Thermodynamics as a Consequence of
Information Conservation
Manabendra Nath Bera
1,2
, Arnau Riera
1,2
, Maciej Lewenstein
1,3
, Zahra Baghali Khanian
1,4
, and
Andreas Winter
3,4
1
ICFO Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain
2
Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany
3
ICREA, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain
4
Departament de Física: Grup d’Informació Quàntica, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona),
Spain
November 30, 2018
Thermodynamics and information have
intricate interrelations. Often thermo-
dynamics is considered to be the logi-
cal premise to justify that information is
physical through Landauer’s principle –,
thereby also linking information and ther-
modynamics. This approach towards in-
formation has been instrumental to under-
stand thermodynamics of logical and phys-
ical processes, both in the classical and
quantum domain. In the present work, we
formulate thermodynamics as an exclusive
consequence of information conservation.
The framework can be applied to the most
assumptions in thermodynamics: we allow
systems and thermal baths to be quan-
tum, of arbitrary sizes and even possessing
inter-system correlations.
Here, systems and baths are not treated
diﬀerently, rather both are considered on
an equal footing. This leads us to in-
troduce a “temperature”-independent for-
mulation of thermodynamics. We rely
on the fact that, for a ﬁxed amount of
information, measured by the von Neu-
mann entropy, any system can be trans-
formed to a state with the same entropy
that possesses minimal energy. This state,
known as a completely passive state, acquires
Boltzmann–Gibbs canonical form with an
intrinsic temperature. We introduce the no-
tions of bound and free energy and use
them to quantify heat and work, respec-
Manabendra Nath Bera: mnbera@gmail.com
Andreas Winter: andreas.winter@uab.cat
tively. Guided by the principle of informa-
tion conservation, we develop universal no-
tions of equilibrium, heat and work, Lan-
dauer’s principle and universal fundamen-
tal laws of thermodynamics. We demon-
strate that the maximum eﬃciency of a
quantum engine with a ﬁnite bath is in
general lower than that of an ideal Carnot
engine. We introduce a resource theo-
retic framework for our intrinsic tempera-
ture based thermodynamics, within which
we address the problem of work extrac-
tion and state transformations. Finally,
the framework is extended to multiple con-
served quantities.
1 Introduction
Thermodynamics constitutes one of the basic foun-
dations of modern science. It not only plays an
important role in modern technologies, but oers a
basic understanding of a vast range of natural phe-
nomena. Initially, thermodynamics was developed
phenomenologically, to address the question of how,
and to what extent, heat can be converted into work.
Later, with the developments of statistical mechanics,
quantum mechanics and relativity, thermodynamics
along with its fundamental laws was put on a formal
and mathematically rigorous footing [1]. Beyond its
original domain of conception, it has been applied in
a wide range of contexts. It has been used to describe
relativistic phenomena in astrophysics and cosmol-
ogy, quantum eects in microscopic systems, or very
complex systems in biology and chemistry, and even
in music [2].
The inter-relation between information and ther-
Accepted in Quantum 2018-11-28, click title to verify 1
arXiv:1707.01750v3 [quant-ph] 29 Nov 2018
modynamics [3] is intricate, and has been studied in
the context of Maxwell’s demon [47], Szilard’s en-
gine [8], and Landauer’s principle [913]. In recent
years, classical and quantum information-theoretic
approaches helped to understand thermodynamics in
the domain of small classical and quantum systems
[1416]. That stimulated a whole new perspective to
tackle and extend thermodynamics beyond the stan-
dard classical domain. In fact, information theory
has recently played an important role in understand-
ing thermodynamics in the presence of inter-system
and system-bath correlations [1719], equilibration
processes [2023], and the foundations of statistical
mechanics [24]. One of the most paradigmatic ex-
amples of this success is the formulation of quantum
thermodynamics within the so-called resource theo-
retic framework [25], which allows to reproduce stan-
dard thermodynamics in the asymptotic limit, when
one processes inﬁnitely many copies of the system
under consideration. In the ﬁnite-copy, or even one-
shot limit, the resource theory reveals that the laws
of thermodynamics have to be modiﬁed to capture
ﬁnite-size, as well as quantum eects in thermody-
namics [2634].
In the present work, elaborating in the interrela-
tions between information and thermodynamics, we
perform an axiomatic construction of thermodynam-
ics and identify the “information conservation” as
the crucial underlying property of any theory that re-
spects it. The cornerstone of our construction is the
notion of bound energy, which we introduce as the
amount of energy locked in a system that cannot be
accessed (extracted) by using the set of allowed op-
erations. The bound energy obviously depends on
the set of allowed operations: the larger this set is,
the smaller the bound energy. We prove that, by tak-
ing (i) global entropy preserving operations as the set
of allowed operations and (ii) inﬁnitely large thermal
baths initially uncorrelated with the system, our for-
malism reproduces standard thermodynamics in its
resource theoretic form.
In information theory, the von Neumann entropy
is the quantity that measures the amount of infor-
mation in a system. In this sense, entropy preserv-
ing operations are transformations that keep this in-
formation constant. All fundamental physical theo-
ries, such as classical and quantum mechanics, share
the property of conserving information at the basic
level. Namely, they have dynamics that is determin-
istic and bijectively maps the set of possible conﬁgu-
rations between any two instants of time. Thus, non-
determinism can only appear when some degrees of
freedom are ignored, leading to apparent information
loss. In classical physics, this information loss is due
to deterministic chaos and mixing in nonlinear dy-
namics. In quantum mechanics, this loss is intrin-
sic, and occurs due to measurement processes and
non-local correlations [35]. Note that the set of en-
tropy preserving operations is larger than the dynam-
ically reversible operations (unitaries) in the sense
that they conserve entropy but, unlike unitaries, not
the individual probabilities. In other words, we de-
mand only coarse-grained, not microscopic informa-
tion conservation. In the limit of many copies both
coarse-grained and ﬁne-grained information conser-
vation become equivalent (see Sec. 2). While it is
known that linear operations that are entropy preserv-
ing for all states are unitary [36], it is an open ques-
tion to what extent coarse-grained information con-
serving operations can be implemented in the single-
copy setting.
