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 Winter3, 4

1ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain2Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany

3ICREA, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain4Departament de Física: Grup d’Informació Quàntica,

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

Thermodynamics and information have intricate inter-relations. Often thermodynamics is considered to bethe logical premise to justify that information is physical – through Landauer’s principle – thereby also linksinformation and thermodynamics. This approach towards information, recently, has been instrumental to under-stand thermodynamics of logical and physical processes, both in classical and quantum domains. In this workwe formulate thermodynamics as an exclusive consequence of information conservation. The framework can beapplied to most general situations, beyond the traditional assumptions in thermodynamics, where systems andthermal-baths could be quantum, of arbitrary sizes and even could posses inter-system correlations. Further, itdoes not require a priori predetermined temperature associated to a thermal-bath, which does not carry muchsense for finite-size cases. Hence, the thermal-baths and systems are not treated here differently, rather both areconsidered on equal footing. This leads us to introduce a “temperature”-independent formulation of thermody-namics. We rely on the fact that, for a fix amount of information, measured by the von Neumann entropy, anysystem can be transformed to a state that possesses minimal energy with same entropy. This state is known asa completely passive state that acquires a Boltzmann–Gibb’s canonical form with an intrinsic temperature. Weintroduce the notions of bound and free energy and use them to quantify heat and work respectively. Guidedby this principle of information conservation, we develop universal notions of equilibrium, heat and work,Landauer’s principle and universal fundamental laws of thermodynamics. We demonstrate that the maximumefficiency of a quantum engine with a finite bath is in general lower than that of an ideal Carnot’s engine. Weintroduce a resource theoretic framework for our intrinsic-temperature based thermodynamics, within which weaddress the problem of work extraction and inter-state transformations. Finally, this framework is extended tothe cases of multiple conserved quantities.

I. INTRODUCTION

Thermodynamics constitutes one of the most basic founda-tions of modern science. It not only plays an important rolein modern technologies, but offers also basic understandingof the vast range of natural phenomena. Initially, thermody-namics was developed in a phenomenological way to addressthe question on how, and to what extent, heat could be con-verted into work. But, in the later course with the develop-ments of statistical mechanics, quantum mechanics and rela-tivity, thermodynamics along with its fundamental laws hasattained quite a formal and mathematically rigorous form [1].It finds applications in ultra-large large systems, where it de-scribes relativistic phenomena in astrophysics and cosmology,in microscopic systems, where it describes quantum effects, orin very complex systems in biology and chemistry.

The inter-relation between information and thermodynam-ics [2] is very intricate, and has been studied in the context ofMaxwell’s demon [3–6], Szilard’s engine [7], and Landauer’sprinciple [8–12]. In the recent years, classical and quantuminformation theoretic approaches help us to understand ther-modynamics in the domain of small classical and quantumsystems [13–15]. That stimulated a whole new perspective totackle and extend thermodynamics beyond the standard clas-sical domain. In fact, information theory has recently playedan important role in understanding thermodynamics in the

∗ mnbera@gmail.com

presence of inter-system and system-bath correlations [16–18], equilibration processes [19–22], or foundations of statis-tical mechanics [23]. One of the most paradigmatic examplesof this success is the formulation of quantum thermodynam-ics within the, so called, resource theoretic framework [24],which allows to reproduce standard thermodynamics in theasymptotic limit, when one processes infinitely many copiesof the system under consideration. In the finite copy limit,commonly known as one-shot limit, the resource theory re-veals that the laws of thermodynamics have to be modified todictate the transformations on the quantum level [25–33].

In this work, elaborating in the inter-relations between in-formation and thermodynamics, we make an axiomatic con-struction of thermodynamics and identify the “informationconservation” as the crucial underlying property of any the-ory that respects it. The corner-stone of our construction is thenotion of bound energy, which we introduce as the amount ofenergy locked in a system that cannot be accessed (extracted)given a set of allowed operations. The bound energy obvi-ously depends on the set of allowed operations: the more pow-erful the allowed operations are, the smaller the bound energy.We prove that, by taking (i) global entropy preserving (EP)operations as the set of allowed operations and (ii) infinitelylarge thermal baths initially uncorrelated from the system, ourformalism reproduces standard thermodynamics.

In information theory, the Von Neumann entropy is thequantity that measures the amount of information hidden in asystem. In this sense, entropy preserving operations are trans-formations that keep this information constant. All fundamen-tal physical theories, such as classical and quantum mechan-

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ics, share the property of conserving information. That is, theyhave dynamics, which are deterministic and they bijectivelymap the set of possible configurations between any two in-stants of time. Thus, non-determinism can only appear, whensome degrees of freedom are ignored, leading to apparent in-formation loss. In classical physics this information loss isdue to deterministic chaos and mixing in nonlinear dynamics.In quantum mechanics this loss is intrinsic, and occurs due tomeasurement processes and non-local correlations [34]. Onecan well argue that the set of entropy preserving operationsis larger than the reversible operations (unitaries) in the sensethat they conserve entropy but, unlike unitaries, not the indi-vidual probabilities. This is the reason why we denote themas coarse-grained information conservation. In the limit ofmany copies both coarse-grained and fine-grained informa-tion conservation become equivalent (see Sec. II). While it isknown that linear operations that are entropy preserving forall states are unitary [40], it is an open question to what ex-tent coarse-grained information conserving operations can beimplemented in the single-copy limit.

The resource theory of thermodynamics can then be seen asa sharpener of the condition (i), i. e. an extension of thermody-namics from coarse-grained to fine-grained information con-servation, where the operations are global unitaries but stillconstrained to infinitely large thermal baths with a well de-fined temperature. Fluctuation theorems can also be seen fromthis perspective. There, the second law is obtained as a conse-quence of reversible transformations on initially thermal statesor states with a well defined temperature [35, 36]. In contrast,the aim of our work is instead to relax condition (ii), that is,to generalize thermodynamics to be valid for arbitrary envi-ronments, irrespective of being thermal, or much larger thanthe system. This idea is illustrated in the table below. Themain obstruction against this generalization relies on the factthat, when allowing for arbitrary states as environment, largeamounts of resources can be pumped into the system lead-ing to trivial “resource” theories. We are able to circumventthis problem thanks to the notion of bound energy, which in-trinsically distinguishes accessible and non accessible energy.Our formalism results in a theory, in which systems and envi-ronments are treated in equal footing, or in other words, in a“temperature”-independent formulation of thermodynamics.

Unitaries(fine-grained IC∗)

EP operations(coarse-grained IC)

Large thermalbath

Resource theory ofThermodynamics

StandardThermodynamics

Arbitraryenvironment ? Our formalism

*IC: Information Conservation

Our work is complementary to other general approaches,where thermodynamics are obtained after inserting some formof thermal state(s) in general mathematical expressions, e.g.[37]. While in these works the mathematical expressions forarbitrary states have no a priori thermodynamic meaning, ourconstruction is build on a physically motivated quantity, thebound energy.

This “temperature”-independent thermodynamics is essen-tial in contexts in which the state of the bath can be affected

by the system after exchange of heat (see [38] for a reviewon the notion of temperature). This can both be due to thefact of having a relatively small environment compared to thesystem, or an environment simply not being thermal. In fact,in the current experiments, environments do not have to benecessarily thermal, but can even possess quantum properties,like coherence or correlations.

The power of the entropy preserving operations makes allthe states with equal energies and entropies thermodynami-cally equivalent. This allows for representing all the statesand thermodynamic processes in a simple energy-entropy di-agram. We exploit this geometric approach and give a dia-grammatic representation for heat, work and other thermody-namic quantities. In this way we are able to reproduce severalresults of the literature, e.g. the resource theory of thermody-namics applicable for arbitrary quantum systems and environ-ments [39]. Our formalism is naturally extended to scenarioswith multiple conserved quantities. Finally, we summarizeour findings and discuss the main obstructions towards an ex-tension of thermodynamics that is valid both in the single-shotlimit and for non-thermal environments.

II. ENTROPY PRESERVING OPERATIONS, ENTROPICEQUIVALENCE CLASS AND INTRINSIC TEMPERATURE

The set of operations that we consider in this frameworkis the set of, so called, entropy preserving (EP) operations.Given a system initially in a state ρ, the set of entropy preserv-ing operations are all the operations that arbitrarily change thestate, but keep its entropy constant

ρ→ σ : S (ρ) = S (σ) , (1)

where S (ρ) B −Tr (ρ log ρ) is the von Neumann entropy. Im-portantly, an operation that acts on ρ and produces a state withthe same entropy, not necessarily preserves entropy when act-ing on other states. In fact, such entropy preserving operationsare in general not linear, since they have to be constraint tosome input state. It was shown in Ref. [40] that a quantumchannel Λ(·) that preserves entropy and, at the same time, re-spects linearity, i. e. Λ(pρ1+(1−p)ρ2) = pΛ(ρ1)+(1−p)Λ(ρ2),has to be a unitary.

However, the entropy preserving operations can be micro-scopically described by global unitaries in the limit of manycopies [39]. Given any two states ρ and σwith equal entropiesS (ρ) = S (σ), there exists an additional system of O(

√n log n)

ancillary qubits and a global unitary U such that

limn→∞‖Tr anc

(Uρ⊗n ⊗ ηU†

)− σ⊗n‖1 = 0 , (2)

where the partial trace is performed on the ancillary qubits.Here ‖X‖1 B Tr

√X†X is the one-norm. As shown in Theo-

rem 4 of Ref. [39], the reverse statement is also true. In otherwords, if two states are related as in Eq. (2), then they alsohave equal entropies.

