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
general situations, beyond the traditional
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 offers 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 effects 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 [4–7], Szilard’s en-
gine [8], and Landauer’s principle [9–13]. In recent
years, classical and quantum information-theoretic
approaches helped to understand thermodynamics in
the domain of small classical and quantum systems
[14–16]. 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 [17–19], equilibration
processes [20–23], 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 effects in thermody-
namics [26–34].
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 [37–40]. 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 affected 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(pρ
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
an additional system of O(
√
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
additional system A with O(
√
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 different 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-
sive (CP) state [17]. Note, in this article, we denote
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
⊗X−1
⊗ H
X
⊗ I
⊗N−X
, 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)
together with the subadditivity property
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
, by the ad-
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 different 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-
not have access to asymptotically large thermal baths,
or in the complete absence of a bath, the equilibra-
tion process is expected to be considerably different.
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
lead to β
A
= β
AB
= β
B
. What we want to show
is that, for β
A
> β
B
, the equilibration leads to
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
leads to the contradiction that
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 differs from the intrinsic temper-
ature β(ρ) unless ρ is thermal. In other words,
the maximum entropy and minimum energy princi-
ples lead to different 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
coordinates (E(ρ), S(ρ)) reads
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 difference
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 different. 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 different 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 difference 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 different 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 different marginals with different
entropies for B, necessarily imply a different 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 different 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 whose sole result is the absorp-
tion of bound energy (heat) from an equilibrium
state and its complete conversion into work.
Proof. Eq. (45) is a consequence of the energy
balance (43). Eq. (46) follows from (45) by con-
sidering reduced states that are initially thermal
and thereby ∆F
A/B
> 0. To prove the final state-
ment is sufficient to notice that entropy preserv-
ing processes on initially uncorrelated systems
fulfill ∆S
A
+ ∆S
B
> 0, which together with (46)
implies that sign(∆Q
A
) = −sign(∆Q
B
).
An alternative formulation of Kelvin-Planck state-
ment, for initial and final equilibrium states, is con-
sidered in Appendix B.
8.4 Carnot statement
A traditional heat engine extracts work from a situ-
ation in which two thermal baths A and B have dif-
ferent temperatures T
A
and T
B
. The work extrac-
tion is usually implemented in practice by means of
a working body S which cyclically interacts with A
and B. Here we are not concerned with how pre-
cisely the working medium interacts with the baths
A and B, but just assume that at the end of every
cycle the working body is left in its initial state and
uncorrelated with the two bath. In other words, the
working body only has the role to move energy from
one bath to the other. From this perspective, the heat
engine can be analyzed by studying the changes of
the environments A and B. In contrast to the stan-
dard situation, here we will not assume that the baths
are infinitely large, but that they have a similar size as
the system, and so the loss or gain of energy changes
their (intrinsic) temperature.
Without loss of generality we assume T
A
< T
B
and ρ
AB
= γ
A
(T
A
) ⊗ γ
B
(T
B
). After operating the
machine for one or several complete cycles, the baths
experience a change ρ
AB
→ ρ
0
AB
.
We define the efficiency of work extraction in a
heat engine as the fraction of energy that is taken
from the hot bath which is transformed into work:
η B
W
−∆E
B
, (47)
where −∆E
B
= E
B
− E
0
B
> 0 is the energy taken
from the hot bath, and W is the work extracted. In
the following, we upper-bound the efficiency of any
heat engine.
Lemma 17 (Second law: Carnot statement).
For an engine working with two initially uncor-
related environments, ρ
AB
= γ
A
(T
A
) ⊗ γ
B
(T
B
),
each in a local equilibrium state with intrinsic
temperatures T
B
> T
A
, the efficiency of work ex-
traction is bounded by
η 6 1 −
∆B
A
−∆B
B
, (48)
where ∆B
A
and ∆B
B
are the changes in bound
energies of the systems A and B, respectively.
In the limit of large baths and under global en-
tropy preserving operations, the Carnot efficiency
is recovered,
η 6 1 −
T
A
T
B
. (49)
Proof. The maximum extractable work due to
the transformation ρ
AB
→ ρ
0
AB
is given by
W = F (ρ
AB
) − F (ρ
0
AB
) = (−∆E
B
) − ∆E
A
> 0.
(50)
The efficiency then reads
η = 1 −
∆E
A
−∆E
B
. (51)
Accepted in Quantum 2018-11-28, click title to verify 13
The condition of A being initially at equilibrium
implies that ∆F
A
> 0, from which it follows that
∆E
A
> ∆B
A
, and analogously for B. Thus,
η 6 1 −
∆B
A
−∆B
B
. (52)
In the limit of large baths, in which ∆B
A
B
A
,
the change in bound energy becomes ∆B
A
=
T
A
∆S
A
. Hence,
η 6 1 −
T
A
∆S
A
−T
B
∆S
B
. (53)
If the process is globally entropy preserving,
i.e. S
0
AB
= S
A
+ S
B
, then ∆S
A
+ ∆S
B
> 0, or
alternatively ∆S
A
> −∆S
B
, which together with
(53) implies (49).
9 Third law?
The third law of thermodynamics establishes the im-
possibility of attaining the absolute zero temperature.