The resource theory of thermodynamics can then
be seen as one where condition (i) is sharpened, i.e.
an extension of thermodynamics from coarse-grained
to ﬁne-grained information conservation, where the
operations are global unitaries but still constrained to
inﬁnitely large thermal baths with a well deﬁned tem-
perature. Fluctuation theorems can also be consid-
ered from this perspective. There, the second law is
obtained as a consequence of reversible transforma-
tions on initially thermal states or states with a well
deﬁned temperature [3740]. In contrast, the aim of
our work is instead to relax condition (ii), that is, to
generalize thermodynamics to be valid for arbitrary
environments, irrespective of being thermal, or much
larger than the system. This idea is illustrated in the
table below.
Unitaries
(ﬁne-grained
information
conservation)
Entropy
preserving
operations
(coarse-grained
information
conservation)
Large
thermal
bath
Resource theory of
thermodynamics
Standard
thermodynamics
Arbitrary
environment
? Present paper
The main obstruction against this generalization
lies in the fact that, when allowing for arbitrary
states as environment, large amounts of resources
Accepted in Quantum 2018-11-28, click title to verify 2
can be pumped into the system leading to trivial
“resource” theories. We are able to circumvent
this problem thanks to the notion of bound energy,
which intrinsically distinguishes accessible and non-
accessible energy. Our formalism results in a theory
in which systems and environments are treated on an
equal footing, or in other words, in a “temperature”-
independent formulation of thermodynamics.
Our work is complementary to other general ap-
proaches, where thermodynamics is obtained after in-
serting some form of thermal state(s) in general math-
ematical expressions, e.g. [41]. While in these works
the mathematical expressions for arbitrary states have
no a priori thermodynamic meaning, our construc-
tion is built on a physically motivated quantity, the
bound energy.
The present temperature-independent thermody-
namics is essential in contexts in which the state of
the bath can be aected by the system after exchange
of heat (see [42] for a review on the notion of temper-
ature). This can be due to the fact of either having a
relatively small environment compared to the system,
or an environment simply not being thermal. In fact,
in current experiments, environments do not have to
be necessarily thermal, but can possess quantum co-
herence or correlations.
The entropy preserving operations make all states
with equal energies and entropies thermodynamically
equivalent. This allows for representing all the states
and thermodynamic processes in a simple energy-
entropy diagram. We exploit this geometric ap-
proach and give a diagrammatic representation for
heat, work and other thermodynamic quantities. In
this way we are able to reproduce several results of
the literature, e.g. the resource theory of thermody-
namics applicable for arbitrary quantum systems and
environments [43]. Our formalism extends naturally
to scenarios with multiple conserved quantities.
The structure of the rest of the paper is as follows:
In the following Section 2, we discuss the class of
entropy preserving operations, on which our theory
is built. Then, in Section 3, we introduce the fun-
damental notions of bound and free energy, and in
Section 4 deﬁne the energy-entropy diagram of a sys-
tem. After that, we develop thermodynamics, from
the zeroth law (Section 5) and a discussion of the
max-entropy vs. min-energy principle (Section 6), to
the ﬁrst law (Section 7) and the second law of ther-
modynamics (Section 8). In Section 9 we discuss if,
and why not, a third law can be formulated in our
setting. Then, in Section 10, we move to the devel-
opment of a resource theory based on the previous
formalism, which allows us to consider transforma-
tion rates between states and visualize the results in
the energy-entropy diagram. In Section 11, we out-
line how the theory so far developed generalizes to
the case of multiple conserved quantities. Finally, we
summarize our ﬁndings in Section 12, and discuss the
main obstructions towards an extension of thermo-
dynamics that would be valid both in the single-shot
scenario and for non-thermal environments.
2 Entropy preserving operations, en-
tropic equivalence class and intrinsic
temperature
The set of operations that we consider in this frame-
work is the set of, so called, entropy preserving oper-
ations. Given a system initially in a state ρ, the set of
entropy preserving operations are all the operations
that arbitrarily change the state, but keep its entropy
constant
ρ σ s.t. S(ρ) = S(σ), (1)
where S(ρ) B Tr ρ log ρ is the von Neumann en-
tropy. Importantly, an operation that acts on ρ and
produces a state with the same entropy, not necessar-
ily preserves entropy when acting on other states. In
fact, such entropy preserving operations are in gen-
eral not linear, since they have to be constrained to
some input state. It was shown in Ref. [36] that a
quantum channel Λ(·) that preserves entropy and, at
the same time, respects linearity, i. e. Λ(
1
+ (1
p)ρ
2
) = pΛ(ρ
1
) + (1 p)Λ(ρ
2
), has to be a unitary.
However, the entropy preserving operations can be
microscopically described by global unitaries in the
limit of many copies [43]. Given any two states ρ
and σ with equal entropies S(ρ) = S(σ), there exists
n log n) ancillary qubits
and a global unitary U such that, as n ,
Tr
anc
Uρ
n
ηU
σ
n
1
0, (2)
where the partial trace is performed on the ancillary
qubits. Here, kXk
1
B Tr
X
X is the one-norm.
As shown in [43, Thm. 4], the reverse statement is
also true. In other words, if two states are related as
in Eq. (2), then they also have equal entropies.
It is important to restrict entropy preserving op-
erations that are also energy preserving, as we shall
see later. The energy and entropy preserving opera-
tions can also be implemented using a global energy
Accepted in Quantum 2018-11-28, click title to verify 3
preserving unitary in the many-copy limit. More ex-
plicitly, in [43, Thm. 1], it is shown that having two
states ρ and σ with equal entropies and energies, i.e.
(S(ρ) = S(σ) and E(ρ) = E(σ)), is equivalent to
the existence of some energy preserving U and an
n log n) of ancillary
qubits with Hamiltonian kH
A
k 6 O(n
2/3
) in some
state η, for which (2) is fulﬁlled. The operator norm
kXk of a Hermitian operator X is the largest of its
eigenvalues in absolute value. Note that the amount
of energy and entropy of the ancillary system per
copy vanishes in the large n limit.
We expect entropy preserving operations to be also
implemented in other ways than taking the limit of
many copies. For instance, in Refs. [44, 45], ther-
mal operations are extended to a class of operations,
in which a catalyst is allowed to build up correla-
tions with the system. For these operations, the stan-
dard Helmholtz free energy singles out as the mono-
tone that establishes the possible transitions between
states, in contrast to the case of strict thermal opera-
tions, in which all the Rényi α-free energies are re-
quired. This suggests that entropy preserving opera-
tions could also be implemented with a single copy
by means of a catalyst that can become correlated
with the system. Further investigation in this direc-
tion is needed.
