Thermodynamics of Minimal Coupling Quantum HeatEnginesMarcin Łobejko1, Paweł Mazurek1,2, and Michał Horodecki1,2

1Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk,80-308 Gdańsk, Poland

2International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

The minimal-coupling quantum heat en-gine is a thermal machine consisting ofan explicit energy storage system, heatbaths, and a working body, which alterna-tively couples to subsystems through dis-crete strokes — energy-conserving two-body quantum operations. Within thisparadigm, we present a general frame-work of quantum thermodynamics, wherea work extraction process is fundamentallylimited by a flow of non-passive energy (er-gotropy), while energy dissipation is ex-pressed through a flow of passive energy.It turns out that small dimensionality ofthe working body and a restriction only totwo-body operations make the engine fun-damentally irreversible. Our main resultis finding the optimal efficiency and workproduction per cycle within the whole classof irreversible minimal-coupling enginescomposed of three strokes and with thetwo-level working body, where we take intoaccount all possible quantum correlationsbetween the working body and the bat-tery. One of the key new tools is the in-troduced “control-marginal state" — onewhich acts only on a working body Hilbertspace, but encapsulates all features regard-ing work extraction of the total workingbody-battery system. In addition, we pro-pose a generalization of the many-strokeengine, and we analyze efficiency vs ex-tracted work trade-offs, as well as workfluctuations after many cycles of the run-ning of the engine.

Microscopic thermal heat engine has been re-cently realised in the lab with a trapped sin-gle calcium ion operating as a working body [1],as well as in superconducting circuits [2], nitro-gen vacancy centers in diamond [3], and elec-

tromechanical [4] settings. Simultaneously, newpropositions for realization of heat quantum en-gines have been put forward in quantum dots [5],nanomechanical [6], cold bosonic atoms [7], su-perconducting circuits [8, 9] and optomechanicalcontexts [10].

Despite these remarkable experimental suc-cesses, as well as vast theoretical studies [11–23], description of these machines still faces manychallenges, such as a proper definition of workand heat, and understanding of the role whichquantum correlations and coherence play in theperformance of these systems. One of the ba-sic questions that remains largely unanswered isabout the optimal performance of possibly small-est quantum engines (see [11, 12, 15] for earlydevelopments).

The problem can be formalized in various ways.Firstly, we may have continuous regime engines[24], where the working body is constantly cou-pled to both heat baths as well as to a work reser-voir, or discrete engines, which are alternatelycoupled to a hot and cold baths. Secondly, thework reservoir can be semiclassical – like an exter-nal classical field, or quantum – e.g. an oscillator.Thirdly, one can have autonomous machines, ornon-autonomous ones, i.e. those that are exter-nally driven.

Furthermore, one can specify the character ofthe contact with the heat bath - it may be givenby interaction Hamiltonian, or in terms of masterequation of GKLS type [25–28]. Recently, a col-lisional model of an engine with heat baths wasalso used where the bath is composed of indepen-dent systems which one by one interact with theworking body [29] (see also [30] for the compre-hensive introduction into the topic and [31–33]for recent developments). As a matter of fact,this kind of modeling of the contact with bathfits into a recently widespread paradigm of ther-

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Figure 1: A graphical representation of the minimal-coupling quantum heat engine – a micro machine con-verting heat into work via a working body operating intwo-body discrete strokes. Here, the minimal versionof the whole class is presented: the lowest dimensionalworking body (a qubit) and thermodynamic cycle con-structed only by three strokes.

mal operations [34–37]. Indeed, the leading ideaof the latter approach is that instead of stickingto a specific interaction Hamiltonian, one allowsfor all unitary transformations that conserve en-ergy (either strictly, or on average). Along thisdirection, in [38, 39] efficiency has been optimizedover all possible engines with a cold bath of afixed size.

An important question arises here - what actu-ally means “the smallest” quantum engine? Thesimplest answer might be: it is the engine withthe working body being an elementary quantumobject – a two level system [40]. However, if sucha two level system is externally driven, then thedriving field should be treated as a constituentof the engine. Note that the driving field usu-ally plays two roles - of the driving force, and ofthe work reservoir. Thus, in order to be sure thatour engine is indeed explicitly minimal, or that wecontrol its size, we should consider explicit workreservoir – e.g. in the form of quantum oscillator,and use no external driving. In other words, weshould consider a fully autonomous setup, withall constituents being explicit quantum systems,as in engines proposed in [16] or [41].

It would be however a formidable task to findan optimal engine in such fully autonomous sce-nario, as we would need to optimize the effi-ciency over all possible interaction Hamiltonianswith the bath, while even for concrete modelswith a fixed interaction only numerical results

are usually available. Indeed, in the literatureone usually considers concrete physical models,and evaluates their efficiency and power, ratherthan searches for the optimal engine. Yet, onecan relax a bit the autonomous character of anal-ysed class of engines, allowing for driving whichconsists of just several discrete steps. In such sce-nario the search for the optimal quantum engine,though still highly nontrivial, seems less hopeless.

In this paper we attempt to substantially ad-vance the above basic problem by considering thefollowing class of engines, which we call minimal-coupling engines: (i) the time evolution consistsof discrete steps, each being an energy preserv-ing unitary acting on two systems only, (ii) anexplicit, translationally invariant battery is in-cluded – the so-called ideal weight [15, 42, 43](see also [44] for the discussion of the physical-ity of the model). Our engines thus consist offour systems: the hot and cold bath, the work-ing body and the battery. The name “minimal-coupling engines” stems from our postulate thatonly two systems are interacting with each otherat a time. The postulated translational symme-try is to assure the Second Law and fluctuationstheorems [15, 42, 43].

Among the minimal-coupling engines, we shallconsider engines with smallest possible workingbody – i.e. two level system – as well as the small-est number of strokes, i.e. three ones (note thatminimal-coupling engine cannot work with justtwo strokes). One of our main results is findingthe optimal engine among such single-qubit, threestroke engines. Let us emphasize that to providethe result we cannot hope for saturating Carnotbound on efficiency, which would greatly simplifythe problem. Of course, if we were to considerefficiency vs power trade-off, we could not evendream about such a possibility, since Carnot ef-ficiency is only attained at zero power. How-ever, in the resource theory approach, we use inthis paper, time remains undefined, and insteadof power, we focus on work production per cycle.In general, obtaining Carnot efficiency with non-zero work production per cycle is allowed. Never-theless, as we show below, in our engines Carnotefficiency can be achieved only with zero workproduction. Thus no simple arguments can beapplied and optimization for efficiency and workproduction for given bath temperatures TH , TCand qubit energy gap ω has to be performed.

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On a technical side, the difficulty lies in theexplicit presence of the battery, so that it is nec-essary to take into account initial coherences ofthe battery’s state as well as the quantum cor-relations between working body and the batterythat build up during subsequent cycles. We over-come this obstacle by introducing a new object– control-marginal state. While it acts solely onthe working body Hilbert space, it equals to theworking body marginal state only in special cases(e.g. when the total battery-working body stateis diagonal in energy eigenbasis).

With this crucial tool at hand, before we turninto engines, we study thermodynamics of theminimal-coupling scenario. We thus first considerthe case of single heat bath and verify that thelaws of thermodynamics are satisfied. Remark-ably, we find that in such paradigm, the basicrole is played by ergotropy [45–47] rather thanby free energy. Namely, ergotropy provides fun-damental bound on an elementary portion of en-ergy that can be passed from the bath to the bat-tery in single step. Next, we show that the worktransferred to the battery equals to the ergotropychange of the control-marginal state rather thanthe marginal state of the working body.

These tools allow us to find the optimal en-gine among all single qubit, three-stroke minimal-coupling engines. We give analytical formulas foroptimal efficiency as well as work production percycle. The optimization is performed over all pos-sible unitaries in any of the three steps, as wellas over arbitrary initial joint states of the workreservoir and the working body.

Note that previously a qubit discrete en-gine with just two steps was considered in [15]which (unlike ours) achieves Carnot efficiency atnonzero work. Yet, unitary transformations overthree rather than two systems at a time were al-lowed, hence it does not belong to the minimal-coupling engine class. Similarly, in [38] a class ofengines was considered where two body unitarywas allowed for a cooler system, but still threebody unitary was applied to hot bath, workingbody and battery. On the other hand, in [20] onlytwo systems can interact at a time (as in our sce-nario). Yet, many steps are allowed, and there isno explicit work reservoir. Moreover, only ther-malization was allowed in the contact with heatbaths.

We compare our optimal engine with a model

which is the closest in spirit - namely the Ottoengine (considered e.g. in [17, 40]). For certainparameter values, the performance of our engineis substantially poorer, which highlights the ther-modynamic significance of the dimension of theHilbert space of the working body. On the otherhand, the optimal minimal-coupling engine canbe shown to be more efficient in other regime ofparameters. This highlights the advantage of fullclass of energy preserving unitaries over thermal-ization present in the Otto case.

We also address the problem of optimal enginewith more steps than three, allowing the workingbody to bounce between hot bath and batterywithin one cycle. We show that this does not in-crease efficiency (while it does increase work pro-duction per cycle).

Our considerations take into account a fullyquantum scenario, in which coherences and corre-lations within the working body and the batterymight be present. Our reasoning shows that theydo not constitute a resource for a cyclic work ex-traction, i.e. that the optimal efficiency and workproduction are obtained in absence of coherences.We also analyse fluctuations of obtained work,and show that (classical) correlations which buildup during engine operations lead to a reductionof fluctuations as compared with a hypotheticalcase of refreshing the working body in each cycle.

The paper is organized as follows. In Section Iwe present a class of operations which constituteminimal-coupling quantum heat engines, and weanalyze thermodynamic properties of these oper-ations in Section II. In Section III we present re-sults of optimal performance of the engines, andconclude with a discussion in Section IV.

1 Model of Minimal Coupling Quan-tum Heat Engine

Our model of a heat engine consists of four mainparts. Hot bath H, which plays the role of theenergy source, cold bath C, used as a sink for theentropy (or passive energy, see further in the ar-ticle), battery B, which plays a role of an energystorage, and a working body S, which steers theflow of the energy between the other subsystems(Fig. 1). The whole engine is treated as an iso-lated system with initial state given by a densitymatrix ρ, and evolving unitarily, i.e. ρ → U ρU †.

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The free Hamiltonian of the engine is given by:

H0 = HS + HH + HC + HB (1)

with local terms corresponding to the subsystems.In this setting we introduce the general

thermodynamic framework characterized by fivedefining properties:

(A1) Energy conserving stroke operations;(A2) Heat baths in equilibrium;(A3) Explicit battery given by the weight;(A4) Two-dimensional working body;(A5) Cyclicity of the heat engine.

The first three properties define class ofminimal-coupling quantum heat engines, where inparticular we establish an idea of stroke opera-tions (A1), and specify the environment (A2) andthe battery (A3), respectively. We will also as-sume (A4) for a special case of a minimal enginewith two-level working body, and (A5) to estab-lish the notion of cyclicity of the machine.

(A1) Energy conserving stroke operationsThe first property constitutes the core idea ofstroke operations: interactions between workingbody and other parts of the engine are turnedon and off in separated time intervals, so-calledstrokes. In other words, the unitary evolution ofan engine can be decomposed into a product of nunitaries:

U = USXnUSXn−1 . . . USX2USX1 , (2)

where the k-th step is an evolution coming fromthe coupling between working body S and sub-system Xk = H,C,B (hot bath, cold bath orbattery).

Furthermore, in the above decomposition weallow only for energy conserving unitaries. Weassume that during each stroke USXk the averagevalue of HS + HXk is a constant of motion, whichis satisfied if

[USXk , HS + HXk ] = 0. (3)

This implies that [U , H0] = 0, which constitutesa strict form of the First Law in our model, validfor arbitrary initial state ρ of the engine.

In the framework of stroke operations there aretwo fundamental blocks from which one can con-struct thermodynamic protocols, namely a work-stroke and heat-stroke (discussed in Section II).

The first one is a coupling of working body withbattery through which the work is extracted, andthe second describes a process of coupling withheat baths (hot or cold), where the heat is ex-changed.

Note that the property (A1) does not lead toa fully autonomous engine, as it requires an ex-ternal implicit system to control the execution ofsteps. Nevertheless, as energy inside the engine isfully conserved, it is a step forward towards an au-tonomous machine. In other words, condition (3)expresses the fact that turning on and off interac-tions does not introduce any energy flow into orout of the system, and thus, work can be definedsolely as the change of energy of the battery.

(A2) Heat baths and initial state

Heat baths are taken in equilibrium Gibbs states:

τA = 1ZA

e−βAHA (4)

where A = H,C and βH = T−1H < βC = T−1

C

are inverse temperatures (throughout the paperwe put Boltzmann constant k = 1), and ZA =Tr[e−βAHA

]is a partition function. In addition

we assume that for each step we have a ‘fresh’part of the bath in a Gibbs state, uncorrelatedfrom the rest of the engine. As a consequence,the initial state of the engine can be written as:

ρ = ρSB ⊗ τ⊗NH ⊗ τ⊗NC , (5)

where N is sufficiently large number providingthat for each stroke involving a heat bath we haveits new Gibbs copy. As a particular realization,later we will consider heat baths as a collection ofN harmonic oscillators, where in each stroke theworking body interacts only with one of them.

Furthermore, in this framework there are noother restrictions on a joint working body-batterystate ρSB. In this sense, the engine is fully quan-tum, e.g. it can involve entanglement or coher-ences both on the battery as well as on the work-ing body state.

(A3) Explicit weight battery

In order to define a closed (i.e. energy-conserving) heat engine, an explicit energy stor-age system (i.e. a battery) is necessary. Theproblem how to explicitly introduce battery

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which is consistent with the laws of thermody-namics is not trivial, i.e. it is equivalent to theproblem of a proper definition of work in thequantum thermodynamics [12, 48–53]. In ourproposal we choose a model of the so called idealweight, recently investigated in research on quan-tum thermodynamics [15, 42, 43, 54].

In contrast to the approaches where particulardynamics leading to the unitary USB is proposedexplicitly, the ideal weight is defined by imposinga symmetry which it has to obey. Specifically,this is a translational invariance symmetry, whichalludes to the intuition that change of the energyshould not depend on how much energy is alreadystored in the battery. It can be expressed in theform:

[USB, Γε] = 0, (6)

where Γε is a shift operator which displaces theenergy spectrum of the weight, i.e. Γ†εHBΓε =HB + ε, and ε is an arbitrary real constant.

As a particular example of the weight model,one can propose the Hamiltonian of the batteryin the form:

HB = Fx (7)

where x is the position operator, and F is a realconstant. This is analogical to a classical defi-nition of the work via an action of the constantforce F , i.e. W = Fδx where δx is a displace-ment of the system. In particular, if we take Fas a gravitational force (in a static and homoge-neous field), it corresponds to the model of thephysical weight.

Motivation behind the translational invariantdynamics of the battery is multiple. Firstly, itwas proven that work defined as a change ofaverage energy of the ideal weight is consistentwith the Second Law of Thermodynamics [15],and that work fluctuations obey fluctuations the-orems [42, 43]. Secondly, we show that workextraction protocol with explicit weight battery(work-stroke) can be understand in terms of theergotropy [55], similarly to the well-known non-autonomous work extraction protocols with cyclicHamiltonians (e.g. [47]). Last, but not least, thetranslational invariant dynamics of the batteryprovides a way to define a notion of ideal cyclic-ity of the heat engine, i.e. an exact periodic run-ning of the heat engine with constant efficiencyand extracted work per cycle, despite the obviouschange of the battery via a charging process, as

well as building up correlations with the workingbody.

(A4) Two-level working body

According to the strict law of energy conservation(3), it is important to stress that in this frame-work the total free Hamiltonian H0 (1) of the en-gine remains constant during the whole protocol.This is essentially different from non-autonomousapproaches with modulated energy levels of aworking body by an external control [40]. Indeed,this implicit external system, a so-called clock, isin fact a part of a ‘bigger’ working body, suchthat protocols with an energy level transforma-tion of a qubit do not apply to a genuinely twodimensional (i.e. minimal) working body. On thecontrary, in this framework we introduce a trulytwo-dimensional working body by the Hamilto-nian:

HS = ω |e〉〈e|S , (8)

where ω is the energy gap, |e〉S is an excited state,and |g〉S is a ground state. Here, and throughoutthe paper, we take h = 1.

2 Thermodynamics of strokes

Having a strict definition of the engine dynamics,in this section we move to its thermodynamics.We start with a definition of the effective state ofthe working body with respect to which we laterdefine all thermodynamic relations, and charac-terize heat engines. Then, we introduce a defini-tion of heat and work and show that the First Lawis satisfied. Further, a general characterizationof stroke operations is provided, namely a work-stroke USB (coupling to the battery), and heat-stroke USH (coupling to the heat bath). Finally,we analyze a work extraction process in contactwith a single heat bath, where the Second Law ofThermodynamics is verified.

