Thermodynamics of Quantum Information Flows
TThermodynamics of Quantum Information Flows
Krzysztof Ptaszy´nski and Massimiliano Esposito Institute of Molecular Physics, Polish Academy of Sciences,ul. M. Smoluchowskiego 17, 60-179 Pozna´n, Poland ∗ Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit,University of Luxembourg, L-1511 Luxembourg, Luxembourg (Dated: March 27, 2019)We report two results complementing the second law of thermodynamics for Markovian openquantum systems coupled to multiple reservoirs with different temperatures and chemical poten-tials. First, we derive a nonequilibrium free energy inequality providing an upper bound for amaximum power output, which for systems with inhomogeneous temperature is not equivalent tothe Clausius inequality. Secondly, we derive local Clausius and free energy inequalities for subsys-tems of a composite system. These inequalities differ from the total system one by the presence of aninformation-related contribution and build the ground for thermodynamics of quantum informationprocessing. Our theory is used to study an autonomous Maxwell demon.
The second law of thermodynamics is one of the mainprinciples of physics. Within equilibrium thermodynam-ics there exist two equivalent formulations of this law.The first, referred to as the
Clausius inequality , statesthat the sum of the entropy change of the system ∆ S and the entropy exchanged with the environment ∆ S env during the transition between two equilibrium states isnonnegative: ∆ S + ∆ S env ≥
0. The exchanged entropycan be further expressed as ∆ S env = − Q/T , where Q is the heat delivered to the system. An alternative for-mulation, referred to as the free energy inequality , statesthat during the transition between two equilibrium states W − ∆ F ≥
0, where W is the work performed on thesystem and F = E − T S is the free energy (here E de-notes the internal energy). The latter formulation canbe obtained from the former by using the first law ofthermodynamics ∆ E = W + Q .Whereas these standard definitions of the second lawapply when considering transitions between equilibriumstates, the last few decades have brought significantprogress towards generalizing them to both classical [1–7] and quantum [8–10] systems far from equilibrium.The most common formulation generalizes the Clau-sius inequality by stating that the average entropy pro-duction σ is nonnegative. For a large class of sys-tems [4–6, 9] the entropy production can be defined as σ ≡ ∆ S − P α Q α β α , where ∆ S is the change of theShannon or the von Neumann entropy of the system(which is well defined also out of equilibrium) and Q α isthe heat delivered to the system from the reservoir α withthe inverse temperature β α ; additionally, in Markoviansystems the entropy production rate ˙ σ is always nonneg-ative [7, 10]. Formulations generalizing the free energyinequality [11–14] are much less common and have beenso far confined mainly to systems coupled to an environ-ment with a homogeneous temperature; for an exception,see Ref. [14].These developments have also brought a deeper under-standing of the relation between thermodynamics and the information theory [12, 15]. One of the most importantachievements is related to the field of thermodynamics offeedback-controlled systems [16]. Following the ground-breaking ideas of Maxwell demon [17] and Szilard en-gine [18], it was verified both theoretically [19–23] and ex-perimentally [24–29] that by employing feedback one canreduce entropy of the system without exchanging heat.In such a case modified Clausius inequalities, which relatethe entropy change to the information flow, have to beapplied [11–14, 30–42]. It was also realized that the feed-back control does not require the presence of any intelli-gent being (as in the original idea of Maxwell), but maybe performed by an autonomous stochastic system cou-pled to the controlled one [43]. A consistent mathemati-cal description of thermodynamics of autonomous infor-mation flow has been, however, so far confined mainlyto classical stochastic systems with a special topology ofnetwork of jump processes, referred to as bipartite [38, 39]or, in general, multipartite [41] systems.Our work adds two new contributions to the field.First, we generalize the free energy inequality to Marko-vian open quantum systems coupled to reservoirs withdifferent temperatures and show that this formulation ofthe second law is in general not equivalent to the Clausiusinequality. Secondly, we formulate a consistent thermo-dynamic formalism describing thermodynamics of contin-uous information flow in a generic composite open quan-tum system and demonstrate the relation between theinformation and the nonequilibrium free energy. The ap-plicability of our results is demonstrated on a quantumautonomous Maxwell demon based on quantum dots. Nonequilibrium Clausius inequality . We consider ageneric open quantum system weakly coupled to N equi-librated reservoirs α with temperatures T α (inverse tem-peratures β α ≡ /T α ) and chemical potentials µ α , de-scribed by the time-independent Hamiltonianˆ H = ˆ H S + ˆ H B + ˆ H I , (1)where ˆ H S , ˆ H B , ˆ H I are, correspondingly, the Hamiltonian a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r of the system, reservoirs and interaction of the systemwith the reservoirs. Within the Markov approximationthe (reduced) density matrix of the system evolves ac-cording to the master equation [44] d t ρ = − i [ ˆ H eff , ρ ] + D ρ, (2)where ρ is the density matrix, d t denotes the total deriva-tive of the function, ˆ H eff is the effective Hamiltonian ofthe system (it may differ from ˆ H S due to coupling to theenvironment [44]), and D is the superoperator describingthe dissipative dynamics. Here and from here on we take ~ = k B = 1. We further assume that the dissipator D is of Lindblad form, thus ensuring a completely-positivetrace-preserving dynamics [45, 46], and that ˆ H eff com-mutes with ˆ H S , which is justified by the perturbationtheory (cf., the Supplementary Material [47]). Further-more, within the Markov approximation the dissipation isadditive, i.e., the superoperator D can be represented as asum of dissipators associated with each reservoir, denotedas D α : D = P α D α [10, 48]. For violation of additivitybeyond the weak coupling regime, see Refs. [49–52].We also assume, that the grand canonical equilibriumstate (Gibbs state) with respect to the reservoir αρ α eq = Z − β α ,µ α e − β α ( ˆ H S − µ α ˆ N ) , (3)is a stationary state of D α , i.e., D α ρ α eq = 0 [10]; here Z β α ,µ α = Tr { exp[ − β α ( ˆ H S − µ α ˆ N )] } is the partition func-tion and ˆ N is the particle number operator. This as-sumption guarantees that for an arbitrary form of dissi-pator the Gibbs state is a stationary state at equilibrium(i.e., for equal temperatures and chemical potentials ofthe reservoirs), which is true for systems weakly coupledto the environment [44]. Let us then apply the Spohn’sinequality [8] − Tr (cid:2) ( D α ρ ) (cid:0) ln ρ − ln ρ α eq (cid:1)(cid:3) ≥ , (4)which is valid for any superoperator D α of Lindblad formwith a steady state ρ α eq (not necessarily a unique steadystate). As a result, one obtains the partial Clausius in-equality for entropy production associated with each dis-sipator [10] ˙ σ α = ˙ S α − β α ˙ Q α ≥ , (5)where ˙ S α = − Tr [( D α ρ ) ln ρ ] , (6)is the rate of change of the von Neumann entropy of thesystem S = − Tr( ρ ln ρ ) due to the dissipator D α and˙ Q α = Tr h ( D α ρ ) (cid:16) ˆ H S − µ α ˆ N (cid:17)i , (7)is the heat current from the reservoir α . Summing all therates ˙ S α one gets the total derivative of the von Neumann entropy: d t S = P α ˙ S α . Therefore, summing up Eq. (5)over the reservoirs α one recovers the standard Clausiusinequality [10]˙ σ ≡ X α ˙ σ α = d t S − X α β α ˙ Q α ≥ , (8)where ˙ σ is the total entropy production rate. We notethat the rates ˙ S α can be non-zero also at the steady state,when d t S = 0 and the total entropy production is fullydetermined by the heat flows. Nonequilibrium free energy inequality.
