Work and entropy production in generalised Gibbs ensembles
M. Perarnau-Llobet, A. Riera, R. Gallego, H. Wilming, J. Eisert
WWork and entropy production in generalised Gibbs ensembles
Mart´ı Perarnau-Llobet, Arnau Riera, Rodrigo Gallego, Henrik Wilming, and Jens Eisert ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain Dahlem Center for Complex Quantum Systems, Freie Universit¨at Berlin, 14195 Berlin, Germany (Dated: June 17, 2016)Recent years have seen an enormously revived interest in the study of thermodynamic notions in the quantumregime. This applies both to the study of notions of work extraction in thermal machines in the quantumregime, as well as to questions of equilibration and thermalisation of interacting quantum many-body systemsas such. In this work we bring together these two lines of research by studying work extraction in a closedsystem that undergoes a sequence of quenches and equilibration steps concomitant with free evolutions. Inthis way, we incorporate an important insight from the study of the dynamics of quantum many body systems:the evolution of closed systems is expected to be well described, for relevant observables and most times, bya suitable equilibrium state. We will consider three kinds of equilibration, namely to (i) the time averagedstate, (ii) the Gibbs ensemble and (iii) the generalised Gibbs ensemble (GGE), reflecting further constants ofmotion in integrable models. For each effective description, we investigate notions of entropy production, thevalidity of the minimal work principle and properties of optimal work extraction protocols. While we keep thediscussion general, much room is dedicated to the discussion of paradigmatic non-interacting fermionic quantummany-body systems, for which we identify significant differences with respect to the role of the minimal workprinciple. Our work not only has implications for experiments with cold atoms, but also can be viewed assuggesting a mindset for quantum thermodynamics where the role of the external heat baths is instead playedby the system itself, with its internal degrees of freedom bringing coarse-grained observables to equilibrium.
Thermodynamics is undoubtedly one of the most success-ful physical theories, accurately describing a vast plethora ofsituations and phenomena. Until not too long ago, the studyof thermodynamic state transformations was mostly confinedto the realm of classical physics, which constitutes a mostmeaningful approach when considering macroscopic situa-tions. Progress on the precisely controlled manipulation ofphysical systems at the nano-scale or at the level of singleatoms, however, has pushed the frontier of the applicability ofthermodynamic notions to the realm of quantum physics. In-deed, the emergent research field of quantum thermodynamicsis concerned with thermodynamics in the quantum regime, aregime in which notions of coherence, strong interactions, andentanglement are expected to play a significant role.Building upon a body of early work [1, 2], recent attemptsof grasping the specifics emerging in the extreme quantumregime have put particular emphasis on notions of thermo-dynamic state transformations for quantum systems. A sim-ilar focus has been put on studying the rates of achievablework extraction of thermodynamic machines [3–17]. In thesenew attempts, a resource-theoretic mindset is often applied,or single-shot notions of work extraction [18, 19] are elab-orated upon. These studies are motivated by foundationalconsiderations—-after all, such thermodynamic state trans-formations are readily available in a number of quantumarchitectures—as well as by technological desiderata: For ex-ample, novel techniques for cooling quantum systems closeto the ground state can be derived from quantum thermody-namical considerations [20, 21]. In these studies of quantumheat engines, heat baths prepared in thermal states are usuallystill taken for granted: This is most manifest in a resource-theoretic language, where such thermal baths in Gibbs statesare considered a free resource.Concomitant with these recent studies of thermal machines,a second branch of quantum thermodynamics is blossom- ing: This is the study of quantum many-body systems out ofequilibrium and the question of thermalisation as such [22–28]. In this context, thermal baths are by no means assumedto be available: Instead it is one of the key tasks of thisfield of research to find out under what precise conditionsclosed many-body systems are expected to thermalise, follow-ing quenches out of equilibrium. This is hence the questionin what precise sense systems—as one often says—-“formtheir own heat bath”. Despite respectable progress in recentyears, many questions on many-body systems out of equi-librium remain open, even when it comes to understandingwhether non-integrable generic systems always thermalise atall [29]. Many-body localised systems are expected to stub-bornly refuse to thermalise, for retaining information of theinitial condition over an infinite amount of time. Integrablemodels, in contrast, are not equilibrating to Gibbs states, butto so-called generalised Gibbs ensembles (GGE) [23, 28, 30–36]. For comprehensive reviews on the subject, see, e.g., Refs.[29, 37–39].It is the purpose of this work to bring these two realms ofstudy closer together and to attempt to formulate a theory ofquantum thermodynamics and notions of work extraction, tak-ing into account these recent insights into the mechanism ofequilibration in many-body systems. More specifically, weconsider work extraction from a closed system that undergoesa sequence of quenches and relaxations to a respective equi-librium state. Importantly, our framework deviates from thestandard realm of thermodynamics, where equilibration to sta-tistical ensembles after each quench occurs through weak cou-pling with an infinite thermal bath. In contrast, we incorporatethe equilibration to such ensembles as an effective descriptionof the unitary evolution of a closed system. This effectivedescription is adequate to capture the system only for a re-stricted, although most relevant, set of observables. We willconsider three kinds of equilibrium states: the time averaged a r X i v : . [ qu a n t - ph ] J un state, the Gibbs ensemble, and the generalised Gibbs ensem-ble for a given set of constants of motion. Entropy productionand the minimal work principle will be studied for these threemodels.The results presented here are expected to be of interest forboth the study of thermal machines in the quantum regime—since new insights for the equilibration of closed quantummany-body is taken into account—as well as for the study ofquantum many-body equilibration itself. Our work highlightsthe importance of investigating not only the equilibration ofsystems after single quenches, but also the equilibration aftersequences of quenches which are the relevant paradigm withinprotocols of work extraction.The structure of this work is as follows. In Sec. I we in-troduce the three models of equilibration that will be consid-ered throughout this work and discuss its physical relevance asa description of the effective evolution of closed many-bodysystems. In Sec. II we turn to presenting our framework ofwork extraction based on quenches and equilibrations. Sec. IIIdiscusses notions of entropy production in each of the modelsof equilibration, where we introduce rigorous conditions forthe absence of entropy production and carefully relate theseconditions to notions of reversible processes. In Sec. IV wediscuss the minimal work principle and the protocols for op-timal work extraction for each of the models of equilibra-tion. Lastly, in Sec. V we study a model of non-interactingfermionic systems, where many of the features throughout ourtheoretical analysis are made concrete. I. EQUILIBRATION MODELS
When referring to equilibration of quantum many-body sys-tems, we relate to finite but large systems. Such closed quan-tum many-body systems cannot truly equilibrate due to theirunitary evolution. What is generically the case, however, isthat expectation values of large restricted sets of observablesequilibrate in time to the value attained for the time average[23, 27, 40, 41], in the sense that they stay close to the timeaverage for most times in an overwhelming majority. This isparticularly true for local observables [42].
A. Time average state or diagonal ensemble
We say that an observable A equilibrates if, after some re-laxation time, its expectation value is for most times the same (cid:104) A ( t ) (cid:105) (cid:39) tr( A Ω TA ) as the expectation value of the infinitetime average Ω TA ( ρ, H ) := lim T →∞ T (cid:90) T e − i Ht ρ e i Ht d t , (1)of an initial state ρ of a system described by a Hamiltonian H . A simple calculation shows that the time averaged statecorresponds to the de-phased state in the Hamiltonian eigen-basis and for this reason is often called diagonal ensemble .More explicitly, given the distinct energies of the Hamiltonian { E k } and the projectors onto their corresponding eigenspaces P k , the time averaged state reads Ω TA ( ρ, H ) = (cid:88) k P k ρP k . (2)The time averaged state corresponds to the maximum entropystate given all the conserved quantities [43]. This observationturns the principle of maximum entropy introduced by Jaynes[44, 45] into a consequence of the quantum dynamics. Theprinciple of maximum entropy states that the probability dis-tribution which best represents the current state of knowledgeof the system is the one with largest entropy given the con-served quantities of the system; this principle will be crucialto define our equilibration models.Although relaxation towards the time averaged state hasbeen proven under very general and naturally fulfilled con-ditions [25–27, 40], in practice, the diagonal ensemble cannotbe used as an equilibration model due to its inefficiency. Thedescription of the equilibrium state by the diagonal ensem-ble requires the specification of as many conserved quantitiesas the dimension of the Hilbert space, which scales exponen-tially in the system size. It is therefore in principle not evenpossible to save all the data in a computer for a large inter-acting many-body system, let alone compute the infinite timeaverage efficiently. B. Canonical or Gibbs ensemble
In practice, the characterisation of the equilibrium state canin many instances be done by specifying only a few quan-tities, e.g., the temperature and the chemical potential. Themost relevant and common such situation is the canonical en-semble or the
Gibbs state , for which only the temperature, orequivalently the energy per particle of the initial state ρ , hasto be specified, Ω Gibbs ( ρ, H ) = e − βH Z , (3)where ρ is the state of the system before undergoing the equi-libration process, Z = tr(e − βH ) is the partition functionand the inverse temperature β > is fixed by imposing that tr( H Ω Gibbs ) = tr( Hρ ) .For generic, non-integrable models, the thermal state is ex-pected to be indistinguishable from the time averaged stateunder very mild assumptions which relate to conditions oneigenstates of the Hamiltonian [24, 29, 46] and on the energydistribution of the initial state [47, 48]. While dynamical ther-malisation in this sense has not yet been rigorously proven, itis highly plausible, and it can be connected to typicality ar-guments [49, 50]. The generality of these conditions explainswhy the canonical ensemble is the corner-stone of the standardthermodynamics. Nevertheless, there are known instances ofsystems that do not thermalise. One central aim of this work isto study how thermodynamic protocols are modified when theGibbs ensemble is not a good equilibration model and doesnot satisfactorily describe the equilibrium state of the system . C. Generalised Gibbs ensemble