2.1 Control-marginal working body state

Analysis of the thermodynamics of the family ofminimal-coupling engines relies on the definitionof the so-called control-marginal state acting onthe Hilbert space of the working body S:

σS = TrB[SρSBS†], (9)

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Figure 2: The role of ergotropy (12) and passive energy (13) in thermodynamics of stroke operations. a) Non-cyclicwork extraction process between hot bath H and battery B, mediated by a two-level working body S. Maximalenergy of the qubit is represented by the volume of the associated square. Interaction with H leads to increase ofergotropy (yellow) and passive energy (purple) of the qubit. Then, ergotropy is transferred to B. As amount ofextracted ergotropy from H is smaller for higher energies of the two-level working body, and passive energy is nevererased, efficiency of the ergotropy extraction falls down, and the process saturates. b) Cyclic work extraction (heatengine). Passive energy of the qubit is dumped into the cold bath C, which enables cyclic energy (ergotropy) transferfrom H to B.

whereS =

∑i

|εi〉〈εi|S ⊗ Γεi (10)

is a kind of control-shift operator, i.e. it trans-lates the battery energy eigenstates according tothe state of the system (6). In particular, for aproduct state ρSB = ρS ⊗ ρB, the channel (9)describes a decoherence process (i.e. it preservesdiagonal elements and decreases the off-diagonalones), such that the control-marginal state σS canbe seen as a ‘dephased version’ of a working bodydensity matrix ρS . Especially, we have equalityσS = ρS for diagonal states ρSB or for prod-uct states with diagonal ρS . Moreover, for non-diagonal state ρS , the decoherence of the workingbody depends on coherences in the battery state,such that only for work reservoirs with big enough‘amount of coherences’ we can have σS ≈ ρS .

Below we show that work and heat can besolely calculated from the control-marginal state.This essentially lowers the dimensionality of theHilbert space, and as a consequence dramaticallysimplifies the problem. Moreover, transforma-tions of the σS according to stroke operations(work- and heat-strokes) can be easily parameter-ized. This makes it possible to define the cyclicityof the whole engine and optimize its performanceover the whole set of stroke operations.

We start with expressing basic thermodynamicfunctions in terms of the state σS . Firstly, weintroduce an average energy :

ES = Tr[HS σS

]. (11)

Notice that [HS , S] = 0, thus the average energyof the control-marginal state σS is also equal tothe average energy of the system S, i.e. ES =Tr[HS ρSB

].

The second state function is ergotropy [46]:

RS = maxU−unitary

(Tr[HS(σS − U σSU †)

]), (12)

where the optimization is done over the set ofall unitaries acting on the S space. Furthermore,we introduce passive energy, which is the rest ofenergy (i.e. non-ergotropy) of the system:

PS = ES −RS . (13)

It quantifies locked energy, being the part ofthe total energy of the system which cannotbe extracted through unitary dynamics [55], orthrough dynamics with the ideal weight (dis-cussed later in the article). States with the wholeenergy being passive are called passive states [45].

Finally, we define the von Neumann entropy forthe state σS :

SS = −Tr[σS log σS ] (14)

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and free energy :

FS = ES − TSS (15)

with respect to the heat reservoir with tempera-ture T .

For the two-level working body (8) we repre-sent the state σS as:

σS = (1− ESω

) |g〉〈g|S + ESω|e〉〈e|S

+ α |g〉〈e|S + α∗ |e〉〈g|S ,(16)

where ES is the energy of the working-body(11) and α is the ‘effective coherence’, which es-sentially encodes the information about workingbody-battery correlations and internal coherenceswithin these subsystems. In general, a non-zerovalue of α corresponds to the entanglement ornon-diagonal product states. Without loss ofgenerality we further assume α to be real, i.e.α = α∗, since the phase plays no role in thermo-dynamics of minimal-coupling engines.

2.2 First Law of Thermodynamics

Let us consider an arbitrary initial state ρ (5),and protocol described by the unitary U (2). Asthe starting point, we define the total heat as achange of the average energy of the heat bath(with a minus sign):

Q = Tr[HH(ρ− U ρU †)

], (17)

and work as a change of the battery average en-ergy:

W = Tr[HB(U ρU † − ρ)

]. (18)

From conditions (2) and (3) we obtain the FirstLaw of Thermodynamics:

Tr[HS(U ρU † − ρ)

]= Q−W, (19)

where the left hand side corresponds to thechange of internal energy of the working body.Later we will see that above definitions obey theSecond Law of Thermodynamics, too.

Further, due to the fact that average energyof control-marginal state (11) is equal to ES =Tr[HS σS

]= Tr

[HS ρSB

], we can formulate the

First Law with respect to the state σS as follow-ing:

∆ES = Q−W. (20)

2.3 Work-stroke characterizationWe begin our considerations with a characteri-zation of the elementary work-stroke USB, whichdescribes the coupling between the working bodyand the battery. From the thermodynamic pointof view it is the process of storing the energy inbattery via the working body, i.e.

∆ES = Tr[HS(USB ρSBU †SB − ρSB)

]= −Tr

[HB(USB ρSBU †SB − ρSB)

]= −W,

(21)

where we used the energy-conservation relation(3) and work definition W (18).

In order to characterize the work-stroke, westart with showing that energy-conservation con-dition (3) and translational invariant dynamics ofthe weight (6) impose a strict form of the unitaryUSB [56], i.e.

USB = S†(VS ⊗ 1B)S, (22)

where VS is an arbitrary unitary operator actingon S, 1B is the identity operator acting on B,and S is given by Eq. (10).

This leads us to the following theorem (see Sec-tion C of Appendix):

Theorem 1. For a transition ρSB → ρ′SB =USB ρSBU

†SB, with energy-conserving (3) and

translational invariant (6) unitary USB, the workis equal to:

W = Tr[HB(USB ρSBU †SB − ρSB)

]= Tr

[HS(σS − VS σSV †S )

]= −∆RS ,

(23)

Furthermore, according to this operation, control-marginal state σS transforms unitarly as follows:

σSW-stroke−−−−−→ σ′S = VS σSV

†S . (24)

Let us note that the last equality in (23) stemsfrom the definition of ergotropy (12). We seethat the work stored in the battery can be calcu-lated solely from the control-marginal state σS .Moreover, the equality (23) reveals that work isequal to a change of the ergotropy of the control-marginal state ∆RS (12), where change of thepassive energy likewise the entropy change is zero,i.e. ∆PS = ∆SS = 0. Thus, we refer to this pro-cess as ergotropy storing. In particular, the max-imal value of the work which can be extracted

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from the state σS is given by its initial ergotropyRS , such that W = RS , and we refer to this ex-tremal case as a maximal ergotropy storing.

One should notice that Eq. (23) and (24) makethe work-stroke equivalent to non-autonomousdynamics of an isolated system in a state σSdriven by the cyclic Hamiltonian [47, 55]. Theonly difference relies on the fact that state σS isaffected by the state of the work reservoir (e.g.coherences and correlations) (9), and in generalσS 6= ρS . Nevertheless, later we optimize heatengines over cyclic evolution of an arbitrary stateσS , thus our results also include the ideal workreservoirs (i.e. with big enough amount of coher-ences) for which σS = ρS , as in a conventionalnon-autonomous approach.

Finally, we stress that the result given by Eq.(23) is valid for an arbitrary finite-dimensionalHilbert space of the working body, and not onlyfor the two-level system, which is generally dis-cussed in this article (see A4).

2.3.1 Ergotropy vs average energy

We have seen that in our setting (i.e. with an ex-plicit battery), it is the ergotropy of the control-marginal state of the system that quantifies theamount of extractable work. Below we demon-strate a connection of this result with a semi-classical setting where no explicit battery is as-sumed, and the work is drawn by a change of theHamiltonian of the working body. In such setting(e.g. for the conventional Carnot or Otto cycles[40]) it is the change of system’s average energyduring energy level transformations that quan-tifies the amount of extractable work, with thedifference of energy compensated by the externaldriving field.

To reconciliate these two pictures we can re-place change of the Hamiltonian of the workingbody by change of state of an extended system.Namely, consider the process where the energygap ω of the two-level working body is changedin N discrete steps, so that ω1 < ω2 · · · < ωN .We now consider the extended system of a qubitand ancilla with the total Hamiltonian givenby H =

∑k Hk ⊗ |k〉〈k| with Hk = ωk |e〉〈e|.

The Hamiltonian has the following eigenstates:H |e〉 |k〉 = ωk |e〉 |k〉. Hence the change of theancilla state from k → k′ effectively mimics thechange ωk → ωk′ in the semi-classical picture.

Such transition will increase or decrease er-

gotropy of the extended system, which could notbe seen explicitly in semi-classical setting, wherein typical engines there is no inversion of popula-tion of the qubit, hence ergotropy of the workingbody vanishes all the time.

2.4 Heat-stroke characterizationThe second elementary block of minimal-couplingengines is the heat-stroke USH , which correspondsto the coupling between working body and heatbath with inverse temperature βH . Firstly, wewould like to stress that (in analogy to the work-stroke) a change of the energy of the workingbody corresponds here to the heat:

∆ES = Tr[HS(USH(ρSB ⊗ τH)U †SH − ρSB ⊗ τH)

]= Q,

(25)

where τH is a Gibbs state (4), and we use a heatdefinition Q (17). Moreover, a transformation ofthe state ρSB via heat-stroke, i.e. a channel:

Λ[ρSB] = TrH [USH(ρSB ⊗ τH)U †SH ] (26)

is a thermal operation [37]. Further, one can showthat corresponding transition of the σS state isthe following (see Section B of Appendix):

σSH-stroke−−−−−→ σ′S = Λ[σS ]. (27)

In particular, a transformation of the diagonalof a density matrix of a two level system underevery thermal operation can be represented as aconvex mixture of two extremal thermal processes[57]: (1− λ)1 + λE, where 0 ≤ λ ≤ 1 and

1 =(

1 00 1

),E =

(1− e−βHω 1e−βHω 0

). (28)

This is accompanied with a decrease of absolutevalue of the coherences by a factor 0 ≤ γ ≤√

1− λe−βω(1− λ). Therefore, thermal opera-tion on (16) can be parametrized as follows :

ESH-stroke−−−−−→ E′S = ES + λ[e−βω(ω − ES)− ES ],

αH-stroke−−−−−→ α′ = γα,

(29)

such that λ ∈ [0, 1] and γ ∈[0,√

(1− λe−βω)(1− λ)] (up to an arbitrary

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phase). The special case λ = 1 refers to anextremal thermal operation, which will play aspecial role in optimal minimal-coupling heatengines.

Furthermore, the heat exchanged through thisprocess can be expressed as:

Q = Tr[HS(Λ[σS ]− σS)

], (30)

such that Clausius inequality is satisfied, i.e.

∆SS ≥ βHQ, (31)

where change of the entropy is defined with re-spect to the state σS (14).

2.4.1 Ergotropy extraction

As we saw in the previous section, charging thebattery is fundamentally connected with changesof ergotropy of the working body. This prop-erty is crucial for the whole thermodynamics ofminimal-coupling engines. It leads us to the fun-damental question: How to extract ergotropyfrom the heat bath in order to store it later inthe battery?

Firstly, we would like to present the followinggeneral relations:

Proposition 1. In the heat-stroke, extraction ofergotropy is accompanied by an increase of thepassive energy and decrease of the free energy:

∆RS > 0 =⇒ ∆PS > 0, (32)∆RS > 0 =⇒ ∆FS < 0. (33)

We refer the reader to Section E and H of Ap-pendix for the proof of the above and Theorem2 below. The main conclusion from the aboveproposition is that ergotropy extraction cannotbe achieved without accumulation of the passiveenergy (32). Specifically, it prevents extractionof work from the single heat bath in a cyclicprocess, since otherwise pure extracted ergotropyfrom the heat bath might be fully stored in thebattery, and then the working body would comeback to the initial state. In other words, with-out accumulation of the passive energy the wholeprocess could be repeated forever and an unlim-ited amount of work would be extracted from asingle heat bath. On the contrary, as we dis-cuss it in more detail below, the passive energylimits the extracted work to the initial free en-ergy, in accordance with the Second Law. Sec-ondly, from the inequality ∆FS < 0 it follows that

for ergotropy extraction Clausius inequality (31)is never saturated. This imposes limitations onthe total amount of possible work extraction andshows that thermodynamics of minimal-couplingheat engines is fundamentally irreversible, as it isdiscussed in more detail in the next section.

Next, we find the maximal value of ergotropywhich can be extracted in the heat-stroke:

Theorem 2. [Optimal ergotropy extrac-tion] In the heat-stroke, the optimal ergotropyextraction is given by

∆RmaxS = max[2(ω−ES)e−βω−ω−RS , 0], (34)

where

RS = 12

[2ES − ω +

√(2ES − ω)2 + ω2α2

](35)

is an initial ergotropy of the state. The optimalvalue is achievable by the extremal thermal oper-ation (λ = 1).

Formula (34) determines the range of parame-ters of the initial state (i.e. ES and α) for which∆RS is nonzero. In particular, necessary condi-tion for positive ergotropy extraction is

ES < ω(1− 12e

βω), (36)

which is also a sufficient condition when thereare no coherences in the initial state (i.e. whenα = 0).

As we see from (34) and (35), ∆RmaxS is a de-creasing function of the initial energy ES . More-over, for fixed ES , the change of ergotropy is max-imised for the state σS with no initial coherences(i.e. α = 0). This is because the optimal er-gotropy extraction is performed by the extremalthermal operation for λ = 1, which, in agree-ment with (29), destroys all coherences. How-ever, final ergotropy for the extremal process isthe same for every α, namely R′S = RS + ∆RS =2(ω − ES)e−βω − ω.

Remark 2.1. Notice that due to the condition(36) one can show that

∆RS > 0 =⇒ ω < T log(2), (37)

i.e. ergotropy extraction is possible if energy gapof the qubit is smaller than Landauer’s erasureenergy.

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2.5 Work extraction processNow we are ready to combine those two thermo-dynamic processes, ergotropy extraction via heat-stroke USH and ergotropy storing via work-strokeUSB, in order to extract work from the single heatbath by a combination USBUSH .

As an extreme example of such a process, themaximal value given by Eq. (34) can be extractedfrom the heat bath H, and then stored in a bat-tery B, which corresponds to the extracted workequal to W = ∆RmaxS . However, any positiveergotropy extraction ∆RS > 0 via the first, heat-stroke, is unavoidably accompanied with passiveenergy accumulation ∆PS > 0 (32). It is a cru-cial property since this additional passive energycorresponds to a dissipation of the working bodystate, so that next ergotropy extraction has to beless efficient or even impossible. In other words,repetition of the work extractions (via pair oper-ations USBUSH) has to get stuck at some point.This idea is graphically represented in the Fig. 2(a). It is nothing else like another formulation ofthe Second Law: work extraction from the singlebath cannot be free, i.e. without any change ina state of the working body. Here, irreversiblechange is quantified by the accumulated passiveenergy, what means that the initial small amountof it (passive energy) can be treated as a resourceused for extraction of work from the bath.

2.5.1 Optimized work extraction

To be more precise, let us consider a work extrac-tion process through the sequence of 2n strokeoperations:

U = U(n)SB U

(n)SH . . . U

(2)SBU

(2)SHU

(1)SBU

(1)SH . (38)

For this we are able to prove (see Section G 5 ofAppendix) the following:

Proposition 2. [Optimized work extrac-tion] If for any heat-stroke U (k)

SH we have positiveergotropy extraction (i.e. ∆RS > 0), and for anywork-stroke U (k)

SB we have positive ergotropy stor-ing (i.e. W > 0), then the maximal work whichcan be extracted is equal to

Wmax = 2e−βω(ω − ES)(1− e−nβω)1− e−βω − nω, (39)

where n = b 1βω log[2(1 − ES/ω)]c, and ES is the

initial average energy of the working body. The

optimal process is achieved if all heat-strokes aregiven by the extremal thermal operations and allwork-strokes are the maximal ergotropy storings.

Remark 2.2. Note that, as discussed before, theassumption of positive ergotropy extraction in thefirst step enforces the inequality (36), hence n >0 and Wmax > 0.

In particular, for two subsequent optimal workextractions, we have:

∆R(k+1)S = ∆R(k)

S − 2e−βω∆P (k)S , (40)

where work stored in the battery via k-th stepis equal to Wk = ∆R(k)

S . This formula quanti-fies the previous observation that repeated workextraction is less and less efficient due to the ac-cumulation of the passive energy (see Fig. 2 (a)).In addition, it is worth to notice that maximalvalue of the extracted work Wmax is neither en-hanced nor diminished by the effective coherenceα. This, as we show later, is not true for cyclicwork extraction.

This example emphasizes that a small dimen-sionality of the two-level working body makeswork extraction process only possible through afinite number of strongly coupled steps (i.e. er-gotropy extractions). Indeed, without access toadditional energy levels or tripartite operations,one cannot split the whole protocol into infinites-imal steps (like in a conventional Carnot cycle)where in each of them dissipation of the work-ing body is minimal. In contrary, a truly two-dimensional working body operating in strokescan only extract work through strong and irre-versible operations, which is justified quantita-tively in the following section.

2.5.2 Work and free energy

For stroke operations, in Appendix H we formu-late the Second Law in a more familiar way interms of non-equilibrium free energy (15) of thecontrol-marginal state σS . For any combinationof strokes USH and USB it holds

W ≤ −∆FS . (41)

This is true whenever change of free energy ispositive or negative, however, from the strong in-equality (33) valid for arbitrary ergotropy extrac-tion, one can further show the following:

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Proposition 3. For a process where ∆FS < 0and initial ergotropy of the working body is zeroRS = 0, the maximal extracted work is alwayssmaller than change of its free energy, i.e.

W < −∆FS . (42)

Remark 2.3. The assumption RS = 0 impliesthat work is solely extracted from the heat bath.In other case (where RS 6= 0), this initial valuecan be stored in a battery without any coupling tothe heat bath, and then and only then work canbe equal to W = −∆FS.