According to Nernst, “it is impossible to reduce the
entropy of a system to its absolute-zero value in a fi-
nite number of operations”. The third law of thermo-
dynamics has been very recently proven in a resource
theoretic setting [50]. The unattainability of absolute
zero entropy is a consequence of the unitary charac-
ter of the transformations considered. For instance,
consider the state transformation
ρ
S
⊗ ρ
B
−→ |0ih0|⊗ ρ
0
B
, (54)
where ρ
B
is the thermal state of the bath B, which
models the cooling down (erasure) of the system S,
initially in a state ρ
S
, which we assume to have
rank(ρ
S
) > 1. Here the dimension of the Hilbert
space of the bath, d
B
, is considered to be arbitrarily
large but finite. As ρ
B
is a full-rank state, the left
hand side and the right hand side of Eq. (54) have
different ranks, and they cannot be transformed via
a unitary transformation. Thus, irrespective of work
supply, one cannot attain the absolute zero entropy
state.
The zero entropy state can only be achieved for in-
finitely large baths and a sufficient work supply. As-
suming a locality structure for the bath’s Hamilto-
nian, this would take an infinitely long time. How-
ever, in the finite dimensional case, a quantitative
bound on the achievable temperature given a finite
amount of resources (e. g. work, time) is derived in
[50].
Note that our set of entropy preserving operations
is more powerful than unitaries. In particular, trans-
formation (54) is always possible as long as the en-
tropy of the system can be allocated in the bath, i.e.
S(ρ
S
) 6 log d
B
− S(ρ
B
) . (55)
The required work to do so will be W = F(|0ih0| ⊗
ρ
0
B
) − F (ρ
S
⊗ ρ
B
). As it has been discussed in
Sec. 2, entropy preserving operations can be imple-
mented as global unitaries acting on asymptotically
many copies. Thus, the attainability of the absolute
zero entropy state by means of entropy preserving op-
erations is in agreement with the cases of infinitely
large baths and unitary operations [50].
In conclusion, the third law of thermodynamics is
a consequence of the microscopy reversibility (uni-
tarity) of the transformation and is not respected by
entropy preserving operations.
10 A temperature-independent re-
source theory of thermodynamics
In this section we connect the framework developed
above with the recently developed “standard” ap-
proach to thermodynamics as a resource theory. The
main ingredients of a resource theory are the state
space, which is usually compatible with a compo-
sition operation —for instance, quantum states to-
gether with the tensor product—, and a set of of al-
lowed operations, which should facilitate state trans-
formations that are in general irreversible. In a first
attempt to have a resource theory of thermodynamics
from our framework, it would seem that we are bound
to transformations within an equivalence class, that
is, between states on the same Hilbert space and with
equal entropies, which is of course a very poor per-
spective. To compare and explore the interconvert-
ibility of states with different entropies, it is neces-
sary to extend the set of operations, and in particular
to add a composition rule. In order to be able to con-
nect states with different entropies and/or number of
particles it is natural to consider different number of
copies, and ask about rates of transformations.
Let us restrict first to the case in which the set of
operations is entropy preserving regardless of the en-
ergy. We ask what is the rate of transforming ρ into σ
and consider first the case that S(ρ) ≤ S(σ). Then,
for large enough n and m we have
S(ρ
⊗n
) ∼ S(σ
⊗m
). (56)
Accepted in Quantum 2018-11-28, click title to verify 14
With such a trick of considering a different number of
copies, we have brought ρ and σ, which had different
entropies, into the same manifold of equal entropy.
There is a subtlety here which is that ρ
⊗n
and σ
⊗m
live in general on different Hilbert spaces. This can
be easily circumvented by adding some product state,
i.e.
ρ
⊗n
−→ σ
⊗m
⊗ |0ih0|
⊗(n−m)
, (57)
where ρ
⊗n
and σ
⊗m
⊗|0ih0|
⊗(n−m)
now live on the
same Hilbert space. Eq. (57) can also be understood
as a process of information compression, in which the
information in n copies of ρ is compressed to m < n
copies of σ, and n − m systems have been erased.
Thus, in the case of having only an entropy preserv-
ing constraint, the transformation rate can be deter-
mined from (56)
r B lim
n→∞
m
n
=
S(ρ)
S(σ)
. (58)
In thermodynamics, however, energy must also be
taken into account. When one does that, it could well
be that the process (57) is not energetically allowed,
since it might happen that E(ρ
⊗n
) , E(σ
⊗m
). In
such a case, we would need more copies of ρ, in order
for the transformation to be energetically favorable.
This would force us to add an entropic ancilla, since
otherwise, the initial and final state of such a process
would not have the same entropy.
Let us thus consider a transformation
ρ
⊗n
−→ σ
⊗m
⊗ φ
⊗(n−m)
, (59)
where the number of copies n and m have to fulfil
the energy and entropy conservation constraints
E(ρ
⊗n
) ∼ E(σ
⊗m
⊗ φ
⊗(n−m)
),
S(ρ
⊗n
) ∼ S(σ
⊗m
⊗ φ
⊗(n−m)
).
(60)
The above conditions can be easily written as a ge-
ometric equation of the points x
ψ
= (E(ψ), S(ψ))
with ψ ∈ {ρ, σ, φ} in the energy-entropy diagram:
x
ρ
= r x
σ
+ (1 − r) x
φ
, (61)
where r B m/n is the conversion rate, and we have
only used the extensivity of both entropy and energy
in the number of copies, e.g. E(ρ
⊗n
) = nE(ρ).