Let us now de-fine energy and work currents to the lead α as˙ E α = Tr h ( D α ρ ) ˆ H S i , (9)˙ W α = µ α Tr h ( D α ρ ) ˆ N i , (10)such that ˙ E α = ˙ Q α + ˙ W α and P α ˙ E α = d t E , where E = Tr( ρ ˆ H S ) is the internal energy. Since we assume theHamiltonian to be time independent, we consider onlychemical and not mechanical work. Multiplying Eq. (5)by T α and replacing ˙ Q α → ˙ E α − ˙ W α one gets T α ˙ σ α = ˙ W α − ˙ F α ≥ , (11)where ˙ F α ≡ ˙ E α − T α ˙ S α is the partial nonequilibrium freeenergy rate . Summing over α one obtains the nonequilib-rium free energy inequality X α T α ˙ σ α = ˙ W − ˙ F ≥ , (12)where ˙ W ≡ P α ˙ W α is the total work rate and˙ F ≡ X α ˙ F α = d t E − X α T α ˙ S α , (13)is the total nonequilibrium free energy rate . Equa-tion (12) is a complementary formulation of the secondlaw of thermodynamics. From a practical point of view,it provides an upper bound for the maximum work out-put. At the steady state d t E = 0 and thus the systemcan perform work ( ˙ W <
0) only when a temperaturedifference between the reservoirs is present.Let us emphasize, that Eqs. (8) and (12) are in gen-eral not equivalent; the former corresponds to the sumof partial Clausius inequalities [Eq. (5)], whereas thelatter to the weighted sum, in which Eq. (5) is multi-plied by a local temperature T α . They become equiv-alent only when the system is attached to an isother-mal environment, i.e., T α = T . Then the rate ˙ F can beidentified as the total derivative of the state function F :˙ F = d t F = d t ( E − T S ). At the steady state d t F = 0and thus ˙ W >
0. This corresponds to the
Kelvin-Planckstatement of the second law , according to which one can-not continuously generate work by cooling an isothermalenvironment.
Local Clausius inequality . Let us now consider a sys-tem made of two coupled subsystems described by theHamiltonian ˆ H S = ˆ H + ˆ H + ˆ H , (14)where ˆ H i is the Hamiltonian of the subsystem i = 1 , H is the interaction Hamiltonian. We also assumethat each subsystem is attached to a separate set of reser-voirs; baths coupled with the subsystem i will be thendenoted as α i . By summing Eq. (5) over the reservoirs α i one obtains˙ σ i ≡ X α i ˙ σ α i = X α i ˙ S α i − X α i β α i ˙ Q α i ≥ . (15)Here ˙ σ i = P α i ˙ σ α i denotes the local entropy production;it is an extensive quantity, i.e., ˙ σ = ˙ σ + ˙ σ . We willnow transform Eq. (15) to a form illustrating the relationbetween entropy and information. Let us remind that thequantum mutual information is defined as I = S + S − S where S i = − Tr( ρ i ln ρ i ) is the von Neumannentropy of the subsystem i [53] (here ρ i is the reduceddensity matrix of the subsystem i ). We can then separatethe total derivative of the mutual information into twocontributions: d t I = ˙ I + ˙ I where˙ I i ≡ d t S i − X α i ˙ S α i . (16)Here we have applied the identity P α ˙ S α = d t S . Therate ˙ I i can be calculated as˙ I i = − Tr ( d t ρ i ln ρ i ) + Tr [( D i ρ ) ln ρ ] , (17)where D i = P α i D α i is the dissipator associated with thesubsystem i . The rate ˙ I i can be further decomposed intocontributions related to the unitary and the dissipativedynamics; see the Supplementary Material [47] for de-tails.Replacing P α i ˙ S α i → d t S i − ˙ I i in Eq. (15) one obtainsthe local Clausius inequlity relating the entropy balanceof the subsystem i to the information flow:˙ σ i = d t S i − X α i β α i ˙ Q α i − ˙ I i ≥ . (18)This inequality is identical in form to the one previouslyderived in Ref. [39]. However, our result is much moregeneral. First, it enables to describe systems undergoinga quantum dynamics formulated in terms of a densitymatrix, whereas the former approach was purely classi-cal and formulated in terms of probabilities. Second, ourresult has a much wider range of applicability even in theclassical limit. Indeed, the approach from Ref. [39] wasrestricted to so-called “bipartite” systems, which excludestochastic transitions generating a simultaneous changeof states of both subsystems. However, two-component open quantum systems are in general not bipartite evenwhen their populations obey a classical master equation.Instead, our only requirement is that the dissipation isadditive, i.e., one can split the dissipator D into contribu-tions D i in a physically meaningful way. The system canbecome bipartite when the total Hamiltonian ˆ H S com-mutes with the subsystem Hamiltonian ˆ H i and one ap-plies the effectively classical description by means of thesecular (rotating wave) approximation; in such a case ourapproach reduces to that from Ref. [39]. We discuss theseissues is detail in the Supplementary Material [47].Let us finally emphasize, that all the previous discus-sion can be easily generalized to the multicomponentsystems consisting of M subsystems. Then P i ˙ I i = d t I ,...,M , where I ,...,M = P i S i − S is the multipartitemutual information [54]. Local free energy inequality . Analogously, summing upEq. (11) over reservoirs α i we derive an inequality de-scribing the nonequilibrium free energy balance for a sin-gle subsystem: X α i T α i ˙ σ α i = ˙ W i − ˙ F i ≥ , (19)where ˙ W i ≡ P α i ˙ W α i and ˙ F i ≡ P α i ˙ F α i .As in the case of the total system, Eqs. (18) and (19)are nonequivalent and complementary formulations ofthe local second law of thermodynamics. They becomeequivalent when the subsystem i is coupled to an isother-mal environment with a single temperature T i . Then T i ˙ I i = − T i P α i ˙ S α i and therefore˙ F i = ˙ E i + T i ˙ I i . (20)The local nonequilibrium free energy rate consists there-fore of the energy-related and the information-relatedcontribution. At the steady state the internal energy