Inequality (42) imposes limits for the maxi-mal work extraction which is less than free en-ergy. Furthermore, this reveals the intrinsic irre-versibility of stroke operations. Literally, if oneconsider a forward process with ∆Ff < 0 and abackward process with ∆Fb = −∆Ff , then from(41) and (42) follows that extracted work Wf isalways smaller than energetic cost of returning tothe initial state, i.e. −Wb > Wf . In other words,the cyclic process with ∆F = 0 has alwaysW < 0(except the trivial identity process whereW = 0)which is another statement of the Second Law.

2.5.3 Free energy vs ergotropy

All these observations give us here, in the frame-work of stroke operations, a natural interpreta-tion of the difference between two state functions:free energy and ergotropy (see also [46]). It is seenthat the maximal value of the work extracted viathe work-stroke is limited by the ergotropy of asystem, i.e. W = −∆RS . Without any accessto the additional heat bath, after extracting allergotropy, the process cannot be repeated, andthe maximal value of the work is restricted to theinitial ergotropy of the working body. However,a protocol with the access to the heat bath canbe repeated, and then the total extracted workcan be much larger, while bound by the changein free energy, i.e. W < −∆FS (42).

In other words, if we consider a particular tran-sition of the working body with a fixed change ofthe entropy ∆SS and energy ∆ES , then the workis bounded by W ≤ T∆SS − ∆ES . However,for the stroke operations, the flow of the energy(from the heat bath to the battery) is limited bythe ergotropy of the system, which for a qubit isnaturally bounded by its energy gap, i.e. RS ≤ ω.

Hence, the working-body ergotropy is a ‘bottle-neck’ of the whole process. As a consequence, avariation of the temperature T effectively changesthe number of possible steps through which thebattery can be charged (or discharged) via ele-mentary portions, such that the sum of them can-not exceed the limit equal to −∆FS .

3 Thermodynamics of Minimal Cou-pling Quantum Heat EngineNow we turn to minimal-coupling quantum heatengines, i.e. a cyclic work extraction protocolconstructed within our paradigm of stroke opera-tions. One of the most important characteristicsof an engine is its efficiency. It is defined as

η = W

QH, (43)

where QH is a (minus) change of the average en-ergy of the hot heat bath (17), i.e. the net inputheat. Secondly, we consider also work extractedper cycle W (18) and refer to it as work produc-tion P (to elucidate that it characterizes a cyclicprocess).

Below, we explain what we mean by cyclic run-ning of the engine.

(A5) Cyclicity of the heat engine

Cyclicity of the heat engine is simply its propertyto retain constant efficiency η and work produc-tion P in the consecutive cycles of the machine,each induced by unitary U (2). Two assumptionsare made in order to ensure cyclicity in this the-oretical framework. The first one is about refre-shability of heat baths (5): in each stroke theworking body couples to an uncorrelated part ofa heat reservoir. Secondly, we impose a transla-tional invariance on the battery (6).

While the assumption of the ‘big and static’heat baths (which do not change during the run-ning of the engine) is natural and convenient, thework reservoir cannot stay in the same state bydefinition (as it is meant to accumulate energy), itcan also correlate with the working body. Never-theless, the remarkable consequence of using thetranslational invariant battery (A3) (and refre-shable heat baths (A2)) is that work and heatare solely defined with respect to the control-marginal state σS (see Eq. (23) and (30)). More-

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over, its transformations during work- and heat-strokes are independent of the state of the sur-rounding (Eq. (24) and (27)). This allows us toimpose cyclicity of the engine by demanding:

σSU−→ σS , (44)

where unitary U (2) generates evolution of theengine during a single cycle. In other words,the work reservoir given by the ideal weight, inconnection with refreshable heat baths, makesit possible to construct a cyclic object, i.e. thecontrol-marginal state σS , which characterizesthe periodic operation of the whole engine andsimultaneously describes changes of the batterystate and formation of the correlations.

Previously, we have seen that with an access toa single heat bath, a working body cannot extractwork periodically due to accumulation of passiveenergy, or, in other words, it cannot extract workin a cyclic process from a single heat bath. Thus,the only way to release passive energy and turnback working body to its initial state is by ex-ploiting another resource, e.g. a second, colderheat bath. Below we show that for some rangeof temperatures (hot and cold) the transition re-leasing all passive energy is possible and workingbody is able to close a cycle after positive workextraction (see Fig. 2 (b)).

All of these observations identify roles playedby each of the three parts of the minimal-coupling heat engine:

(i) Hot bath is used for ergotropy extrac-tion (as a side effect, passive energy is extractedas well);(ii) Battery is used for ergotropy storage;(iii) Cold bath is used for releasing passive energy.

However, fundamental irreversibility of strokeoperations, expressed by (42), also has an im-pact on maximal efficiency. Indeed, the maxi-mal efficiency given by the Carnot efficiency ηCis only attainable for reversible engines. Thus, forstep heat engine (with non-zero work productionP > 0) we always have

η < 1− βHβC≡ ηC . (45)

If Carnot efficiency is not achievable for aminimal-coupling quantum heat engine, then a

question about how close it can be approached isnatural. We discuss it in the next section.

3.1 Three-stroke heat engine

Minimal step heat engine is the one which con-sists of only three strokes, i.e. with hot bath H,battery B and cold bath C. Then, the roles ofengine elements (i-iii) characterize them uniquelyonly if efficiency of the engine is to be positive.From this follows one of the main result of thiswork (see Section F of Appendix for details of theproof):

Theorem 3. Consider a class of three-stroke en-gines with dynamics obeying conditions (A1-A5),for a fixed hot bath temperature βH , cold bathtemperature βC and working body energy gap ω.

• such engines can operate with positive effi-ciency only if

eβHω + e−βCω < 2, (46)

• optimal efficiency and maximal work pro-duction per cycle are given by

η1 = 1− eβHω − 11− e−βCω ,

P1 = ω[ 2e−βHω

1 + e−(βC+βH)ω − 1],(47)

and they are achieved simultaneously in theoptimal engine,

• the optimal protocol is unique and consistsof the extremal thermal operations with bathsand the maximal ergotropy storing with thebattery,

• steady control-marginal state σS (16) of theoptimal engine after each stroke is diagonal(α = 0), and is determined by the followingenergy transformations

E0S = ω e−(βH+βC)ω

Z1

H−→ E1S = ω e−βHω

Z1

B−→ E2S = ω[1− e−βHω

Z1] C−→ E0

S

(48)

where Z1 = 1 + e−(βH+βC)ω.

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Figure 3: Characterization of the optimal three-stroke minimal-coupling engine for given bath temperatures βH ,βC and qubit splitting ω, as described by Theorem 3, in terms of optimal efficiency (left plot and the inset) andwork production per cycle (right plot). Operational region (46) lies above 0 contour lines, and is contained within0 ≤ βHω ≤ log 2, 0 ≤ βCω < ∞. Solid contour lines of optimal efficiency η1 (left plot) and solid contour lines ofoptimal work production per cycle in units of bath temperature βHP1 (right plot) are presented, with dashed linescorresponding to solid lines on the neighboring plot. The black area corresponds to a regime βH > βC . In the insetof the left plot, Carnot efficiency contour dotted lines 1 − βH

βCare presented for values corresponding to efficiency

contour lines. Notice that in the limit βHω → 0 and βCω → 0 the efficiency η1 → 1− βHβC

, which graphically meansthat constant-efficiency Carnot lines are tangent with respect to the efficiency η1 contour lines, in the origin of thecoordinate system.

Sketch of the proof. The basic idea is that maxi-mal efficiency η1 arises through optimization forgiven bath temperatures and energy splitting ofa two-level working body, over all energies E0

S

of the working body (i.e. energy just before er-gotropy extraction USH), as well as over all pos-sible unitaries USH , USC and USB, such that ηis maximal and the working body returns to itsinitial state.For simplicity, let us consider a diagonal state

with energy evolving as follows:

E0S

H−→ E1S

B−→ E2S

C−→ E0S . (49)

In particular, the maximal ratio W/QH can beachieved for extremal ergotropy extraction USHwith heat:

QH = E1S − E0

S = ∆RS(E0S) + ∆PS(E0

S), (50)

and maximal ergotropy storing USB with workgiven by:

W = E2S − E1

S = −∆RS(E0S), (51)

where

∆RS(E0S) = 2(ω − E0

S)e−βHω − ω,∆PS(E0

S) = (ω − E0S)(1− e−βHω).

(52)

In this case efficiency of the engine is equal to:

η = W

QH= 2(ω − E0

S)e−βHω − ω(ω − E0

S)e−βHω − E0S

. (53)

We see that η is a decreasing function with re-spect to initial energy E0

S . This suggests that forE0S = 0 we obtain the maximal possible ratio.

However, it does not mean that the cycle of theengine can be closed for given bath temperaturesβH , βC and splitting ω.Indeed, after extremal ergotropy extraction

and maximal ergotropy storing, the energy ES2of the working body has an additional contribu-tion given by passive energy coming from the hotbath, i.e.

E2S = E0

S + ∆PS(E0S) (54)

where ∆PS(E0S) is once again a decreasing func-

tion with respect to E0S . Finally, since working

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body has to return to the initial state with en-ergy E0

S , it must release all accumulated passiveenergy through the cold bath stroke USC , namely

QC = E0S − E2

S = −∆PS(E0S). (55)

However, this operation is more efficient forstates with higher initial energy E0

S , and the fol-lowing inequality has to be satisfied:

E2S ≥ ω − E0

SeβCω. (56)

In other words, more passive energy accumulatedin the state σS helps with closing the cycle.

Then, we have a trade-off between these two:higher ratio (53) is for smaller initial energy E0

S ,and closing of the cycle is more efficient for higherenergies E0

S . The solution of this optimizationproblem leads to unique protocol operating atmaximal efficiency η1 (and P1). We include a de-tailed proof including coherences, non-extremalprocesses and different order of applied steps inSection F of Appendix.

Let us stress that the optimal engine simul-taneously operates with maximal efficiency andmaximal work production per cycle (47) for spe-cific values of βH , βC and ω. Nevertheless, thereis still a trade-off between them if we modulatethese parameters. Now, we would like to analyzeit in more detail.

In Fig. 3 we visualized the optimal engineperformance in the space of coordinates βHωand βCω. In particular, we showed the workingregime of an arbitrary minimal-coupling engine(46). This regime is stricter than the fundamen-tal constraint for heat engines given by the rela-tion βH < βC . Furthermore, it is seen that inthe limit βHω → 0 and βCω → 0 the efficiencyη1 → 1− βH

βC, i.e. the optimal efficiency (47) tends

to the Carnot’s one, what in the figure is visual-ized by the fact that constant-efficiency Carnot’slines are tangent with respect to the same valuecontour lines of the efficiency η1, in the origin ofthe coordinate system. The similar analysis, fora fixed ω, is visualized in the Fig. 4, where wereveal how the efficiency changes with increasingratio βC/βH .

Next, we focus on the relation between ef-ficiency and work production per cycle in ourparadigm. If we fix temperatures and start tomodulate energy gap ω of the working body, weobserve a trade-off between efficiency η1 and workproduction P1 (see Fig. 6). Moreover, for ω → 0,

Figure 4: Efficiency η1 of the engine for different valuesof βHω. In the limit βHω → 0, Carnot efficiency 1− βH

βCis achieved.

the engine also reaches Carnot efficiency, but op-erates at zero work production, i.e. P1 → 0.Furthermore, from Fig. 5 we see that minimal-coupling engine always achieves maximum workproduction per cycle below the Curzon-Ahlbornvalue 1−

√βHβC

[58] (unless working in the regimeof ideally cold bath βC → ∞, when Curzon-Ahlborn and Carnot efficiencies coincide). Ar-bitrarily high work production per cycle can beachieved in the limit βH → 0, at the cost ofω →∞.

Finally, we would like to discuss the role of co-herences. We proved that the unique optimalprocess with energy transformation (48) forcesthe state σS to have no coherences at the begin-ning of each step, α = 0. This leads to the con-clusion that coherences have a diminishing roleboth on efficiency and extracted work per cycleof minimal-coupling engines. The intuition be-hind this behavior is that coherences can only becreated through work-stroke unitary V (24) (at acost of additional energy), while heat baths canonly suppress them (29).

3.1.1 Comparison with the Otto cycle

How well does the performance of the optimal en-gine within minimal-coupling engines class rankwhen compared to the performance of schemestaking advantage of higher dimensionality of theworking body? We address this question com-paring our model with that of a qubit workingbody in the Otto cycle, where work is performedby an external field. There, energy levels of thequbit are ε0 and ε1 ∈ {εC , εH}, εH > εC > ε0,and the engine works in 4 strokes: (i) shift of ex-

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0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

η1

βHP1

0.10.2

0.30.4

0.1

0.2

0.3

0.4

0.5

0.60.1

0.2

0.30.4

0.5

0.6

0.1

0.2

0.3

0.4

0.5

0.6

βC /βH=2 βC /βH=5

βC /βH=10 βC /βH=100

Figure 5: Efficiency vs work production per cycle (inunits of hot bath inverse temperature βH) for differenttemperature ratios of the baths. Vertical lines indicaterespective values of Curzon-Alhborn efficiency 1−

√βHβC

,while numbers correspond to values of βHω.

cited energy level εC → εH , (ii) thermalization incontact with hot reservoir at inverse temperatureβH , (iii) shift of excited energy level εH → εC ,(iv) thermalization in contact with cold reservoirat inverse temperature βH In stroke operationsframework, the Otto cycle on a qubit with time-dependent Hamiltonian can be equivalently de-scribed on a qutrit working body with energy lev-els ε0, εC and εH .

As a figure of merit in our comparison wechoose work production per cycle expressed inunits of the energy gap εH − εC , i.e. gap modu-lated via the adiabatic segments (i) and (iii), dur-ing which work is extracted. In this way, the com-parison between the engines is based on how ef-fectively they use energy gap of the working bodyto extract work.

For the Otto engine, we arrive with maximalwork production given by

POttoεH − εC

= maxz

[(1 + e

z1−ηOtto )−1 − (1 + ezy)−1

],

(57)

for a fixed y = βC/βH , where optimization isperformed over a parameter z = βHεC , and weexploit the fact that

ηOtto = 1− εCεH. (58)

In Fig. 6 we see that, while both engines reachthe same Carnot efficiency at zero work produc-tion per energy gap, the minimal three-stroke en-gine performs better than the Otto engine for a

Figure 6: Comparison with the Otto cycle. Dashed lines:Relation between efficiency η1 and extracted work percycle and energy gap P1/w (47) of the minimal three-stroke engine, for different values of βC/βH . The plotwas obtained through parametrization of (47) with x =βHω and y = βC

βH, and for a given line (value of y), x

increases to the left. The performance of the engine,i.e. the usage of energy gap of the working body, iscompared with the Otto cycle. Solid lines of a givencolor correspond to the maximal work production perenergy gap POtto/(εH − εC) (57) for a given efficiencyηOtto (58) of the Otto engine.

region of high efficiencies. In principle, for ratioβC/βH high enough, the minimal three-stroke en-gine surpasses the bound 1/2 of the Otto engine.The reason for this is that we allow for the arbi-trary thermal operation to describe an interactionbetween the working body and a bath, while theOtto engine is restricted to thermalisation. Nev-ertheless, the fact that the working body in theOtto cycle can be effectively defined on a higher,three-dimensional space, is reflected in higher val-ues of work production per energy gap for smallerefficiencies.

3.2 Many-stroke generalization

Analysis of the many-stroke engine is much morecomplicated than the simplest three-stroke one.The reason for this is that the roles of differentstrokes (i-iii) of the minimal-coupling engine areno longer unique. In this case, it remains truethat any positive efficiency requires performingat least one ergotropy extraction, ergotropy stor-ing and releasing passive energy. However, themany-stroke protocol is able to involve also otheroperations, like spending work or heat-flow fromthe system to the hot bath.

Here we consider the most natural generaliza-tion of the three-stroke engine to many-stroke en-

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Figure 7: Relation between efficiency ηn and work pro-duction Pn/ω (61) of the multi-stroke engine for dif-ferent number of steps n (indicated by points). Num-ber of steps is a natural number satisfying the conditionn ≤ ln 2−e−βCω

βHω. Lines are drawn to guide the eye.

gine, given by the unitary

Un = USCU(n)SB U

(n)SH . . . U

(2)SBU

(2)SHU

(1)SBU

(1)SH , (59)

where we assume that any hot bath step U(k)SH

is ergotropy extraction (i.e. ∆RS > 0) andany work-stroke U

(k)SB is ergotropy storing (i.e.

W > 0). It is fully analogous to work extractionprotocol from a single heat bath which we haveconsidered previously (38). However, here, coldbath operation USC appears at the end in orderto make the process cyclic. In other words, we in-vestigate a subclass of minimal-coupling engineswhich are hybrids of engines performing work ex-traction from a single heat bath, described in Fig.2 (a), and the simplest cyclic three-stroke workextraction presented in Fig. 2 (b). With such adefinition of the many-stroke engine we are ableto generalize the previous result, with the three-stroke engine being a special case.

