Entropy Production in Quantum Systems: A Unifying Picture

Philipp Strasberg1 and Andreas Winter1,2

1Fısica Teorica: Informacio i Fenomens Quantics, Departament de Fısica,Universitat Autonoma de Barcelona, 08193 Bellaterra (Barcelona), Spain and

2ICREA – Institucio Catalana de Recerca i Estudis Avancats,Pg. Lluis Companys, 23, 08010 Barcelona, Spain

(Dated: October 2, 2020)

Deriving the laws of thermodynamics from an underlying microscopic picture is a central questof statistical mechanics. The present article focuses on the derivation of the first and second lawof thermodynamics for closed and open quantum systems, where such foundational questions inaddition become practically relevant for emergent nanotechnologies. Recent progress is presentedin a self-contained way, with an emphasis on general ideas and useful tools instead of particularapplications. We identify limits and conceptual shortcomings of previous approaches. Then, wepropose a unifying perspective, which starts from a microscopic definition of nonequilibrium ther-modynamic entropy. The change of this entropy is identified with the entropy production, whichsatisfies a fluctuation theorem for a large class of initial states. Moreover, this entropy productioncan be naturally separated into a quantum and a classical component. Within this framework, weintroduce the notions of recoverable work and remaining heat. In the case of a weakly perturbedideal thermal bath, this approach is in quantitative agreement with previous ones, thus ensuringtheir thermodynamic consistency.

I. INTRODUCTION TO NONEQUILIBRIUMTHERMODYNAMICS: PHENOMENOLOGY

The theory of thermodynamics arose out of the de-sire to understand transformations of matter in chem-istry and engineering in the 19th century.1 The systemsunder investigation were macroscopic and described byvery few variables; for instance, temperature T , pres-sure p and volume V . These macroscopic bodies couldexchange heat Q with their surroundings and externalwork W could be supplied to them. Note that the ideato partition the ‘universe’ into a system and an environ-ment (also called a ‘bath’ or a ‘reservoir’ in the sequel) isan idea inherent to thermodynamic itself. A prototypicalexample of a thermodynamic setup is sketched in Fig. 1.

To apply the theory in practice, for instance, to com-pute the efficiency of a heat engine, two axioms areneeded, which are called the first and second law of ther-modynamics (there is also a zeroth and a third law ofthermodynamics, which are, however, not the topic ofthis paper) [1, 2]. The first law concerns the energy bal-ance of the system. It states that the change ∆US in

1 Sometimes it is asserted that thermodynamics played an impor-tant role for the industrial revolution to design efficient heatengines. Historically speaking, this is incorrect. The industrialrevolution is associated with the period from 1760 to (at most)1840 (the steam engine of Watt was introduced in 1776). Thefirst modern work on thermodynamics is perhaps due to Carnotin 1824, who, however, was not read by his contemporaries. Thefirst law of thermodynamics was established around 1850 andthe modern formulation of the second law goes back to Clausiusin 1865. Even then, however, engineers did not seem to be veryinspired by the theory of thermodynamics. To the best of ourknowledge, Diesel (at the end of the 19th century) patented thefirst engine which was based on the insight that a high temper-ature gradient increases the efficiency of the engine.

internal energy of the system is balanced by heat andwork:

∆US = Q+W. (1)

Note that we define heat and work to be positive when-ever they increase the internal energy of the system. Thefirst law is a consequence of conservation of energy ap-plied to the system, the heat bath and the work reservoir.However, the fundamental distinction between heat andwork becomes only transparent by considering the secondlaw.

The second law, perhaps in its most general form,states that “the entropy of the universe tends to a max-imum” [3]. In equations, for any physical process

∆Suniv ≥ 0, (2)

where Suniv denotes the thermodynamic entropy of theuniverse, which should be clearly distinguished from any

Figure 1. Thermodynamic setup where the system is a gas ina container. By pushing a piston, the thermodynamic vari-ables (such as T , p or V ) can be changed in a mechanicallycontrolled way, which is abstracted as the action of a ‘workreservoir’. Furthermore, through the walls of the containerthe gas is in simulatenoeus contact with a heat bath, withwhich it can exchange energy. As we will see below, this ex-change of energy is accompanied with an exchange of entropy,which is the defining property to call this energy exchange‘heat’.

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information theoretic notion of entropy at this point.Note that the terminology ‘universe’ does not necessar-ily refer to the entire universe in the cosmological sense,but to any system which is sufficiently isolated from therest of the world. For our purposes here, this includes,e.g., a gas of ultracold atoms or the system and the bathwithin the open quantum systems paradigm. The changein entropy of the universe is often called the entropy pro-duction and denoted by Σ = ∆Suniv. If Σ = 0, theprocess is called reversible, otherwise irreversible.

Focusing on the system-bath setup, e.g., as sketchedin Fig. 1, the entropy of the universe Suniv = SS + Senv

decomposes into the entropy of the system and the en-vironment. This decomposition implicitly assumes thatthe system and environment can be clearly distinguishedsuch that the entropy becomes additive. Then, the sec-ond law becomes

Σ = ∆SS + ∆Senv ≥ 0. (3)

Furthermore, the environment is typically assumed to bewell-described by an equilibrium state at a (in generalvarying) temperature T such that its change in entropycan be computed via ∆Senv = −

∫dQ/T . Here, dQ de-

notes an infinitesimal heat flow into the system. Then,the second law takes on the traditional form

Σ = ∆SS −∫dQ

T≥ 0, (4)

which was already introduced by Clausius, who called Σuncompensated transformations (“unkompensierte Ver-wandlungen”) [3]. In fact, the word ‘entropy’ was chosenby Clausius based on the ancient greek word for ‘transfor-mation’ (τροπή). Equation (4) is also often referred to asClausius’ inequality, in particular in its infinitesimal formdS ≥ dQ/T . Finally, if the bath gets only slightly per-turbed away from its initial temperature, here denotedby T0, then Eq. (4) reduces to

Σ = ∆SS −Q

T0≥ 0 (5)

with Q =∫dQ the total flow of heat from the bath.

The first law (1) together with the second law (4)or (5), which can be extended to multiple heat bathsat different temperatures, constitute the basic buildingblocks of phenomenological nonequilibrium thermody-namics [2]. The larger Eqs. (4) or (5), the smaller is theefficiency of a real-world heat engine compared to theefficiency of the idealized Carnot cycle. Contrary to fre-quently made statements that traditional thermodynam-ics dealt only with equilibrium transformations, Clausiusand his contemporaries knew that the above formalismcan be applied to irreversible nonequilibrium processes,but an assumption was that even after a nonequilibriumprocess the system can be still described by its basic ther-modynamic variables such as T , p and V .

It is clear that the last assumption becomes less ad-equate the smaller the system becomes. How to define

the basic building blocks of phenomenological nonequi-librium thermodynamics is then no longer clear. If thesystem is not too small, one way out of this dilemma is toassume that the system looks locally equilibrated, albeitits overall state is out of equilibrium [2]. Another pos-sibility, which works even for very small systems, is toassume that the system stays close to equilibrium by ap-plying only small thermodynamic forces to it (e.g., smalltemperature biases or slow variations of external param-eters). This is the regime of linear reponse theory [4].

In the present article we are interested in very smallsystems that can show quantum effects, which are drivenfar away from equilibrium, and which are in contactwith an environment or bath. Such systems are calledopen quantum systems [5, 6]. For many potential futuretechnologies—such as thermoelectric devices, solar cells,energy efficient computers, refrigerators that cool downto almost zero Kelvin, or quantum computing, sensing orcommunication devices—these are very interesting sys-tems. Furthermore, our theory also applies to isolatedquantum many-body systems such as ultracold quantumgases. Clearly, neither the local equilibrium assumptionnor linear response theory can be applied in these cases.This calls for a more advanced thermodynamic theory,which is called quantum thermodynamics.

II. SCOPE OF THIS ARTICLE

In our view, the theory of quantum thermodynamicsaims at answering the following three main questions.

First, given that nature behaves quantum mechanicalat small scales, how can one derive the laws of thermo-dynamics from a microscopic picture under plausible as-sumptions?

Second, how does one measure the basic thermody-namic quantities such as internal energy, heat, work andentropy in a laboratory? That is to say, a formal deriva-tion of the laws of thermodynamics should ideally be ac-companied with an experimentally meaningful descrip-tion to apply and verify them.

Third, assuming that it is possible to construct a con-ceptually satisfying and experimentally meaningful the-ory of quantum thermodynamics, what are its conse-quences and applications?

Here, we address the first question in detail, in the hopeto provide a satisfactory answer to the second question,too. We do not touch the third question about applica-tions.

The first question has indeed a long history in physicsgoing back to the famous ‘H-theorem’ of Boltzmann [7],who realized the statistical character of the second law(as also Maxwell did). However, whereas there seems tobe a consensus on the qualitative origin of the second lawand the macrosopic behaviour of matter [8], no univer-sal agreement on how to quantify dissipation and irre-versibility exists. This is in particular true in context ofour increased theoretical and experimental capabilities

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to model and manipulate small systems, which can nolonger be adequately described in terms of macroscopicvariables such as T , p or V .

Nevertheless, under certain idealized assumptionsthere exists a consensus on how to derive and quantifythe laws of thermodynamics for an open quantum sys-tem. These assumptions are justified when the system-bath coupling can be treated as weak, the heat bath(s)as memoryless, and the resulting system dynamics us-ing a ‘rotating wave approximation’. As a consequencethe system dynamics is described by the so-called ‘Born-Markov secular master equation’ [5, 6], whose consistentthermodynamic interpretation was established some timeago [9–13], see Refs. [14, 15] for introductions.

Unfortunately, the validity of the Born-Markov secularmaster equation quickly breaks down in practice. Thiscan occur even at weak coupling, for instance, if the bathis not macroscopic or if the system is sufficiently complexsuch that the rotating wave (or secular) approximationis not justified. Under these circumstances it remainsa challenge to derive the laws of thermodynamics froman underlying microscopic picture, which is the startingpoint of the present article.

For this purpose we proceed as follows. First, after in-troducing some basic concepts of statistical mechanics inSec. III, we present in Sec. IV a short, but self-containedreview of previous approaches to derive the laws of ther-modynamics for open systems. These approaches provideuseful tools and present remarkable progress compared tothe approximations, on which the validity of the Born-Markov secular master equation rests. Nevertheless, wewill see that their applicability is still limited and further-more, we will also identify conceptually open problems,which were not yet addressed in a satisfactory way.

In the second part (Secs. V to VII) of this article wetherefore propose a novel and unifying paradigm basedon the recently (re)introduced concept of ‘observationalentropy’ [16]. By starting with the latter as a defini-tion for the nonequilibrium thermodynamic entropy ofthe ‘universe’ (i.e., the system and the bath), we are ableto overcome various conceptual shortcomings of previousapproaches. Furthermore, it is a very flexible conceptallowing us to analyze a plethora of different situationswithin one approach.

Finally, while we believe that the second part reportson significant progress, it seems safe to conjecture thatthis is not the end of the story. Therefore, we finish ourarticle with an outlook and open questions.

III. PERTINENT CONCEPTS FROMSTATISTICAL MECHANICS

In this section we consider an isolated system. Thestate of the system is described by its density matrix ρand the Hamiltonian of the system is H(λt). Here, λtdenotes some externally specified driving protocol (e.g.,a changing electromagnetic field or the moving piston

in Fig. 1). The validity of modelling the dynamics of aquantum system via a time-dependent Hamiltonian restson the assumption that the work reservoir can be con-sidered as a classical and macroscopic device. While itis desirable to overcome this assumption, it neverthelessprovides a good tool to probe the nonequilibrium dy-namics of a quantum system and it is well justified formany practical applications. One might question whysystems with a time-dependent Hamiltonian are called‘isolated’, as they are clearly coupled to a work reservoir.Nevertheless, the dynamics of such a system is very dif-ferent compared to a system coupled to a heat bath andhence, it is useful not to call them ‘open’ either. Namely,the dynamics is governed by the Liouville-von Neumannequation (~ ≡ 1 throughout)

∂

∂tρ(t) = −i[H(λt), ρ(t)], (6)

where [A,B] = AB −BA denotes the quantum mechan-ical commutator. The time-evolution starting from aninitial state ρ(0) (we always set the initial time to bet = 0) is therefore unitary:

ρ(t) = U(t, 0)ρ(0)U†(t, 0). (7)

Here, the unitary time-evolution operator is defined asthe time-ordered exponential of the Hamiltonian, abbre-

viated as U(t, 0) = exp+[−i∫ t

0dsH(λs)].

A. Internal energy, heat and work

For an isolated system we identify the expectationvalue of its Hamiltonian with the internal energy knownfrom phenomenological thermodynamics,

U(t) ≡ U [ρ(t), λt] ≡ trH(λt)ρ(t), (8)

where tr denotes the trace operation. Note that we usu-ally write U(t) instead of U [ρ(t), λt] if the state ρ(t) isclear from the context. We wish to remark that the def-inition (8) is an assumption, but we are not aware ofany attempt to define internal energy differently. If thesystem is not driven (λt = 0), its internal energy is con-served, ∆U(t) = 0, where here and in the following weuse the notation ∆X(t) = X(t) − X(0) to denote thechange of any time-dependent function X(t). If the sys-tem is driven, its internal energy can change in time:

∆U(t) = trH(λt)ρ(t) − trH(λ0)ρ(0). (9)

Since the system is isolated (i.e., only coupled to a workreservoir), no heat is flowing (Q = 0) and we identify itschange in internal energy with the work supplied to thesystem:

∆U(t) = W (t). (10)

This is the first law of thermodynamics for an isolatedsystem. A straightforward calculation, using Eq. (6) and

4

that the trace is cyclic, shows that we can express thework also as

W (t) =

∫ t

0

dsd

dstrH(λs)ρ(s)

=

∫ t

0

dstr

∂H(λs)

∂sρ(s)

=

∫ t

0

dsW (s)

(11)

with the instantaneously supplied power W (s). There-fore, the identifcation of work in an isolated quantumsystem is quite general and rests solely on two assump-tions: Eqs. (8) and (10).