ofthe system is constant ( d t E = 0) and thus E i can be in-terpreted as the energy flow to the subsystem j = i . Thesubsystem attached to an isothermal environment maytherefore perform work either due to the energy flow fromthe other subsystem or due to the information flow; thelatter case corresponds to the operation of information-powered devices. Example . The applicability of our approach will nowbe demonstrated on a recently proposed [55] model ofautonomous quantum Maxwell demon. Here we describethe device only briefly; for more details we refer to theoriginal paper.The analyzed setup [Fig. 1 (a)] is composed of twoquantum dots coupled by the XY exchange interaction,each attached to two electrodes with equal temperatures T . The Hamiltonian of the system is defined asˆ H S = X iσ (cid:15) i d † iσ d iσ + X i U i n i ↑ n i ↓ (21)+ J (cid:16) d † ↑ d ↓ d † ↓ d ↑ + d † ↓ d ↑ d † ↑ d ↓ (cid:17) , (a ) (b) J ε ,U ε ,U μ L μ R μ L μ R Γ L Γ R Γ L Γ R ↑ ↑↑↑ FIG. 1. (a) Scheme of the autonomous quantum Maxwell de-mon described in the main text. (b) Schematic representationof the spin exchange induced by the XY interaction. where d † iσ ( d iσ ) is the creation (annihilation) operator ofan electron with spin σ ∈ {↑ , ↓} in the dot i ∈ { , } , n iσ = d † iσ d iσ is the particle number operator, (cid:15) i is theorbital energy, U i is the intra-dot Coulomb interactionin the dot i and J is the exchange coupling. The bathHamiltonian reads ˆ H B = P α i kσ (cid:15) α i kσ c † α i kσ c α i kσ , where c † α i kσ ( c † α i kσ ) is the creation (annihilation) operator ofan electron with spin σ , wave number k and energy (cid:15) α i kσ in the reservoir α i ; here α i = L i ( R i ) denotesthe left (right) reservoir attached to the dot i . Finally,the system bath-interaction Hamiltonian is expressed asˆ H I = P iα i kσ t α i c † α i kσ d iσ + h.c., where t α i is the tunnelcoupling of the dot i to the reservoir α i . We also definethe coupling strength Γ σα i = 2 π | t α i | ρ σα i , where ρ σα i is thedensity of states of electrons with spin σ in the bath α i .To describe the dynamics of the device we apply amicroscopically derived Lindblad equation which couplespopulations to coherences and is thermodynamically con-sistent in the weak coupling limit (it is equivalent to thephenomenological approach proposed in Ref. [59]). De-tails of the method, discussion of its limits of validity andcomparison with the secular Lindblad equation (whichis by construction thermodynamically consistent but ne-glects genuine quantum coherent effects) are presented inthe Supplementary Material [47].The device works as follows: The baths are taken tobe fully spin polarized, i.e., either Γ ↑ α i or Γ ↓ α i is equal to0. As a result, the electrodes act as spin filters whichforbid the tunneling of electrons with a spin opposite tothe polarization [56, 57]. Additionally, the polarizationsof reservoirs attached to a single dot are arranged in ananti-parallel way, such that the electron may be trans-ferred between the electrodes only when it changes itsspin. This is enabled by the XY interaction which ex-changes spins between the dots [Fig. 1 (b)]. Since thisinteraction conserves the total spin, the spin flips occursimultaneously in both subsystems and thus the steady-state currents through both dots have to be equal. Let us (a)(b) FIG. 2. Steady state work, free energy, heat, information flowand energy flow for the second (a) and the first (b) quantumdot. Results for T = 100, (cid:15) i = 0, U i → ∞ (strong Coulombblockade), µ L = − µ R = 60, µ L = − µ R = −
30, Γ ↓ L =Γ ↑ R = Γ ↑ L = Γ ↓ R = 0, all other coupling strengths Γ σα i equalto Γ = 1. now apply a high positive bias V = µ L − µ R > V = µ L − µ R < | V | < | V | ) to the other one. Then, the voltage-drivencurrent through the first dot will pump electrons in thesecond dot against the bias, which is due to a nonequilib-rium spin population induced by spin flips. This can beinterpreted as the operation of a Maxwell demon: Thehigh positive voltage in the first dot tends to reset theupper dot to the state ↑ (i.e., the singly occupied statewith a spin up). As a result, the spin dynamics generatedby the XY interaction (equivalent to the operation of thequantum iSWAP gate [58]) flips the spin in the seconddot if it is in the state ↓ , whereas leaves it unchangedwhen it is in the state ↑ [cf. Fig. 1 (b)], thus creatingan excess populations of spins ↑ . This feedback mecha-nism induces the information flow between the dots, thusenabling a conversion of heat into work.Our local Clausius and free energy inequalities[Eqs. (18), (19)] for this system are demonstrated inFig. 2. As one can see, for J / T ˙ I < − ˙ Q < F < ˙ W < J /
50Γ the work is performed in the second dot onlydue to feedback-induced information flow ( ˙ F ≈ T ˙ I )and not due to energy flow, which is negligible ( E ≈ J ’
50Γ onecan observe a noticeable energy flow from the first to thesecond dot which results from the splitting of energy lev-els. As a consequence, for J ’ − ˙ Q >
0) and the setup ceasesto work as a Maxwell demon. However, the dot 2 stillperforms work thanks to the negativity of the nonequilib-rium free energy rate ˙ F , which now includes a significantenergy-related contribution ˙ E . Conclusions . Our inequality (12), by providing a com-plement to the second law, may have novel implicationsfor the design of quantum heat engines [60] which needto be explored. In turn, our inequalities (18) and (19)provide the basis for thermodynamics of quantum infor-mation processing. We hope that, as happened with theclassical counterpart of Eq. (18), numerous applicationsand experiments [25] will also ensure it in the quantumrealm.
