well as being in a local thermal state, the system is uncorrelated with the reference. We argue that this is necessary in order to apply standard statistical mechanics to the study of the interaction between a thermalized system and a reference. We then derive a condition for relative thermalization of quantum systems interacting with an arbitrary environment. This condition has two components: the first is state-independent, reflecting the structure of invariant subspaces, like energy shells, and the relative sizes of system and environment; the second depends on the initial correlations between reference, system and environment, measured in terms of conditional entropies. Intuitively, a small system interacting with a large environment is likely to thermalize relative to a reference, but only if, initially, the reference was not highly correlated with the system and environment. Our statement makes this intuition precise, and we show that in many natural settings this thermalization condition is approximately tight. Established results on thermalization, which usually ignore the reference, follow as special cases of our statements.