While this identification looks meaningful, one cannevertheless question its usefulness. Consider, for in-stance, our daily lunch box put into the microwave fora couple of minutes. Now, let us neglect the surround-ings of the microwave chamber, which is not needed forits functioning and which is only weakly coupled to theinside. Thus, to a good degree of approximation the mi-crowave chamber (consisting of the lunch box as well asthe air and the electromagnetic field inside the cham-ber) is an isolated system. After subjecting it to themicrowave radiation, its energy, including the one of ourlunch box, has increased. But after taking out the lunchbox, one has a sensation of an increased temperature:nobody calls this process “charging the food”, but ev-erybody says “heating the food”.

This (perhaps oversimplified) example demonstratesthe problem with identifying ∆U(t) = W (t) in an iso-lated many-body system: the work typically gets quicklydegraded into ‘useless’ heat as the excitations spreadout over many degrees of freedom. Therefore, while no-body would doubt that one has spent an amount of work∆U = W (t) > 0 to heat the food (see your electricitybill), the question is how much of this work is recoverableor still useful in a macrocopic sense. We will come backto this topic in Sec. VI.

B. Equilibrium states

We now consider only time-independent Hamiltoni-ans H, i.e., λt = constant. We write the stationarySchrodinger equation as

H|Ei, `i〉 = Ei|Ei, `i〉, (12)

where |Ei, `i〉 denotes an energy eigenstate with eigenen-ergy Ei and `i labels possible exact degeneraries. Fur-thermore, we are for the rest of this section only inter-ested in equilibrium states of the dynamics, i.e., states ρthat do not change in time. By the Liouville-von Neu-mann equation this implies that the state has to commutewith the Hamiltonian: [H, ρ] = 0. This is satisfied by allstates of the form

ρ =∑Ei,`′i

pEi,`′i |Ei, `′i〉〈Ei, `′i|, (13)

where pEi,`′i denotes the probability to find the system in

state |Ei, `′i〉. Note that the set |Ei, `′i〉`′i denotes anybasis is the subspace associated to energy Ei, not neces-sarily identical to the basis |Ei, `i〉`i chosen in Eq. (12).We now review three particular important classes of equi-librium states in statistical mechanics. How they can ef-fectively emerge in an isolated system at first place isaddressed elsewhere [17–21].

The first class is usually introduced via the equal apriori postulate where one assumes the following. Sup-pose the internal energy of the system presents the onlymacroscopic property we know about it. We denote itsvalue by E and we assume that it is known up to an un-certainty δ, which is experimentally more realistic for alarge system. We denote the corresponding projector as

ΠE =∑

Ei∈[E,E+δ)

∑`i

|Ei, `i〉〈Ei, `i|. (14)

One then postulates that, given the knowledge of E, allstates are equally likely within that energy shell of widthδ. This means that the equilibrium state is assumed tobe

ω(E) ≡ ΠE

VE, (15)

where the dimension or volume VE = trΠE of the en-ergy shell has two contributions: one coming from theimprecision of the measurement and one due to the ex-act degeneracy. Note that we leave the dependence on δimplicit in the notation. The state (15) is known as themicrocanonical ensemble.

For the second class of equilibrium states one also as-sumes the expectation value U = trHρ of the internalenergy to be fixed to some value E, but nothing moreis assumed. In particular, one does not fix the projec-tor ΠE , which was above assumed to be the only onecompatible with the measurement result E. Instead, theequilibrium state is postulated to be

π(β) ≡ e−βH

Z(β), (16)

where Z(β) = tre−βH is known as the partition func-tion and the inverse temperature β = T−1 (we set kB ≡ 1throughout) is fixed via the relation

E = trHπ(β) = − ∂

∂βlnZ(β). (17)

The state (16) is known as the canonical ensemble orGibbs state and to make it more plausible one usuallyinvokes one of the following two arguments. Either oneuses the maximum entropy principle [22, 23] to say thatthe canonical ensemble maximizes our ignorance (as mea-sured by the Gibbs-Shannon-von Neumann entropy in-troduced properly in the next section) or one refers to thecanonical ensemble as being completely passive [24, 25],which means that it is impossible to extract any work

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from it deterministically. More precisely, the state (16)is characterized by the fact that there exists no unitaryV such that

∆U = trHV ρV † − trHρ < 0 (18)

and this remains true even for a multipartite Gibbs stateπ(β) ⊗ · · · ⊗ π(β). Note that the argument of completepassivity does not fix the value of the inverse tempera-ture.

For many practical purposes, it turns out that the mi-crocanonical and canonical ensemble give identical pre-dictions if one adjusts the temperature associated to themicrocanonical ensemble correctly (see below). This isknown as the equivalence of ensembles. It holds gener-ically for short-range macroscopic systems, where thevolume VE grows quickly enough with increasing energyE [26]. Furthermore, it is well known that a local sub-system, which is weakly coupled to the rest of the overallisolated system, is described by a Gibbs state if the iso-lated system is described by a microcanonical ensemble.Remarkably, an even stronger statement holds, namelythat the majority of wave functions randomly picked fromthe microcanonical ensemble look locally identical to thereduced state of the microcanonical ensemble. This isknown as canonical typicality [27, 28].

Finally, a third class of equilibrium states arises whenone also includes fluctuations in particle numbers. Thecanonical ensemble is then generalized to

Ξ(β, ν) ≡ e−β(H−µN)

Z(β, µ), (19)

where N denotes the particle number operator (whichusually commutes with H), µ denotes the chemi-cal potential and the partition function Z(β, µ) =tre−β(H−µN) is again fixed by normalization. Thestate (19) is known as the grand canonical ensemble.

In principle, other ensembles can be important in equi-librium statistical mechanics, for instance, the pressureor volume ensembles, depending on the parameters whichare assumed to be fixed or fluctuating. Also for quantumthermodynamics it turns out that considering more gen-eralized Gibbs ensemble can give various insights, see,e.g., Refs. [29–31]. Although we will also consider gener-alized equilibrium states below, we will mostly focus onenergy and particle number as the fluctuating quantities.

C. Equilibrium entropies

Entropy is an important thermodynamic state func-tion, which was introduced in Sec. I in a phenomenolog-ical way. How to define it microscopically still remains asubject of debate, even at equilibrium. However, at leastif the equivalence of ensembles applies, the various no-tions of equilibrium entropies coincide numerically. Here,we review some basic notions of entropy, which are im-portant for the remainder of this paper.

Perhaps the most popular choice for entropy is

SvN(ρ) ≡ −trρ ln ρ. (20)

An adequate name for Eq. (20) is probably Gibbs-Shannon-von Neumann entropy, but—since we are inter-ested in quantum systems throughout this manuscript—we use a subscript ‘vN’ and mostly call it von Neumannentropy for brevity. It is defined for any quantum stateρ, even if it is out of equilibrium, but it has to be dis-tinguished from the notion of thermodynamic entropyin general. In fact, von Neumann himself confessed thatEq. (20) is “not applicable” for problems in statistical me-chanics [32] (for an English translation see [33]). The gen-eral reason is that Eq. (20) is too fine-grained, “computedfrom the perspective of an observer who can carry outall measurements that are possible in principle” [32, 33].This is also reflected in the fact that the von Neumannentropy is preserved for any isolated system:

SvN[ρ(t)] = SvN[U(t, 0)ρ(0)U†(t, 0)] = SvN[ρ(0)]. (21)

Therefore, if the von Neumann entropy were a legitimatecandidate for the thermodynamic entropy of an isolatedsystem, it would predict that all processes are reversible,which conflicts with the observation that most (perhapsall) processes observed in nature are irreversible. Never-theless, Eq. (20) is a useful concept at equilibrium (not tomention its many important applications in informationtheory [34, 35]). Together with the definition of inter-nal energy, Eq. (8), it yields to the standard textbookdefinition of equilibrium free energy,

F (β) ≡ U(β)− TSvN(β) = −T lnZ(β). (22)

Here, we used the shorthand notation U(β) = trHπ(β)and SvN(β) = SvN[π(β)] to denote the internal energyand von Neumann entropy of the Gibbs state π(β). Fur-thermore, the maximization of entropy, which yields thecanonical ensemble (16) for a fixed expectation value ofthe internal energy, is carried out with respect to the vonNeumann entropy. Finally, when applied to the micro-canonical ensemble (15), it becomes

SvN[ω(E)] = lnVE . (23)

The last equation brings us to a second commonlyemployed notion of equilibrium thermodynamic entropy,which we call the Boltzmann entropy. Focusing on thecase where only the value E of the internal energy isknown, the Boltzmann entropy

SB(E) ≡ lnVE (24)

counts all possible microstates compatible with E andtakes the logarithm of it. Clearly, there is a subjectivechoice involved when we say that a microstate is ‘compat-ible’ with a ‘known’ E. We here use the convention thatthe dimension of the Hilbert space spanned by the pro-jector (14) equals the number of compatible microstates,

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which equals VE . Although the von Neumann entropyfor a microcanonical state [Eq. (23)] equals the Boltz-mann entropy, both concepts should be clearly distin-guished because the Boltzmann entropy is only based ona counting argument. In particular, it is non-zero evenfor a pure state |ψ〉 = ΠE |ψ〉 contained in the energywindow [E,E + δ) whereas the von Neumann entropythen always equals zero: SvN[|ψ〉〈ψ|] = 0. Using theBoltzmann entropy, one can associate a temperature toa state with energy E via the definition

β =1

T=∂SB(E)

∂E. (25)

Using the equivalence of ensembles, this is identical to theinverse temperature of the canonical ensemble. The rea-son is that for macroscopic systems the volume VE , whichis related to the density of states in the limit δ → 0, of-ten grows very fast with E. When multiplied with e−βE

this results in a canonical distribution strongly peakedaround the value E = U(β). Finally, and in contrast toEq. (21), the Boltzmann entropy is in general not pre-served for unitary dynamics unless the Hamiltonian istime-independent. Nevertheless, if more fine-grained in-formation is available, the Boltzmann entropy conceptshows weaknesses. It can behave discontinuously in con-tinuous processes [36] and it is not clear how it can be ap-plied to the plethora of small systems studied in stochas-tic and quantum thermodynamics.

Finally, we mention a variant of the Boltzmann en-tropy, which is defined as

SG(E) = ln

E∑E′=0

VE′ , (26)

where, without loss of generality, we assumed that thelowest energy of the Hamiltonian is zero. In contrastto Eq. (24), which takes the logarithm of the number ofstates in an energy shell, Eq. (26) considers the numberof all states up to a fixed energy value E. For this rea-sons, Eq. (24) is also called the ‘surface entropy’, whereasEq. (26) is called the ‘volume entropy’ or the ‘Gibbs en-tropy’. The difference between SB(E) and SG(E) wasrecently debated in context of apparent negative tem-perature states because the definition (25) can give riseto negative temperatures whereas ∂SG(E)/∂E > 0 al-ways (for a non-exhaustive list of references on this topicsee [37–40]). We note that this problem is absent forsystems for which the equivalence of ensembles applies.

To conclude this section, even at equilibrium there arevarious candidates of entropy and the situation does notbecome simpler out of equilibrium (see, e.g., Ref. [41] fora recent discussion). In our point of view none of theabove candidates seems satisfactory out of equilibrium.Whereas the von Neumann entropy is too fine-grained,the Boltzmann entropy is too coarse-grained. The lastproblem is also shared by the Gibbs entropy, where itis furthermore less clear how to define it for observablesother than energy. In Sec. V we will therefore introduce

a different concept to microscopically define thermody-namic entropy out of equilibrium.

D. Early microscopic derivations of the second law:The Kelvin-Planck statement

As emphasized in Sec. II, providing exact microscopic(i.e., Hamiltonian) identities related to the second lawof thermodynamics has a long history. In connectionto the Hamiltonian dynamics of isolated systems, wewant to mention two early approaches to derive theKelvin-Planck formulation of the second law. One ap-proach is the statement of complete passivity, derived in1978 [24, 25]: Eq. (18) states that it is impossible to ex-tract work from a single heat bath in a cyclic process, i.e.,in a process where the Hamiltonian is the same at the be-ginning and at the end of the protocol, provided that theheat bath is initially modeled with a canonical equilib-rium state. Even one year earlier an exact identity forwork fluctuations was derived, which implies Eq. (18) forclassical systems [42]. In addition, another almost forgot-ten approach by Bassett [43] even includes a derivationof the second law for multiple heat reservoirs similar tothe approach reviewed below.

Probably since statistical mechanics was assumed toapply to macroscopic systems only, these identities cre-ated little experimental attention back at that time.However, with the recent progress in the design and con-trol of small quantum (and classical) systems, this ques-tion has gained new attention, which we are going toreview in the next section and extend afterwards.

IV. DISSIPATION IN OPEN QUANTUMSYSTEMS: PREVIOUS APPROACHES

We now turn our attention to open quantum systems.Following the standard system-bath paradigm, we modelthe bath as another (typically large) quantum systemcoupled to the system of interest. The global Hamilto-nian is taken to be of the form

HSB(λt) = HS(λt)⊗ 1B + 1S ⊗HB + VSB

= HS(λt) +HB + VSB ,(27)

where we suppressed tensor products with the identity inthe notation of the second line. Thus, HS (HB) denotesthe Hamiltonian of the unperturbed system (bath) andVSB denotes their interaction. Furthermore, in view ofthe work reservoir of Fig. 1 we allowed some externallyspecified driving protocol λt in the system part of theHamiltonian. Sometimes one also considers the interac-tion VSB = VSB(λt) to be time-dependent. This models,e.g., the effect of coupling and decoupling the system andthe bath. All results in this manuscript continue to holdin that case, but for ease of presentation we assume VSBto be time-independent in the notation. Furthermore,HB could denote not a single, but multiple baths and

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VSB then describes the interaction of the system witheach bath, respectively. Again, for simplicitiy in the pre-sentation we here assume that there is only a single bathalbeit many results can be extended to multiple baths.An explicit treatment of the multiple baths case is givenin Sec. VII B 1.