As we see later it is important to restrict entropy preserv-ing operations that are also be energy preserving. The energy

3

and entropy preserving operations can also be implementedusing a global energy preserving unitary in the many copylimit. More explicitly, in Theorem 1 of Ref. [39], it is shownthat having two states ρ and σ with equal entropies and ener-gies, i.e. (S (ρ) = S (σ) and E(ρ) = E(σ)), is equivalent to theexistence of some energy preserving U and an additional sys-tem A with O(

√n log n) of ancillary qubits with Hamiltonian

‖HA‖ 6 O(n2/3) in some state η, for which (2) is fulfilled. Theoperator norm ‖X‖ of a Hermitian operator X corresponds tothe largest of its eigenvalues in absolute value. Note that theamount of energy and entropy of the ancillary system per copyvanishes in the large n limit.

We expect entropy preserving operations to be also imple-mented in other ways than taking the limit of many copies. Forinstance, in Refs. [41, 42], thermal operations are extended toa class of operations, in which a catalyst is allowed to build upcorrelations with the system. For these operations, the stan-dard Helmholtz free energy singles out as the monotone thatestablishes the possible transitions between states, in contrastto the case of strict thermal operations, in which all the Rényiα-free energies are required. This suggests that entropy pre-serving operations could also be implemented with a singlecopy by means of a catalyst that can become correlated withthe system. Further investigation in this direction is needed.

Once the the entropy preserving operations have been mo-tivated and introduced, let us classify the set of states of asystem in different equivalence classes depending on their en-tropy. Thereby, we establish a hierarchy of states according totheir information content. More formally,

Definition 1 (Entropic equivalence class). Two states ρ andσ on any quantum system of dimension d are equivalent andbelong to the same entropic equivalence class if and only ifboth have the same Von Neumann entropy,

ρ ∼ σ iff S (ρ) = S (σ) . (3)

Assuming that the system has some fixed Hamiltonian H,one can take as a representative element of every class thestate that minimizes the energy within it, i.e.,

γ(ρ) B arg minσ : S (σ)=S (ρ)

E(σ), (4)

where E(σ) B Tr (Hσ) is the energy of the state σ.The maximum-entropy principle [43, 44] identifies the ther-

mal state as the state that maximizes the entropy for a givenenergy. Conversely, one can show that, for a given entropy,the thermal state also minimizes the energy. We refer to thiscomplementary property as min-energy principle [16, 45, 46].Thus, the min-energy principle identifies thermal states as therepresentative 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 theequivalence class, to which the state ρ belongs. We denoteβ(ρ) as the intrinsic inverse temperature associated to ρ.

The state γ(ρ) is, often, termed as the completely passive(CP) state [16] with the minimum internal energy, for the same

information content. The CP state, with the form γ(HS , βS ),has the following interesting properties [45, 46]:

(P1) For a given entropy, it minimizes the energy.

(P2) Both energy and entropy monotonically increase (de-crease) with the decrease (increase) in βS , and viceversa.

(P3) For non-interacting Hamiltonians, HT =∑N

X=1 I⊗X−1 ⊗

HX ⊗ I⊗N−X , the joint complete passive state is ten-

sor product of individual ones, i.e, γ(HT , βT ) =

⊗NX=1γ(HX , βT ), for identical βT [45, 46].

III. BOUND AND FREE ENERGIES

Let us now identify two relevant forms of internal energy:the free and the bound energy. The bound energy is definedas the amount of internal energy that cannot be accessed inthe form of work. Note that it is a notion that is dependingon the set of allowed operations. For the set of entropy pre-serving operations, in which the entropic classes and CP statesemerge, it is quantified in the following.

Definition 2 (Bound energy). For a state ρ with the systemHamiltonian H, the bound energy in it is

B(ρ) B minσ : S (σ)=S (ρ)

E(σ) = E(γ(ρ)), (6)

where γ(ρ) is the CP state, with minimum energy, within theequivalence class, to which ρ belongs.

Indeed, B(ρ) is the amount of energy that cannot be ex-tracted further, by exploiting any entropy preserving opera-tions, as guaranteed by the min-energy principle. The abovedefinition of bound energy also has strong connection with in-formation content in the state. It can be easily seen that onecould have access to this energy (in the form of work) only, ifit allows an outflow of information from the system and viceversa.

In contrast to bound energy, free energy is the part of theinternal energy that can be accessed with entropy preservingoperations:

Definition 3 (Free energy). For a system ρwith system Hamil-tonian HS , the free energy stored in the system is given by

F(ρ) B E(ρ) − B(ρ), (7)

where B(ρ) is bound energy in ρ.

Note that the free energy as defined in Eq. (7) does not havea preferred temperature, unlike the standard out of equilibriumHelmholtz free energy FT (ρ) B E(ρ) − T S (ρ), where thetemperature T is decided beforehand with choice of a ther-mal bath. Nevertheless, our definition of free energy can bewritten in terms of both the relative entropy and the out ofequilibrium free energy as

F(ρ) = T (ρ)D(ρ||γ(ρ)) = FT (ρ)(ρ) − FT (ρ) (γ(ρ)) , (8)

4

for the intrinsic temperature T (ρ) B β(ρ)−1 that labels theequivalence class that contains ρ. Here the relative entropyis defined as D(ρ||σ) = Tr

(ρ log ρ − ρ logσ

). Let us mention

that the standard out of equilibrium free energy is also denotedby Fβ(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 considerabove, can be extended beyond single systems. If fact, inmulti-particle (multipartite) systems they exhibit several inter-esting properties. For example, they can capture the presenceand/or absence of inter-party correlations. To highlight thesefeatures, we consider a bipartite system below.

Lemma 4 (Bound and free energy properties). Given a bipar-tite system with non-interacting Hamiltonian HA ⊗ I + I ⊗ HBin an arbitrary state ρAB and a product state ρA ⊗ ρB withmarginals ρA/B B Tr B/A(ρAB). Then, the bound and the freeenergy fulfill the following properties:

(P4) Bound energy and correlations:

B(ρAB) ≤ B(ρA ⊗ ρB) . (9)

(P5) Bound energy of composite systems:

B(ρA ⊗ ρB) ≤ B(ρA) + B(ρB) . (10)

(P6) Free energy and correlations:

F(ρA ⊗ ρB) ≤ F(ρAB) . (11)

(P7) Free energy of composite systems:

F(ρA) + F(ρB) ≤ F(ρA ⊗ ρB) . (12)

In Eqs. (9) and (11) the equality is saturated if and only if Aand B are uncorrelated ρAB = ρA ⊗ ρB. Equations (10) and(12) become an equality if and only if β(ρA) = β(ρB).

Proof. To prove (P4), we have used the fact that, for a fixedHamiltonian, the bound energy is monotonically increasingwith entropy, i.e.

B(ρ) < B(σ) iff S (ρ) < S (σ) , (13)

together with the subaddivity property S (ρAB) ≤ S (ρA ⊗ ρB).The proof of (P5) relies on the definition of the bound en-

ergy, where one exploits entropy preserving (EP) operationsto minimize energy. For composite systems, the set of jointEP operations when applied on product states ρA ⊗ ρB arestrictly larger than the independent local EP operations, i.e.{Λ

epAB

}!

{Λ

epA ⊗ Λ

epB

}. This immediately implies Eq. (10). To

see that the inequality (10) is saturated iff the states pose iden-tical internal temperature, i.e., β(ρA) = β(ρB), we refer thereader to Definition 6 and Lemma 7.

The other properties can be easily proven by noting that thetotal internal energy is sum of individual ones and E(ρAB) =

E(ρA) + E(ρB), irrespective of inter-system correlations. Thenwith the definition of free energy is F(ρX) = E(ρX) − B(ρX),(P4) and (P5), we could easily arrive at properties (P6) and(P7). �

The above properties allow us to give an additional opera-tional meaning to the free energy F(ρ). Let us remind first thatthe work that can be extracted from a system in a state ρ whenhaving at our disposal and infinitely large bath at inverse tem-perature β and global entropy preserving operations is givenby

W = Fβ(ρ) − Fβ(γ(β)) , (14)

where Fβ(ρ) = E(ρ) − β−1S (ρ) is the standard free energywith respect to the inverse temperature β and γ(β) is the ther-mal state, at which the system is left once the work has beenextracted.

Lemma 5 (Free energy vs. β-free energy). The free energyF(ρ) corresponds to work extracted by using a bath at theworst possible temperature,

F(ρ) = minβ

(Fβ(ρ) − Fβ(γ(β))

). (15)

The inverse temperature achieves the minimum is the inverseintrinsic temperature β(ρ).

Proof. The proof is a corollary of Lemma 14 together with(P7). �

IV. ENERGY-ENTROPY DIAGRAM

Let us describe here the energy-entropy diagram whichappears often in thermodynamics literature and in particu-lar has recently been exploited in Ref. [39]. Given a sys-tem described by a Hamiltonian H, a state ρ is representedin the energy-entropy diagram by a point with coordinatesxρ B (E(ρ), S (ρ)), as shown in Fig. 1. All physical statesreside in a region that is lower bounded by the horizontalaxis (i.e., S = 0) corresponding to the pure states, and up-per bounded by the convex curve (E(β), S (β)) which repre-sents the thermal states of both positive and negative temper-atures. Let us denote such a curve as the thermal boundary.The inverse temperature associated to one point of the ther-mal boundary is given by the slope of the tangent line in sucha point, since

dS (β)dE(β)

= β . (16)

In Fig. 1, the free energy and the bound energy are plottedgiven a state ρ. Its free energy F(ρ) can be seen from the di-agram as the horizontal distance from the thermal boundary.This is the part of internal energy which can be extracted with-out altering system’s entropy. The slope of the tangent line ofthe thermal boundary in that point is the intrinsic temperature,β(ρ), of the state ρ. The bound energy B(ρ) is the distance inthe horizontal direction between the thermal boundary and theenergy reference and it can no way be extracted with entropypreserving operations.