Eq. (61) implies that the three points x
ρ,σ,φ
need
to be aligned. In addition, the fact that 0 ≤ r ≤
1 implies that x
ρ
lies in the segment connecting x
σ
and x
φ
(see Fig. 6). The conversion rate r has then a
geometric interpretation. It is the relative Euclidean
S
E
ρ
⊗n
→ σ
⊗m
⊗ φ
⊗(n−m)
x
φ
x
σ
x
ρ
a
b
r =
m
n
=
a
b
Figure 6: Representation in the energy-entropy diagram
of how the energy and entropy conservation constraints
in the transformation ρ
⊗n
→ σ
⊗m
⊗ φ
⊗n−m
imply x
ρ
,
x
σ
and x
φ
to be aligned and x
ρ
to lie in between x
σ
and
x
φ
.
distance between x
φ
and x
ρ
over the total distance
kx
σ
− x
φ
k, or in other words, it is the fraction of the
path that one has run when going from x
φ
to x
σ
and
reaches x
ρ
(see Fig. 6).
In order for the conversion rate r from ρ and σ to
be maximum, the state φ needs to lie in the boundary
of the energy-entropy diagram, that is, it needs to be
either a thermal or a pure state. This allows us to
quantitatively determine the conversion rate r by first
finding the temperature of the thermal state φ which
satisfies
S(σ) − S(φ)
S(σ) − S(ρ)
=
E(σ) − E(φ)
E(σ) − E(ρ)
, (62)
and then
r =
S(ρ) − S(φ)
S(σ) − S(φ)
. (63)
See the geometrical interpretation in Fig. 6.
Note how Eq. (63) becomes (58) in the case of φ
being pure. Let us also mention that, although at the
first sight the inter-convertibility rate (63) looks dif-
ferent from the one obtained in [43], it can be proven
that they are the same. Eq. (63) is more compact than
the one given there, and its derivation much less tech-
nical.
11 Thermodynamics with multiple
conserved quantities
The formalism for thermodynamics developed above
can easily be extended to situations with multiple
conserved quantities besides the energy, which we
call “charges”. These could be particle numbers
of certain chemical species, or physically conserved
quantities such as angular momentum. Such exten-
sions to quantum thermodynamics, especially in the
resource setting, have been considered previously by
several authors [51–55].
Accepted in Quantum 2018-11-28, click title to verify 15
Figure 7: Schematic of the charges-entropy diagram of
a system with two conserved quantities E and L.
The setting is that each system comes with q con-
served quantities
ˆ
L
k
for k = 0, . . . , q −1, with
ˆ
Q
0
=
H the Hamiltonian. Here we will assume that these
quantities are all pairwise commuting. As before,
when we compose systems, the quantities are simply
added: for another Hilbert space H
0
with charges
ˆ
L
0
k
(k = 0, . . . , q − 1), the joint Hilbert space H ⊗ H
0
carries the global (or total) charges
ˆ
L
k
⊗ + ⊗
ˆ
L
0
k
.
11.1 The charges-entropy diagram
The energy-entropy diagram from the previous dis-
cussion is extended to include the additional con-
served quantities. Let us introduce the charges-
entropy diagram as:
Definition 18 (Charges-entropy diagram). The
charges-entropy diagram is the subset of points
in R
q+1
produced by all the states ρ ∈ B(H) with
coordinates
x
ρ
= (L
0
(ρ), L
1
(ρ), . . . , L
q−1
, S(ρ)), (64)
where L
k
(ρ) B Tr
ˆ
L
k
ρ for k = 0, . . . , q − 1. Note
that in this section, to avoid confusion between
the observable (the operator
ˆ
L
k
) and the coordi-
nate in the charges-entropy diagram (the number
L
k
), we mark the former always by a hat.
Using the mutual commutation of the conserved
quantities, we can easily show that the charges-
entropy diagram forms a convex set in R
q+1
. To do
so, let us first prove convexity for the zero entropy
hyperplane, which corresponds to the base of the dia-
gram. The zero entropy hypersurface of the charges-
entropy diagram is the set of points in R
q
with co-
ordinates x
ψ
= (L
0
(ψ), L
1
(ψ), . . . , L
q−1
(ψ)) for all
normalized pure states ψ = |ψihψ|. Let us consider
in R
q
the family of hyperplanes perpendicular to the
unit vector ~µ = (µ
0
, . . . , µ
q−1
)
~µ ·
~
L =
q−1
X
k=0
µ
k
L
k
= C, (65)
where
~
L = (L
0
, . . . , L
q−1
) is a point in R
q
and C
determines the hyperplane. All points with coordi-
nates
~
L that fulfil Eq. (65) belong to the hyperplane
defined by ~µ and C. Given a direction ~µ, there is an
hyperplane that is particularly relevant, since it corre-
sponds to the minimum possible value of C
C
min
(~µ) = min
ψ∈H
hψ|~µ ·
~
ˆ
L|ψi
hψ|ψi
= min
ψ∈H
~µ ·
~
L(ψ)
hψ|ψi
.