The joint system-bath state ρSB(t) evolves in time ac-cording to the Liouville-von Neumann equation (6) withrespect to the global Hamiltonian (27). In contrast, theevolution of the reduced system state

ρS(t) = trBρSB(t) (28)

is strikingly different: it is no longer unitary and the vonNeumann entropy of ρS(t) is no longer conserved. Find-ing a way to describe the dynamics of ρS(t) presents aformidable theoretical challenge [6]. Luckily, we are herenot concerned with this problem per se. Instead, we aimat unraveling some universal structure in the evolution ofopen quantum systems, which are related to the laws ofthermodynamics and hold for any open quantum system.

A. First and second law-like relations

During the past decades various exact identities havebeen derived to describe the dissipation in open classicaland quantum systems [43–78]. While differing in severalaspects, two common ingredients are the information-preserving character of the microscopic dynamics (con-servation of the Gibbs-Shannon-von Neumann entropy;sometimes also expressed in terms of Liouville’s theoremor time-reversal invariance) and a special choice of ini-tial state (typically related to a canonical Gibbs state),which breaks the time-reversal invariance. Furthermore,a common starting point is the definition of mechan-ical work (11), which—when applied to the Hamilto-nian (27)—becomes

W (t) =

∫ t

0

dstrS

∂HS(λs)

∂sρS(s)

. (29)

Note that the work is fully determined by knowledgeof the reduced system state only. Including a time-dependent interaction VSB(λt) is trivially possible basedon the definition (11), but it then no longer holds truethat the work can be computed using only ρS(t).

The question how to correctly identify heat, which fixesthe definition of internal energy via the first law, is moresubtle and requires a valid second law. Here we exem-plarily review the approach of Refs. [55, 56, 58], on whichalso the work of Refs. [59, 61–64, 67, 74, 76] is built. Theadvantage of this approach is that its derivation is rel-atively easy to follow while providing at the same timea quite powerful and general identity to describe dissi-pation in open quantum systems. Despite these merits,the approach also has conceptual and technical disadvan-tages, which we point out below and aim to eventuallyovercome in the second part of this paper. We note that

these shortcomings also remain for the other approachesmentioned above.

The only assumption we will add now is that the initialstate of the universe is given by

ρSB(0) = ρS(0)⊗ πB(β0), (30)

where ρS(0) is arbitary and πB(β0) denotes the canon-ical equilibrium state of the bath at inverse temper-ature β0, compare with Eq. (16). This assumptionis essential for the following and conventionally usedin open system theory [5, 6], attempts to overcomeit from a thermodynamic perspective can be found inRefs. [48, 53, 65, 66, 68, 70, 72, 75, 77, 78].

Next, we use Eq. (21), which guarantees SvN[ρSB(t)] =SvN[ρSB(0)]. Using furthermore Eq. (30) together withthe fact that SvN(ρ⊗ σ) = SvN(ρ) + SvN(σ) for any twostates ρ and σ [35], we confirm that

∆SvN[ρS(t)] + ∆SvN[ρB(t)] = I[ρSB(t)] ≥ 0. (31)

Here, we have introduced the always positive mutual in-formation I(ρSB) ≡ SvN(ρS) +SvN(ρB)−SvN(ρSB) [35].It is tempting to view Eq. (31) already as the entropyproduction: at least it is always positive and given bya change in a state function, namely the sum of the lo-cal von Neumann entropies. Indeed, if the ‘bath’ itselfis microscopically small, it was proposed to identify theheat flux directly via −β0Q = ∆SvN[ρB(t)] [68]. Then,Eq. (31) formally looks identical to the phenomenologicalexpression (5) if we identifiy the thermodynamic entropyof the system with SvN[ρS(t)]. However, for a mesoscopicor macrocopic heat bath this does not provide a satisfac-tory resolution as the mutual information can always bebounded by I(ρSB) ≤ 2 ln dim(HS), where dim(HS) de-notes the dimension of the system Hilbert space (whichwe assumed to be smaller than the bath dimension).Thus, e.g., for a two-level system the ‘entropy produc-tion’ (31) would be bounded from above for all times by2 ln 2, which is in general not the case. As a counterexam-ple consider, e.g., an open system subject to an intenselaser field constantly dissipating energy into its environ-ment. The entropy production in this case should ratherscale extensively with time (meaning that Σ ∼ t aftersome transient time).

Therefore, one employs a second important step bynoting the exact identity

SvN[ρB(t)]−SvN(β0) = β0∆EB−D[ρB(t)‖πB(β0)], (32)

where ∆EB ≡ trBHB [ρB(t) − ρB(0)] is the change inenergy of the bath alone andD[ρ‖σ] ≡ trρ(ln ρ−lnσ) isthe always positive relative entropy. Then, one identifies

Σ ≡ ∆SvN[ρS(t)] + β0∆EB ≥ 0 (33)

as the entropy production, here denoted by Σ to distin-guish it from our approach put forward later on. Thepositivity of Eq. (33) follows by confirming that

Σ = I[ρSB(t)] +D[ρB(t)‖πB(β0)]

= D[ρSB(t)‖ρS(t)⊗ πB(β0)].(34)

8

Upon further identifying (minus) the change in bath en-

ergy with the heat flux into the system, Q ≡ −∆EB ,which then also fixes the internal energy ∆U = W + Qvia the first law, one obtains Σ ≡ ∆SS − β0Q ≥ 0 inunison with Eq. (5). Note that Eq. (33) is bounded byterms of the order of ln dB , where dB denotes the bathdimension, and therefore it can grow extensively for allexperimentally relevant times.

The present approach has a number of advantages.Most importantly, it is quite general as it holds for ar-bitary system-bath dynamics and arbitary system andbath sizes based solely on assumption (30). Furthermore,

Σ has a transparent information theoretic interpretationand it has found widespread application in quantum ther-modynamics. While having these advantages, the herereviewed approach nevertheless fails to provide a fullysatisfactory picture for at least four reasons:

(A) Missing definition of thermodynamic en-tropy. The approach fails to relate the positivity ofEq. (33) to the traditional statement of the second law ofthermodynamics, namely the ‘increase of entropy’. Whentrying to express Σ = ∆Suniv(t) as a change in a state

function Suniv, which is supposed to define the thermo-dynamic entropy of the universe, one ends up only withdoubtful choices. Indeed, neglecting any additive con-stants, we get in order to ensure Σ(t) = ∆Suniv(t) thechoices

Suniv(t) = SvN[ρS(t)] + β0trHBρB(t) (35)

or

Suniv(t) = SvN[ρS(t)] + SvN[ρB(t)] +D[ρB(t)‖πB(β0)].(36)

But these definitions do not look very meaningful for athermodynamic entropy, see also the discussion aboveor Refs. [16, 41]. For instance, they depend explicitlyon the initial inverse bath temperature β0, which seemsawkward as the formal definition of any known thermo-dynamic entropy does not depend on it (clearly, if ρ it-self depends on β0, then so does also, e.g., the Gibbs-Shannon-von Neumann entropy, but a priori the defi-nition −trρ ln ρ is free from any temperature depen-dence). Furthermore, the system entropy is identifiedwith its von Neumann entropy independent of the factwhether, e.g., the system contains 1 or 1023 spins. Thatthis can not be a satisfactory definition in general wasdiscussed in Sec. III C.

(B) Restricted initial state. While the initial sys-tem state can be arbitrary, the bath is strictly required tobe a canonical equilibrium state. While this is a conve-nient mathematical choice, we believe that the second lawshould also hold for more general states of the bath (e.g.,a microcanonical state). The question whether it shouldalso hold for pure initial states was recently discussed inRefs. [79–81] and will be also addressed in Sec. VII C.

(C) Questionable identification of heat. While Σcan be connected to energy changes ∆EB in the bath,it should be noted that additional care is required when

identifying it with ‘heat’, which is still a much debatedconcept at the moment (for additional model-dependentstudies on this topic see Refs. [82–94]).

(D) Unwanted generality. In phenomenologicalthermodynamics, the second law (5) is a consequence ofthe more general second law (4), which again comes fromthe more general form (3), which is ultimatively impliedby Eq. (2). But here we have directly derived Eq. (5)based on almost no assumption.

To conclude, while the derivation of Eq. (33) is elegant,its literal physical interpretation as the entropy produc-tion seems doubtful in general. Put differently, despitebeing a very useful inequality, its mathematical general-ity comes at the cost of a resticted physical scope. Thegoal of this paper is to propose a framework, which con-tains the above approach as a particular case (which willturn out to be the case of a small system coupled to aweakly perturbed bath), but overcomes the problems (A)to (D). This requires a conceptually different approach tothe problem. Before we come to it, we shortly review an-other powerful identity related to the fluctuations of Σ.Readers less interested in fluctuations theorems, whichonly appear again in Secs. VII A 4 and briefly at the endof Sec. V C, can skip this subsection.

B. Fluctuation theorems

Fluctuation theorems can be seen as refinements of thesecond law or related dissipation inequalities by providingexact identities for the fluctuations in dissipation and en-tropy production. They play an important role in classi-cal nonequilibrium statistical mechanics [95–97], stochas-tic thermodynamics [98, 99], and also in the quantumregime based on the so-called ‘two-point measurementscheme’ [100, 101].

In general, fluctuation theorems are of the followingform. Suppose that Σ ≥ 0 describes the dissipation ofsome process. In every single run of an experiment, thedissipation can, however, fluctuate around its averageΣ = 〈σ〉. Here, 〈. . .〉 denotes an ensemble average ofthe stochastic dissipation σ along a single stochastic re-alization of the same process (examples are given below).Let p(σ) denote the associated probability distribution ofthe stochastic dissipation. We say that p(σ) obeys an in-tegral fluctuation theorem if⟨

e−σ⟩

=∑σ

p(σ)e−σ = 1. (37)

Note that the inequality Σ = 〈σ〉 ≥ 0 follows fromEq. (37) by use of the inequality ex ≥ 1 + x. Further-more, it often turns turns out that one can associate aconjugate process to the original process, denoted by a‘dagger’ †, with a probability distribution of dissipationp†(σ). This conjugate process is often (but not always)related via time-reversal to the original process. Then,

9

the so-called detailed fluctuation theorem asserts that

p(σ)

p†(−σ)= eσ. (38)

Obviously, Eq. (38) implies Eq. (37), but the reverse isnot true. The detailed fluctuation theorem is therefore astronger statement than the integral fluctuation theorem.

We now derive the integral fluctuation theorem associ-ated to Eq. (33), see also Refs. [46, 102–104]. We empha-size that the derivation is rather formal and requires pre-cise knowledge of the system and bath state. However,similar manipulations are used throughout the literatureand also turn out to be useful below, where we derive aquantum fluctuation theorem that does not require pre-cise knowledge of the system-bath state and therefore, itshould be experimentally easier to confirm.

We start by writing the system state in its eigenbasis

ρS(t) =∑α

pα(t)|αt〉〈αt|S (39)

at time t. Furthermore, the Hamiltonian of the bathis decomposed as HB =

∑k εk|εk〉〈εk|B , where we as-

sumed for simplicity that the energy eigenstates arenon-degenerate. In spirit of the two-point measurementscheme [100, 101], we now assume to measure the system-bath composite initially in its eigenbasis |α0〉S⊗|εk〉B ≡|α0, εk〉 and finally at time t in the basis |αt, εk〉. Notethat this is in general not the eigenbasis of the jointsystem-bath state at time t. The process is describedby the probability distribution

p[αt, ε′k;α0, εk]

= |〈αt, ε′k|USB(t, 0)|α0, εk〉|2〈α0, εk|ρSB(0)|α0, k〉

= |〈αt, ε′k|USB(t, 0)|α0, εk〉|2pα(0)e−βεk

ZB.

(40)

The fluctuations of Σ are defined as

σ = −[ln pα(t)− ln pα(0)] + β(ε′k − εk) (41)

such that Σ = 〈σ〉. Furthermore, they satisfy

e−σpα(0)e−βεk = pα(t)e−βε′k . From this relation the in-

tegral fluctuation theorem follows:⟨e−σ

⟩=

∑αt,ε′k;α0,εk

p[αt, ε′k;α0, εk]e−σ (42)

=∑

αt,ε′k;α0,εk

|〈αt, ε′k|USB(t, 0)|α0, εk〉|2pα(t)e−βε

′k

ZB

=∑αt,ε′k

pα(t)e−βε′k

ZB= 1.

More on integral fluctuation theorems can be found inSecs. V C and VII A 4. The detailed fluctuation theoremwill appear in Appendix A again.

V. OBSERVATIONAL ENTROPY

We now come to a central part of this paper, where weintroduce a microscopic notion of entropy, which we willidentify with the thermodynamic entropy from Sec. VIon, even for out-of-equilibrium situations. In this andthe next section we will forget the system-bath paradigmfor a moment and simply consider an arbitrary isolatedsystem in a state ρ with Hamiltonian H as in Sec. III.

A. Basic definition

Let Πx denote a set of projectors associated tosome observable X =

∑x xΠx. The projectors satisfy

ΠxΠx′ = δx,x′Πx, and∑x Πx = 1 (where 1 is the iden-

tity operator). Furthermore, we denote by Vx ≡ trΠxthe dimension of the projector Πx, which can be largerthan one and, in fact, will often be much larger than onefor thermodynamically relevant observables. The obser-vational entropy with respect to Πx is then defined as

SXobs(ρ) ≡∑x

px(− ln px + lnVx), (43)

where px = trΠxρ is the probability to observe out-come x. Equation (43) describes the usual Shannon en-tropy with respect to the probability distribution px plusthe average remaining uncertainty

∑x px lnVx reflecting

our ignorance about not knowing the precise microstateafter receiving the outcome x. In that sense, Eq. (43)interpolates between the standard Gibbs-Shannon-vonNeumann entropy (20), obtained by choosing X = ρ,and the Boltzmann entropy (24), obtained when the sys-tem is restricted to an energy shell and our measurementdoes not reveal any finer information about it. In general,observational entropy is bounded by [16]

SvN(ρ) ≤ SXobs(ρ) ≤ ln dimH, (44)

where dimH denotes the dimension of the Hilbert spaceof the isolated system. Below, whenever it is clear fromcontext, we will write SXobs(ρ) = SXobs. We remark that weleave the choice of the observable X completely generalhere. Which observables are of relevance to quantumthermodynamics will be explored from Sec. VI on.

