Lower Bound on Irreversibility in Thermal Relaxation of Open Quantum Systems
LLower Bound on Irreversibility in Thermal Relaxation of Open Quantum Systems
Tan Van Vu ∗ and Yoshihiko Hasegawa † Department of Information and Communication Engineering,Graduate School of Information Science and Technology,The University of Tokyo, Tokyo 113-8656, Japan (Dated: February 16, 2021)We consider thermal relaxation process of a quantum system attached to a single or multiple reser-voirs. Quantifying the degree of irreversibility by entropy production, we prove that the irreversibil-ity of the thermal relaxation is lower-bounded by a relative entropy between the unitarily-evolvedstate and the final state. The bound characterizes the state discrepancy induced by the non-unitarydynamics, thus reflecting the dissipative nature of irreversibility. Intriguingly, the bound can beevaluated solely in terms of the initial and final states and the system Hamiltonian; hence, pro-viding a feasible way to estimate entropy production without prior knowledge of the underlyingcoupling structure. Our finding refines the second law of thermodynamics and reveals a universalfeature of thermal relaxation processes.
Introduction. —The last two decades have witnessedsubstantial progress in thermodynamics of nonequilib-rium systems subject to significant fluctuations. Variousproperties of small systems have been elucidated with theadvent of comprehensive frameworks such as stochasticthermodynamics [1, 2] and quantum thermodynamics [3–5]. One of the prominent universal relations is the cele-brated fluctuation theorem [6–11], from which the secondlaw of thermodynamics and the fluctuation-dissipationtheorem can be immediately derived [12–14]. Beyondthe fluctuation theorem, much recent attention has beenfocused on thermodynamic uncertainty relations [15–34],speed limits [35–43], and refinements of the second law[44–54]. These findings are not only theoretically impor-tant but also provide powerful tools for thermodynamicinference, for instance, in the estimation of free energy[55] and dissipation [56–59].Central to most established relations is thermody-namic irreversibility, which is quantified by irreversibleentropy production. The positivity of entropy produc-tion is universally captured by the second law of thermo-dynamics, imposing fundamental limits on the computa-tional cost via Landauer’s principle [60–62] and on theperformance of physical and biological systems such asheat engines [63, 64] and molecular motors [65]. The im-portance of entropy production has triggered intense re-search to formulate and investigate its properties [66–70],across from classical to quantum (see [71] for a review).Nonetheless, restricting to a specific class of nonequilib-rium processes, one may find rich features of thermo-dynamic irreversibility. One of the interesting classes isthermal relaxation, which is ubiquitous in nature andplays crucial roles in condensed matter [72], heat en-gines [73], and quantum state preparation. Any systemcoupled to thermal reservoirs unavoidably exchanges en-ergy with the surrounding environment and relaxes toa stationary state. Notably, Ref. [51] proved that theentropy production during relaxation of classical Marko-vian processes is bounded from below by the classical relative entropy between the initial and final distribu-tions. For relaxation to equilibrium in open quantumsystems, it has been shown that the entropy productionis lower-bounded by a lag between states in terms of atime-reversed map [48] and a geometrical distance on theRiemannian manifold [54]. In addition, entropy produc-tion is also a relevant quantifier of nonequilibrium degreein a hypothesis stating that nanoscale warming is fasterthan cooling [74].In this Letter, we deepen our understanding of ther-modynamic irreversibility in thermal relaxation processesof open quantum systems. Specifically, we derive a fun-damental bound on irreversibility for systems that arein contact with thermal reservoirs and described by theLindblad master equations. We prove that the irre-versible entropy production during relaxation is lower-bounded by a relative entropy between the final statesobtained with the unitary dynamics and the original dy-namics [cf. Eq. (4)]. Since the Lindblad dynamics com-prise the unitary and non-unitary parts, the lower-boundquantifies the state discrepancy induced by the dissi-pative non-unitary dynamics, thus intuitively reflectingthe nature of thermodynamic irreversibility (shown inFig. 1). Remarkably, the derived bound is saturable,experimentally accessible, and stronger than the conven-tional second law of thermodynamics. Furthermore, thebound provides a feasible way to estimate irreversibleentropy production with the help of the quantum statetomography technique.
Main result. —We consider thermal relaxation processof a Markovian open quantum system. The system canbe simultaneously coupled to multiple thermal reservoirsat different temperatures. The dynamics of the densitymatrix ρ ( t ) are governed by the Lindblad master equa-tion [75]: ∂ t ρ ( t ) = L [ ρ ( t )] := − i [ H, ρ ( t )] + X k D k [ ρ ( t )] , (1)where H is the time-independent Hamiltonian and the a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Unitary manifold