Note that in general a point in the energy-entropy diagramrepresents multiple states, since different quantum states can

5

S

E

β = 0

B(ρ) F (ρ)

β(ρ)

xρ := (E(ρ), S(ρ))

E(ρ) EmaxEmin

S(ρ)

β → −∞

Figure 1. Energy-entropy diagram. Any quantum state ρ is repre-sented in the diagram as a point with coordinates xρ B (E(ρ), S (ρ)).The free energy F(ρ) is the distance in the horizontal direction fromthe thermal boundary. The bound energy B(ρ) is the distance in thehorizontal direction between the thermal boundary and the energyreference.

have identical entropy and energy. As it is pointed out in [39],the energy-entropy diagram establishes a link between the mi-croscopic and the macroscopic thermodynamics, in the sensethat, in the macroscopic limit of many copies all the thermo-dynamic quantities only rely on the energy and entropy perparticle. More precisely, all the states with same entropy andenergy are thermodynamically equivalent and, as it was shownin [39], they can be unitarily transformed into each other in thelimit of many copies n → ∞ with an ancilla at our disposal,which is of sub-linear size O(

√n log n) with a Hamiltonian

upper bounded by sub-linear bound O(n2/3).

V. EQUILIBRIUM AND ZEROTH LAW

The zeroth law of the thermodynamics establishes the ab-solute scaling of temperature, in terms of thermal equilibrium.It says that if a system A is in thermal equilibrium with B andagain, B is in thermal equilibrium with C, then A will alsobe in thermal equilibrium with C. All the systems that arein mutual thermal equilibrium can be classified to thermody-namically equivalent class and each state in the class can beassigned with a unique parameter called temperature. In otherwords, at thermal equilibrium, temperatures of the individualsystems will exactly be equal to each other and also to theirarbitrary collections, as there would be no spontaneous heator energy flow in between. Further, when a non-thermal stateis brought in contact with a large thermal bath, then the sys-tem tends to acquire a thermally equilibrium state with thetemperature of the bath. In this equilibration process, the sys-tem could exchange both energy and entropy with bath andthereby minimizes its Helmholtz free energy.

However, in the new setup, where a system cannot haveaccess to considerably large thermal bath or in absence of athermal bath, the equilibration process is expected to be con-siderably different than that of the cases of large baths. In thefollowing, we introduce a formal definition of equilibration,based on information preservation and intrinsic temperature.

Definition 6 (Equilibrium and zeroth law). Given a collec-tion of systems A1, . . . , An with non-interacting HamiltoniansH1, . . .Hn in a joint state, ρA1...An , we call them to be mutuallyat equilibrium if and only if they “jointly” minimize the freeenergy as defined in (7), i.e.,

F(ρA1...An ) = 0 . (17)

Let us consider the states ρA and ρB, with correspondingHamiltonians HA and HB respectively. The equilibrium isachieved when they jointly attain an iso-informatic state withminimum energy. Then the corresponding equilibrium stateis a completely passive (CP) state γ(HAB, βAB) with the jointHamiltonian HAB = HA⊗ I+ I⊗HB. Further the joint CP state,following (P3), is

γ(HAB, βAB) = γ(HA, βAB) ⊗ γ(HB, βAB), (18)

where the local systems also assume the completely passivestates but with same βAB. Here we recall that γ(HX , βY ) =

e−βY HX/Tr (e−βY HX ). It also says that if two arbitrary CP statesγ(HA, βA) and γ(HB, βB) have βA , βB, then their combinedequilibrium state γ(HAB, βAB) can, still, jointly reduce bound-energy without altering the total information content. There-fore joint CP state acquires a unique βAB, i.e.,

E(γ(HA, βA)) + E(γ(HB, βB)) > E(γ(HAB, βAB)), (19)

which is immediately followed from min-energy principle and(P5). Moreover if βA ≥ βB, then Eq. (19) dictates a bound onβAB, as expressed in Lemma 7:

Lemma 7. Any iso-informatic equilibration process betweenγ(HA, βA) and γ(HB, βB), with βA ≥ βB, leads to a mutuallyequilibrium state γ(HAB, βAB), irrespective of non-interactingsystems’ Hamiltonians, where βAB satisfies

βA ≥ βAB ≥ βB. (20)

Proof. Note for βA = βB, (P3) and (P5) immediately lead toβA = βAB = βB. What we need to show is that, for initialβA > βB, the equilibration leads to βA > βAB > βB. It canbe seen from information preservation condition on the equi-libration process. Say after the equilibration the final temper-ature is βAB = βA > βB. That implies the state γ((HA, βA))remains unchanged, while γ(HB, βB) changed to a state withhigher entropy as a consequence of (P2). Therefore, it de-mands S (γ(HA, βA) ⊗ γ(HB, βB)) < S (γ(HAB, βAB)), which isforbidden by the information preservation condition. Same isalso true for the case with βAB > βA ≥ βB. Now consider theother extreme where one has βA > βB = βAB. Then the (P2)leads to S (γ(HA, βA) ⊗ γ(HB, βB)) > S (γ(HAB, βAB)), which isalso true for βA > βB > βAB. In both the cases the informa-tion preservation condition is not respected. Therefore onlypossibility would be βA > βAB > βB. �

Now with the clear notion of equilibration and equilibriumstate which has minimum internal energy for a fixed informa-tion content, we could restate zeroth law in terms of internaltemperature. By definition, the global CP state has minimum

6

internal energy and one cannot extract free energy by usingglobal EP operations. Further a global CP state not only as-sures that the individual states are also CP but also confirmsthat they share the same intrinsic temperature, i.e., β and van-ishing inter-system correlations. Therefore we may argue thatthe individual systems are in mutual equilibrium, but deter-mined by an intrinsic temperature.

We may recover the traditional notion of thermal equilib-rium, as well as, zeroth law. In traditional thermodynamics,the thermal bath is considerably large (compared to the sys-tems under consideration), and assumes CP state with a prede-fined temperature. Mathematically a thermal bath, with bathHamiltonian HB, can be expressed as γB = e−βBHB/Tr (e−βBHB ),where |γB| → ∞. Now, if a finite state ρS (with |ρS | � ∞) withHamiltonian HS is brought in contact with thermal bath, thenthe global thermal equilibrium state will be a CP state, i.e.,

γB⊗ρSΛep

−−→ γ′B⊗γS , with a global inverse equilibrium temper-ature, say βe. With |γB| � |ρS | and |γB| → ∞, one could easilysee that γ′B → γB, βe → βB and γS → e−βBHS /Tr (e−βBHS ).

VI. MAX-ENTROPY PRINCIPLE VS. MIN-ENERGYPRINCIPLE

Let us mention that our framework based on informationconservation is suited for the work extraction. It assumes thatthe system has mechanisms to release energy to some bat-tery or classical field, but it cannot interchange entropy withit. This contrasts with other situations in spontaneous equi-libration, in which the system is assumed to evolve keepingconstant the conserved quantities. In such scenario, the equi-librium state is described by the principle of maximum en-tropy, that is, the state the maximizes the entropy given theconserved quantities.

In the context of the maximum entropy principle, let us in-troduce the absolute athermality as:

Definition 8 (Athermality). The athermality of a system in astate ρ and Hamiltonian H is the amount of entropy (informa-tion) that the system can still accommodate without increasingits energy, i. e.

A(ρ) B maxσ : E(σ)=E(ρ)

S (σ) − S (ρ), (21)

where the maximization is made over all states σ with energyE(ρ).

The state σ that maximizes (21) is obviously a thermalstate. In particular, it is the equilibrium state of an equilibra-tion process guided by the maximum entropy principle. Let uscall the temperature of such thermal state as the spontaneousequilibration temperature and denote it by β(ρ). The ather-mality corresponds to the amount of entropy produced duringsuch equilibration.

Note that the spontaneous equilibration temperature β(ρ)always differs from the intrinsic temperature β(ρ) unless ρis thermal. In other words, the maximum entropy and min-imum energy principles lead to different equilibrium tempera-tures. As both entropy and energy are monotonically increas-

ing functions of temperature, the equilibrium temperature de-termined by the minimum energy principle is always smallerthan the one determined by the maximum entropy principle.This is represented in the energy entropy diagram of Fig. 2,where the athermality can be seen as the vertical distance froma state ρ to the thermal boundary. The point in which the ther-mal boundary intersects with the vertical line E = E(ρ) repre-sents the equilibrium thermal state given by the max-entropyprinciple. Its temperature β(ρ) corresponds to the slope of thetangent line of the thermal boundary at that point.

S

E

F (ρ)

A(ρ)β(ρ)

β(ρ)β(ρ) > β(ρ)

xρ

E(ρ)

S(ρ)

Figure 2. Representation of the free energy F(ρ) and the athermalityA(ρ) of a state ρ in the energy-entropy diagram. The intrinsic temper-ature β(ρ) and the spontaneous equilibration temperature β(ρ) fulfillβ(ρ) > β(ρ) due to the concavity of the thermal boundary.

We have chosen the name “athermality” for the quantitydefined in (8) in agreement with Ref. [39]. There, a quantitycalled β-athermality and defined by

Aβ(ρ) B βE(ρ) − S (ρ) + log Zβ (22)

with Zβ = Tr (e−βH) the partition function, is introduced tocharacterise the energy-entropy diagram as the set of all pointswith positive entropy S (ρ) ≥ 0 and positive β-athermalityAβ(ρ) ≥ 0 for all β ∈ R.

Lemma 9 (Athermality vs. β-athermality). The athermalityof a state ρ can be written in terms of the β-athermalities as

A(ρ) = minβ

Aβ(ρ) , (23)

where the minimum is attained by the spontaneous equilibra-tion temperature β(ρ).