B. Historical remarks

Definition (43) or, more often, similar but less generalforms of it appear scattered throughout the literature onstatistical mechanics. Not seldomly Eq. (43) is used invarious computations without, however, explicitly iden-tifying it with thermodynamic entropy, in particular notout of equilibrium. Our efforts to trace back the ori-gin and use of definition (43) has yielded the followingresults, which shall not imply that the given list is ex-haustive.

10

First, in a quantum mechanical context, Eq. (43) canbe traced back to von Neumann, who attributes it toa personal communication from Wigner and clearly ac-knowledges its usefulness for problems in statistical me-chanics [32, 33]. For classical systems, where the mea-surement can be regarded as dividing the phase spaceinto small cells, variants of Eq. (43) appear already in thework of Gibbs [105] and Lorentz [106], see also Sec. 23aof the treatise about statistical mechanics of the Ehren-fests [107]. In this context, Eq. (43) is also known as“coarse-grained entropy” (see, e.g., Wehrl [108], who con-nects it to ergodicity and mixing and cites further ref-erences). Coming back to the quantum case, we notethat an increase of the quantity defined in Eq. (43) wasproven in §106 of Tolman’s book [109] and Sec. 1.3.1of the book by Zubarev et al. [110]. This statement isbasically identical to our Lemma V.4 below and in theclassical case it also seems to date back to Gibbs, seeagain the Ehrenfests [107]. It is, however, interesting tonote that both books refuse to use Eq. (43) as a defi-nition of thermodynamic entropy for out-of-equilibriumprocesses: Tolman discusses the connection to thermo-dynamic entropy only at equilibrium and prefers to usethe Gibbs-Shannon-von Neumann entropy [compare withEq. (122.10) therein] and Zubarev et al. prefer the Gibbs-Shannon-von Neumann entropy of a generalized out-of-equilibrium Gibbs ensemble. Moreover, definition (43)was used in Ref. [111, 112] and it also appears in Sec. 3.5of the review article by Penrose [113] alongside other pos-sible definitions. Finally, an increase of the quantity (43)was noted for various Pauli-like master equations describ-ing the coarse-grained Markovian dynamics of an isolatedsystem [108, 109, 113–118]. Despite these various contri-butions, Eq. (43) was only recently put back into the fo-

cus by Safranek, Deutsch and Aguirre [16], who proposedit as a generally valid definition of thermodynamic en-tropy for isolated nonequilibrium quantum systems andcoined the terminology “observational entropy.” Fur-ther compelling arguments for it are also given in theirsubsequent work [36, 119–121]. In there, one can alsofind extensions of Eq. (43) to multiple measurementsΠx,Πy, . . . . In this case, the order of the measurements

matters since [Πx,Πy] 6= 0 in general. However, Safraneket al. argue that for typical situations encountered inthermodynamics the effect of the non-commutativity be-comes very small. We will not be concerned with thisproblem here.

Instead, we are interested in applying the concept ofobservational entropy to the emerging field of quantumthermodynamics, where it was not yet applied. In con-trast to Safranek et al. [16, 36, 119, 120], who study iso-lated quantum systems, which are initialized out of equi-librium and then relax back to equilibrium (in a macro-scopic sense) for a constant Hamiltonian H, we look atthe opposite situation. We consider isolated systems andsystem-bath setups initialized in equilibrium (at least insome particular sense, see below), which are then drivenout of equilibrium, either by changing a time-dependent

field or by having multiple heat baths initialized at dif-ferent temperatures (or both). We use the concept ofobservational entropy to define (ir)reversibility, to iden-tify heat and work and to connect it to the notion ofentropy production.

C. Useful observations about observational entropy

We here list a couple of facts as Lemmas, which addfurther appeal to the definition of observational entropyand which will be repeatedly used below. These Lemmashold for any set of projectors Πx and therefore might beof interest even outside thermodynamic considerations.

First, observational entropy is extensive in the limitwhere one expects it to be extensive.

Lemma V.1. Consider a composite system in the stateρ = ρ1⊗· · ·⊗ρn. If we measure on each subsystem j theobservable Xj, then

SXobs(ρ) =

n∑j=1

SXjobs(ρj). (45)

Proof. See Ref. [119].

Next, we note a useful rewriting of observational en-tropy:

Lemma V.2. Let ρ(x) ≡ ΠxρΠx/px be the post-measurement state associated to outcome x and letω(x) ≡ Πx/Vx denote the ‘microcanonical ensemble’given the constraint x. Then,

SXobs = SvN

[∑x

pxρ(x)

]+∑x

pxD[ρ(x)‖ω(x)]. (46)

Proof. Since the states ρ(x) have support on orthogonalsubspaces, it follows from Theorem 11.10 in Ref. [35] that

SvN

[∑x

pxρ(x)

]=∑x

px SvN[ρ(x)]− ln px . (47)

Using this insight in Eq. (46) yields

SXobs = −∑x

px [ln px + trρ(x) lnω(x)]

= −∑x

px [ln px − trρ(x) lnVx] ,(48)

which is identical to Eq. (43) since trρ(x) = 1.

The next lemma characterizes the states ρ which havethe same von Neumann and observational entropy (seealso Ref. [119]).

Lemma V.3. We have SvN(ρ) = SXobs(ρ) if and only if

ρ =∑x

pxω(x) (49)

for an arbitary set of probabilities px.

11

Proof. Using Eq. (46), we can write

SXobs − SvN(ρ) = SvN

[∑x

pxρ(x)

]− SvN(ρ)

+∑x

pxD[ρ(x)‖ω(x)].

(50)

Thus, SXobs − SvN(ρ) is given as the sum of two non-negative terms. It follows from Theorem 11.9 in Ref. [35]that

SvN

[∑x

pxρ(x)

]− SvN(ρ) ≥ 0 (51)

with equality if and only if ρ =∑x pxρ(x). Furthermore,

D[ρ(x)‖ω(x)] = 0 if and only if ρ(x) = ω(x). Hence,Eq. (49) follows.

The fourth lemma can be already seen as a precursorof the second law of thermodynamics, albeit the projec-tors Πx are still arbitrary and not necessarily of ther-modynamic relevance. Furthermore, since we are nowinterested in changes in observational entropy, we explic-itly write SXtobs(t) for the observational entropy at time tand indicate that also the chosen observable X = Xt candepend on time.

Lemma V.4. If SX0

obs(0) = SvN[ρ(0)], then

∆SXtobs(t) = SXtobs(t)− SX0

obs(0) ≥ 0. (52)

Moreover, the change in observational entropy

∆SXtobs(t) = Σqm(t) + Σcl(t) (53)

can be divided into a quantum and classical part, whichare both separately non-negative:

Σqm(t) = SvN

[∑xt

pxtρ(xt, t)

]− SvN[ρ(t)] ≥ 0, (54)

Σcl(t) =∑xt

pxtD[ρ(xt, t)‖ω(xt)] ≥ 0. (55)

Here, we defined ρ(xt, t) ≡ Πxtρ(t)Πxt/pxt .

Proof. Since SvN[ρ(0)] = SvN[ρ(t)] due to the unitarytime evolution, we have

∆SXtobs(t) = SXtobs(t)− SvN[ρ(t)]. (56)

This term is easily shown to be positive using Eqs. (46)and (51), see also Ref. [32] and Theorem 3 in Ref. [119].The splitting into Eqs. (54) and (55) follow from Eqs. (46)and their positivity is evident [cf. Eq. (51)].

We remark that the splitting into the quantum andclassical part is unambiguous. In fact, the quantum partcoincides with what is known as the ‘relative entropyof coherence’, which plays an important role in resource

theories of quantum coherence [122]. If we denote thedephasing operation by DXρ ≡

∑x ΠxρΠx, we can write

Σqm(t) = D [ρ(t) ‖DXtρ(t) ] . (57)

This confirms that Σqm(t) quantifies a true quantum fea-ture, namely the distance between the actual state of thesystem and its dephased (‘classical’) counterpart. Fur-thermore, by evaluating the trace in Eq. (55) in the eigen-basis of ρ(xt, t) =

∑i(xt)

pi(xt)|i(xt)〉〈i(xt)|, we see that

D[ρ(xt, t)‖ω(xt)] =∑i(xt)

pi(xt)

(ln pi(xt) − ln

1

Vxt

).

(58)Thus, Eq. (55) describes a weighted average of the rela-tive entropies between ρ(xt, t), seen as a classical mixtureof microstates, and the maximally mixed probability dis-tribution 1/Vxt . This uncertainty is entirely classical inorigin.

The final lemma presents a constraint on how observa-tional entropy can fluctuate along single realizations of anexperiment. We call it the integral fluctuation theoremfor observational entropy.

Lemma V.5. If SX0

obs(0) = SvN[ρ(0)], then⟨e−∆s

Xtobs(t)

⟩= 1. (59)

Here, ∆sXtobs(t) = sXtobs(t) − sX0

obs(0) is the change instochastic observational entropy

sXtobs(t) ≡ − ln pxt + lnVxt . (60)

Furthermore, 〈. . .〉 ≡∑xt,x0

. . . pxt,x0denotes an ensem-

ble average over the joint probability distribution

pxt,x0= trΠxtU(t, 0)Πx0

ρ(0)Πx0U†(t, 0) (61)

to get the measurement results x0 and xt.

Proof. From −∆sXtobs(t) = ln(pxtVx0/Vxtpx0

) and theassumption ρ(0) =

∑x0px0

Πx0/Vx0

, which implies

Πx0ρ(0)Πx0

= px0Πx0

/Vx0, we get the following chain

of equalities:⟨e−∆s

Xtobs(t)

⟩=∑xt,x0

trΠxtU(t, 0)Πx0U†(t, 0) pxt

Vxt

=∑xt

trΠxtU(t, 0)U†(t, 0) pxtVxt

=∑xt

trΠxtpxtVxt

= 1.

(62)

For the last steps we used∑x0

Πx0 = 1, U(t, 0)U†(t, 0) =

1, trΠxt = Vxt , and∑xtpxt = 1.

Remember that the integral fluctuation theorem (59)implies the formal second law (52). Furthermore, an evenmore general class of integral fluctuation theorems, whichalso imply Eq. (59), was recently derived in Ref. [123].Finally, it is also possible to derive an associated detailedfluctuation theorem. As we are in the main text onlyconcerned with integral fluctuation theorems, we shiftits derivation to Appendix A.

12

VI. RECOVERABLE WORK ANDREMAINING HEAT

In this section, we still focus on the case of an isolatedsystem with Hamiltonian H(λt) and state ρ(t). Indeed,it is instructive to consider this preliminary step first be-fore turning to open systems, which additionally assumea tensor product structure between the system and thebath. Note that deriving the laws of thermodynamicsfor isolated systems has attracted less attention withinthe recent context of quantum thermodynamics, but seeRefs. [24, 25, 42, 43, 124–128] for a selection of various ap-proaches. These approaches were not based on the notionof observational entropy, but we remark that in their re-spective regime of validity (e.g., initial Gibbs ensembles,macroscopic systems able to effectively self-equilibrate)our results are in unison with them. In addition, how-ever, we introduce the concepts of recoverable work andremaining heat, which we will also use in Sec. VII.

Let us now reconsider the problem mentioned inSec. III A in light of observational entropy. We foundthere that the identification ∆U(t) = W (t) has to bemet with care. For instance, if ∆U(t) > 0, we knowhow much work was spent in the process, but it does nottell us how much of this work can be recovered after theprocess, i.e., how much of the energy ∆U(t) is stored asaccessible and valuable work in the system. Vice versa, if∆U(t) < 0, one concludes that one has extracted work,but this identification requires that the amount ∆U(t)transfered to the macroscopically large work reservoir canbe easily accessed there. If ∆U(t) is tiny, this can becomepractically impossible. In the following we focus on thecase W (t) ≥ 0 for definiteness.

To substantiate our reasoning, remember the stan-dard argument of how to revover the work for an arbi-trary state change by using time-reversal symmetry. Letρ(t) = U(t, 0)ρ(0)U†(t, 0) be the time-evolved state in theforward process with an associated work cost W (t) ≥ 0.Now, let Θ denote the anti-unitary time-reversal oper-ator and let UΘ(t, 0) = ΘU†(t, 0)Θ denote the unitarytime evolution operator generated by the HamiltonianHΘ(λs, B) = H(λt−s,−B) with a time-reversed drivingprotocol and, perhaps, a reversed magnetic field B (seeAppendix B for a brief introduction to time-reversal sym-metry). Finally, let Θρ(t)Θ denote the time-reversed fi-nal state of the forward process. Then, by definition ofthe time-reversed process we have the identity

ρ(0) = ΘUΘ(t, 0)Θρ(t)ΘU†Θ(t, 0)Θ. (63)

Thus, operationally speaking, we recover the sameamount of internal energy W (t) = ∆U(t) spent in theforward process if we time-reverse the final state ρ(t),let the protocol λt run backward (and perhaps reverse amagnetic field) and time-reverse the state again. Clearly,the easy part here is to implement a time-reversed drivingprotocol λt and a reversed magnetic field. The hard partinstead corresponds to time-reversing the state ρ(t) be-cause most states are not symmetric under time-reversal,

i.e., ρ(t) 6= Θρ(t)Θ. Thus, an implementation of Eq. (63)remains experimentally out of reach in most situations.

Note that this argument provides another demonstra-tion why the Gibbs-Shannon-von Neumann entropy isnot a legitmate candidate for thermodynamic entropyout of equilibrium. It does not change during the pro-cess and therefore, one would conclude that the processis reversible, which implies that it should be easy to time-reverse.

To circumvent the above problem, it seems that anincreasing part of the quantum thermodynamics com-munity favors a more inclusive approach. This is basedeither on an explicitly modeled work storage device (i.e.,a quantum battery) [129–131] or on autonomous heat en-gines such as thermoelectric devices or quantum absorp-tion refrigerators [132–134]. While this is an importantresearch direction, we here rather want to show an alter-native way offered by observational entropy to identifythe ‘useful’ part of work. This has the benefit that itdoes not require us to explicitly model a work storagesystem, which can be theoretically and experimentallydemanding.