(66)
The minimum is reached by the eigenstate with
smallest eigenvalue of the Hermitian operator ~µ ·
ˆ
L
that we denote by |ψ(~µ)i,
X
k
µ
k
ˆ
L
k
!
|ψ(~µ)i = C
min
(~µ)|ψ(~µ)i. (67)
Assume for the moment that there is a unique ground
state of ~µ ·
~
ˆ
L. This means that the hyperplane defined
by C
min
(~µ) is tangent to the charges-entropy diagram
since it passes through one point of the diagram (cor-
responding to the single ground state |ψ(~µi)) leav-
ing the rest of the diagram on the same side, i.e.
~µ ·
~
L(ψ) > C
min
(~µ) for all |ψi ∈ H.
In case that there is a degenerate ground space de-
scribed by a basis whose elements have different co-
ordinates in the space of charges, the contact between
the hyperplane C = C
min
and the zero entropy hy-
persurface is not a point but a simplex of some higher
dimension, up to the dimension of the ground space
(i.e. a segment in dimension 2, triangle in dimen-
sion 3, etc). As the conserved quantities are mutu-
ally commuting, there is a common eigenbasis for
all the q charges. Thus, there exists a basis of the
ground space of ~µ ·
~
L which is also eigenbasis of
all the charges L
k
individually. If |ψi and |ψ
0
i are
two elements of that basis with different coordinates
~
L and
~
L
0
, the state |ψ(θ)i = cos θ|ψi+sin θ|ψ
0
iwith
θ ∈ [0, π/2] has coordinates
~
L(ψ(θ)) = (cos
2
θ)
~
L + (sin
2
θ)
~
L
0
, (68)
that is, the coordinates of ψ(θ) in the diagram are
simply the corresponding convex combinations of the
coordinates of
~
L and
~
L
0
.
Accepted in Quantum 2018-11-28, click title to verify 16
Property (68) holds for any two eigenstates of
~
ˆ
L,
in particular, for any two states |ψi and |ψ
0
i that
lie in the boundary of the zero-entropy hypersurface.
Hence, as any point in the bulk of the zero-entropy
hypersurface can be obtained as a convex combina-
tion of two points of the boundary, all the points in
the bulk belong to the zero entropy surface. Alto-
gether, this proves that the zero-entropy surface is a
convex set.
Once we understand the zero entropy hypersur-
face of the charges-entropy diagram, let us study
its upper boundary. The upper boundary of the
charges-entropy diagram is described by the General-
ized Gibbs Ensemble (GGE) instead of the canonical
ensemble, i.e.
γ(
~
β) B
1
Z
~
β
e
−
P
k
β
k
ˆ
L
k
, (69)
where Z
~
β
B Tr
e
−
P
k
β
k
ˆ
L
k
is the generalized par-
tition function. These states are precisely the states
of maximum entropy among all those with prescribed
expectation values h
ˆ
L
k
i, k = 0, . . . , q − 1. We call
this boundary the equilibrium boundary.
The equilibrium boundary is mathematically de-
scribed by all the points
~
L(ρ) for which there exists a
~
β ∈ R
q
such that
q−1
X
k=0
β
k
L
k
(ρ) − S(ρ) = −log Z
β
. (70)
Note that, given some
~
β, the points
~
L(ρ) that fulfil
q−1
X
k=0
β
k
L
k
(ρ) − S(ρ) = C, (71)
belong to a hyperplane of dimension q. The hyper-
plane with C = −log Z
β
is the one tangent to the
equilibrium boundary. The normal vector to this fam-
ily of hyperplanes (71) is precisely (
~
β, −1). This
agrees with the fact that the tangent plane has a slope
β
k
in the k-th direction, i. e.
∂S
∂L
k
= β
k
∀ k = 0, . . . , q − 1. (72)
Eq. (72) can also be written in a vectorial form as
~
β =
~
∇S, (73)
and
~
β corresponds to the direction of maximal varia-
tion of the entropy. This is shown in Fig. 8.
Note that in order to microscopically justify the
charges-entropy diagram, it must be shown that states
S
Q
k
β
k
=
∂S
∂Q
k
(β
k
, −1)
x
ρ
A
~
β
(ρ)
x
σ
A(σ)
Figure 8: Transverse section of the charges-entropy dia-
gram in the L
k
-S plane. The normal vector to the tan-
gent plane has coordinates (
~
β, −1), which in the section
appears as (β
k
, −1). The
~
β-athermality, A
~
β
(ρ), and the
absolute athermality, A(σ), are respectively represented
for two states ρ and σ.
with the same expectation values for the conserved
quantities can be connected to each other using uni-
taries that commute with all charges, and using sub-
linear ancillas, in the limit of many copies. To
state the theorem, we fix some notation. Consider
a quantum system with mutually commuting charges
ˆ
L
0
= H,
ˆ
L
1
, . . . ,
ˆ
L
q−1
on a finite dimensional
Hilbert space H. The corresponding description of
n copies of this system has Hilbert space H
⊗n
and
charges
ˆ
L
(n)
k
=
n
X
i=1
H
⊗i−1
⊗ (
ˆ
L
k
)
i
⊗
H
⊗n−i
. (74)
Theorem 19. Two states ρ and σ of the above
quantum system have the same entropy S(ρ) =
S(σ) and the average charge values Tr (
ˆ
Q
k
ρ) =
Tr (
ˆ
Q
k
σ) for all k = 0, . . . , q − 1 if and only if
for all n there exists an ancillary quantum sys-
tem with the Hilbert space K of dimension 2
o(n)
,
with charges
ˆ
L
0
0
= H’,
ˆ
L
0
1
, . . . ,
ˆ
L
0
q−1
, and a charge
conserving unitary U acting on H
⊗n
⊗ K, such
that
lim
n→∞
U(ρ
⊗n
⊗ ω
0
)U
†
− σ
⊗n
⊗ ω
1
= 0, (75)
ˆ
L
0
k
∞
≤ o(n) and [U,
ˆ
L
(n)
k
+
ˆ
L
0
k
] = 0 ∀k,
(76)
where ω and ω
0
are suitable states of the ancillary
system.
