Efficiency Versus Speed in Quantum Heat Engines: Rigorous Constraint

Eﬃciency versus speed in quantum heat engines: Rigorous constraint from Lieb-Robinson bound

Naoto Shiraishi Department of Physics, Keio University, 3-14-1 Hiyoshi, Yokohama 223-8522, Japan

Hiroyasu Tajima Center for Emergent Matter Science (CEMS), RIKEN, 2-1 Hirosawa, Wako, 351-0198 Japan (Dated: August 18, 2017) A long standing open problem whether a heat engine with finite power achieves the Carnot efficiency is investigated. We rigorously prove a general trade-off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines. In a first step, employing the Lieb-Robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable. This inequality provides a rigorous character- ization of the following intuitive picture that most of the energy emitted from the engine to the cold bath remains near the engine when the cyclic process is finished. Using the above description, we finally prove an upper bound on efficiency with the aid of quantum information geometry. Our result generally excludes the possibility of a process with finite speed at the Carnot efficiency in quantum heat engines. In particular, the obtained constraint covers engines evolving with non-Markovian dynamics, which almost all previous studies on this topics fail to address. PACS numbers: 03.67.-a, 05.30.-d, 05.70.Ln, 87.10.Ca,

I. INTRODUCTION the relation between finite power and the Carnot efficiency. One frequently-used setup to tackle the problem Around 200 years ago, Carnot revealed that thermody- is a mesoscopic conducting system with non-interacting namic efficiency in a cyclic process with two heat baths electrons under a magnetic field, where the incompati- with inverse temperatures βH and βL (βH < βL) is bility between finite power and the Carnot efficiency is bounded by a universal function of these two temper- shown through deriving a novel restriction on the On- atures, which was turned to be the celebrated Carnot sager matrix [22–27]. Owing to the newly-derived restric- efficiency [1–3] tion, it is shown both theoretically [23, 27] and numeri- cally [25, 26] that this system never attains the Carnot ef- βH ficiency at finite power in the linear response regime. An- η = 1 − . (1) C βL other frequently-used setup is a periodically driven system, for which the Onsager matrix can be defined and is At the same time, Carnot also showed that infinitely in general asymmetric. Utilizing the detailed information slow processes (i.e., zero power) realize the maximum ef- on microscopic dynamics, Refs. [28, 29] and Refs. [30, 31] ficiency. An infinitely slow heat engine is however useless respectively demonstrate the incompatibility for under- in practical sense, and thus a question naturally arises damped Langevin particles with two heat baths and for whether finite power engines attain the Carnot efficiency, isothermal driven systems. On the other hand, some re- which is the opposite statement of the aforementioned searches proposed ideas to realize these two simultane- one. Contrary to its apparent triviality, this has still ously [32–36] (some comments on these studies are seen been an open problem in spite of enormous effort for in- in Ref. [37]). Recently, inspired by the idea of partial en- vestigation of large power [4–6], high efficiency [7–15], tropy production [38–40], one of the authors has shown and both of them [16–19]. We emphasize that although this incompatibility for general classical Markovian heat the question seems to be negative by intuition, it is hard engines beyond the linear response regime [41]. However, to prove this rigorously in general setups. In fact, con- non-Markovian heat engines have not been addressed in ventional thermodynamics provides no restriction on the the previous works though real experimental heat engines speed of processes, and even in the linear response regime are inevitably non-Markovian. On the basis of the re- the linear irreversible thermodynamics neither prohibits cent experimental development of small quantum heat arXiv:1701.01914v2 [cond-mat.stat-mech] 17 Aug 2017 engines at the Carnot efficiency with finite power if time- engines [42–45], a general result covering non-Markovian reversal symmetry is broken [20]. These findings eluci- heat engines is highly desired. date the fact that the connection between heat exchange and dissipation is, surprisingly, not