A. Equilibrium states and reversibletransformations

We now take the energy H(λt) as our observable, butto take into account the fact that it is impossible to mea-sure the single microstates |Ei, `i〉 defined in Eq. (12) fora many-particle system, we consider the coarse-grainedprojectors ΠEt from Eq. (14). The subscript t indicatesthat they parametrically change in time as H = H(λt)does and we tacitly assume that the measurement un-certainty δ > 0, which we leave implicit in the notation,is chosen small enough to avoid trivial situations and tomake contact with traditional thermodynamics. We willdiscuss in Sec. VIII how small one has to choose δ in areal experiment, but for the moment we assume it to be atheoretical parameter chosen at our convenience. Thus,the observational entropy becomes

SEtobs(t) = −∑Et

pEt lnpEtVEt

(64)

with VEt = trΠEt. This will be our microscopic defini-tion for the thermodynamic entropy of an isolated many-body system in this section.

Next, we introduce the set of equilibrium states with re-spect to SEtobs by demanding that they are precisely those

states ρ that satisfy SEtobs = SvN(ρ). Notice that thesestates are characterized by the fact that any measure-ment backaction with respect to ΠEt is absent. We de-note this set of states by Ω(λt) and from Lemma V.3 weknow that it can be written as

Ω(λt) =

∑Et

pEtω(Et)

∣∣∣∣∣ pEt arbitrary

. (65)

13

This corresponds to a somewhat larger set of equilibriumstates than conventionally considered in statistical me-chanics, but the states share the same feature: they areinvariant in time for a fixed Hamiltonian H(λt) and rep-resent maximum ignorance given the distribution pEt .

Now, consider that we start with such an equilibriumstate ρ(0) ∈ Ω(λ0) at time t = 0. We call the processthermodynamically reversible, if its change in observa-tional entropy with respect to the energy observable nul-lifies:

∆SEtobs(t) = SEtobs(t)− SE0

obs(0) = 0. (66)

By Lemma V.3 and V.4 this implies that also the fi-nal state must be an equilibrium state: ρ(t) ∈ Ω(λt).The probability distribution pEt must obey the con-straint (66), but it is otherwise arbitrary. Our claim isnow that any energy change ∆U(t) > 0 due to a re-versible process as defined by Eq. (66) can be easily re-covered and under these circumstances it is meaningfulto speak about ∆U(t) = W (t) as the work stored in thestate Ω(t).

For this purpose we come back to Eq. (63). We con-cluded there that reversing a driving protocol λt can beconsidered as an easy experimental task, but reversingthe final state from ρ(t) to Θρ(t)Θ is experimentally hard.However, there is one set of states for which this oper-ation is easy to achieve and this is precisely the set ofequilibrium states characterized by Eq. (65), which fur-ther justifies their terminology. Symbolically, we can de-note this by

ΘΩ(λt)Θ = Ω(λt) (67)

or ΘΩ(λt, B)Θ = Ω(λt,−B) in presence of a magneticfield. In words, an equilibrium state is invariant undertime-reversal and hence, there is actually no need to im-plement the cumbersome time-reversal operation (apartfrom perhaps flipping B). Thus, our approach based onobservational entropy precisely shows that reversible pro-cesses are characterized by the fact that they are simpleto time-reverse from a macrocopic point of view. Thus,any amount of work W (t) ≥ 0 invested in a reversible for-ward process can be easily recovered in the time-reversedprocess.

Note that this statement is not a trivial tautology. Ifwe had started with a different notion of thermodynamicentropy, it is far from clear whether this would imply thesame statement (for the Gibbs-Shannon-von Neumannentropy this is, for instance, not the case). Furthermore,while we will be mostly interested in nonequilibrium sit-uations, the reversible limit is not unimportant. One im-portant class of transformations, which falls into this cat-egory, are adiabatic transformations achieved by chang-ing the protocol λt very slowly.

B. Irreversible transformations and remaining heat

In general, even if we start from an equilibrium stateρ(0) ∈ Ω(λ0), the final state ρ(t) will not be an equi-librium state and the process will be irreversible, i.e.,characterized by an increase of observational entropy,∆SEtobs > 0, see Lemma V.4. Our goal is now to splitthe change in internal energy (9) into a part Wrec, whichis recoverable work in a macroscopic sense, and some re-maining heat Qrem:

∆U(t) = Wrec(t) +Qrem(t). (68)

Here, by “recoverable in a macrocopic sense” we meanthe following. First, the only information we are allowedto use about the system state ρ(t) is encoded in the ob-

servational entropy SEtobs(t) (if we had more information,that would be equivalent to choosing a different, morefine-grained observational entropy). Second, the onlyway we can control the Hamiltonian of the system is viathe specified driving protocol λt and perhaps some otherexternal fields such as a global magnetic field. Third, wealso allow that we can bring the system into weak contactwith an ideal superbath at temperature T , which inducesequilibration on the system and causes heat exchanges.

For any nonequilibrium state ρ(t) we now introducean effective nonequilibrium temperature T ∗t = T ∗[ρ(t)],which, roughly speaking, corresponds to the temperatureof its respective equilibrium state with the same internalenergy. More precisely, we define T ∗t to be the temper-ature T of a superbath such that no net heat exchangewill take place when coupling the system to it. This isan operationally well-defined temperature and it is alsounique if the heat exchanged between the system and thesuperbath is a monotonic function of the temperatureT . Note that similar constructions (called “nonequilib-rium contact temperatures”) are used for a long time inphenomenological nonequilibrium thermodynamics [135],also see, e.g., Ref. [128, 136].

In certain pathological situations, the nonequilibriumtemperature might not be defined, for instance, for anassembly of spins all aligned anti-parallel to an exter-nal magnetic field (also called a ‘negative temperature’state). However, by flipping the magnetic field B 7→ −B,we can deterministically change the energy of this stateand end up in a situation where T ∗t is again well-defined.In general, we believe that the construction of counterex-amples requires precise microscopic control over the ini-tial state ρ(0) or the Hamiltonian H(λt), which is beyondthe scope of our general (rather coarse-grained) discus-sion in this section. Thus, we here assume that our exper-imental capabilities are limited. In particular, we assumethat we can not implement any unitary V we like in or-der to deterministically extract any possible amount ofwork trH(λt)[V ρ(t)V † − ρ(t)] from the final nonequi-librium state. We will return, however, to this situationin Sec. VII A 2.

14

We proceed by defining

Wrec(t) ≡ ∆U(t)−∫ t

0

T ∗s dSEsobs(β

∗s ), (69)

Qrem(t) ≡∫ t

0

T ∗s dSEsobs(β

∗s ), (70)

where SEsobs(β∗s ) denotes the observational entropy of

an equilibrium state at effective inverse temperatureβ∗s = 1/T ∗s . Remember that the observational entropySEobs(β

∗s ) coincides with the equilibrium entropies from

Sec. III C if the measurement uncertainty δ is smallenough. To be rigorous, we assume from here on thatthe equivalence of ensembles applies.

It remains to be shown that Wrec(t) is really recover-able in the macroscopic sense as specified above. This canbe done by coupling the system at time t to a superbathwith the temperature T ∗t . This brings the final nonequi-librium state ρ(t) to its associated equilibrium stateπ(β∗t ). Since by definition there is no net heat exchangebetween the system and the superbath, the system’s in-ternal energy does not change, i.e., U [ρ(t)] = U [π(β∗t )].Now, we implement a process in which we change theprotocol λt back to its initial value and successively cou-ple the system to superbaths at the reverse order of thetemperatures T ∗t . This is called a Clausius-Duhem pro-cess, see Ref. [45] for a detailed microscopic analysis andFig. 2 for a sketch. In the reversible limit, equilibriumthermodynamics predicts [cf. Eq. (4) in the limit Σ = 0]

SEobs(β∗0)− SEobs(β

∗t ) =

∫ t

0

dQCD(s)

T ∗s, (71)

where dQCD(s) is an infinitesimal heat flux at time sabsorbed from the system during the Clausius-Duhemprocess according to the reversed protocol λs 7→ λt−s.By comparison with Eq. (70) we find that the remainingheat introduced above is identical to minus the heat ab-sorbed in this ideal, reversible Clausius-Duhem process.Consequently, the maximum amount of extractable workfollows from the first law and is given by minus Eq. (69).

We have checked above already that Qrem(t) = 0 fora reversible process. We now show that Qrem(t) > 0 for

an irreversible process with ∆SEtobs > 0 if we start withan equilibrium state with initial temperature T0. Thisguarantees that we cannot extract more work than whatwe have invested in a cyclic process in accordance withthe Kelvin-Planck statement of the second law. From∆SEtobs(t) > 0 together with definition (70) we get

∆SEtobs(t) = SEtobs(t)− SEtobs(β

∗t ) + SEtobs(β

∗t )− SEtobs(β0),

= SEtobs(t)− SEtobs(β

∗t ) +

∫ t

0

dQrem(s)

T ∗s> 0,

(72)

where the difference SEtobs(t)−SEtobs(β

∗t ) takes into account

the nonequilibrium nature of the final state. Since by

Figure 2. Sketch of the Clausius-Duhem process. First, thesystem evolves under an arbitrary nonequilibrium processρ(0) 7→ ρ(t) = U(t, 0)ρ(0)U†(t, 0) (thick line). To computethe recoverable work and remaining heat, the isolated systemafter time t is put into contact with an ideal superbath attemperature Tt = T ∗t , which causes changes in entropy butnot in internal energy (dotted arrow). Afterwards, followingan ideal Clausius-Duhem process, the system is returned toits initial equilibrium state (dashed arrow). This process pro-ceeds entirely within the manifold of equilibrium states (greyarea).

construction β∗t is defined by the requirement of zero en-

ergy changes, we must have SEtobs(t) − SEtobs(β

∗t ) ≤ 0 be-

cause entropy gets maximized at equilibrium. We there-fore get the chain of inequalities

Qrem(t)

mins∈[0,t]T ∗s ≥∫ t

0

dQrem(s)

T ∗s

> SEtobs(β∗t )− SEtobs(t) ≥ 0.

(73)

This proves Qrem(t) > 0 since the absolute temperatureT ∗s > 0 is positive for all s ∈ [0, t].

We briefly summarize this section. If we can only mea-sure the coarse-grained energy of an isolated many-bodysystem, then SEobs as defined in Eq. (64) provides a goodcandidate for thermodynamic entropy. For the large classof initial states (65) its change is never non-negative. A

reversible process satisfies ∆SEtobs(t) = 0, which implies

ρ(t) ∈ Ω(λt) ⇔ Wrec(t) = W (t) ⇔ Qrem(t) = 0. (74)

An irreversible process starting at a well-defined temper-ature T0 is characterized by Qrem(t) > 0.

VII. SECOND LAW FOR OPEN SYSTEMS

A. Single heat bath

We now return to the situation reviewed in Sec. IVof a driven system in contact with a single heat bath.

15

Multiple heat baths are discussed in Sec. VII B 1. Fur-thermore, although our identities remain valid for a sys-tem and a bath of any size, we have in mind the typicalsituation of open quantum system theory, which is alsoof relevance for quantum thermodynamics and many ex-periments. This means that we have a small system,which we can control precisely, in contact with a largebath about which we have only coarse-grained informa-tion. Here, the notions “small” and “large” as well as“precise” and “coarse-grained” have to be understoodin a relative sense, of course, and crucially depend onthe experimental capabilities. A major advantage of thepresent approach based on observational entropy is thatit is naturally formulated in a way, which allows to takedifferent levels of knowledge into account.

As the thermodynamically relevant observables wechoose an arbitrary system observable St, which can de-pend on time, and the bath energy EB , which is time-independent by construction (see also Sec. IV). To mea-sure the energy EB , we assume as in Sec. VI A a coarse-grained measurement with uncertainty δ. Then, the ob-servational entropy reads in general

SSt,EBobs (t) = −∑st,EB

pst,EB (t) lnpst,EB (t)

Vst,EB(75)

with pst,EB (t) = trSBΠstΠEBρSB(t) and Vst,EB =trSBΠstΠEB. Note that, since we measure the twoobservables St and EB on two different Hilbert spaces,the time-ordering of the measurement does not matter.Furthermore, in view of the agreement above, we assumethat the observable St =

∑stst|st〉〈st| is fine-grained,

i.e., composed out of rank-1 projectors. This is not nec-essary, but it makes the following exposition easier tofollow. Hence, Vst,EB = VEB = trBΠEB and therefore,

SSt,EBobs (t) = −∑st,EB

pst,EB (t) lnpst,EB (t)

VEB. (76)

The initial state is assumed to be

ρSB(0) = ρS(0)⊗ ρB(0), ρB(0) ∈ ΩB . (77)

Note that this initial state is considerably more generalthan the initial state (30). In particular, ρB(0) need notbe a (completely) passive state, yet our second law alsoholds in that case. Correlated initial states are treatedin Sec. VII B 2 and pure initial states are discussed inSec. VII C. Finally, we take the initial system observableS0 to satisfy [S0, ρS(0)] = 0, which implies SvN[ρSB(0)] =

SS0,EBobs (0) (Lemma V.3). Again, this also implies that the

initial measurement does not disturb the dynamics.Based on the previous considerations the second law

follows directly from Lemma V.4:

Σ(t) ≡ ∆SSt,EBobs (t) ≥ 0. (78)

Here, we defined the entropy production Σ, which is pos-itive and characterizes the increasing uncertainty about

the state of the universe, i.e., the system and the bath,with respect to the observables St and EB . It also followsfrom Lemma V.4 that the second law can be divided intoa classical and a quantum part via Σ(t) = Σcl(t)+Σqm(t).If we take into account that the measurement of St isgiven by rank-1 projectors, we obtain for the classicalpart

Σcl(t) =∑EB

pEB (t)D[ρB(EB , t)‖ωB(EB)], (79)

where ρB(EB , t) = ΠEB trSρSB(t)ΠEB/pEB (t) isthe post-measurement bath state and pEB (t) =∑stpst,EB (t). Interestingly, only the state of the bath

influences the classical contribution of the entropy pro-duction. For the quantum part we obtain

Σqm(t) = SvN

∑st,EB

pst,EB (t)|st〉〈st| ⊗ ρB(EB , t)

− SvN[ρSB(t)].