Examples of systems which do not fully thermalise to Gibbsstates are constituted by integrable systems . The infinite-timeaveraged states are not well described by the Gibbs ensemblebecause of the existence of (quasi) local integrals of motion,i.e. conserved quantities Q i , that retain information about theinitial state over an infinite amount of time. Instead, thereis strong evidence that they can be well-described by the so-called generalised Gibbs ensemble (GGE) defined as Ω GGE ( ρ, H, { Q i } ) ∝ e − βH + (cid:80) qj =1 λ j Q j , (4)where the generalised chemical potential λ j is a Lagrangemultiplier associated with the specific conserved quantity Q j , j = 1 , . . . , q , such that its expectation value is the same as theone of the initial state tr (Ω GGE ( ρ, H, { Q j } ) Q k ) = tr( ρQ k ) , (5)for each k = 1 , . . . , q . The GGE can be understood as aninterpolation between the diagonal and the canonical ensem-bles. The diagonal ensemble maximises the von Neumannentropy S ( ρ ) = − tr( ρ log ρ ) given all the conserved quanti-ties (CQ). The Gibbs ensemble maximises the von Neumannentropy considering only the energy as a conserved quantity.The GGE is situated in between. For a given state ρ and a setof operators (conserved quantities) { Q i } , it is natural to de-fine the set of states compatible with the values the conservedquantities E ( ρ, { Q i } ) := { σ | tr( ρQ i ) = tr( σQ i ) } . (6)The GGE is the state that maximises the von Neumann en-tropy within E ( ρ, { Q i } ) . From this perspective, the ensem-bles introduced so far can be summarised as Ω TA := argmax σ ∈E ( ρ, { all CQ } ) S ( σ ) , (7) Ω GGE ( ρ, H, { Q i } ) := argmax σ ∈E ( ρ, { H,Q i } ) S ( σ ) , (8) Ω Gibbs ( ρ, H ) := argmax σ ∈E ( ρ, { H } ) S ( σ ) . (9)A relevant question in the construction of GGEs is how theconserved quantities have to be chosen, which is discussed inAppendix A. In general, there is a certain degree of ambiguityof what constants of motion to pick in order to arrive at theappropriate GGE. This discussion is not relevant for the gen-eral study pursued in this work, however. It is the aim of thiswork to study the thermodynamical behaviour of the GGE infull generality, hence we will not have to make any precise as-sumption about the conserved quantities, unless it is explicitlyspecified. D. Example: Equilibration of a quadratic fermionic model
To illustrate the above considerations, let us consider aquadratic Hamiltonian of fermions in a one dimensional lat-tice H (0) = n (cid:88) i =1 (cid:15) i a † i a i + g n − (cid:88) i =1 (cid:16) a † i a i +1 + a † i +1 a i (cid:17) , (10) where n is the total number of sites and a i ( a † i ) are the cre-ation (annihilation) operators at the i -site which satisfy thefermionic anti-commutation relations { a i , a † j } = δ i,j , { a i , a j } = { a † i , a † j } = 0 . (11)We would like to study how an initially out of equilibriumstate relaxes to equilibrium and see that the Gibbs ensemblefails to describe the equilibrium state.The initial state of the system is taken to be in thermal equi-librium, ρ (0) = e − βH (0) / Z . A quench is then performed to anew Hamiltonian H (1) , H (0) (cid:55)→ H (1) , (12)in which the energy of the first fermion is modified, H (1) = H (0) + ∆ a † a . After the quench, the population of the firstfermion evolves in time t > as n ( t ) = tr( a † a ρ ( t )) (13)with ρ ( t ) = e − i H (1) t ρ (0) e i H (1) t . As the Hamiltonian isquadratic, it is a problem involving free fermions and can benumerically simulated for very long times and system sizes(see Appendix G 1).In Fig. 1, we plot the time evolution of the occupation ofthe first site n ( t ) . As expected, we see that after some re-laxation time t , n ( t ) equilibrates to the value predicted bythe GGE—which is relatively far from the one given by theGibbs equilibration model. The situation described in this ex-ample, a quench and the characterisation of the equilibriumstate, is extensively studied in the literature, see for a recentreview ref. [29]. In order to study thermodynamic processesin which many quenches and equilibrations are performed, itwill be necessary to promote the suitability of effective de-scriptions in terms of GGE states for equilibration processesbeyond a single quench. II. FRAMEWORK FOR THERMODYNAMICPROTOCOLS
In the previous section we have introduced the differentequilibration models, given by equations (7)-(9), that describethe equilibrium state that is reached when a system initiallyout of equilibrium in a state ρ evolves under a Hamiltonian H . One way to bring a system out of equilibrium is to quenchits Hamiltonian. More explicitly, a system initially at equi-librium with initial Hamiltonian H ( ini ) undergoes a quench H ( ini ) (cid:55)→ H ( fin ) and starts to evolve non-trivially under thenew Hamiltonian H (fin) . The models of equilibration intro-duced above can be used to describe the new equilibrium statethat is reached after a single quench and a posterior suffi-ciently long time evolution under H (fin) . However, thermody-namic processes (for instance a protocol of work extraction)often involve a series of quenches and equilibrations. We nowextend our previous considerations to such processes involv-ing sequences of quenches and equilibrations. FIG. 1. Time evolution of the occupation of the first site of the lat-tice n = a † a for a quadratic Hamiltonian of n fermions in a onedimensional lattice. For the example we take n = 100 , (cid:15) i = 1 , ∆ = 0 . , β = 2 , g = 0 . and time is measured in units of / (10 g ) .An equilibration around the GGE is observed, even for this moder-ately sized quantum system. A. Equilibration under repeated quenches
Consider a sequence of changes of the Hamiltonian, asdefined by a list of N + 1 Hamiltonians, H ( m ) , where m = 0 , , . . . , N denotes the step in the protocol and H (0) is the initial Hamiltonian. These Hamiltonian transforma-tions H ( m − (cid:55)→ H ( m ) are considered to be quenches , in thesense that they are performed sufficiently fast such that thestate of the system ρ is unchanged. Let us denote the time atwhich the quench H ( m − (cid:55)→ H ( m ) is performed by t m with t m < t m +1 for all m . After a quench, the system evolves un-der the Hamiltonian H ( m ) for a time t m +1 − t m until a newquench H ( m ) (cid:55)→ H ( m +1) is performed at time t m +1 . Thistime interval is taken to be much longer than the equilibrationtime such that the system can be considered to be in equilib-rium. The exact state of the system ρ ( t ) when m quencheshave taken place ( t m < t < t m +1 ) is given by, ρ ( t ) = e − i( t − t m ) H ( m ) ρ ( t m )e i( t − t m ) H ( m ) , (14)where ρ ( t m ) is the state of the system at t = t m when theHamiltonian H ( m ) starts to dictate the evolution. The state ρ ( t m ) is given by the recursive expression ρ ( t k ) = e − i( t k − t k − ) H ( k − ρ ( t k − )e i( t k − t k − ) H ( k − , (15)with ρ ( t ) the initial state and k = 1 , . . . , m .Now, our aim is to construct an effective description of thewhole evolution of ρ , in such a way that the state after the m -thquench and its posterior equilibration, ρ ( t ) , can be describedby an appropriate equilibrium state. We denote such equilib-rium state that approximates the real state after m quenches, ρ ( t ) , as ω ( m )( ··· ) where ( · · · ) is the place holder for one of thethree models of equilibration: time-average (TA), GGE orGibbs. The effective description of (14) is then built in a re- cursive way as follows, ω ( m ) TA = Ω TA (cid:0) ω ( m − TA , H ( m ) (cid:1) ,ω ( m ) GGE = Ω
GGE (cid:0) ω ( m − GGE , H ( m ) , { Q ( m ) i } (cid:1) , (16) ω ( m ) Gibbs = Ω
Gibbs (cid:0) ω ( m − Gibbs , H ( m ) (cid:1) . Here, ω (0)( ··· ) = ρ ( t ) is the intial state, before any quenchor evolution has taken place. Note that, when constructingthe GGE description, the set of conserved quantities { Q ( m ) i } changes for every Hamiltonian H ( m ) , as well as the La-grange multipliers { λ ( m ) j } qj =1 , or simply the inverse tempera-ture β ( m ) in the case of equilibration to the Gibbs ensemble.In order to provide a motivation and interpretation of eq.(16), together with the implicit assumptions that come intoplay, let us illustrate it with a simple example. Suppose a sys-tem initially in state ρ (0) and with Hamiltonian H (0) . At time t , we perform a first quench H (0) (cid:55)→ H (1) and let the sys-tem evolve under H (1) ; at time t we perform second quench H (1) (cid:55)→ H (2) and let the system evolve under H (2) until itequilibrates at time t . For both evolutions, we now considereffective descriptions in terms of GGE states. After the evolu-tion under H (1) and immediately before performing the sec-ond quench, the system is exactly described by ρ ( t ) as givenby eq. (15). For a set of conserved quantities { Q (1) i } , the cor-responding GGE equilibrium state is given by, ω (1) GGE = Ω
GGE (cid:0) ρ ( t ) , H (1) , { Q (1) i } (cid:1) (cid:39) ρ ( t ) , (17)where the symbol “ (cid:39) ” means in this context that the averagevalue of relevant observables is well approximated by ω (1) GGE ,that is tr( Aρ ( t )) (cid:39) tr( Aω (1) GGE ) . (18)Now, when describing the equilibrium state after the sec-ond quench, one can simply apply the same recipe. That is,the state ρ ( t (1) ) is the initial state when the evolution under H (2) starts. Then, assuming that the new conserved quanti-ties { Q (2) i } i are chosen appropriately and applying the samereasoning one obtains an approximation by taking Ω GGE (cid:0) ρ ( t ) , H (2) , { Q (2) i } (cid:1) (cid:39) ρ ( t ) , (19)with t longer than the t plus the subsequent equilibrationtime. Importantly, note that this effective description is notefficient, in the sense that it requires keeping track of the ex-act state ρ ( t ) to obtain the equilibrium state at time t . If thisis extended to N quenches, having to keep track of the ex-act evolution until the ( N − -th quench is as demanding askeeping track of the whole exact evolution over the process.It is here when the effective description (16) becomes handy,as it can be constructed by keeping track of the value of theconserved quantities only. First of all, coming back to the firstevolution, note that by applying (16) with m = 1 we recover(17), i.e., the standard result for single quenches. Now, in or-der to construct the GGE state corresponding to ρ ( t ) , we as-sume that the conserved quantities { Q (2) i } are within the set ofphysically relevant observables A in (18). That is, we assumethat tr( Q (2) i ρ ( t )) (cid:39) tr( Q (2) i ω (1) GGE ) (20)for all i . In this way, in order to obtain the equilibrium GGEensemble after the second quench, it is not necessary to keeptrack of the exact state ρ ( t ) , but one can simply use ω (1) GGE instead. Using (20) we then obtain, ω (2) GGE := Ω
GGE (cid:0) ω (1) GGE , H (2) , { Q (2) i } (cid:1) (cid:39) Ω GGE (cid:0) ρ ( t ) , H (2) , { Q (2) i } (cid:1) (21) (cid:39) ρ ( t ) . (22)Extending the same reasoning to the case of N quenches andother models of equilibration other than the GGE, we arriveto an effective description of the form (16).In the rest of this work we will always use the effectivedescription (16) for the full process consisting on a sequenceof quenches and equilibrations. We do not claim by this thatthis model will accurately describe the real dynamics of anysystem or protocol, and indeed we explicitly leave here as anopen question to identify for which Hamiltonians and con-served quantities condition (20) is satisfied for each quench.Nonetheless, and in exactly the same way as equilibration tothe Gibbs state is assumed in the usual scenario in thermody-namics, we will assume that equilibration to statistical ensem-bles of the form (16) occurs over any protocol, so that we cantackle questions about entropy production and work extrac-tion.To examine the validity of our model, we will later providea numerical comparison of the real exact evolution and themodel of eq. (16) for the case of free fermions. We will seefor this example that the model predicts with great accuracythe amount of work that is extracted in a protocol involving asequence of quenches. B. Work cost of quenches
Concatenations of quenches and equilibrations constitute aframework to describe thermodynamic processes -see, e.g.,Refs. [12, 18, 51]. Within this framework, work is associ-ated with the input energy under quenches, whereas heat isassociated with the exchange of energy under equilibrationprocesses. At the level of average quantities, the work costof a single quench, H ( m − (cid:55)→ H ( m ) , reads W ( m ) := tr (cid:16) ρ ( t m )( H ( m ) − H ( m − ) (cid:17) , (23)where ρ ( t m ) is given in (14). The main assumption of thisstudy is precisely that the work cost of a quench is very wellapproximated by the effective description of the equilibriumstate, i. e. W ( m ) = tr (cid:16) ω ( m − ··· ) ( H ( m ) − H ( m − ) (cid:17) , (24) where ω ( m − ··· ) is its effective description (16). While we fo-cus our attention on average quantities, primarily for simplic-ity of the analysis, one could also conceive a study of workextraction under GGE for other work quantifiers [10, 52, 53].As the equilibration processes happen spontaneously and haveno work cost, the total work extracted in the entire protocol issimply given by the sum of the steps W := N (cid:88) m =1 W ( m ) . (25) C. The system - bath setting beyond the weak coupling andinfinite bath limits