Now that the entropy preserving operations have
been motivated and introduced, let us classify the set
of states of a system in dierent equivalence classes
depending on their entropy. Thereby, we establish
a hierarchy of states according to their information
content.
Deﬁnition 1 (Entropic equivalence class). Two
states ρ and σ on any quantum system of dimen-
sion d are equivalent and belong to the same en-
tropic equivalence class if and only if both have
the same von Neumann entropy,
ρ σ iﬀ S(ρ) = S(σ). (3)
We call such two states, ρ and σ iso-entropic. As-
suming that the system has some ﬁxed Hamiltonian
H, one can take as a representative element of every
class the state that minimizes the energy within it, i.e.,
γ(ρ) B arg min
σ : S(σ)=S(ρ)
E(σ), (4)
where E(σ) B Tr Hσ is the energy of the state σ.
The maximum-entropy principle [46, 47] identiﬁes
the thermal state as the state that maximizes the en-
tropy for a given energy. Conversely, one can show
that, for a given entropy, the thermal state also min-
imizes the energy. We refer to this complementary
property as min-energy principle [17, 48, 49]. To be
precise, this duality holds for all ﬁnite inverse tem-
peratures, β < , corresponding to energies strictly
above the ground state energy and entropies strictly
larger then the ground state entropy. Thus, the min-
energy principle identiﬁes thermal states as the repre-
sentative elements of every class, which is
γ(ρ) =
e
β(ρ)H
Tr
e
β(ρ)H
. (5)
The inverse temperature β(ρ) is the parameter that
labels the equivalence class, to which the state ρ be-
longs. We denote β(ρ) as the intrinsic inverse tem-
perature associated to ρ.
The state γ(ρ) is often termed the completely pas-
the CP states, synonymously, with the γ(H, β), γ(ρ),
γ(β) and γ(T ), where H is the Hamiltonian of sys-
tem in the state ρ, and β = 1/T is the inverse of the
temperature T . The γ(H, β) is exclusively used in
the cases where the Hamiltonian H of the system is
ﬁxed. Moreover, the baths, in thermal equilibrium at
certain temperature, are also CP states, and we some-
time denote these thermal baths with γ(T ) and γ(β).
The completely passive state of the form
γ(H
S
, β
S
) has the following interesting proper-
ties [48, 49]:
(P1) For a given entropy, it minimizes the energy.
(P2) Energy (entropy) monotonically increases (de-
crease) with the decrease (increase) in β
S
.
(P3) For non-interacting Hamiltonians and identical
β
T
, H
T
=
P
N
X=1
I
X1
H
X
I
NX
, the
joint complete passive state is the tensor prod-
uct of the individual ones, i.e., γ(H
T
, β
T
) =
N
X=1
γ(H
X
, β
T
). [48, 49].
3 Bound and free energies
Let us now identify two relevant forms of internal en-
ergy: the free and the bound energy. The bound en-
ergy is motivated as the amount of internal energy
that cannot be accessed in the form of work. We em-
phasize once more that it is a notion that depends
on the set of allowed operations. For the set of en-
tropy preserving operations, from which the entropic
classes and completely passive states emerge, it is
quantiﬁed as follows.
Accepted in Quantum 2018-11-28, click title to verify 4
Deﬁnition 2 (Bound energy). For a state ρ with
the system Hamiltonian H, the bound energy in
it is
B(ρ) B min
σ : S(σ)=S(ρ)
E(σ). (6)
By the previous discussion, B(ρ) = E(γ(ρ)),
where γ(ρ) is the completely passive state, with
minimum energy, within the equivalence class to
which ρ belongs.
Indeed, B(ρ) is the amount of energy that cannot
be extracted further, by exploiting any entropy pre-
serving operations, as guaranteed by the min-energy
principle. The above deﬁnition of bound energy also
has strong connection with information content in the
state. It can be easily seen that one could have access
to this energy (in the form of work) only, if it allows
an outﬂow of information from the system and vice
versa.
In contrast to bound energy, free energy is the part
of the internal energy that can be accessed with en-
tropy preserving operations.
Deﬁnition 3 (Free energy). For a system ρ with
system Hamiltonian H, the free energy stored in
the system is given by
F (ρ) B E(ρ) B(ρ), (7)
where B(ρ) is the bound energy in ρ.
Note that the free energy as deﬁned in Eq. (7) does
not have a preferred temperature, unlike the standard
out-of-equilibrium Helmholtz free energy F
T
(ρ) B
E(ρ) T S(ρ), where the temperature T is decided
beforehand with the choice of a thermal bath. Nev-
ertheless, our deﬁnition of free energy can be written
in terms of both the relative entropy and the out-of-
equilibrium free energy as
F (ρ) = T (ρ)D(ρkγ(ρ)) = F
T (ρ)
(ρ)F
T (ρ)
(γ(ρ)) ,
(8)
for the intrinsic temperature T (ρ) B β(ρ)
1
that labels the equivalence class that contains ρ.
Here the relative entropy is deﬁned as D(ρkσ) =
Tr (ρ log ρ ρ log σ). Let us mention that the stan-
dard out-of-equilibrium free energy is also denoted
by F
β
(ρ) B E(ρ) β
1
S(ρ) in the rest of the
manuscript, wherever we ﬁnd it more convenient.
The notions of bound and free energy, as we con-
sider above, can be extended beyond single systems.
If fact, in composite systems they exhibit several in-
teresting properties. For example, they can capture
the presence and/or absence of inter-party correla-
tions. To highlight these features, we consider a bi-
partite system below.
Lemma 4 (Bound and free energy properties).
Given a bipartite system with non-interacting
Hamiltonian H
A
I + I H
B
in an arbitrary
state ρ
AB
with marginals ρ
A/B
B Tr
B/A
(ρ
AB
).
Then, the bound and the free energy satisfy the
following properties:
(P4) Bound energy and correlations:
B(ρ
AB
) B(ρ
A
ρ
B
). (9)
(P5) Bound energy of composite systems:
B(ρ
A
ρ
B
) B(ρ
A
) + B(ρ
B
). (10)
(P6) Free energy and correlations:
F (ρ
A
ρ
B
) F (ρ
AB
). (11)
(P7) Free energy of composite systems:
F (ρ
A
) + F (ρ
B
) F (ρ
A
ρ
B
). (12)
Ineqs. (9) and (11) are saturated with equality if
and only if A and B are uncorrelated, i.e. ρ
AB
=
ρ
A
ρ
B
. Ineqs. (10) and (12) become equalities
if and only if β(ρ
A
) = β(ρ
B
).