Acknowledgements . K. P. is supported by the Na-tional Science Centre, Poland, under the project Opus11 (No. 2016/21/B/ST3/02160) and the doctoral scholar-ship Etiuda 6 (No. 2018/28/T/ST3/00154). M. E. is sup-ported by the European Research Council project Nan-oThermo (ERC-2015-CoG Agreement No. 681456). ∗ [email protected][1] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Proba-bility of second law violations in shearing steady states,Phys. Rev. Lett. , 2401 (1993).[2] G. Gallavotti and E. G. D. Cohen, Dynamical Ensem-bles in Nonequilibrium Statistical Mechanics, Phys. Rev.Lett. , 2694 (1995).[3] J. L. Lebowitz and H. Spohn, A Gallavotti–Cohen-TypeSymmetry in the Large Deviation Functional for Stochas-tic Dynamics, J. Stat. Phys. , 333 (1999).[4] C. Jarzynski, Microscopic Analysis of Clausius–DuhemProcesses, J. Stat. Phys. , 415 (1999).[5] U. Seifert, Entropy Production along a Stochastic Tra-jectory and an Integral Fluctuation Theorem, Phys. Rev.Lett. , 040602 (2005).[6] M. 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Rep. , 1 (2017). upplementary material to: Thermodynamics of Quantum Information Flows Krzysztof Ptaszy´nski and Massimiliano Esposito Institute of Molecular Physics, Polish Academy of Sciences,ul. M. Smoluchowskiego 17, 60-179 Pozna´n, Poland ∗ Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg (Dated: March 27, 2019)
SUPPLEMENTARY MATERIAL
This appendix contains in the following order: • General derivation of the non-secular Lindbladequation from the Redfield equation [Sec. 1] • Derivation of the non-secular Lindblad equation forthe autonomous Maxwell demon [Sec. 2] • Discussion of the thermodynamic consistency of thenon-secular Lindblad equation [Sec. 3] • Comparison with the results obtained by means ofthe secular approximation [Sec. 4] • Discussion of the non-bipartite structure of dynam-ics generated by the secular master equation [Sec. 5] • Demonstration of the equivalence of Eq. (11) fromRef. [1] with Eq. (18) from the main text for thecase of bipartite dynamics [Sec. 6] • Calculation of the unitary and the dissipative con-tribution to the information flow [Sec. 7]
1. Derivation of the non-secular Lindblad equation
In the main text it was assumed that one can describethe dynamics of the system by the Lindblad equation pro-viding the Gibbs state at equilibrium. However, a mostcommon approach to the dynamics of open quantum sys-tems, namely, the perturbatively derived Redfield equa-tion, is not of Lindblad form and thus does not guaranteethe complete positivity of the dynamics [2]. Furthermore,it does not provide an exact Gibbs state, and thus may beinconsistent with the second law of thermodynamics [3].The non-completely positive dynamics arises due tothe presence of terms in the Redfield equation couplingthe state populations to the coherences between the non-degenerates eigenstates of the system (referred to as thenon-secular terms). The usual way to ensure the com-plete positivity is to apply the secular approximationwhich neglects such terms. In such a way one obtains athermodynamically consistent master equation of Lind-blad form [2]. However, as a result one may miss some ∗ [email protected] genuine quantum coherent effects, such as a finite time-scale of coherent oscillations between states of the sys-tem [4, 5].To deal with such a problem, Refs. [4] and [5] proposed,on purely phenomenological grounds, a Lindblad equa-tion including the non-secular terms. Here we provide arigorous theoretical justification of such an approach byderiving a corresponding Lindblad equation directly fromthe Redfield equation. This is done by applying a set ofapproximations which modifies the form of non-secularterms instead of canceling them. We will first describea general framework; the derivation of the master equa-tion for the autonomous Maxwell demon analyzed in themain text is presented in the next section.Following Ref. [2], we express a generic form of theinteraction Hamiltonian asˆ H I = X α A α ⊗ B α , (S1)where A α and B α are the operators acting in the Hilbertspace of the system and the reservoirs, respectively. Afterapplying the Born-Markov approximation, dynamics