A particularly relevant scenario is the system - bath setting.We call system S to the part of the total system upon whichone has control and it is possible to quench its Hamiltonian H S . The bath B contains the degrees of freedom upon one hasno control and it is the responsible for equilibrating the system S . In order for this equilibration to happen, the dimension ofthe Hilbert-space of S , dim( H S ) , is considered to be muchsmaller than that of the bath, dim( H S ) (cid:28) dim( H B ) (26)and the total Hamiltonian to be of the form [54], H ( m ) = H ( m ) S ⊗ B + S ⊗ H B + V , (27)where the interaction V is supported on S and B and couplesthe two subsystems. Unlike the standard assumptions in ther-modynamics, note that we do not assume that the interaction V is weak or that bath size is infinite. Let us be more explicitabout what we mean by that.Usually, within thermodynamics, it is assumed that the sys-tem S equilibrates, upon contact with a bath B , according to tr B (cid:16) ω ( m ) β (cid:17) = Ω β ( H ( m ) S ) := e − βH ( m ) S Z , (28)where β > is fixed throughout all the protocol. In con-trast, in the model that we consider, given by ω ( m ) Gibbs in (16),the inverse temperature changes along the protocol and theGibbs states describe the whole compound SB . Nonetheless,let us note that the model of equilibration Ω β in (28) repre-sents a particular case of our Gibbsian model Ω Gibbs in thelimit of weak coupling and infinite bath. In the limit of aninfinite bath , the total energy of SB in (16) will not be sub-stantially affected by the energy pumped or subtracted in allthe quenches H ( m ) SB (cid:55)→ H ( m +1) SB and the parameter β ( m ) willremain constant throughout the protocol, β ( m ) ≈ β for all m . In the limit of weak coupling V between S and B , then Ω β ( m ) ( H ( m ) SB ) ≈ Ω β ( m ) ( H ( m ) S ) ⊗ Ω β ( m ) ( H ( m ) B ) .In sum, the model of equilibration ω ( m ) Gibbs should be regardedas a correction to the usual setup in thermodynamics given byeq. (28). This correction incorporates the fact that the bath isof finite size, which introduces a dependence of the inversetemperature β ( m ) and also allows for strong couplings be-tween S and B . III. ENTROPY PRODUCTION AND REVERSIBLEPROCESSES
An important quantity in thermodynamic processes is theentropy production on system and bath during the protocol.Of course, the exact unitary dynamics on SB does not changethe von Neumann entropy in the system. However, we areusing an effective description on SB , given by (16), and inthis effective description the entropy in the system SB mightwell change. Indeed, due to the fact the equilibration modelscan all be understood as a maximisation of the entropy givensome constraints, it follows that the entropy of the states ω ( m ) in (16) is non-decreasing during a protocol S ( ω ( m ) ) ≥ S ( ω ( m − ) ∀ m = 1 , . . . , N. (29)where S is the von Neumann entropy defined as S ( ρ ) = − tr( ρ log ρ ) . (30)Therefore, sequences of quenches followed by equilibrationsare in general irreversible: if we start with the final state ofthe protocol and then run the protocol backwards, we will ingeneral not end up with the original initial state.From phenomenological thermodynamics we would expectthat the protocols become reversible if they are done in aquasi-static way. In the context of our set of operations,a quasi-static process is defined by considering N → ∞ quenches H ( m ) (cid:55)→ H ( m +1) such that H ( m +1) − H ( m ) is oforder /N , followed each by an equilibration process as givenby eq. (16). In this limit of an infinite number of quencheswe can simply describe the quasi-static process by definingthe continuous path of Hamiltonians as u (cid:55)→ H ( u ) with u ∈ [0 , . This corresponds to the Hamiltonian H ( m ) = H ( u = m/N ) , and equivalently for the equilibrium state ω ( u = m/N ) := ω ( m ) in the limit of N → ∞ (where ω isan effective description of the equilibirum state given by TA,GGE or Gibbs). We will be concerned with the von Neumannentropy of the equilibrium state along the trajectory S ( u ) = − tr (cid:0) ω ( u ) log ω ( u ) (cid:1) . (31)We now discuss in detail under which conditions the en-tropy remains constant over the quasi-static process, i.e. S (1) = S (0) , for the three models of equilibration. Impor-tantly, note that we are concerned with the entropy productionin a given quasi-static process. Hence, as the quasi-static pro-cess requires an arbitrarily large number of quenches and sub-sequent equilibrations, it is by definition an arbitrarily slowprocess. We will see that the fact that the process is arbitrarilyslow alone (by definition as it is a quasi-static process) doesnot guarantee that there is no entropy production. A. Entropy production for time averaged ensembles
We start by analysing the entropy production of a quasi-static process when all conserved quantities are taken into ac-count. In this case the equilibrium state is given by ω TA ( u ) . Our first result shows that there is no entropy production ina quasi-static process if the trajectory of Hamiltonians u (cid:55)→ H ( u ) is smooth. Result 1 (Absence of entropy production within the TAmodel) . Consider a Hamiltonian trajectory u (cid:55)→ H ( u ) = (cid:80) k E k ( u ) | E k ( u ) (cid:105)(cid:104) E k ( u ) | , and a quasi-static process alongthis trajectory which induces a family of time-average states ω TA ( u ) . Then, if the trajectory is continuous and the eigen-vectors | E k ( u ) (cid:105) are differentiable, there is no entropy produc-tion in such a quasi-static process, that is, S (0) = S (1) = 0 . The proof and discussion can be found in Appendix B. Notethat this result is independent of the state which is evolvingunder H ( u ) . In fact, for a given state, there exist quenchesthat are not quasi-static but preserve its entropy, such as anyquench to a Hamiltonian with the same eigenbasis as the state.This is for instance the case of raising and lowering energylevels. B. Entropy production for generalised Gibbs ensembles
Now, we consider the case of a generic GGE equilibrationwhere not all the conserved quantities are taken into account.In this case, the equilibration model (16) satisfies the relation, tr (cid:16) ω ( m ) GGE Q ( m ) i (cid:17) = tr (cid:16) ω ( m − GGE Q ( m ) i (cid:17) , (32)for all i = 1 , . . . , q . Here the { Q ( m ) i } correspond to the q conserved quantities of H ( m ) , and eq. (32) determines the cor-responding Lagrange multipliers λ ( m ) i in (4). For such equi-librium states, we also identify conditions so that there is noentropy production. More precisely, we find the following: Result 2 (Absence of entropy production within the GGEmodel) . Consider a quasi-static process along a Hamiltoniantrajectory u → H ( u ) described by a family of equilibriumstates ω GGE ( u ) . Then, the entropy of ω GGE ( u ) is preservedin such a quasi-static process, provided that the Lagrange-multipliers as determined by (32) , form in the limit N → ∞ aset of smooth functions u (cid:55)→ λ j ( u ) for j = 1 , . . . , q . This result is shown simply by taking the continuum limitof eq. (32) which yields tr (cid:18) d ω GGE ( u )d u Q j ( u ) (cid:19) = 0 , ∀ j = 1 , . . . , m (33)which can be in turn used to show that the entropy productionvanishes, d S d u = m (cid:88) j =1 λ j ( u ) tr (cid:18) d ω GGE ( u )d u Q j ( u ) (cid:19) = 0 . (34)Hence, we see that, if the conditions of Result 2 are satisfied,the entropy of the effective description in terms of GGE statesis also preserved in the limit of a quasi-static process.Let us now discuss heuristically under which conditions thepremise that { u (cid:55)→ λ j ( u ) } q are smooth functions is expectedto be fulfilled. This can be well illustrated by the followingexample: Example 1 (Quasi-static process with entropy productionwithin Gibbs and GGE) . Consider the case of a two dimen-sional system for which we take q = 1 , that is, the only con-served quantity is the Hamiltonian Q = H itself (the Gibbsequilibration model). Consider initially a non-degenerateHamiltonian H (0) = E | (cid:105)(cid:104) | and an arbitrary initial state ρ (0) with an inverse temperature β (0) > and thus theentropy is smaller than log(2) . Now suppose that the finalHamiltonian H (1) = 0 has degenerate energy levels. We nowshow that:(i) there is a quasi-static trajectory without a smooth be-haviour of the Lagrange-multipliers (in this case β ( u ) ),(ii) this results in a positive entropy production, and(iii) how this implies that taking only a single conservedquantity—in this case the energy—does not provide agood approximation of the time-averaged state. To see the above points, take as Hamiltonian path H ( u ) = E (1 − u ) | (cid:105)(cid:104) | = H (0)(1 − u ) and an initial Gibbs statewith inverse temperature β (0) . Then the eigenbasis in the en-tire process does not change. Now note that the condition (32)implies that the energy is preserved in every equilibration. Butsince we are dealing with a two-dimensional system, as longas H ( u ) is non-degenerate, the state itself will remain con-stant ω ( u ) = ρ ( u ) for any u ∈ [0 , . This requires that theinverse temperature β ( u ) → ∞ as u → : Along the path,the inverse temperature needs to fulfil β ( u ) = β (0) / (1 − u ) to keep the state constant. Therefore, it necessarily divergesas u → . To show ii), simply note that when one reaches H (1) , the final state is a maximally mixed state with entropy log(2) , which is larger than the one of the initial state by as-sumption. To show iii), observe that the time averaged statewould remain constant throughout the protocol, thus it differsfrom the GGE at u = 1 . Similar reasoning as for this exampleholds true for higher dimensional systems, where the groundstate degeneracy of H (1) is higher than that of H (0) .The previous example shows that in some cases the premiseof Result 2 is not fulfilled, however, these pathological casesoften imply that the chosen GGE description is not accurate.For example, in the case of encountering a ground state de-generacy, any conserved quantity in the GGE that discerns theground states would be enough to fix the problem. However,we leave in general open whether one can find smooth trajec-tories for u (cid:55)→ λ j ( u ) for a given set of conserved quantitiesand trajectory of Hamiltonians—this may well depend on thespecifics of the model and on the ambiguity of what constantsof motion to pick in the first place [29]. C. Entropy production for Gibbs ensembles
As discussed above in the case of the GGE ensemble, it isin general necessary to ensure that the Lagrange multipliers u (cid:55)→ λ j ( u ) follow a smooth trajectory in order to certify thatthere is no entropy production in a quasi-static process. Thisrequires to compute the Lagrange multipliers following themodel of (16) and keeping track of the conserved quantities. We will see now that the situation simplifies substantially forthe case of the Gibbs model of equilibration (where the energyis the only conserved quantity). Result 3 (Absence of entropy production within Gibbsmodel) . Consider an initial and final Hamiltonian H (0) and H (1) and initial state ω Gibbs (0) = e − β (0) H (0) /Z with finite β (0) > . There exist a quasi-static trajectory H ( u ) so thatthere is no entropy production if and only if there exist β ∗ > so that S ( ω Gibbs (0)) = S ( e − β ∗ H (1) /Z ) (35)Note that one of the implications is trivial. The final state is e − β (1) H (1) /Z , hence if there exist no β (1) = β ∗ so that (35) isfulfilled, then it is clearly impossible to keep the entropy con-stant. This can happen if H (1) does not admit any Gibbs statewith the initial entropy. The non trivial implication of the pre-vious result is that as long as H (1) admits a Gibbs state withthe initial entropy, one can always find a quasi-static trajec-tory that keeps the entropy constant. Indeed, we find that thequasi-static trajectory achieving it does not need to be fine-tuned. We discuss in Appendix C, together with the proof ofResult 3, that any quasi-static process where the degeneracyof the ground state does not increase along the protocol willindeed keep the entropy constant. This condition is expectedto be satisfied for trajectories of generic local Hamiltonians,which have non-degenerate ground spaces for typical choicesof the Hamiltonian parameters [55]. D. Entropy production and reversibility
