Contributions from populations and coherences in non-equilibrium entropy production
Adalberto D. Varizi, Mariana A. Cipolla, Martí Perarnau-Llobet, Raphael C. Drumond, Gabriel T. Landi
CContributions from populations and coherences in non-equilibrium entropy production
Adalberto D. Varizi,
1, 2
Mariana A. Cipolla, Mart´ı Perarnau-Llobet, Raphael C. Drumond, and Gabriel T. Landi Instituto de F´ısica da Universidade de S˜ao Paulo, 05314-970 S˜ao Paulo, Brazil Departamento de F´ısica, Instituto de Ciˆencias Exatas,Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, Minas Gerais, Brazil D´epartement de Physique Appliqu´ee, Universit´e de Gen`eve, 1211 Geneva, Switzerland Departamento de Matem´atica, Instituto de Ciˆencias Exatas,Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, Minas Gerais, Brazil (Dated: February 23, 2021)The entropy produced when a quantum system is driven away from equilibrium can be decomposed in twoparts, one related with populations and the other with quantum coherences. The latter is usually based on theso-called relative entropy of coherence, a widely used quantifier in quantum resource theories. In this paperwe argue that, despite satisfying fluctuation theorems and having a clear resource-theoretic interpretation, thissplitting has shortcomings. In particular, we find that at low temperatures it predicts the entropy production iscompletely dominated by the classical term, even though quantum e ff ects become more relevant in this regime.To amend for this, we provide an alternative splitting of the entropy production, in which the contributions frompopulations and coherences are written in terms of a thermal state of a specially dephased Hamiltonian. Thephysical interpretation of our proposal is discussed in detail. We also contrast the two approaches by studyingwork protocols in a transverse field Ising chain and a macrospin of varying dimension. I. INTRODUCTION AND PRELIMINARY RESULTS
Quantum coherence and quantum correlations play a keyrole in the thermodynamics of microscopic systems [1, 2].They can be exploited to extract useful work [3–9], speed-up energy exchanges [10–13], and improve heat engines [14–19]. On a more fundamental level, they alter the possiblestate transitions in thermodynamic processes [20–22], leadto new forms of work and heat fluctuations [23–28], modifythe fluctuation-dissipation relation for work [29–31] and mayeven generate heat flow reversals [32–35]. Understanding therole of coherence in the formulation of the laws of quantumthermodynamics is therefore a major overarching goal in thefield, which has been the subject of considerable recent inter-est.When a system relaxes to equilibrium, in contact with aheat bath, quantum coherences are known to contribute an ad-ditional term to the entropy production [21, 36, 37], whichquantifies the amount of irreversibility in the process. A sim-ilar e ff ect also happens in unitary work protocols [38, 39]. Tobe concrete, we focus on the latter and consider a scenariowhere a system is described by a Hamiltonian H t = H ( g t ), de-pending on a controllable parameter g t . The system is initiallyprepared in thermal equilibrium at a temperature T , such thatits initial state is the thermal state ρ th0 ≡ ρ th ( g ) = e − β H / Z ,where β = / T and Z = tr (cid:110) e − β H (cid:111) is the partition function.At t =
0, a work protocol g t , that lasts for a total time τ , isapplied to the system, driving it out of equilibrium [40, 41].Letting U denote the unitary generated by the drive, the stateof the system after a time τ will be ρ τ = U ρ th0 U † . (1)In general, ρ τ will be very di ff erent from the correspondingequilibrium state ρ th τ = e − β H τ / Z τ . This di ff erence is capturedby the entropy production (also called non-equilibrium lag in this context) [42–45], Σ = S (cid:0) ρ τ || ρ th τ (cid:1) , (2)where S ( ρ || σ ) = tr { ρ (ln ρ − ln σ ) } (cid:62) Σ = β (cid:0) (cid:104) W (cid:105) − ∆ F (cid:1) ,where (cid:104) W (cid:105) = tr (cid:0) H τ ρ τ − H ρ th0 (cid:1) is the work performed in theprocess and ∆ F = F ( g τ ) − F ( g ) is the change in equilib-rium free energy, F ( g ) = tr (cid:110) H ( g ) ρ th ( g ) (cid:111) − T S ( ρ th ( g )) (with S ( ρ ) = − tr( ρ ln ρ ) being the von Neumann entropy). Dueto its clear thermodynamic interpretation, Σ has been widelyused as a quantifier of irreversibility, both theoretically [45–53] and experimentally [54–65].The entropy production Σ in Eq. (2) contains contributionsof both a classical and quantum nature. This is linked withthe fact that the work protocol g t can modify the Hamiltonian H ( g ) in two ways. On the one hand, it may alter the spacing ofthe energy levels; and, on the other, it may rotate the eigenvec-tors (Fig. 1(a)). The latter is directly associated with quantumcoherence and to the fact that [ H ( g t ) , H ( g t )] (cid:44)
0, for twodi ff erent times t , t . It therefore has no classical counterpart,and corresponds to a fundamental feature distinguishing clas-sical and quantum processes. Identifying how each physicalprocess contributes to Σ is in general a challenging task, as wewill show in this article. In the literature, a popular choice isthe splitting put forward in [20, 21, 36, 38]: Σ = Γ cl + Γ qu , (3)where Γ cl = S (cid:0) D H τ ( ρ τ ) || ρ th τ (cid:1) , (4) Γ qu = S (cid:0) ρ τ || D H τ ( ρ τ ) (cid:1) = S (cid:0) D H τ ( ρ τ ) (cid:1) − S ( ρ τ ) , (5)with D H ( ρ ) being the super-operator that completely dephasesthe state ρ in the eigenbasis of H (explicitly defined below, a r X i v : . [ qu a n t - ph ] F e b (a) (b) Eq. (3)Eq. (10) (c) (d)
FIG. 1. Di ffi culties in identifying the classical and quantum contributions to the entropy production. (a) A work protocol H ( g t ) canmodify the Hamiltonian in two ways: altering the energy level spacings, which can be viewed as a semi-classical e ff ect, and rotating theenergy eigenbasis, which is a fully quantum property with no classical counterpart. (b) The entropy production (2) compares the final state ρ τ with the reference thermal state ρ th τ . To understand its classical and quantum contributions, the splitting (3) uses an intermediate state D H τ ( ρ τ ).Conversely, in this paper we introduce the splitting (10), which uses the state ˜ ρ th τ , in Eq. (17). (c) The contributions Γ qu and Γ cl of Eq. (3) in aminimal qubit model, as a function of βω , plotted using Eqs. (6) and (7) with θ = . in Eq. (12)). The first term, Γ cl , measures the entropic dis-tance between the populations of the actual final state ρ τ andthose of the reference thermal state ρ th τ , and is generally iden-tified with the classical contribution. The term Γ qu , in turn,is known as the relative entropy of coherence and comparesthe final state ρ τ with the dephased state D H τ ( ρ τ ). It hencecaptures the contribution from coherences in the energy basis.By construction, Γ cl and Γ qu in Eq. (3) are both non-negative,which shows that coherences increase the entropy productionin the process, as compared to a fully classical (incoherent)scenario.The splitting (3), first analyzed in [20], has been studiedin the context of the resource theory of thermodynamics [21],relaxation towards equilibrium [36, 37], and in work proto-cols in the absence of a bath [9, 38, 39]. At the stochas-tic level, both Γ qu and Γ cl satisfy individual fluctuation the-orems [38], which is a very desirable property. Moreover, Γ cl has a resource-theoretic interpretation within the resourcetheory of athermality [66, 67], while Γ qu is a natural mono-tone in the resource theory of coherence [68, 69]. These factsmake the splitting (3) a valuable tool in understanding the rel-ative contribution of classical and quantum features to non-equilibrium processes. However, working with various mod-els, we have observed that this splitting behaves strangely,even in some simple protocols. More specifically, we iden-tify two main shortcomings.The first concerns the relative magnitudes of Γ qu and Γ cl :At low temperatures, Γ cl is always found to be much largerthan Γ qu . This is highly counter-intuitive, as we usually ex-pect that low temperatures should make quantum e ff ects moresalient. And the reason behind it is quite simple: The classicalterm Γ cl is a relative entropy with respect to ρ th τ , which tends toa pure state when β → ∞ . Since the relative entropy divergeswhen the support of ρ τ is not contained in that of ρ th τ [70], Γ cl will grow unbounded when β → ∞ . Conversely, Γ qu is aspecial kind of relative entropy, since it can be expressed asa di ff erence between two von Neumann entropies, as in thesecond equality of (5). As β → ∞ , S ( ρ τ ) tends to zero (sinceit tends to a pure state), while S ( D H τ ( ρ τ )) ∈ [0 , ln d ], where d is the dimension of the Hilbert space. As a consequence, Γ qu will always remain finite. Thus, when β → ∞ , the entropyproduction will always be dominated by the classical contri-bution Γ cl .This issue can be neatly illustrated by a minimal qubitmodel. Consider a qubit which starts at H = ωσ z and issuddenly quenched ( U =
1) to H τ = ω ( σ z cos θ + σ x sin θ )(where σ α are Pauli matrices). In this quench the energy lev-els remain intact and all that happens is that the eigenbasis isrotated by an angle θ . This is thus, by all accounts, a highlyquantum process. The entropy production (2) for this modelreads Σ = t tanh − ( t ) sin ( θ / ) , (6)where t = tanh( βω ) ∈ [0 , Γ qu in Eq. (5), reads Γ qu = t tanh − ( t ) − t cos θ tanh − ( t cos θ ) (7) −
12 ln (cid:16) + sinh ( βω ) sin θ (cid:17) . A plot of Γ qu and Γ cl = Σ − Γ qu is shown in Fig. 1(c) as a func-tion of βω , for θ = .