Proof. We prove the lemma by giving a geometric interpreta-tion to the β-athermality in the energy-entropy diagram. In theenergy entropy diagram represented in Fig. 3, let us considerthe tangent line to the thermal boundary with slope β. Suchstraight line is formed the set of all points which fulfill

S − S β = β(E − Eβ) , (24)

where S β and Eβ are respectively the entropy and the energyof the the point in the thermal boundary that belongs to theline. The vertical distance (difference in entropy) of such linefrom a point with coordinates (E(ρ), S (ρ)) reads

S ∗ − S (ρ) = β(E(ρ) − Eβ) − S (ρ)= βE(ρ) − S (ρ) − β(Eβ − TS β)

(25)

7

where S ∗ is the entropy of the line at E = E(ρ) and T = β−1.By noticing that log Zβ = β(Eβ − TS β), we get that the β-athermality of ρ, A(ρ), is the vertical distance of xρ from theline with slope β tangent to the thermal boundary. Equation(23) trivially follows from this. �

S

E

β

xρ

Aβ(ρ)

Fβ(ρ)− Fβ(γ(β))

Figure 3. Representation of the β-free energy difference Fβ(ρ) −Fβ(γ(β)) and the β-athermality Aβ(ρ) for a state ρ with coordinatesxρ in the energy-entropy diagram.

By identifying the β-free energy in Eq. (22), the β-athermality can be written as

Aβ(ρ) = β(Fβ(ρ) − Fβ(γ(β))

), (26)

that is, it is β times the work that can be potentially extractedfrom a state ρ when having an infinite bath uncorrelated fromthe system at temperature β (see Eq. (14)). From Eq. (26), wesee that the difference in β-free energies Fβ(ρ) − Fβ(γ(β)) canbe represented in the energy-entropy diagram as the horizon-tal distance from the point xρ to the tangent line with slope β.This is represented in Fig. 3. Hence, given an infinitely largebath at some inverse temperature β, all the states with the samework potential lie on a line with slope β in the energy-entropydiagram. The work extracted from a state ρ can be decom-posed into its free energy F(ρ), the part of energy that couldhave been extracted without bath, and the rest, that is, the partof energy that has been accessed thanks to the bath. This latterpart is closely related to heat. The definition of work and heatare discussed in the next section.

Furthermore, thanks to Lemmas 5 and 9, both the free en-ergy and the athermality can be expressed in terms of a mini-mization over standard β-free energies

F(ρ) = minβ

(Fβ(ρ) − Fβ(γ(β))

),

A(ρ) = minβ

(β(Fβ(ρ) − Fβ(γ(β))

)),

(27)

where the minimum is respectively attained by the intrinsictemperature β(ρ) and the spontaneous equilibration tempera-ture β(ρ).

Note that, while for positive temperatures both the ather-mality and the free energy are a measure of out of equilib-rium, for negative temperatures, states with small athermalityare highly active and have huge free energies.

Let us finally show that in situations as the equilibration ofa hot body in contact with a cold one, the intrinsic temperature

S

E

βA

βB

EBEA E

SA

SB

SβS

βE

E := 12 (EA + EB)

S := 12 (SA + SB)

Figure 4. Energy-entropy diagram of a system with Hamiltonian H.Two systems with the same Hamiltonian H and initially at differenttemperatures βA and βB equilibrate to different temperature depend-ing on the approach taken: minimum energy principle vs. maximumentropy principle. The equilibrium temperature when entropy is con-served β−1

S is always smaller than the equilibrium temperature whenenergy is conserved β−1

E , i. e. βS > βE .

and the spontaneous equilibration one are not so much differ-ent. In Fig. 4, we represent in the energy-entropy diagram forthe particular case of the equilibration of a system composedof two identical subsystems initially at different temperaturesβA and βB. As expected and due to the concavity of the thermalboundary, the equilibrium temperature given by the constantenergy constrain (max. entropy principle) β−1

E is larger thatthe one given by the constant entropy constrain (min. energyprinciple) β−1

S , i. e. βE < βS . The difference in bound energiesof these two thermal states corresponds precisely to the workextracted in the constant entropy scenario.

VII. WORK, HEAT AND THE FIRST LAW

In thermodynamics, the first law deals with the conserva-tion of energy. In addition, it dictates the distribution of en-ergy over work and heat, that are the two forms of thermody-namically relevant energy transfer.

Let us define a thermodynamic process involving a systemA and a bath B as a transformation ρAB → ρ′AB that conservesthe global entropy S (ρAB) = S (ρ′AB). In standard thermody-namics, the bath is by construction assumed to be initiallyin a thermal state and completely uncorrelated from the sys-tem. In such scenario, the heat dissipated in a process is usu-ally defined as the change in the internal energy of the bath∆Q = E(ρ′B) − E(ρB) where ρ(′)

B = Tr Aρ(′)AB is the reduced

state of the bath. This definition, however, may have somelimitations and an alternative definitions have been discussedrecently [18]. In particular, in [18], an information theoreticapproach suggests heat to be defined as ∆Q = TB∆S B with TBbeing the temperature and ∆S B = S (ρ′B)−S (ρB) the change invon Neumann entropy of the bath. Note that ∆S B = −∆S (A|B)is also the conditional entropy change in system A, condi-tioned on the bath B, defined as S (A|B) = S (ρAB) − S (ρB),which can be understood as the information flow from the sys-tem to the bath in the presence of correlations.

In the present approach, we go beyond the restriction that

8

S

E

xρB

x′ρB

EB E′B

SB

S′B

TB∆SB

∆Q

∆EB

Figure 5. Representation in the energy-entropy diagram of the dif-ferent notions of heat: ∆Q as change of the bound energy of the bath,∆EB as the change in internal energy, and TB∆S B. In this example,the bath is initially thermal but this does not have to be necessarilythe case.

the environment has to have a definite predefined temperature,and be in the state of the Boltzmann-Gibbs form. For an ar-bitrary environment, which could even be athermal, the heattransfer is defined in terms of bound energies, as in the fol-lowing.

Definition 10 (Heat). Given a system A and its environment B,

the heat dissipated by the system A in the process ρABΛep

−−→ ρ′ABis defined as the change in the bound energy of the environ-ment B, i.e.

∆Q B B(ρ′B) − B(ρB) (28)

where B(ρ(′)B ) is the initial (final) bound energy of the bath B.

Note that heat is a process dependent quantity in the sensethat there might be different processes with the same initialand final marginal state for A, but different marginals for B.Because the global process is entropy preserving, processesthat lead to the same marginal for A, but different marginalswith different entropies for B, necessarily imply enabling adifferent amount of correlations between A and B measuredby the mutual information I(A : B) B S A + S B − S AB. That is,∆S A + ∆S B = ∆I(A : B).

Lemma 11 (Connections among heat definitions). Given asystem A and its environment B, the heat absorbed by A in

a globally entropy preserving process ρABΛep

−−→ ρ′AB can bewritten as

∆Q =

∫ S ′B

S B

T (s)ds, (29)

where S (′)B = S (ρ(′)

B ) is the initial (final) entropy of B, andT (S ) is the intrinsic temperature of the thermal state of B withentropy S . Because of the continuity of T (S ), there exists as∗ ∈ [S B, S ′B] such that

∆Q = T (s∗)∆S B . (30)

Thus, definition 10 becomes ∆Q = T∆S B in the limit of smallentropy changes in which T (S B) ' T (S ′B), while it differs fromthe common definition and ∆Q , ∆EB.

In the case that the initial state of the environment is ther-mal ρB = e−HB/T /Tr (e−HB/T ) with Hamiltonian HB at temper-ature T , then the heat definition is lower and upper boundedby

T∆S B ≤ ∆Q ≤ ∆EB , (31)

where the three quantities have been represented in theenergy-entropy diagram in Fig. 5. If, on top of that, the pro-cess slightly perturbs the environment ρ′B = ρB +δρB, then, thethree definitions of heat coincide, in the sense that

T∆S B + O(δρ2B) = ∆Q = ∆EB − O(δρ2

B) . (32)

That is, in the limit of large thermal baths, the three definitionsbecome equivalent.

Proof. Equation (29) is a consequence of the definitions ofheat (28) and bound energy (6), together with Eq. (5). Equa-tion (30) is proven by using the mean value theorem for thefunction T (s). The lower bound in (31) is due to the concav-ity of the thermal boundary together with the reminder theo-rem of a first order Taylor expansion of the thermal bound-ary (see Fig. 5). The upper bound in (31) is a consequenceof the positivity of the free-energy. Finally, Eq. (32) followsfrom the fact that the thermal state ρB is a minimum of thestandard free-energy FT (ρ) = E(ρ) − TS (ρ), which implies∆FT,B = O(δρ2

B), and ∆EB = T∆S B + O(δρ2B). �

Once the heat has been introduced and related to the stan-dard definitions of heat, let us define work.

Definition 12 (Work). For an arbitrary entropy preservingtransformation involving a system A and its environment B,ρAB → ρ′AB, with fixed non-interacting Hamiltonians HA andHB, the worked performed on the system A is defined as,

∆WA B W − ∆FB (33)

where W = ∆EA + ∆EB is the work cost of implementing theglobal transformation and ∆FB = F(ρ′B) − F(ρB).

Now equipped with the notions of heat and work, the firstlaw takes the form of a mathematical identity.

Lemma 13 (First law). For an arbitrary entropy preservingtransformation involving a system A and its environment B,ρAB → ρ′AB, with fixed non-interacting Hamiltonians HA andHB, the change in energy for A is distributed as

∆EA = ∆WA − ∆Q . (34)

Proof. The proof follows from the definitions of work andheat. �

Clearly, in a process, −∆WA is the amount of “pure” energythat can be transferred and stored in a battery. While, the ∆QAis the change in energy due to flow of information from/intothe system.

9

VIII. SECOND LAW

The second law of thermodynamics is formulated in manydifferent forms: an upper bound on the extracted work, the im-possibility of converting heat into work completely etc. In thissection we show how all these formulations are a consequenceof the principle of entropy conservation.

A. Work extraction

Major concern in thermodynamics is to convert any form ofenergy into work, which can be used for any application withcertainty.