FIG. 1. Geometrical illustration of the main result. The man-ifold of density matrices that can be generated from ρ (0) vianondissipative unitary transforms is denoted by M ρ (0) . Thetime evolution of the density matrix ρ ( t ) under the Lindbladdynamics and the unitary dynamics is described by the solidline and the dashed line, respectively. The irreversible entropyproduction Σ τ is bounded from below by the information-theoretical distance S M [ ρ u ( τ ) || ρ ( τ )] — a relative entropy be-tween the unitarily-evolved state ρ u ( τ ) := U τ ρ (0) U † τ and thefinal state ρ ( τ ). dissipator is given by D k [ ◦ ] := L k ( ◦ ) L † k − { L † k L k , ( ◦ ) } .The jump operators come in pairs ( L k , L k ) with en-ergy changes ( ω k , ω k ), which satisfy [ L k , H ] = ω k L k and ω k = − ω k . This condition implies that the jump op-erators account for transitions between different energyeigenbasis with the same energy change [43, 76]. Notethat [ ◦ , ◦ ] and {◦ , ◦} are, respectively, the commutatorand the anticommutator of two operators, and the Planckconstant and the Boltzmann constant are both set tounity throughout this Letter, ~ = k B = 1. We assumethe local detailed balance condition L k = e ∆ s k env / L † k ,which is fulfilled in most cases of physical interest [69].Here ∆ s k env is the change in the environment entropy dueto the jump of type k . After a sufficiently long time, thesystem reaches a stationary state which may no longerbe a Gibbs state when multiple reservoirs are attached.The degree of irreversibility of the relaxation process dur-ing time τ can be quantified by the irreversible entropyproduction Σ τ , defined asΣ τ := ∆ S sys + ∆ S env , (2)where ∆ S sys := tr { ρ (0) ln ρ (0) } − tr { ρ ( τ ) ln ρ ( τ ) } is thechange in the system entropy (characterized by the vonNeumann entropy) and ∆ S env corresponds to the entropychange of the environment, given by [77]∆ S env := Z τ X k tr n L k ρ ( t ) L † k o ∆ s k env dt. (3)Within this definition, one can prove that the entropyproduction is always nonnegative, Σ τ ≥ τ in terms of a relative entropy between the initial and final states,Σ τ ≥ S M [ U τ ρ (0) U † τ || ρ ( τ )] , (4)where U t := e − iHt is the unitary operator and S M ( ρ || σ ) := S ( ρ || P n Π n σ Π n ) is the projectively mea-sured relative entropy between ρ and σ with the eigen-basis { Π n } of ρ . Here, S ( ρ || σ ) := tr { ρ (ln ρ − ln σ ) } ≥ ρ and σ .The inequality (4) indicates that the entropy productionis bounded from below by an information-theoretical dis-tance between the initial and final states; thus, strength-ening the Clausius inequality in the conventional secondlaw of thermodynamics for thermal relaxation processes.Some remarks regarding the main result are in or-der. First, the bound is geometrically and intuitivelyunderstandable. The system state is governed by theLindblad dynamics consisting of the non-dissipative uni-tary part and the dissipative non-unitary part. Sincethe lower-bound is the distances between the unitarily-evolved state and the final state, it quantifies how far thesystem is driven by the non-unitary dynamics; thereby,intuitively reflecting the nature of entropy production,namely a dissipative term. When the system is uncou-pled to the reservoirs and governed by a unitary dynam-ics, both the entropy production and relative entropyvanish, and the derived relation becomes a trivial equal-ity. Second, the bound is tight and can be saturated,for example, in the long-time regime; consequently, itcan be applied to thermodynamic inference. Particu-larly, given the initial and final states and the systemHamiltonian, a lower-bound of entropy production can beestimated without requiring prior knowledge of the un-derlying dynamics. Third, the bound can be interpretedas a quantum speed limit, τ ≥ S M [ U τ ρ (0) U † τ || ρ ( τ )] / Σ,where Σ := Σ τ /τ is the average entropy production rate.An important implication of this speed limit is that afast state-transformation requires a high dissipation rate[41, 54]. Last, in the classical limit (e.g., when the initialdensity matrix has no coherence in the energy eigenbasisof the Hamiltonian), the lower-bound reduces exactly tothe classical relative entropy between the initial and finaldistributions; therefore, our result recovers the classicalbound reported in Ref. [51] for a single-reservoir case.Note also that the relation (4) is valid even for systemswith broken time-reversal symmetry, such as electronicsystems with Peierls phase.As coupled to a single thermal reservoir at the inversetemperature β , the system relaxes toward an equilibriumstate π := e − βH /Z , irrespective of the initial state. Inthis case, a Hamiltonian-free lower-bound on the entropyproduction can be obtained,Σ τ ≥ S E [ ρ (0) || ρ ( τ )] , (5)where S E ( ρ || σ ) := P n a n ln( a n /b n ) =: D ( a n || b n ) is ex-actly the classical relative entropy between distributions Time Time o r FIG. 2. Numerical illustration of the main result. (a) Schematic diagram of the two-reservoir machine. (b) The ratio S M [ U τ ρ (0) U † τ || ρ ( τ )] / Σ τ is plotted as a function of time τ , and each solid line depicts the result obtained with κ rangedfrom 1 to 4. Parameters are fixed as ω = ω = 0 . γ = 0 .
01, and β = 1. (c) Plotted are the ratios S M [ U τ ρ (0) U † τ || ρ ( τ )] / Σ τ (solid line) and D [ p n (0) || p n ( τ )] / Σ τ (dash-dotted line) with κ ranged from 1 to 4. Parameters are fixed as ω = 0 . ω = 0 . γ = 0 .