The proof of this theorem (which generalises the
asymptotic equivalence theorem of [43]) is presented
in our forthcoming work [56]. There, we exten-
sively study this issue, and show how to generalize it
Accepted in Quantum 2018-11-28, click title to verify 17
even further from commuting to non-commuting con-
served charges. In the following, we limit ourselves
to the main points.
11.2 Athermality and free entropy
Proceeding as before, let us introduce some quantities
that will be relevant in the following.
Definition 20 (
~
β-athermality). The
~
β-
athermality of a state ρ is defined as
A
~
β
(ρ) B
q−1
X
k=0
β
k
L
k
(ρ) − S(ρ) + log Z
~
β
. (77)
The
~
β-athermality of a point with coordinates
(
~
L(ρ), S(ρ)) can be interpreted geometrically in the
charges-entropy diagram as the vertical distance from
the hyperplane tangent to the equilibrium boundary
with normal vector (
~
β, −1). This is represented in
Fig. 8. Note that the
~
β-athermality is also introduced
in [54] under the name free entropy.
The
~
β-athermality can also be written as
A
~
β
(ρ) = D(ρkγ(
~
β)), (78)
with D(ρkσ) B Tr (ρ log ρ −ρ log σ) being the rela-
tive entropy. Hence, because of the non-negativity of
the relative entropy D(ρkσ) > 0, we have that
S(ρ) 6
q−1
X
k=0
β
k
L
k
(ρ) + log Z
~
β
. (79)
Note now that the right hand side corresponds to the
entropy coordinate of the hyperplane with normal
vector
~
β tangent to the equilibrium boundary. Thus,
any tangent hyperplane with normal vector
~
β leaves
all the points of the charges-entropy diagram below
it. This is consistent with what we saw before, that
the charges entropy diagram is a convex set.
In a similar way we define the absolute athermality
or simply athermality in the following.
Definition 21 (Absolute athermality). The ab-
solute athermality of a state ρ is defined as
A(ρ) B min
~
β∈R
q
A
~
β
(ρ). (80)
The athermality of a state ρ can be understood ge-
ometrically as the vertical distance from the thermal
boundary (see Fig. 8).
11.3 Charge extraction
A first relevant scenario of thermodynamics with
multiple conserved quantities is the extraction of a
charge of the system while keeping the other charges
constant. For this subset of operations, the system is
restricted to move along a straight line in the direc-
tion of the extracted charge within the iso-entropic
hyperplane (see Fig. 9).
The bound charge for such a scenario is given by
B
k
(ρ) B min
σ : S(σ)=S(ρ),
L
i
(σ)=L
i
(ρ) ∀i,k
L
k
(σ), (81)
where γ
k
(ρ) is the GGE state that attains the min-
imum and corresponds to the point of the equilib-
rium boundary which is intersected by the straight-
line with direction k which passes through ρ. As
in the case of only energy, it is easy to see that
B
k
(ρ) = L
k
(γ
k
(ρ)). This is represented in Fig. 9
(left). The direction
~
β corresponding to γ
k
can be
geometrically determined as the normal vector of the
tangent plane to the equilibrium surface in that point.
The free charge F
k
(ρ) is now analogously defined
as
F
k
(ρ) = L
k
(ρ) − B
k
(ρ), (82)
and corresponds to the maximum amount of charge
that can be extracted given the restrictions of constant
entropy and charges. This is represented in Fig. 9
(left) for a case of 2 conserved quantities.
A detailed and full rigorous analysis of the charge
extraction by means of unitary conserving operations
in the many copy limit (for both commuting and non-
commuting charges) will be made in our forthcoming
work [56].
11.4 Extraction of a generalized potential
An different, but also relevant scenario is when the set
of operations allowed for the variation of the charges
is unrestricted, and the aim is the extraction of a gen-
eralized potential; cf. [54, 55]
V
~µ
(ρ) B
q−1
X
k=0
µ
k
L
k
(ρ), (83)
where ~µ B (µ
0
, . . . , µ
q−1
) specifies the weight µ
k
of every charge L
k
in the linear combination V
~µ
. For
convenience, ~µ is assumed to be normalized k~µk = 1
in the Euclidean norm and its components to be pos-
itive µ
k
> 0. (An alternative normalization could be
to take the coefficient of the Hamiltonian µ
0
= 1 as
it is usually done in the grand canonical ensemble.)
Accepted in Quantum 2018-11-28, click title to verify 18
E
L
γ
ρ
F (
ρ
)
E
L
μ
γ
ρ
F (
ρ
)
Figure 9: Constant entropy section of a charges-entropy
diagram with 2 charges E and L. (Left) Scenario of
extraction of a single charge while keeping constant the
other charges. The system is constrained to move in
one dimensional line and free-charge F (ρ) of a state
ρ is the distance from the boundary in such direction.