ofthe density matrix of the system in the interaction pictureis given by the Redfield equation [2] d t ρ I ( t ) = X ω,ω X α,β e i ( ω − ω ) C αβ ( ω ) (S2) × (cid:2) A β ( ω ) ρ I ( t ) A † α ( ω ) − A † α ( ω ) A β ( ω ) ρ I ( t ) (cid:3) + h.c.Here A α ( ω ) = X E l − E k = ω | l ih l | A α | k ih k | , (S3)are the operators describing jumps between eigenstates | k i and | l i of the Hamiltonian ˆ H S differing by the energygap ω = E l − E k . The coefficients C αβ ( ω ) = Z ∞ dτ e iωτ Tr B (cid:2) B † α ( τ ) B β (0) ρ B (cid:3) , (S4)are the one sided Fourier transforms of the bath correla-tion functions. Here Tr B denotes the partial trace overdegrees of freedom of the reservoirs, ρ B is the densitymatrix of the bath and B α ( τ ) = e i ˆ H B τ B α e − i ˆ H B τ , (S5)are the interaction picture operators of the environment. a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r The Redfield equation can be rewritten in theSchr¨odinger picture as d t ρ ( t ) = − i h ˆ H S , ρ ( t ) i (S6)+ X ω,ω X α,β C αβ ( ω ) (cid:2) A β ( ω ) ρ ( t ) A † α ( ω ) − A † α ( ω ) A β ( ω ) ρ ( t ) (cid:3) + X ω,ω X α,β C ∗ βα ( ω ) h A α ( ω ) ρ ( t ) A † β ( ω ) − ρ ( t ) A † β ( ω ) A α ( ω ) i . One can also express the Fourier transform of the bathcorrelation functions as a sum of their real and imaginarypart C αβ ( ω ) = 12 γ αβ ( ω ) + iS αβ ( ω ) , (S7)with coefficients γ αβ ( ω ) being positive and S αβ ( ω ) form-ing a real symmetric matrix [2].Within the secular approximation one assumes thata typical time of intrinsic evolution of the system ismuch smaller then the relaxation time of the system τ R : | ω − ω | − (cid:28) τ R . One can therefore neglect therapidly oscillating terms of the Redfield equation by tak-ing A α ( ω = ω ) = 0. Here, following similar way of think-ing, we assume that for | ω − ω | − (cid:28) τ R the non-secularterms of the Redfield equation corresponding to ω = ω does not influence much the dynamics of the populations,such that they may be kept into the master equation;moreover, their exact form is not very important andthus may be modified without much consequences. Wealso assume that the functions C αβ ( ω ) are smooth, i.e., C αβ ( ω ) ≈ C αβ ( ω ) for | ω − ω | − ≈ τ R . One may thenreplace γ αβ ( ω ) , γ αβ ( ω ) → γ αβ ( ω, ω ) , (S8) S αβ ( ω ) , S αβ ( ω ) → S αβ ( ω, ω ) , (S9)where γ αβ ( ω, ω ) = q γ αβ ( ω ) γ αβ ( ω ) , (S10) S αβ ( ω, ω ) = S αβ ( ω ) + S αβ ( ω )2 , (S11)such that γ αβ ( ω ) ≈ γ αβ ( ω ) ≈ γ αβ ( ω, ω ) and S αβ ( ω ) ≈ S αβ ( ω ) ≈ S αβ ( ω, ω ) for | ω − ω | − ≈ τ R . One can nowperform the following replacement in Eq. (S6): C αβ ( ω ) → γ αβ ( ω, ω ) + iS αβ ( ω, ω ) , (S12) C ∗ βα ( ω ) → γ βα ( ω, ω ) − iS βα ( ω, ω ) . (S13)Changing indexes α ↔ β and ω ↔ ω in the second sumin Eq. (S6) one obtains d t ρ ( t ) = − i h ˆ H S + ˆ H LS , ρ ( t ) i + X ω,ω X α,β γ αβ ( ω, ω ) × (cid:20) A β ( ω ) ρ ( t ) A † α ( ω ) − (cid:8) A † α ( ω ) A β ( ω ) , ρ ( t ) (cid:9)(cid:21) , (S14) where the last part of the equation defines the dissipator D . Hereˆ H LS = X ω,ω X α,β S αβ ( ω, ω ) A † α ( ω ) A β ( ω ) , (S15)is the Lamb shift Hamiltonian commuting with ˆ H S . Thecommutation property can be verified by taking into ac-count the explicit form of operators A α and expressingˆ H LS asˆ H LS = X klmn X α,β S αβ ( ω mn , ω lk ) (S16) × | n ih n | A † α | m ih m | l ih l | A β | k ih k | = X kln X α,β S αβ ( ω ln , ω lk ) | n ih n | A † α | l ih l | A β | k ih k | . The commutator can be then expressed as h ˆ H S , ˆ H LS i (S17)= X kln X α,β S αβ ( ω ln , ω lk ) E n | n ih n | A † α | l ih l | A β | k ih k |− X kln X α,β S αβ ( ω ln , ω lk ) E k | n ih n | A † α | l ih l | A β | k ih k | . Since each pair E k , E l will appear two times and E l − E k = − ( E k − E l ) the terms will cancel each other,and thus [ ˆ H S , ˆ H LS ] = 0.Equation (S14) is of the first standard form, and thusensures the completely-positive trace-preserving dynam-ics [2]. Since the matrix γ αβ ( ω, ω ) is positive, the equa-tion can be brought to the Lindblad form by diagonaliz-ing the matrix γ αβ ( ω, ω ). We will demonstrate this inthe next subsection for a particular case.