1. As can be seen, in general both quan-tities are comparable in magnitude. But, as the temperaturegoes down ( β goes up), the classical contribution becomes in-creasingly larger and eventually dominates. Thus, at very lowtemperatures, most of Σ comes from the population term Γ cl and very little from coherences.The second issue with the splitting (3) concerns infinites-imal quenches . This is a very important scenario, widelystudied in the context of critical systems [71–74] and quasi-isothermal processes [29, 30]. The idea is to analyze theentropy production perturbatively, for a small instantaneousquench of the work parameter, from g to g + δ g . The prob-lem with Γ qu and Γ cl in this case is that, as will be shown, theparameter δ g appears multiplied by a factor that increases ex-ponentially with β . Hence, the radius of convergence of Γ qu and Γ cl , in δ g , tends to zero exponentially fast as β → ∞ . For Σ , no such issue arises.This is again well illustrated by the qubit example inEqs. (6) and (7), where the quench parameter is now the an-gle θ . We see that Σ in (6) can be readily expanded in powersof θ , for any temperature β (or any t = tanh( βω )). The same isnot true for Γ qu , however. The problem is in the third term ofEq. (7), which is a function of x = sinh ( βω ) sin θ . This quan-tity appears inside a logarithm, in the form ln(1 + x ). How-ever, a series expansion of ln(1 + x ) only converges if | x | < ( βω ) grows exponentially with β ,at low temperatures, extremely small values of θ are requiredto validate a series expansion.More generally, one can readily show that for Σ this issuedoes not arise. If we use Σ = β (cid:0) (cid:104) W (cid:105) − ∆ F (cid:1) , we find in the caseof infinitesimal quenches that Σ = β tr (cid:8) ∆ H ρ th ( g ) (cid:111) − β ∆ F , (8)where ∆ H = H ( g + δ g ) − H ( g ) and ∆ F = F ( g + δ g ) − F ( g ).A series expansion of Σ in δ g therefore amounts to two things.First, an expansion of ∆ H in powers of δ g , which is entirelyindependent of β . And second, an expansion of F ( g ), which isan analytic and generally smooth function (except possibly ata critical point [72]). Indeed, if H ( g ) is linear in g , the leadingorder contribution to the expansion becomes [73] Σ (cid:39) − βδ g ∂ F ∂ g , (9)showing that Σ is simply proportional to the equilibrium sus-ceptibility, a textbook quantity used throughout equilibriumstatistical mechanics.The above results show that, despite its interesting prop-erties (individual fluctuation theorems and resource-theoreticinterpretation), the splitting (3) is not without flaws. In or-der to overcome these shortcomings, in this paper we proposea di ff erent splitting, which is inspired by the recent resultsof [30]. We label it as Σ = Λ cl + Λ qu . (10)The actual definitions of Λ qu and Λ cl will be given below inSec. II and a stochastic trajectories formulation will be givenin Sec. III. A comparison in the case of the minimal qubitexample is also presented in Fig. 1(d). In this case, using theresults of Sec. II, one finds the following elegant expressionfor Λ qu (to be contrasted with Eq. (7)): Λ qu =
12 ln (cid:32) − tanh ( βω cos θ )1 − tanh ( βω ) (cid:33) . (11)As seen in Fig. 1(d), the behavior of Λ qu and Λ cl in this caseis much more reasonable: Since the process is highly coher-ent, Λ cl is very small; and as the temperature goes down, Λ qu grows monotonically, showing that cold processes have highercontributions from the coherences.The features discussed in Fig. 1 are not restricted toquenches. To illustrate that we show in Fig. 2 another qubitexample, where the process is assumed to be cyclic, with H τ = H = ωσ z , and the unitary is taken to be generated by Γ �� Γ �� Σ � � � � � ���������������� βω ( � ) Λ �� Λ �� Σ � � � � � ���������������� βω ( � ) Γ �� Γ �� Σ � � � � � �������� βω ( � ) Λ �� Λ �� Σ � � � � � �������� βω ( � ) FIG. 2. Splitting of the entropy production in a cyclic qubit model, H τ = H = ωσ z . The unitary is generated by an x -pulse with aduration τ ; that is, U = e − i τ ( H + h x σ x ) , where h x is the pulse intensity.The curves were computed using Eqs. (14), (15), (16), (19) and (18),with ω = h x = . ff erent values of τ : in (a),(b) τ = . τ = an x -pulse with a duration τ ; that is, U = e − i τ ( H + h x σ x ) , where h x is the pulse intensity. Fig. 2 illustrates the results for ω = h x = . τ : in the upper panels τ = . τ =
1. The results show that for (3) thebehavior is always roughly the same, with Γ cl always eventu-ally dominating at low temperatures. Conversely, for the newsplitting (10) a richer competition is observed. Depending onthe parameters we may either have Λ qu dominating, or Λ cl , orboth.As we will show in this paper, our new splitting (10) cor-rects both of the aforementioned shortcomings of Eq. (3). As adownside, however, Λ qu and Λ cl do not share some of the niceproperties of Γ qu and Γ cl . First, Λ qu cannot be directly linkedwith a monotone for coherence or asymmetry [69]. Second,while Λ cl always satisfies an individual fluctuation theorem, Λ qu only does so in the case of infinitesimal quenches. Thepros and cons of each splitting are highlighted in Table I. Wealso show that for infinitesimal quenches at high temperatures,both splittings coincide - see Sec. III C.To illustrate the usefulness of our results, we analyze ournew splitting in two quantum many-body problems. Previ-ous works have focused on the behaviour of the statistics ofwork and entropy production Σ for quantum quenches [71–76], with emphasis in quantum phase transitions [72, 77–84].Motivated by this, we analyze in Sec. IV the transverse fieldIsing model (TFIM), and discuss the behavior of (10) in thevicinity of the quantum critical point. This is complementaryto the analysis put forth in [39], which studied Eq. (3). Then, TABLE I. Comparison between Λ qu , Λ cl , Γ qu and Γ cl . Λ qu Λ cl Γ qu Γ cl Fluctuation Theorem (cid:55) (cid:51) (cid:51) (cid:51)
Fluctuation Theorem when ∆ H → (cid:51) (cid:51) Analytic when ∆ H → T (cid:51) (cid:55) Resource-theoretic interpretation (cid:55) (cid:51)
Vanishing for commuting protocols (cid:51) - (cid:51) - Dominant for highly coherent protocols (cid:51) - (cid:55) - Dominant at low temperatures (cid:51) - (cid:55) - in Sec. V, we consider a macrospin of varying size and focuson the full statistics of Λ qu and Λ cl , including their probabil-ity distributions and their first four cumulants. We finish withconclusions and future perspectives in Sec. VI. II. SPLITTINGS OF THE ENTROPY PRODUCTION
In this section we introduce our alternative splitting of theentropy production [Eq. (10)]. We focus for now at the levelof averages; the corresponding stochastic formulation will bepresented in Sec. III.Let O denote any Hermitian observable and decompose itas O = (cid:80) α o α Π α , where Π α are projectors onto the subspaceswith eigenvalues o α . We define the dephasing operation D O ( • ) = (cid:88) α Π α • Π α . (12)The rationale of the splitting Eq. (3) was to introduce an in-termediate step, associated with the state D H τ ( ρ τ ) (Fig. 1(b)).This represents the final state ρ τ dephased in the eigenbasisof the final Hamiltonian. If the process generates coherences,this state will di ff er from the actual final state ρ τ and their en-tropic distance will be precisely Γ qu in Eq. (5).For convenience, we introduce the non-equilibrium free en-ergy, associated with the final Hamiltonian H τ F ( ρ ) = tr (cid:110) H τ ρ (cid:111) − T S ( ρ ) . (13)Non-equilibrium free energies depend on two parameters, H and ρ . However, in this paper, we will henceforth only needfree energies defined with respect to H τ , so we write it moresimply as F ( ρ ). In terms of F , the entropy production (2) canbe written as Σ = β (cid:110) F ( ρ τ ) − F ( ρ th τ ) (cid:111) , (14)Similarly, one can also express Γ qu and Γ cl in terms of freeenergy di ff erences. Since tr (cid:8) H τ D H τ ( ρ τ ) (cid:9) = tr (cid:8) H τ ρ τ (cid:9) , one findsthat Γ qu = β (cid:110) F ( ρ τ ) − F (cid:0) D H τ ( ρ τ ) (cid:1)(cid:111) , (15) Γ cl = β (cid:110) F (cid:0) D H τ ( ρ τ ) (cid:1) − F ( ρ th τ ) (cid:111) , (16)which clearly add up to Σ . The splitting (3) uses D H τ ( ρ τ ) as intermediate state. Ournew splitting (10) follows a similar logic, but in reverse: In-stead of working with ρ τ dephased in the basis of H τ , we workwith H τ dephased in the basis of ρ τ . More precisely, we define˜ ρ th τ = exp {− β D ρ τ ( H τ ) } tr (cid:16) exp {− β D ρ τ ( H τ ) } (cid:17) , (17)which is a thermal state based only on the incoherent part of H τ , in the basis of ρ τ (as a consequence, [ ˜ ρ th τ , ρ τ ] = Λ cl = β (cid:110) F ( ρ τ ) − F ( ˜ ρ th τ ) (cid:111) , (18) Λ qu = β (cid:110) F ( ˜ ρ th τ ) − F ( ρ th τ ) (cid:111) , (19)which add up to Σ , as in Eq. (10). The first term, Λ cl , comparesthe two commuting states ρ τ and ˜ ρ th τ and is hence associatedwith their population mismatch. The nonnegativity of Λ cl be-comes evident by noting that it can also be written as Λ cl = S ( ρ τ || ˜ ρ th τ ) . (20)The term Λ qu , on the other hand, compares ρ th τ ∝ e − β H τ with˜ ρ th τ ∝ e − β D ρτ ( H τ ) . Unlike Λ cl , the contribution Λ qu cannot bewritten as a relative entropy. In fact, written down explicitly,it reads Λ qu = tr (cid:110) ρ τ (cid:16) ln ˜ ρ th τ − ln ρ th τ (cid:17)(cid:111) . (21)Notwithstanding, as shown in Appendix A, it turns out that Λ qu is still non-negative, and zero if and only if [ H τ , ρ τ ] = U = ρ τ = ρ th0 . Hence,all we need to do in order to compute Λ qu is to dephase thefinal Hamiltonian H τ = ω ( σ z cos θ + σ x sin θ ) in the basis of ρ th0 . Or, what is equivalent, in the basis of H . The result isthus simply D ρ τ ( H τ ) = ω cos( θ ) σ z . Using this in (19) yieldsEq. (11), which is the result plotted in Fig. 1(d) and discussedin Sec. I. A. Infinitesimal quenches