established even in In this paper, we establish a trade-off relation between the linear response regime when the framework of en- efficiency and speed of operation, which leads to this no- doreversible thermodynamics [21] no longer holds. go theorem for general quantum heat engines. We fo- The latter work attracted renewed interest in this prob- cus on the fact that most of the energy emitted from lem, and many researches are devoted to analyze specific the engine to the cold heat bath remains in the region models to find clues for capturing general properties on close to the engine, which leads to finite dissipation in 2 the bath. To prove this intuitive picture rigorously, we employ the Lieb-Robinson bound [46–51], which claims that a commutator of two observables with different time acting on different regions far from each other cannot be large. The Lieb-Robinson bound has recently used in various problems including problems on structure of a We cannot control We can gapped ground state [49, 52, 53], the Lieb-Shultz-Mattis here. control here. theorem in higher dimensions [47], the quantum Hall effect [54], and thermalization [55, 56]. Using this bound, we establish a useful upper bound on the change in a local observable caused by an operation with finite time- interval far from the area on which the local observable acts. Applying this bound, we find that the expectation value of energy in the cold bath far from the engine does not largely change. With help of the Pythagorean theo- FIG. 1. Schematic of the total system. Three subsets of rem for quantum relative entropy and some relations on lattices ΛE,ΛH,ΛL correspond to the regions of the engine the quantum Fisher information [57, 58], we finally arrive (yellow), the hot bath (orange), and the cold bath (pink). The subset Λ is a composite set of Λ and its nearest-neighbor at an upper bound for efficiency with the time interval E˜ E sites. A time-dependent Hamiltonian acts only on Λ . of a cyclic process. This bound clearly manifests that E the faster the engine is operated the less the maximum efficiency becomes. is why many rigorous results on thermodynamic proper- This paper is organized as follows. In Sec. II, we show ties of nonequilibrium dynamics are studied in such sys- the setup of quantum heat engines and state our main tems [55, 56, 59–62]. We remark that recent sophisticated inequality on efficiency and the time interval of a cyclic techniques indeed realize such lattice systems experimen- process. Before going to the derivation, we sketch the tally [63–65]. outline of the proof in Sec. III. The derivation is con- We in particular consider a composite system of an structed with two part. The former part is discussed in engine E and two heat baths H and L with inverse tem- H L H L Sec. IV, where we first introduce a useful lemma which peratures β and β (β < β ) (see Fig. 1). Let ΛE,ΛH, bounds the effect of time-dependent operation, and using ΛL be subsets of the sites in a region corresponding to the this we rigorously show the lower bound for the energy engine, the hot bath, and the cold bath, respectively. ΛE increase in a region near the engine. The latter part is is attached to both ΛH and ΛL, and ΛH is not attached discussed in Sec. V, where we connect the relative en- to ΛL. We also define ΛE˜ as a set of sites in ΛE and its tropy to the energy increase, which leads to the desired nearest-neighbor ones. For convenience, we assume that inequality. To examine how the obtained trade-off rela- the size of the engine |ΛE˜ | is finite. The treatment of tion works, in Sec. VI we demonstrate our inequality in thermodynamic limit is briefly discussed in Sec. VIII. a simple concrete example. In Sec. VII, we apply our We now describe the dynamics of the system. We sup- analysis to a transient process and derive an extended pose that baths are not driven, and interact only with the version of the principle of maximum work. engine. This is because our interest is on engines satisfy- We note that the obtained inequality turns to be no ing conventional thermodynamics, where a bath itself is more than the second law of thermodynamics in the an isolated system, not externally-driven [66]. If a bath Markovian limit, in which the Lieb-Robinson velocity is driven, the system may violate the zeroth law and the diverges. Therefore, we treat Markovian heat engines second law of