We now connect entropy production to reversibility of pro-cesses. First, let us note that for the GGE equilibration model(similarly for the Gibbs model since it is a particular case ofthe former), condition (33) is invariant if one reverses the pro-cess. More specifically, given H ( u ) and ω GGE (0) as initialstate, condition (33) determines the trajectory of states (if thepremises of Result 2 are met) ω GGE ( u ) , with u from to .Now, we can consider the trajectory H (˜ u ) with initial state ω GGE (˜ u = 0) with ˜ u = 1 − u . One can easily verify that tr (cid:18) d ω GGE (˜ u )d˜ u Q j (˜ u ) (cid:19) = 0 , ∀ j = 1 , . . . , m. (36)Hence, the equilibrium state for the trajectory H (˜ u ) is givenexactly by ω GGE (˜ u = 1 − u ) and thus, the protocol is re-versible. In other words, we have seen that for the GGE equi-libration model reversible protocols correspond to arbitrarilyslow protocols where no entropy is produced on the systemand bath together, exactly as is the case for phenomenologi-cal thermodynamics.A well-known feature of phenomenological thermodynam-ics is that reversible transformations are always beneficial inwork-extraction protocols, a phenomenon which is referred toas the minimum work principle . We will later see that thisprinciple naturally holds when the model of equilibration isgiven by Gibbs states, but its range of applicability is consid-erably reduced when the equilibrium states are described byGGE. Indeed, we will see explicitly that when the equilibra-tion model is given by a GGE ensemble of free fermions, itcan well be beneficial to go through a given protocol quicklyand thereby producing entropy.Before we go on to discuss explicit work extraction pro-tocols, let us stress that the entropy in SB , which can onlyincrease or remain constant, is not simply the sum of the en-tropies of S and B . This happens because we are consideringinteracting quantum systems that show correlations between S and B . This is true both in the exact and the effective de-scription. Indeed, in general the von Neumann entropy in SB is smaller than or equal to the sum of local entropies S ( ω ) ≤ S (tr B ( ω )) + S (tr S ( ω )) , (37)with equality if and only if ω = tr B ( ω ) ⊗ tr S ( ω ) , i.e., when S and B are completely uncorrelated. Thus, entropy-productionin our set-up does not always mean that entropy is locally pro-duced in the system and the bath. The generation of entropy isnot always associated with the generation of correlations, asin ref. [56], but rather to the mixing induced by equilibrationprocesses. The global entropy may, for example, increase dueto a decrease of correlations, but entirely without changing thelocal states of the system.As a final remark, note that in the so-called Isothermal Re-versible Process (IRP) the entropy of the system S does notremain constant, while the entropy of the whole compound SB does, as we discuss in Examples 1 and 2 in Appendix C). IV. THE MINIMUM WORK PRINCIPLE AND WORKEXTRACTION
In order to study work extraction, we first focus on the min-imum work principle , which is intimately related to work ex-traction and other tasks in thermodynamics such as, e.g., theerasure of information (Landauer’s Principle). We take as thedefinition of the minimum work principle that, given an initialequilibrium state and a path of Hamiltonians, the work per-formed on the system is minimal for the slowest realisation ofthe process [57]. More precisely, we consider a trajectory ofHamiltonians u (cid:55)→ H ( u ) with u ∈ [0 , . Consider now pro-tocols with N quenches (each followed by an equilibration).That is, we choose N values ( u , . . . , u N ) , so that the proto-col is determined by H ( m ) = H ( u m ) and ω ( m )( ··· ) as determinedby (16). The minimal work principle states that the optimalprotocol maximizing W in (25) is the one where N → ∞ and u m = m/N . Note that here, as we generically take the con-vention that work is extracted from the system, minimisingthe work cost corresponds to maximising W in (25).We note that, while being the most relevant notion of theminimum work principle for our set-up (see also ref. [57]),this definition differs from the one usually found in ther-modynamics text-books, where the content of the minimalwork principle reads: among all the possible paths betweentwo fixed equilibrium states , reversible protocols are optimal.Here, we fix instead a given trajectory between an initial andfinal Hamiltonian and question whether the quasi-static real-isation is also the optimal. Note that both notions—where the initial and final states are fixed or where the trajectory isfixed instead—coincide in the model of equilibration of eq.(28), which is the standard one in text-book thermodynamics.The reason is that in the model (28) Hamiltonians and statesare in one to one correspondence and all the quasi-static tra-jectories between two Hamiltonians provide the same work.However, when other models of equilibration are considered( ω ( ··· ) in (16)) the equivalence breaks down since Hamiltoni-ans and states are not in one-to-one correspondence: the finalstate depends on the specific trajectory.It seems then natural to ask what justifies our definition ofthe minimal work principle. The answer lies in the fact thatthe notion of the minimal work principle considered here canbe easily connected with the second law of thermodynamics,formulated as: no positive work can be extracted in a cyclicprocess from states initially in thermal equilibrium (a Gibbsstate), or more generally, in a passive state. In this contextcyclic refers to the fact that the initial and final Hamiltoniancoincide, which does not imply that the initial and final statecoincide, unless we would be using the model of equilibra-tion (28). This relation with the minimal work principle andthe second law will be made explicit in the following sectionswhere we study the Gibbs and the time-average ensembles. A. The minimum work principle and work extraction forGibbs ensembles
Let us consider the same setup as the one laid out in SectionII A, with an initial state ω (0) Gibbs and a protocol that performs N quenches according to a certain trajectory u (cid:55)→ H ( u ) , where H ( m ) := H ( m/N ) . Let us stress that here we do not take thelimit of N → ∞ and we keep it general by considering finite N . Let us recall from eq. (25) that the total work performedis given by the sum of the individual work W ( m ) in the m -thstep, W = N (cid:88) m =1 W ( m ) = N (cid:88) m =1 tr (cid:16) ω ( m − Gibbs ( H ( m − − H ( m ) ) (cid:17) = tr (cid:16) ω (0) Gibbs H (0) (cid:17) − tr (cid:16) ω ( N ) Gibbs H ( N ) (cid:17) + N (cid:88) m =2 tr (cid:16) ( ω ( m − Gibbs − ω ( m ) Gibbs ) H ( m ) (cid:17) , (38)where in eq. (38) we have simply reorganised the terms andadded and subtracted the quantity tr( ω ( N ) Gibbs H ( N ) ) . We cannow use our model of equilibration, as given by eq. (16) thatwe recall here for completeness, ω ( m ) Gibbs = Ω
Gibbs ( ω ( m − Gibbs , H ( m ) ) = e − β ( m ) H ( m ) Z ( m ) (39)for all m ≥ , where Z ( m ) = tr( e − β ( m ) H ( m ) ) and β ( m ) > is determined by the conservation of average energy: tr( ω ( m − Gibbs H ( m ) ) = tr( ω ( m ) Gibbs H ( m ) ) . One can easily checkthat energy conservation implies that the last sum in (38) van-ishes, which implies that W = tr (cid:16) ω (0) Gibbs H (0) (cid:17) − tr (cid:16) ω ( N ) Gibbs H ( N ) (cid:17) , (40)where ω ( N ) Gibbs depends on N and the trajectory H ( u ) .From eq. (40) we see that given a fixed final Hamiltonian H (1) , the protocol that costs the minimum amount of work(and maximises the extracted work W ) is given by the onethat leaves the final state with the least average energy. Sincethe average energy is monotonic with the entropy for Gibbsstates of positive temperature, we conclude that the optimalprotocol is the one minimising the entropy of the final state ω ( N ) Gibbs . Furthermore, as the entropy can only increase through-out the protocol (see Sec. III), a protocol creating no entropyis optimal.It has to be stressed that this holds true only as long as thefinal temperature of the Gibbs state is positive, which happensif tr( ω ( N ) Gibbs H N ) ≤ d tr( H ( N ) ) , (41)where d is the dimension of the Hilbert space. Note that theright hand side of the equation typically (e.g., for many bodysystems with short range interactions) grows linearly with thenumber of particles. Therefore, if the total system is bigenough, we expect condition (41) to be satisfied, and thus theminimum work principle to hold.Taking together the facts that a protocol creating no en-tropy is optimal and the results of Appendix C–which showconditions so that the quasi-static entropy has no entropyproduction—one can conclude that the minimal work prin-ciple is satisfied for any trajectory so that d g ( H (0)) ≥ d g ( H ( u )) ≥ d g ( H (1)) (see Result 5). As mentioned inSec. III C, this condition is satisfied for trajectories of genericHamiltonians, which have non-degenerate ground spaces.Let us now comment on the relation between the minimalwork principle and the second law of thermodynamics. First,note that if we fix a trajectory H ( u ) so that H (0) = H (1) ,then the final state is a Gibbs state Ω β N ( H (0)) . The inversetemperature β N at the end of the protocol certainly dependson the particular trajectory and the number of quenches per-formed. However, it is clear by the discussion of Sec. III that S (Ω β N ( H (0))) ≥ S ( ω (0)) . Hence, since the final state is aGibbs state with respect to H (0) and with more entropy thanthe initial Gibbs state and the energy is monotonic with the en-tropy for Gibbs state, the extracted work is negative. Note thatthis depends crucially on having Gibbs states as equilibriumstates and it will not be reproduced by time-average or GGEmodels of equilibration as we discuss in the next sections.Also, the minimum work principle can be used to studywork-extraction protocols from non-equilibrium states. Asan example, consider as initial conditions a pair of state andHamiltonian ρ (0) and H (0) respectively. The goal is to ex-tract work from ρ (0) by performing a cyclic protocol, where H ( N ) = H (0) . Note that here the initial state is not in a Gibbsstate with respect to the initial Hamiltonian, H (0) . Never-theless, after the first quench, it does thermalise to ω (1) Gibbs = Ω
Gibbs ( ρ (0) , H (1) ) . From that moment onwards, the minimumwork principle can be used, implying that it is always optimalto come back to H (0) by a protocol that does not create en-tropy. The only remaining question is in fact to which Hamil-tonian the first quench is performed, an issue that is discussedin Appendix D. B. Work extraction and the minimum work principle for timeaveraged states
We now discuss the minimum work principle for protocolsof work extraction when the model of equilibration that is usedis the time-average Ω TA . Let us assume a smooth trajectory ofHamiltonians H ( u ) and some initial equilibrium state ω TA (0) .Since the trajectory of Hamiltonians is smooth we know thatthe final state in the quasi-static protocol ω q.s.TA (1) has the sameentropy as the initial state ω TA (0) , indeed even the same eigen-values as ω TA (0) (see Appendix B). The question is, whetherthis also implies that the quasi-static protocol is optimal interms of the average work-cost. We will show that this is ingeneral only true if this final state in the quasi-static proto-col is also a passive state, meaning that it is diagonal in theenergy-eigenbasis and the energy-populations decrease withincreasing energy: tr( H (1) ω q.s.TA (1)) = (cid:88) k ( ω q.sTA (1)) ↓ k E k (1) , (42)where ( ω TA (1)) ↓ is the vector of eigenvalues of ω TA (1) , or-dered such that ( ω TA (1)) ↓ k ≥ ( ω TA (1)) ↓ l if E k (1) ≤ E l (1) . Result 4 (Passiveness of optimal protocols) . Given an initialstate and a smooth trajectory of Hamiltonians, if the final statein the quasi-static realisation of the protocol is passive, thenthe the quasi-static realisation of the protocol is optimal.