(Right) Scenario in which the charges are allowed to
change and the aim is the extraction of a potential
V
ˆµ
(ρ) =
P
q−1
k=0
µ
k
L
k
(ρ). The parallel purple lines rep-
resent equipotential surfaces of V
ˆµ
. The GGE state γ is
the state which minimizes V and belongs to the hyper-
plane which is tan tangent to the equilibrium boundary.
In this setting, the bound potential is defined as
B
~µ
(ρ) B min
σ : S(σ)=S(ρ)
V
~µ
(σ), (84)
where γ
~µ
(ρ) is the state that attains the mini-
mum which is again a GGE. As before, B
~µ
(ρ) =
V
~µ
(γ
~µ
(ρ)). Note that the
~
β values of the state γ
~µ
(ρ)
need to be proportional to the unit vector ~µ
~
β = β~µ, (85)
with β a scalar that is determined by the condition of
equal entropies, S(ρ) = S(γ
~µ
(ρ)). This can be seen
geometrically in Fig. 9 (right) or analytically by mak-
ing the simple observation that, regarding
˜
H B V
~µ
as
a new Hamiltonian, the min-energy principle singles
out e
−β
˜
H
as the state that attains the minimum.
Analogously, the free potential is given by
F
~µ
(ρ) = V
~µ
(ρ) − B
~µ
(ρ), (86)
and corresponds to the maximum amount of general-
ized potential V
~µ
that can be extracted under entropy
preserving operations. This situation has been dia-
grammatically represented in Fig. 9.
The scenario with a generalized potential V
~µ
is
analogue to the single charge situation. Both first and
second laws can be stated as in Secs. 7 and 8, replac-
ing the free energy F(ρ) by F
~µ
(ρ).
11.5 Second law
Any process (not necessarily entropy preserving) that
brings an initial generalized Gibbs state γ
B
(
~
β) out of
equilibrium fulfils
X
k
β
k
∆L
B
k
> ∆S
B
, (87)
where the inequality is only saturated in the limit of
large baths (small variations of entropy) and when fi-
nal state is also at equilibrium (GGE). Geometrically,
Eq. (87) is a trivial consequence that the charges en-
tropy diagram is upper bounded by any of its tangent
planes.
In the particular case of an entropy preserving pro-
cess on an initial bipartite state ρ
A
⊗ γ
B
(
~
β), the
change in mutual information between the subsys-
tems A and B is
∆I = ∆S
A
+ ∆S
B
> 0, (88)
which together with Eq. (87) implies
X
k
β
k
∆L
B
k
> −∆S
A
. (89)
Note that the above equation (89) is equivalent to the
second law formulated in [54]. We see here that such
a law is also valid for baths of arbitrary small sizes.
Eq. (89) is asymptotically saturated with equality in
the limit of large system sizes and in the absence of
correlations in the final state ρ
0
AB
= ρ
0
A
⊗ γ
0
B
.
11.6 Interconvertibility rates
The rate of interconversion from a quantum state ρ to
a state σ, in the same sense as in Section 10, i.e.
ρ
⊗n
−→ σ
⊗m
⊗ φ
⊗(n−m)
, (90)
is given by the conservation of the entropy and the
charges L
k
:
L
k
(ρ
⊗n
) ∼ L
k
(σ
⊗m
⊗ φ
⊗(n−m)
) ∀k = 0, . . . , q − 1,
S(ρ
⊗n
) ∼ S(σ
⊗m
⊗ φ
⊗(n−m)
).
(91)
The above conditions can again be written
as a geometric equation for the points x
ξ
=
(L
0
(ξ), . . . , L
q−1
(ξ), S(ξ)) ∈ R
q+1
in the energy-
entropy diagram, with ξ ∈ {ρ, σ, φ}:
x
ρ
= r x
σ
+ (1 − r) x
φ
, (92)
Accepted in Quantum 2018-11-28, click title to verify 19
where we have merely used the extensivity of both
entropy and the conserved quantities in the number
of copies, e.g. S(ρ
⊗n
) = nS(ρ).
Thus, by means of the same argument used in the
previous section, the convertibility rate from ρ to σ
reads
r = lim
n→∞
m
n
=
S(ρ) − S(φ)
S(σ) − S(φ)
, (93)
where the state φ is determined from the following
set of q equations
S(σ) − S(φ)
S(σ) − S(ρ)
=
L
k
(σ) − L
k
(φ)
L
k
(σ) − L
k
(ρ)
∀k = 0, . . . , q−1.
(94)
The state φ can be found geometrically in the
charges-entropy diagram as the point in the bound-
ary of the charges-entropy diagram that intersects the
straight half-line that goes through ρ from σ (see
Fig. 6 for the single charge example).
12 Discussion
In the recent years, thermodynamics, and in par-
ticular work extraction from non-equilibrium states,
has been studied in the quantum domain, giving rise
to radically new insights into quantum thermal pro-
cesses. However, in the standard treatment of thermo-
dynamics, be it classical or quantum, thermal baths
are considered to be asymptotically large in size.
That is why if a system is attached to a bath and al-
lowed to exchange energy and entropy, the bath stays
intact and its temperature remains unchanged. That is
also why the equilibrated system finally acquires the
same temperature as the bath. However, if one goes
beyond this assumption and considers the bath to be
a finite system, too, then traditional thermodynamics
breaks down. This situation is very much relevant for
thermodynamics in the quantum regime, where both
system and bath may be small. The first problem that
appears in such a situation is the notion of tempera-
ture itself, since the finite bath may go out of ther-
mal equilibrium due to the exchange of energy with
the system. Therefore, it is absolutely necessary to
develop a temperature independent universal thermo-
dynamics, in which the bath could be small or large,
and would not get any special status.