(80)

It is instructive to also consider the change in marginalobservational entropies, which, by virtue of the decorre-lated initial state (77), is given by

Σ′(t) ≡ ∆SStobs(t) + ∆SEBobs (t)

= Σ(t) + ISt,EBobs (t) ≥ 0.(81)

Here,

ISt,EBobs (t) =∑st,EB

pst,EB (t) lnpst,EB (t)

pst(t)pEB (t)(82)

is the classical mutual information characterizing the cor-relations in the final measurement results of St and EB .The relevance of this term for the thermodynamic de-scription of open quantum systems still needs further elu-cidation. Preliminary results suggest the following [137].If st denotes measurements of the system energy and ifthe system is undriven (λt = constant), then—as a resultof the microscopic conservation of energy—strong corre-lations can build up, in particular at weak coupling. Butif the system is driven, correlations seem to be smaller.Coming back to our example in Sec. IV of a two-level sys-tem subjected to a laser field, it seems that the one bit ofinformation associated to measuring St can tell us onlyvery little about the energetics of the bath after manydriving cycles.

1. Connection to the first law

In this section we are asking how far we can link thechange in observational entropy of the bath ∆SEBobs (t) tothe traditional notion of heat identified as the change inenergy of the bath ∆EB(t). In general, since we allow

16

for baths of any size, possibly strongly coupled to thesystem, and initialized in states beyond the canonicalGibbs paradigm, it might not be possible to establish thislink. We therefore assume in this section that the initialstate of the bath has a well-defined initial temperatureT0. Note that we do not consider the weak coupling limithere, which is treated in the next section.

We start by recalling the definition of remaining heatfrom Sec. VI B, originally established for a driven isolatedsystem. We then adapt Eq. (70) to define the remainingheat of the bath via

QBrem(t) ≡∫ t

0

T ∗s dSEBobs (β∗s ). (83)

Here, T ∗s is the effective nonequilibrium temperatureof the bath with the same operational meaning as inSec. VI B. Furthermore, consider that we are now im-plementing the same Clausius-Duhem process as above,but only for the bath. Since there is no driving protocolλt with which we can change the bath Hamiltonian, thisClausius-Duhem process simply describes a very slow, re-versible change in the bath temperature from T ∗t backto the initial temperature T0. Consequently, there isno mechanical work applied to the system and the re-maining heat in the bath is equal to its change in en-ergy QBrem(t) = ∆EB(t). Furthermore, we can split thechange in observational entropy of the bath as follows[cf. Eq. (72)]:

∆SEBobs (t) = SEBobs (t)− SEBobs (β∗t ) +

∫ t

0

dQBrem(s)

T ∗s. (84)

Note that this expression explicitly takes into accountthat the bath temperature can change, which seems rea-sonable when the effect of the system on the bath is notnegligible, compare with Eq. (4). Since for a fixed en-ergy the equilibrium state maximizes the entropy, we get

∆SEBobs (t) ≤∫ t

0dQBrem(s)/T ∗s . From the second law (81)

we can then deduce that

∆SStobs(t) +

∫ t

0

QBrem(s)

T ∗sds ≥ 0. (85)

We summarize our main results so far. Based on ourdefinition (76) for the entropy of the universe, we derivedthe hierarchy of inequalities

0 ≤ ∆SSt,EBobs (t)

≤ ∆SStobs(t) + ∆SEBobs (t)

≤ ∆SStobs(t) +

∫ t

0

QBrem(s)

T ∗sds,

(86)

where the last inequality requires the initial state of thebath to have a well-defined temperature, otherwise theinitial state only needs to be of the form (77). This hi-erarchy has not yet been derived within the open quan-tum systems context and it should be compared with thephenomenological second laws (2), (3) and (4), keeping

in mind that QBrem is positive if it increases the bath en-ergy. The first inequality in Eq. (86) becomes an equalityfor reversible transformations, the second if system-bathcorrelations are negligible, and the third if the bath iswell approximated by an equilibrium state with a time-dependent temperature throughout the evolution. Tomake the picture complete, it remains to derive the in-equality (5).

2. Weak coupling regime

We now focus on the weak coupling regime, albeit weemphasize that the weak coupling assumption here is lessrestrictive than the traditional one used in open quantumsystem theory [5, 6, 14, 15]: we do not invoke the Marko-vian or secular approximation or a van Hove-like weakcoupling limit. Instead, our weak coupling regime can bedefined as follows. We start by writing the probabilitiespEB (t) to measure the bath energy EB at time t as

pEB (t) = pEB (0)[1 + εqEB (t)], (87)

where qEB (t) is a correction, which, due to normaliza-tion, satisfies

∑EB

pEB (0)qEB (t) = 0. Now, our weak-coupling treatment is restricted to considerations wherethe parameter ε is small enough such that terms of or-der O(ε2) can be neglected. This should be a reasonableapproximation if the bath is large enough such that itspopulations only get slighlty perturbed due to the pres-ence of the system. Hence, one could call this regimemore precisely the weak bath perturbation regime.

Furthermore, we now assume the initial bathstate to be approximately canonical, i.e., ρB(0) ≈πB(β0) = e−β0HB/ZB(β0) such that pEB (0) ≈VEBe

−β0EB/ZB(β0). A straightforward computationthen reveals

∆SEBobs (t) = β0∆EB(t) +O(ε2), (88)

where the change in coarse-grained bath energy is

∆EB(t) =∑EB

EB [pEB (t)− pEB (0)]

= ε∑EB

EBpEB (0)qEB (t),(89)

which can in this regime safely equated with (minus) theheat flow into the system: Q = −∆EB . Since QBrem(t) =∆EB(t) = O(ε), we can also write∫ t

0

QBrem(s)

T ∗sds = −Q(t)

T0+O(ε2). (90)

Here, we took only the initial temperature T0 of the bathinto account because the final effective temperature devi-ates only slighly from it, i.e., T ∗t = T0(1 + ε). Note that εis numerically not identical to the parameter in Eq. (87),

17

but the underlying philosophy is the same, namely a weakperturbation of the bath caused by the system.

Finally, to recover the second law usually derivedin the Born-Markov-secular approximation, we considerthe case where we measure the final state of the sys-tem in its eigenbasis such that [St, ρS(t)] = 0. Then,the change in observational entropy of the system be-comes identical to its change in von Neumann entropy:∆SStobs(t) = ∆SvN[ρS(t)]. It then follows that

Σ ≥ Σ′ = ∆SvN[ρS(t)]− β0Q(t) ≥ 0, (91)

as desired. Now, however, we have microscopically de-rived that this second law is identical to the traditionalstatement about the increase in thermodynamic entropyof the universe. Furthermore, note that a measurementof St, which satisfies [St, ρS(t)] = 0, is ‘optimal’ in the

sense that it minimizes ∆SStobs(t) and hence, also Σ′.

In the remainder of this subsection, we return to thenotion of recoverable work and remaining heat in detailand finish with a discussion on fluctuation theorems.

3. Recoverable work and remaining heat in open systems

We reconsider our notions of recoverable work and re-maining heat from Sec. VI B in the weak coupling regime.Within the open system paradigm there are different lev-els of control over the system, which one can imagine,and we will consider two opposite cases here. The high-est degree of control assumes that ρS(t) is known at alltimes and that we have complete control about the sys-tem Hamiltonian. In particular, we can implement anyunitary operation we want on the system. The lowestlevel of control instead assumes that the changes HS(λt)that we can implement are fixed and constrained suchthat we cannot generate any unitary operation we wish.These two levels of control will indeed correspond to twowell-known second-law-like identities as we are going todemonstrate now. Different levels of control and knowl-edge are also imaginable, see Refs. [138, 139] for furtherresearch in this direction.

We start with the lowest level of control (as we actu-ally also did in Sec. VI B). To compute the recoverablework, we imagine that we bring the final system-bathstate ρSB(t) into contact with a superbath such that itrelaxes to an equilibrium Gibbs state πSB(β′) without anet heat exchange. Furthermore, since the bath is verylarge and the driven system very small, πSB(β′) describesan equilibrium state with respect to the inverse temper-ature β′ = β0(1 + ε), where ε describes a very smallcorrection to the initial inverse temperature β0 with asimilar meaning as above. Then, in the first step of theClausius-Duhem process we connect πSB(β′) to a super-bath at inverse temperature β0. This induces a slightchange in the system-bath temperature and its change in

energy and entropy is given to lowest order in ε by

USB(β′)− USB(β0) = −εβ0Varβ0(HSB) +O(ε2), (92)

SSB(β′)− SSB(β0) = −εβ20Varβ0

(HSB) +O(ε2). (93)

Here, Varβ(H) ≡ trH2(λt)π(β) − trH(λt)π(β)2 de-notes the energy variance for a thermal state. Thus, dur-ing the first step the recoverable work (69) becomes

Wrec = USB(β′)− USB(β0)− T0[SSB(β′)− SSB(β0)]

= O(ε2), (94)

i.e., it is negligible. In the second stage of the protocol,we adiabatically change the system Hamiltonian from itsfinal value HS(λt) back to its initial value HS(λ0) whilethe system is in weak contact with the bath, which inturn is still in weak contact with the superbath at inversetemperature β0. Since the bath Hamiltonian is fixed andsince we assume the system-bath interaction to be weak,we can approximate in this case

∆USB(β0) ≈ US(β0, λ0)− US(β0, λt), (95)

∆SSB(β0) ≈ SS(β0, λ0)− SS(β0, λt), (96)

where we now denoted the dependence on the control pa-rameter λt explicitly. Thus, the recoverable work equalsin this case the change in equilibrium free energy of thesystem, which was defined in Eq. (22):

Wrec = FS(β0, λt)− FS(β0, λ0) = ∆FS(β0). (97)

Now, imagine that the forward process started with anequilibrium system state ρS(0) = πS(β0, λ0). Then, ourwork recovery protocol has finished as we returned theglobal system-bath state back to its initial state. Therecoverable work is given by Eq. (97) and the remainingheat becomes

Qrem(t) = ∆USB(t)−Wrec = W (t)−∆FS(β0), (98)

where we used Eq. (10). Remarkably, Eq. (98) is well-known in the literature as the dissipated work [44, 49, 52,53], which is always positive as also follows from Eq. (91).Thus, what we have previously identified with the re-maining heat in the system-bath composite turns out tobe identical to the dissipated work for the case of low-level control as considered here.

Let us now consider the case of high-level control,where we are also able to extract work from the nonequi-librium nature of the final (and possibly also initial)system state. How this works in principle is well-known [140]. One instantaneously rotates the systemHamiltonian to the eigenbasis of ρS(t) and instanta-neously shifts the energy eigenvalues such that the result-ing Hamiltonian H ′S(λt) satisfies −β′H ′S(λt) ∼ ln ρS(t).Then, we follow the same two steps as above and returnthe system to an equilibrium state with respect to theHamiltonian H ′S(λ0). If the system started in equilib-rium, then H ′S(λ0) = HS(λ0). If the system started out

18

of equilibrium, then we choose −β0H′S(λ0) ∼ ln ρS(0)

and again instantaneously change the system Hamilto-nian H ′S(λ0) to HS(λ0) without changing the systemstate. The recoverable work becomes then the changein nonequilibrium free energy, Wrec = ∆F noneq

S (t), whereF noneqS (t) ≡ US [ρS(t), λt] − TSvN[ρS(t)]. The remaining

heat instead becomes

Qrem(t) = W (t)−∆F noneqS (t). (99)

Again, this expression is well-known from the litera-ture [55, 58, 59] and it is always positive due to Eq. (91).

4. Fluctuation theorems

From Lemma V.5 it follows that various fluctuationtheorems hold, whose precise form depends on the systemobservable St and the initial system-bath state. Here wepoint out a couple of interesting observations.

First, if we choose that [St, ρS(t)] = 0 and if the bathbehaves as an ideal, weakly coupled thermal bath (asassumed in Sec. VII A 2), then it holds that

∆sSt,EBobs ≈ ∆sS + ∆sB ≈ − lnpstps0− β0q. (100)

Here, pst (ps0) is the final (initial) probability to findthe system in state |st〉 (|s0〉) and q ≡ EB(0) − EB(t)is (minus) the change in bath energy given the initialand final outcomes EB(0) and EB(t). In this case, theintegral fluctuation theorem⟨

e−(∆sS−β0q)⟩

=⟨e−β(w−∆fnoneq

S )⟩

= 1 (101)

holds. Note that the stochastic work w as well as thechange in stochastic nonequilibrium free energy ∆fnoneq

Sare here determined by measuring the system in its finaleigenbasis, e.g., w = 〈st|HS(λt)|st〉 − 〈s0|HS(λ0)|s0〉 −q. The fluctuation theorem (101) is the counterpart ofEq. (42), already known for a classical Markov processobeying local detailed balance [141, 142].

On the other hand, within our approach based on ob-servational entropy, there is no straightforward derivationof the quantum work fluctuation theorem [46, 124, 143]:⟨

e−β0wδ=0⟩

= e−β0∆FS(β0). (102)

Here, in contrast to the stochastic work appearing inEq. (101), wδ=0 ≡ Eδ=0

t − Eδ=00 is determined by per-

forming two perfect error-free projective measurementsof the global energy at the beginning and at the end ofthe process giving outcomes Eδ=0

0 and Eδ=0t , respectively.

Hence, the superscript δ = 0. It is known that the in-tegral fluctuation theorem (101) does not directly implythe work fluctuation theorem (102). Only if the systemis allowed to relax back to equilibrium for a fixed value ofλt after the driving protocol, then Eq. (101) implies fora sufficiently large time t′ > t the result (102). This, of

course, assumes that the bath is ideal in the sense thatit is large enough to induce equilibration on the system.