The physics of the problem becomes particularly simpler inthe case of infinitesimal quenches. We therefore now special-ize the above results to this scenario. This will provide strongjustifications in favor of the new splitting (10). Furthermore,in this limit the splitting (10) becomes equivalent to the one re-cently put forward in [30]. More precisely, in [30] the authorsdescribe quasi-isothermal processes as a series of infinitesi-mal quenches, and in particular consider how Σ splits into aclassical and quantum contribution. Focusing on a single in-finitesimal quench, both approaches become directly compa-rable and, as we will show, agree with each other.We thus analyze what happens if we take U =
1, and as-sume that H changes only by a small amount ∆ H (i.e., wewrite H τ = H + ∆ H ). Since U =
1, the state of the systemremains unchanged: ρ τ = ρ th0 . Therefore, dephasing H τ in thebasis of ρ τ is equivalent to dephasing in the basis of H : D ρ τ ( H τ ) = D ρ th0 ( H τ ) = D H ( H τ ) . (22)Let us define the dephased (incoherent) and coherent partsof the perturbation ∆ H , in the initial energy basis, ∆ H d = D H ( ∆ H ) and ∆ H c = H τ − D H ( H τ ). Then, following a pro-cedure detailed in Appendix B of Ref. [30], one may showthat, ˜ ρ th τ = ρ th0 − β J ρ th0 [ ∆ H d − (cid:104) ∆ H d (cid:105) ] + O ( ∆ H ) , (23) ρ th τ = ρ th0 − β J ρ th0 [ ∆ H − (cid:104) ∆ H (cid:105) ] + O ( ∆ H ) , (24)where (cid:104) . . . (cid:105) = tr { . . . ρ th0 } and J ρ is a super-operator defined as J ρ [ • ] = (cid:90) ρ t • ρ − t d t . (25)We see that both ρ th τ and ˜ ρ th τ can be expanded essentially in apower series in β ∆ H . Conversely, the same is not true for thestate D H τ ( ρ th0 ) entering (16) and (15). In fact, one may showthat to order ∆ H [85] D H +∆ H ( ρ th0 ) = ρ th0 + lim s →∞ is (cid:90) s d t (cid:90) d x t e − ixH t [ ρ th0 , ∆ H ] e ixH t . (26)Even though this is an expansion in ∆ H , the dependence on β enters in a highly non-trivial way. This explains the non-analytic behavior of Γ cl and Γ qu at low temperatures, discussedin Sec. I.Plugging (23)-(24) in Eqs. (2), (20) and (21) leads, up tosecond order, to Σ = β (cid:110) ∆ H J ρ th0 [ ∆ H − (cid:104) ∆ H (cid:105) ] (cid:111) = Λ cl + Λ qu , (27) Λ cl = β (cid:110) ∆ H d J ρ th0 [ ∆ H d − (cid:104) ∆ H d (cid:105) ] (cid:111) , (28) Λ qu = β (cid:110) ∆ H c J ρ th0 [ ∆ H c ] (cid:111) , (29)where we used the fact that (cid:104) ∆ H d (cid:105) = (cid:104) ∆ H (cid:105) . The interestingaspect of these results is that, within this infinitesimal quenchlimit, Λ cl and Λ qu are found to be related to Σ via the simpleseparation of the perturbation, ∆ H = ∆ H d + ∆ H c , into a de-phased and a coherent part. These results also coincide withthe splitting proposed in [30].An additional justification for the splitting (10) can begiven in terms of the fluctuation-dissipation relation (FDR).As shown in Refs. [29, 30], Eq. (27) can also be written as Σ = β Var [ ∆ H ] − β Q , (30)where Var [ ∆ H ] = (cid:104) ∆ H (cid:105) − (cid:104) ∆ H (cid:105) , is the variance of theperturbation, and Q = β (cid:90) d y I y ( ρ th0 , ∆ H ) (cid:62) , (31) is a measure of quantum coherence, associated with the so-called Wigner-Yanase-Dyson skew information [86] I y ( (cid:37), X ) = −
12 tr (cid:110) [ (cid:37) y , X ][ (cid:37) − y , X ] (cid:111) . (32)For incoherent processes one recovers the usual FDR Σ = β Var [ ∆ H ] [46]. But when the process is coherent, the FDRis broken by a term − β Q . Repeating the same procedure for Λ cl and Λ qu , one readily finds that Λ cl = β [ ∆ H d ] , Λ qu = β [ ∆ H c ] − β Q . (33)Whence, Λ cl always satisfies a standard FDR, and all viola-tions are associated to Λ qu . This provides additional justifica-tion as to why Λ qu is referred to as a quantum contribution. III. STOCHASTIC TRAJECTORIES
We now discuss how to formulate the splittings (3) and(10) at the level of stochastic trajectories, based on a stan-dard two-point measurement (TPM) scheme [48]. Since
Σ = β ( (cid:104) W (cid:105) − ∆ F ), the statistics of Σ can be obtained solely frommeasurements in the eigenbasis of the initial and final Hamil-tonians. As first shown in [38], a major advantage of the orig-inal splitting (3) is that this remains true when assessing theindividual contributions Γ cl and Γ qu ; that is, no additional mea-surements are necessary. As we will now show, the same isalso true for Λ cl and Λ qu [Eq. (10)]. This means that both split-tings can be assessed, at the stochastic level, with the sameamount of information as a standard TPM.Irrespective of the splitting one is interested in, the protocolmay therefore be described as follows. Initially the systemis in the thermal state ρ th0 , associated with the Hamiltonian H = (cid:80) i (cid:15) i | i (cid:105)(cid:104) i | . The first measurement is performed in thebasis | i (cid:105) , which occurs with probability p i = e − β(cid:15) i / Z . Con-versely, the second measurement is performed at time τ , afterthe map (1), and in the eigenbasis of the final Hamiltonian H τ = (cid:80) j (cid:15) τ j | j τ (cid:105)(cid:104) j τ | . The bases {| i (cid:105)} and {| j τ (cid:105)} are, in general,not compatible.The conditional probability of finding the system in | j τ (cid:105) given that it was initially in | i (cid:105) is |(cid:104) j τ | U | i (cid:105)| . The probabil-ity associated with the forward protocol | i (cid:105) → | j τ (cid:105) is thus P F [ i , j ] = |(cid:104) j τ | U | i (cid:105)| p i . The dynamics is defined as be-ing incoherent when |(cid:104) j τ | U | i (cid:105)| = δ i , j , which means U isnot able to generate transitions between states of the initialand final Hamiltonians. Similarly, in the backward protocolthe system starts in ρ th τ and one measures first in the basis of H τ , yielding | j τ (cid:105) with probability p τ j = e − β(cid:15) τ j / Z τ . The time-reversed unitary U † is then applied, after which one measuresin the basis | i (cid:105) of H . This yields the backward distribution P B [ i , j ] = |(cid:104) i | U † | j τ (cid:105)| p τ j .The entropy production associated to the trajectory | i (cid:105) →| j τ (cid:105) is now defined as usual: σ [ i , j ] = ln P F [ i , j ] P B [ i , j ] = ln p i / p τ j . (34)The second equality follows from the fact that |(cid:104) i | U † | j τ (cid:105)| = |(cid:104) j τ | U | i (cid:105)| . As a consequence, σ [ i , j ] depends only on theequilibrium populations p i and p τ j , associated with the initialand final Hamiltonians. As can be readily verified, (cid:104) σ [ i , j ] (cid:105) = (cid:80) i , j σ [ i , j ] P F [ i , j ] = Σ , returns precisely Eq. (2). In addition, σ [ i , j ] also satisfies an integral fluctuation theorem (cid:104) e − σ (cid:105) = A. Stochastic definitions for the splittings (3) and (10)