Lemma 14 (Work extraction). For an arbitrary compositesystem ρ, the extractable work by any entropy preserving pro-cess ρ → ρ′, W = E(ρ) − E(ρ′) is upper-bounded by the freeenergy

W ≤ F(ρ) (35)

where the equality is saturated if and only if ρ′ = γ(ρ).If the system has the particular structure ρ = ρA ⊗ γB(TB)

with γB(TB) being thermal at temperature TB, then

W ≤ FTB (ρA) − FTB (γA(TB)) (36)

where FT (·) is the standard out of equilibrium free energy andthe equality is saturated in the limit of an infinitely large bath(infinite heat capacity).

Proof. By using E(ρ′) ≥ B(ρ′), one gets

W = E(ρ) − E(ρ′) ≤ E(ρ) − B(ρ′) = F(ρ), (37)

where note that B(ρ′) = B(ρ).When the system has a composite structure ρ = ρA⊗γB(TB),

then

F(ρA ⊗ γB(TB)) = E(ρA) + E(γB(TB))− (B(γA(TAB)) + B(γB(TAB)))

(38)

where TAB is the intrinsic temperature of the joint state ρA ⊗

γB(TB). By considering that thermal states have zero free en-ergy, E(γ) = B(γ), and reshuffling the terms above, we recoverthe first law of thermodynamics

F(ρA ⊗ γB(TB)) = −∆EA − ∆QA (39)

where we have identified ∆EA = E(γA(TAB)) − E(ρA) and∆QA = B(γB(TAB)) − B(γB(TB)). From Eq. (31) in Lemma11, we have that ∆Q ≥ TB∆S B. By using now that the wholeprocess is entropy preserving and that initially and finallythe subsystems A and B are in a product state, we have that∆S A = −∆S B, and thus,

F(ρA⊗γB(TB)) ≤ −∆EA−TB∆S B = − (∆EA − TB∆S A) (40)

which proves (36). Finally, in the limit of an infinitely largebath, the intrinsic temperature TAB tends to the bath temper-ature TB and ∆Q = TB∆S B, which saturates the bound inEq. (40).

To see that the intrinsic temperature TAB tends to TB in thelimit of large baths, let us increase the bath size by addingseveral copies of it. The entropy change of such bath reads

∆S B = S (γ⊗nB (TAB)) − S (γ⊗n

B (TB))= n (S (γB(TAB)) − S (γB(TB))) .

(41)

By considering the bound ∆S B = −∆S A ≤ |A|, we get

S (γB(TAB)) − S (γB(TB) ≤|A|n, (42)

which, together with the the continuity of S (γ(T )), proveslimn→∞ TAB = TB. �

One of the most important questions in thermodynamics isif heat can be converted into work. It was very much funda-mental to understand, not only qualitatively but also quanti-tatively, to what extent and efficiency the heat can be trans-formed into work. These quests led to various other formula-tions of second law in standard thermodynamics, like Clausiusstatement, Kelvin-Planck statement and Carnot statement, tomention a few.

Similar questions can be posed in the frame-work consid-ered here, in terms of bound energy. In the following, analo-gous forms of second laws that consider these questions qual-itatively as well as quantitatively.

B. Clausius statement

Second law, in thermodynamics, not only dictates the direc-tion of state transformations, but also put fundamental boundon extractable work from such transformations. Here we firstconcentrate on the bounds on extractable work and introduceanalogous versions of second law in our setup.

Lemma 15 (Clausius statement). Any iso-informatic processinvolving two bodies A and B in an arbitrary state with intrin-sic temperatures TA and TB respectively fulfills the followinginequality

(TB − TA)∆S A > ∆FA + ∆FB + TB∆I(A : B) −W , (43)

where ∆FA/B is the change in the free energy of the bodyA/B,∆I(A : B) is the change of mutual information and W = ∆EA+

∆EB is the amount of external work performed on the globalsetting. In absence of initial correlations between two bodiesA and B, the states being initially thermal, and no externalwork bein performed,

(TB − TA)∆S A > 0 , (44)

meaning that no iso-informatic equilibration process is possi-ble whose sole result is the transfer of heat from a cooler to ahotter body.

Proof. The definition of free and bound energy implies that

W = ∆FA + ∆FB + ∆QA + ∆QB , (45)

10

where have used the definition of heat as the change of boundenergy of the environment. From Eq. (31), one gets,

∆QA + ∆QB > TB∆S B + TA∆S A , (46)

where TA/B is the initial intrinsic temperature of the body A/B.Due to the conservation of the total entropy, the change inmutual 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 in the right hand side of Eq. (43) show the threereasons for which the standard Clausius statement can be vi-olated. Either because of the process not being spontaneous(external work is performed W > 0), or due to initial stateshaving free energy which is consumed, or the presence of cor-relations [18]. An alternative formulation of Clausius state-ment, for initial and final equilibrium states, is considered inthe Appendix A.

C. Kelvin-Planck statement

If the Clausius statement tells us that spontaneously heatcannot flow from a hotter to a colder body, the Kelvin-Plankformulation of second law states that, when heat going froma hotter to a colder body, it cannot be completely transformedinto work.

Lemma 16 (Kelvin-Planck statement). Any iso-informaticprocess involving two bodies A and B in an arbitrary statesatisfies the following energy balance

∆QB + ∆QA = −(∆FA + ∆FB) + W (47)

where ∆FA/B is the change in the free energy of the body A/B,∆QA/B the heat dissipated by the body A/B, and W = ∆EA +

∆EB is the amount of external work performed on the globalsetting.

In the case of the reduced states being thermal, and for awork extracting process W < 0, the above equality becomes

∆QB + ∆QA 6 W < 0 . (48)

Finally, in absence of initial correlations, Eq. (48) implies thatno iso-informatic equilibration process is possible whose soleresult is the absorption of bound energy (heat) from an equi-librium state and its complete conversion into work.

Proof. Equation (47) is a consequence of the the energy bal-ance (45). Equation (48) follows from (47) by considering re-duced states that are initially thermal and thereby ∆FA/B > 0.To prove the final statement is sufficient to notice that en-tropy preserving processes on initially uncorrelated systemsfulfill ∆S A + ∆S B > 0, which together with (48) implies thatsign(∆QA) = −sign(∆QB). �

An alternative formulation of Kelvin-Planck statement, forinitial and final equilibrium states, is considered in the Ap-pendix B.

D. Carnot statement

A heat engine extracts work from a situation in which twobaths A and B have different temperatures TA and TB. Thework extraction is usually implemented in practice by meansof a working body S which cyclically interacts with A and B.Here we do not mind on how the working medium interactswith the baths A and B, but just assume that at the end of everycycle the working body is left in its initial state and uncorre-lated with the bath(s). In other words, the working body onlyabsorbs heat from a bath and releases to the other one, and atthe end of the cycle it comes back to its initial state. Fromthis perspective, the analysis of a heat engine can be made bystudying the changes of the environments A and B. In contrastto the standard situation, here we will not assume that bathsare infinitely large, but that can have a similar size as the sys-tem, and their loss or gain of energy changes their (intrinsic)temperature.

Let us consider that initially the environments A and B areuncorrelated and at equilibrium with temperatures TA and TBrespectively, where without loss of generality TA < TB, i. e.ρAB = γA ⊗ γB. After operating the machine for one or severalcomplete cycles, the environments change to ρAB → ρ′AB.

We define the efficiency of work extraction in a heat engineas the fraction of energy that is taken from the hot bath whichis transformed into work

η BW−∆EB

(49)

where −∆EB = EB − E′B > 0 is the energy absorbed from thehot environment, and W is the work extracted. In the follow-ing, we upper-bound the efficiency of any heat engine.

Lemma 17 (Carnot statement). For an engine working withtwo initially uncorrelated environments γA⊗γB each in a localequilibrium state with intrinsic temperatures TB > TA, theefficiency of work extraction is bounded by

η 6 1 −∆BA

−∆BB, (50)

where ∆BA and ∆BB are the change in bound energies of thesystems A and B respectively.

In the limit of large baths and under global entropy pre-serving operations, the Carnot efficiency is recovered,

η 6 1 −TA

TB. (51)

Proof. Here the systems could be small in size. Then the max-imum extractable work due to the transformation ρAB → ρ′ABis given by

W = F(ρAB) − F(ρ′AB) = (−∆EB) − ∆EA > 0 . (52)

The efficiency then reads

η = 1 −∆EA

−∆EB. (53)

11

The condition of A being initially at equilibrium implies that∆FA > 0, from where it follows ∆EA > ∆BA, and analogouslyfor B. Thus,

η 6 1 −∆BA

−∆BB. (54)

In the limit of large environments in which ∆BA ≪ BA, thechange in bound energy becomes ∆BA = TA∆S A. Hence,

η 6 1 −TA∆S A

−TB∆S B. (55)

If the process is globally entropy preserving, i. e. S ′AB = S A +

S B, then ∆S A + ∆S B > 0 or alternatively ∆S A > −∆S B, whichtogether with (55) implies (51). �

IX. THIRD LAW

The third law of thermodynamics establishes the impossi-bility of attaining the absolute zero temperature, or accordingto Nernst, is stated as: “it is impossible to reduce the entropyof a system to its absolute-zero value in a finite number ofoperations”. The third law of thermodynamics has been veryrecently proven in Ref. [47]. The unattainability of absolutezero entropy is a consequence of the unitarity character of thetransformations considered. For instance, consider the trans-formation

ρS ⊗ ρB → |0〉〈0| ⊗ ρ′B (56)

where ρB is a thermal state of the bath B and S is the system tobe cooled down (erased), initially in a state ρS with rank(ρS ) >1. Here the dimension of the Hilbert space of the bath, dB, isconsidered to be arbitrarily large but finite. As ρB is a full-rankstate, the left hand side and the right hand side of Eq. (56) havedifferent ranks, and they cannot be transformed via a unitarytransformation. Thus, irrespective of work supply, one cannotattain the absolute zero entropy state.