This result follows, because passive states can only in-crease their average energy under any unitary transformation[58, 59]: tr( Hρ ) ≤ tr( HU ρU † ) , ρ passive w.r.t. H. (43)In particular the final state of the quasi-static protocol ω q.s.TA (1) is related to the initial state by some unitary transformation U ∗ since their spectra are identical. To see that the quasi-static realisation of the protocol is optimal in this case, letus now consider any realisation of the protocol with only afinite number of quenches N and let us denote the final statein a protocol with N quenches as ω N TA . Since the time-averageequilibration model can be thought of as applying a mixture ofunitaries (evolving the system for some random time) in anyfinite realisation including N quenches, the final state ω N TA isrelated to the initial state by: ω N TA = (cid:88) i p i U i ω TA (0) U † i = (cid:88) i p i ( U i U ∗ ) ω TA (1)( U i U ∗ ) † , where p i is some probability distributions of unitaries. Butsince ω q.s.TA (1) is passive, we henceforth have tr( H (1) ω N TA ) ≥ tr( H (1) ω TA (1)) , (44)0which proves the claim.The minimum work principle for cyclic unitary processeswas studied in ref. [57] where it was shown that the minimumwork-principle holds if: i) the initial state is passive with re-spect to the initial Hamiltonian H (0) and ii) the trajectoryof Hamiltonians is such that the initial and final Hamiltoni-ans H (0) and H (1) , respectively, do not have a level-crossingw.r.t to each other. Here, by an absence of level-crossing wemean that if E i (0) ≥ E j (0) , then also E i (1) ≥ E j (1) (notethat the labelling of the energy-basis is fixed since we requirethe Hamiltonian trajectory to be smooth). It is now easy to seethat under the premise that the initial state is passive, the con-dition that the final state in the quasi-static realisation is alsopassive is indeed equivalent to the absence of such a level-crossings. Thus, our result naturally generalises that of [57].Finally, let us note that given two Hamiltonians H (0) and H (1) and an initial equilibrium state, it is always possible toconstruct a smooth trajectory of Hamiltonians that connectsthe two Hamiltonians and such that the final state in the quasi-static protocol is passive and has the same spectrum as it hadinitially. This can be done with the protocol presented in Ap-pendix F. However, note that this protocol requires global con-trol over the Hamiltonians. Once we can only control somepart of the Hamiltonian, all the available smooth trajectoriesmight lead to a non-passive final state in the quasi-static re-alisation, so that it can become beneficial to use a protocolwith a finite number of quenches which results in entropy-production.As in the case of the Gibbs equilibration model, one caneasily relate the analysis above to discuss the second law ofthermodynamics. First, note that the optimal protocol be-tween H (0) and H (1) is such the final state has the samespectrum and it is passive. Hence, if H (0) = H (1) we con-clude that one can extract positive work from the initial equi-librium state ω TA (0) if and only if it is not passive. Of coursethis fact is well-known if we consider protocols of work ex-traction that just apply a unitary transformation to the initialstate. Here, we are deriving a similar behaviour with familiesof protocols that are instead quenches and equilibrations to thetime-average state.In summary, we have identified conditions that ensure thatthe quasi-static realisation of a given protocol is optimal. Thiscondition generalise the ones found in ref. [57]. Also, we haveshown that any state can be brought to its passive form—keeping the same spectrum—by applying a quasi-static pro-tocol over a specific trajectory of Hamiltonians. Altogether,this show that quasi-static protocols are as powerful for workextraction as they can conceivably be. C. Work extraction and the minimum work principle for GGEstates
In this section we briefly analyse notions of work extrac-tion in the case of GGE models of equilibration. Although itis difficult to provide general results for the case of the GGE,without having specified the particular form of the conservedquantities, we do include here a general formulation of the problem at hand as an introduction to particular example offree fermions that we study later. In this situation, the equili-brated states are maximum entropy states Ω GGE ( ρ, H ( m ) , { Q ( m ) j } ) := argmax σ ∈E ( ρ, { H ( m ) ,Q ( m ) j } ) S ( σ ) . (45)for a collection of constants of motion { Q ( m ) j } that are rele-vant at a given step m of the protocol. For a given protocol,the work extracted is again W = N (cid:88) m =1 W ( m ) = N (cid:88) m =1 tr (cid:16) ω ( m − GGE ( H ( m − − H ( m ) ) (cid:17) , (46)so that in order to compute the extracted work for a given pro-tocol, one has to keep track of the Lagrange multipliers alongthat protocol. The optimal work extraction is attained as thesupremum of this expression over such protocols. In agree-ment with our considerations for time-averaged states, herewe will find that the minimum work principle is in general notsatisfied for many-body models that equilibrate to a GGE. Ul-timately, this result is linked to the fact that for GGEs there isin general no direct link between entropy and energy , in strongcontrast to the case of Gibbs states. We show this statementby considering specific classes of models for which the GGEis relevant, namely the class of physical systems described byfree fermions, a most relevant type of systems that are knownto be well described by the generalised Gibbs ensemble. Inparticular we will show an example where a fast protocol out-performs a slow protocol despite the fact that an effective de-scription by Gibbs states would suggest the opposite. V. FREE FERMIONIC SYSTEMS
On top of showing the validity of the above result, thereason for largely focusing on quadratic fermionic models isthree-fold. First, they can be efficiently simulated, allowingus to test how well the effective description of the systemapproximates its real (exact) dynamics. Also, they are inte-grable, which implies that a GGE description is in generalnecessary to capture their equilibration behaviour [29, 38]. Fi-nally, they can be simulated with ultra-cold atoms in opticallattices in and out of equilibrium [60–63]. While the discus-sion presented here is focused on non-interacting fermionicsystems, it should be clear that their bosonic lattice instances[62, 64, 65] and even bosonic continuous systems [66, 67]can be captured in an analogous framework with very similarpredictions. The latter situation is specifically interesting asmodelling the physics of ultra-cold atoms on atom chips thatis expected to provide an experimental platform probing thesituation explored here where a GGE description is relevant.1
A. Hamiltonian, covariance matrix and GGE construction
We consider quadratic fermionic Hamiltonians of the form H = n (cid:88) i,j =1 c i,j a † i a j , (47)where n is the number of different modes and the fermionicoperators satisfy the anti-commutation relations { a i , a † i } = δ i,j , { a i , a j } = { a † i , a † j } = 0 . The Hamiltonian H can betransformed into H = n (cid:88) k =1 (cid:15) k η † k η k , (48)where η ( † ) k is the annihilation (creation) operator correspond-ing to the k -th eigenmode of the Hamiltonian.It is well known that equilibrium states of Hamiltonians ofthe form (48) are not well described by Gibbs states, but ratherby generalised Gibbs ensembles, with the conserved quanti-ties being the occupations of the energy modes Q k = η † k η k , k = 1 , . . . , n [29]. Notice that the number of conserved quan-tities used for the construction of the GGE is the number ofdistinct modes n and, hence, is linear (and not exponential) inthe system size.We define the correlation matrix γ ( ρ ) of a state ρ as thesymmetric matrix having entries γ i,j ( ρ ) = Tr( η † i η j ρ ) . (49)If the state ρ is Gaussian, then γ ( ρ ) contains all informationabout ρ , and its time evolution under Hamiltonians of the type(48) keeps it Gaussian. In other words, the full density ma-trix ρ can be reconstructed from just knowing the correlationmatrix.The correlation matrix of the GGE Ω GGE ( ρ, H, { η † k η k } ) is found by maximising the entropy while preserving all Q k = η † k η k , which simply reduces to dephasing the corre-lation matrix defined in (49) to the diagonal (see AppendixG 1 for details). This provides a simple method for obtaining γ (Ω GGE ( ρ, H, { η † k η k } )) .Note that these GGE descriptions are also Gaussian states.Hence, in the following we can always restrict to Gaussianstates. Even when the initial state is not Gaussian, all the re-sults are unchanged if the initial state is replaced by a Gaus-sian state that has the same correlation matrix. Consequently,in the following, the discussion is reduced to the level of cor-relation matrices instead of the full density matrices. Thisallows us to perform numerical simulations of the real time-evolution as well as the effective description of large systems,since they have dimension n × n instead of the n × n neededto describe the full density matrix. B. Work extraction and minimum work principle for freefermions
First we consider optimal protocols for work extraction ina scenario where the Hamiltonian can be transformed to any quadratic Hamiltonian of the form (48). The discussion issimilar to that of Sec. IV B, but in the context of GGE equi-librium states. As in the previous sections, the optimal pro-tocol is the one minimising the final energy, tr( ω ( N ) GGE H (0) ) = (cid:80) k n ( N ) k (cid:15) (0) k , where we assume the process to be cyclic and n ( N ) k = tr( η (0) k † η (0) k ω ( N ) GGE ) . In Appendix G 2, we show thatthis minimisation yields tr( ω ( N ) GGE H (0) ) ≥ tr( ω ∗ GGE H (0) ) , with tr( ω ∗ GGE H (0) ) = n (cid:88) k =1 ( d (0) ) ↓ k ( (cid:15) (0) ) ↑ k , (50)where d (0) k are the eigenvalues of γ ( ρ (0) ) and the symbols ↑ and ↓ indicate that the lists are ordered in increasing and de-creasing order, respectively. An explicit protocol saturatingthis bound is constructed in Appendix G 2. The optimal pro-tocol is found to be reversible , so that no entropy is gener-ated, and one needs to perform an arbitrarily large amount ofquenches to reach optimality.In the optimal final state ω ∗ GGE , the diagonal elements ofthe correlation matrix, corresponding to the population of theenergy modes, decay as the energy of the modes increases.This form is reminiscent of the passive states previously in-troduced. However, in general, states of the form ω ∗ GGE donot need to be passive: While in passive states the occupa-tion probabilities of the energy eigenstates are decreasing withincreasing energy, here only the occupations of the differentfermionic modes decrease with the energy of the mode. Thetotal energies are however obtained by combinations of dif-ferent modes. An example of a state that is non-passive, butwhere the mode-populations are decreasing with increasingmode-energy is provided in Appendix G 2.Regarding the minimum work principle, one can use a sim-ilar line of reasoning as in Sec. IV B. For a fixed process, theminimum work principle is guaranteed to hold true as long asthe possible final states—which are realised by implementingthe process at different speeds—have the form (50), i.e., theirpopulations decrease with the energy of the modes. If thiscondition is not satisfied, the minimum work principle doesnot hold in general.