Following [38], we now define stochastic quantities asso-ciated to Γ cl and Γ qu . In order to do that, we first write thedephased state D H τ ( ρ τ ) as D H τ ( ρ τ ) = (cid:80) j q τ j | j τ (cid:105)(cid:104) j τ | , where q τ j = (cid:104) j τ | ρ τ | j τ (cid:105) = (cid:88) i |(cid:104) j τ | U | i (cid:105)| p i . (35)In passing, we note that q τ j = (cid:80) i P F [ i , j ], so q τ j can also beinterpreted as the marginal distribution of the final measure-ment. As shown in [38], we may now define γ cl [ i , j ] = ln q τ j / p τ j , (36) γ qu [ i , j ] = ln p i / q τ j . (37)Clearly γ cl [ i , j ] + γ qu [ i , j ] = σ [ i , j ], which is the stochasticanalog of (3). Moreover, (cid:104) γ cl [ i , j ] (cid:105) = Γ cl and (cid:104) γ qu [ i , j ] (cid:105) = Γ qu .Similarly, we construct stochastic quantities for the newquantities Λ cl and Λ qu in Eq. (10). The central object now isthe thermal state ˜ ρ th τ , defined in Eq. (17) and associated withthe Hamiltonian D ρ τ ( H τ ). Since the system evolves unitarily, ρ τ = U ρ th0 U † = (cid:80) i p i | ψ i (cid:105)(cid:104) ψ i | , where | ψ i (cid:105) = U | i (cid:105) . That is, ρ τ has the same populations p i as ρ th0 , but a rotated eigenbasis.Based on this, we can now write Eq. (17) as˜ ρ th τ = (cid:88) i ˜ p τ i | ψ i (cid:105)(cid:104) ψ i | , ˜ p τ i = e − β (˜ (cid:15) τ i − F (˜ ρ th τ )) , (38)where ˜ (cid:15) τ i = (cid:104) ψ i | H τ | ψ i (cid:105) are the eigenvalues of the dephasedHamiltonian D ρ τ ( H τ ) and F ( ˜ ρ th τ ) is the same free energy asthat appearing in Eq. (18). We then define λ cl [ i , j ] = ln p i / ˜ p τ i , (39) λ qu [ i , j ] = ln ˜ p τ i / p τ j . (40)These quantities satisfy λ cl [ i , j ] + λ qu [ i , j ] = σ [ i , j ], as well as (cid:104) λ cl [ i , j ] (cid:105) = Λ cl and (cid:104) λ qu [ i , j ] (cid:105) = Λ qu . B. Cumulant generating functions
For all stochastic quantities in the previous section, we candefine their corresponding probability distributions or, what ismore convenient, their cumulant generating functions (CGFs).For instance, from (34) we define P ( σ ) = (cid:88) i , j P F [ i , j ] δ ( σ − σ [ i , j ]) , (41) from which we may compute the CGF, K σ ( v ) = ln (cid:104) e − v σ (cid:105) .With some manipulations, this can be neatly written as [53,87] K σ ( v ) = ln tr (cid:110) ( ρ th τ ) v ( ρ τ ) − v } (42) = ( v − S v ( ρ th τ || ρ τ ) . The second equality expresses the CGF in terms of the R´enyidivergences S v ( ρ || σ ) = ( v − − ln tr (cid:110) ρ v σ − v (cid:111) , which may beconvenient in some situations. Setting v = K σ (1) = (cid:10) e − σ (cid:11) = . (43)In addition, from the CGF we may compute any cumulant of σ as κ n ( σ ) = ( − n ∂ n K σ ∂ v n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v = , (44)with κ ( σ ) = Σ being the mean in Eq. (2).We may also compute the joint CGF of γ cl and γ qu , definedas K γ cl ,γ qu ( v , u ) = ln (cid:104) e − v γ cl − u γ qu (cid:105) . With similar manipulations, itmay be written as K γ cl ,γ qu ( v , u ) = ln tr (cid:110) ( ρ th τ ) v (cid:2) D H τ ( ρ τ ) (cid:3) u − v ( ρ τ ) − u (cid:111) . (45)The CGF of σ = γ cl + γ qu , Eq. (42), is recovered by setting u = v ; that is K σ ( v ) = K γ cl ,γ qu ( v , v ). The reduced CGFs of γ cl and γ qu are found by setting u = v =
0, respectively: K γ cl ( v ) = ln tr (cid:110) ( ρ th τ ) v (cid:2) D H τ ( ρ τ ) (cid:3) − v ρ τ (cid:111) , (46) K γ qu ( u ) = ln tr (cid:110)(cid:2) D H τ ( ρ τ ) (cid:3) u ( ρ τ ) − u (cid:111) . (47)From this one may verify that γ cl and γ qu individually satisfyfluctuation theorems (cid:104) e − γ cl (cid:105) = (cid:104) e − γ qu (cid:105) = . (48)Note also that, except in certain particular cases, Eq. (45) can-not be written as a sum of two CGFs, which means γ cl and γ qu are statistically dependent.Similarly, we compute the joint CGF of λ cl and λ qu , definedas K λ cl ,λ qu ( v , u ) = ln (cid:104) e − v λ cl − u λ qu (cid:105) . It reads K λ cl ,λ qu ( v , u ) = ln tr (cid:110) ( ρ th τ ) u ( ˜ ρ th τ ) v − u ( ρ τ ) − v (cid:111) . (49)The reduced CGFs of λ cl and λ qu are again found by setting u = v = K λ cl ( v ) = ln tr (cid:110) ( ˜ ρ th τ ) v ( ρ τ ) − v (cid:111) (50) K λ qu ( u ) = ln tr (cid:110) ( ρ th τ ) u ( ˜ ρ th τ ) − u ρ τ (cid:111) , (51)Once again, λ cl and λ qu are, in general, statistically dependent.Eq. (50) implies that λ cl satisfies a fluctuation theorem, (cid:104) e − λ cl (cid:105) = . (52)But the same is not true for λ qu . Notwithstanding, as we willshow, this property is recovered in the limit of infinitesimalquenches. C. Infinitesimal quenches
As before, we now specialize the above expressions to thecase of infinitesimal quenches. Since U =
1, the path proba-bility reduces to P F [ i , j ] = |(cid:104) j τ | i (cid:105)| p i . Moreover, since ∆ H is assumed to be small, | j τ (cid:105) will be close to | i (cid:105) and (cid:15) τ j willbe close to (cid:15) j . For concreteness, we assume that the spectraof H is non-degenerate. Standard perturbation theory thenyields, to order ∆ H , (cid:15) τ j = (cid:15) j + ∆ H j j + E (2) j , (53)where ∆ H i j = (cid:104) i | ∆ H | j (cid:105) and E (2) j = (cid:80) l (cid:44) j | ∆ H jl | / ( (cid:15) j − (cid:15) l ).Note that if we split ∆ H = ∆ H d + ∆ H c , the first non-trivialcontribution of the former is ∆ H j j , while that of the latter is E (2) j . Similarly, the eigenstates | j τ (cid:105) of the final Hamiltoniancan be expanded as |(cid:104) j τ | i (cid:105)| = | ∆ H i j | ( (cid:15) j − (cid:15) i ) , (54)for i (cid:44) j , while |(cid:104) j τ | j (cid:105)| = − (cid:80) (cid:96) (cid:44) j |(cid:104) j τ | (cid:96) (cid:105)| .Using this, we can expand all relevant probabilities { ˜ p τ j } , { q τ j } and { p τ j } entering in the stochastic definitions (34), (36),(37), (39) and (40): ˜ p τ i = p i (1 − ˜ f i ) , (55) p τ j = p j (1 − f j ) , (56) q τ j = p j (1 − s j ) (57)where˜ f j = β ( ∆ H j j − (cid:104) ∆ H d (cid:105) ) + β (cid:104) ∆ H d (cid:105) ( ∆ H j j − (cid:104) ∆ H d (cid:105) ) (58) + β [ ∆ H j j − (cid:104) ( ∆ H d ) (cid:105) ] , f j = ˜ f j + β [ E (2) j − (cid:104) E (2) (cid:105) ] , (59) s j = (cid:88) (cid:96) (cid:44) j − e − β ( (cid:15) (cid:96) − (cid:15) j ) ( (cid:15) j − (cid:15) (cid:96) ) | ∆ H (cid:96) j | , (60)and (cid:104) E (2) (cid:105) = (cid:80) i p i E (2) i . Note how ˜ f j depends only on thediagonal part of the perturbation, ∆ H d . This is in line withEq. (23). Conversely, f j , which is associated with the fullprobabilities p τ j , also has an additional contribution from E (2) j ,which is the term associated to coherences.Inserting Eqs. (55)-(56) into Eqs. (34), (36), (37), (39) and(40) we obtain, σ [ i , j ] = ln p i / p j − ln (cid:16) − f j (cid:17) , (61) γ cl [ i , j ] = ln (cid:16) − s j (cid:17) − ln (cid:16) − f j (cid:17) , (62) γ qu [ i , j ] = ln p i / p j − ln (cid:16) − s j (cid:17) , (63) λ cl [ i , j ] = − ln (cid:16) − ˜ f i (cid:17) , (64) λ qu [ i , j ] = ln p i / p j + ln (cid:16) − ˜ f i (cid:17) − ln (cid:16) − f j (cid:17) . (65) We are now in the position to discuss the analyticity of theentropy production and its splittings, at the stochastic level. Aseries expansion of ln(1 − x ) is convergent only for | x | < p i / p j = e − β ( (cid:15) i − (cid:15) j ) is a well behaved function,the analyticity of σ , λ cl and λ qu are all conditioned on hav-ing | f j | < | ˜ f j | <
1, which is satisfied if β | ∆ H i j | (cid:46) | s j | <
1. Each s j in (60) is a weighted contribution from all energies (cid:15) (cid:96) , with (cid:96) (cid:44) j . At low temperatures, those energies for which (cid:15) (cid:96) < (cid:15) j will yield an exponentially large contribution 1 − e − β ( (cid:15) (cid:96) − (cid:15) j ) tothe sum. Conversely, those with (cid:15) (cid:96) > (cid:15) j will contribute negli-gibly. The expansion is thus not in powers of β ∆ H , which isalso visible from (26). Instead, it is an expansion in powersof ∆ H , with coe ffi cients that depend exponentially in β . Vi-olating the condition | s j | < H = ωσ z , and after an instantaneous quenchit becomes H = ω ( σ z cos θ + σ x sin θ ), where we consider θ to be small. The problematic term in this case is s [Eq. (60)],which is given by s = (cid:16) − e βω (cid:17)(cid:32) sin θ (cid:33) . (66)In comparison, we have˜ f = βω sin ( θ/ + tanh βω ) (67) × (cid:2) + βω sin ( θ/
2) tanh βω (cid:3) f = ˜ f + βω sin θ (1 − tanh βω ) . (68)In Fig. 3 we plot Eqs. (66)-(68) as a function of βω , for θ = .
1. The condition for analyticity of γ cl and γ qu in thiscase, | s | <
1, is rapidly violated with increasing βω . For σ , λ cl and λ qu , instead, | f | < | ˜ f | < p , ˜ p τ and p τ ∝ e − βω all tend to zero exponentially as e − βω . Conversely, q τ tends to (sin θ/ .We now move on to discuss what becomes of the CGFsof Sec. III B in the infinitesimal quench regime. We start wethe CGFs of σ , λ cl and λ qu in Eqs. (42), (49)-(51). UsingEqs. (61), (64) and (65), together with the path probability P F [ i , j ] = p i |(cid:104) j τ | i (cid:105)| , we find to order ∆ H , that K λ cl ,λ qu ( v , u ) = K λ cl ( v ) + K λ qu ( u ) , (69) | s || f || f | βω FIG. 3. Analyticity of thermodynamic quantities at the stochasticlevel, for the minimal qubit model of Fig. 1. The figure comparesEqs. (66) (red-solid), (67) (blue-dashed) and (68), as a function of βω for θ = .