Firstly, temperature regimes at which the en-gine can operate with positive efficiency general-ize to

enβHω + e−βCω < 2 (60)

(see Section G of Appendix for details of thederivations). Further, maximal efficiency andmaximal work production are given by:

ηn = 1− (1− aH)(1− anH)(1− anH)(1 + aH)− n(1 + aCanH)(1− aH) ,

Pn = ω[ 2aH(1− anH)(1− aH)(1 + aCanH) − n],

(61)

where aH,C = e−βH,Cω. As previously, the op-timal protocol is the one where all heat-bathstrokes are extremal thermal operations and anywork-stroke is maximal ergotropy storing process,such that energy of the working body in each stepis given by formulas:

E0S = ω e−(nβH+βC)ω

Zn, E2k−1

S = ω e−kβHω

Zn,

E2kS = ω[1− e−kβHω

Zn],

(62)

where Zn = 1 + e−(nβH+βC)ω and k = 1, 2, . . . , n.One can further show that η1 < ηm (for m >

1), i.e. the simplest three-stroke engine is the onewith maximal efficiency. However, work produc-tion Pn of the engine increases with number ofsteps, i.e. Pn > Pm (for n < m). Once again weobserve here a thermodynamic trade-off betweenefficiency and work production (see Fig. 7), whichin this case is related to how many work extrac-tions we perform within a single cycle of the en-gine. In other words, we see that an increasingnumber of work extractions within a cycle givesus more work, but the transformation of heat intowork is less efficient.

We prove that the three-stroke engine has max-imal possible efficiency within the class of many-step engines defined by Eq. (59). Neverthe-less, the question about the optimal two-levelminimal-coupling engine with an arbitrary num-ber of steps remains open.

3.3 Many cycle analysis3.3.1 Realization

In this section, we propose a particular unitaryUn which realizes the maximal efficiency ηn andwork production Pn. It allows us to analyze thebehavior of the engine over many cycles.

Firstly, we assume a specific form of the heatbath Hamiltonians. We propose a well-knownmodel of a heat bath given as a collection of har-monic oscillators, i.e.

HH,C =( ∞∑k=0

kω |k〉〈k|H,C

)⊗N, (63)

such that the working body couples to a singleoscillator in each of N steps. Then, maximal ef-ficiency ηn can be achieved by a unitary:

Un = USC(USBUSH)n, (64)

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where extremal bath operations are given by thefollowing swaps of states:

|g〉S |0〉H,CH-stroke←−−−−→ |g〉S |0〉H,C ,

|g〉S |k〉H,CH-stroke←−−−−→ |e〉S |k − 1〉H,C ,

(65)

for k > 0. ( Note that these transitions can be(imperfectly) simulated via Jaynes-Cummings in-teraction [59]). Analogously, maximal ergotropystoring via battery operation USB is realized by:

|g〉S |k〉BW-stroke←−−−−→ |e〉S |k − 1〉B , (66)

where |k〉B is an eigenstate of the HamiltonianHB with an eigenvalue kω.

For such Un there exists a unique diagonal sta-tionary state of the working body ρS = TrB[ρSB]with energy E0

S (Eq. (62)), such that

Tr[HS ρ

]= Tr

[HSUnρU

†n

], (67)

i.e. the working body returns to the same ener-getic state (see section I of Appendix). For thisstationary state, the engine operates with maxi-mal efficiency ηn and work production Pn. Whatis more, after many cycles arbitrary initial diag-onal state ρSB converges to the stationary one,i.e.

limN→∞

Tr[HSU

Nn ρU

Nn†]

= E0S . (68)

3.3.2 Work fluctuations

Let us now concentrate on the optimal three-stroke minimal-coupling engine with a unitaryU1 = USCUSBUSH (64) and stationary state ofa qubit

ρS = (1− E0S

ω) |g〉〈g|S + E0

S

ω|e〉〈e|S (69)

with energy E0S (48). Periodicity of the engine

means that cycle after cycle the marginal stateof the working body during any step is the same(in this case σS = ρS). Specifically, any quantitysolely dependent on the state of a working bodyis also stationary, like efficiency η1 and extractedwork P1.

Nevertheless, correlations between the batteryand working body are not stationary and affectthe final state of the battery. In fact, thanksto the cyclicity of the working body we are ableto extract information encoded in these correla-tions. Basically, we can compare the final state

Figure 8: Work distribution (upper panel) and variance(lower panel), measured in the final battery state after2m work extractions. P2m(2k) line corresponds to ascenario of independent realization of the three-strokecycle on 2m qubits, where the final energy storage isperformed with respect to the same battery. Fluctua-tions of work can be reduced due to possible correlationsbetween a system and a battery: P2m(2k) gives corre-sponding profiles for a single working body and a bat-tery, with the three-stroke cycle run on them 2m times.(βHω = 0.2, βCω = 0.8)

of the battery, firstly, after N cycles of running ofthe three-stroke heat engine, and secondly, aftercharging of the battery through N uncorrelatedqubits, such that each of them is subjected to thesame work-stroke operation USB (65). Moreover,we take uncorrelated qubits in the same state

%S = (1− E1S

ω) |g〉〈g|S + E1

S

ω|e〉〈e|S (70)

with energy E1S (48), equal to the marginal state

of the working body just before USB couplingduring running of the three-stroke engine.

Then, we initialize battery in the ‘zero state’|0〉〈0|B and consider its final state after N = 2mcycles of the three-stroke engine with unitaryU1 = USCUSBUSH :

ρB = TrS,H,C [U2m1 (|0〉〈0|B ⊗ ρS ⊗ τ

⊗2mH ⊗ τ⊗2m

C )U2m1†]

=m∑

k=−mP2m(2k) |2k〉〈2k|B

(71)

(for even number of cycles only even eigenergies,i.e. 2kω, of the battery are occupied), and com-

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pare it with a battery charged through 2m inde-pendent couplings USB with uncorrelated qubitsin a state %S :

%B = TrS [U2mSB (|0〉〈0|B ⊗ %

⊗2mS )U2m

SB†]

=m∑

k=−mP2m(2k) |2k〉〈2k|B .

(72)

The formulas for functions P2m(k) and P2m(k)are presented in section I of Appendix. From con-servation of energy, total extracted work is equalin both cases, i.e.

W = Tr[HB ρB

]= Tr

[HB %B

]= 2mP1. (73)

However, the state ρB (71) is different from thestate %B (72) due to the accumulated correlationsthrough repeated coupling with a single workingbody. As it is can be seen in Fig. 8, correla-tions between working body and battery reducethe fluctuations, i.e. the work distribution is morenarrow than the one resulting from coupling withcollection of uncorrelated systems.

4 Conclusions and discussionThe main achievement of this work is the estab-lishment of new fundamental limits for the perfor-mance of quantum heat engines, which similarlyto Carnot result are independent of microscopicdetails of engine dynamics. The new boundscome from additional restrictions on realizationof heat engines via two-dimensional working body,operating only in two-body discrete strokes. Thisleads to an intrinsic irreversibility of thermody-namics processes, and, as a consequence, theminimal-coupling micro engine defined in thisway operates at efficiency smaller that of that ofthe Carnot engine.

This opens a new field of research on minimalmicro engines, i.e. these which are restricted bythe dimension of the working-body and/or heatbaths, or the number of subsystems which can in-teract with each other at a time. In particular, inorder to obtain a better understanding of the roleswhich multi-body interactions and dimensional-ity of the system play in behavior of engines,one could diverge from our description by gradu-ally taking into account multi-body interactions,and/or design protocols for low-dimensional qu-dits acting as the working body. The challenge in

the latter would be to find the optimal protocol,as we have done for the minimal-coupling engine.The difficulty of this task comes from the factthat structure of the set of thermal operationsbecomes complex quickly with increasing dimen-sion of the working body, and different thermaloperations may be needed for a specific choice ofenergy splittings of system Hamiltonian, temper-atures of a baths and initial state of the work-ing body in order to optimally extract work in acyclic process. It would be of primary interest tofind maximal ergotropy increase possible in thisgeneral case.

Optimal usage of minimal-coupling engineswith two-level working body should also be fur-ther investigated. One would expect that in-creasing number of steps in a cycle can lead toimproved efficiency of these engines. Therefore,studies of cycles which do not belong to the sub-class of multi-step engines characterized in thisarticle should be carried on. Especially, the re-versed heat-flows from the heat baths and partialusage of the energy of the battery may turn ben-eficial for the operation of these engines.

Finally, the tools used in our analysis, control-marginal state σS (9) and ergotropy RS (12), de-serve separate discussions of their own. Identi-fication of work extractable from a system withits ergotropy is a consequence of the ideal weightmodel of the battery. As it is shown, this is equiv-alent to a cyclic dynamics of an isolated systemdriven by an external force, which makes a strongconnection between theoretical frameworks withimplicit and explicit work reservoirs. Neverthe-less, the definition of the control-marginal stateallows for description of additional effects comingfrom coherences and correlations, which are ab-sent when the battery is treated implicitly. More-over, the ideal weight applied for heat engines asan energy storage naturally establishes the no-tion of cyclicity. Remarkably, this holds even inthe presence of coherences and formation of cor-relations between the working body and battery,which occurs during cyclic operation of an engine.Studies of different possible notions of cyclicity,together with establishment of necessary and suf-ficient conditions for ergotropy to be a measureof extractable work, constitute subject for futureresearch.

Accepted in Quantum 2020-12-08, click title to verify. Published under CC-BY 4.0. 18

Acknowledgements

M.Ł. acknowledges support from the Na-tional Science Centre, Poland, through grantSONATINA 2 2018/28/C/ST2/00364. M.H.

and P.M. acknowledge support from Na-tional Science Centre, Poland, grant OPUS 92015/17/B/ST2/01945, the Foundation for Pol-ish Science through IRAP project co-financed byEU within the Smart Growth Operational Pro-gramme (contract no. 2018/MAB/5).

References[1] Johannes Roßnagel et al. “A single-atom heat engine”. Science 352.6283 (2016), pp. 325–329.

doi: https://doi.org/10.1126/science.aad6320.[2] Nathanaël Cottet et al. “Observing a quantum Maxwell demon at work”. Pro-

ceedings of the National Academy of Sciences 114.29 (2017), pp. 7561–7564. doi:https://doi.org/10.1073/pnas.1704827114.

[3] James Klatzow et al. “Experimental Demonstration of Quantum Effects in the Oper-ation of Microscopic Heat Engines”. Phys. Rev. Lett. 122 (11 2019), p. 110601. doi:https://doi.org/10.1103/PhysRevLett.122.110601.

[4] Daniel Goldwater et al. “Levitated electromechanics: all-electrical cooling of chargednano- and micro-particles”. Quantum Science and Technology 4.2 (2019), p. 024003. doi:https://doi.org/10.1088/2058-9565/aaf5f3.

[5] Christian Bergenfeldt et al. “Hybrid Microwave-Cavity Heat Engine”. Phys. Rev. Lett. 112 (72014), p. 076803. doi: https://doi.org/10.1103/PhysRevLett.112.076803.

[6] Andreas Dechant, Nikolai Kiesel, and Eric Lutz. “All-Optical Nanome-chanical Heat Engine”. Phys. Rev. Lett. 114 (18 2015), p. 183602. doi:https://doi.org/10.1103/PhysRevLett.114.183602.

[7] O. Fialko and D. W. Hallwood. “Isolated Quantum Heat Engine”. Phys. Rev. Lett. 108.8, 085303(2012), p. 085303. doi: https://doi.org/10.1103/PhysRevLett.108.085303.

[8] Jonatan Bohr Brask et al. “Autonomous quantum thermal machine for generating steady-state entanglement”. New Journal of Physics 17.11, 113029 (2015), p. 113029. doi:https://doi.org/10.1088/1367-2630/17/11/113029.

[9] Jukka P. Pekola. “Towards quantum thermodynamics in electronic?circuits”. Nature Physics 11(2015), pp. 118–123. doi: https://doi.org/10.1038/nphys3169.

[10] Keye Zhang, Francesco Bariani, and Pierre Meystre. “Quantum Optomechan-ical Heat Engine”. Phys. Rev. Lett. 112.15, 150602 (2014), p. 150602. doi:https://doi.org/10.1103/PhysRevLett.112.150602.

[11] H. E. D. Scovil and E. O. Schulz-DuBois. “Three-Level Masers as Heat Engines”. Phys. Rev.Lett. 2 (6 1959), pp. 262–263. doi: https://doi.org/10.1103/PhysRevLett.2.262.

[12] R Alicki. “The quantum open system as a model of the heat engine”. Journal of PhysicsA: Mathematical and General 12.5 (1979), pp. L103–L107. doi: https://doi.org/10.1088/0305-4470/12/5/007.

[13] Marlan O. Scully. “Extracting Work from a Single Thermal Bath via Quantum Negentropy”.Phys. Rev. Lett. 87 (22 2001), p. 220601. doi: https://doi.org/10.1103/PhysRevLett.87.220601.

[14] Marlan O. Scully. “Improving the Efficiency of an Ideal Heat Engine: The Quan-tum Afterburner”. Phys. Rev. Lett 88, quant-ph/0105135 (2002), p. 050602. doi:https://doi.org/10.1103/PhysRevLett.88.050602.

[15] Paul Skrzypczyk, Anthony J. Short, and Sandu Popescu. “Work extraction and thermo-dynamics for individual quantum systems”. Nature Communications 5.4185 (2014). doi:https://doi.org/10.1038/ncomms5185.

Accepted in Quantum 2020-12-08, click title to verify. Published under CC-BY 4.0. 19

[16] D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki. “Work and energy gain of heat-pumped quantized amplifiers”. EPL (Europhysics Letters) 103.6, 60005 (2013), p. 60005. doi:https://doi.org/10.1209/0295-5075/103/60005.

[17] Raam Uzdin, Amikam Levy, and Ronnie Kosloff. “Equivalence of Quantum Heat Ma-chines, and Quantum-Thermodynamic Signatures”. Phys. Rev. X 5 (3 2015), p. 031044. doi:https://doi.org/10.1103/PhysRevX.5.031044.

[18] Raam Uzdin, Amikam Levy, and Ronnie Kosloff. “Quantum Heat Machines Equivalence, WorkExtraction beyond Markovianity, and Strong Coupling via Heat Exchangers”. Entropy 18.4(2016), p. 124. doi: https://doi.org/10.3390/e18040124.

[19] Arnab Ghosh et al. “Two-level masers as heat-to-work converters”. Proceed-ings of the National Academy of Science 115.40 (2018), pp. 9941–9944. doi:https://doi.org/10.1073/pnas.1805354115.

[20] Janet Anders and Vittorio Giovannetti. “Thermodynamics of discrete quantum processes”.New Journal of Physics 15.3, 033022 (2013), p. 033022. doi: https://doi.org/10.1088/1367-2630/15/3/033022.

[21] Krzysztof Szczygielski, David Gelbwaser-Klimovsky, and Robert Alicki. “Markovian masterequation and thermodynamics of a two-level system in a strong laser field”. Phys. Rev. E 87.1,012120 (2013), p. 012120. doi: https://doi.org/10.1103/PhysRevE.87.012120.

[22] Vasco Cavina, Andrea Mari, and Vittorio Giovannetti. “Slow Dynamics and Thermodynam-ics of Open Quantum Systems”. Phys. Rev. Lett. 119.5, 050601 (2017), p. 050601. doi:https://doi.org/10.1103/PhysRevLett.119.050601.

[23] M. Perarnau-Llobet et al. “Strong Coupling Corrections in Quantum Ther-modynamics”. Phys. Rev. Lett. 120.12, 120602 (2018), p. 120602. doi:https://doi.org/10.1103/PhysRevLett.120.120602.

[24] Ronnie Kosloff and Amikam Levy. “Quantum Heat Engines and Refrigerators: Continuous De-vices”. Annual Review of Physical Chemistry 65.1 (2014). PMID: 24689798, pp. 365–393. doi:https://doi.org/10.1146/annurev-physchem-040513-103724.

[25] E. B. Davies. “Markovian master equations”. Commun. Math. Phys. 39 (2 1974), pp. 91–110.doi: https://doi.org/10.1007/BF01608389.

[26] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan. “Completely positive dynamicalsemigroups of N-level systems”. Journal of Mathematical Physics 17.5 (1976), pp. 821–825. doi:https://doi.org/10.1063/1.522979.

[27] G. Lindblad. “On the generators of quantum dynamical semigroups”. Communications in Math-ematical Physics 48.2 (1976), pp. 119–130. doi: https://doi.org/10.1007/BF01608499.

[28] R Alicki and K Lendi. Quantum dynamical semigroups and applications. Lectures Notes inPhysics. Berlin: Springer, 2007. doi: 10.1007/3-540-70861-8.

[29] Raam Uzdin and Ronnie Kosloff. “The multilevel four-stroke swap engine and its environ-ment”. New Journal of Physics 16.9 (2014), p. 095003. doi: https://doi.org/10.1088/1367-2630/16/9/095003.

[30] Philipp Strasberg et al. “Quantum and Information Thermodynamics: A UnifyingFramework Based on Repeated Interactions”. Physical Review X 7.2 (2017). doi:https://doi.org/10.1103/physrevx.7.021003.

[31] Marco Pezzutto, Mauro Paternostro, and Yasser Omar. “An out-of-equilibrium non-Markovianquantum heat engine”. Quantum Science and Technology 4.2 (2019), p. 025002. doi:https://doi.org/10.1088/2058-9565/aaf5b4.

Accepted in Quantum 2020-12-08, click title to verify. Published under CC-BY 4.0. 20

[32] Stefano Cusumano et al. “Entropy production and asymptotic factorization via ther-malization: A collisional model approach”. Physical Review A 98.3 (2018). doi:https://doi.org/10.1103/physreva.98.032119.

[33] Franklin L.S. Rodrigues et al. “Thermodynamics of Weakly Coherent Collisional Models”. Phys-ical Review Letters 123.14 (2019). doi: https://doi.org/10.1103/physrevlett.123.140601.