Proof. To prove (P4), we have used the fact that,
for a ﬁxed Hamiltonian, the bound energy is
monotonically increasing with the entropy, i.e.
B(ρ) < B(σ) iﬀ S(ρ) < S(σ) , (13)
S(ρ
AB
) S(ρ
A
ρ
B
) of the von Neumann
entropy.
The proof of (P5) relies on the deﬁnition of
the bound energy, where one exploits entropy
preserving operations to minimize the energy.
Namely, the iso-entropic equivalence relation is
stable under tensor product: if ρ
A
σ
A
and
ρ
B
σ
B
, then ρ
A
ρ
B
σ
A
σ
B
ditivity of the entropy. This immediately implies
Eq. (10). To see that the inequality (10) is sat-
urated iﬀ the states have identical intrinsic tem-
peratures β(ρ
A
) = β(ρ
B
), we appeal to Deﬁnition
6 and Lemma 7.
The other properties can be easily proven by
noting that the total internal energy is sum of lo-
cal ones: E(ρ
AB
) = E(ρ
A
) + E(ρ
B
), irrespective
Accepted in Quantum 2018-11-28, click title to verify 5
of inter-system correlations. Then, with the def-
inition of free energy F (ρ
X
) = E(ρ
X
) B(ρ
X
),
and properties (P4) and (P5), we easily arrive at
(P6) and (P7).
The above properties allow us to give an additional
operational meaning to the free energy F (ρ). Let us
recall ﬁrst that the work that can be extracted from a
system in a state ρ when having at our disposal an in-
ﬁnitely large bath at inverse temperature β and global
entropy preserving operations, is given by
W = F
β
(ρ) F
β
(γ(β)), (14)
where F
β
(ρ) = E(ρ) β
1
S(ρ) is the standard free
energy with respect to the inverse temperature β and
γ(β) is the thermal state.
Lemma 5 (Free energy vs. β-free energy). The
free energy F (ρ) corresponds to work extracted by
using a bath at the “worst” possible temperature:
F (ρ) = min
β
(F
β
(ρ) F
β
(γ(β))) . (15)
In words, the free energy of ρ is the minimum ex-
tractable work with respect to arbitrary tempera-
ture baths. The inverse temperature that achieves
the minimum is the inverse intrinsic temperature
β(ρ).
Proof. It follows from Lemma 14 together with
(P7).
4 Energy-entropy diagram
Let us describe here the energy-entropy diagram
which appears often in the thermodynamics litera-
ture and in particular has recently been exploited in
Ref. [43]. Given a system described by a Hamiltonian
H, a state ρ is represented in the energy-entropy dia-
gram by a point with coordinates x
ρ
B (E(ρ), S(ρ)),
as shown in Fig. 1. All physical states reside in a
region that is lower bounded by the horizontal axis
(i.e. S = 0), corresponding to the pure states, and
upper bounded by the convex curve (E(β), S(β)),
which represents the thermal states of both positive
and negative temperatures. Let us denote such a
curve as the thermal boundary. The inverse temper-
ature associated to one point of the thermal bound-
ary is given by the slope of the tangent line in such a
point,
β =
dS(β)
dE(β)
. (16)
S
E
β = 0
B(ρ) F (ρ)
β(ρ)
x
ρ
:
= (E(ρ), S(ρ))
E(ρ)
E
max
E
min
S(ρ)
β −∞
Figure 1: Energy-entropy diagram. Any quantum state ρ
is represented in the diagram as a point with coordinates
x
ρ
B (E(ρ), S(ρ)). The free energy F (ρ) is the distance
in the horizontal direction from the thermal boundary.
The bound energy B(ρ) is the distance in the horizontal
direction between the thermal boundary and the energy
reference.
In Fig. 1, the free energy and the bound energy are
plotted given a state ρ. Its free energy F (ρ) can be
seen from the diagram as the horizontal distance from
the thermal boundary. This is the part of the internal
energy which can be extracted without altering sys-
tem’s entropy. The slope of the tangent line of the
thermal boundary in that point is the inverse intrinsic
temperature β(ρ), of the state ρ. The bound energy
B(ρ) is the distance in the horizontal direction be-
tween the thermal boundary and the energy reference
and it can in no way be extracted with entropy pre-
serving operations.
Note that in general a point in the energy-entropy
diagram represents multiple states, in fact, an entire
equivalence class, since dierent quantum states can
have identical entropy and energy. As it is pointed
out in [43], the energy-entropy diagram establishes
a link between the microscopic and the macroscopic
thermodynamics, in the sense that in the macroscopic
limit of many copies all the thermodynamic quan-
tities only rely on the energy and entropy per par-
ticle. More precisely, all the states with same en-
tropy and energy are thermodynamically equivalent:
as was shown in [43], they can be transformed into
each other using energy-conserving unitaries in the
limit of many copies n , and using an ancil-
lary system, which is of sublinear size O(
n log n)
and with a Hamiltonian with its Hamiltonian O(n
2/3
)
sublinearly bounded.
5 Equilibrium and zeroth law
The zeroth law of thermodynamics establishes the
temperature as an intensive quantity, and its the abso-
Accepted in Quantum 2018-11-28, click title to verify 6
lute scaling, in terms of mutual thermal equilibrium.
It says that if a system A is in thermal equilibrium
with B and again, B is in thermal equilibrium with
C, then A will also be in thermal equilibrium with C.
All the systems that are in mutual thermal equilib-
rium can be classiﬁed in a thermodynamical equiv-
alence class, and each class can be labelled with a
unique parameter called temperature. In other words,
at thermal equilibrium, temperatures of the individ-
ual systems will be equal to each other and also to
their arbitrary collections, as there would be no spon-
taneous heat or energy ﬂow in between. When a non-
thermal state is brought in contact with a large ther-
mal bath, then the system tends to acquire a thermally
equilibrium state with the temperature of the bath. In
this equilibration process, the system can exchange
both energy and entropy with the bath and thereby
minimize its Helmholtz free energy.
However, in the new setup, where a system can-
or in the complete absence of a bath, the equilibra-
tion process is expected to be considerably dierent.
In the following, we introduce a formal deﬁnition of
equilibrium, based on information preservation and
intrinsic temperature.
Deﬁnition 6 (Equilibrium and zeroth law).