01, and β = 1. { a n } and { b n } , which are the increasing eigenvalues of ρ and σ , respectively. Any unitary transform does notchange the magnitude of the eigenvalues of a density ma-trix but only the eigenbasis. Hence, the term S E quan-titatively characterizes the state change caused by thenon-unitary dynamics. The lower-bound now dependsonly on the initial and final states, which is an inherentfeature of thermal relaxation processes. For time-drivensystems, one can prove that there does not exist such auniversal metric that bounds irreversible entropy produc-tion from below (see Supplemental Material [78]). Examples. —To demonstrate the main result [Eq. (4)],we consider an autonomous thermal machine [76] withthree levels {| (cid:15) g i , | (cid:15) A i , | (cid:15) B i} . Such machines can oper-ate as refrigerators and are also the building blocks forquantum clocks [79, 80]. The Hamiltonian of the sys-tem is H = ω | (cid:15) A ih (cid:15) A | + ω | (cid:15) B ih (cid:15) B | , where ω , ω , and ω := ω − ω are frequency gaps between | (cid:15) g i ↔ | (cid:15) A i , | (cid:15) g i ↔ | (cid:15) B i , and | (cid:15) A i ↔ | (cid:15) B i , respectively. The machineis powered by two reservoirs at different inverse temper-atures β ≥ β , which mediate transitions between theenergy levels [see Fig. 2(a)].First, we examine the case in which ω = ω andthe Lindblad equation has four jump operators, L = √ η ( | (cid:15) A ih (cid:15) g | + | (cid:15) B ih (cid:15) A | ), L = √ η ( | (cid:15) g ih (cid:15) A | + | (cid:15) A ih (cid:15) B | ), L = √ η | (cid:15) B ih (cid:15) g | , and L = √ η | (cid:15) g ih (cid:15) B | , where η = γn th1 ( ω ), η = γ [ n th1 ( ω ) + 1], η = γn th2 ( ω ), and η = γ [ n th2 ( ω ) + 1]. Here, n th r ( ω ) := ( e β r ω − − denotes thePlanck distribution and γ is the decay rate. Let π ss be thesystem’s stationary state. We set β = κβ with κ ≥ ρ (0) = | ϕ ih ϕ | , where | ϕ i = √ − κ | (cid:15) g i / √ κ − | (cid:15) B i /
3. Themagnitude of κ characterizes coherence in the initial stateand the nonequilibrium degree of π ss . As κ = 1, ρ (0)is diagonal in the energy levels, and the system relaxesto the equilibrium Gibbs state. As κ >
1, coherenceemerges in the initial state, and π ss becomes a nonequi-librium steady state. We vary κ from 1 to 4 and plot the ratio S M / Σ τ as a function of time τ in Fig. 2(b). As canbe seen, the bound is tight and can be saturated for along time as κ = 1. Note that entropy production rate isalways positive for κ >
1, and hence S M / Σ τ goes to zeroin the long-time limit. However, the relation (4) is use-ful, since it shows the meaningful bound for initial rapidrelaxation processes.Next, to compare our result with a classical bound,we consider the case that each jump operator charac-terizes a single jump between two energy levels, L mn = √ η mn | (cid:15) m ih (cid:15) n | ( m = n ). The transition rates are η Ag = γn th1 ( ω ), η gA = γ [ n th1 ( ω ) + 1], η BA = γn th1 ( ω ), η AB = γ [ n th1 ( ω ) + 1], η Bg = γn th2 ( ω ), and η gB = γ [ n th2 ( ω ) + 1]. In this case, the time evolution of thediagonal terms p n ( t ) := h (cid:15) n | ρ ( t ) | (cid:15) n i follows a classicalmaster equation with time-independent transition rates[81]. Thereby, applying Eq. (4) to the diagonal dynam-ics gives Σ cl τ ≥ D [ p n (0) || p n ( τ )], where Σ cl τ is the entropyproduction associated with the classical master equation.In the long-time regime (i.e., when coherence in ρ ( τ )vanishes), one can prove that Σ τ ≥ Σ cl τ ; consequently,Σ τ ≥ D [ p n (0) || p n ( τ )], which is referred to as the classi-cal bound. The temperatures and the initial state are thesame as in the previous case. Analogously, we vary κ from1 to 4 and plot the ratios S M / Σ τ and D/ Σ τ as functionsof time τ in Fig. 2(c). As κ = 1, two bounds coincidesince there is no coherence in ρ ( t ) for all t . However, as κ increases, our bound is tighter than the classical bound.This is because our bound captures the coherence con-tribution in the initial state, whereas the classical bounddoes not. Proof of Eq. (4) . —We first rewrite the dynamics ofthe density matrix ρ ( t ) in the interaction picture. Define ρ I ( t ) := U † t ρ ( t ) U t , the time evolution of ρ I ( t ) obeys theequation (see Supplemental Material [78]) ∂ t ρ I ( t ) = X k D k [ ρ I ( t )] (6)with the initial condition ρ I (0) = ρ (0). Our approach isbased on unraveling the dynamics described by Eq. (6)in terms of quantum trajectories. In what follows, wedemonstrate that the irreversible entropy production Σ τ can be mathematically linked to the level of individualtrajectories.The framework of quantum trajectories [82, 83] wasoriginally developed in the field of quantum optics as ameans of numerically simulating open quantum systems[81]. Within this approach, the master equation is unrav-eled into stochastic time evolutions of the pure state ofthe system | ψ ( t ) i , conditioned on measurement outcomesobtained from continuous monitoring of the environment.Each individual trajectory of a stochastic realization canbe described by a smooth evolution with discontinuouschanges caused by quantum jumps in the state at randomtimes. A quantum jump is associated with the detectionof an event in the environment (e.g., emission or absorp-tion of photons). The time evolution of the pure statecan be described by the stochastic Schr¨odinger equation[81]: d | ψ ( t ) i = S [ | ψ ( t ) i ] dt + X k L k | ψ ( t ) i q h L † k L k i ψ ( t ) − | ψ ( t ) i dN k ( t ) , (7)where S ( | ψ i ) := (1 / P k ( h L † k L k i ψ − L † k L k ) | ψ i and h A i ψ := h ψ | A | ψ i . The stochastic increment dN k ( t ) iseither 0 or 1 (when the jump of type k is detected),and its ensemble average at time t is E [ dN k ( t )] = h L † k L k i ψ ( t ) dt . Under appropriate initial conditions, theaverage of | ψ ( t ) ih ψ ( t ) | over all possible trajectories re-duces exactly to the density matrix in the interactionpicture, E [ | ψ ( t ) ih ψ ( t ) | ] = ρ I ( t ).Now we define the stochastic entropy production on asingle trajectory. We employ a two-point measurementscheme on the system, where projective measurementsare performed at the beginning and at the end of any sin-gle trajectory [69]. Let ρ I (0) = P n p n | n ih n | and ρ I ( τ ) = P m p m | m ih m | be the spectral decompositions of ρ I (0)and ρ I ( τ ), the forward process is operated as follows.The state | ψ (0) i is sampled from the ensemble {| n i} with probabilities { p n } . The selected state is confirmedby the first measurement in the {| n i} eigenbasis at time t = 0. The pure state then evolves in time according