In this work, we have formulated temperature in-
dependent thermodynamics as an exclusive conse-
quence of information conservation. We have relied
on the fact that, for a given amount of information,
measured by the von Neumann entropy, any system
can only be transformed to states with the same en-
tropy. Given this constraint of information conser-
vation, there is a distinguished state within the con-
stant entropy manifold, which is the state that pos-
sesses minimal energy. This state is known as a com-
pletely passive state and acquires the Boltzmann-
Gibbs canonical form with an intrinsic temperature.
We call the energy of the completely passive state
the bound energy, since no further energy can be ex-
tracted by means of entropy preserving operations.
Thus, for a given state, the difference between its
energy and bound energy corresponds to the maxi-
mum amount of energy that can be extracted in form
of work and we call it free energy. In fact, in this
framework, two states that have the same entropy and
energy are thermodynamically equivalent [43]. The
thermodynamic equivalence between equal entropy
and energy states has allowed us to use of the energy-
entropy diagram to illustrate the notions of bound and
free energies in a geometric way. We have introduced
a new definition of heat for arbitrary systems in terms
of the bound energy of the bath.
We have seen that the laws of thermodynamics are
a consequence of the reversible dynamics of the un-
derlying physical theory. In particular:
• Zeroth law emerges as the consequence of infor-
mation conservation.
• First and second law emerge as the consequence
of energy conservation, together with informa-
tion conservation.
• Third law emerges as the consequence of fine-
grained information conservation (microscopic
reversibility or unitarity). Therefore, there is no
third law if one considers coarse-grained infor-
mation conservation, as we did here.
We have demonstrated that the maximum effi-
ciency of a quantum engine with a finite bath is in
general lower than that of an ideal Carnot engine. We
have introduced a resource theoretic framework for
our intrinsic thermodynamics, within which we ad-
dress the problem of work extraction and inter-state
transformations. All these results have been illus-
trated by means of the energy-entropy diagram. Fur-
thermore, we gave a geometric interpretation in the
diagram of the relevant thermodynamic quantities, as
well as the inter-convertibility rate between quantum
states under entropy and energy preserving opera-
tions.
The information conservation based framework for
thermodynamics, as well as the resource theory and
Accepted in Quantum 2018-11-28, click title to verify 20
the energy-entropy diagram, is also extended to mul-
tiple conserved quantities, as long as they commute
with each other. In this case, the energy-entropy di-
agram becomes the charges-entropy diagram and al-
lows us to understand thermodynamics in a geometri-
cal way. In particular, we have studied the extraction
of a single charge while keeping the other charges
conserved as well as the extraction of a generalized
potential. In the first scenario, we have seen that the
maximum work extractable from any state by opera-
tions that asymptotically conserve the given charges
is the difference between the free energy of the state
and that of the iso-entropic generalized grand canoni-
cal Gibbs state. Concerning the extraction of a gener-
alized potential (a linear combination of charges), we
have shown that it is analogous to the work extraction
(the single charge case). Finally, we have determined
the interconvertibility rates between states with dif-
ferent entropy and charges. A deeper investigation of
the setting with multiple charges, including the gen-
eralisation to non-commuting conserved quantities, is
left to our forthcoming paper [56].
In general, thermodynamics can be studied in three
different scenarios:
• One-shot or single-copy setting, where only one
copy of the joint system-environment is avail-
able. In this case, even the notion of expecta-
tion value is not meaningful, as well as von Neu-
mann entropy.
• Limit of many runs, where there are many
copies, but operation are restricted to single
copies of the system. In this scenario, the no-
tions of expectation value and von Neumann en-
tropy are well defined.
• Limit of many copies, where one has access to
arbitrarily many copies of the system and an an-
cilla sub-linear in the number of copies that can
globally be processed.
A first observation is that our formalism cannot be
applied in the single-shot setting, since the notion of
expectation value (say energy) cannot in general be
used.
A relevant point to discuss is what happens when
operations are not entropy preserving but unitaries.
In the limit of many copies, state transformations un-
der unitaries converge to entropy preserving opera-
tions and our formalism is recovered. The limit of
many runs is a bit more subtle. On the one hand,
all the thermodynamic inequalities of our formalism
are respected, since unitaries form a subset of entropy
preserving operations. On the other hand, these ther-
modynamic inequalities will not be in general sat-
urated. For instance, our formalism states that the
work that can be extracted per system in a state ρ is
upper bounded by its free energy W 6 F (ρ). In the
single-copy or many-run settings with fine-grained
information conservation, the law will be respected,
but there will not be in general a unitary for which
W = F (ρ).
A natural open question is to what extent our for-
malism can be extended from considering coarse-
grained information conservation operations as the
set of allowed operations to unitaries. In that case,
the notion of bound energy would be different and
many more equivalence classes of states would ap-
pear. Something similar already happens in the re-
source theory of thermodynamics, where instead of
having a single monotone as an “if and only if”
condition for state transformation, infinitely many
are required [30]. It is far from clear whether un-
der fine-grained information conservation restriction
the energy-entropy diagrams (or a generalization of
them) would still be useful.