Given a collection of systems A
1
, . . . , A
n
with
non-interacting Hamiltonians H
1
, . . . , H
n
in a
joint state ρ
A
1
...A
n
, we say that they are mutually
at equilibrium if and only if they jointly minimize
the free energy as deﬁned in (7):
F (ρ
A
1
...A
n
) = 0. (17)
Let us consider two systems A and B in the states
ρ
A
and ρ
B
, with corresponding Hamiltonians H
A
and
H
B
, respectively. The equilibrium is achieved when
they jointly attain an iso-entropic state with minimum
energy. The corresponding equilibrium state is then a
completely passive state γ(H
AB
, β
AB
) with the joint
Hamiltonian H
AB
= H
A
I + I H
B
. Further, the
joint completely passive state, following (P3), is
γ(H
AB
, β
AB
) = γ(H
A
, β
AB
) γ(H
B
, β
AB
),
(18)
where the local systems assume their respec-
tive completely passive states, importantly with
the same β
AB
. Recall that γ(H
X
, β
Y
) =
e
β
Y
H
X
/Tr (e
β
Y
H
X
). Note, for interaction sys-
tems, i.e. H
AB
, H
A
I + I H
B
, one could
minimize global energy using global entropy preserv-
ing operation too. This minimum energy state will
give rise to a global equilibrium (completely passive
) state with intrinsic global temperature. However,
the reduced state, after tracing out one sub-system,
may not be a completely passive state.
Another consequence of the deﬁnition is that if
two arbitrary completely passive states γ(H
A
, β
A
)
and γ(H
B
, β
B
) have β
A
, β
B
, then their combined
equilibrium state γ(H
AB
, β
AB
) = γ(H
A
, β
AB
)
γ(H
B
, β
AB
) can still have a smaller bound energy
without altering the total information content. There-
fore, the associated joint completely passive state ac-
quires a unique β
AB
such that
E(γ(H
A
, β
A
)) + E(γ(H
B
, β
B
)) > E(γ(H
AB
, β
AB
)),
(19)
which is follows immediately from the min-energy
principle and (P5). Moreover, if β
A
β
B
, then
Eq. (19) dictates a bound on β
AB
, as expressed in
the following lemma.
Lemma 7. Any iso-entropic equilibration pro-
cess between γ(H
A
, β
A
) and γ(H
B
, β
B
), with
β
A
β
B
, leads to a state γ(H
AB
, β
AB
) of mu-
tual equilibrium, where β
AB
satisﬁes
β
A
β
AB
β
B
. (20)
Proof. For β
A
= β
B
, (P3) and (P5) immediately
A
= β
AB
= β
B
. What we want to show
is that, for β
A
> β
B
β
A
> β
AB
> β
B
. This can be seen from the
information preservation condition on the equili-
bration process. Assume the contrary, say that
after the equilibration the ﬁnal inverse tempera-
ture is β
AB
β
A
> β
B
. But according to (P2),
the entropy is monotonically decreasing with β,
hence we would conclude that the total entropy
of the system has decreased, contradiction to the
assumption of an iso-entropic process. Likewise,
β
A
> β
B
β
AB
the total entropy of the system would have in-
creased. Therefore, the only possibility remain-
ing is β
A
> β
AB
> β
B
.
Now, with a clear notion of equilibration and equi-
librium state, which has minimum internal energy for
a ﬁxed information content, we can restate the zeroth
law in terms of the intrinsic temperature. By deﬁ-
nition, the global completely passive state has mini-
mum internal energy and one cannot extract free en-
ergy by using global entropy preserving operations.
Further, a global completely passive state not only
Accepted in Quantum 2018-11-28, click title to verify 7
assures that the individual states are also completely
passive but also enforces that they share the same in-
trinsic temperature β and has vanishing inter-system
correlations.
We can recover the traditional notion of ther-
mal equilibrium as well as zeroth law. In tradi-
tional thermodynamics, the thermal bath is consid-
erably much larger than the system under consider-
ation, and is initially in a completely passive state
with a predeﬁned temperature. With respect to the
bath Hamiltonian H
B
, it can be expressed as γ
B
=
e
β
B
H
B
/Tr (e
β
B
H
B
). Now, if the state ρ
S
of the
system S (with |S| |B|) with Hamiltonian H
S
is brought in contact with the thermal bath, then the
global thermal equilibrium state will be a completely
passive state, i.e. γ
B
ρ
S
Λ
ep
γ
0
B
γ
0
S
, with
a global inverse equilibrium temperature β
e
. With
|B| |S|, one now easily sees that β
e
β
B
and
γ
0
S
γ(H
S
, β
B
) = e
β
B
H
S
/Tr (e
β
B
H
S
).
6 Max-entropy principle
vs. min-energy principle
Let us mention that our framework, which is based
on the principle of information conservation, is suited
for the work extraction problem. It assumes that the
system has mechanisms to release energy to some
battery or classical ﬁeld, but it cannot interchange
entropy with it. This contrasts with other situations
in spontaneous equilibration, in which the system is
assumed to evolve keeping constant the conserved
quantities. In such scenario, the equilibrium state is
described by the principle of maximum entropy, that
is, the state the maximizes the entropy given the con-
served quantities.
In the context of the maximum entropy principle,
let us introduce the absolute athermality as (cf. [43]):
Deﬁnition 8 (Athermality). The athermality of
a system in a state ρ and Hamiltonian H is the
amount of entropy (information) that the system
can still accommodate without increasing its en-
ergy, i.e.
A(ρ) B max
σ : E(σ)=E(ρ)
S(σ) S(ρ), (21)
where the maximization is over all states σ with
energy E(ρ).
The state σ that maximizes (21) is obviously a
thermal state. In particular, it is the equilibrium state
of an equilibration process guided by the maximum
entropy principle, where the energy of the system re-
mains constant. Let us call the temperature of such
thermal state the spontaneous equilibration temper-
ature, and denote it by
˜
β(ρ). The athermality cor-
responds to the amount of entropy produced during
such an equilibration process.
Note that the spontaneous equilibration tempera-
ture
˜
β(ρ) always diers from the intrinsic temper-
ature β(ρ) unless ρ is thermal. In other words,
the maximum entropy and minimum energy princi-
ples lead to dierent equilibrium temperatures. As
both entropy and energy are monotonically increas-
ing functions of temperature, the equilibrium temper-
ature determined by the minimum energy principle is
always smaller than the one determined by the maxi-
mum entropy principle. This is represented in the en-
ergy entropy diagram of Fig. 2, where the athermality
can be seen as the vertical distance from a state ρ to
the thermal boundary. The point in which the thermal
boundary intersects with the vertical line E = E(ρ)
represents the equilibrium thermal state given by the
max-entropy principle. Its temperature
˜
β(ρ) corre-
sponds to the slope of the tangent line of the thermal
boundary at that point.