The zero entropy state can only be achieved for infinitelylarge baths and a sufficient work supply. Assuming a localitystructure for the bath’s Hamiltonian, this would take an in-finitely long time. However, in the finite dimensional case, aquantitative bound on the achievable temperature given a fi-nite amount of resources (e. g. work, time) is derived in [47].

Note that our set of entropy preserving operations is morepowerful than unitaries. In this case, transformation (56) ispossible given that the

S (ρS ) 6 log dB − S (ρB) (57)

the required work to do so will be W = F(|0〉〈0| ⊗ρ′B)−F(ρS ⊗

ρB). As it has been discussed in Sec. II, entropy preservingoperations can be implemented as global unitaries acting oninfinitely many copies. Thus, the attainability of the absolutezero entropy state by means of entropy preserving operationsis in agreement with the cases of infinitely large baths andunitary operations [47].

In conclusion, the third law of thermodynamics is a conse-quence of the microscopy reversibility (unitarity) of the trans-formation and is not respected by entropy preserving opera-tions.

X. A TEMPERATURE INDEPENDENT RESOURCETHEORY OF THERMODYNAMICS

In this section we connect the framework developed abovewith the standard approach of thermodynamics as a resourcetheory. The main ingredients of a resource theory are the statespace, which is usually compatible with a composition oper-ation – for instance, quantum states together with the tensorproduct– and a set of of allowed state transformations. Ina first attempt to have a resource theory of thermodynamicsfrom our framework, we look like to be bounded to transfor-mations within an equivalence class, that is, between stateswith equal entropies and same Hilbert spaces. In that case,the set of allowed operations seem reasonable to be definedby the non-increasing energy transformations and the mono-tone to be given by both energy and entropy. To compareand explore inter-convertibility between states with differententropies or dimensions, it is necessary to extend the set ofoperations, and in particular to add a composition rule. In or-der to be able to connect states with different entropies and/ornumber of particles it is natural to consider different numberof copies.

Let us restrict first to the case in which the set of operationsis entropy preserving regardless the energy. We wonder whatis the rate of transforming ρ into σ and consider without lossof generality that S (ρ) ≤ S (σ). Then, there is a large enoughn such that

S (ρ⊗n) = S (σ⊗m). (58)

With such a trick of considering a different number of copies,we have brought ρ and σ which had different entropies intothe same manifold of equal entropy. There is a subtlety herewhich is that ρ⊗n and σ⊗m live in general in spaces of differentdimension. This can be easily circumvented by adding someproduct state, i. e.

Λ(ρ⊗n

)= σm ⊗ |0〉〈0|n−m (59)

where ρ⊗n and σ⊗m ⊗ |0〉〈0|n−m lie now in spaces of the samedimension. Equation (59) can also be understood as a pro-cess of randomness compression in which the information inn copies of ρ is compressed to m < n copies of σ and n − msystems have been erased. Thus, in the case of having only anentropy preserving constraint, the transformation rate can bedetermined from (58)

r Bmn

=S (ρ)S (σ)

. (60)

In thermodynamics, however, energy must also be takeninto account. When one does that, it could well be that theprocess (59) is not energetically favorable since 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 other-wise, the initial and final state of such a process would nothave the same entropy

Λ(ρ⊗n) = σ⊗m ⊗ φ⊗n−m (61)

12

S

E

ρ⊗n → σ⊗m ⊗ φ⊗(n−m)

xφ

xσxρ

ab

r = mn = a

b

Figure 6. Representation in the energy-entropy diagram of howthe energy and entropy conservation constraints in the transformationρ⊗n → σ⊗m ⊗ φ⊗n−m imply xρ, xσ and xφ to be aligned and xρ to lie inbetween xσ and xφ.

where the number of copies n and m have to fulfill the energyand entropy conservation constraints

E(ρ⊗n) = E(σ⊗m ⊗ φ⊗n−m)S (ρ⊗n) = S (σ⊗m ⊗ φ⊗n−m) .

(62)

The above conditions can be easily written as a geometricequation of the points xψ = (E(ψ), S (ψ)) with ψ ∈ {ρ, σ, φ}in the energy-entropy diagram

xρ = r xσ + (1 − r) xφ , (63)

where r B m/n is the conversion rate, and we have only usedthe extensivity of both entropy and energy in the number ofcopies, e.g. E(ρ⊗n) = nE(ρ).

Equation (63) implies that the three points xρ,σ,φ need to bealigned. In addition, the fact that 0 ≤ r ≤ 1 implies that xρ liesin the segment xσ and xφ (see Fig. 6). The conversion rate rhas then a geometric interpretation. It is the relative Euclideandistance between xφ and xρ over the total distance ‖xσ − xφ‖,or in other words, is the fraction of the path that one has runwhen going form xφ to xσ and reaches xρ (see Fig. 6).

In order for the conversion rate r from ρ and σ to be max-imum, the state φ needs to lie in the boundary of the energy-entropy diagram, that is, it needs to be either a thermal or apure state. This allows us to quantitatively determine the con-version rate r by first finding the temperature of the thermalstate φ which satisfies

S (σ) − S (φ)S (σ) − S (ρ)

=E(σ) − E(φ)E(σ) − E(ρ)

(64)

and then

r =mn

=S (ρ) − S (φ)S (σ) − S (φ)

. (65)

See the geometrical interpretation in Fig. 6.Note how (65) becomes (60) in the case of φ being pure.

Let us also mention that, although at the first sight the inter-convertibility rate (65) looks different from the one obtainedin [39], it can be proven that they are the same. Equation (65)is more compact than the one given in [39] and its derivationmuch less technical.

Figure 7. Charges-entropy diagram of a system with 2 conservedquantities E and L.

XI. THERMODYNAMICS WITH MULTIPLECONSERVED QUANTITIES

The formalism for thermodynamics developed above can beeasily extended to situations with multiple conserved quanti-ties.

A. The charges-entropy diagram

The energy-entropy diagram then is extended to include theadditional conserved quantities. Let us introduce the charges-entropy diagram as:

Definition 18 (Charges-entropy diagram). Let us consider asystem with q conserved quantities (not necessarily commut-ing) Qk for k = 0, . . . , q − 1 and Q0 = H the Hamiltonian.The charges-entropy diagram is the subset of points in Rq+1

produced by all the states ρ ∈ B(H) with coordinates

xρ = (Q0(ρ),Q1(ρ), . . . ,Qq−1, S (ρ)) , (66)

with Qk(ρ) B Tr (Qkρ) for k = 0, . . . , q − 1.

In the case of mutually commuting conserved quantities,we can easily show that the charges-entropy diagram formsa convex set in Rq+1. To do so, let us first prove convexityfor the zero entropy hyper-surface, which corresponds to thebase of the diagram. The zero-entropy hyper-surface of thecharges-entropy diagram is the set of points in Rq with coor-dinates xψ = (Q0(ψ),Q1(ψ), . . . ,Qq−1(ψ)) for all pure statesψ = |ψ〉〈ψ| with norm one 〈ψ|ψ〉 = 1. Let us consider inRq the family of hyper-planes perpendicular to the unit vec-tor ~µ = (µ0, . . . , µq−1)

~µ · ~Q =

q−1∑k=0

µkQk = C , (67)

where ~Q = (Q0, . . . ,Qq−1) is a point in Rq and C determinesthe hyperplane. All the points with coordinates ~Q that fulfil

13

Eq. (67) belong to the hyper-plane defined by ~µ and C. Givena direction ~µ, there is an hyper-plane that is particularly rel-evant, since it corresponds to the minimum possible value ofC

Cmin(~µ) = minψ∈H

〈ψ|~µ · ~Q|ψ〉〈ψ|ψ〉

= minψ∈H

~µ · ~Q(ψ)〈ψ|ψ〉

. (68)

The minimum is reached by the eigenstate with smallesteigenvalue of the Hermitian operator ~µ · Q that we denote by|ψ(~µ)〉, ∑

k

µkQk

|ψ(~µ)〉 = Cmin(~µ)|ψ(~µ)〉 . (69)

Let us consider now that there is a unique ground state of ~µ · ~Q.This means that the hyper-plane defined by Cmin(~µ) is tangentto the charges entropy-diagram since it passes through onepoint of the diagram (the corresponding to the single groundstate |ψ(~µ〉) leaving the rest of the diagram on the same side,i.e. ~µ · ~Q(ψ) > Cmin(~µ) for all |ψ〉 ∈ H .

In case that there is a degenerate ground space described bya basis whose elements have different coordinates, the contactbetween the hyper-plane C = Cmin and the zero entropy hyper-surface is not a point but a simplex of dimension equal to thedimension of ground space (e. g. a segment in dimension 2,triangle in dimension 3). As the conserved quantities are mu-tually commuting, there is a common eigenbasis for all the qcharges. Then, there is a basis of the ground space of ~µ · ~Qwhich is also eigenbasis of all the charges Qk individually. If|ψ〉 and |ψ′〉 are two elements of that basis with different coor-dinates ~Q and ~Q′, the state |ψ(θ)〉 = cos θ|ψ〉 + sin θ|ψ′〉 withθ ∈ [0, π/2] has coordinates

〈ψ(θ)| ~Q|ψ(θ)〉 = cos2 θ ~Q + sin2 θ ~Q′, (70)

that is, the coordinate of |ψ(θ)〉 in the diagram are simply thecorresponding convex combination of ~Q and ~Q′.

Property (70) holds for any two eigenstates of ~Q, in partic-ular, for any two states |ψ〉 and |ψ′〉 that lie in the boundaryof the zero-entropy hyper-surface. Hence, as any point in thebulk of the zero-entropy hyper-surface can be gotten as a con-vex combination of two points of the boundary, all the pointsin the bulk belong to zero entropy surface. Altogether, thisproves that the zero-entropy surface is a convex set for whichany point within it there exists at least a quantum state thatreproduces its expectation vales.

Once we understand the zero entropy hyper-surface of thecharges-entropy diagram, let us study its upper boundary. Theupper boundary of the charges-entropy diagram is describedby the Generalized Gibbs Ensemble (GGE) instead of thecanonical ensemble, i. e.