2. Lindblad equation for the Maxwell demon
Following the general discussion from the previous sub-section, let us now derive the Lindblad equation for theautonomous Maxwell demon considered in the main text.Let us first note, that in Eq. (S1) it was assumed thatthe interaction Hamiltonian can be expressed as a tensorproduct of operators acting on the Hilbert spaces of thesystem and the bath; this requires that the operators A α and B α commute. On the other hand, in the consideredsystem creation and annihilation operators included inthe interaction Hamiltonianˆ H I = X iα i kσ t α i c † α i kσ d iσ + t ∗ α i c α i kσ d † iσ , (S18)follow the fermionic anticommutation relations. How-ever, this problem can be solved by employing theJordan-Wigner transformation [6, 7]. Let us first replaceeach set of quantum numbers iσ or α i kσ with a singlequantum number: iσ → κ , α i kσ → λ , with κ ∈ { , K } and λ ∈ { , L } . Annihilation operators can be then rep-resented by tensor products of Pauli matrices d κ = κ − Y κ =1 σ zκ ⊗ σ − κ ⊗ K Y κ = κ +1 I κ ⊗ L Y λ =1 I λ , (S19) c λ = K Y κ =1 σ zκ ⊗ λ − Y λ =1 σ zλ ⊗ σ − λ ⊗ L Y λ = λ +1 I λ , (S20)where σ − = ( σ z − iσ y ) / b Y n = a σ jn = σ ja ⊗ σ ja +1 ⊗ ... ⊗ σ jb . (S21)Creation operators are defined by replacing the loweringoperator σ − with the raising operator σ + = ( σ − ) † =( σ z + iσ y ) /
2. One can verify, that the operators definedin such a way follow the fermionic anicommutation rela-tions.Let us now introduce the modified operators˜ d κ = − κ − Y κ =1 I κ ⊗ σ − κ ⊗ K Y κ = κ +1 σ zκ ⊗ L Y λ =1 I λ , (S22)˜ c λ = K Y κ =1 I κ ⊗ λ − Y λ =1 σ zλ ⊗ σ − λ ⊗ L Y λ = λ +1 I λ , (S23)which are obtained from d κ and c λ by replacing σ zκ → I κ , I κ → σ zκ and σ − κ → − σ − κ . These operators may beinterpreted as creation and annihilation operators whichact separately on the Hilbert spaces of the system andthe bath, and therefore follow the commutation relationssuch as [ ˜ d κ , ˜ c λ ] = 0. Their fermionic character is retaineddue to the fact that pairs of operators acting on eitherthe system or the bath still anticommute; for example { ˜ d κ , ˜ d κ } = 0. One may then verify, that the Hamiltonianwill not change upon replacing d κ , c λ → ˜ d κ , ˜ c λ . Let usdemonstrate this on the example of a pair of operators c † λ d κ appearing in the interaction Hamiltonian c † λ d κ = κ − Y κ =1 ( σ zκ ) ⊗ σ zκ σ + κ ⊗ K Y κ = κ +1 σ zκ I κ (S24) ⊗ λ − Y λ =1 σ zλ I λ ⊗ σ − λ I λ ⊗ L Y λ = λ +1 I λ I λ = κ − Y κ =1 ( I κ ) ⊗ ( − I κ σ + κ ) ⊗ K Y κ = κ +1 I κ σ zκ ⊗ λ − Y λ =1 σ zλ I λ ⊗ σ − λ I λ ⊗ L Y λ = λ +1 I λ I λ = ˜ c † λ ˜ d κ , where we have used the properties ( σ z ) = I and σ z σ ± = − σ ± . The interaction Hamiltonian can be therefore rewrittenas ˆ H I = X α i σ X + , − A α i σ ± ⊗ B iα i σ ± , (S25)where A α i σ + = ˜ d † iσ , (S26) A α i σ − = ˜ d iσ , (S27) B α i σ + = X k ˜ c α i kσ t ∗ α i , (S28) B α i σ − = X k ˜ c † α i kσ t α i . (S29)In the considered system, the reduced density matrixof the bath can be written as ρ B = Y α i Z − α i e − β αi ( ˆ H αi − µ αi ˆ N αi ) , (S30)where ˆ H α i = X kσ (cid:15) α i kσ c † α i kσ c α i kσ , (S31)ˆ N α i = X kσ c † α i kσ c α i kσ , (S32)are the Hamiltonian and the particle number oper-ator of the reservoir α i , respectively, and Z α i =Tr { exp[ − β α i ( ˆ H α i − µ α i N α i )] } is the partition function.One may then verify that the Fourier transform of thebath correlation function defined by Eq. (S4) vanishesfor α = β . In the continuous limit, one may also replacetrace in the definition of the bath correlation functionsby an integral. As a result, the Redfield equation in theSchr¨odinger picture [Eq. (S6)] can be expressed as d t ρ ( t ) = − i h ˆ H S , ρ i (S33)+ X ω,ω X iα i σ X + , − C α i σ ± ( ω ) × h A α i σ ± ( ω ) ρA † α i σ ± ( ω ) − A † α i σ ± ( ω ) A α i σ ± ( ω ) ρ i + h.c. o , where A α i σ ± ( ω ) = X E l − E k = ω | l ih l | A α i σ ± | k ih k | , (S34)are the jump operators, and the Fourier transforms ofthe bath correlations functions read as C α i σ ± ( ω ) = (S35) ∞ Z dτ e iωτ ∞ Z −∞ dEρ σα i ( E ) | t α i | e ∓ iEτ f [ ± β α i ( E − µ α i )] , where ρ σα i ( E ) is the density of states in the lead α i forthe spin σ . We will further assume that ρ σα i ( E ) = const.and define the coupling strength Γ σα i = 2 π | t α i | ρ σα i . Thefunction C α i σ ± ( ω ) reads then C α i σ ± ( ω ) = 12 γ α i σ ± ( ω ) + iS α i σ ± ( ω ) , (S36)where γ α i σ ± ( ω ) = Γ σα i f [ β α i ( ω ∓ µ α i )] , (S37)is the tunneling rate and S α i σ ± ( ω ) = Γ σα i π P ∞ Z −∞ dEf [ β α i ( E ∓ µ α i )] ω − E (S38)= − Γ σα i π ReΨ (cid:20)