[34] D. Janzing et al. “Thermodynamic Cost of Reliability and Low Temperatures: Tightening Lan-dauer’s Principle and the Second Law”. International Journal of Theoretical Physics 39.12(2000), pp. 2717–2753. doi: https://doi.org/10.1023/A:1026422630734.

[35] R. Streater. Statistical Dynamics: A Stochastic Approach to nonequilibrium Thermodynamics.Imperial College Press, London, UK, 1995. doi: https://doi.org/10.1007/BF02174220.

[36] Ernst Ruch and Alden Mead. “The principle of increasing mixing character andsome of its consequences”. Theoretica chimica acta 41.2 (1976), pp. 95–117. doi:https://doi.org/10.1007/BF01178071.

[37] Michał Horodecki and Jonathan Oppenheim. “Fundamental limitations for quan-tum and nanoscale thermodynamics”. Nature Communications 4.2059 (2013). doi:https://doi.org/10.1038/ncomms3059.

[38] Mischa P. Woods, Nelly Huei Ying Ng, and Stephanie Wehner. “The maximum efficiency of nanoheat engines depends on more than temperature”. Quantum 3 (2019), p. 177. doi: 10.22331/q-2019-08-19-177.

[39] Nelly Huei Ying Ng, Mischa Prebin Woods, and Stephanie Wehner. “Surpassing the Carnotefficiency by extracting imperfect work”. New Journal of Physics 19.11 (2017), p. 113005. doi:https://doi.org/10.1088/1367-2630/aa8ced.

[40] H. T. Quan et al. “Quantum thermodynamic cycles and quantum heat engines”. Phys. Rev. E76 (3 2007), p. 031105. doi: https://doi.org/10.1103/PhysRevE.76.031105.

[41] Alexandre Roulet et al. “Autonomous rotor heat engine”. Phys. Rev. E 95.6, 062131 (2017),p. 062131. doi: https://doi.org/10.1103/PhysRevE.95.062131.

[42] Álvaro M. Alhambra et al. “Fluctuating Work: From Quantum Thermodynamical Iden-tities to a Second Law Equality”. Phys. Rev. X 6 (4 2016), p. 041017. doi:https://doi.org/10.1103/PhysRevX.6.041017.

[43] Johan Åberg. “Fully Quantum Fluctuation Theorems”. Phys. Rev. X 8 (1 2018), p. 011019. doi:https://doi.org/10.1103/PhysRevX.8.011019.

[44] Patryk Lipka-Bartosik, Paweł Mazurek, and Michał Horodecki. “Second law of thermodynamicsfor batteries with vacuum state” (2019), arXiv:1905.12072.

[45] W. Pusz and S. L. Woronowicz. “Passive states and KMS states for general quantum systems”.Comm. Math. Phys. 58.3 (1978), pp. 273–290. doi: https://doi.org/10.1007/BF01614224.

[46] A. E Allahverdyan, R Balian, and Th. M Nieuwenhuizen. “Maximal work extractionfrom finite quantum systems”. Europhysics Letters (EPL) 67.4 (2004), pp. 565–571. doi:https://doi.org/10.1209/epl/i2004-10101-2.

[47] Robert Alicki and Mark Fannes. “Entanglement boost for extractable workfrom ensembles of quantum batteries”. Physical Review E 87.4 (2013). doi:https://doi.org/10.1103/physreve.87.042123.

[48] Peter Talkner, Eric Lutz, and Peter Hänggi. “Fluctuation theorems: Work is not an observable”.Phys. Rev. E 75 (5 2007), p. 050102. doi: https://doi.org/10.1103/PhysRevE.75.050102.

[49] Peter Talkner and Peter Hänggi. “Aspects of quantum work”. Phys. Rev. E 93 (2 2016),p. 022131. doi: https://doi.org/10.1103/PhysRevE.93.022131.

Accepted in Quantum 2020-12-08, click title to verify. Published under CC-BY 4.0. 21

[50] Martí Perarnau-Llobet et al. “No-Go Theorem for the Characterization of Work Fluctu-ations in Coherent Quantum Systems”. Phys. Rev. Lett. 118 (7 2017), p. 070601. doi:https://doi.org/10.1103/PhysRevLett.118.070601.

[51] Johan Åberg. “Truly work-like work extraction via a single-shot analysis”. Nature Communica-tions 4.1 (2013), p. 1925. doi: https://doi.org/10.1038/ncomms2712.

[52] Masahito Hayashi and Hiroyasu Tajima. “Measurement-based formulation of quantum heat en-gines”. Phys. Rev. A 95 (3 2017), p. 032132. doi: https://doi.org/10.1103/PhysRevA.95.032132.

[53] R. Sampaio et al. “Quantum work in the Bohmian framework”. Phys. Rev. A 97 (1 2018),p. 012131. doi: https://doi.org/10.1103/PhysRevA.97.012131.

[54] Lluís Masanes and Jonathan Oppenheim. “A general derivation and quantification ofthe third law of thermodynamics”. Nature Communications 8.1 (2017), p. 14538. doi:https://doi.org/10.1038/ncomms14538.

[55] A. E Allahverdyan, R Balian, and Th. M Nieuwenhuizen. “Maximal work extractionfrom finite quantum systems”. Europhysics Letters (EPL) 67.4 (2004), pp. 565–571. doi:https://doi.org/10.1209/epl/i2004-10101-2.

[56] Álvaro M. Alhambra et al. “Fluctuating Work: From Quantum Thermodynamical Iden-tities to a Second Law Equality”. Phys. Rev. X 6 (4 2016), p. 041017. doi:https://doi.org/10.1103/PhysRevX.6.041017.

[57] Piotr Ćwikliński et al. “Limitations on the Evolution of Quantum Coherences: Towards FullyQuantum Second Laws of Thermodynamics”. Phys. Rev. Lett. 115 (21 2015), p. 210403. doi:https://doi.org/10.1103/PhysRevLett.115.210403.

[58] F. L. Curzon and B. Ahlborn. “Efficiency of a Carnot engine at maximum power output”.American Journal of Physics 43.1 (1975), pp. 22–24. doi: https://doi.org/10.1119/1.10023.

[59] Matteo Lostaglio, Álvaro M. Alhambra, and Christopher Perry. “Elementary Thermal Opera-tions”. Quantum 2 (Feb. 2018), p. 52. doi: https://doi.org/10.22331/q-2018-02-08-52.

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A Preliminaries

The full information of the thermal engine in the framework of stroke operations is encoded in thejoint battery and working body state:

ρSB =∫dEdE′

∑i,j

%ij(E,E′) |εi〉〈εj |S ⊗∣∣E − εi⟩⟨E′ − εj∣∣B , (74)

where in general it is assumed a continuous and unbounded energy spectrum of the battery. However,the average quantities, like extracted work or exchanged heat, can be solely deduced from the effective,so called control-marginal state, defined as:

σS = TrB[SρSBS†] =∫dE

∑i,j

ρij(E,E) |εi〉〈εj |S , (75)

whereS =

∑i

|εi〉〈εi|S ⊗ Γεi (76)

is a unitary operator, and Γε is a shift operator

Γε =∫dE |E + ε〉〈E|B (77)

such that Γε |E〉B = |E + ε〉B. For the two-level working body we further represent the state σS as

σS = 12

(1− z αα∗ 1 + z

), (78)

and describe it by corresponding quantities, i.e. energy, passive energy and ergotropy:

ES = ω

2 (1 + z), PS = ω

2 (1− r), RS = ω

2 (z + r) (79)

where r =√z2 + |α|2 ∈ [0, 1] and z ∈ [−1, 1].

B Heat-stroke characterization

B.1 Thermal operation

Let us consider a general heat-stroke USH , obeying condition

[USH , HH + HS ] = 0. (80)

If τH is Gibbs state with respect to the Hamiltonian HH , then, the following channel

Λ[|εi〉〈εj |

]= TrH [USH(|εi〉〈εj | ⊗ τH)U †SH ] (81)

is a thermal operation. In general thermal operation can be parameterized as:

Λ[|εi〉〈εi|

]=∑j

pij |εj〉〈εj | , Λ[|εi〉〈εj |

]=

∑m,n

ωmn=ωij

ξmn |εm〉〈εn| , (82)

where the second sum is over all frequencies ωmn = εm − εn such that ωmn = ωij .

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B.2 Transformation of the control-marginal stateWe would like to analyze how state σS evolves according to the heat-stroke. In general the total stateof system and battery evolves as:

ρSBH-stroke−−−−−→ ρ′SB = TrH [USH(ρSB ⊗ τH)U †SH ]

=∫dEdE′

∑i,j

%ij(E,E′) TrH [USH(|εi〉〈εj |S ⊗ τH)U †SH ]⊗∣∣E − εi⟩⟨E′ − εj∣∣B . (83)

According to the relation (82), we obtain:

ρ′SB =∫dEdE′

[∑i,j

%ii(E,E′)pij |εj〉〈εj | ⊗∣∣E − εi⟩⟨E′ − εi∣∣

+∑i 6=j

%ij(E,E′)∑m,n

ωmn=ωij

ξmn |εm〉〈εn| ⊗∣∣E − εi⟩⟨E′ − εj∣∣ ]. (84)

Finally, the corresponding state σS transform as:

σSH-stroke−−−−−→ σ′S = TrB[Sρ′SBS†]

=∫dE

∑i,j

%ii(E,E)pij |εj〉〈εj |+∑i 6=j

%ij(E,E)∑m,n

ωmn=ωij

ξmn |εm〉〈εn|

= Λ[σS].

(85)

In particular, for the two-level working body we obtain

σS = 12

(1− z αα∗ 1 + z

)H,C−−→ 1

2

(1− z′ e−iδγαeiδγα∗ 1 + z′

)= σ′S . (86)

In our framework such a transformation corresponds to the hot bath step H or cold bath step C, andcan be fully characterized by the parameter λ ∈ [0, 1] and

γ ∈ [0,√

(1− λak)(1− λ)], (87)

such thatz′ = z − λ [z(1 + ak) + 1− ak] , (88)

where ak = e−βkω and k = H,C. The phase δ can be arbitrary, however, it plays no role in ther-modynamics of the engine since quantities given by Eq. (79) depends only on the magnitude of theoff-diagonal elements. That is why we further assume that α is real, i.e. α = α∗, and δ = 0.

Furthermore, one can easily show that heat defined for this process is equal to:

Q = −Tr[HH(USH ρU †SH − ρ)

]= Tr

[HS(Λ

[σS]− σS)

]. (89)

C Work-stroke characterizationC.1 Translational invariance and energy conservationWe start with showing that any unitary USB which obeys conditions

[HS + HB, USB] = 0 and [Γε, USB] = 0, (90)

where Γε is the shift operator (77), can be expressed in a general form:

USB =∫dE

∑i,j

Vij |εi〉〈εj |S ⊗ |E − εi〉〈E − εj |B , (91)

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where Vij are some complex entries such that the following operator

V =∑i,j

Vij |εi〉〈εj |S (92)

is unitary. In order to prove this, let us consider a general energy conserving unitary, such that it isblock-diagonal in energy basis, and within each energy block E we have arbitrary unitary Vij(E). Byfollowing calculations one can show that:

[Γε, USB] =∫dEdE′

∑i,j

Vij(E) |εi〉〈εj |S ⊗[∣∣E′ + ε

⟩⟨E′∣∣B , |E − εi〉〈E − εj |B

]=∫dE

∑i,j

Vij(E) |εi〉〈εj |S ⊗(|E − εi + ε〉〈E − εj |B − |E − εi〉〈E − εj − ε|B

)=∫dE

∑i,j

[Vij(E)− Vij(E + ε)] |εi〉〈εj |S ⊗ |E − εi + ε〉〈E − εj |B = 0 ⇐⇒ Vij(E) = Vij(E + ε) ≡ Vij .

(93)

By means of operator S (76), the unitary USB can be rewritten in the form:

USB = S†(V ⊗ 1B)S, (94)

where 1B is the identity operator acting on battery Hilbert space.

C.2 Transformation of the control-marginal stateLet us now analyze how state ρSB transform under the action of USB operation, i.e.

ρSBW-stroke−−−−−→ ρ′SB = USB ρSBU

†SB = S†(V ⊗ 1B)SρSBS†(V † ⊗ 1B)S. (95)

From this follows that transformation of the corresponding state σS is given by

σSW-stroke−−−−−→ σ′S = TrB[SUSB ρSBU †SBS

†] = TrB[V⊗1BSρSBS†V †⊗1B] = V TrB[SρSBS†]V † = V σSV

†.(96)

C.3 Work and ergotropyWe prove that the change of the average battery energy (i.e. work W ) is equal to the change of theergotropy of the state σS . From the definition of work and the structure of unitary USB (94), we have:

W = Tr[HB(USB ρSBU †SB − ρSB)

]= −Tr

[HS(S†V SρSBS†V †S − ρSB)

]= Tr

[(HS − V †HSV )SρSBS†

],

(97)

where we used a fact that [HS , S] = 0 (for simplicity we omitted the identity operators). SinceHS − V †HSV is operator acting only on the system Hilbert space S, we obtain finally:

W = Tr[(HS − V †HSV )σS

]= −∆RS , (98)

where the last equality follows from the fact that any change of the energy via the unitary transfor-mation V is equal to the change of the ergotropy of the state. In particular, if we consider a two-levelsystem (78), then the maximal work which can be extracted is equal to:

W = −R′S +RS ≤ RS = ω

2 (z + r), (99)

where we put ∆RS = R′S −RS .

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D Characterization of stroke operations (summary)To summarize, for the heat-stroke we present following relations:

Q = ∆ES , σSH-stroke−−−−−→ σ′S = Λ

[σS], (100)

and analogous for the work-stroke:

W = −∆RS , σSW-stroke−−−−−→ σ′S = V σSV

†. (101)

It is seen that quantities like exchanged heat Q and work W solely depend on the state σS , and wederive the rules how it transforms under stroke operations, where Λ[·] is arbitrary thermal operation,and V is arbitrary unitary operator.

Especially it shows that arbitrary function f(W,Q) (e.g. efficiency or extracted work per cycle) canbe derived solely from the evolution of the σS . In particular, any optimization problem based on thefunction f(W,Q) can be defined on the domain of all possible transformations of the state σS .

E Characterization of the ergotropy extraction processThe following section is about ergotropy extraction process via the heat-stroke, i.e. coupling with heatbath in inverse temperature β. In this section a = e−βω (for simplicity we also put ω = 1), and werefer to quantities given by Eq. (79) and state transformation (85).

E.1 Ergotropy extraction and passive energy accumulationWe would like to show that whenever a < 1, we have

∆RS > 0 =⇒ ∆PS > 0. (102)

In order to prove this, firstly we reveal that

∆PS ≤ 0 =⇒ z′ ≤ 0. (103)

Let us assume that z′ > 0 (88), what leads us to the formula:

|z′| − |z| = z − |z| − λ[z(1 + a) + 1− a]. (104)

Then, it is enough to observe that |z′| < |z|, since whenever

|z′| < |z| =⇒ r′ < r =⇒ ∆PS > 0, (105)

what according to the assumption z′ > 0 implies Eq. (103). The conclusion is straightforward if z > 0(note that z 6= 0 since otherwise z′ ≤ 0), i.e. in this case we obtain:

|z′| − |z| = −λ(|z|(1 + a) + 1− a) < 0, (106)

since |z| ∈ [0, 1]. On the other hand, for z < 0 we have following formula:

|z′| − |z| = −2|z| − λ(−|z|(1 + a) + 1− a). (107)

The maximum value of this difference is given by:

maxa,λ

[|z′| − |z|

]= max

a,λ[−2|z| − λ(−|z|(1 + a) + 1− a)] = −2|z| −min

a,λ[λ(−|z|(1 + a) + 1− a)]. (108)

However, above minimum is achieved for λ = 1 and a = 1, and equal to −2|z|, what reveals that|z′| ≤ |z|. Furthermore, according to our assumption that a < 1, we proved that |z′| < |z|.

Finally, whenever ∆PS ≤ 0 it implies z′ ≤ 0 (103), and in this case ∆RS can be rewritten in theform:

∆RS = 12(r′ − |z′| − r − z). (109)

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E.1.1 No coherences α = 0

For the state without coherences, i.e. α = 0, it implies that α′ = γα = 0, and we have r′ = |z′|. Thisleads us straightforwardly to conclusion that whenever

∆PS ≤ 0 =⇒ ∆RS = 12(r′ − |z′| − r − z) = −1

2(r + z) = −RS ≤ 0. (110)

E.1.2 Non-zero coherences α 6= 0

On the other hand, for coherences we have the following chain of implications:

∆PS ≤ 0 =⇒ r ≤ r′ =⇒ |z| < |z′| =⇒ γ|z| < |z′| =⇒ γ2z2 < z′2 =⇒ γ2α2z2 < α2z′2

=⇒ (z′2 + γ2α2)z2 < (z2 + α2)z′2 =⇒ r′2z2 < r2z′2 =⇒ 2r′|z| < 2r′|z|

=⇒ 2r′|z|+ z2 + z′2 + γ2α2 < 2r′|z|+ z2 + z′2 + α2 =⇒ (r′ + |z|)2 < (r + |z′|)2 =⇒ r′ + |z| < r + |z′|=⇒ r′ − z < r + |z′| =⇒ r′ − |z′| − r − z < 0 =⇒ ∆RS < 0.

(111)

Finally, from (110) and (111) follows (102).