γ(~β) B1Z~β

e−∑

k βk Qk , (71)

where Z~β B Tr (e−∑

k βk Qk ) is the generalized partition func-tion. These states are precisely the states of maximum en-tropy among all with prescribed expectation values for 〈Qk〉,

S

Qk

βk = ∂S∂Qk

(βk,−1) xρ

A~β(ρ)

xσ

A(σ)

Figure 8. Transverse section of the charges-entropy diagram inthe plane Qk − S . The normal vector to the tangent plane has co-ordinates (~β,−1), which in the section appears as (βk,−1). The ~β-athermality, A~β(ρ), and the absolute athermality, A(σ), are respec-tively represented for two states ρ and σ.

k = 0, ..., q − 1. We call this boundary the equilibrium bound-ary.

The equilibrium boundary is mathematically described byall the points ~Q(ρ) for which there exists a ~β ∈ Rq such that

q−1∑k=0

βkQk(ρ) − S (ρ) = − log Zβ . (72)

Note that, given some ~β, the points ~Q(ρ) that fulfil

q−1∑k=0

βkQk(ρ) − S (ρ) = C (73)

belong to a hyperplane of dimension q. The hyperplane withC = − log Zβ is the one that is tangent to the equilibriumboundary. The normal vector to this family of hyperplanes(73) is precisely (~β,−1). This agrees with the fact that thetangent plane has a slope βk in the k-th direction, i. e.

∂S∂Qk

= βk ∀ k ∈ {0, . . . , q − 1} , (74)

where here the Qk need to be understood as coordinates inthe charges-entropy diagram (not matrices). This is shown inFig. 8. Equation (74) can also be written in a vectorial formas

~β = ~∇S , (75)

and ~β corresponds to the direction of maximal variation of theentropy.

Note that in order to microscopically justify the charges-entropy diagram, it must be shown that states with the sameexpectation values for the conserved quantities can be unitar-ily connected in the limit of many copies. We extensivelystudy this particular issue in a separate forthcoming work forthe cases of both commuting and non-commuting conservedquantities.

14

B. Athermality and free entropy

Proceeding as above, let us introduce some quantities thatwill be relevant in the following.

Definition 19 (~β-athermality). The ~β-athermality of a state ρis defined as

A~β(ρ) Bq−1∑k=0

βkQk(ρ) − S (ρ) + log Z~β . (76)

The ~β-athermality of a point with coordinates ( ~Q(ρ), S (ρ))can be interpreted geometrically in the charges-entropy dia-gram as the vertical distance from the hyperplane tangent tothe equilibrium boundary with normal vector (~β,−1). This isrepresented in Fig. 8. Note that the ~β-athermality is also intro-duced in [48] as as the free entropy.

The ~β-athermality can also be written as

A~β(ρ) = D(ρ||γ(~β)) , (77)

with D(ρ||σ) B Tr (ρ log ρ − ρ logσ) being the relative en-tropy. Hence, because of the positivity of the relative entropyD(ρ||σ) > 0, we have that

S (ρ) 6q−1∑k=0

βkQk(ρ) + log Zβ . (78)

Note now that the right hand side corresponds to the entropycoordinate of the hyperplane with normal vector ~β tangent tothe equilibrium boundary. Thus, any tangent hyperplane withnormal vector ~β leaves all the points of the charges entropy di-agram below it. This proves that the charges entropy diagramis a convex set.

In a similar way we define the absolute athermality or sim-ply athermality in the following.

Definition 20 (Absolute athermality). The absolute athermal-ity of a state ρ is defined as

A(ρ) B min~β∈Rq

A~β(ρ) . (79)

The athermality of a state ρ can geometrically be under-stood as the vertical distance from the thermal boundary (seeFig. 8).

C. Charge extraction

A first relevant scenario of thermodynamics with multipleconserved quantities is the extraction of a charge of the systemwhile keeping constant the other charges. For such set of op-erations the system is restricted to move along a straight linein the direction of the extracted charge within the iso-entropichyperplane (see Fig. 9).

The bound charge for such scenario is given by

Bk(ρ) B minσ : S (σ)=S (ρ);

Qi(σ)=Qi(ρ)∀i,k

Qk(σ) = Qk(γk(ρ)) , (80)

E

L

γ

ρ

F (ρ)

E

L

μ

γρ

F (ρ)

Figure 9. Constant entropy section of a charges-entropy diagramwith 2 charges E and L. (Left) Scenario of extraction of a singlecharge while keeping constant the other charges. The system is con-strained to move in one dimensional line and free-charge F(ρ) of astate ρ is the distance from the boundary in such direction. (Right)Scenario in which the charges are allowed to change and the aim isthe extraction of a potential Vµ(ρ) =

∑q−1k=0 µkQk(ρ). The parallel pur-

ple lines represent equipotential surfaces of Vµ. The GGE state γ isthe state which minimizes V and belongs to the hyperplane which istan tangent to the equilibrium boundary.

where γk(ρ) is the GGE state that attains the minimum andcorresponds to the point of the equilibrium boundary which isintersected by the straight-line with direction k which passesthrough ρ. This is represented in Fig. 9 (left). The ~β corre-sponding to γk can be geometrically determined by the normalvector of the tangent plane to the equilibrium surface in thatpoint.

The free charge Fk(ρ) is then defined by

Fk(ρ) = Qk(ρ) − Bk(ρ) (81)

and corresponds to the maximum amount of charge that canbe extracted given the restrictions of constant entropy andcharges. This is represented in Fig. 9 (left) for a case of 2conserved quantities.

A detailed and full rigorous analysis of the charge extrac-tion by means of unitary conserving operations in the manycopy limit for both commuting and non-commuting chargeswill be made in our forthcoming work.

D. Extraction of a Generalized Potential

An alternative but also relevant scenario is when the set ofoperations allow for the variation of the charges and the aimis the extraction of a generalized potential

Vµ(ρ) Bq−1∑k=0

µkQk(ρ) (82)

where µ B (µ0, . . . , µq−1) specifies the weight µk of everycharge Qk in the generalized potential Vµ. For convenience,µ is assumed to be normalized ‖µ‖ = 1 in the Euclidean normand its components to be positive µk > 0. An alternative nor-malization could be to take the coefficient of the Hamiltonianµ0 = 1 as it is usually made in the Grand-canonical ensemble.

15

In this setting, the bound potential is defined as

Bµ(ρ) B minσ : S (σ)=S (ρ)

Vµ(σ) = Vµ(γµ(ρ)) , (83)

where γµ(ρ) is the state that attains the minimum which isagain a GGE. Note that the ~β values of the state γµ(ρ) need tobe proportional to the unitary vector µ

~β = βµ (84)

with β a scalar that is determined by the equal entropies condi-tion S (ρ) = S (γµ(ρ)). This can be seen geometrically in Fig. 9(right) or analytically by making the simple observation that,given a new Hamiltonian H = Vµ, the min energy principlesingles out e−βH as the state that attains the minimum.

Analogously, the free potential is given by

Fµ(ρ) = Vµ(ρ) − Bµ(ρ) , (85)

and corresponds to the maximum amount of generalized po-tential Vµ that can be extracted under entropy preserving oper-ations. This situation has been diagrammatically representedin Fig. 9.

The scenario with a generalized potential Vµ is analogue tothe single charge situation. Both first and second laws can bestated as in Secs. VII and VIII but replacing the free energyF(ρ) by Fµ(ρ).

E. Second law

In order to give express the second law irrespective of aparticular choice of the generalized potential, let us formulateit as an inequality that

Any process (not necessarily entropy preserving) thatbrings an initial generalized Gibbs state γB(~β) out of equilib-rium fulfills ∑

k

βk∆QBk > ∆S B , (86)

where the inequality is only saturated in the limit of largebaths (small variations of entropy) and when final state is alsoat equilibrium (GGE). Geometrically (86) is a trivial conse-quence that the charges entropy diagram is upper bounded byany of its tangent planes.

In the particular case of an entropy preserving process onan initial bipartite state ρA ⊗ γB(~β), the change in mutual in-formation between the subsystems A and B

∆I = ∆S A + ∆S B > 0 , (87)

which together with Eq. (86) implies∑k

βk∆QBk > −∆S A . (88)

Note that the above equation (88) is equivalent to the secondlaw formulated in [48]. We see here that such a law is alsovalid for baths of arbitrary small sizes. Equation (88) is onlysaturated in the limit of large system sizes and in absence ofcorrelations in the final state ρ′AB = ρ′A ⊗ γ

′B.

F. Inter-convertibility rates

The convertibility rate from a quantum state ρ’s to a state σ

Λ(ρ⊗n) = σ⊗m ⊗ φ⊗n−m (89)

given the constraints on the conservation of the entropy andthe charges Qk

Qk(ρ⊗n) = Qk(σ⊗m ⊗ φ⊗n−m) ∀ k = 0, . . . , q − 1S (ρ⊗n) = S (σ⊗m ⊗ φ⊗n−m) .

(90)

The above conditions can be again written as a geometricequation of the points xψ = (Q0(ψ), . . . ,Qq−1(ψ), S (ψ)) withψ ∈ {ρ, σ, φ} in the energy-entropy diagram

xρ = r xσ + (1 − r) xφ , (91)

where we have merely used the extensivity of both entropyand 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 previoussection, the convertibility rate from ρ to σ reads

r =mn

=S (ρ) − S (φ)S (σ) − S (φ)

. (92)

where the state φ is determined from the following set of qequations

S (σ) − S (φ)S (σ) − S (ρ)

=Qk(σ) − Qk(φ)Qk(σ) − Qk(ρ)

k = 0, . . . , q − 1 . (93)

The state φ can be found geometrically in the charges-entropydiagram as the point in the equilibrium boundary that is inter-sected by the straight line that goes from σ to ρ (see Fig. 6 forthe single charge example).