S
E
F (ρ)
A(ρ)
β(ρ)
˜
β(ρ)
β(ρ) >
˜
β(ρ)
x
ρ
E(ρ)
S(ρ)
Figure 2: Representation of the free energy F (ρ) and
the athermality A(ρ) of a state ρ in the energy-entropy
diagram. The intrinsic temperature β(ρ) and the spon-
taneous equilibration temperature
˜
β(ρ) satisfy β(ρ) >
˜
β(ρ) due to the concavity of the thermal boundary.
We have chosen the name “athermality” for the
quantity deﬁned in (8) in agreement with Ref. [43].
There, a quantity called β-athermality is deﬁned by
A
β
(ρ) B βE(ρ) S(ρ) + log Z
β
, (22)
with Z
β
= Tr (e
βH
) the partition function. It is in-
troduced to characterise the energy-entropy diagram
as the set of all points with positive entropy S(ρ) 0
and positive β-athermality A
β
(ρ) 0 for all β R.
Lemma 9 (Athermality vs. β-athermality). The
athermality of a state ρ can be written in terms
Accepted in Quantum 2018-11-28, click title to verify 8
of the β-athermalities as
A(ρ) = min
β
A
β
(ρ), (23)
where the minimum is attained by the sponta-
neous equilibration temperature
˜
β(ρ).
Proof. We prove the lemma by giving a geometric
interpretation to the β-athermality in the energy-
entropy diagram. In Fig. 3, let us consider the
tangent line to the thermal boundary with slope
β. Such a straight line is formed by the set of all
points satisfying
S S
β
= β(E E
β
), (24)
where S
β
and E
β
are respectively the entropy and
the energy of the point on the thermal boundary
that lies on the line. The vertical distance (dif-
ference in entropy) of this line from a point with
S
S(ρ) = β(E(ρ) E
β
) S(ρ)
= βE(ρ) S(ρ) β(E
β
T S
β
),
(25)
where S
is the entropy of the line at E = E(ρ)
and T = β
1
. By noticing that log Z
β
= β(E
β
T S
β
), we get that the β-athermality A
β
(ρ) of ρ is
the vertical distance of x
ρ
from the line with slope
β tangent to the thermal boundary. Eq. (23) triv-
ially follows from this.
S
E
β
x
ρ
A
β
(ρ)
F
β
(ρ) F
β
(γ(β))
Figure 3: Representation of the β-free energy diﬀerence
F
β
(ρ) F
β
(γ(β)) and the β-athermality A
β
(ρ) for a
state ρ with coordinates x
ρ
in the energy-entropy dia-
gram.
By identifying the β-free energy in Eq. (22), the
β-athermality can be written as
A
β
(ρ) = β (F
β
(ρ) F
β
(γ(β))) , (26)
that is, it is β times the work that can be potentially
extracted from a state ρ when having an inﬁnite bath
uncorrelated from the system at temperature β (see
Eq. (14)). From Eq. (26), we see that the dierence
in β-free energies F
β
(ρ) F
β
(γ(β)) can be repre-
sented in the energy-entropy diagram as the horizon-
tal distance from the point x
ρ
to the tangent line with
slope β. This is represented in Fig. 3. Hence, given
an inﬁnitely large bath at some inverse temperature
β, all the states with the same work potential lie on
a line with slope β in the energy-entropy diagram.
The work extracted from a state ρ can be decomposed
into its free energy F (ρ), the part of energy that could
have been extracted without bath, and the rest, i.e. the
part of the energy that has been accessed thanks to the
bath. This latter part is closely related to heat. The
deﬁnitions of work and heat are discussed in the next
section.
Furthermore, thanks to Lemmas 5 and 9, both the
free energy and the athermality can be expressed in
terms of a minimization over standard β-free energies
F (ρ) = min
β
(F
β
(ρ) F
β
(γ(β))) ,
A(ρ) = min
β
(β (F
β
(ρ) F
β
(γ(β)))) ,
(27)
where the minimum is attained by the intrinsic tem-
perature β(ρ) and the spontaneous equilibration tem-
perature
˜
β(ρ), respectively.
Note that, while for positive temperatures both the
athermality and the free energy are a measure of out
of equilibrium, for negative temperatures, states with
small athermality are highly active and have huge free
energies.
Let us ﬁnally show that in the equilibration of a
hot body in contact with a cold one, the intrinsic
temperature and the spontaneous equilibration one
are not so much dierent. In Fig. 4, we sketch the
energy-entropy diagram for the particular case of the
equilibration of a system composed of two identical
subsystems initially at dierent temperatures β
A
and
β
B
. As expected and due to the concavity of the
thermal boundary, the equilibrium temperature β
1
E
given by the constant energy constraint (max-entropy
principle) is larger that the one given by the con-
stant entropy constrain (min-energy principle), β
1
S
,
i.e. β
E
< β
S
. The dierence in bound energies of
these two thermal states corresponds precisely to the
work extracted in the constant entropy scenario.
7 Work, heat and the ﬁrst law
In thermodynamics, the ﬁrst law deals with the con-
servation of energy. In addition, it dictates the distri-
Accepted in Quantum 2018-11-28, click title to verify 9
S
E
β
A
β
B
E
B
E
A
¯
E
S
A
S
B
¯
S
β
S
β
E
¯
E
:
=
1
2
(E
A
+ E
B
)
¯
S
:
=
1
2
(S
A
+ S
B
)
Figure 4: Energy-entropy diagram of a system with
Hamiltonian H. Two systems with the same Hamilto-
nian H and initially at diﬀerent temperatures β
A
and
β
B
equilibrate to diﬀerent temperature depending on
the approach taken: minimum energy principle vs. max-
imum entropy principle. The equilibrium temperature
when entropy is conserved β
1
S
is always smaller than
the equilibrium temperature when energy is conserved
β
1
E
, i. e. β
S
> β
E
.
bution of energy over work and heat, that are the two
forms of thermodynamically relevant energy.