However and quite interestingly, it turns out that wecan somewhat adapt the derivation to also cover thecase of the work fluctuation theorem (102), even in pres-ence of a measurement uncertainty δ. For this purposeimagine a driven isolated system with an initial state ofthe form (65), where the initial probabilities are givenby pE0 = e−β0E0VE0/Z(λ0) with the partition function

Z(λ0) =∑E0e−βE0VE0 . Such an initial state corre-

sponds to a coarse-grained version of a canonical Gibbsensemble with a finite uncertainty δ > 0, i.e., we as-sume ρ(0) =

∑E0e−βE0ΠE0

/Z(λ0). Now, we define thestochastic work w ≡ Et−E0 similar to the standard two-point measurement approach, but note that this work isnot identical to the stochastic work obtained from perfect(δ = 0) projective measurements. Nevertheless, follow-ing very similar steps as in the proof of Lemma V.5, it iseasy to confirm that

⟨e−β0w

⟩=Z(λt)

Z(λ0). (103)

This corresponds to an effective work fluctuation theo-rem at a coarse-grained level, which, to the best of ourknowledge, was not noted before. It effectively inter-polates between the perfect measurement limit and themicrocanonical case, where the state is entirely restrictedto one energy window with finite δ [144, 145]. Of course,due to the finite measurement uncertainty δ > 0 we haveZ(λt) 6= Z(λt) = tre−βH(λt) and therefore, Eq. (103)is strictly speaking not equivalent to Eq. (102). However,if we return to the open system paradigm, it is naturalto assume that we can measure the system energy per-fectly. Furthermore, within the weak coupling regime weapproximate Z(λt) ≈ ZS(λt)ZB as in Ref. [44]. Since wedo not drive the bath Hamiltonian, we can conclude

⟨e−β0w

⟩=Z(λt)

Z(λ0)=ZS(λt)

ZS(λ0). (104)

Thus, the work fluctuation theorem remains valid evenin presence of measurement errors in the bath providedthat the bath initially looks microcanonical within themeasurement uncertainty δ > 0.

B. Further Extensions

In the following, we detail the steps necessary to applythe above framework to the presence of multiple bathsand initially correlated states.

1. Multiple heat baths and exchange of particles

The extension to multiple heat baths, labeled by ν ∈1, . . . , n, is straightforward by measuring the energy Eν

19

of each bath (we treat the measurement of particle num-bers below). The observational entropy from Eq. (76) isconsequently generalized to

SSt,EBobs (t) = −∑st,EB

pst,EB (t) lnpst,EB (t)

VEB, (105)

where we used a boldface notation to denote the listof measurement results EB = (E1, . . . , En). Here,pst,EB (t) = tr|st〉〈st|ΠE1

. . .ΠEnρ(t) and VEB =∏ν VEν =

∏ν trBνΠEν. Furthermore, the initial

state (77) is generalized to

ρSB(0) = ρS(0)⊗ν

ρBν (0), ρBν (0) ∈ ΩBν , (106)

where B = (B1, . . . , Bn) denotes the presence of multipleheat baths.

Then, from Lemma V.4 the second law follows,

Σ ≡ ∆SSt,EBobs (t) ≥ 0, (107)

and an obvious generalization of Eqs. (79) and (80) showshow to split it into a classical and quantum part. Fur-thermore, in analogy to Eq. (81), we can also confirmthat the sum of the changes in marginal observationalentropy is positive:

Σ′ ≡ ∆SStobs(t) +∑ν

∆SEνobs(t) ≥ 0. (108)

The difference Σ′ − Σ characterizes the total amount ofcorrelations between the system and all heat baths andwe suspect that it can be large in general. For instance,consider ‘pure transport’, i.e., an undriven system (λt =constant) coupled to two baths labeled 1 and 2. Then,due to the conservation of total energy, strong correla-tions between E1 and E2 can build up.

Nevertheless, in the weak coupling regime Eq. (87) stillremains a good approximation for each bath (given thatthe baths are large enough). Then, Eq. (108) reduces tothe standard expression

Σ′ = ∆SStobs(t)−∑ν

βνQν(t) ≥ 0 (109)

with Qν(t) =∑EνEν [pEν (0)− pEν (t)].

We further comment on how to generalize the theoryto include particle transport by measuring the particlenumber Nν of some species (e.g., electrons) in each bathν. We assume the particle number operator to commutewith the bath Hamiltonian such that the generalization ofEq. (105) is straightforward by replacing pst,EB (t) withpst,EB ,NB

(t) = trSB|st〉〈st|ΠE1,N1 . . .ΠEn,Nnρ(t) andVEB with VEB ,NB

=∏ν VEν ,Nν =

∏ν trBνΠEν ,Nν.

Furthermore, the class of initial states (106) is now char-acterized by initial bath states ρBν ∈ ΩBν with

ΩBν =

∑Eν ,Nν

pEν ,NνΠEν ,Nν

VEν ,Nν

∣∣∣∣∣∣ pEν ,Nν arbitrary

.

(110)

The second law then follows immediately:

Σ = ∆SSt,EB ,NB

obs (t) ≥ 0. (111)

Furthermore, in the weak coupling regime we can use thesame argumentation as in Sec. VII A 2 by assuming that

pEν ,Nν (t) = pEν ,Nν (0)[1 + εqEν ,Nν (t)] (112)

for a sufficiently small ε. If the initial state of the bathcan be approximated by a grand canonical ensemble (19),then

∆SEν ,Nνobs (t) = β∆EB − βµ∆Nν(t) +O(ε2) (113)

follows. Here, ∆Nν(t) = ε∑Eν ,Nν

NνpEν ,Nν (0)qEν ,Nν (t)is the average change in particle number of bath ν. Equa-tion (109) is consequently generalized to

Σ′ = ∆SStobs(t)−∑ν

βν [∆Eν(t)−µν∆Nν(t)] ≥ 0. (114)

Finally, we remark that derivations of various fluctua-tion theorems are possible as in Sec. VII A 4.

2. Initial system-bath correlations

Despite of its major importance for the theory ofopen quantum systems [5, 6] and quantum thermody-namics [14, 15], completely decorrelated states of theform (30) or (77) seem to be a rather artificial as-sumption for many situations. So far, only a few ref-erences showed how to treat initial system-bath cor-relations and even then, these correlations were re-stricted to a very particular form (mostly global Gibbsstates) [48, 53, 65, 66, 70, 72, 73]. Very recently, it waspossible to partially overcome this difficulty and to treata larger class of correlated and decorrelated states withinone framework [68, 73, 75, 77, 78]. However, even thenthe presence of only one heat bath was considered.

In our framework based on observational entropy, clas-sical correlations between the chosen system and bathobservables are naturally included. Consider a general-ization of Eq. (106) to initial states of the form

ρSB(0) =∑s0,EB

ps0,EB (0)|s0〉〈s0|⊗ν

ω(Eν), (115)

where ps0,EB (0) is arbitary. This is simply Eq. (49) ap-plied to a joint measurement of multiple observables ondifferent Hilbert spaces. Then, it is clear that the secondlaw expressed as a change in observational entropy is notviolated:

∆SSt,EBobs (t) ≥ 0. (116)

Therefore, our second law and even our fluctuation the-orem (59) remain valid in presence of arbitrary classicalinitial correlations in the measurement basis. Note that,

20

in principle, quantum correlations could be also includedby choosing a ‘hybridized’ measurement basis, which isnot of the system-bath tensor product form.

Out of curiosity, we take a look at more general correla-tions. For this purpose we split the initial von Neumannentropy for an arbitary state ρSB(0) as follows:

SvN[ρSB(0)] = SvN[ρS(0)] +∑ν

SvN[ρBν (0)]

− IvN[ρSB(0)].

(117)

Here, the first line describes the von Neumann entropy ofthe decorrelated state ρSB(0) = ρS(0)ρB1(0) . . . ρBn(0)obtained from the reduced states of ρSB(0). The sec-ond line defines the remaining correlations in the stateρSB(0) and, by subadditivity of entropy, the total infor-mation IvN[ρSB(0)] is positive. Now, let us apply thesame philosophy to the initial observational entropy:

SSt,EBobs (0) = SStobs(0) +∑ν

SEνobs(0)− ISt,EBobs (0). (118)

We add that the following inequality between the amountof correlations always holds:

IvN[ρSB(t)] ≥ ISt,EBobs (t). (119)

Now, we consider the set of initial states ρSB(0), whosereduced states ρX(0) with X ∈ S,B1, . . . , Bn satisfy

SvN[ρX(0)] = SX0

obs(0). This means that only the reducedstates look ‘equilibrated’, i.e., are of the form (49) or (65),while the joint state can have arbitrary correlations. Thisset of initial states is even larger than those defined byEq. (115). From Eqs. (117) and (118) we obtain

SvN[ρSB(0)] + IvN[ρSB(0)] = SSt,EBobs (0) + ISt,EBobs (0).(120)

Now, using Eq. (120), we can write the change in ob-servational entropy as follows:

∆SSt,EBobs (t) = SSt,EBobs (t)− SvN[ρSB(0)]

− IvN[ρSB(0)] + ISt,EBobs (0).(121)

Since SvN[ρSB(0)] = SvN[ρSB(t)], and after splitting thefinal von Neumann and observational entropy into itsmarginal parts and correlations, we reveal that

∆SSt,EBobs (t) = SStobs(t)− SvN[ρS(t)]

+∑ν

SEνobs(t)− SvN[ρBν (t)]

+ IvN[ρSB(t)]− ISt,EBobs (t)

− IvN[ρSB(0)] + ISt,EBobs (0).

(122)

The first three lines of this expression are always non-negative, whereas the last line is never positive. However,if the correlations do not drastically change during thecourse of the evolution, then

IvN[ρSB(t)]−ISt,EBobs (t) ≈ ISt,EBobs (0)−IvN[ρSB(0)], (123)

which implies ∆SSt,EBobs (t) ≥ 0. However, precise con-ditions on the system-bath dynamics, which ensureEq. (123), are unknown at present.

C. Initial pure states

We now turn to a final subtle question, which we like tobriefly address within the present framework. This ques-tion is related to the observation that our choice of initialstates, albeit being more general than previous ones, isnonetheless special and the second laws introduced so farcease to hold for, e.g., a pure initial state. For ease ofpresentation we consider the case of one big isolated sys-tem in the following, the arguments given here do notchange if we imagine the isolated system to be split intoa system and a bath.

We first discuss the physical meaning of having a pureinitial state. Clearly, in every single run of the experi-ment the isolated system will be in some pure state. Butthis state typically changes from run to run and thereis no way to say which pure initial state one actuallysees in a single run. This ultimatively justifies the useof an ensemble of states as we have done here. Askingwhat is the impact of a particular pure initial state |ψ(0)〉on the entropy production, therefore requires to assumethat one has the experimental capabilities—at least inprinciple—to reliably and repeatedly prepare exactly thesame initial state |ψ(0)〉, even if one performs the ex-periment only a single time. Let us assume for a mo-ment that this is possible, i.e., we know |ψ(0)〉 precisely.Then, this state has no further classical information forus to offer. Therefore, if we want to study the impactof an initial pure state on the second law, we can choosethe initial projectors entering the observational entropyto be Π0 = |ψ(0)〉〈ψ(0)| and Π1 = 1 − |ψ(0)〉〈ψ(0)|.Clearly, in this case SX0

obs(0) = SvN[|ψ(0)〉〈ψ(0)|] = 0and from Lemma V.4 it follows that the second lawholds. In fact, even if the initial state is known to bein the pure state |ψ(0)〉, we might loose track of it dur-ing the (perhaps complicated) unitary time-evolution to|ψ(t)〉 = U(t, 0)|ψ(0)〉. It then makes sense again to con-sider other projectors than Π0,Π1 at time t, which, ingeneral, will lead to an increase of thermodynamic en-tropy even for pure initial states.

If it is not possible to reliably and repeatedly preparethe same initial state |ψ(0)〉, it no longer seems to bemeaningful to even ask about the second law for a pureinitial state. In particular, for many relevant open systemscenarios it is not possible to have precise control aboutthe global initial state. The question what is the correctinitial ensemble is nevertheless subtle and should there-fore be considered jointly with the present approach andthe recent progress reviewed in Refs. [17–21]. After all,

instead of insisting that ∆SXtobs(t) ≥ 0, negative changes

in observational entropy ∆SXtobs(t) < 0 should be viewedas a welcome signature that some interesting and not yetunderstood physics is waiting for us.

21

VIII. CONCLUDING REMARKSAND OPEN QUESTIONS

In this tutorial we introduced an overarching perspec-tive for the laws of thermodynamics in isolated and openquantum systems, which was based on starting with amicroscopic definition of nonequilibrium thermodynamicentropy. This approach allows to derive the hierarchyof second laws (2), (3), (4), and (5), and overcomes theshortcomings (A) to (D) mentioned in Sec. IV. Its va-lidity solely relies on the acceptance of observational en-tropy as a valid microscopic definition of nonequilibriumthermodynamic entropy for a suitablly chosen set of ob-servables. This point of view is in unison with, e.g.,Refs. [16, 32, 33, 36, 111, 112, 115, 118–121, 137]. Im-portantly, our approach is not in contradiction with pre-vious proposals from Sec. IV, but reaffirms their validityin their respective limit of applicability. Besides its con-ceptual clarity, we believe that the benefit of the presentapproach certainly lies in its versatility as it allows totreat various scenarios (one or multiple heat baths, withor without initial correlations, baths prepared in Gibbsstates or beyond, small and large bath, etc.) within onecommon framework. The present paper was devoted to ageneral and conceptual understanding of this frameworktogether with its connection to the previous literature. Inthis last section we comment on interesting future per-spectives, also in relation to experiments.

A. Experimental challenges

Up to now, we have viewed observational entropy as acomputational tool, we have focused only on mathemat-ical identities and kept the actual value of the measure-ment uncertainty δ out of the discussion—tacitly assum-ing that we can theoretically choose it at our convenience.We now view it from an operational perspective, askingunder which conditions our approach can be experimen-tally verified or falsified.