XII. DISCUSSION

In the recent years, thermodynamics, and in particular workextraction from non-equilibrium states, has been studied inthe quantum domain, giving rise to radically new insights intoquantum thermal processes. However, in standard endeavorof thermodynamics, be it classical or quantum, thermal bathsare considered to be considerably large in size. That is whyif a system is attached with a bath, and allowed to exchangeenergy and entropy, the bath stays intact and its temperatureremains unchanged. That is also why the equilibrated systemfinally acquires the same temperature of the bath. However, ifone goes beyond this assumption and considers bath to be a fi-nite and small system, then traditional thermodynamics breaksdown. This situation is very much relevant for thermodynam-ics in the quantum regime, where both system and bath maybe small. The first problem that appears in such a situation isthe notion of temperature itself, since the finite bath may goout of thermal equilibrium due to the exchange of energy withthe system. Therefore, it is absolutely necessary to develop atemperature independent universal thermodynamics, in which

16

the bath could be small or large, and would not get any specialstatus.

In this work, we have formulated temperature independentthermodynamics as an exclusive consequence of informationconservation. We have relied on the fact that, for a givenamount of information, measured by the von Neumann en-tropy, any system can only be transformed to states with thesame entropy. Given this constraint of information conserva-tion, there is a state that singles out within the constant en-tropy manifold which is the state that possesses minimal en-ergy. This state is known as a completely passive state andacquires a Boltzmann–Gibb’s canonical form with an intrin-sic temperature. We call the energy of the completely passivestate as the bound energy, since no further energy can be ex-tracted by means of entropy preserving operations. Thus, fora given state, the difference between its energy and bound en-ergy corresponds to the maximum amount of energy that canbe extracted in form of work and we have denoted it as freeenergy. In fact, in this framework, two states that have thesame entropy and energy are thermodynamically equivalent[39]. The thermodynamic equivalence between equal entropyand energy states has allowed us to use of the energy-entropydiagram to illustrate the notions of bound and free energies ina geometric way. We have introduced a new definition of heatfor arbitrary systems in terms of bound energy.

We have seen that the laws of thermodynamics are a conse-quence of the reversible dynamics of the underlying physicaltheory. In particular:

• Zeroth law emerges as the consequence of informationconservation.

• First and second laws emerge as the consequence ofenergy conservation, together with information conser-vation.

• Third law emerges as the consequence of "strict" infor-mation conservation (microscopic reversibility or uni-tarity). Therefore there is no third law if one considers"average" information conservation.

We have demonstrated that the maximum efficiency of aquantum engine with a finite bath is in general lower than thatof an ideal Carnot’s engine. We have introduced a resourcetheoretic framework for our intrinsic thermodynamics, withinwhich we address the problem of work extraction and inter-state transformations. All these results have been illustratedby means of the energy-entropy diagram. Furthermore, wegive a geometric interpretation in the diagram to the relevantthermodynamic quantities as well as the inter-convertibilityrate between quantum states under entropy and energy pre-serving operations.

The information conservation based framework for ther-modynamics, as well as the resource theory and the energy-entropy diagram, is also extended to multiple conserved quan-tities. In this case, the energy-entropy diagram becomes thecharges-entropy diagram and allows us to understand thermo-dynamics in a geometrical way. In particular, we have stud-ied the extraction of a single charge while keeping the othercharges conserved as well as the extraction of a generalized

potential. In the first scenario, we have seen that the maximumwork extractable from any state by operations that asymptoti-cally conserve the given charges is the difference between thefree energy of the state and that of the iso-entropic general-ized grand canonical Gibbs state. Concerning the extractionof a generalized potential (a linear combination of charges),we have shown that it is analogous to the work extraction (thesingle charge case). Finally, we have determined the inter-convertibility rates between states with different entropy andcharges.

In general, thermodynamics can be studied in three differentscenarios:

• One-shot or single-copy limit, where only one copy ofjoint system-environment is available. Albeit, in thiscase, even the notion of expectation value is not mean-ingful, as well as von Neumann entropy.

• Limit of many-runs, where there are many copies butoperation are restricted to one-copy operations. In thisscenario, the notions of expectation value and von Neu-mann entropy are well defined.

• Limit of many-copies, where one has access to arbitrar-ily many copies of the system and an ancilla sub-linearin the number of copies that can globally be processed.

A first observation is that our formalism cannot be appliedin the single-shot limit, since the notion of expectation value(say energy) cannot in general be used.

A relevant point to discuss is what happens when opera-tions are not EP but unitaries. In the limit of many copies,unitaries converge to EP operations and our formalism is re-covered. The limit of many runs is a bit more subtle. On onehand, all the thermodynamic inequalities of our formalism arerespected, since unitaries form a subset of EP operations. Onthe other hand, these thermodynamic inequalities will not bein general saturated. For instance, our formalism states thatthe work that can be extracted per system in a state ρ is upperbounded by its free energy W 6 F(ρ). In the many run casewith fine-grained information conservation, the law will berespected, but there will not be in general a unitary for whichW = F(ρ).

A natural open question is to what extent our formalismcan be extended from considering coarse-grained informationconservation operations as the set of allowed operations tounitaries. In that case, the notion of bound energy would bedifferent and many more equivalence classes of states wouldappear. Something similar already happens in the resourcetheory of thermodynamics, where instead of having a singlemonotone as an “if and only if” condition for state transfor-mation, infinitely many are required [29]. It is far from clearwhether under fine-grained information conservation restric-tion the energy-entropy diagrams (or a generalization of them)would still be useful.

Let us finally point out that, in the extension of our workto unitary operations in the settings of single-shot and many-runs, a consistent formulation of zeroth law would not be pos-sible. Note that zeroth law states that a collection of systemsare in mutual thermal equilibrium if and only if their arbitrary

17

combinations are also in equilibrium. It is well known thatpassive states that are not thermal do not remain passive whensufficiently many copies are considered. Hence, for establish-ing a consistent zeroth law, one has to consider operations be-yond unitaries on a single copy.

ACKNOWLEDGEMENTS

We thank P. Faist, K. Gawecdzki, N. Y. Halpern, R. B. Har-vey, J. Kimble, S. Maniscalco, Ll. Masanes, J. Oppenheim,V. Pellegrini, M. Polini, C. Sparaciari, A. Vulpiani and R.Zambrini for useful discussions and comments in both the-

oretical and experimental aspects of our work. We also thankunknown referees for constructive comments. We acknowl-edge financial support from the European Commission (FET-PRO QUIC H2020-FETPROACT-2014 No. 641122), the Eu-ropean Research Council (AdG OSYRIS and AdG IRQUAT),the Spanish MINECO (grants no. FIS2008-01236, FISI-CATEAMO 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 deCatalunya (grants no. 2017 SGR 1341, and SGR 875 and 966),CERCA Program/Generalitat de Catalunya and Fundació Pri-vada Cellex. AR also thanks support from the CELLEX-ICFO-MPQ fellowship.

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Appendix A: Alternative formulation of the Clausius statementfor the second law

Lemma 21 (Clausius statement). No iso-informatic equili-bration process is possible whose sole result is the transferof bound energy (i.e., heat) from an equilibrium state with βCto another equilibrium state with βH , where βC > βH .

Proof. In order to prove, we need to show that the iso-informatic equilibration process cannot lead to an increase of∆β = βH − βC . Further any iso-informatic process that leadsto such increase must requires a supply of energy. First con-sider the case where βC = βH (∆β = 0), which means thecorresponding states γC and γH are in equilibrium betweeneach other and jointly acquire the min-energy state. The min-energy principle says that any iso-informatic transformationto make βC , βH is bound to increase the sum of their indi-vidual min-energy. Therefore it requires a inflow of energy,which is nothing but introduction of work.

Note, for a completely passive state γ with given Hamilto-nian H and β, the change in bound energy, which is nothingbut heat in our definition, due to an infinitesimal change inentropy dS (γ), is given by

dE(γ) =1β

dS (γ). (A1)

Therefore larger the β, smaller be the change in internal en-ergy for a fixed infinitesimal change in entropy in the state.Now consider the case where βC > βH . Any flow of heat fromγC to γH will lead to a reduction of entropy in the former. Thatwill also lead to an equal increase of of the same in the latter.A very small amount of bound energy will result in a infinites-imal entropy flow, say dS from γC to γH . However the changein internal energy (dE(γC/H) = dS

βC/H) would not be equal and

that is −dE(γC) < dE(γH), for βC > βH . As a consequence,in this iso-informatic process the overall energy is bound toincrease, which is not possible without an influx of energy orwork. Therefore, without external work the process will nevertake place spontaneously.

�

Appendix B: Alternative formulation of the Kelvin-Planckstatement for the second law

Lemma 22 (Kelvin-Planck statement). No iso-informaticequilibration process is possible whose sole result is the ab-sorption of bound energy (heat) from an equilibrium state andits complete conversion into work.

Proof. To prove Kelvin-Planck statement, we consider fol-lowing example of two CP states γA and γB with βA and βBrespectively, which undergo an iso-informatic equilibrationtransformation as

γA ⊗ γB ⊗ |0〉〈0|WΛEP

ABW−−−−→ γAB ⊗ |W〉〈W |W , (B1)

γAB is the final joint equilibrium state with βAB. Since the ini-tial states are CP states, a non-zero W could only result fromheat transfer. Let say βA < βB. Then βA < βAB < βB. Inthis case the heat is flown out from γA, say ∆QA, with an as-sociated decrease in its entropy. Again the information con-servation of whole process guarantees an entropy increase inγB. Therefore, there has to be an associated increase in boundenergy (heat) content in γB. As a result, a part of ∆QA couldbe converted to work and that is

W 6 ∆QA − ∆QB. (B2)

Note any transfer of heat, i.e., −∆QA > 0, also bounds∆QB > 0. As a consequence, heat cannot be converted intowork completely. �