toEq. (7), and the second measurement in the {| m i} eigen-basis is executed at time t = τ . This procedure results ina stochastic trajectory Γ = { n, ( k , t ) , . . . , ( k J , t J ) , m } ,where n and m are, respectively, measurement outcomesof the first and second measurements, and ( k j , t j ) de-notes a jump of type k j occurred at time t j ( j = 1 , . . . , J and 0 ≤ t ≤ · · · ≤ t J ≤ τ ). To define the time-reversed (backward) process, we introduce the antiuni-tary time-reversal operator Θ, which satisfies Θ i = − i Θand ΘΘ † = Θ † Θ = I . This operator changes the signof odd variables under time reversal such as angular mo- mentum or magnetic fields [11]. In the backward pro-cess, the initial state | ˜ ψ (0) i is sampled from the ensem-ble {| ˜ m i = Θ | m i} with probabilities { p m } and is verifiedwith the projective measurement in the {| ˜ m i} eigenba-sis. The pure state | ˜ ψ ( t ) i analogously obeys Eq. (7),in which the jump operators are replaced by the time-reversed counterparts [76]˜ L k = e − ∆ s k env / Θ L † k Θ † = Θ L k Θ † . (8)At time t = τ , the second projective measurement inthe {| ˜ n i = Θ | n i} eigenbasis is performed. Let ˜Γ = { m, ( k J , τ − t J ) , . . . , ( k , τ − t ) , n } be the time-reversedtrajectory corresponding to Γ, the stochastic entropy pro-duction associated with the trajectory Γ is defined as∆ s tot [Γ] := ln P (Γ)˜ P (˜Γ) = ln p n p m + J X j =1 ∆ s k j env , (9)where P (Γ) and ˜ P (˜Γ) are the probabilities of observingthe trajectories Γ and ˜Γ, respectively. The first term inthe right-hand side of Eq. (9) represents the stochasticchange in the von Neumann entropy, whereas the secondterm corresponds to the stochastic entropy production ofthe reservoirs. Notably, we can prove that the average of∆ s tot is exactly the irreversible entropy production of theoriginal dynamics [Eq. (1)] (see Supplemental Material[78]), Σ τ = h ∆ s tot [Γ] i = D [ P (Γ) || ˜ P (˜Γ)] . (10)Note that the classical relative entropy monotonicallydecreases under information processing. Applying thecoarse-graining operation Λ : Γ n , which leaves onlythe first measurement outcome, to D [ P (Γ) || ˜ P (˜Γ)], weobtain a lower-bound on the entropy production fromEq. (10) as Σ τ ≥ D ( p n || ˜ p n ) , (11)where ˜ p n is the probability to observe the measurementoutcome | ˜ n i at the end of the backward process. Equa-tion (11) can also be derived using an integral fluctuationtheorem (see Supplemental Material [78]). It is crucial tonotice from Eq. (8) that the (inverted) jump operatorsin the backward process are identical with those in theforward process. As a consequence, the density matrixin the backward process right before performing the sec-ond projective measurement can be explicitly expressedas ˜ ρ I ( τ ) = Θ ρ I (2 τ )Θ † . The probability distribution ˜ p ( n )can be calculated as˜ p n = h ˜ n | ˜ ρ I ( τ ) | ˜ n i = h n | U † τ ρ (2 τ ) U τ | n i . (12)Since the inequality (11) holds for an arbitrary time τ > τ ≥ Σ τ/ ), we immediately obtain Eq. (4). Proof of Eq. (5) . —In the single-reservoir case, irre-versible entropy production can be explicitly written asΣ τ = S [ ρ (0) || π ] − S [ ρ ( τ ) || π ] ≥
0. It is worth noting thatthe straight-forward extension of the classical bound,Σ τ ≥ S [ ρ (0) || ρ ( τ )], does not generally hold. Its viola-tion can be intuitively confirmed in the limit of vanishingcoupling to the reservoir [54] and has also been experi-mentally verified for a single-atom system [84]. Since theinteraction-picture Lindblad master equation [Eq. (6)]has no unitary part, one can prove that [48] S [ ρ I (0) || π ] − S [ ρ I ( τ / || π ] ≥ S [ ρ I (0) || ρ I ( τ )] . (13)Due to the invariance of the relative entropy under uni-tary transforms, the term in the left-hand side of Eq. (13)equals Σ τ/ . Since Σ τ ≥ Σ τ/ and S [ ρ I (0) || ρ I ( τ )] = S [ U τ ρ (0) U † τ || ρ ( τ )], a bound on the entropy productioncan be obtained, Σ τ ≥ S [ U τ ρ (0) U † τ || ρ ( τ )]. The depen-dence of this lower-bound on the Hamiltonian can beeliminated by taking the minimum over all unitary oper-ators, yielding the following bound:Σ τ ≥ min V † V = I S [ V ρ (0) V † || ρ ( τ )] . (14)The variational term in the right-hand side of Eq. (14) isexactly S E ( ρ, σ ) (see Supplemental Material [78]). Con-sequently, Eq. (5) is obtained. Conclusion. —In this Letter, we derived the fundamen-tal bound on irreversibility for thermal relaxation pro-cesses of Markovian open quantum systems. The boundrefines the second law of thermodynamics and can beevaluated without knowing details of the underlying dy-namics, thereby, it is applicable to the estimation of irre-versible entropy production. Since thermal relaxation isthe basis for heat engines, our study result is expected tolay the foundations for obtaining useful thermodynamicbounds on relevant physical quantities such as power andefficiency [85].We thank Keiji Saito for many insightful discussionsand for invaluable comments on the manuscript. Thiswork was supported by Ministry of Education, Culture,Sports, Science and Technology (MEXT) KAKENHIGrant No. JP19K12153. ∗ [email protected] † [email protected][1] K. 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Lett. , 040602(2020). upplemental Material for“Lower Bound on Irreversibility in Thermal Relaxation of Open Quantum Systems” Tan Van Vu ∗ and Yoshihiko Hasegawa † Department of Information and Communication Engineering,Graduate School of Information Science and Technology,The University of Tokyo, Tokyo 113-8656, Japan
This supplemental material describes the details of calculations introduced in the main text. The equations andfigure numbers are prefixed with S [e.g., Eq. (S1) or Fig. S1]. Numbers without this prefix [e.g., Eq. (1) or Fig. 1]refer to items in the main text.