1. The condition | s | <
1, for the analyticity of γ cl and γ qu , is quickly violated. For σ , λ cl and λ qu , on the other hand, theconditions | f | < | ˜ f | < where K λ cl ( v ) = β v − v )Var [ ∆ H d ] , (70) K λ qu ( u ) = β u − u )Var [ ∆ H c ] + β (cid:90) u d x (cid:90) − xx d yI y ( ρ th0 , ∆ H c ) . (71)These results are quite illuminating. Eq. (69) implies λ cl and λ qu become statistically independent in this limit. Moreover,since K σ ( v ) = K λ cl ,λ qu ( v , v ), we now find that K σ ( v ) = K λ cl ( v ) + K λ qu ( v ) . (72)This means that all cumulants of σ can be split as a sum ofthe cumulants of λ cl and λ qu : κ n ( σ ) = κ n ( λ cl ) + κ n ( λ qu ). For allintents and purposes, the two channels of entropy production, Λ cl and Λ qu , may thus be regarded as stemming from indepen-dent processes: Λ cl gives the entropy production associatedwith a quench from H → D H ( H τ ), while Λ qu is associatedwith a second quench from D H ( H τ ) → H τ . We also note that,from Eq. (72), it can now be seen that in this limit λ qu satisfiesan integral fluctuation theorem: (cid:104) e − λ qu (cid:105) = K γ cl ,γ qu ( v , u ) = ln (cid:88) i , j ( p j / p i ) u (1 − s j ) u − v (1 − f j ) v p i |(cid:104) j τ | i (cid:105)| . (73)Once again, a series expansion of (1 − s j ) − x is convergent onlyif | s j | <
1. However, if | s j | < ffi ciently high temperatures, one may show that, to or-der ∆ H , we can also split K γ cl ,γ qu ( v , u ) = K γ cl ( v ) + K γ qu ( u ).And, what is more important, K γ cl and K γ qu coincide with K λ cl and K λ qu respectively. Whence, at su ffi ciently high tempera-tures the splittings (3) and (10) coincide, even at the stochasticlevel. IV. TRANSVERSE FIELD ISING MODEL
We now turn to discuss applications of our framework.We begin with the behavior of the splitting (10) in the one-dimensional transverse field Ising model (TFIM), which is aprototypical model of a quantum critical system. An analysisof (3) for the same model was recently made in [39]. Herewe aim to contrast those results with our new splitting (10).We thus restrict the analysis to quench protocols, and studythe problem at the level of the averages Λ cl and Λ qu (Eqs. (18)and (19)). Non-trivial unitaries and higher order statistics willbe studied in Sec V, for a di ff erent model.We begin by introducing the model and delineating thesteps to compute Λ cl and Λ qu . To make the paper self-consistent, some additional details are provided in Appen-dices B and C. Consider a linear chain of N spins, each de-scribed by Pauli operators σ x , y , zj and interacting via the Hamil-tonian H ( g ) = − N (cid:88) j = (cid:18) J σ xj σ xj + + g σ zj (cid:19) , (74)where g is the applied magnetic field and J is the couplingstrength, which we henceforth set to unity ( J = σ α N + = σ α . This modelpresents critical points at g = ±
1, where the system changesfrom a ferromagnetic phase, for | g | <
1, to a paramagneticphase, for | g | > H ( g ) = (cid:88) k (cid:15) k ( g ) (cid:0) η † k η k − (cid:1) , (75)where { η k } are fermionic operators and (cid:15) k ( g ) = (cid:113) ( g − cos k ) + sin k , (76)are the single-particle energies. Here, k = ± (2 n + π/ N , with n = , , ..., N / −
1, denotes the system’s quasi-momenta.We consider that the system initially has a transverse field g and is prepared in the thermal state ρ th0 = e − β H / Z . The fullexpression for ρ th0 can be found in Appendix C. Due to thestructure of (75), it can be decomposed as a product over in-dividual k modes, which greatly facilitates the calculation ofall thermodynamic quantities.The system is then decoupled from the reservoir and un-dergoes an instantaneous quench, where the field is changedto g τ = g + δ g . Since the quench is instantaneous, the stateof the system remains the same, but its Hamiltonian changes,from H to H τ = H + ∆ H , where ∆ H = − δ g (cid:80) j σ zj . Fulldetails on the computation of Λ cl and Λ qu are provided inAppendices B and C. The overall contributions of the diag-onal vs. o ff -diagonal is described by the Bogoliubov anglecos θ k = ( g − cos k ) /(cid:15) k and sin θ k = sin k /(cid:15) k . And thestate (17), associated with the dephased final Hamiltonian, isdescribed by the modified energies ˜ (cid:15) τ k = (cid:15) k + δ g cos θ k .We are interested in the thermodynamic limit ( N verylarge), where k sums can be converted to integrals and allquantities become extensive in N . In this case, we ultimatelyfind that Λ cl = N (cid:90) π d k π (cid:40) ln (cid:34) cosh (cid:16) β ˜ (cid:15) τ k (cid:17) cosh (cid:16) β(cid:15) k (cid:17) (cid:35) + β (cid:16) (cid:15) k − ˜ (cid:15) τ k (cid:17) tanh (cid:16) β(cid:15) k (cid:17)(cid:41) , (77)and Λ qu = N (cid:90) π d k π (cid:34) cosh (cid:16) β(cid:15) τ k (cid:17) cosh (cid:16) β ˜ (cid:15) τ k (cid:17) (cid:35) . (78)Adding both contributions recovers the full entropy produc-tion Σ , which was computed in [39, 72, 82] and reads Σ = N (cid:90) π d k π (cid:40) ln (cid:34) cosh (cid:16) β(cid:15) τ k (cid:17) cosh (cid:16) β(cid:15) k (cid:17) (cid:35) + β (cid:16) (cid:15) k − ˜ (cid:15) τ k (cid:17) tanh (cid:16) β(cid:15) k (cid:17)(cid:41) . (79)For comparison, in Appendix D we also present the formulasfor the splitting (3), which were developed in [39]. We alsomention, in passing, that Eqs. (77) and (78) are not perturba-tive in the quench magnitude. That is, they hold for arbitraryquenches δ g . The only assumption is that U =
1. For com-pleteness, their behavior in the infinitesimal case is presentedin Eqs. (C10) and (C11). Γ �� Λ �� � � � � ������������� β × �� - � ( � ) Γ �� Λ �� � � � � ���������������������� β × �� - � ( � ) FIG. 4. Comparison between (a) Γ cl and Λ cl , and (b) Γ qu and Λ qu ,for the TFIM, as a function of β , with g = .
75 and δ g = . Fig. 4 compares the two splittings (3) and (10) as a functionof β , with fixed g = .
75 (outside criticality) and δ g = .
01. At high temperatures, one clearly sees how both splittings co-incide. But as the system is cooled, they eventually begin todi ff er. In particular, Γ cl tends to grow linearly with β , while Λ cl tends to zero. For the quantum components the opposite isobserved: Γ qu tends to saturate while Λ qu tends to grow. Thus,at very low temperatures Λ qu becomes the dominant contribu-tion in (10), while Γ cl becomes the dominant one in (3).Next we turn to the behavior near criticality. In Fig. 5 weplot Γ cl , Γ qu , Λ cl and Λ qu as a function of the initial field g ,for di ff erent values of β (focusing on low temperatures) andfixed quench magnitude of δ g = .
01. The full entropy pro-duction Σ behaves similarly to Λ qu in Fig. 5(a); for any finite T it presents a peak at g =
1, which eventually tends to a di-vergence as β → ∞ (note also that all curves are scaled by β ).As is clear by comparing Figs. 5(a) and (b), the dominant con-tribution to the splitting (10) is Λ qu . Moreover, Λ qu is found toalways grow (and eventually diverge) with β at g =
1, while Λ cl sharply drops to zero. Conversely, for the splitting (3), wefind in Figs. 5(c), (d) that the dominant contribution is insteadthat of the populations Γ cl . Crucially, we find that in this case Γ qu remains finite as β → ∞ , while Γ cl diverges [39]. Thisshows that the critical properties of these quantities dependcrucially on the type of splitting used. V. MACROSPIN MODEL
Finally, we analyze our framework from the stochastic per-spective developed in Sec. III. To emphasize the generality ofour results, we also focus on non-quench scenarios ( U (cid:44) d = S + S x , S y , S z [88]. We consider a scenario sim-ilar to that of Fig. 2: The initial and final Hamiltonians aretaken to coincide, being given by H = H τ = − h z S z . Andthe unitary is driven by a magnetic pulse in the x direction, so U = exp {− i ( H − h x S x ) τ } . Since the Hamiltonian is cyclic, theeigenbases | i (cid:105) and | j τ (cid:105) coincide. However, since the unitaryis now non-trivial, the final state ρ τ = U ρ th0 U † will containcoherences in the S z -basis.A panel summarizing the results for the splitting (10) isshown in Fig. 6, where we plot the first four cumulants of λ cl (images (a)-(d)) and those of λ qu (images (f)-(i)), as a func-tion of the Hilbert space dimension d and for di ff erent valuesof β . In Figs. 6(e) and (j), we also show exemplary plots ofthe full distributions P ( λ cl ) and P ( λ qu ), for fixed d =
200 andtwo values of β . For comparison, a similar panel, but for thequantities in (3), is shown in Fig. 7. Note also that some cu-mulants are scaled by either d or β , whenever a simple scalingrule could be found.From these plots the following conclusions can be drawn.Concerning Fig. 6, all cumulants of λ cl are found to be inten-sive, saturating at a finite value when d → ∞ . Conversely,all cumulants of λ qu are extensive, scaling proportionally to d .The cumulants of λ qu also scale with powers of β at low tem-peratures (Figs. 6(f)-(i)), but for higher order cumulants thisscaling only becomes good at very low temperatures. For thesplitting (3) the situation is reversed: now the cumulants of γ cl become extensive (and quite similar to those of λ qu ), while0 �������� ��� ��� ��� ��� ������������������ � � Λ �� / � β δ � � ( � ) ��� ��� ��� ��� ����������������������������������� � � Λ � � / � β δ � � ( � ) ��� ��� ��� ��� ������������������ � � Γ �� / � β δ � � ( � ) ��� ��� ��� ��� ������������������������ � � Γ � � / � β δ � � ( � ) FIG. 5. Λ qu , Λ cl , Γ qu and Γ cl for the TFIM as a function of g , fordi ff erent values of temperatures (in the low temperature regime) and δ g = .