E.2 Maximal ergotropy extraction

Let us consider a positive ergotropy extraction, i.e. ∆RS > 0. From the previous considerations weobtain

∆RS > 0 =⇒ ∆PS > 0 =⇒ r > r′ =⇒ z′ > z, (112)

where the last inequality implies that

h = −z(1 + a)− 1 + a > 0 =⇒ z < −1− a1 + a

, (113)

and we used abbreviation z′ = z + λh (88). The last inequality in the above formula means that theinitial state is less excited than the Gibbs state.

In the following consideration we assume that h > 0, as a necessary condition for the positiveergotropy extraction. We will prove that for all such protocols, the maximal value of ∆RS and minimalvalue of ∆PS is for λ = 1.

E.2.1 No initial coherences (α = 0)

Maximal change of the ergotropy

Due to the fact that z = −|z|, the initial state has no ergotropy, i.e. RS = 12(z + |z|) = 0.

In accordance, the change of ergotropy is solely dictated by the final value:

∆RS = 12(z′ + |z′|), (114)

and it is positive whenever z′ > 0, what is fulfilled if and only if λ ∈ (λ0, 1] (where λ0 = − zh). If this

is true, we can then rewritten formula (114) as follows

∆RS = z + λh, (115)

what indicates that it is an increasing linear function with maximum at the point λ = 1, and given by

maxλ∈(λ0,1]

[∆RS ] = z + h ≡ ∆R0. (116)

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Minimal change of the passive energy.

Similarly, a change of the passive energy for the diagonal state is given by

∆PS = 12(|z| − |z′|), (117)

what in the regime where ∆RS > 0 gives us

∆PS = |z| − λh2 , (118)

which is a decreasing linear function, and reaches the minimum in the point λ = 1:

minλ∈(λ0,1]

[∆PS ] = |z| − h

2 = ∆P0. (119)

E.2.2 With initial coherences (α 6= 0)

Maximal dumping factor

Firstly, we calculate a derivative with respect to γ:

d

dγ

∆RS∆PS

= ∆Eα2γ

2∆P 2r′> 0. (120)

From that follows that the ratio is maximal for the highest γ for any λ, thus we further only consideran extremal case where γ =

√(1− λ)(1− aλ) (see (87)).

Maximal change of the ergotropy

The derivative of ∆RS with respect to λ is equal to

d

dλ∆RS = 1

2d

dλ(z′ + r′) = 1

4r′ [2h(r′ + z′)− α2(1 + a− 2aλ)]. (121)

Thus, it is an increasing function whenever

∆RS >α2

4h(1 + a)− 12(z + r)− λaα

2

2h ≡ A− λB. (122)

Let us suppose that exist such λ0 that above inequality is satisfied. Then, the derivative of the lefthand side is positive, i.e. d

dλ∆RS∣∣λ=λ0

> 0, and the derivative of right hand side is negative, i.e.ddλ(A− λB)

∣∣λ=λ0

= −B < 0. This implies that this inequality is also satisfied for all λ > λ0, and as aconsequence ∆RS is an increasing function with respect to λ in the interval λ ∈ [λ0,∞).

Next, we solve the equation ∆RS = 0, which gives us

r′ = r + z − z′ = r − λh ⇐⇒ (r′)2 = r2 − 2λh+ λ2h2 ⇐⇒ (z + λh)2 + (1− aλ)(1− λ)α2

= r2 − 2λh+ λ2h2 ⇐⇒ aα2λ2 + (2h(r + z)− (1 + a)α2)λ = 0 ⇐⇒ λ = 0 ∨ λ = 1 + r − z − 2ha(r − z) .

(123)

Later we use an abbreviation:λ0 = 1 + r − z − 2h

a(r − z) . (124)

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The derivative in a point λ0 is then equal to:

d

dλ∆RS

∣∣λ=λ0

= 14r′ [2h(r′ + z′)− α2(1 + a− 2aλ0)] = 1

4r′ [−2h(r + z) + (r2 − z2)(1 + a)], (125)

and it is seen that

d

dλ∆RS

∣∣λ=λ0

> 0 ⇐⇒ −2h(r + z) + (r2 − z2)(1 + a) > 0 ⇐⇒ r < z + 2h1 + a

∨ r > −z. (126)

However r =√z2 + α2 > |z| ≥ −z, thus we proved that ∆RS > 0 whenever λ ∈ (λ0, 1], and in this

interval ddλ∆RS > 0. Finally, it shows that maximal positive value of ∆RS is in the point λ = 1, and

it is equal to:

maxλ∈(λ0,1]

[∆RS ] = h+ 12(z − r) = ∆R0 −RS , (127)

where RS = 12(z + r) is initial ergotropy of the system, and ∆R0 is given by Eq. (116).

Minimal change of the passive energy

Next, we analyze the function ∆PS . The derivative with respect to λ is equal to:

d

dλ∆PS = − 1

2r′ [(h2 + aα2)λ+ zh− 1

2α2(1 + a))], (128)

what gives us two intervals of monotonicity, i.e.

d

dλ∆PS > 0 ⇐⇒ λ <

α2(1 + a)− 2hz2(h2 + aα2) ,

d

dλ∆PS < 0 ⇐⇒ λ >

α2(1 + a)− 2hz2(h2 + aα2) , (129)

what proves that ∆PS has at most one extremum, which is a maximum. Due to this fact in the intervalof positive ergotropy extraction, i.e. λ ∈ (λ0, 1], the function ∆PS has minimum either in λ = λ0 (124)or λ = 1.

Next, we prove that ∆PS∣∣λ=1 < ∆PS

∣∣λ=λ0

. We have

∆PS∣∣λ=1 = 1

2(r − |z + h|), ∆PS∣∣λ=λ0

= ∆ES∣∣λ=λ0

= 12hλ0, (130)

thus∆PS

∣∣λ=1 < ∆PS

∣∣λ=λ0

⇐⇒ r − |z + h| < hλ0. (131)

Let us firstly exclude situation where z + h < 0, which implies that ∆RS ≤ 0 (see proof in the nextsubsection). Then, we consider an opposite case where z + h ≥ 0:

r − |z + h| − hλ0 = r − z − 2h− hr − z − 2ha(r − z) < 0 ⇐⇒ (a(r − z)− h)(r − z − 2h) < 0

⇐⇒ a(r − z)− h > 0 ⇐⇒ r > z + h

a,

(132)

were we used a fact that r − z − 2h < 0 in order to have λ0 < 1. We can further estimate that

z + h

a= z + 1

a(−z(1 + a)− 1 + a) = −1

a(1 + z) + 1 ≥ −z = |z|. (133)

However, r > |z| what finally proves that for λ ∈ (λ0, 1] the minimal value of ∆PS is in the pointλ = 1, and equal to:

minλ∈(λ0,1]

[∆PS ] = 12(r − z − h) = ∆P0 +RS , (134)

where ∆P0 is given by Eq. (119).

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0.2 0.4 0.6 0.8 1.0a

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0z

Figure 9: We plot curves that set bounds on the parameter of the initial state 1 ≥ z ≥ −1 (78) that enablespositivity and convexity of ergotropy extration ∆R. For all z below z0(a,B) (142), there exists a thermal processgiven by 1 ≥ λ ≥ λ0 such that ∆R is positive. For all z below z+(a,B) (141), ∆R is convex (for all λ). Forno coherences present B = 0, the bounds coincide z0(a, 0) = z+(a, 0) = − 1−a

a (red dashed line). For maximalcoherences B = 1, we plot z0(a, 1) (black dashed line) and z+(a, 1) (green dashed line). The bound a > 1/2 isvisible. In this regime, both z0(a,B) and z+(a,B) are monotonically decreasing functions of B, and z+ ≥ z0. Wesee that presence of coherences has a detrimental effect on the range of parameters that enable positive ergotropyextraction. If z = z0(a,B), only extremal thermal process leads to positive ergotropy extraction.

E.2.3 Positive ergotropy extraction

We would like to summarize conditions for positive ergotropy extraction. Whenever α = 0 or α 6= 0the necessary condition is that h > 0 from which follows that z < −1−a

1+a . Specifically for the case α = 0we have a constraint:

λ0 = − zh< 1 ⇐⇒ z + h > 0 ⇐⇒ z < −1− a

a, (135)

what in terms of the energy ES = ω2 (1 + z) is equivalent to

∆RS > 0 =⇒ ES < ω(1− 12a). (136)

For the case α 6= 0 the necessary condition is

λ0 = 1 + r − z − 2ha(r − z) < 1 ⇐⇒ r − z − 2h < 0 ⇐⇒ r + z + 2(az + 1− a) < 0. (137)

One can show that also for the case α 6= 0 it is necessary that z+h > 0 (i.e. z < −1−aa ), since in other

case we have

r + z + 2(az + 1− a) ≥ 2(az + 1− a) ≥ 0 =⇒ λ0 > 1 =⇒ ∆RS ≤ 0, (138)

where we also used a fact that r ≥ |z| = −z. This proves that (136) is valid for arbitrary α.Further, we can derive bounds on the parameter of the initial state 1 ≥ z ≥ −1 that enables positivity

and convexity of ergotropy extraction ∆R (Fig. 9). From the definition of ergotropy change (109), andputting α2 = B(1 − z2), where B ∈ [0, 1], direct calculation leads to the conclusion that the secondderivative is non-negative iff

d2

dλ2 ∆RS ≥ 0 =⇒[(1 + z)(4−B(1− z)) + a2(1− z)(4−B(1 + z))− 2a(4−B − z2(2−B))

]≥ 0,(139)

which is satisfied in two regimes, i.e.

d2

dλ2 ∆RS ≥ 0 =⇒ z ≤ z+ ≤ 0 and z ≥ z− ≥ z+, (140)

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with

z±(a,B) = −(1− a)(2 + 2a±√

(2−B)2(1 + a2) + 2a(−B2 + 6B − 4))2a(2−B) +B(1 + a2) . (141)

Further, one can show that λ0 (124) is a monotonously increasing function of z, and thereforeachieves maximum at z = 1. Therefore, the maximum z allowable is calculated from the conditionλ0 = 1, and we have

∆RS > 0 =⇒ z ≤ z0(a,B) = −(1− a)(1 + 2a)−√

4a2 + 4a(3B − 2) + (2−B)2

2a(1 + a) +B/2 ≤ 0. (142)

F Three-stroke engineF.1 Order of stepsThree-step engine is composed of three unitary operations USH , USC and USB. The state of the workingbody σS can be parametrized by the energy E and coherence α, such that it evolves as follows

(E0, α0) 1−→ (E1, α1) 2−→ (E2, α2) 3−→ (E3 = E0, α3 = α0), (143)

where we do not yet assume in which order we have used operations. For each energy En one candefine corresponding ergotropy Rn and passive energy Pn, such that En = Rn + Pn.

Let us write changes of the working body energy (ergotropy and passive energy) for each step:

∆EBS = ∆RBS < 0,∆EHS = ∆RHS + ∆PHS ,∆ECS = ∆RCS + ∆PCS ,

(144)

where the first inequality is necessary in order to have a positive efficiency. From the conservation ofstate functions we have further

∆PHS = −∆PCS ,∆RHS + ∆RCS = −∆RBS > 0.

(145)

The labels H and C at that moment just distinguishes between two different heat baths and so far wedo not assume that TH > TC .

We see that ∆RHS > 0 or ∆RCS > 0, what implies that ∆PHS 6= 0 and ∆PCS 6= 0 (see Eq.(102)).Without loss of generality we can put ∆PCS < 0, what further implies that ∆RCS ≤ 0, and as aconsequence ∆ECS < 0. On the other hand, we conclude also that ∆RHS > 0, ∆PHS > 0 and ∆EHS > 0.Furthermore, we have a freedom to assume that E0 is the lowest energy. Then, the H step has to be thefirst one since ∆EBS < 0 and ∆ECS < 0. Let us further suppose that the second step is C. This howevercomes back the working body to the initial state, due to the fact that P0 → P0 + ∆PHS + ∆PCS = P0.Thus, in order to close the cycle, the last B step has to be the identity, what results with zero efficiency.

Finally, we deduct an unique order of steps for positive efficiency defined as:

η = W

QH= −∆EBS

∆EHS, (146)

which is given by:

(E0, α0) H−→ (E1, γ1α0) B−→ (E2, γ−12 α0) C−→ (E0, α0), (147)

where E0 is the lowest energy, and we used a fact that H and C are thermal operations (whereγ1 < 1, γ2 < 1). Let us now split the problem to two cases.

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F.2 No initial coherences (α0 = 0)F.2.1 Hot bath step (heat-stroke)

In this case ∆RHS > 0, i.e. it is an ergotropy extraction process. This implies that initial ergotropyR0 = 0 and P0 = E0 < ω(1− 1

2aH ) (136). According to this we can write:

∆RHS = 2aH(ω − E0)− ω − 2h(λ) = ∆R0(E0)− 2h(λ),∆PHS = (ω − E0)(1− aH) + h(λ) = ∆P0(E0) + h(λ),

(148)

where λ ∈ (λ0, 1] (see Eq. (124)) such that h(λ) ∈ [0,∆R0(E0)/2). According to (115) and (118),h(λ) = ωh

2 (1 − λ). Function h(λ) is just another parametrization of all possible protocols for which∆RHS > 0, specifically h(1) = 0 corresponds to the extremal process and h(λ0) = ∆R0(E0)/2.

F.2.2 Battery step (work-stroke)

For the work-stroke USB, the energy transfer is limited by the ergotropy (99), i.e.

W = −∆RBS = R1 −R2 ≤ R1 = R0 + ∆RHS = ∆RHS , (149)

what implies that∆EBS = −∆RHS + ξ = −∆R0(E0) + 2h(λ) + ξ, (150)

where ξ ∈ [0,∆R0(E0)− 2h(λ)). A non-zero ξ is for protocols that creates coherences in the state σSsuch that α2 6= 0.

F.2.3 Cold bath step (heat-stroke)

The last step C is used to bring the system back to the initial state such that

∆ECS = E0 − E2 < 0. (151)

Since step C is a thermal operation we have

∆ECS = λ[ωaC − E2(1 + aC)] < 0, (152)

what implies that E2 > ωaC1+aC . If this is satisfied we can further formulate necessary condition for

closing the cycle in the form:

E0 − E2 = λ[ωaC − E2(1 + aC)] ⇐⇒ E0 − E2 ≥ ωaC − E2(1 + aC) ⇐⇒ E2 ≥ ω −E0aC

⇐⇒ h(λ) + ξ ≥ ω − E0(1 + a−1C )−∆P0(E0) = ωaH − E0(aH + a−1

C ) ≡ K(E0).(153)

F.2.4 Temperature regimes

Now, we are able to derive temperature regimes for which η > 0, and we close the cycle. Firstly, weobserve that in order to have positive efficiency (136), we need

E0 < ω(1− 12aH

), (154)

which is the necessary condition for ergotropy extraction. From this we easily obtain aH > 12(1−E0

ω)≥ 1

2 ,

what gives us the possible range of hot temperatures, i.e.

aH ∈ (12 , 1]. (155)

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In order to derive range for the cold temperature, firstly let us observe that

h(λ) + ξ < ∆R0(E0), (156)

and thus we can estimate that

E2 = E0 + ∆P0(E0) + h(λ) + ξ < E0 + ∆P0(E0) + ∆R0(E0) = aH(ω − E0). (157)

Finally, from closing the cycle condition (153), we obtain

E2 ≥ ω −E0aC⇐⇒ aC ≤

E0ω − E2

<E0

ω − aH(ω − E0) < 2− a−1H . (158)

Finally, the range of cold temperature (with a fixed hot temperature) is given by

aC ∈ [0, 2− a−1H ). (159)

This implies that aC < aH what means that TC < TH . Further, the temperature intervals can beexpressed as a single inequality, namely:

eβHω + e−βCω < 2. (160)

F.2.5 Maximal efficiency and work production

We can proceed now with estimation of the efficiency η and work production P . From the definitionwe have:

η(E0, λ, ξ) = ∆R0(E0)− [2h(λ) + ξ]∆R0(E0) + ∆P0(E0)− h(λ) , (161)

andP (E0, λ, ξ) = ∆R0(E0)− [2h(λ) + ξ]. (162)

For a fixed aH ∈ (12 , 1] and aC ∈ [0, 2 − a−1

H ), the problem reduces to the maximization over allE0 ∈ [0, ω(1− 1

2aH )) and λ ∈ (λ0, 1], ξ ∈ [0,∆R0(E0)− 2h(λ)), such that h(λ) + ξ ≥ K(E0).Let us now split the problem into two parts: 1) E0 ≥ ε0, and 2) E0 < ε0 where

ε0 = ωaCaH

1 + aCaH(163)

1) For the first case we have

K(E0) = ωaH − E0(aH + a−1C ) ≤ ωaH − ε0(aH + a−1

C ) = 0, (164)

what shows that condition (153) is satisfied for all λ and ξ. It leads us to the maximal efficiency forλ = 1, such that h(1) = 0, ξ = 0, and E0 = ε0 i.e.

maxE0,λ,ξ

[η(E0, λ, ξ)] = maxE0

[ ∆R0(E0)∆R0(E0) + ∆P0(E0) ] = ∆R0(ε0)

∆R0(ε0) + ∆P0(ε0) = 2aH(ω − ε0)− ωaHω − ε0(1 + aH)

= aH(1− aC) + aH − 1aH(1− aC) = 1− 1− aH

aH(1− aC) ≡ η1.