Entropy production in time-driven systems cannot be bounded by a universal metric
Here we prove that irreversible entropy production cannot be bounded from below by a system-independent distancebetween the initial and final states. Since it is enough to prove for the classical case, we consider time-dependentMarkov jump processes described by the master equation˙ p n ( t ) = X m ( = n ) [ R nm ( t ) p m ( t ) − R mn ( t ) p n ( t )] , (S1)where R mn ( t ) ≥ n to state m . Let p ( t ) := [ p ( t ) , . . . , p N ( t )] > be the time-dependent probability distribution. With proof by contradiction, we assume that there exists a metric ‘ that is independent on the system parameters and satisfiesΣ τ ≥ ‘ [ p (0) , p ( τ )] (S2)for all Markov jump processes. The irreversible entropy production can be written asΣ τ = 12 Z τ X m = n [ R mn ( t ) p n ( t ) − R nm ( t ) p m ( t )] ln R mn ( t ) p n ( t ) R nm ( t ) p m ( t ) dt. (S3)We consider the nontrivial case: ‘ [ p (0) , p ( τ )] >
0. Let δ > ‘ [ p (0) , p ( τ )] > δ . Wedefine an auxiliary dynamics with transition rates:˜ R mn ( t ) := R mn ( t ) + αp n ( t ) , (S4)where α > p (0) = p (0). Then, one can prove that the probability distribution of the auxiliary dynamics is the same asthe original for all times, i.e., ˜ p ( t ) = p ( t ) ∀ t ≥
0. Specifically, we show that if ˜ p ( t ) = p ( t ) then ˜ p ( t + ∆ t ) = p ( t + ∆ t )for arbitrarily small ∆ t >
0. Indeed,˜ p n ( t + ∆ t ) − ˜ p n ( t ) = ∆ t X m = n h ˜ R nm ( t )˜ p m ( t ) − ˜ R mn ( t )˜ p n ( t ) i (S5a)= ∆ t X m = n h ˜ R nm ( t ) p m ( t ) − ˜ R mn ( t ) p n ( t ) i (S5b)= ∆ t X m = n [ R nm ( t ) p m ( t ) − R mn ( t ) p n ( t )] (S5c)= p n ( t + ∆ t ) − p n ( t ) , (S5d)which implies that ˜ p n ( t + ∆ t ) = p n ( t + ∆ t ) for all n . Namely, the original and auxiliary dynamics have the samedistributions for all times 0 ≤ t ≤ τ . Therefore, the entropy production ˜Σ τ in the auxiliary dynamics is also boundedfrom below by the distance ‘ as ˜Σ τ ≥ ‘ [ p (0) , p ( τ )] > δ. (S6) a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b However, ˜Σ τ = 12 Z τ X m = n [ ˜ R mn ( t )˜ p n ( t ) − ˜ R nm ( t )˜ p m ( t )] ln ˜ R mn ( t )˜ p n ( t )˜ R nm ( t )˜ p m ( t ) dt (S7a)= 12 Z τ X m = n [ R mn ( t ) p n ( t ) − R nm ( t ) p m ( t )] ln R mn ( t ) p n ( t ) + αR nm ( t ) p m ( t ) + α dt (S7b) ≤ α Z τ X m = n [ R mn ( t ) p n ( t ) − R nm ( t ) p m ( t )] dt. (S7c)To obtain Eq. (S7c), we have applied the inequality( x − y ) ln x + zy + z ≤ ( x − y ) z (S8)for x, y, z ≥
0. Now, we choose a sufficiently large value of α such that12 α Z τ X m = n [ R mn ( t ) p n ( t ) − R nm ( t ) p m ( t )] dt < δ ⇔ α > δ Z τ X m = n [ R mn ( t ) p n ( t ) − R nm ( t ) p m ( t )] dt. (S9)Then we obtain the following inequality from Eq. (S7c):˜Σ τ < δ. (S10)This is inconsistent with Eq. (S6), which completes our proof. Derivation of Eq. (6) in the main text
For any operators A and B , one can prove that e − λA Be λA = ∞ X n =0 ( − λ ) n n ! [ A, B ] n , (S11)where the nested commutator is recursively defined as [ A, B ] n = [ A, [ A, B ] n − ] and [ A, B ] = B . From the relation[ L k , H ] = ω k L k , one can readily obtain [ H, L k ] n = ( − ω k ) n L k . (S12)Consequently, e − λH L k e λH = ∞ X n =0 ( − λ ) n ( − ω k ) n n ! L k = e λω k L k ⇒ e − λH L k = e λω k L k e − λH . (S13)Now, taking the time derivative of ρ I ( t ) = e iHt ρ ( t ) e − iHt , we can derive Eq. (6) as ∂ t ρ I ( t ) = e iHt i [ H, ρ ( t )] e − iHt + e iHt ∂ t ρ ( t ) e − iHt (S14a)= X k e iHt (cid:20) L k ρ ( t ) L † k − { L † k L k , ρ ( t ) } (cid:21) e − iHt (S14b)= X k (cid:20) L k e iHt ρ ( t ) e − iHt L † k − { L † k L k , e iHt ρ ( t ) e − iHt } (cid:21) (S14c)= X k (cid:20) L k ρ I ( t ) L † k − { L † k L k , ρ I ( t ) } (cid:21) (S14d)= X k D k [ ρ I ( t )] . (S14e)Here, we have used the following relations: e iHt L k = e − iω k t L k e iHt , (S15) L † k e − iHt = e iω k t e − iHt L † k , (S16)[ L † k L k , H ] = 0 . (S17) Quantum trajectories and entropy production
Here we show that the irreversible entropy production Σ τ can be mapped to the stochastic entropy production onthe level of individual trajectories. First, Eq. (6) can be rewritten as ∂ t ρ I ( t ) = − i [ H eff ρ I ( t ) − ρ I ( t ) H † eff ] + X k L k ρ I ( t ) L † k , (S18)where H eff := − ( i/ P k L † k L k is the skew-Hermitian effective Hamiltonian (i.e., H † eff = − H eff ). In the forwardprocess, the evolution of the pure state between jumps is described by the deterministic equation ddt | ψ ( t ) i = S [ | ψ ( t ) i ] . (S19)The formal solution of Eq. (S19) gives the state at time t ( > s ) as | ψ ( s ) i 7→ | ψ ( t ) i = U eff ( t, s ) | ψ ( s ) i q h U eff ( t, s ) † U eff ( t, s ) i ψ ( s ) , (S20)where the effective time-evolution operator U eff ( t, s ) = e − iH eff ( t − s ) is the solution of the differential equation ∂ t U eff ( t, s ) = − iH eff U eff ( t, s ) , (S21)with the initial condition U eff ( s, s ) = I . The smooth evolution of the pure state is interrupted by sudden jumps, whichalter the state as | ψ ( t ) i 7→ L k | ψ ( t ) i q h L † k L k i ψ ( t ) . (S22)This discontinuous change is induced by the jump operator J k := L k √ dt . Given the stochastic trajectory Γ = { n, ( k , t ) , . . . , ( k J , t J ) , m } , the probability to observe Γ is encoded into the unnormalized state | ψ τ (Γ) i = | m ih m | U eff ( τ, t J ) J k J . . . J k U eff ( t , | n i (S23a)= | m ih m |P (Γ) | n i , (S23b)where P (Γ) := U eff ( τ, t J ) J k J . . . J k U eff ( t ,