The first important observation concerns the fact thatthe measurement uncertainty δ is an intrinsic elementof our theory. This distinguishes it from previous theo-retical proposals, which can be experimentally confirmedonly by perfectly measuring the universe, i.e., the systemand the bath. This is evident for the two-point mea-surement scheme [100, 101], but also for the approachreviewed in Sec. IV A, where the change in bath energy∆EB = trBHB [ρB(t)− ρB(0)] was defined in terms ofthe microscopic bath Hamiltonian and state. Yet, howsmall do we have to choose δ in a real experiment?

Evidently, this requires in general a case-by-case study,but it is interesting to point out a few observations. First,δ should be small enough such that it is possible to resolvechanges in the distribution of the bath, otherwise theobservational entropy of the bath remains insensitive tothe dynamics. Therefore, as a rule of thumb, we believe

that the experimental requirement for δ is

trBHB [ρB(t)− ρB(0)] ≈∑EB

EB [pEB (t)− pEB (0)],

(124)where pEB (t) = trBΠEBρB(t) is the probability tomeasure EB given an arbitary bath state ρB(t). Whetheror not the second law (52) or fluctuation theorem (59)hold, then depends on the question whether the initialstate assumption [see, e.g., Eqs. (49) or (77)] is satisfiedfor a given δ. As discussed in Sec. VII C, observing ‘viola-tions’ of the second law can give us valuable informationabout the initial state.

Meeting experimentally the requirement (124) can stillturn out to be challenging, in particular if the bath ismacroscopically large such that it remains virtually un-affected by the system. In this case, it is beneficial toinfer the change in bath energy rather indirectly by mon-itoring changes in the open quantum system. There are,however, also an increasing number of promising exper-imental platforms, where the bath has to be consideredfinite and can be measured precisely enough. One in-teresting group of candidates are trapped ion systemsor ultracold quantum gases [146, 147]. These systemscan still be precisely measured, but they are at the sametime already large enough to show clear signatures ofequilibration and thermalization and they can be tailoredto study specific open quantum systems [148–152]. An-other interesting experimental platform are mesoscopicsystems used in circuit quantum electrodynamics or elec-tronic transport, where very sensitive thermometers forcalormetry have been demonstrated [153, 154]. Indeed, ina very recent experiment resolution on the level of singleenergy quanta has been reported for a mesoscopic bathof electrons [155].

We also note that in an actual experiments, where onereally performs the initial measurements, the results pre-sented in this manuscript hold for an even larger class ofinitial states than previously anticipated. Returning toour general notation of Sec. V C, we recognize that theinitial state ρ(0) can be any state provided that the postmeasurement state

ρ′(0) ≡∑x

Πxρ(0)Πx (125)

has the special form (49), (65) or (77). This includesinitial states ρ(0) with arbitrary coherences with respectto the measurement basis Πx. Thus, we conclude thatthe present theoretical framework seems to be promisingalso from an experimental perspective.

B. Open theoretical problems

While we argued that the present framework is veryflexible, much more remains to be understood from a the-oretical perspective. We here provide a non-exhaustivelist of open questions.

22

First, one might wonder why the energy observable isso special. Within our framework there are two reasonsfor it. First, equilibrium statistical mechanics predictsthat states of the form (65) or (77) are good candidatesfor initial states, see also Refs. [17–21]. Second, observ-ing the energy has the outstanding advantage that it islinked to the first law of thermodynamics. Nevertheless,it seems worthwile to pursue the question what happensif we use other observables than energy. This question isexperimentally relevant because it might not be alwayseasy to measure the total energy. Furthermore, to a largeextend we also left the case of multiple observables open.It is not clear what is the relevance and consequenceof observing non-commuting observables X1, X2, . . . inquantum thermodynamics, where the product of projec-tors Πx1

Πx2. . . is no longer a projector.

This brings us to our second open point, namely how toextend the present framework to more generalized mea-surements characterized by arbitrary positive operator-valued measures (‘POVMs’) [156]. In fact, strict pro-jective measurements are hard to realize in an experi-ment. More likely is that a measurement result x′ cor-responds to applying a Gaussian weight of projectorsΠx fixed around x ≈ x′. Interestingly, for an arbi-trary set of POVM elements Px, which always satisfy∑x Px = 1, the main definition (43) of observational en-

tropy remains: the probability to observe outcome x isgiven by px = trPxρ and the volume term becomesVx = trPx. At the moment it remains open, however,whether variants of the Lemmas in Sec. V C can be es-tablished.

Third, here we have focused only on two measurements(an initial and a final one). In many experiments, how-ever, one repeatedly probes the system and it is not clearwhat happens to our framework in the situation of multi-ple sequential measurements, but see Ref. [117] for somepreliminary results.

Fourth, it remains open to study the behaviour of ob-servational entropy for various concrete open quantumsystems. One way to pursue would be (analytically or

numerically) exactly solvable system-bath models. How-ever, it could be more interesting, but also more chal-lenging, to aim at studying the evolution of observationalentropy for ‘typical’ open quantum system dynamics byestablishing bounds on the quantum and classical partsΣqm and Σcl valid for a large class of evolutions underreasonable assumptions. Another possibility is to developefficient approximate methods to study the evolution ofobservational entropy as recently done in Ref. [137].

Fifth and finally, it is desirable to understand the po-tential for applications of the present framework better.For instance, what is the amount of extractable workfrom a generic quantum system given a set of allowedtransformations and a set of measurement outcomes xtof one or multiple observables? What are the optimalmeasurements to perform in order to extract the maxi-mum amount of work?

To conclude, observational entropy is a versatile con-cept, which provides a link between problems studied inthe field of equilibration and thermalization in isolatedquantum systems [17–21] and quantum thermodynam-ics and open quantum systems theory. We therefore be-lieve that it provides an overarching framework for manyproblems studied in nonequilibrium quantum statisticalmechanics.

Acknowledgements

We are grateful to Kavan Modi, Juan Parrondo, FelixPollock, Andreu Riera-Campeny, Dominik Safranek andJoan Vaccaro for many interesting discussions and use-ful comments. PS also acknowledges various stimulatingdiscussions with Massimiliano Esposito about the natureof entropy production over the years. The authors werepartially supported by the Spanish MINECO (projectFIS2016-86681-P) with the support of FEDER funds,and the Generalitat de Catalunya (project 2017-SGR-1127). PS is financially supported by the DFG (projectSTR 1505/2-1).

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Appendix A: Detailed fluctuation theorem for observational entropy

In this section we make use of the time-reversal operator Θ, see Appendix B. We denote ∆sobs = ∆sXtobs for brevityand start with the probability p(∆sobs) to observe a change in observational entropy ∆sobs in the ‘forward’ process:

p(∆sobs) =∑xt,x0

trΠxtU(t, 0)Πx0ρ(0)Πx0U†(t, 0)δ

[∆sobs −

(− ln

pxtVxt

+ lnpx0

Vx0

)], (A1)

where δ(·) denotes the Dirac-delta function. Furthermore, as in Lemma V.4 and V.5 we assume that SX0

obs(0) =SvN[ρ(0)]. Hence, with the help of ρ(0) =

∑x0px0

Πx0/Vx0

(Lemma V.3), we can express p(∆sobs) as

p(∆sobs) =∑xt,x0

trΠxtU(t, 0)Πx0U†(t, 0) pxt

Vxtexp

(lnVxtpx0

pxtVx0

)δ

[∆sobs −

(− ln

pxtVxt

+ lnpx0

Vx0

)]= e∆sobs

∑xt,x0

trΠx0U†(t, 0)ΠxtU(t, 0) pxt

Vxtδ

[∆sobs −

(− ln

pxtVxt

+ lnpx0

Vx0

)],

(A2)

where the Dirac-delta function allowed us to pull out the factor e∆sobs in the second equation. Now, we use time-reversal symmetry. For simplicity, we assume that the time-reversal operator obeys Θ2 = 1 and denote the time-reversed observable by XΘ

t ≡ ΘXtΘ =∑xtxtΠ

Θx . From Eqs. (B2) and (B5), we obtain

p(∆sobs) = e∆sobs

∑xt,x0

tr

ΘΠx0ΘUΘ(t, 0)ΘΠxtΘU†Θ(t, 0)

pxtVxt

δ

[∆sobs +

(− ln

px0

Vx0

+ lnpxtVxt

)]

= e∆sobs

∑xt,x0

tr

ΠΘx0UΘ(t, 0)ΠΘ

xt

∑x′t

px′tΠΘx′t

Vx′tΠΘxtU

†Θ(t, 0)

δ

[∆sobs +

(− ln

px0

Vx0

+ lnpxtVxt

)]= e∆sobspΘ(−∆sobs).

(A3)

Here, pΘ(∆sobs) denotes the probability to observe a change in observatonal entropy of ∆sobs with respect to thefollowing time-reversed process: we start with the initial state

∑xtpxtΠ

Θxt/Vxt (where pxt are the probabilities to

obtain outcome xt in the forward process), measure XΘt , evolve it under a time-reversed driving protocol (perhaps

with an inverted magnetic field), and finally we measure the time-reversed observable XΘ0 . Equation (A3) is the

detailed fluctuation theorem for observational entropy.

Appendix B: The time-reversal operator

In classical mechanics, it is clear from intuition that the trajectory of a particle is traversed in the opposite directionif one flips the momentum p of the particle to −p. More precisely, if one follows the trajectory in phase space duringa time window [0, t], then flips the momentum at time t and follows the trajectory in phase space further during thetime window [t, 2t], one ends up with the same initial state at time 2t after flipping the momentum again. This is

at least true for all classical Hamiltonian systems in absence of any driving protocol (λt = 0) and in absence of anymagnetic field B. If a magnetic field is present, the above statement remains true if we also flip the magnetic fieldfrom B to −B during the time window [t, 2t]. This is intuitively appealing if one recalls that a magnetic field is

caused by moving charges. The correct treatment of time-dependent Hamiltonians (λt 6= 0) is revealed below. Thepicture above describes the essence of time-reversal symmetry, which might be better called “reversal of the directionof motion” according to Wigner [157].

In quantum mechanics, one introduces a time-reversal operator Θ [158]. Quite strangely, it turns out that thetime-reversal operator has the property of being anti-unitary, which means that

〈Θψ|Θφ〉 = 〈ψ|φ〉∗ for all |ψ〉, |φ〉 ∈ H. (B1)

26

Therefore, Θ is not an operator in the conventional sense (one should not intend to write it as a matrix). However,it follows from the above property that Θ nevertheless leaves all probabilities unchanged since |〈Θψ|Θφ〉| = |〈ψ|φ〉|,which is required for a symmetry operator. It is also easy to show that anti-unitarity implies trace conjugation:

trΘXΘ−1 = trX∗ for all X. (B2)

Furthermore, if X is an observable, then ΘXΘ−1 is also an observable with the same eigenvalues as X but potentiallydifferent eigenvectors.

For simplicity, we now focus on the quantum mechanical treatment of particles without spin, which is in closeanalogy to the classical case. More complicated systems are treated, e.g., in Ref. [159]. It then turns out that Θ can beidentified with complex conjugation of the wavefunction in position representation. In equations, if |ψ〉 =

∫drψ(r)|r〉,

where |r〉 are the eigenstates of the position operator r (here denoted with a hat to be unambiguous), then

Θ|ψ〉 =

∫drψ∗(r)|r〉. (B3)

Without too much effort, one confirms that

Θ2 = I, ΘrΘ = r, ΘpΘ = −p, (B4)

where p denotes the momentum operator. The properties (B4) are the ones we expect by analogy with the classicalcase.

From what we said initially, we expect |ψ(0)〉 = ΘUΘ(t, 0)ΘU(t, 0)|ψ(0)〉 for any initial state |ψ(0)〉. In words: ifwe propagate any initial state forward in time using U(t, 0), then time-reverse it, then propagate it forward in timewith respect to the time-reversed propagator UΘ(t, 0), and finally time-reverse it again, then we end up with the sameinitial state. Written as an operator identity, we have

ΘUΘ(t, 0)Θ = U†(t, 0). (B5)

This is obviously the result one would expect mathematically, but its physical interpretation reveals an importantsymmetry. In fact, directly implementing the right hand side of this equation, i.e., U†(t, 0), in a lab is not possibleas it requires to map t 7→ −t. In contrast, as demonstrated below, UΘ(t, 0) corresponds to a legitimate ‘forward’evolution of a physical system. Unfortunately, however, the operator Θ, being anti-unitary, can not be implementedin a lab in general. We now turn to the question how to define UΘ(t, 0) microscopically.

We first consider the case of a time-independent Hamiltonian H and set U(t, 0) = e−iHt and UΘ(t, 0) = e−iHΘt withHΘ still unknown. To infer HΘ, we use the fact that anti-unitarity implies anti-linearity, which means Θλ|ψ〉 = λ∗Θ|ψ〉for any complex number λ. From UΘ(t, 0) = ΘU†(t, 0)Θ and Θ2 = 1, it is then easy to deduce that HΘ = ΘHΘ. If Hdenotes a Hamiltonian of interacting particles in absence of any magnetic field, then HΘ = H, i.e., the time-reversedmotion is generated by the same Hamiltonian. This follows from the fact that the momentum enters quadraticallythe Hamiltonian: Θp2Θ = p2. If H = H(B) depends on an external magnetic field, then HΘ = H(−B), which followsfrom the fact that for a particle with charge q a term (p − qA/c)2 enters the Hamiltonian, where c is the speed oflight and A the vector potential, which gives rise to the magnetic field.

Finally, we consider the case with driving protocol λs, s ∈ [0, t], and approximate the time-evolution operator as

U(t, 0) ≈ e−iH(λN−1)δt/~ . . . e−iH(λ0)δt/~, (B6)

where we divided the time interval into steps of size δt = t/N and implicitly keep in mind the limit N →∞ in whichEq. (B6) becomes exact. We can then infer for the time-reversed time evolution operator

UΘ(t, 0) = ΘU†(t, 0)Θ = ΘeiH(λ0)δt/~ . . . eiH(λN−1)δt/~Θ = e−iHΘ(λ0)δt/~ . . . e−iHΘ(λN−1)δt/~, (B7)

where we again set HΘ(λt) = ΘH(λt)Θ. Thus, the time-reversed dynamics are defined by changing the protocolbackwards in time from λt to λ0 with respect to the time-reversed Hamiltonian.