01. All quantities are scaled by βδ g . those of γ qu tend to saturate. The only exception is κ ( γ qu ),which is found to grow logarithmically with d . Notice thatthis dependence on the dimensionality is fundamentally dif-ferent from what was found in the TFIM (Sec. IV), whereall quantities were extensive in the number of particles. Wealso note that the statistics of λ cl (Fig. 6(e)) has significantlysmaller support than that of λ qu . This is a consequence of thefact that, from its definition in Eq. (39), λ cl depends only onthe initial points | i (cid:105) , while λ qu depends on both | i (cid:105) and | j τ (cid:105) . VI. CONCLUSION
In this article, we studied how entropy production can bedivided into a classical and quantum contribution, when a sys-tem is driven out of equilibrium. A popular choice in the liter-ature is given in Eq. (3), see in particular [38]. This splittinghas several interesting properties, including individual fluc-tuation theorems for each term [38] and a resource-theoretic interpretation [20, 21, 36]. However, we here noted it also hastwo major shortcomings. First, we showed that the classicalcontribution Γ cl in Eq. (3) dominates at low temperatures andfor highly coherent processes, in contrast with what might beexpected. We observed this undesired behaviour in all con-sidered systems, from a simple driven qubit to a many-bodyIsing model at criticality, and identified the divergence of therelative entropy in (4), at low temperatures, as the underlyingcause. Second, given a perturbation δ g of the Hamiltonian,the radius of convergence of Γ qu and Γ cl tends to zero expo-nentially fast as β → ∞ , making this splitting impractical tocharacterise the entropy production of quenched systems atlow temperatures.In order to overcome these shortcomings, we suggested anew splitting for the entropy production given in Eq. (10),which was motivated by the developments of Ref. [30] forinfinitesimal quenches. The definition is valid arbitrarily out-of equilibrium. We also provided a formulation in terms ofstochastic trajectories and a physical interpretation, highlight-ing how it can be obtained following a similar logic to theone behind (3). Indeed, both (3) and (10) can be understoodby introducing intermediate state for comparison. The di ff er-ent choices, however, turn out to have crucial consequences,especially for highly coherent processes. Indeed, in the low-temperature regime the quantum term Λ qu dominates in Eq.(10), but the classical one does in (3). For high temperatures,both splittings coincide. A comparison between the two ap-proaches is summarized in Table I.More generally, our considerations illustrate that it is non-trivial to identify the classical and quantum contributions inentropy production for an arbitrarily out-of-equilibrium pro-cess. In analogy with the definition of work for coherent pro-cesses [26, 89], the splitting of Σ in a classical and quantumterm may not be unique, and will depend on the specific con-text into consideration. Nevertheless, there are some relevantscenarios where such a splitting is unambiguous. One is in athermalization process described by either a Markovian mas-ter equation or as a resource-theoretic state transformation; inboth cases, such a distinction seems to be very well capturedby Eq. (3) [20, 21, 36]. On the other hand, when an equi-librium state is slightly moved out of equilibrium (e.g. by aninfinitesimal quench), the splitting (10) provides a more accu-rate description of the quantum and classical contributions. Infact, in such a scenario, the entropy production can be decom-posed into a classical and quantum contribution at all levelsof the statistics, as shown in Sec. III B (see also [30, 31, 90]).For general out-of-equilibrium processes, however, classicaland quantum contributions become inevitably mixed. Still,our results show that the splitting (10) has a more reasonablebehaviour (i.e., the quantum term dominates at low tempera-tures and for highly coherent processes).In a second part of the article, we applied these ideas to atransverse field Ising model, and to a macrospin undergoingfinite time dynamics. For the Ising model, we found that thebehavior close to criticality is fundamentally di ff erent for bothsplittings, with the quantum component playing a predomi-nant role for (10) and the classical component being dominantin (3). For the macrospin model, we focused not only on the1 �������������������������������� κ � ( λ � � ) ( � ) ������������������������ κ � ( λ � � ) ( � ) � �� �� �� �� - ���� - ���� - ���� - ���� - ���� - ���� - �������� � κ � ( λ � � ) ( � ) � �� �� �� ���������������������������������� � κ � ( λ � � ) ( � ) - � - � � � ������������������� λ �� � ( λ � � ) ( � ) � ���� ��� ���������������������������� κ � ( λ �� ) / β � ( � ) ������������������ κ � ( λ �� ) / β � � ( � ) �� �� �� ���������������������������������� � κ � ( λ �� ) / β � � ( � ) � �� �� �� �� - ��� - ��� - ��� - ��� - ��� - ������ � κ � ( λ �� ) / β � � ( � ) � �� ��� ����������������������������������� λ �� � ( λ �� ) ( � ) FIG. 6. Statistics of the splitting (10) for the macrospin model, as a function of the Hilbert space dimension d and di ff erent values of β , asshown in image (f). (a)-(d) First four cumulants of λ cl . (e) P ( λ cl ) for d = β = β = . λ qu .Some of the cumulants are scaled by powers of β and d , whenever such a scaling law exists. Additional parameters: h z = h x = . τ = �������������������������������� κ � ( γ � � ) / β � ( � ) ��������������� κ � ( γ � � ) / β � � ( � ) �� �� �� �� - ���� - ���� - ���� - ���� - ���������������� � κ � ( γ � � ) / β � � ( � ) �� �� �� �� - ��� - ��� - ��� - ��� - ��� - ��������� � κ � ( γ � � ) / β � � ( � ) � �� ��� ����������������������������������� γ �� � ( γ � � ) ( � ) ��������������������� κ � ( γ �� ) / � � ( � ) ( � ) ������������������������ κ � ( γ �� ) ( � ) �� �� �� �� - ��� - ��� - ��������� � κ � ( γ �� ) ( � ) �� �� �� �������� � κ � ( γ �� ) ( � ) - � � � � ������������������������� γ �� � ( γ �� ) ( � ) FIG. 7. Same as Fig. 6, but for the splitting (3). average, but on the full statistics, including the first four cumu-lants and the corresponding probability distributions. We havefound that di ff erent cumulants scale with the Hilbert space di-mension d in non-trivial ways, some being extensive, othersintensive or even logarithmic.We hope that these results help to motivate further investi-gations on the non-trivial way in which populations and co-herences intermix in quantum thermodynamic processes. Weare particularly interested in further understanding how thisunfolds for many-body systems in general. In particular, theanalysis of higher order cumulants for these models has beenseldom explored in the literature, even for Σ itself. It wouldalso be interesting to generalize the present results for opensystems, undergoing generic interactions with a heat bath.This can be done for quasi-static processes, following the ap-proach in [30]. Or it can be constructed in a controllable wayusing collisional models [91]. Finally, these ideas could alsobe extended to describe quantum correlations in bipartite sys-tems. For instance, instead of studying the entropy productionin a work protocol, one may analyze it in the context of heatexchange between two quantum correlated systems, which arelocally thermal, as studied in [35]. ACKNOWLEDGEMENTS
We thank M. Scandi for insightful discussions. We ac-knowledge financial support from the Brazilian agencies Con-selho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogicoand Coordenac¸ ˜ao de Aperfeic¸oamento de Pessoal de N´ıvelSuperior. GTL acknowledges the financial support of the S˜aoPaulo Funding Agency FAPESP (Grants No. 2017 / / / Appendix A: Nonnegativity of Λ qu In this appendix we show that Λ qu defined in Eq. (19) isalso non-negative, even though it cannot be written as a rela-tive entropy. The proof is essentially based on the Bogoliubovvariational theorem [92]. The first term in Eq. (19) reads ex-plicitly F ( ˜ ρ th τ ) = tr (cid:110) ˜ ρ th τ H τ (cid:111) − T S ( ˜ ρ th τ ). At first sight, this isnot an equilibrium free energy, because the Hamiltonian H τ is not the same as the one appearing in the exponent of ˜ ρ th τ [Eq. (17)]. However, due to the presence of the trace, we canequivalently write this as F ( ˜ ρ th τ ) = tr (cid:110) ˜ ρ th τ D ρ τ ( H τ ) (cid:111) − T S ( ˜ ρ th τ ),which shows that it is actually an equilibrium free energy.Next, we note that the final Hamiltonian can be rewritten as H τ = D ρ τ ( H τ ) + H c τ , where H c τ = H τ − D ρ τ ( H τ ). The Bogoli-2ubov variational theorem [92] then yields F ( ρ th τ ) (cid:54) F ( ˜ ρ th τ ) + tr (cid:110) ˜ ρ th τ H c τ (cid:111) . (A1)But, by construction, H c τ = H τ − D ρ τ ( H τ ) has only o ff -diagonalelements in the common eigenbasis of ˜ ρ th τ and ρ τ . Thus, thesecond term in Eq. (A1) vanishes. Plugging the resulting in-equality back into Eq. (19), we finally conclude that Λ qu (cid:62) Λ qu is zeroif and only if ρ τ is incoherent in the eigenbasis of H τ . The if part of this statement is easy: when ρ τ is incoherent in the finalenergy eigenbasis, D ρ τ ( H τ ) = D H τ ( H τ ) = H τ , which leads to Λ qu ( ρ τ ) =
0. Conversely, if we assume that Λ qu ( ρ τ ) = β >
0, we must have F ( ˜ ρ th τ ) − F ( ρ th τ ) =
0. This implies that + ∞ (cid:88) k = ( − β ) k k ! tr (cid:110) H k τ − D ρ τ ( H τ ) k (cid:111) = , (A2)which means tr (cid:110) H k τ − D ρ τ ( H τ ) k (cid:111) = , ∀ k ∈ N . The case k = k = D ρ τ ( H τ ). For the case k =