Let us deﬁne a thermodynamic process involving a
system A and a bath B as a transformation ρ
AB
ρ
0
AB
that conserves the global entropy S(ρ
AB
) =
S(ρ
0
AB
). In standard thermodynamics, the bath is
assumed to be initially in a thermal state and com-
pletely uncorrelated from the system. In such a sce-
nario, the heat dissipated in the process is usually de-
ﬁned as the change in the internal energy of the bath,
Q = E(ρ
0
B
) E(ρ
B
), where ρ
(0)
B
= Tr
A
ρ
(0)
AB
is
the reduced state of the bath. Here, and in the fol-
lowing, primed quantities refer to the state after the
process, whereas their unprimed versions refer to the
state before the process. This deﬁnition, however,
may have some limitations and alternative deﬁnitions
have been discussed recently [19]. In particular the
information theoretic approach from [19] suggests
that heat be deﬁned as Q = T
B
S
B
with T
B
be-
ing the temperature and S
B
= S(ρ
0
B
) S(ρ
B
) the
change in von Neumann entropy of the bath. Note
that S
B
= S(A|B) is also the conditional en-
tropy change in system A, conditioned on the bath
B, where S(A|B) = S(ρ
AB
) S(ρ
B
), which can be
understood as the information ﬂow from the system
to the bath in the presence of correlations.
In the present approach, we go beyond the restric-
tion that the environment has to have a deﬁnite prede-
ﬁned temperature, and be in a state of the Boltzmann-
Gibbs form. For an arbitrary environment, which
could even be athermal, we propose to quantify the
heat transfer in terms of bound energies, as follows.
S
E
x
ρ
B
x
0
ρ
B
E
B
E
0
B
S
B
S
0
B
T
B
S
B
Q
E
B
Figure 5: Representation in the energy-entropy diagram
of the diﬀerent notions of heat: Q as change of the
bound energy of the bath, E
B
as the change in internal
energy, and T
B
S
B
. In this example, the bath is initially
thermal but this does not have to be necessarily the case.
Deﬁnition 10 (Heat). Given a system A and its
environment B, the heat dissipated by the system
A in the process ρ
AB
Λ
ep
ρ
0
AB
is deﬁned as the
change in the bound energy of the environment
B, i.e.
Q B B(ρ
0
B
) B(ρ
B
), (28)
where B(ρ
B
) and B(ρ
(0)
B
) are the initial and ﬁnal
bound energy of the bath B, respectively.
Note that heat is a somewhat process dependent
quantity from the point of view of the work system A,
in the sense that there can be dierent processes with
the same initial and ﬁnal marginal state for A, but dif-
ferent marginals for B. Because the global process is
entropy preserving, processes that lead to the same
marginal for A, but dierent marginals with dierent
entropies for B, necessarily imply a dierent amount
of correlations between A and B, as measured by the
mutual information I(A : B) B S
A
+ S
B
S
AB
.
Namely, S
A
+ S
B
= ∆I(A : B).
Lemma 11 (Connections among deﬁnitions of
heat). Given a system A and its environment B,
the heat disspated by A in a globally entropy pre-
serving process ρ
AB
Λ
ep
ρ
0
AB
can be written as
Q =
Z
S
0
B
S
B
T (s)ds, (29)
where S
(0)
B
= S(ρ
(0)
B
) is the initial (ﬁnal) entropy
of B, and T(S) is the intrinsic temperature of
the thermal state of B with entropy S. Because
of the continuity of T (S), there exists hence an
s
[S
B
, S
0
B
] such that
Q = T (s
)∆S
B
. (30)
Thus, Deﬁnition 10 becomes Q = T S
B
in the
limit of small entropy changes along the process,
Accepted in Quantum 2018-11-28, click title to verify 10
in which case T (S
B
) T (S
0
B
) T (s
), while
it diﬀers from the common deﬁnition in general,
and Q , E
B
.
In the case that the initial state of the
environment is thermal ρ
B
= γ(H
B
, T
1
) =
e
H
B
/T
/Tr (e
H
B
/T
) with Hamiltonian H
B
at
temperature T , the heat is lower and upper
bounded by
T S
B
Q E
B
, (31)
where the three quantities have been represented
in the energy-entropy diagram in Fig. 5. If, in
addition, the process perturbs the environment
only slightly, ρ
0
B
= ρ
B
+ δρ
B
, then the three deﬁ-
nitions of heat coincide, in the sense that
T S
B
+ O(δρ
2
B
) = ∆Q = ∆E
B
O(δρ
2
B
). (32)
That is, in the limit of large thermal baths, the
three deﬁnitions become equivalent.
Proof. Eq. (29) is a consequence of the deﬁni-
tions of heat (28) and bound energy (6), together
with Eq. (5). Next, Eq. (30) is proven by using
the mean value theorem for the function T(s).
The lower bound in (31) is due to the concav-
ity of the thermal boundary together with the
reminder theorem of a ﬁrst order Taylor expan-
sion of the thermal boundary (see Fig. 5). The
upper bound in (31) is a consequence of the pos-
itivity of the free-energy. Finally, Eq. (32) fol-
lows from the fact that the thermal state ρ
B
is
a minimum of the standard free energy F
T
(ρ) =
E(ρ) T S(ρ), which implies F
T,B
= O(δρ
2
B
),
and E
B
= T S
B
+ O(δρ
2
B
).
Now that heat has been unambiguously deﬁned,
we can pass to the deﬁnition of work (cf. [2]).
Deﬁnition 12 (Work). For an arbitrary entropy
preserving transformation involving a system A
and its environment B, ρ
AB
ρ
0
AB
, with ﬁxed
non-interacting Hamiltonians H
A
and H
B
, the
worked performed on the system A is deﬁned as
W
A
B W F
B
, (33)
where W = E
A
+ E
B
is the work cost of im-
plementing the global transformation and F
B
=
F (ρ
0
B
)F (ρ
B
) is the change in free energy of the
bath.
Equipped with the notions of heat and work, the
ﬁrst law takes the form of a mathematical identity.
Lemma 13 (First law). For an arbitrary entropy
preserving transformation involving a system A
and its environment B, ρ
AB
ρ
0
AB
, with ﬁxed
non-interacting Hamiltonians H
A
and H
B
, the
change in energy for A is distributed as
E
A
= ∆W
A
Q. (34)
Proof. The proof follows from the deﬁnitions of
work and heat.
The interpretation of this identity is that under a
given process, W
A
is the amount of energy that
can be transferred to and stored in a battery. While
the heat Q
A
is the change in energy due to ﬂow of
information from/into the system.