(165)

The maximal work production in this case is also straightforward:

maxE0,λ,ξ

[P (E0, λ, ξ)] = maxE0

[∆R0(E0)] = ∆R0(ε0) = 2aH(ω − ε0)− ω = ω[ 2aH1 + aCaH

− 1] ≡ P1. (166)

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2) For the second case, where K(E0) > 0, one can estimate that

η(E0, λ, ξ) = ∆R0(E0)− (2h(λ) + ξ)∆R0(E0) + ∆P0(E0)− h(λ) ≤

∆R0(E0)− (h(λ) + ξ)∆R0(E0) + ∆P0(E0) ≤

∆R0(E0)−K(E0)∆R0(E0) + ∆P0(E0)

= 1− 1aH

ω − E0(1 + a−1C )

ω − E0(1 + a−1H )≡ f(E0).

(167)

The function f(E0) is increasing whenever

aC < 2− a−1H =⇒ df(E0)

dE0= − ω

aH

a−1H − a

−1C

[ω − E0(1 + a−1H )]2

> 0, (168)

what is satisfied if engine works in the cyclic mode (159). Finally, since we consider situation whereE0 < ε0, then

η(E0, λ, ξ) ≤ f(E0) < f(ε0) = 1− 1aH

1− aH1− aC

= η1. (169)

In analogy, for the extracted work, one can estimate:

P (E0, λ, ξ) = ∆R0(E0)− (2h(λ) + ξ) ≤ ∆R0(E0) + ∆P0(E0)− ω + E0(1 + a−1C )− h(λ)

≤ ω(aH − 1) + E0(a−1C − aH)− h(λ) ≤ ω(aH − 1) + E0(a−1

C − aH) < ω(aH − 1) + ε0(a−1C − aH)

= ω[ 2aH1 + aCaH

− 1] = P1.

(170)

Finally, the maximum over all possible protocols which close the cycle is given by

maxE3=E0

[η]

= η1, maxE3=E0

[P]

= P1. (171)

The maximum efficiency and work production is simultaneously achieved for the unique protocol, suchthat E0 = ε0, λ = 1 and ξ = 0.

F.3 With initial coherences (α0 6= 0)F.3.1 Hot bath step (heat-stroke)

For the state with initial coherences there is some non-zero initial ergotropy, i.e. R0 = 12(z+r), however,

in order to have positive efficiency it is still necessary that E0 ∈ [0, ω(1− 12aH )) (136). Further, for the

H step we proved that

∆RHS = ∆R0(E0)−R0 − g(λ),∆PHS = ∆P0(E0) +R0 + g(λ)− h(λ),

(172)

for λ ∈ (λ0, 1], where g(λ) ∈ [0,∆R0(E0)−R0), and as previously g(1) = 0 correspond to the extremalprocess and g(λ0) = ∆R0(E0) − R0. In comparison to previous consideration for α0 = 0 we haveg(λ)

∣∣α0=0 = 2h(λ) and R0

∣∣α0=0 = 0. It was proven previously (134) that for λ ∈ (λ0, 1] the minimal

value of ∆PHS is given by ∆P0(E0) +R0, what implies that in this interval g(λ) ≥ h(λ).

F.3.2 Battery step (work-stroke)

In analogy to the previous case, for the work-stroke we have (99):

W ≤ R1 = R0 + ∆RHS , (173)

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thus we can represent change of the energy as

∆EBS = −∆R0(E0) + g(λ) + δ, (174)where δ ∈ (0,∆RHS − g(λ)). The important thing is that in this case parameter δ cannot be zero.It follows from the fact that α0 = γ2α2 6= 0, and as a consequence also α2 6= 0. However, δ = 0corresponds to the maximal ergotropy storing such that W = R1 and R2 = 0, what implies thatα2 = 0.

F.3.3 Cold bath step (heat-stroke)

For the C step we can derive an analogous condition:

E2 ≥ ω −E0aC⇐⇒ δ + g(λ)− h(λ) ≥ ω − E0(1 + 1

aC)−∆P0(E0) = K(E0). (175)

F.3.4 Temperature regimes

In analogy to the previous case, for α0 6= 0 we have a necessary condition for the positive ergotropyextraction (and positive efficiency) in the form:

E0 < ω(1− 12aH

). (176)

Moreover, the following inequality has to be fulfilled:

g(λ)− h(λ) + δ < ∆R0, (177)and

E2 = E0 + ∆P0(E0) + g(λ)− h(λ) + δ < E0 + ∆P0(E0) + ∆R0(E0) = aH(ω − E0). (178)Finally, the temperature regimes are the same as for the engine with α0 = 0, i.e.

eβHω + e−βCω < 2. (179)

F.3.5 Maximal efficiency and work production

The efficiency of the engine is given by

η(E0, λ, δ) = ∆R0(E0)− g(λ)− δ∆R0(E0) + ∆P0(E0)− h(λ) ≤

∆R0(E0)− [δ + g(λ)− h(λ)]∆R0(E0) + ∆P0(E0) , (180)

and work productionP (E0, λ, δ) = ∆R0(E0)− g(λ)− δ. (181)

Once again we split the problem into two parts: 1) E0 ≥ ε0, and 2) E0 < ε0.1) For the first case due to the fact that g(λ) − h(λ) ≥ 0, and we can put g(λ) = h(λ) = 0, since

K(E0) ≤ 0 (such that condition (175) is always fulfilled), it straightforwardly leads us to the followingbound:

η(E0, λ, δ) ≤∆R0(E0)− δ

∆R0(E0) + ∆P0(E0) ≤∆R0(ε0)− δ

∆R0(ε0) + ∆P0(ε0) < η1, (182)

since we have shown that δ > 0. In analogy, the work production in this case is bounded by

P (E0, λ, δ) = ∆R0(E0)− g(λ)− δ ≤ ∆R0(ε0)− δ < P1. (183)2) For the second range of energies, i.e. E0 < ε0, from (175) we obtain exactly the same estimation

as previously

η(E0, λ, δ) ≤∆R0(E0)− [δ + g(λ)− h(λ)]

∆R0(E0) + ∆P0(E0) ≤ f(E0) < η1,

P (E0, λ, δ) ≤ ω(aH − 1) + E0(a−1C − aH) < ω(aH − 1) + ε0(a−1

C − aH) = P1,

(184)

what proves that the maximal efficiency η1 (and work production P1) cannot be reached for the enginewith non-zero initial coherence α0 6= 0.

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G Many-stroke generalization

In this section we consider a particular heat engine which is consisted of n subsequent ergotropyextractions such that working body evolution is following:

(E0, α0) H−→ (E′1, α′1) B−→ (E1, α1) H−→ (E′2, α′2) B−→ (E2, α2) H−→ . . .H−→ (E′n, α′n) B−→ (En, αn) C−→ (E0, α0).

(185)We assume further that each step H is the ergotropy extraction, i.e. ∆RS > 0, and each step B isergotropy storing such that W > 0.

G.1 Extremal protocol

Let us firstly consider a particular (extremal) protocol such that any heat-stroke U (k)SH is the extremal

thermal process and any work-stroke U (k)SB is the maximal ergotropy storing. For this case energies of

the working body are equal to:

E′k = akH(ω − E0) ≡ E′k, Ek ≡ ω − akH(ω − E0) = Ek, (186)

and the total sum of energy changes are given by:

∆RHS =n∑k=0

∆R0(Ek) =n∑k=0

[2aH(ω − Ek)− ω] = 2aH(ω − E0)(1− anH)1− aH

− nω ≡ ∆Rn0 (E0),

∆PHS =n∑k=0

∆P0(Ek) =n∑k=0

(1− aH)(ω − Ek) = (ω − E0)(1− anH) ≡ ∆Pn0 (E0),

∆EBS = −∆Rn0 (E0).

(187)

G.2 General protocol

G.2.1 Heat- and work-stroke

From the assumptions that all hot bath steps are the ergotropy extractions, from which follows thatin general each of them can be parameterized as follows

∆RHkS = ∆R0(Ek)−Rk − g(λk),∆PHkS = ∆P0(Ek) +Rk + g(λk)− h(λk).

(188)

In order to fulfill this condition, any energy Ek < ω(1− 12aH ) for k = 1, 2, . . . , n− 1.

For work-strokes we assume that each of them leads to the positive work, i.e. Wk = −∆EBkS > 0,then one can write down:

∆EBkS = −∆R0(Ek) + g(λk) + δk. (189)

We assume as previously that each λk ∈ (λ0, 1] and δk ∈ [0,∆R0(Ek)−g(λk)), however, we notice thatthe condition Ek < ω(1 − 1

2aH ) imposes here some additional constraints. Nevertheless, for arbitraryprotocol:

Ek = Ek−1 + ∆RHk−1S + ∆PHk−1

S + ∆EBk−1S = Ek−1 + ∆P0(Ek−1) + [g(λk−1)−h(λk−1) + δk−1, (190)

where the last term is always non-negative. For k = 1 we get

E1 = E0 + ∆P0(E0) + [g(λ0)− h(λ0) + δ0] = E1 + [g(λ0)− h(λ0) + δ0] ≥ E1. (191)

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Further, if Ek−1 ≤ Ek−1 then

Ek = Ek−1 + ∆P0(Ek−1) ≤ Ek−1 + ∆P0(Ek−1) + [g(λ0)− h(λ0) + δ0]≤ Ek−1 + ∆P0(Ek−1) + [g(λ0)− h(λ0) + δ0] = Ek,

(192)

since ∆P0(E) is a decreasing function with respect to E. Finally, we prove that Ek ≤ Ek for k =1, 2, . . . , n, where equality is for all λk = 1 and δk = 0. Having this we further assume that conditionEk < ω(1− 1

2aH ) is at least fulfilled for the extremal protocol (i.e. when Ek = Ek), and we put:

Ek = Ek + sk(~λ,~δ), (193)

where sk(~λ, ~δ) ≥ 0. Finally, we can write down

∆RHS =n∑k=1

∆RHkS =n∑k=1

[∆R0(Ek − sk(~λ, ~δ))−Rk − g(λk)]

= ∆Rn0 (E0)− 2aHs(~λ, ~δ)−n∑k=1

[Rk + g(λk)],

∆PHS =n∑k=1

∆PHkS =n∑k=1

[∆P0(Ek − sk(~λ, ~δ)) +Rk + g(λk)− h(λk)]

= ∆Pn0 (E0)− (1− aH)s(~λ,~δ) +n∑k=1

[Rk + g(λk)− h(λk)],

∆EBS =n∑k=1

∆EBkS =n∑k=1

[−∆R0(Ek − sk(~λ, ~δ)) + g(λk) + δk]

= −∆Rn0 (E0) + 2aHs(~λ,~δ) +n∑k=1

[g(λk) + δk],

(194)

where s(~λ,~δ) =∑nk=1 sk(~λ,~δ) ≥ 0. For the extremal protocol, such that each λk = 1 and δk = 0, then

s(~λ,~δ) = 0.

G.2.2 Closing the cycle condition

For the many-step engine necessary condition for closing the cycle in this case generalize to:

En ≥ ω −E0aC⇐⇒ F (~λ,~δ) ≥ ω − E0(1 + 1

aC)−∆Pn0 (E0) + (1− aH)s(~λ,~δ) ≡ K(E0, ~λ, ~δ), (195)

where

F (~λ,~δ) =n∑k=1

[g(λk)− h(λk) + δk] ≥ 0. (196)

G.2.3 Temperatures regimes

We have following constraints for energies: Ek < ω(1− 12aH ) for all k = 1, 2, . . . , n−1 and Ek ≤ Ek. In

particular, the energy En−1 just before the last ergotropy extraction has to satisfied those inequalities,from which follows that En−1 < ω(1 − 1

2aH ). From this one can derive the minimal possible value ofE0 which is given by

E0 < ω(1− 12anH

). (197)

On the other hand anH > 12− 2E0

ω

≥ 12 , what constitutes the possible range of hot temperature at which

engine can operate, i.e.aH ∈ (2−n, 1]. (198)

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The range for cold temperature can be derive as follows. Firstly, let us estimate an upper bound forthe energy En, i.e.

En = En−1 + ∆P0(En−1) + g(λn−1)− h(λn−1) + δn−1

< En−1 + ∆P0(En−1) + ∆R0(En−1) = aH(ω − En−1) ≤ aH(ω − En−1) = anH(ω − E0),(199)

where we used a fact that g(λk) − h(λk) + δk < ∆R0(Ek). Further, in order to close the cycle thefollowing has to be satisfied:

En ≥ ω −E0aC⇐⇒ aC ≤

E0ω − En

. (200)

Although, we haveE0

ω − En<

E0ω − anH(ω − E0) < 2− a−nH , (201)

where we used Eq. (197). Finally, the possible range of cold temperatures for a fixed aH is given bythe set

aC ∈ [0, 2− a−nH ), (202)

what gives us the condition

enβHω + e−βCω < 2. (203)

G.3 Maximal efficiency and work production

The upper bound for the many-step efficiency can be estimated as follows

η(E0, ~δ, ~λ) = ∆Rn0 (E0)− 2aHs(~λ,~δ)−∑nk=1[g(λk) + δk]

∆Rn0 (E0) + ∆Pn0 (E0)− (1 + aH)s(~λ,~δ)−∑nk=1 h(λk)

≤ ∆Rn0 (E0)− 2aHs(~λ,~δ)− F (~λ,~δ)∆Rn0 (E0) + ∆Pn0 (E0)− (1 + aH)s(~λ,~δ)

.

(204)

Furthermore, one can prove that for any x ≥ 0

aH ≤ 1⇒ (1 + aH)∆Rn0 (E0) ≤ 2aH(∆Rn0 (E0) + ∆Pn0 (E0))

⇒ ∆Rn0 (E0)− 2aHx∆Rn0 (E0) + ∆Pn0 (E0)− (1 + aH)x ≤

∆Rn0 (E0)∆Rn0 (E0) + ∆Pn0 (E0) ,

(205)

what leads to the algebraic bound for the efficiency, i.e.

η(E0, ~δ, ~λ) ≤ ∆Rn0 (E0)− 2aHs(~λ, ~δ)− F (~λ, ~δ)∆Rn0 (E0) + ∆Pn0 (E0)− (1 + aH)s(~λ, ~δ)

≤ ∆Rn0 (E0)− 2aHs(~λ,~δ)∆Rn0 (E0) + ∆Pn0 (E0)− (1 + aH)s(~λ,~δ)

≤ ∆Rn0 (E0)∆Rn0 (E0) + ∆Pn0 (E0) .

(206)

Work production of the engine is given by:

P (E0, ~δ, ~λ) = ∆Rn0 (E0)− 2aHs(~λ, ~δ)−n∑k=1

[g(λk) + δk]. (207)

Once again we split the problem into two parts: 1) E0 ≥ εn0 , and 2) E0 < εn0 , however, in this case

εn0 = ωaCanH

1 + aCanH. (208)

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1) For the first situation if each λk = 1 and δk = 0, we have Kn(E0, ~λ, ~δ) ≤ 0 and F (~λ,~δ) = 0, whatmakes the condition (195) fulfilled, and leads to the maximal value of efficiency:

maxE0,~δ,~λ

[η(E0, ~δ, ~λ)] = maxE0

[ ∆Rn0 (E0)∆Rn0 (E0) + ∆Pn0 (E0) = ∆Rn0 (εn0 )

∆Rn0 (εn0 ) + ∆Pn0 (εn0 ) ≡ ηn, (209)

where

∆Rn0 (εn0 ) = ω[ 2aH(1− anH)(1 + aCanH)(1− aH) − n],

∆Rn0 (εn0 ) + ∆Pn0 (εn0 ) = ω[ (1− anH)(1 + aH)(1 + aCanH)(1− aH) − n],

(210)

such thatηn = 1− (1− aH)(1− anH)

(1− anH)(1 + aH)− n(1 + aCanH)(1− aH) . (211)

According to above formula, the maximal value of the work production is given by:

maxE0,~δ,~λ

[P (E0, ~δ, ~λ)] = ∆Rn0 (εn0 ) = ω[ 2aH(1− anH)(1 + aCanH)(1− aH) − n] ≡ Pn. (212)

2) For the second subset of possible initial energy we obtain:

η(E0, ~δ, ~λ) ≤ ∆Rn0 (E0)− 2aHs(~λ,~δ)− F (~λ,~δ)∆Rn0 (E0) + ∆Pn0 (E0)− (1 + aH)s(~λ,~δ)

≤ ∆Rn0 (E0)− F (~λ, ~δ) + (1− aH)s(~λ, ~δ)∆Rn0 (E0) + ∆Pn0 (E0)

≤ 1−ω − E0(1 + 1

aC)

∆Rn0 (E0) + ∆Pn0 (E0) = 1− (ω − E0(1 + 1/aC))(1− aH)(ω − E0)(1− anH)(1 + aH)− ωn(1− aH) ≡ fn(E0).

(213)

One can further show that the function fn(E0) is increasing with respect to E0 if and only if

aC <(1− anH)(1 + aH)

n(1− aH) − 1. (214)

However, in order to close the cycle we have

aC < 2− a−nH =⇒ aC <(1− anH)(1 + aH)

n(1− aH) − 1. (215)

Finally, one can show that whenever E0 < εn0 , then

η(E0, ~δ, ~λ) ≤ fn(E0) < fn(εn0 ) = 1− (1− aH)(1− anH)(1− anH)(1 + aH)− n(1 + aCanH)(1− aH) = ηn. (216)

In analogy, the work production can be estimated by the condition (195), i.e.