C. Numerical results: comparison between exact dynamicsand effective descriptions
In this section we compute the work extracted in differentscenarios by (i) a numerical simulation of the exact unitaryevolution of the system, (ii) using the effective description interms of Gibbs states, and (iii) in terms of GGE states. Asphysical system we consider a chain of fermions, taking as aninitial Hamiltonian, H (0) = n (cid:88) i =1 (cid:15) i a † i a i + g n − (cid:88) i =1 (cid:16) a † i a i +1 + a † i +1 a i (cid:17) . (51)First we study the optimal protocol for the case unrestrictedHamiltonian case derived in Appendix IV C, and next we con-sider the case of local changes of the Hamiltonian. In all cases2we find a very good agreement between the real dynamics andthe GGE effective description.Besides comparing the effective descriptions with the realdynamics, we also study the applicability of the minimumwork principle. We give an explicit example of a processin which producing entropy is beneficial for work extraction,hence showing that the minimum work principle is violated inthis example.
1. Work extraction with unrestricted Hamiltonians and freefermions
Here, we take as the initial state ρ (0) a GGE state whosepopulations γ ( ρ (0) ) i,i ∈ (0 , in (49) are chosen i.i.d. froma Gaussian distribution. Again, note that this state is Gaus-sian. We then apply the protocol described in Appendix G 2for maximal work extraction, and compare the results ob-tained by the exact dynamics and the GGE model of equilibra-tion. The exact dynamics are computed by, after the the i -thquench, letting the system unitarily evolve under the Hamil-tonian H ( i ) for a time much longer than the time scale ofequilibration. Fig. 2 shows the results obtained using both ap-proaches. It shows a very good agreement, as long as the num-ber of fermions is sufficiently large (in the figure n = 100 ).Yet small discrepancies are observed, which is due to the factthat we implement global quenches, for which the state maynot equilibrate. Note that, when performing local quenchesand starting with a Gibbs state, as in Fig. 3, equilibration oflocal observables is expected (see Sec. I) and the agreement isindeed excellent.We can also see in Fig. 2 how work increases as the processbecomes slower, becoming maximal in the limit N → ∞ ,when reversibility is achieved. This is in agreement with ourconsiderations in Sec. V B.
2. Work extraction with restricted free fermionic Hamiltonianswith a Gibbs initial state
Let us now assume that the Hamiltonian can only be locallymodified, as discussed in Sec. II. The Hamiltonian (51) is splitin three components. H S = (cid:15) a † a ( S is a single fermion), V = g ( a † a + a † a ) and H B = H − V − H S . Our capabilityto change the Hamiltonian is thus reduced to a single param-eter: the local energy (cid:15) . Note that the coupling between the S and the B is not assumed to be weak. The initial state takesthe form, ρ (0) = ρ S ⊗ Ω β ( H B ) = ρ S ⊗ e − βH B Z , (52)where ρ S is initially out of thermal equilibrium; for example,in Fig. 3, it is set to a lower temperature than the bath. As dis-cussed before, we do not need to assume that the initial state ρ S (and hence ρ (0) ) is Gaussian, but the work extracted willonly depend on its correlation matrix and not on the full den-sity matrix, since the energy is a sum of second moments ofthe fermionic creation and annihilation operators and all the FIG. 2. Extracted work in the optimal protocol with unrestrictedHamiltonians. As an initial state, we take a diagonal state in the ba-sis H (0) , with the populations { p (0) k } chosen at random between and . We take (cid:15) = 1 , g = 0 . and N = 100 . In order to simulatethe real dynamics, after every quench, we let the system evolve for atime chosen at random between /g and /g . In green, we showthe results using the actual unitary dynamics, in yellow our effectivedescription in terms of GGE states, and in dashed lines the analyticalresult leading to eq. (50). The inset figure shows the entropy gen-erated in the effective description using GGE states. As the numberof quenches increases (i.e., the process becomes slower), the gener-ated entropy tends to zero and the extracted work tends to the upperbound. GGE states constructed in the process are Gaussian automati-cally.Fig. 3 shows the extracted work from ρ S as a function of thenumber of quenches N , which is computed using the real ex-act unitary evolution, and the effective description in terms ofboth GGE and Gibbs states. The agreement between the uni-tary dynamics and the GGE description is excellent, for anyvalue of N and the parameters, but the Gibbs states fail to de-scribe the process. Even if the bath is initially in a Gibbs state,see eq. (52), the posterior evolution of SB can not be correctlydescribed by them. Although the description in terms of Gibbsensembles is quantitatively incorrect, it is fair to say that it de-scribes some qualitative features of the results. In particular,the exact dynamics satisfies the minimum work principle, andso does the effective description with Gibbs states. This fol-lows because condition (41) is satisfied during the process.However, as we show in the next section, condition (41) canfail to predict the applicability of the minimum work princi-ple.
3. Work extraction with free fermionic restricted Hamiltonianswith a GGE initial state
Equilibrium states when dealing with Hamiltonians of thetype (48) are well described by GGE states, it is therefore nat-ural to generalise the initial state (52) to ρ = ρ S ⊗ ω ( B ) GGE , (53)3 FIG. 3. Extracted work with only local transformations on the stateof the system. The different points correspond to the exact unitaryevolution (in green), to the effective evolution in terms GGE states(in yellow), and the effective evolution using Gibbs states (in blue).The continuous lines correspond to transformations with N → ∞ .As an initial state we take, β = 1 / , tr( a † a ρ S ) = 0 . , n = 100 .For the initial Hamiltonian, (cid:15) = 0 . , (cid:15) i = 1 ∀ i (cid:54) = 1 , g = 0 . . As aprotocol we perform a first quench to (cid:15) = 4 . , followed by N − equidistant quenches back to the original Hamiltonian. As in Fig. 2,the exact evolution is obtained by letting system and bath interact fora time much larger than the equilibration time ( t Eq ∝ /g ). where ω ( B ) GGE is a GGE state with respect to the local Hamil-tonian of B , H B = (cid:80) nk =1 (cid:15) ( B ) k η ( B ) † k η ( B ) k . Let us now pick avery particular initial state given by
Tr( ω ( B ) GGE η ( B ) † k η ( B ) k ) = (cid:26) k ≥ K k < K , (54)for some K < n . That is, only the K most energetic modesare populated. No actual thermal state with positive temper-ature would have such properties due to the population inver-sion of the fermionic modes. It is important to acknowledge,however, that if we would chose an effective description as aGibbs state for such initial states, we would nevertheless ob-tain a positive effective temperature provided that condition(41) is satisfied. This will be the case as long as the number ofpopulated energy-levels K is small enough. Indeed, for anyfinite K , but large n , the energy-density in the state is muchlower than the critical energy-density needed for negative ef-fective temperatures.The work extracted in a particular protocol with initial state(53) is plotted in Fig. 4. The results clearly show how theextracted work decreases with the time spent in the process.Therefore, more work is extracted when more entropy is pro-duced, and the minimum work principle does not apply in thissituation. In fact, this is to be expected because both the ini-tial and the final state of the protocol are highly non-passive,and thus the conditions described in Sec. V B are not satisfied.However, when using an effective description in terms Gibbsstates, we would have predicted that it is always beneficial touse a quasi-static, reversible protocol since condition (41) issatisfied for the case described in Fig. 4. FIG. 4. The extracted work achieved with only local transformationson the state of the system. As an initial state we take the one specifiedby K = 32 , tr( a † a ρ S ) = 0 . , and n = 150 . For the initial Hamil-tonian, we take (cid:15) = 0 . , (cid:15) i = 1 ∀ i (cid:54) = 1 , g = 0 . . As a protocolwe perform a first quench to (cid:15) = 1 . , followed by N − equidis-tant quenches back to the original Hamiltonian. The different pointscorrespond to the exact unitary evolution (in green), to the effectiveevolution in terms GGE states (in yellow), and to infinitesimally slowprotocol ( N → ∞ ). As in 2, the real evolution is obtained by let-ting system and bath interact for a sufficiently long time (chosen atrandom). VI. CONCLUSIONS
In this work, we have brought together the fields of researchon equilibration and quantum heat engines. The main contri-bution of this work is to go beyond the usual paradigm of ther-modynamics where work is extracted from a system in weakthermal contact with an infinite heat bath at a given fixed tem-perature. Instead, we consider closed quantum many-bodysystems of finite size and with strong coupling between itsconstituents. We make use of recent insights into the studyof states out of equilibrium: closed many body systems donot equilibrate, but can be effectively described as if they hadequilibrated when looking at a restricted, although most rel-evant, class of observables. The effective equilibrium statethat describes the system for these observables is, however,not necessarily given by a Gibbs state; and even if so, its tem-perature will not remain constant under repeated quenches. Inthis case the effective equilibrium state is given by the timeaveraged state, the GGE or the Gibbs state, depending on theparticular kind of system considered, as well as the family ofobservables that are taken into account.With this in mind, we have put forward a framework thatstudies work extraction of closed many body systems, incor-porating Hamiltonian quenches as well as equilibrations ac-cording to the three models mentioned before. We do not onlyassume that effective equilibrium state is a good description ofthe state evolving after a single quench, but also that such anequilibrium state can be taken as the initial state to describefurther evolutions under subsequent quenches. This model,which is successfully tested for the model of free fermions,is what allows us to describe a closed system similarly to theway open systems (in contact with baths) are described in con-4ventional thermodynamics. Thus, we can formulate similarquestions regarding work and entropy production and indeedrecover many of the phenomena present for open systems.In particular, we provide stringent conditions for the ab-sence of entropy production in quasi-static protocols. Thisturns out to be intimately related to the optimal protocolsfor work extraction and the minimum work principle, whichroughly speaking states that the work performed on the sys-tem is minimal for the slowest realisation of a given process.We find that the minimum-work principle can break down inthe presence of a large number of conserved quantities, whileit remains intact if system and bath together can be well de-scribed by a Gibbs ensemble, even in the strongly interactingregime. This is shown numerically with the paradigmatic ex-ample of free fermions for which the extracted work decreaseswith the time spent in the process if we consider the GGE asequilibration model, but the minimum work principle still ap-plies when the Gibbs description is assumed. It is the hopethat the present work stimulates further studies at the intersec-tion of the theory of quantum thermal machines and quantummany-body systems.