8 Second law
The second law of thermodynamics is formulated in
many dierent forms: as an upper bound on the ex-
tracted work, the impossibility of converting heat into
work completely, etc. In this section we show how
all these formulations are a consequence of the prin-
ciple of information conservation. Note that the for-
mulation of the second law following from global en-
tropy conservation is not new (e.g. in [39]), but it was
thought to require a bath with well-deﬁned tempera-
ture. What is new in the present approach is that we
do not need even a bath, let alone that it is initially in
a thermal state with pre-deﬁned temperature.
One of the most important questions in thermo-
dynamics, both historically and conceptually, is to
which extent the energy in a heat bath can be con-
verted into work. The second law puts a limit on
the amount of work extracted. Over the years, re-
search has led to various formulations of second law
in standard thermodynamics, like the Clausius state-
ment, the Kelvin-Planck statement and the Carnot
statement, to mention a few.
Similar questions can be posed in the framework
considered here, in terms of bound energy. In the fol-
lowing, we present analogous forms of second laws
that consider these questions qualitatively as well as
quantitatively.
8.1 Work extraction
One of the major concerns in thermodynamics is the
conversion of any form of energy into work, which
can be used for any application with certainty.
Accepted in Quantum 2018-11-28, click title to verify 11
Lemma 14 (Second law: work extraction). For
an arbitrary composite system ρ, the extractable
work by any entropy preserving process ρ ρ
0
,
W = E(ρ) E(ρ
0
) is upper-bounded by the free
energy
W F (ρ), (35)
which is saturated with equality if and only if ρ
0
=
γ(ρ).
If the system initially is in the particular state
ρ = ρ
A
γ
B
(T
B
) with γ
B
(T
B
) being thermal at
temperature T
B
, then
W F
T
B
(ρ
A
) F
T
B
(γ
A
(T
B
)) (36)
where F
T
(·) is the standard out-of-equilibrium
free energy, and the inequality is saturated in the
limit of an inﬁnitely large bath (inﬁnite heat ca-
pacity).
Proof. By using E(ρ
0
) B(ρ
0
), one gets
W = E(ρ) E(ρ
0
) E(ρ) B(ρ
0
) = F (ρ), (37)
where note that B(ρ
0
) = B(ρ).
When the initial state has the composite struc-
ture ρ = ρ
A
γ
B
(T
B
), we get
W F (ρ
A
γ
B
(T
B
))
F
T
(ρ
A
γ
B
(T
B
)) F
T
(γ
A
(T
B
) γ
B
(T
B
))
= F
T
(ρ
A
) F
T
(γ
A
(T
B
)),
(38)
we have used Lemma 5.
Finally, in the limit of an inﬁnitely large bath,
the intrinsic temperature T
AB
tends to the bath
temperature T
B
and Q = T
B
S
B
, which satu-
rates the bound in Eq. (38).
To see that the intrinsic temperature T
AB
tends to T
B
in the limit of large baths, let us
increase the bath size by adding several copies of
it. The entropy change of such bath reads
S
B
= S(γ
n
B
(T
AB
)) S(γ
n
B
(T
B
))
= n (S(γ
B
(T
AB
)) S(γ
B
(T
B
))) .
(39)
By considering the bound S
B
= S
A
|A|,
we get
S(γ
B
(T
AB
)) S(γ
B
(T
B
))
|A|
n
, (40)
which, together with the continuity of S(γ(T )),
proves lim
n→∞
T
AB
= T
B
.
8.2 Clausius statement
The second law of thermodynamics not only dictates
the direction of state transformations, but also estab-
lishes a fundamental bound on the amount of work
that can be extracted by them. Here we ﬁrst concen-
trate on the bounds on extractable work and introduce
analogous versions of the second law in our setup.
Lemma 15 (Second law: Clausius statement).
Any iso-entropic process involving two bodies A
and B in an arbitrary state, with intrinsic tem-
peratures T
A
and T
B
, respectively satisﬁes the fol-
lowing inequality:
(T
B
T
A
)∆S
A
> F
A
+F
B
+T
B
I(A : B)W,
(41)
where F
A/B
is the change in the free energy of
the body A/B, I(A : B) is the change of mu-
tual information, and W = E
A
+ ∆E
B
is the
amount of external work performed on the total
system.
In the absence of initial correlations between
the two bodies A and B, the states being initially
thermal, and no external work performed, this
implies
(T
B
T
A
)∆S
A
> 0 , (42)
meaning that no iso-entropic equilibration process
is possible whose sole result is the transfer of heat
from a cooler to a hotter body.
Proof. The deﬁnition of free and bound energy
implies that
W = ∆F
A
+ F
B
+ Q
A
+ Q
B
, (43)
where have used the deﬁnition of heat as the
change of bound energy of the environment.
From Eq. (31), one gets,
Q
A
+ Q
B
> T
B
S
B
+ T
A
S
A
, (44)
where T
A/B
is the initial intrinsic temperature of
the body A/B. Due to the conservation of the to-
tal entropy, the change in mutual information can
be written as I(A : B) = ∆S
A
+ S
B
. Putting
everything together and noting that sign(∆B) =
sign(∆S) completes the proof.
The terms on the right hand side of Eq. (41) show
the three reasons for which the standard Clausius
statement can be violated. Either because of the pro-
cess not being spontaneous (external work is per-
formed, W > 0), or due to initial states having free
Accepted in Quantum 2018-11-28, click title to verify 12
energy which is consumed, or the presence of corre-
lations [19].
An alternative formulation of the Clausius state-
ment, for initial and ﬁnal equilibrium states, is con-
sidered in Appendix A.
8.3 Kelvin-Planck statement
While the Clausius statement tells us that heat cannot
ﬂow spontaneously from a hotter to a colder body, the
Kelvin-Plank formulation of second law states that,
when heat going from a hotter to a colder body, it
cannot be completely transformed into work.
Lemma 16 (Second law: Kelvin-Planck state-
ment). Any iso-entropic process involving two
bodies A and B in an arbitrary state satisﬁes the
following energy balance
Q
B
+ Q
A
= (∆F
A
+ F
B
) + W, (45)
where F
A/B
is the change in the free energy of
the body A/B, Q
A/B
the heat dissipated by the
body A/B, and W = ∆E
A
+ E
B
is the amount
of external work performed on the total system.
In the case of the reduced states being thermal,
and for a work extracting process W < 0, the
above equality becomes
Q
B
+ Q
A
6 W < 0. (46)
Finally, in the absence of initial correlations,
Eq. (46) implies that no iso-entropic equilibration
process is possible </