0) is the forward propagator. In other words, given that the initial stateis | n i , the probability of observing Γ is the norm of | ψ τ (Γ) i , P (Γ | n ) = h ψ τ (Γ) | ψ τ (Γ) i = |h m |P (Γ) | n i| . (S24)With the backward process defined as in the main text, the effective Hamiltonian is ˜ H eff = − ( i/ P k ˜ L † k ˜ L k =Θ( i/ P k L † k L k Θ † = − Θ H eff Θ † and the corresponding effective time-evolution operator is ˜ U eff ( t, s ) = e − i ˜ H eff ( t − s ) .In addition, the backward jump operators are ˜ J k := ˜ L k √ dt = Θ J k Θ † . (S25)Analogously, the probability of observing the time-reversed trajectory ˜Γ = { m, ( k J , τ − t J ) , . . . , ( k , τ − t ) , n } in thebackward process is encoded into the unnormalized state | ˜ ψ τ (˜Γ) i = | ˜ n ih ˜ n | ˜ U eff ( τ, τ − t ) ˜ J k . . . ˜ J k J ˜ U eff ( τ − t J , | ˜ m i (S26a)= | ˜ n ih ˜ n | ˜ P (˜Γ) | ˜ m i , (S26b)where ˜ P (˜Γ) := ˜ U eff ( τ, τ − t ) ˜ J k . . . ˜ J k J ˜ U eff ( τ − t J ,
0) the backward propagator. The probability to observe ˜Γ giventhat the initial state is | ˜ m i is the norm of | ˜ ψ τ (˜Γ) i ,˜ P (˜Γ | ˜ m ) = h ˜ ψ τ (˜Γ) | ˜ ψ τ (˜Γ) i = |h ˜ n | ˜ P (˜Γ) | ˜ m i| . (S27)It can be confirmed thatΘ † ˜ U † eff ( τ − s, τ − t )Θ = Θ † e i ˜ H † eff ( t − s ) Θ = Θ † e i Θ H eff Θ † ( t − s ) Θ = e − iH eff ( t − s ) = U eff ( t, s ) . (S28)Consequently, the propagators in the forward and backward processes can be related as P (Γ) = Θ † ˜ P † (˜Γ)Θ e P Jj =1 ∆ s kj env / ⇒ h m |P (Γ) | n i = h ˜ n | ˜ P (˜Γ) | ˜ m i ∗ e P Jj =1 ∆ s kj env / . (S29)From the relations P (Γ) = P (Γ | n ) p n and ˜ P (˜Γ) = ˜ P (˜Γ | ˜ m ) p m , we immediately have∆ s tot = ln P (Γ)˜ P (˜Γ) = ln p n p m + J X j =1 ∆ s k j env . (S30)Finally, we show that the ensemble average of ∆ s tot is equal to the irreversible entropy production Σ τ . Noting that ρ I ( t ) = e iHt ρ ( t ) e − iHt and the von Neumann entropy is invariant under unitary transforms, we have h ∆ s tot i = h− ln p m + ln p n + J X j =1 ∆ s k j env i (S31a)= − X m p m ln p m + X n p n ln p n + Z τ X k tr n L † k L k ρ I ( t ) o ∆ s k env dt (S31b)= − tr { ρ I ( τ ) ln ρ I ( τ ) } + tr { ρ I (0) ln ρ I (0) } + ∆ S env (S31c)= − tr { ρ ( τ ) ln ρ ( τ ) } + tr { ρ (0) ln ρ (0) } + ∆ S env (S31d)= ∆ S sys + ∆ S env (S31e)= Σ τ . (S31f) Derivation of Eq. (11) based on an integral fluctuation theorem
Here we derive Eq. (11) using a fluctuation theorem. To this end, we prove the following equality: (cid:28) exp (cid:20) − ∆ s tot + ln p n ˜ p n (cid:21)(cid:29) = 1 , (S32)where the average is taken with respect to distribution P (Γ). Indeed, plugging ∆ s tot = ln[ P (Γ) / ˜ P (˜Γ)], we have (cid:28) exp (cid:20) − ∆ s tot + ln p n ˜ p n (cid:21)(cid:29) = (cid:28) exp (cid:20) − ln P (Γ | n ) p n ˜ P (˜Γ | ˜ m ) p m + ln p n ˜ p n (cid:21)(cid:29) (S33a)= X Γ P (Γ | n ) p n ˜ P (˜Γ | ˜ m ) p m P (Γ | n )˜ p n (S33b)= X n p n ˜ p n X Γ | n ˜ P (˜Γ | ˜ m ) p m (S33c)= X n p n ˜ p n ˜ p n (S33d)= X n p n (S33e)= 1 . (S33f)To obtain Eq. (S33d), we have used the fact P Γ | n ˜ P (˜Γ | ˜ m ) p m = ˜ p n . Applying Jensen’s inequality to Eq. (S32), weobtain Σ τ = h ∆ s tot i ≥ (cid:28) ln p n ˜ p n (cid:29) = D ( p n || ˜ p n ) . (S34) Proof of the equality S E ( ρ, σ ) = min V † V = I S ( V ρV † || σ ) Here we prove that S E ( ρ, σ ) = min V † V = I S ( V ρV † || σ ). Note that S E ( ρ, σ ) = D ( a n || b n ), where { a n } n and { b n } n are the increasing eigenvalues of ρ and σ . To this end, we will show that min V † V = I S ( V ρV † || σ ) ≤ D ( a n || b n ) andmin V † V = I S ( V ρV † || σ ) ≥ D ( a n || b n ). First, we prove the former. Let ρ = P n a n | a n ih a n | and σ = P n b n | b n ih b n | be thespectral decompositions of ρ and σ , respectively. Noticing that the matrix V = P n | b n ih a n | is a unitary matrix, wereadily obtain min V † V = I S ( V ρV † || σ ) ≤ S ( V ρV † || σ ) = S ( X n a n | b n ih b n | || X n b n | b n ih b n | ) = D ( a n || b n ) . (S35)Next, we prove the later. For arbitrary unitary matrix V , we have S ( V ρV † || σ ) = tr { ρ ln ρ } − tr (cid:8) V ρV † ln σ (cid:9) (S36a)= X n a n ln a n − X n,m a n ln b m |h b m | V | a n i| (S36b)= X n a n ln a n − X n,m c nm a n ln b m , (S36c)where c nm := |h b m | V | a n i| ≥
0. Note that P m c nm = P n c nm = 1.Before proceed further, we prove the following Lemma. Lemma 1.
Let ≤ a ≤ · · · ≤ a N and ≤ b ≤ · · · ≤ b N be two arrays of nonnegative numbers and C = [ c nm ] ∈ R N × N be a doubly stochastic matrix (i.e., c nm ≥ and P m c nm = P n c nm = 1 ). Then, X n,m c nm a n ln b m ≤ X n a n ln b n . (S37) Proof.