Acknowledgements
We acknowledge funding from the A.-v.-H., the BMBF(Q.com), the EU (RAQUEL, SIQS, AQuS), the DFG (EI519/7-1, CRC 183), the ERC (TAQ) and the Studienstiftungdes Deutschen Volkes. M. P.-L. acknowledges funding fromthe Spanish MINECO (Severo Ochoa grant SEV-2015-0522)and Grant No. FPU13/05988. A. R. is supported by the Beat-riu de Pinos fellowship (BP-DGR 2013), the Spanish projectFOQUS, and the Generalitat de Catalunya (SGR 874). Allauthors thank the EU COST Action No. MP1209,
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Here we discuss which are the physical arguments that jus-tify the choice of a given set of conserved quantities that leadto a GGE. This question can be argued from two different ap-proaches. On the one hand, one can argue that the relevantconserved quantities are the ones that are experimentally ac-cessible and, hence, must be given beforehand. This was thespirit of the seminal work of Jaynes [44, 45]. The objectionagainst this approach is that it is subjective , in the sense thatthe set of experimentally accessible observables depends onthe experimentalist. On the other hand, one could take an ob-jective perspective and think that the relevant conserved quan-tities are precisely the ones that make the GGE as close aspossible to the diagonal ensemble independently of the ca-pabilities of the experimentalist [71]. Within this approach,the notion of physically relevant is provided by how much anobservable is able to reduce the distance between the GGEand the diagonal ensemble by being added into the set of con-served quantities that defines the GGE. More specifically, in[71] the distance between the time averaged state and the GGEis taken by the Kullback-Leibler (KL) distance (relative en-tropy) leading to D (Ω TA ( H ) , Ω GGE ( H, { Q i } )) = S (Ω GGE ( H, { Q i } )) − S (Ω TA ( H )) , which is always positive and where we have omitted the initialstate ρ for brevity.In practice, given an ε > , the conserved quantities aresuccessively added to the set of conserved quantities, until D (Ω TA ( ρ, H ) , Ω GGE ( ρ, H, { Q i } )) ≤ ε . By the Pinsker’s in-equality, this guarantees the physical indistinguishability be-tween the two ensembles, i.e., (cid:88) (cid:96) | tr( B (cid:96) (Ω TA − Ω GGE ) | ≤ √ ε , (A1)for any positive operator valued measure (POVM) B . Theaddition of operators to the set of conserved quantities is doneas follows. Given a set of j conserved quantities, the newconserved quantity j + 1 is introduced such that reduces asmuch as possible the entropy min Q j +1 S (Ω GGE ( ρ, H, { Q i } j +1 i =1 )) . (A2)In the subsequent sections we will study what are differencesbetween the thermodynamics given the Gibbs and the GGE asequilibration models. Appendix B: Time-average equilibration model - dissipationand reversibility
In this section we show that it is possible to have dissi-pation, i.e., entropy-production, in an infinitely slow processwithin the time average equilibration model. Let us introducethe following example. We consider the Hamiltonians givenby H ( λ x , λ z ) = λ x σ x + λ z σ z (B1) and the continuous trajectory for − ≤ u ≤ λ ( u ) = ( λ x ( u ) , λ z ( u )) = (cid:26) ( − u, if − ≤ u < , u ) if ≤ u < (B2)starting from an eigenstate of σ x .For − ≤ u < , the equilibration processes do not doanything to the state since the eigenbasis of the Hamiltonianis the eigenbasis of σ x and the system is left in its initial statewith zero entropy. But then, from u = 0 on, the system is de-phased in the eigenbasis of σ z which is mutually orthogonalto the one of σ x and the entropy suddenly jumps to log 2 . Thereason for that is that although the Hamiltonians H ( (cid:15), and H (0 , (cid:15) ) , with (cid:15) > arbitrarily small, are very close in theHamiltonian space, their eigenbasis are totally different.To avoid such effects, it is sufficient that not only the Hamil-tonian trajectory is continuous, but also that the eigenvectorscan be chosen in a smooth manner, i.e., so that each eigen-vector | E k ( u ) (cid:105) is a smooth curve parametrized by u . Moreexplicitly, the eigenvalues p k ( u + δu ) of the density matrix atparameter u + δu can be written in terms of the eigenvaluesof the density matrix ω ( u ) at time u , as p k ( u + δu ) = (cid:104) E k ( u + δu ) | Ω TA ( u ) | E k ( u + δu ) (cid:105) = (cid:88) k (cid:48) p k (cid:48) ( u ) |(cid:104) E k (cid:48) ( u ) | E k ( u + δu ) (cid:105)| , (B3)where we have used that the eigenvalues of Ω TA ( u + δu ) aresimply the diagonal elements of Ω TA ( u ) in the basis givenby | E k ( u + δu ) (cid:105) . Let us now assume differentiability of theeigenbasis, i.e., | E k ( u + δu ) (cid:105) = | E k ( u ) (cid:105) + | X k ( u ) (cid:105) δu + O ( δu ) , (B4)with Re (cid:104) E k (cid:48) ( u ) | X k ( u ) (cid:105) = 0 due to ortho-normalisation.Then we get p k ( u + δu ) = (cid:88) k (cid:48) p k (cid:48) ( u ) ( δ k (cid:48) k + δu (cid:104) E k (cid:48) ( u ) | X k ( u ) (cid:105) )+ δu (cid:88) k (cid:48) p k (cid:48) ( u ) |(cid:104) E k (cid:48) ( u ) | X k ( u ) (cid:105)| = p k ( u ) + O ( δu ) . (B5)This implies that the populations of the density matrix of thesystem are constant in the slow process limit δu → .A natural way to guarantee that the Hamiltonian eigenbasischanges continuously along the Hamiltonian trajectory is torestrict ourselves to smooth trajectories, in the sense that thetangent vectors to the curve in the Hamiltonian space are alsocontinuous. Appendix C: Physically relevant situation of quasi-staticprocesses for the Gibbs ensemble
In this Appendix we discuss the entropy production ofquasi-static processes with the model of equilibration givenby ω Gibbs . In particular we show Result 3 and provide otherlemmas that are used in the proof and that are interesting onits own.7
Lemma 2 (General condition for entropy production withinGibbs model) . Consider a quasi-static process along a tra-jectory of Hamiltonians H ( u ) and an initial state ρ (0) = e − β (0) H (0) /Z ; if there exists any smooth function u (cid:55)→ f ( u ) (cid:54) = 0 ∀ u with f (0) = β (0) such that S (cid:18) e − f ( u ) H ( u ) Z (cid:19) = S ( ρ (0)) (C1) then the quasi-static process along u (cid:55)→ H ( u ) has no entropyproduction.Proof. Defining the family of states Ω f ( u ) := e − f ( u ) H ( u ) Z , (C2)Lemma 2 can be shown by noting that eq. (C1) implies that d S ( ω f )d u = f ( u ) tr (cid:18) dΩ f ( u )d u H ( u ) (cid:19) = 0 . (C3)Taking the equality at the r.h.s., one sees that the state ω f ( u ) fulfils condition (33) and hence, Ω f ( u ) = ω ( u ) and in turn, S ( ω (0)) = S ( ω ( u )) . In other words, any function f ( u ) that—playing the role of the inverse temperature β ( u ) —keepsthe entropy constant, will also fulfill the energy conservationcondition given by (33), so that f ( u ) = β ( u ) .Lemma 2 can be used to answer whether there is entropyproduction given a quasi-static process defined by H ( u ) with ≤ u ≤ and initial state ω Gibbs (0) . We provide now twoexamples.
Result 5.
Let us refer to the ground state degeneracy ofa Hamiltonian H as d g ( H ) . Consider an initial and fi-nal Hamiltonian H (0) and H (1) such that d g ( H (0)) ≥ d g ( H (1)) and initial state ω Gibbs (0) = e − β (0) H (0) /Z ( H (0)) with β (0) > . Then, any quasi-static trajectory H ( u ) thatsatisfies d g ( H (0)) ≥ d g ( H ( u )) ≥ d g ( H (1)) for all u ∈ [0 , will keep the entropy constant.Proof. First, let us invoke the fact I) that for any Hamiltonian H and any entropy S ∈ (log d g , log D ) , there is a finite β S such that the Gibbs state of inverse temperature β S has en-tropy S . Now, let us consider the premise given above of atrajectory u (cid:55)→ H ( u ) , so that the ground state degeneracy sat-isfies d g ( H (0)) ≥ d g ( H ( u )) ≥ d g ( H (1)) for all u ∈ [0 , .This implies that can choose a function u (cid:55)→ f ( u ) such that S (Ω f ( u )) = S ( ω (0) Gibbs ) for all u ∈ [0 , . That this is thecase can be seen at u = 0 just using that β (0) > and hence,the entropy of the initial state is at least log( d g ( H (0)) . Hence,it lies within the limits where fact I) applies. For any other u > we just apply the same reasoning and the premise that d g ( H (0)) ≥ d g ( H ( u )) ≥ d g ( H (1)) for all u ∈ [0 , . Sincethe path of Hamiltonians is smooth, it follows that the function f is also smooth.Lastly, by Lemma this function satisfies f ( u ) = β ( u ) ,where β ( u ) is the inverse temperature of the quasi-static pro-cess. Hence, such a process keeps the entropy constant. Result 6 (Formal version of Resut 3 in the main text) . Con-sider an initial and final Hamiltonian H (0) and H (1) andinitial state ω Gibbs (0) = e − β (0) H (0) /Z with finite β (0) > . If there exist finite β ∗ > so that S ( ω Gibbs (0)) = S ( e − β ∗ H (1) /Z ) , then any quasi-static trajectory H ( u ) with d g ( H ( u )) = 1 for all u in the open interval u ∈ (0 , , issuch there is no entropy production.Proof. Using the same argument as in the proof of Example5, we find that thermal states of non-degenerate Hamiltonianscan take any entropy between and log d . This implies thatwe can then find a smooth function f ( u ) such that S (Ω f ( u ) ) = S ( ω Gibbs (0)) for all u < . But since we assume that a suitable β ∗ exists we can smoothly rescale the Hamiltonians along thetrajectory to make sure that f (1) = β ∗ , obtaining S (Ω f ( u ) ) = S ( ω Gibbs (0) for all u ∈ [0 , . This ensures by Lemma 2 thatsuch quasi-static processes exhibit no entropy production. Appendix D: Optimal protocols for work extraction with Gibbsensembles1. The case of unrestricted Hamiltonians
First we consider an idealised scenario where one has fullcontrol over the global Hamiltonian H . That is, the Hamil-tonians H ( i ) at the i -th step of the protocol can be chosen tobe any Hamiltonian. Given this maximal level of control, wewould like to identify the optimal protocol for work extrac-tion.We have initial conditions described by a pair of state andHamiltonian ρ (0) and H (0) respectively. The goal is to ex-tract work by performing a cyclic protocol, where H ( N ) = H (0) . Importantly, we will no longer assume that the initialstate is in a Gibbs state with respect to the initial Hamilto-nian H (0) . After the first quench, the state thermalises to ω (1) Gibbs = Ω
Gibbs ( ρ (0) , H (1) ) . Hence, from that moment on-wards, the minimum work principle can be applied implyingthat it is optimal to come back to H (0) by a protocol that doesnot create entropy.The only remaining question concerning the optimal pro-tocol is to which Hamiltonian H (1) the first quench is to beperformed. This can be straightforwardly answered by ex-pressing the total work, as in (40), W = tr (cid:16) ( ρ (0) − ω ( N ) Gibbs ) H (0) (cid:17) , (D1)where by eq. (39), we see that ω ( N ) Gibbs is a Gibbs state with in-verse temperature β ( N ) . Arguing in the same way as in theminimum work principle, we obtain that the optimal protocolis the one which has no entropy production. Note that a proto-col creating zero entropy is only possible for initial states ρ (0) such that S ( ρ (0) ) = S ( e − β ( N ) H (0) /Z ) for some β ( N ) > , asdiscussed in Result 3. Here we provide the steps of a proto-col that achieves zero entropy production if that condition ismet, which is, as discussed above, the protocol that extractsthe maximum amount of work. This protocol reads:81. Apply first a quench from H (0) to H (1) = k ln( ρ (0) ) for any k ∈ R − .2. Let the system equilibrate to ω (1) Gibbs :=Ω
Gibbs ( ρ (0) , H (1) ) given by (39). The condition ofaverage energy conservation implies that β (1) = − /k ,and thus, ω (1) Gibbs = ρ (0) .3. Apply a quasi-static process (a sequence of infinitesi-mal quenches and equilibrations) from H (1) to H (0) .Such process keeps the entropy constant S ( ρ (1) ) = S ( ρ ( N ) ) , as discussed in Sec. III.This protocol resembles the optimal protocol of work extrac-tion for the model of equilibration of eq. (28) [18, 72]; how-ever, the first quench is chosen to a different Hamiltonian H (1) .