2, we use thattr (cid:110) H τ (cid:111) = tr (cid:26)(cid:16) D ρ τ ( H τ ) + H c τ (cid:17) (cid:27) = tr (cid:110) D ρ τ ( H τ ) + D ρ τ ( H τ ) H c τ + ( H c τ ) (cid:111) . Again, using the definition of D ρ τ ( H τ ) one may verify thattr (cid:110) D ρ τ ( H τ ) H c τ (cid:111) =
0. Therefore we are left withtr (cid:110) H τ − D ρ τ ( H τ ) (cid:111) = tr (cid:26)(cid:16) H τ − D ρ τ ( H τ ) (cid:17) (cid:27) = . (A3)But since H τ − D ρ τ ( H τ ) is also Hermitian, we must have H τ − D ρ τ ( H τ ) =
0. Then, since D ρ τ ( H τ ) = H τ , for k > (cid:110) H k τ − D k ρ τ ( H τ ) (cid:111) = ρ τ must be inco-herent in the eigenbasis of H τ , i.e., we must have [ ρ τ , H τ ] = Appendix B: Diagonalization of the transverse field Ising model
The TFIM Hamiltonian in Eq. (74) can be diagonalized bya series of transformations, as shown in [93]. Our notation fol-lows closely that of Ref. [39], which contains a self-containedderivation of these results. The first step is the introduction ofa Jordan-Wigner transformation, that maps the spin chain ontoan equivalent system of spinless fermions, σ xj = ( c † j + c j ) (cid:89) i < j (1 − c † i c i ) , σ zj = − c † j c j , (B1)where c † j and c j are canonical creation and annihilationfermionic operators. We assume N is large and even. We maythen ignore boundary terms [94], and introduce the Fouriertransform c j = e − ıπ/ √ N (cid:88) k c k e ı k j , (B2) where k = ± (2 n + π N and n = , , ..., N / −
1. Eq. (74) isthen transformed to H ( g ) = (cid:88) k > (cid:104) ( g − cos k )( c † k c k − c − k c †− k ) + sin k ( c † k c †− k + c − k c k ) (cid:105) . (B3)Next, we introduce a new set of Fermionic operators η k through the Bogoliubov transformation η k = cos( θ k / c k + sin( θ k / c †− k . (B4)With the definitions (cid:15) k ( g ) = (cid:113) ( g − cos k ) + sin k , (sin θ k , cos θ k ) = (cid:18) sin( k ) (cid:15) k , g − cos( k ) (cid:15) k (cid:19) , (B5)we then finally obtain H ( g ) = (cid:88) k (cid:15) k ( g ) (cid:0) η † k η k − (cid:1) , (B6)Exploring the fact that (cid:15) − k ( g ) = (cid:15) k ( g ), we can rewrite H ( g ) asa sum over only positive values of k , H ( g ) = (cid:88) k > (cid:15) k ( g )( η † k η k + η †− k η − k − . (B7)This is useful because, as we will see, a perturbation δ g cou-ples pairs of modes + k and − k . Finally, if we let | n − k n k (cid:105) bethe joint eigenstates of η †− k η − k and η † k η k , where n ± k = ,
1, wemay also write H ( g ) = (cid:88) k > (cid:15) k ( g ) (cid:0) − | − k k (cid:105)(cid:104) − k k | + | − k k (cid:105)(cid:104) − k k | (cid:1) . (B8)If we consider now a perturbation δ g in the field, we have ∆ H = − δ g N (cid:88) j = σ zj = δ g N (cid:88) j = (2 c † j c j − = δ g (cid:88) k (2 c † k c k − , (B9)where we used Eqs. (B1) and (B2). Finally, using Eqs. (B4)and (B5) we obtain ∆ H = δ g (cid:88) k > (cid:2) cos θ k ( η † k η k + η †− k η − k − + sin θ k ( η †− k η † k − η − k η k ) (cid:3) , (B10)where the coupling between + k and − k modes is clear fromthe second term. Alternatively, this can be written as ∆ H =∆ H d + ∆ H c , with ∆ H d = δ g (cid:88) k > cos θ k (cid:0) − | − k k (cid:105)(cid:104) − k k | + | − k k (cid:105)(cid:104) − k k | (cid:1) , (B11) ∆ H c = δ g (cid:88) k > sin θ k (cid:16) | − k k (cid:105)(cid:104) − k k | + | − k k (cid:105)(cid:104) − k k | (cid:17) , (B12)where ∆ H d and ∆ H c are the dephased and coherent parts ofthe perturbation, respectively.3 Appendix C: Λ cl and Λ qu for the TFIM Using the results from (B), we now show how to compute Λ cl and Λ qu using Eqs. (18) and (19). Since we consider theinitial field to be g , the initial Hamiltonian is given by H = (cid:88) k (cid:15) k (2 η † k η k − , = (cid:88) k > (cid:15) k (cid:0) − | − k k (cid:105)(cid:104) − k k | + | − k k (cid:105)(cid:104) − k k | (cid:1) (C1)where (cid:15) k = (cid:15) k ( g ) and | n − k n k (cid:105) are the joint eigenstates of η † k η k and η †− k η − k . Thus, the initial state ρ th0 = e − β H / Z can be writ-ten as ρ th0 = (cid:89) k > ρ th0 |± k , (C2a) ρ th0 |± k = (cid:88) n k = , , n − k = , e β(cid:15) k (1 − n k − n − k ) ( β(cid:15) k ) | n − k n k (cid:105)(cid:104) n − k n k | . (C2b)After the instantaneous quench, which changes the field toits final value g τ = g + δ g , we have the final Hamiltonian, H τ = (cid:88) k (cid:15) τ k (2 ξ † k ξ k − , = (cid:88) k > (cid:15) τ k (cid:0) − | τ − k τ k (cid:105)(cid:104) τ − k τ k | + | τ − k τ k (cid:105)(cid:104) τ − k τ k | (cid:1) (C3)where (cid:15) τ k = (cid:15) k ( g τ ) and | n τ − k n τ k (cid:105) are the joint eigenstates of thepost-quench fermionic operators ξ † k ξ k and ξ †− k ξ − k , which arerelated to the pre-quench operators { η k } according to [72] ξ k = cos( ∆ k / η k + sin( ∆ k / η †− k , (C4)where sin ∆ k = − δ g sin( k ) /(cid:15) τ k (cid:15) k . As discussed in Appendix B,Eq. (C4) shows us that the perturbation couples pairs of modes + k and − k . This is why it is more convenient to write allquantities as ρ th0 = (cid:81) k > ρ th0 |± k and H = (cid:80) k > H |± k , instead ofa product / sum over the negative values of k .The corresponding final equilibrium state ρ th τ = e − β H τ / Z τ isgiven by ρ th τ = (cid:89) k > ρ th τ |± k , (C5a) ρ th τ |± k = (cid:88) n τ k = , , n τ − k = , e β(cid:15) τ k (1 − n τ k − n τ − k ) ( β(cid:15) τ k ) | n τ − k n τ k (cid:105)(cid:104) n τ − k n τ k | . (C5b) We can proceed now to calculate ˜ ρ th τ in Eq. (17). We firstcompute the dephased Hamiltonian H + ∆ H d , where ∆ H d isgiven in Eq. (B11). That is, D ρ th0 ( H τ ) = (cid:88) k > (cid:15) τ k (cid:0) − | − k k (cid:105)(cid:104) − k k | + | − k k (cid:105)(cid:104) − k k | (cid:1) , (C6)where ˜ (cid:15) τ k = (cid:15) τ k cos ∆ k = (cid:15) k + δ g cos θ k . From this, one thenfinds the associated thermal state [Eq. (17)]˜ ρ th τ = (cid:89) k > ˜ ρ th τ |± k , (C7a)˜ ρ th τ |± k = (cid:88) n k = , , n − k = , e β ˜ (cid:15) τ k (1 − n k − n − k ) ( β ˜ (cid:15) τ k ) | n − k n k (cid:105)(cid:104) n − k n k | . (C7b)We have all we need to compute Λ cl and Λ qu now. We justhave to plug Eqs. (C2), (C5) and (C7) into (18) and (19).Because all states ρ th0 , ˜ ρ th τ and ρ th τ are separable in terms of ± k modes, Λ cl and Λ qu will be given as sums over k . Hence, wefind Λ cl = (cid:88) k > (cid:40) ln (cid:34) cosh (cid:16) β ˜ (cid:15) τ k (cid:17) cosh (cid:16) β(cid:15) k (cid:17) (cid:35) + β (cid:16) (cid:15) k − ˜ (cid:15) τ k (cid:17) tanh (cid:16) β(cid:15) k (cid:17)(cid:41) , (C8)and Λ qu = (cid:88) k > (cid:34) cosh (cid:16) β(cid:15) τ k (cid:17) cosh (cid:16) β ˜ (cid:15) τ k (cid:17) (cid:35) . (C9)Finally, in the limit of very large N , all k -sums can be con-verted to integrals and all quantities become extensive in N .In particular, we can substitute (cid:80) k > → N (cid:82) π k π in Eqs. (C8)and (C9) to obtain Eqs. (77) and (78).We note that Eqs. (77) and (78) do not assume that thequench is infinitesimal. All they assume is that U =
1. If,in particular, we are interested in infinitesimal quenches, thenwe may series expand these expressions in powers of δ g , lead-ing to Λ cl = N β δ g (cid:90) π d k π sech ( β(cid:15) k ) cos θ k , (C10) Λ qu = N β δ g (cid:90) π d k π tanh (cid:16) β(cid:15) k (cid:17) β(cid:15) k sin θ k , (C11)where it is clear the relation of Λ cl and Λ qu with the dephasedand coherent parts of the perturbation in Eqs. (B11) and (B12).Furthermore, it is easy to check that they satisfy Eq. (33).4 Appendix D: Γ cl and Γ qu for the TFIM For completeness, in this appendix we write down the expressions for Γ cl and Γ qu for the TFIM, computed in [39]: Γ qu = N (cid:90) π d k π (cid:40)
12 tanh (cid:0) β(cid:15) k (cid:1)(cid:34) ln (cid:20) + tanh (cid:0) β(cid:15) k (cid:1) − tanh (cid:0) β(cid:15) k (cid:1) (cid:21) − cos( ∆ k ) ln (cid:20) + tanh (cid:0) β(cid:15) k (cid:1) cos( ∆ k )1 − tanh (cid:0) β(cid:15) k (cid:1) cos( ∆ k ) (cid:21)(cid:35) − cosh (cid:0) β(cid:15) k (cid:1) (cid:0) β(cid:15) k (cid:1) × ln (cid:104) + sinh (cid:0) β(cid:15) k (cid:1) sin ( ∆ k ) (cid:105)(cid:41) , (D1) Γ cl = N (cid:90) π d k π (cid:40) (cid:20) cosh (cid:0) β(cid:15) τ k (cid:1) cosh (cid:0) β(cid:15) k (cid:1) (cid:21) −
12 tanh (cid:0) β(cid:15) k (cid:1) cos( ∆ k ) (cid:34) ln (cid:20) + tanh (cid:0) β(cid:15) τ k (cid:1) − tanh (cid:0) β(cid:15) τ k (cid:1) (cid:21) − ln (cid:20) + tanh (cid:0) β(cid:15) k (cid:1) cos( ∆ k )1 − tanh (cid:0) β(cid:15) k (cid:1) cos( ∆ k ) (cid:21)(cid:35) + cosh (cid:0) β(cid:15) k (cid:1) (cid:0) β(cid:15) k (cid:1) ln (cid:104) + sinh (cid:0) β(cid:15) k (cid:1) sin ( ∆ k ) (cid:105)(cid:41) . (D2)These expressions were used in plotting Figs. 5(c) and (d). The problem in the analyticities of these quantities stem from thelast term in both integrals: In order to series expand them we need to satisfy the condition sinh (2 β(cid:15) k ) sin ( ∆ k ) <