We prove by induction. The N = 1 case is trivial. Assume that the result holds for N = k − k ≥
2, weshow it also holds for N = k . For convenience, we define F ( C ) := P kn =1 P km =1 c nm a n ln b m as the function associatedwith the matrix C . We need only prove F ( C ) ≤ P kn =1 a n ln b n . If c kk < ≤ i, j ≤ k − c ik > c kj >
0. Set (cid:15) = min( c ik , c kj ) and note that a i ln b j + a k ln b k ≥ a i ln b k + a k ln b j [ ∵ ( a k − a i )(ln b k − ln b j ) ≥ . (S38)Define a new matrix C = [ c nm ] ∈ R k × k as c kk = c kk + (cid:15), c ij = c ij + (cid:15), c ik = c ik − (cid:15), c kj = c kj − (cid:15), (S39)and c nm = c nm otherwise. As can be seen, all elements of C are nonnegative and satisfy P m c nm = P n c nm = 1. Thisimplies that the above procedure generates a new doubly stochastic matrix that yields a larger value of the function F , i.e., P n,m c nm a n ln b m = F ( C ) ≤ F ( C ) = P n,m c nm a n ln b m . Repeating this procedure a finite number of times,we eventually obtain a doubly stochastic matrix C fin with c kk = 1. Then, c kn = c nk = 0 for all n = 1 , . . . , k − C sub = [ c nm ] ∈ R ( k − × ( k − , which is obtained by eliminating the k th row and the k th column of C fin ,is also a doubly stochastic matrix; thus, F ( C sub ) ≤ P k − n =1 a n ln b n . Consequently, F ( C ) ≤ F ( C fin ) = F ( C sub ) + a k ln b k ≤ k X n =1 a n ln b n , (S40)which completes our proof.According to Lemma 1, we have P n,m c nm a n ln b m ≤ P n a n ln b n . Consequently, Eq. (S36c) implies S ( V ρV † || σ ) ≥ X n a n ln a n − X n a n ln b n = D ( a n || b n ) . (S41)Since Eq. (S41) holds for an arbitrary unitary matrix V , we obtain min V † V = I S ( V ρV † || σ ) ≥ D ( a n || b n ). Another proof of Eq. (5) for a two-level atom system
We consider the thermal relaxation process of a two-level atom, which is weakly coupled to a thermal reservoir.The Hamiltonian of the system is H = ωσ z /
2. The time evolution of the density matrix obeys the Lindblad equation: ∂ t ρ = − i [ H, ρ ] + γ ¯ n ( ω )( σ + ρσ − − { σ − σ + , ρ } ) + γ (¯ n ( ω ) + 1)( σ − ρσ + − { σ + σ − , ρ } ) , (S42)where σ ± = ( σ x ± iσ y ) / γ is a positive damping rate, and ¯ n ( ω ) = ( e βω − − is the Planck distribution. Thedensity operator ρ ( t ) during the relaxation process is analytically solvable and the irreversible entropy production canbe explicitly evaluated as Σ τ = S [ ρ (0) || π ] − S [ ρ ( τ ) || π ], where τ denotes the process time.The density matrix can be represented using the Bloch sphere ρ ( t ) = 12 [ I + r ( t ) · σ ] , (S43)where r ( t ) := [ r x ( t ) , r y ( t ) , r z ( t )] > is the Bloch vector and σ := [ σ x , σ y , σ z ] > denotes the vector of Pauli matrices. Notethat r ( t ) := r x ( t ) + r y ( t ) + r z ( t ) ≤
1. The density matrix can be explicitly calculated as r x ( t ) = e − γτ/ [ r x (0) cos( ωτ ) − r y (0) sin( ωτ )] , (S44) r y ( t ) = e − γτ/ [ r x (0) sin( ωτ ) + r y (0) cos( ωτ )] , (S45) r z ( t ) = r z (0) e − γτ + tanh( βω/ e − γτ − , (S46)where γ := γ coth( βω/ ρ ( t ) are [1 ± r ( t )] /
2, the von Neumman entropy S ( t ) = − tr { ρ ( t ) ln ρ ( t ) } can be expressed in terms of the magnitude of the Bloch vector as S ( t ) = − − r ( t )2 ln 1 − r ( t )2 − r ( t )2 ln 1 + r ( t )2 . (S47)The irreversible entropy production can be written asΣ τ = S ( τ ) − S (0) + β tr { H [ ρ (0) − ρ ( τ )] } (S48a)= S ( τ ) − S (0) + βω [ r z (0) − r z ( τ )] / . (S48b)In what follows, we prove the inequality [Eq. (5)]:Σ τ ≥ D [ p n (0) || p n ( τ )] , (S49)where { p n ( t ) } Nn =1 are increasing eigenvalues of ρ ( t ).Equation (S49) is equivalent to βω [ r z (0) − r z ( τ )] ≥ [ r ( τ ) − r (0)] ln 1 + r ( τ )1 − r ( τ ) . (S50)For convenience, we set a := e − γτ ∈ (0 , r eq := tanh( βω/ ∈ (0 , κ := r x (0) + r y (0) ≥
0. Then, r (0) = p κ + r z (0) and r ( τ ) = p aκ + [ ar z (0) + ( a − r eq ] . To prove Eq. (S50), we divide into two cases: r ( τ ) ≤ r (0) and r ( τ ) > r (0).(a) r ( τ ) ≤ r (0): Setting f ( κ ) := [ r ( τ ) − r (0)] ln { [1 + r ( τ )] / [1 − r ( τ )] } as a function of κ , Eq. (S50) can be rewrittenas f ( κ ) ≤ βω [ r z (0) − r z ( τ )] . (S51)Since f ( κ ) ≤