Let us finally point out that, in the extension of our
work to unitary operations in the settings of single-
shot and many-runs, a consistent formulation of the
zeroth law would not be possible. Recall that the
zeroth law states that a collection of systems are in
mutual thermal equilibrium if and only if their ar-
bitrary combinations are also in equilibrium. It is
well known that passive states that are not thermal do
not remain passive when sufficiently many copies are
considered. Hence, for establishing a consistent ze-
roth law, one has to consider operations beyond uni-
taries on a single copy.
Acknowledgements
We thank Ph. Faist, K. Gawe¸dzki, R. B. T. Harvey, J.
Kimble, S. Maniscalco, Ll. Masanes, J. Oppenheim,
V. Pellegrini, M. Polini, C. Sparaciari, A. Vulpiani,
N. Yunger Halpern and R. Zambrini for useful dis-
cussions and comments in both theoretical and ex-
perimental aspects of our work. We also thank the
referees for constructive comments.
We acknowledge financial support from the
European Commission (FETPRO QUIC H2020-
FETPROACT-2014 No. 641122), the European Re-
search Council (AdG OSYRIS and AdG IRQUAT),
the Spanish MINECO (grants no. FIS2008-01236,
Accepted in Quantum 2018-11-28, click title to verify 21
FISICATEAMO FIS2016-79508-P, FIS2013-40627-
P, FIS2016-86681-P, and Severo Ochoa Excellence
Grant SEV-2015-0522) with the support of FEDER
funds, the Generalitat de Catalunya (grants no. 2017
SGR 1341, and SGR 875 and 966), CERCA Pro-
gram/Generalitat de Catalunya and Fundació Privada
Cellex. AR thanks for support from the CELLEX-
ICFO-MPQ fellowship.
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Accepted in Quantum 2018-11-28, click title to verify 25
A Alternative formulation of the
Clausius statement for the second law
Lemma 22 (Clausius statement). No iso-
entropic equilibration process is possible whose
sole result is the transfer of bound energy (i.e.
heat) from an equilibrium state with inverse tem-
perature β
C
to another equilibrium state with in-
verse temperature β
H
< β
C
.
Proof. In order to prove it, we need to show that
the iso-entropic equilibration process cannot lead
to a negative ∆β = β
H
− β
C
. Further, that
any iso-entropic process that leads to such in-
crease must require a supply of energy. First
consider the case where β
C
= β
H
(∆β = 0),
which means the corresponding states γ
C
and γ
H
are in equilibrium between each other and jointly
in the min-energy state. The min-energy princi-
ple says that any iso-entropic transformation to
make β
C
, β
H
is bound to increase the sum of
their individual min-energy. Therefore it requires
an inflow of energy, which is nothing but intro-
duction of work.
Note that for a completely passive state γ with
given Hamiltonian H and β, the change in bound
energy, which is nothing but heat in our defini-
tion, due to an infinitesimal change in entropy
dS(γ), is given by
dE(γ) =
1
β
dS(γ). (95)
Therefore the larger β, the smaller the change in
internal energy for a fixed infinitesimal change in
entropy of the state. Now assume by contradic-
tion that β
C
> β
H
. Any flow of heat from γ
C
to
γ
H
will lead to a reduction of entropy in the for-
mer. That will also lead to an equal increase of
of the same in the latter. A very small amount of
bound energy will result in a infinitesimal entropy
flow, say dS from γ
C
to γ
H
. However the change
in internal energy (dE(γ
C/H
) =
dS
β
C/H
) would not
be equal and that is −dE(γ
C
) < dE(γ
H
), for
β
C
> β
H
. As a consequence, in this iso-entropic
process the overall energy is bound to increase,
which is not possible without, again, an influx
of energy or work. Therefore, without external
work the process will never take place sponta-
neously.
B Alternative formulation of the
Kelvin-Planck statement for the second
law
Lemma 23 (Kelvin-Planck statement). No iso-
entropic equilibration process is possible whose
sole result is the absorption of bound energy
(heat) from an equilibrium state and its complete
conversion into work.
Proof. To prove the Kelvin-Planck statement,
we consider the following two completely pas-
sive states γ
A
and γ
B
with inverse temperatures
β
A
and β
B
, respectively, which undergo an iso-
entropic equilibration transformation as
γ
A
⊗ γ
B
⊗ |0ih0|
W
Λ
EP
ABW
−−−−→ γ
AB
⊗ |W ihW |
W
,
(96)
where γ
AB
is the final joint equilibrium state
with inverse temperature β
AB
. Since the initial
states are completely passive states, a non-zero
W can only result from heat transfer. Let us say
β
A
< β
B
, thus β
A
< β
AB
< β
B
. In this case
the heat flows out from A, say by an amount
∆Q
A
, with an associated decrease in its entropy.
Again the information conservation of whole pro-
cess guarantees an entropy increase in B. There-
fore, there has to be an associated increase in
bound energy (heat) content in γ
B
. As a result,
a part of ∆Q
A
could be converted to work and
that is
W 6 ∆Q
A
− ∆Q
B
. (97)
Note that any transfer of heat, i.e. −∆Q
A
> 0,
also bounds ∆Q
B
> 0. As a consequence, heat
cannot be converted into work completely.
Accepted in Quantum 2018-11-28, click title to verify 26