1. This isprohibitive at low temperatures, since this function scales exponentially with β , through sinh (2 β(cid:15) k ), but only polynomially withthe perturbation, through sin ( ∆ k ). [1] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk,1505.07835 , 31 (2015), arXiv:1505.07835.[2] S. Vinjanampathy and J. Anders, Contemporary Physics ,545 (2016).[3] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Eu-rophysics Letters (EPL) , 565 (2004).[4] M. O. Scully, M. S. Zubairy, G. S. Agarwal, and H. Walther,Science , 862 (2007).[5] K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings,New Journal of Physics , 023045 (2016).[6] G. Manzano, F. Plastina, and R. Zambrini, Physical ReviewLetters , 120602 (2018), arXiv:1805.08184.[7] N. L¨orch, C. Bruder, N. Brunner, and P. P. Hofer, QuantumScience and Technology , 035014 (2018).[8] F. L. S. Rodrigues, G. De Chiara, M. Paternostro, andG. T. Landi, Physical Review Letters , 140601 (2019),arXiv:1906.08203.[9] G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison,J. Goold, and F. Plastina, Physical Review Letters , 180603(2020), arXiv:2006.05424.[10] K. V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, andA. Ac´ın, Physical Review Letters (2013), 10.1103 / phys-revlett.111.240401.[11] F. Campaioli, F. A. Pollock, F. C. Binder, L. C´eleri, J. Goold,S. Vinjanampathy, and K. Modi, Physical Review Letters ,150601 (2017), arXiv:1612.04991.[12] F. Campaioli, F. A. Pollock, and S. Vinjanampathy, in Ther-modynamics in the Quantum Regime (Springer, 2018) pp. 207–225.[13] S. Juli`a-Farr´e, T. Salamon, A. Riera, M. N. Bera, andM. Lewenstein, Phys. Rev. Research , 023113 (2020).[14] L. A. Correa, J. P. Palao, D. Alonso, and G. Adesso, Scientificreports , 3949 (2014), arXiv:arXiv:1308.4174v1.[15] J. Rossnagel, O. Abah, F. Schmidt-Kaler, K. Singer, andE. Lutz, Physical Review Letters , 030602 (2014), arXiv:1308.5935.[16] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, andS. Wehner, Reviews of Modern Physics , 419 (2014),arXiv:1303.2849.[17] R. Uzdin, A. Levy, and R. Koslo ff , Physical Review X ,031044 (2015).[18] G. Manzano, F. Galve, R. Zambrini, and J. M. R. Parrondo,Physical Review E , 052120 (2016).[19] K. Hammam, Y. Hassouni, R. Fazio, and G. Manzano, arXivpreprint arXiv:2101.11572 (2021).[20] D. Janzing, Journal of statistical physics , 761 (2006).[21] M. Lostaglio, D. Jennings, and T. Rudolph, Nature communi-cations , 6383 (2015), arXiv:1405.2188.[22] P. Cwiklinski, M. Studzinski, M. Horodecki, and J. Op-penheim, Physical Review Letters , 210403 (2015),arXiv:1405.5029.[23] A. E. Allahverdyan, Physical Review E (2014),10.1103 / physreve.90.032137.[24] P. Talkner and P. H¨anggi, Physical Review E , 022131 (2016),arXiv:1512.02516.[25] P. P. Hofer, J. B. Brask, M. Perarnau-Llobet, and N. Brunner, ,23 (2017), arXiv:1703.03719.[26] E. B¨aumer, M. Lostaglio, M. Perarnau-Llobet, and R. Sam-paio, in Fundamental Theories of Physics (Springer Interna-tional Publishing, 2018) pp. 275–300.[27] A. Levy and M. Lostaglio, PRX Quantum , 010309 (2020).[28] K. Micadei, G. T. Landi, and E. Lutz, Phys. Rev. Lett. ,090602 (2020).[29] H. J. D. Miller, M. Scandi, J. Anders, and M. Perarnau-Llobet, Physical Review Letters , 230603 (2019),arXiv:1905.07328.[30] M. Scandi, H. J. D. Miller, J. Anders, and M. Perarnau-Llobet, Physical Review Research , 023377 (2020),arXiv:1911.04306.[31] H. J. Miller, M. H. Mohammady, M. Perarnau-Llobet, and G. Guarnieri, arXiv preprint arXiv:2011.11589 (2020).[32] S. Lloyd, Physical Review A , 5378 (1989).[33] D. Jennings and T. Rudolph, Physical Review E - Statisti-cal, Nonlinear, and Soft Matter Physics , 061130 (2010),arXiv:1002.0314.[34] S. Jevtic, T. Rudolph, D. Jennings, Y. Hirono, S. Nakayama,and M. Murao, Physical Review E , 042113 (2015),arXiv:1204.3571.[35] K. Micadei, J. P. S. Peterson, A. M. Souza, R. S. Sarthour, I. S.Oliveira, G. T. Landi, T. B. Batalh˜ao, R. M. Serra, and E. Lutz,Nature Communications , 2456 (2019), arXiv:1711.03323.[36] J. P. Santos, L. C. C´eleri, G. T. Landi, and M. Paternostro, npjQuantum Information , 23 (2019), arXiv:1707.08946.[37] M. H. Mohammady, A. Au ff `eves, and J. Anders, Communica-tions Physics , 89 (2020), arXiv:1907.06559.[38] G. Francica, J. Goold, and F. Plastina, Physical Review E ,042105 (2019), arXiv:1707.06950.[39] A. D. Varizi, A. P. Vieira, C. Cormick, R. C. Drumond, andG. T. Landi, Physical Review Research , 033279 (2020),arXiv:2004.00616.[40] M. Esposito, U. Harbola, and S. Mukamel, Reviews of ModernPhysics , 1665 (2009).[41] M. Campisi, P. H¨anggi, and P. Talkner, Reviews of ModernPhysics , 771 (2011).[42] R. Kawai, J. M. Parrondo, and C. Van Den Broeck, PhysicalReview Letters , 080602 (2007), arXiv:0701397 [cond-mat].[43] S. Vaikuntanathan and C. Jarzynski, EPL (Europhysics Letters) , 60005 (2009).[44] J. M. Parrondo, C. Van Den Broeck, and R. Kawai, New Jour-nal of Physics , 073008 (2009), arXiv:0904.1573.[45] S. De ff ner and E. Lutz, Physical Review Letters , 170402(2010), arXiv:1005.4495.[46] C. Jarzynski, Physical Review Letters , 2690 (1997).[47] J. Kurchan, Journal of Physics A: Mathematical and General , 3719 (1998).[48] P. Talkner, E. Lutz, and P. H¨anggi, Physical Review E ,050102 (2007).[49] B. Derrida and J. L. Lebowitz, Physical Review Letters , 209(1998), arXiv:9809044 [cond-mat].[50] G. E. Crooks, Journal of Statistical Physics , 1481 (1998).[51] J. Lebowitz and H. Spohn, Journal of Statistical Physics , 333(1999).[52] S. Mukamel, Physical Review Letters , 170604 (2003),arXiv:0302190 [cond-mat].[53] G. Guarnieri, N. H. Y. Ng, K. Modi, J. Eisert, M. Paternos-tro, and J. Goold, Physical Review E , 050101 (2018),arXiv:1804.09962.[54] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Busta-mante, Science (New York, N.Y.) , 1832 (2002).[55] F. Douarche, S. Ciliberto, a. Petrosyan, and I. Rabbiosi, Euro-physics Letters (EPL) , 593 (2005).[56] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, andC. Bustamante, Nature , 231 (2005).[57] T. Speck, V. Blickle, C. Bechinger, and U. Seifert, EurophysicsLetters (EPL) , 30002 (2007).[58] O. P. Saira, Y. Yoon, T. Tanttu, M. M¨ott¨onen, D. V. Averin,and J. P. Pekola, Physical Review Letters , 180601 (2012),arXiv:1206.7049.[59] J. V. Koski, T. Sagawa, O. P. Saira, Y. Yoon, A. Kutvonen,P. Solinas, M. M¨ott¨onen, T. Ala-Nissila, and J. P. Pekola, Na-ture Physics , 644 (2013), arXiv:1303.6405.[60] T. B. Batalh˜ao, A. M. Souza, L. Mazzola, R. Auccaise, R. S.Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro,and R. M. Serra, Physical Review Letters , 140601 (2014). [61] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q.Yin, H. T. Quan, and K. Kim, Nature Physics , 193 (2014),arXiv:1409.4485.[62] T. B. Batalh˜ao, A. M. Souza, R. S. Sarthour, I. S. Oliveira,M. Paternostro, E. Lutz, and R. M. Serra, Physical Review Let-ters , 190601 (2015), arXiv:1502.06704v1.[63] M. A. Talarico, P. B. Monteiro, E. C. Mattei, E. I. Duzzioni,P. H. Souto Ribeiro, and L. C. C´eleri, Physical Review A ,042305 (2016), arXiv:1604.07237.[64] Z. Zhang, T. Wang, L. Xiang, Z. Jia, P. Duan, W. Cai, Z. Zhan,Z. Zong, J. Wu, L. Sun, Y. Yin, and G. Guo, New Journal ofPhysics , 085001 (2018), arXiv:1805.10879.[65] A. Smith, Y. Lu, S. An, X. Zhang, J.-N. Zhang, Z. Gong, H. T.Quan, C. Jarzynski, and K. Kim, New Journal of Physics ,013008 (2018), arXiv:1708.01495.[66] F. G. S. L. Brand˜ao, M. Horodecki, J. Oppenheim, J. M. Renes,and R. W. Spekkens, Physical Review Letters , 250404(2013), arXiv:1111.3882.[67] M. Horodecki and J. Oppenheim, Nature communications ,2059 (2013), arXiv:arXiv:1111.3834v1.[68] T. Baumgratz, M. Cramer, and M. B. Plenio, Physical ReviewLetters , 140401 (2014), arXiv:1311.0275.[69] A. Streltsov, G. Adesso, and M. B. Plenio, Reviews of ModernPhysics , 041003 (2017), arXiv:1609.02439.[70] M. A. Nielsen and I. L. Chuang, Quantum Computation andQuantum Information (Cambridge University Press, 2000).[71] A. Gambassi and A. Silva, “Statistics of the work in quantumquenches, universality and the critical casimir e ff ect,” (2011),arXiv:1106.2671 [cond-mat.stat-mech].[72] R. Dorner, J. Goold, C. Cormick, M. Paternostro, and V. Ve-dral, Physical Review Letters , 160601 (2012).[73] L. Fusco, S. Pigeon, T. J. G. Apollaro, A. Xuereb, L. Mazzola,M. Campisi, A. Ferraro, M. Paternostro, and G. De Chiara,Physical Review X , 031029 (2014).[74] J. Goold, F. Plastina, A. Gambassi, and A. Silva, in Thermo-dynamics in the quantum regime - Recent Progress and Out-look , edited by F. Binder, L. A. Correa, G. C, J. Anders, andG. Adesso (Springer, 2018) pp. 317–336, arXiv:1804.02805.[75] S. Sharma and A. Dutta, Phys. Rev. E , 022108 (2015).[76] E. Vicari, Phys. Rev. A , 043603 (2019).[77] E. Mascarenhas, H. Braganc¸a, R. Dorner, M. Fran c¸a Santos,V. Vedral, K. Modi, and J. Goold, Phys. Rev. E , 062103(2014).[78] F. Cosco, M. Borrelli, P. Silvi, S. Maniscalco, andG. De Chiara, Phys. Rev. A , 063615 (2017).[79] S. Paganelli and T. J. G. Apollaro, InternationalJournal of Modern Physics B , 1750065 (2017),https: // doi.org / / S0217979217500655.[80] Q. Wang, D. Cao, and H. T. Quan, Phys. Rev. E , 022107(2018).[81] A. Bayat, T. J. G. Apollaro, S. Paganelli, G. De Chiara, H. Jo-hannesson, S. Bose, and P. Sodano, Phys. Rev. B , 201106(2016).[82] F. A. Bayocboc and P. N. C. Paraan, Physical Review E ,032142 (2015).[83] A. Pelissetto, D. Rossini, and E. Vicari, Phys. Rev. E ,052148 (2018).[84] D. Nigro, D. Rossini, and E. Vicari, Journal of Statistical Me-chanics: Theory and Experiment , 023104 (2019).[85] This is done by noting that the dephasing D H ( ρ ) can be alsogiven by D H ( ρ ) = lim s →∞ s (cid:90) s d te − iHt ρ e iHt . We then use that e t ( H +∆ H ) = e tH + t J e tH [ ∆ H ] + O ( ∆ H ) and[ ρ th0 , H ] =
0. To order ∆ H this gives Eq. (26).[86] D. Petz, Journal of Physics A: Mathematical and General ,929 (2002), arXiv:0106125 [quant-ph].[87] B. B. Wei and M. B. Plenio, New Journal of Physics , 023002(2017), arXiv:1509.07043.[88] J. J. Sakurai and J. J. Napolitano, Modern Quantum Mechanics ,2nd ed. (Addison-Wesley, 2010) p. 550.[89] W. Niedenzu, M. Huber, and E. Boukobza, , 1 (2019),arXiv:1907.01353.[90] H. J. D. Miller, G. Guarnieri, M. T. Mitchison, and J. Goold, Phys. Rev. Lett. , 160602 (2020).[91] G. De Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro,A. J. Roncaglia, and M. Antezza, New Journal of Physics ,113024 (2018), arXiv:1808.10450.[92] H. B. Callen, Thermodynamics and an introduction to Thermo-statistics , 2nd ed. (Wiley, 1985) p. 493.[93] E. H. Lieb, T. Schultz, and D. Mattis, Annals of Physics ,407 (1961).[94] B. Damski and M. M. Rams, Journal of Physics A: Mathemati-cal and Theoretical47