0, we need only consider the r z (0) < r z ( τ ) case. To prove Eq. (S51), we first prove that f ( κ ) is adecreasing function. Specifically, we show that df ( κ ) /dκ ≤
0. Taking the derivative of f ( κ ) with respect to κ , oneobtains ddκ f ( κ ) = ar (0) − r ( τ ) r (0) ln 1 + r ( τ )1 − r ( τ ) + 2 a [ r ( τ ) − r (0)]1 − r ( τ ) . (S52)If ar (0) − r ( τ ) ≤
0, then it is obvious that df ( κ ) /dκ ≤
0. If ar (0) − r ( τ ) >
0, then applying the inequalityln 1 + r ( τ )1 − r ( τ ) ≤ r ( τ )1 − r ( τ ) (S53)to Eq. (S52), we have ddκ f ( κ ) ≤ r ( τ )[ ar (0) − r ( τ )] r (0)[1 − r ( τ ) ] + 2 a [ r ( τ ) − r (0)]1 − r ( τ ) (S54a)= 2 r ( τ )[ ar (0) − r ( τ )] + 2 ar (0)[ r ( τ ) − r (0)] r (0)[1 − r ( τ ) ] . (S54b)Since r ( τ ) ≤ r (0) and ar (0) − r ( τ ) ≤ a [ r (0) − r ( τ )], one readily obtains 2 r ( τ )[ ar (0) − r ( τ )] + 2 ar (0)[ r ( τ ) − r (0)] ≤ df ( κ ) /dκ ≤ r z (0) < ar z (0) + ( a − r eq = r z ( τ ), one obtains r z (0) + r eq <
0. In addition, from r (0) = κ + r z (0) ≥ aκ + [ ar z (0) + ( a − r eq ] = r ( τ ) , one can derive κ ≥ max { , [ r eq + r z (0)][(1 − a ) r eq − (1 + a ) r z (0)] } = 0 . (S55)Since f ( κ ) is a decreasing function, we have f ( κ ) ≤ f (0), which is equivalent to f ( κ ) ≤ [ | r z ( τ ) | − | r z (0) | ] ln 1 + | r z ( τ ) | − | r z ( τ ) | (S56a)= [ r z (0) − r z ( τ )] ln 1 + | r z ( τ ) | − | r z ( τ ) | . (S56b)Here, we have used the facts that r z (0) < r z ( τ ) = ar z (0)+( a − r eq ≤
0. Since | r z ( τ ) | = (1 − a ) r eq − ar z (0) ≥ r eq and ln[(1 + x ) / (1 − x )] is an increasing function of x ∈ [0 , f ( κ ) ≤ [ r z (0) − r z ( τ )] ln 1 + r eq − r eq = βω [ r z (0) − r z ( τ )] , (S57)which is exactly Eq. (S51).(b) r ( τ ) > r (0): The condition r ( τ ) > r (0) is equivalent to0 ≤ κ ≤ ( r z (0) + r eq )[(1 − a ) r eq − ( a + 1) r z (0)] , (S58)from which one can derive r z (0) + r eq ≥
0. Consequently, r z (0) − r z ( τ ) = (1 − a )[ r z (0) + r eq ] ≥
0. Therefore, Eq. (S50)can be rewritten as βω ≥ r ( τ ) − r (0) r z (0) − r z ( τ ) ln 1 + r ( τ )1 − r ( τ ) =: g ( a ) . (S59)To prove this, we first show that g ( a ) is a decreasing function with respect to a . Specifically, we prove that g ( a ) :=ln { [1 + r ( τ )] / [1 − r ( τ )] } and g ( a ) := [ r ( τ ) − r (0)] / [ r z (0) − r z ( τ )] are decreasing functions with respect to a . Notingthat dda r ( τ ) = κ + 2[ ar z (0) + ( a − r eq ][ r z (0) + r eq ]2 r ( τ ) (S60a) ≤ ( a − r z (0) + r eq ] r ( τ ) ≤ . (S60b)The monotonicity of g ( a ) can be verified as dda g ( a ) = 21 − r ( τ ) dr ( τ ) da ≤ . (S61)Taking the derivative of g ( a ) with respect to a , we have dda g ( a ) = c − r (0) r ( τ )2(1 − a ) [ r eq + r z (0)] r ( τ ) , (S62) Time Entropy production
FIG. S1. Numerical illustration of Eq. (5) in the two-level atom. (a) The irreversible entropy production Σ τ and the lower-bound S E are plotted as functions of time τ . Parameters are fixed as ω = 0 . β = 0 . γ = 0 .
01, and the initial density matrixis ρ (0) = ( I − . σ z ) /
2. (b) Random verification of the bound. The dashed line depicts the upper-bound of the ratio S E / Σ τ .Each circle represents the ratio S E / Σ τ calculated with a random initial density matrix and τ ∈ [10 , ]. where, c := ( a + 1) κ + 2 r z (0)[ ar z (0) + ( a − r eq ]. If c ≤ r (0) r ( τ ), then dg ( a ) /da ≤ c > r (0) r ( τ ), then we have dda g ( a ) = c − r (0) r ( τ ) − a ) r ( τ )[ c + 2 r (0) r ( τ )][ r eq + r z (0)] (S63a)= κ ( κ − r eq [ r eq + r z (0)])2 r ( τ )[ c + 2 r (0) r ( τ )][ r eq + r z (0)] . (S63b)Since κ ≤ [ r z (0) + r eq ][(1 − a ) r eq − ( a + 1) r z (0)] and r eq + r z (0) ≥
0, one can easily derive that κ ≤ r eq [ r eq + r z (0)].Consequently, we obtain dg ( a ) /da ≤ a as0 ≤ a ≤ r eq − r z (0) r eq + r z (0) − κ [ r eq + r z (0)] . (S64)Since g ( a ) is a decreasing function, we obtain g ( a ) ≤ g (0), which is equivalent to r ( τ ) − r (0) r z (0) − r z ( τ ) ln 1 + r ( τ )1 − r ( τ ) ≤ r eq − r (0) r z (0) + r eq ln 1 + r eq − r eq (S65a) ≤ ln 1 + r eq − r eq = βω. (S65b)Here, we have used the fact that r eq − r (0) ≤ r z (0) + r eqeq