2. Work extraction with restricted Hamiltonians and Gibbsensembles
We now consider the restricted case where H ( i ) ∈ H and H is a given set of Hamiltonians. While we will later be inter-ested in the case where restriction are such that we can onlychange the initial Hamiltonian locally on the subsystem S , sothat H Local = { H | H = H (0) + H S ⊗ B } , (D2)we will keep the discussion completely general.In the same way as in the case of unrestricted Hamiltonians,a maximum amount of work will be extracted if we minimisethe final energy, as expressed by eq. (40). Since the final stateis by assumption a Gibbs state, it is therefore optimal to end upwith a Gibbs state with minimal possible positive temperature(every state with negative temperature has higher energy thanall states with positive temperature). This is clearly possibleif the initial state already has an effective positive temperaturewith respect to any Hamiltonian in H . We will assume fromnow on that this is the case.Considering steps . − . of protocol in Sec. D 1, one caneasily see that step . cannot be applied if k ln( ρ (0) ) / ∈ H .Instead, we will make a quench H (0) (cid:55)→ H (1) with H (1) = argmin H ∈H S (cid:0) Ω Gibbs ( ρ (0) ,H ) (cid:1) , (D3)while steps . − . are not modified by the restrictions on H . Appendix E: Optimal protocol of work extraction for timeaverage equilibration and unrestricted Hamiltonians
We now construct an explicit protocol that saturates thebound W ≤ tr( ρ (0) H (0) ) − tr( ω ∗ TA H (0) ) (E1)in the limit of N → ∞ , where N is the number of quenchesperformed. Here, ω ∗ TA is a state with the following proper-ties: i) it has the same eigenvalues as ρ (0) , ii) it is diagonal in the basis of H (0) , iii) it is passive, i.e., its eigenvalues are or-dered in non-increasing order with increasing energy. Giventhe initial state ρ (0) , let us denote by U the unitary that diag-onalises the initial state, such that U ρ (0) U † = D . The firststep of the protocol is to make a quench H (0) (cid:55)→ H (1) with H (1) = U † H (0) U . Since ρ (0) is diagonal in the eigenbasis of H (1) , it follows that the first equilibration process to the timeaveraged state will not alter the state, that is, ω (1) TA = ρ (0) . Thesecond step is to perform N/ quenches (followed each by anequilibration process) in a given trajectory from H (1) back tothe initial Hamiltonian H (0) . Note that in the limit of N → ∞ this is a quasi-static process, thus the state ω ( N/ TA is diagonalwith respect to H (0) and with the same eigenvalues as D . Thenext step is to find some unitary V that orders the eigenvaluesof ω N/ TA , in such a way that we have V ω N/ TA V † = ω ∗ TA := (cid:88) k ( ρ (0) ) ↓ k P (0) k , (E2)where ( ρ (0) ) ↓ k denotes the list of eigenvalues of ρ (0) ordered ina non-increasing manner with increasing energy and the P (0) k are the energy-eigenprojectors of H (0) . As in the previousstep, now first perform a quench to H ( N/ = V † H (0) V and return to H ( N ) = H (0) in a quasi-static process, so thatin the limit of N → ∞ we obtain ω N TA = ω ∗ TA . Appendix F: Work extraction with time-average equilibration
In this section we present the optimal protocol of work ex-traction between an initial and final Hamiltonian H (0) and H (1) respectively, from an initial state ω TA (0) . This pro-tocol consists on the quasi-static realisation of the follow-ing trajectory H ( u ) : Let us denote the initial Hamiltonian as H (0) = (cid:80) i E i (0) | E i (0) (cid:105)(cid:104) E i (0) | and equivalently for the fi-nal H (1) . Let us assume no degenerate eigenspaces for sim-plicity (the generalisation to the case with degenerate sub-spaces is straightforward) so that the initial state is simplygiven by ω TA (0) = (cid:80) i p i | E i (0) (cid:105)(cid:104) E i (0) | . Then, the quasi-static realisation of the following trajectory of Hamiltonianleaves the final state ω q.s.TA with the same spectrum and passivewith respect to H (1) :1. Change the eigenvalues smoothly from { E i (0) } i to { E i ( u ) } i while leaving the eigenstates invariant. Notethat in this part of the protocol the state remains also in-variant, so that ω TA ( u ) = ω TA (0) . The final eigenval-ues E i ( u ) are chosen so that ω TA ( u ) is passive withrespect to H ( u ) = (cid:80) i E i ( u ) | E i (0) (cid:105)(cid:104) E i (0) | and thatthe spectrum of H ( u ) coincides with the one of H (1) .2. Given the conditions on the spectrum of H ( u ) and H (1) , one can identify E j (1) = E i ( u ) . In this secondpart of the protocol we define a smooth trajectory from u to u where only the eigenvectors are changed as | E i ( u ) (cid:105) → | E i ( u ) (cid:105) = | E j ( u ) (cid:105) . By definition, afterthis second step the final Hamiltonian H ( u ) is indeedthe desired final Hamiltonian so that H ( u ) = H (1) .9Also, this second step from u to u keeps the statepassive, so that the final state ω TA ( u ) is passive withrespect to the desired final Hamiltonian.However, note that this protocol requires global control overthe Hamiltonians. Once we can only control some part of theHamiltonian, all the available smooth trajectories might leadto a non-passive final state in the quasi-static realisation, sothat it can become beneficial to use a protocol with a finitenumber of quenches which results in entropy-production. Appendix G: Free fermionic systems1. Correlation matrices, time evolution, and entropy
We consider Hamiltonians of the type H = (cid:88) i,j c i,j a † i a j (G1)where the operators a i , a † i satisfy the fermionic anti-commutation relations, { a i , a † j } = δ i,j , (G2) { a i , a j } = { a † i , a † j } = 0 . (G3)Since the matrix c in (G1) is Hermitian, it can be diagonalisedby a unitary operator, c = ADA † , where AA † = 1 and D =diag { (cid:15) , . . . , (cid:15) n } . The Hamiltonian then can be expressed as, H = (cid:88) k (cid:15) k η † k η k , (G4)with η k = (cid:88) j A ∗ j,k a j , (G5) η † k = (cid:88) j A j,k a † j . (G6)The fermionic operators η † k , η k are usually referred to as nor-mal modes . The unitarity of A ensures that the transformationpreserves the commutation relations, { η k , η † l } = (cid:88) i,j A k,i A ∗ l,j { a i , a † j } = δ k,l . (G7)where we used (G3).In the following, we will describe states within the frame-work of correlation matrices. Define the entries of the corre-lation matrix γ ( ρ ) corresponding to ρ as γ a ( ρ ) i,j = Tr( a † i a j ρ ) . (G8)Notice that the diagonal elements represent the occupationprobabilities, or populations, of each physical fermion. Thecorrelation matrix in the diagonal basis γ η ( ρ ) i,j = Tr( η † i η j ρ ) is related to γ a through γ η = A T γ a A ∗ . The diagonal ele-ments of γ η , corresponding to the populations of the normalmodes, play an important role, and we denote them by p k , p k = Tr( η † k η k ρ ) . (G9)The time evolution of γ ( ρ ) under H , ρ ( t ) = e − i Ht ρe i Ht , canbe easily computed in the Heisenberg picture, ˙ η k = i[ H, η k ] = − i E k η k , (G10) η k ( t ) = e − i E k t η k , (G11)where we have used { η i , η † j } = δ i,j and η k = 0 . Therefore,on the one hand, it follows that γ η ( ρ ( t )) = e i tD γ η ( ρ ) e − i tD (G12)with D = diag { E , . . . , E n } . In the original basis it reads, γ a ( ρ ( t )) = U γ a ( ρ ) U † (G13)with U = A ∗ e i tD A T . On the other, the time averaged state,which is defined as, (cid:104) ρ (cid:105) t = lim T →∞ T (cid:90) T ρ ( t ) , (G14)is represented simply by γ η ( (cid:104) ρ (cid:105) t ) = (cid:104) γ η ( ρ ) (cid:105) t = Γ [ γ η ( ρ ( t ))] , (G15)where Γ corresponds to a de-phasing operation. In fact, thiscorrelation matrix is the same as the one of the GGE wherethe conserved quantities are the normal modes η † k η k , i.e., γ (cid:16) Ω GGE ( ρ, H, { η † k η k } ) (cid:17) = γ ( (cid:104) ρ (cid:105) t ) . (G16)Note however, that this does not imply that the full quantumstate of the GGE is the same as the time-averaged state.A particularly important class of fermionic states is givenby Gaussian states . In this situation of fixed particle number,such Gaussian states are completely determined from the cor-relation matrix. In particular, eigenstates and thermal states offree fermionic Hamiltonians are Gaussian states, but clearlyalso the GGEs given above, as they are obtained by maximiz-ing the entropy given the expectation values of the operators η † k η k .If a state ρ is Gaussian, the entropy of ρ can be calculatedas S ( ρ ) = (cid:88) k H( d k ) , (G17)where d k are the eigenvalues of γ ( ρ ) , and H( p ) = − p ln p − (1 − p ) ln(1 − p ) . This fact allows us to study entropy-production numerically for large systems.0
2. Work extraction for free fermions