12 + i β α i ( ω ∓ µ α i )2 π (cid:21) , is the principal part of the Cauchy integral of the tun-neling rate describing the level renormalization [6, 8, 9];here Ψ denotes the digamma function.Following the approach presented in the previous sec-tion, one obtains the non-secular Lindblad equation d t ρ = − i h ˆ H S + ˆ H LS , ρ i + X i D i ρ, (S39)where the dissipator D i is defined as D i ρ = X ω,ω X α i σ X + , − γ α i σ ± ( ω, ω ) (S40) × (cid:20) A α i σ ± ( ω ) ρA † α i σ ± ( ω ) − n A † α i σ ± ( ω ) A α i σ ± ( ω ) , ρ o(cid:21) , with γ α i σ ± ( ω, ω ) = p γ α i σ ± ( ω ) γ α i σ ± ( ω ), and the Lambshift Hamiltonian readsˆ H LS = 12 X ω,ω X iα i σ X + , − [ S α i σ ± ( ω ) + S α i σ ± ( ω )] × A † α i σ ± ( ω ) A α i σ ± ( ω ) . (S41)The secular master equation is obtained if onefurther takes γ α i σ ± ( ω, ω ) = 0 in Eq. (S40) and A † α i σ ± ( ω ) A α i σ ± ( ω ) = 0 in Eq. (S41) for ω = ω .The dissipator D i can be rewritten directly in the Lind-blad form D i ρ = X α i σ X + , − (cid:18) L α i σ ± ρL † α i σ ± − n L † α i σ ± L α i σ ± , ρ o(cid:19) , (S42)with the Lindblad operators defined as L α i σ ± = X ω p γ α i σ ± ( ω ) A iα i σ ± ( ω ) . (S43) J = Γ J = Γ J = Γ LindbladRed fi eld ∼ ( Γ / T ) - - - - - - Γ / T - F FIG. S1. Deviation from the Gibbs state 1 − F for the non-secular Lindblad equation (solid lines) and the Redfield equa-tion (dotted lines) for Γ ↓ L = Γ ↑ R = Γ ↑ L = Γ ↓ R = 0, allother coupling strengths Γ σα i equal to Γ, (cid:15) i = 0, µ α i = 0 and U i → ∞ (strong Coulomb blockade). Explicitly, the Lindblad operators read as L α i σ + = X kl q Γ σα i f + α i ( ω kl ) | k ih k | c † iσ | l ih l | , (S44) L α i σ − = X kl q Γ σα i f − α i ( ω kl ) | k ih k | c iσ | l ih l | , (S45)where f ± α i ( ω ) = f [ β α i ( ω ∓ µ α i )] and ω kl = E k − E l .As this form clearly illustrates, the operators describethe jumps between different superposition of eigenstatesof the system with probability amplitudes of related tothe tunneling rates. We note, that the same form ofthe Lindblad operators may be obtained by using thephenomenological approach proposed in Ref. [5].
3. Relaxation to equilibrium
The thermodynamic formalism described in the maintext was based on the assumption, that the Gibbs stateis a stationary state at equilibrium, i.e., for β α = β , µ α = µ . This is ensured by the Redfield equation withinthe secular approximation [2]. Unfortunately, this is notby construction guaranteed by the newly derived non-secular Lindblad equation (the similar observation hasbeen already made for a phenomenological approach fromRef. [5]). However, the steady state converges to theGibbs state when the coupling to the bath is weak. Weverify this numerically on the example of the autonomousquantum Maxwel demon. In particular, we calculate theconvergence to the Gibbs state characterized by the fi-delity defined as [10] F = h Tr q √ ρ eq ρ st √ ρ eq i , (S46)where ρ st denotes the stationary state of the master equa-tion and the density matrix of the Gibbs state is definedas ρ eq = Z − β,µ e − β ( ˆ H S − µ ˆ N ) , (S47)with Z β,µ = Tr { exp[ − β ( ˆ H S − µ ˆ N )] } being the partitionfunction. The fidelity takes values within the interval[0 ,
1] with F = 1 indicating the perfect convergence ofthe density matrices [10]. As shown in Fig. S1 the fidelityconverges to 1 in the weak coupling regime with the di-vergence from the Gibbs state scaling as 1 − F ∝ (Γ /T ) for Γ / T . As one can also observe, deviation from theGibbs state is not a result of approximations made whenderiving Lindblad equation from the Redfield equation,but is rather inherent to the perturbative nature of theRedfield equation itself.
4. Comparison with the secular approximation
The thermodynamic consistency is by constructionprovided by the Redfield equation within the secular ap-proximation. However, except from the case when theeigenstates are fully degenerate (analyzed in Ref. [11]),the secular approximation decouples the populationsfrom the coherences and thus neglects the genuine quan-tum coherent effects. This is demonstrated in Fig. S2. Asshown, the secular master equation agrees with both theRedfield equation and the non-secular Lindblad equationfor J (cid:29) Γ. In this regime the assumption that the char-acteristic time of the intrinsic evolution of the system ismuch smaller than the relaxation time is valid. On theother hand, the secular approximation diverges from theother approaches when the ratio J/ Γ is relatively small.In particular, it predicts non-vanishing particle and heatcurrents for J →
0, when the quantum dots become de-coupled, which is clearly a non-physical result. This isa consequence of the fact that the secular approxima-tion neglects the finite timescale of the coherent oscilla-tions between the spin states. With this qualification,as demonstrated in Fig. S3 the secular master equationcan also describe the operation of the system as an au-tonomous Maxwell demon (cf. Fig. 2 in the main text).
5. Non-bipartite structure of the secular masterequation
We emphasize, that in the system we consider the sec-ular approximation generates a classical rate equationdescribing the dynamics of eigenstate population, which,however, does not possess a bipartite structure as definedin Ref. [1]. The term “bipartite dynamics” refer here tothe situation when there are no transition generating thesimultaneous change of states of both subsystems. As aresult, the information flow cannot be analyzed withinthe approach proposed in Ref. [1] and can only be de-scribed by our method. This clearly demonstrates thewider range of applicability of our approach even whenthe dynamics is effectively classical.