P (E0, ~δ, ~λ) = ∆Rn0 (E0)− 2aHs(~λ,~δ)−n∑k=1

[g(λk) + δk]

≤ ∆Rn0 (E0) + ∆Pn0 (E0)− ω + E0(1 + 1aC

)− (1 + aH)s(~λ,~δ)−n∑k=1

h(λk)

≤ (ω − E0)(1− anH)1 + aH1− aH

− nω − ω + E0(1 + 1aC

)

= ω(1− anH)1 + aH1− aH

− nω − ω + E0[1 + 1aC− (1− anH)(1 + aH)

1− aH≤ ω[ 2aH(1− anH)

(1− aH)(1 + aCanH) − n].

(217)

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G.4 Maximal efficiency according to the number of stepsWe start by rewriting the general formula for the efficiency as

ηn = A+B

C +D, (218)

where A = 2aH − (1 + aHaC), C = A + (1 − aH), B = 2S(n) − (n − 1) − aHac(nan−1H − 1) and

D = B + S(n)1−aHaH

, with S(n) = a2H(1−an−1

H )1−aH for n > 1, and S(1) = 0.

Note that, for n = 1, B = D = 0. Therefore, to obtain ηn ≤ η1 it is enough to show thatAC ≥

A+BC+D , which is equivalent to A

C ≥BD for non-zero C and D. This in turn demands that (2aH −

(1 − aHaC))S(n)1−aHaH

≥ (1 − aH)(2S(n) − (n − 1) − aHaC(nan−1H − 1)). It is equivalent to showing

that

f(n, aH , aC) = n− 1 + aHaC(nan−1H − 1)− aH(1− an−1

H )1− aH

(1− aHaC) ≥ 0, (219)

where we exploited the form of S(n).First, let us notice that f(n, aH , aC) ≥ f(n, aH , 0). It is because ∂f(n,aH ,aC)

∂aC= a2

H(1−an−1H )

1−aH +aH(an−1

H n − 1). The first term is always positive, while the second is positive for all n > 1. Tosee this, let us point to the necesarry conditions for ergotropy extractions: 2aH − 1 ≥ 0 for the firstextraction, 2a2

H − 1 ≥ 0 for the second, up to 2anH − 1 ≥ 0 for the n-th one. Therefore, ∂f(n,aH ,aC)∂aC

≥ 0,and we have f(n, aH , aC) ≥ f(n, aH , 0).

Finally, let us note that f(n, aH , 0) = n − 1 − a2H(1−an−1

H )1−aH is non-negative for all n ≥ 1. Therefore,

we have ηn ≤ η1 for every n ≥ 1.

G.5 Maximal work extractionFor the work extraction protocol with the single heat bath, such that

(E0, α0) H−→ (E′1, α′1) W-stroke−−−−−→ (E1, α1) H−→ (E′2, α′2) W-stroke−−−−−→W-stroke−−−−−→ (E2, α2) H−→ . . .

H−→ (E′n, α′n) W-stroke−−−−−→ (En, αn).(220)

The maximal value of work can be deduced from the optimization over the function

max~λ,~δ

[W (~λ,~δ)] = max~λ,~δ

[∆Rn0 (E0)−

n∑k=1

[2aHsk(~λ,~δ) + g(λk) + δk]], (221)

where straightforwardly we obtain the maximum for λk = 1 and δk = 0, such that

Wmax = ∆Rn0 (E0) = 2aH(ω − E0)(1− anH)1− aH

− nω. (222)

H Stroke operations and Free energyH.1 Second Law proofLet us consider an arbitrary process with subsequent steps USH and USB. The energy and entropychange of the working body through the heat-stroke is ∆EHS and ∆SHS , and for the work-stroke wehave ∆EBS and ∆SBS . Let us define, a relative entropy between two states ρ and σ, as

S(ρ|σ) = Tr[ρ log ρ]− Tr[ρ log σ]. (223)

For arbitrary CPTP map Λ[·] it is valid an inequality:

S(ρ|σ) ≥ S(Λ[ρ]|Λ[σ]). (224)

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For the heat-stroke we have ∆EHS = Tr[HS(σ′S − σS)

]= Q (100), where σ′S = Λ[σS ], and Gibbs state

τH is invariant under the thermal operation, i.e. Λ[τH ] = τH . As a consequence we obtain:

S(σS |τH) ≥ S(σ′S |τH) =⇒

− Tr[σ′S log σ′S

]+ Tr[σS log σS ] ≥ Tr

[(σS − σ′S) log τH

]= β Tr

[HS(σ′S − σS)

],

(225)

what can be rewritten as a well-known Clausius inequality:

T∆SHS ≥ ∆EHS = Q. (226)

On the other hand, according to the relation for work-stroke (101), we have ∆SBS = 0. From these onecan easily show that

∆FS = ∆EHS + ∆EBS − T∆SHS ≤ ∆EBS . (227)Then, due to the energy conservation, i.e. ∆EBS = −∆EBB = −W , we finally obtain

W ≤ −∆FS . (228)

H.2 Free energy and ergotropy extractionLet us consider the ergotropy extraction via the heat-stroke, i.e. ∆RHS > 0. We will prove that forany such a process ∆FS < 0. Firstly, let us observe that state σS with passive energy PS has entropyequal to:

SS = S(σS) = −PS log[PS ]− (1− PS) log[1− PS ], (229)and since PS ∈ [0, 1

2 the entropy is an increasing function with respect to the passive energy of thestate. Especially, due to the result given by Eq. (134), the minimal change of the passive energy ∆PHSfor any ergotropy extraction (i.e. when ∆RHS > 0) is for the extremal process with λ = 1, and for astate without initial coherences such that α = 0, what implies also the minimal change of the entropy∆SHS . Furthermore, the change of the energy ∆EHS is maximal for the extremal process what showsthat if inequality T∆SHS > ∆EHS is fulfilled for λ = 1 and α = 0 it is also fulfilled for any otherergotropy extraction.

Let us then analyzed only this extremal case. If the initial energy is E0, then

β∆EHS = βω[x(1 + e−βω)− 1] ≡ f(x),∆SHS = S(xe−βω)− S(x) ≡ g(x),

(230)

where x = 1− E0/ω, andS(x) = −x log(x)− (1− x) log(1− x) (231)

Firstly, we show that g(x) is a convex function, i.e.

g′′(x) = 1− e−βω

x(1− x)(1− xe−βω) > 0 (232)

for x ∈ (0, 1). Next, we prove that function f(x) is a tangent line of the function g(x) in the pointx0 = 1

1+e−βω . Indeed, it is seen that

g′(x0) = e−βω[log(x0)− log(x0e−βω

)]− [log

(x0e−βω

)− log(x0)] = βω(1 + e−βω) = f ′(x), (233)

and f(x0) = g(x0) = 0. It proves that solution x0 is the only solution of the equation f(x) = g(x).From these follows that equation

T∆SHS = ∆EHS , (234)for the extremal thermal process can be only satisfied if qubit is in a Gibbs state, i.e. with energyE0 = 1− x0/ω = ωe−βω

1+e−βω , however as a consequence, it cannot be a work extraction process. Thus, forany ergotropy extraction we have

T∆SHS > ∆EHS , (235)what finally proves inequality ∆FS < 0.

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H.3 Free energy and work extractionLet us consider an arbitrary sequence of USH and USB where the total change of the free energy isequal to

∆FS = ∆FH1 + ∆FB1 + ∆FH2 + ∆FB2 + · · · =∑k

(∆FHk + ∆FBk ). (236)

Moreover, for each work-stroke we have ∆SBS = 0, thus∑k

∆FBk = ∆ES = −W, (237)

and as a consequenceW = −∆FS +

∑k

∆FHk , (238)

where each ∆FHk ≤ 0 (226). Then, we will prove the following: whenever ∆FS < 0 and state σS hasno initial ergotropy R0 = 0, it implies that W < −∆FS .

Firstly, let us observer that this is trivially obeyed if W ≤ 0. Otherwise, since for any work-stroke∆FBm = ∆RBm, we have ∑

k

∆RBk < 0. (239)

Next, since ergotropy is non-negative state function, we obtain the following:

R0 +∑k

(∆RBk + ∆RHk ) ≥ 0, (240)

and according to the assumption that R0 = 0, it implies that∑k

∆RHk ≥ −∑k

∆RBk > 0. (241)

It is seen that at least one heat-stroke is the ergotropy extraction, i.e. ∆RHm > 0 for some m, whatfurther implies ∆FHm < 0 (235). Finally, this proves that

W < −∆FS . (242)

I Many cycle evolutionI.1 Stationary and asymptotic stateIn order to analyze the engine after many cycles we define the following map

Tn(ρSB) = TrH,C(UnρU †n) (243)

where the action of the map on basis states is following:

|g,m〉〈g,m| T−→ anH(1− aC) |g,m+ n〉〈g,m+ n| ,

+n−1∑k=0

akH(1− aH) |g,m− n+ 2k〉〈g,m− n+ 2k|+ anHaC |e,m+ n〉〈e,m+ n|

|e,m〉〈e,m| T−→ |g,m− n〉〈g,m− n| .

(244)

If we trace out the battery we obtain the S map:

|g〉〈g| S−→ (1− anHaC) |g〉〈g|+ anHaC |e〉〈e| ,

|e〉〈e| S−→ |g〉〈g| .(245)

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As it seen, the marginal state does not depend on a battery state at all. Further, eigenvectors of theS map are equal to

~v1 = |g〉〈g| − |e〉〈e| S−→ −anHaC ~v1,

~v2 = |g〉〈g|+ anHaC |e〉〈e|S−→ ~v2,

(246)

and arbitrary qubit state can be decomposed in the basis ~v1, ~v2, i.e.

p |g〉〈g|+ (1− p) |e〉〈e| = (p− 11 + anHaC

)~v1 − p~v2. (247)

This leads us to the formula for a qubit state after m cycles, i.e.

p |g〉〈g|+ (1− p) |e〉〈e|~v2Sm−−→

[1 + (anHaC)m

1 + anHaC− p(anHaC)m

]|g〉〈g|

+[anHaC − (anHaC)m

1 + anHaC+ p(anHaC)m

]|e〉〈e| ,

(248)

what in the limit m→∞ gives

p |g〉〈g|+ (1− p) |e〉〈e| Sm−−−−→m→∞

11 + anHaC

|g〉〈g|+ anHaC1 + anHaC

|e〉〈e| . (249)

It also proves that above state is a fixed point under the transformation S.

I.2 Work fluctuationsI.2.1 Work distribution for three-stroke heat engine

We consider a final state of a battery after N = 2n cycles of running the three-stroke heat engine:

ρB = TrS,H,C [U2n1 (|0〉〈0|B ⊗ ρS ⊗ τ

⊗2nH ⊗ τ⊗C )U2n

1†] =

∑k

P2n(2k) |2k〉〈2k|B . (250)

For the simplest three-stroke case, a T map (244) of a single cycle is given by:

|g, k〉〈g, k| T−→ aH(1− aC) |g, k + 1〉〈g, k + 1|+ (1− aH) |g, k − 1〉〈g, k − 1|+ aHaC |e, k + 1〉〈e, k + 1| ,

|e, k〉〈e, k| T−→ |g, k − 1〉〈g, k − 1| .(251)

Let us imagine this process as a random walk with three different transitions: right R is given bythe transition |g, k〉〈g, k| → |g, k + 1〉〈g, k + 1| with probability p+ = aH(1 − aC), and left L is thetransition |g, k〉〈g, k| → |g, k − 1〉〈g, k − 1| with probability p− = (1 − aH). The last step is ‘doublezero’ transition OO which is a composition of two: |g, k〉〈g, k| → |e, k + 1〉〈e, k + 1| with probabilityp0 = aHaC , and second (deterministic) transition |e, k + 1〉〈e, k + 1| → |g, k〉〈g, k| which brings thestate back to the initial one. It means that transition OO does not change the position of a walker,however it has length of two iterations.

Let us now consider a process with 2n iterations where we have n+ right steps R, n− left steps L,and n0 zeros O, and we start in a state ρSB = (p |g〉〈g|S + (1− p) |e〉〈e|S)⊗ |0〉〈0|B. The probability ofoccupying the state |2k〉〈2k|B at the end of the protocol we define as:

P2n(2k) = p pg(2k, 2n) + (1− p) pe(2k, 2n), (252)

where pg(2k, 2n) = pgg(2k, 2n) +pge(2k, 2n), and pgg(2k, 2n) is a probability of transition |g, 0〉〈g, 0| →|g, 2k〉〈g, 2k|, and pge(2k, 2n) is a probability of transition |g, 0〉〈g, 0| → |e, 2k〉〈e, 2k|, and analogouslyfor pe(2k, 2n).

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The probability of occupying the state |g, 2k〉〈g, 2k| at the end of the protocol is given by the sumover all trajectories with even number of zeros n0, where the associated probability is given by thetrinomial distribution, i.e.

pgg(2k, 2n) =∑

n++n−+n0=2nn0 - even

δ2k,n+−n− f(n+, n−,n02 ) (253)

where

f(n+, n−, n0) = (n+ + n− + n0)!n+!n−!n0! p

n++ p

n−− pn0

0 . (254)

Furthermore, for odd values of n0 all trajectories always end up in the same final state |e, 2k〉〈e, 2k|.Then, it is enough to realize that the last step is always given by the O-transition |g, k〉〈g, k| →|e, k + 1〉〈e, k + 1| with probability p0 = aHaC , and the rest can be once again calculated from thetrinomial distribution, namely

pge(2k, 2n) = p0∑

n++n−+n0=2nn0 - odd

δ2k,n+−n−+1 f(n+, n−,n0 − 1

2 ). (255)

Finally, we obtain:

pg(2k, 2n) =

=m∑i=0

2(m−i)∑n+=0

δk,n+−m+i f(n+, 2n− n+ − 2i, i) + p0

m−1∑i=0

2(m−i)−1∑n+=0

δk,n+−m+i+1 f(n+, 2n− n+ − 2i− 1, i)

=m∑i=0

θ(k +m− i)θ(−k +m− i)[f(k +m− i,−k +m− i, i)

+ p0

m−1∑i=0

θ(k +m− i− 1)θ(−k +m− i)f(k +m− i− 1,−k +m− i, i)]

=m∑i=0

θ(k + i)θ(−k + i)f(k + i,−k + i,m− i) + p0

m∑i=1

θ(k + i− 1)θ(−k + i)f(k + i− 1,−k + i,m− i)

=m∑

i=|k|f(k + i,−k + i,m− i) +

m∑i=1

(k + i)p0(m+ i)p+

θ(k + i− 1)θ(−k + i)f(k + i,−k + i,m− i)

=m∑

i=|k|(1 + (i+ k)p0

(i+m)p+)f(i+ k, i− k,m− i).

(256)

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Similarly to previous considerations we have

pg(2k + 1, 2n+ 1) =

=∑

n++n−+n0=2n+1n0 - even

δ2k,n+−n−−1 f(n+, n−,n02 ) + p0

∑n++n−+n0=2n+1

n0 - odd

δ2k,n+−n− f(n+, n−,n0 − 1

2 )

=n∑i=0

2(n−i)+1∑n+=0

[δk,n+−n+i−1 f(n+, 2n+ 1− n+ − 2i, i) + p0δk,n+−n+i f(n+, 2n− n+ − 2i, i)]

=n∑i=0

2i+1∑n+=0

[δk,n+−i−1 f(n+, 2i+ 1− n+, n− i) + p0δk,n+−i f(n+, 2i− n+, n− i)]

=n∑i=0

[θ(k + i+ 1)f(k + i+ 1,−k + i, n− i) + p0θ(k + i)f(k + i,−k + i, n− i)]θ(i− k)

=n∑i=0

[p+n+ i+ 1k + i+ 1θ(k + i+ 1) + p0θ(k + i)]f(k + i,−k + i, n− i)θ(i− k)

= θ(−k − 1)f(|k|+ k, |k| − k − 1, n− |k|+ 1) +n∑

i=|k|(p+

n+ i+ 1k + i+ 1 + p0)f(k + i,−k + i, n− i).

(257)

Moreover, if we start in a state |e, 0〉〈e, 0|, then each realization starts with a step |e, 0〉〈e, 0| →|g,−1〉〈g,−1|. This straightforwardly leads to the formula:

pe(2k, 2n) = pg(2k + 1, 2n− 1). (258)

Finally, for arbitrary state ρS = p |g〉〈g|S + (1− p) |e〉〈e|S , we have

P2n(2k) = p pg(2k, 2n) + (1− p) pg(2k + 1, 2n− 1). (259)

I.2.2 Work distribution for charging protocol via uncorrelated qubits

Let us start with a definition of the map T :

T (ρSB) = USB(ρB ⊗ %S)USB, (260)

where%S = (1− aH

1 + aHaC) |g〉〈g|S + aH

1 + aHaC|e〉〈e|S . (261)

The action of the map on basis states is following:

|g, n〉〈g, n| T−→ |e, n− 1〉〈e, n− 1| , |e, n〉〈e, n| T−→ |g, n+ 1〉〈g, n+ 1| (262)

We then consider a battery state after the charging process by N = 2n uncorrelated qubits, where ineach step the battery and particular qubit evolve according to the map T , namely we define the state:

%B = TrS [U2nSB(|0〉〈0|B ⊗ %

⊗2nS )U2n

SB†] =

∑k

P2n(2k) |2k〉〈2k| . (263)

In analogy to the previous consideration we have here once again a random walk process, however withonly left and right transition. For the specific state %S ,the left transition L is observe with probabilityp− = 1 − aH

1+aHaC , and right transition R with probability p+ = aH1+aHaC . As a consequence, the final

distribution of the battery is simply given by the binomial distribution:

P2n(2k) =(

2nn− |k|

)pn−k+ pn+k

− . (264)

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