FIG. S2. Particle current (a) and cooling power (b) in thesecond dot calculated with the secular approximation (bluedashed line), non-secular Lindblad equation (red solid line)and the Redfield equation (black dots) for T = 100, (cid:15) i = 0, U i → ∞ (strong Coulomb blockade), µ L = − µ R = 60, µ L = − µ R = −
30, Γ ↓ L = Γ ↑ R = Γ ↑ L = Γ ↓ R = 0, all othercoupling strengths Γ σα i equal to Γ = 1.FIG. S3. Steady state work, free energy, heat, informationflow and energy flow for the second (a) and the first (b) quan-tum dot calculated within the secular approximation. Param-eters as in Fig. S2. In the system we consider, the non-bipartite struc-ture of the dynamics results from the fact that the totalHamiltonian ˆ H S [Eq. (21) in the main text] does notcommute with the Hamiltonian of a single dot ˆ H i = (cid:15) i d † iσ d iσ + U i n i ↑ n i ↓ and thus the eigenstates of the to-tal system cannot be expressed as products of statesof the subsystems. More specifically, there exist eigen-states of the form | + i = ( αc † ↑ c † ↓ + βc † ↓ c † ↑ ) | i and |−i = ( βc † ↑ c † ↓ − αc † ↓ c † ↑ ) | i (with | i being the emptystate) which are the entangled states of the first and thesecond dot. This leads to the presence of non-local cor-relations between the subsystems; for example, tunnel-ing from one dot acts as a quantum measurement which“fixes” the spin state of another dot. The presence ofsuch non-local correlations breaks the assumption of bi-partite dynamics. However, since every change of state isrelated to the act of electron tunneling to or from a sin-gle dot, one can still separate the dissipator D into thecontributions D and D in a physically meaningful way[cf. Eqs. (S39) and (S40)]. This enables us to describethe information flow within our approach.
6. Equivalence with the classical approach forbipartite systems
In the previous section we have shown that our ap-proach can be applied to systems with a non-bipartitedynamics, for which the method proposed in Ref. [1] isnot applicable. Here, on the other hand, we demonstratethat when the dynamics is bipartite both approaches be-comes equivalent; this can take place when the Hamilto-nians of both subsystems commute with the total Hamil-tonian. Therefore, our method can be seen as a directgeneralization of the one proposed in Ref. [1].Let us consider a two-component open quantum sys-tem consisting of subsystems X and Y , with dynamicsdescribed by an effectively classical secular master equa-tion. Furthermore, let us also assume that the Hamilto-nian of the total systemˆ H S = ˆ H X + ˆ H Y + ˆ H XY , (S48)commutes with the Hamiltonians of the subsystems ˆ H X and ˆ H Y . Then the eigenstates of the Hamiltonian ˆ H S are products of the eigenstates of the Hamiltonians ofsubsystems: | i i = | x i| y i , where | i i is an eigenstate of thetotal system whereas | x i ( | y i ) is an eigenstate of the sub-system X ( Y ). The state of the system can be expressedby the diagonal density matrix ρ = X x,y p ( x, y ) | x i| y ih y |h x | , (S49)where p ( x, y ) is the probability of the state | x i| y i . Thereduced density matrix of the subsystems X and Y will analogously read as ρ X = X x p ( x ) | x ih x | , (S50) ρ Y = X y p ( y ) | y ih y | , (S51)where p ( x ) = X y p ( x, y ) , (S52) p ( y ) = X x p ( x, y ) , (S53)are the probabilities of the states | x i and | y i , respectively.The dynamics of the state probabilities generated bythe dissipator D is given by the rate equation (Eq. (1)from Ref. [1]) d t p ( x, y ) = X x ,y h W y,y x,x p ( x , y ) − W y ,yx ,x p ( x, y ) i , (S54)where W y,y x,x is the transition rate of the jump ( x , y ) → ( x, y ).Let us now assume that the dynamics is bipartite, i.e.,that there is no jumps simultaneously changing statesof the subsystems X and Y . Mathematically this reads(Eq. (2) from Ref. [1]) W y,y x,x = w yx,x x = x ; y = y w y,y x x = x ; y = y . (S55)Here the rates w yx,x and w y,y x correspond to the oper-ation of the dissipator D X and D Y , respectively. Dy-namics of the subsystem X is then described by the rateequation d t p ( x ) = X x ,y h w yx,x p ( x , y ) − w yx ,x p ( x, y ) i . (S56)An analogous equation can be written for the subsystem Y .Following Eq. (17) from the main text, the informationrate associated with the subsystem X can be calculatedas˙ I X = − Tr ( d t ρ X ln ρ X ) + Tr [( D X ρ ) ln ρ ] (S57)= − X x d t p ( x ) ln p ( x )+ X x,x ,y h w yx,x p ( x , y ) − w yx ,x p ( x, y ) i ln p ( x, y )= X x,x ,y h w yx,x p ( x , y ) − w yx ,x p ( x, y ) i ln p ( x, y ) p ( x )= X x ≥ x ,y J yx,x ln p ( x, y ) p ( x ) p ( x ) p ( x , y ) = X x ≥ x ,y J yx,x ln p ( y | x ) p ( y | x ) . FIG. S4. The unitary and the dissipative contribution to theinformation flows ˙ I and ˙ I calculated using the non-secularLindblad equation for parameters as in Fig. S2. Here we have denoted w yx,x p ( x , y ) − w yx ,x p ( x, y ) = J yx,x and used the definition of the conditional probability p ( y | x ) = p ( y, x ) /p ( x ). The result is equivalent toEq. (10) from Ref. [1]. Upon putting on the obtained˙ I X into Eq. (18) from the main text one obtains Eq. (11)from Ref. [1]. This demonstrates the equivalence of bothapproaches for systems with the bipartite dynamics.
7. The unitary and the dissipative contribution tothe information flow