thermodynamics [67, 68]. Therefore, we in a completely different way. We extend the result of do not consider externally-driven heat baths, and safely classical Markov processes shown in Ref. [41] to quan- assume that the Hamiltonian of the composite system tum Markov processes, which denies the compatibility Htot(t) is short-range interaction. For simplicity, we con- between finite power and the Carnot efficiency. In Ap- sider the case of the sum of one-body Hamiltonians and pendix.G, we derive an inequality on efficiency and power nearest-neighbor two-body interaction Hamiltonians in for quantum Markovian heat engines. the main part. The case with general Hamiltonians with short-range interaction is analyzed in Appendix.F. In a cyclic process in 0 ≤ t ≤ τ, the Hamiltonian is changed II. SETUP AND MAIN RESULT with time as satisfying Htot(0) = Htot(τ). Suppose that the Hamiltonian is time-dependent only on ΛE˜ . We then We first describe our setup of heat engines. In the most decompose the total Hamiltonian Htot(t) into five parts part of this paper, we restrict our attention to a quantum as lattice system. A lattice system is known as a manage- E EH EL H L Htot(t) = H (t) + H (t) + H (t) + H + H , (2) able stage to investigate rigorous thermodynamic properties of nonequilibrium dynamics, and in most cases lattice where HX (X = E, H, L) acts only on X, and HEX (X = systems keep universal thermodynamic properties. This H, L) is the sum of all interaction Hamiltonians between 3 a site in ΛE and that in ΛX . The Hamiltonians of the Engine E baths are set to be time-independent because we consider the situation that the external operation acts only on the engine and the bath is not driven as explained above. R We also suppose that the engine is initially separated Cold bath L from baths: HEH(0) = HEL(0) = 0, which is a widely- used setup in the context of nonequilibrium statistical t=0 t=τ mechanics [15, 69–71]. This setup corresponds to the situation that the initial state of the engine and that FIG. 2. The key fact to derive the bound on efficiency. Most of the baths (i.e., canonical distributions) are prepared of the heat emitted from the engine to the cold bath does not escape from the region close to the engine by the end of the independently. We do not impose any restriction on the operation t = τ. Hamiltonian except the aforementioned ones. We denote the density matrix of the total system at time t by ρ (t), and write ρ := ρ (0) and ρ := ρ (τ). tot i tot f tot corresponds to the maximum speed of information prop- The partial trace of ρ to X (X = E, H, L) is de- agation in the bath L, which is sometimes called a coun- noted by ρX . Using the canonical distribution of baths X −βX HX X terpart of the speed of light in non-relativistic quantum X expressed as ρcan := e /Z (X =H,L) with theory. C is a correction term depending on the details of X −βX HX Z := Tr[e ], we set the initial state as a product the bath and the engine, which is irrelevant for the case E H L E state ρi = ρi ⊗ ρcan ⊗ ρcan, where ρi is arbitrary. The of large τ. In this case, the obtained bound is roughly cyclicity of the process requires that the final and the regarded as initial state of the engine have the same energy expec- E E E E E E tation value Tr[H (τ)ρf ] = Tr[H (0)ρf ] = Tr[H (0)ρi ] b E E and the same von Neumann entropy S(ρ ) = S(ρ ) [72]. η ≤ ηC − D . (8) f i (2vLRτ) We remark that if the above conditions are satisfied, the initial and final density matrices can be different. The The precise definition of these constants are given in law of energy conservation tells that the extracted work Sec. IV, and a concrete example is demonstrated in W is given by Sec. VI. The inequality (6) (or (8)) exhibits the existence of a trade-off between speed and efficiency, and in partic- W := Tr[H (0)ρ (0)] − Tr[H (τ)ρ (τ)]. (3) tot tot tot tot ular shows that a finite power heat engine (i.e., τ < ∞) The heat released from the bath H and that absorbed by never attains the Carnot efficiency as long as the coeffi- the bath L are respectively written as cients are finite. We finally confirm the finiteness of coefficients. First, H H H H the Lieb-Robinson velocity is finite (v < +∞) if the Q := Tr[H (ρi − ρf )], (4) LR operator norm of local Hamiltonians of the bath is fi- QL := Tr[HL(ρL − ρL)]. (5) f i nite. The case of Markovian limit, where vLR diverges, is considered in Appendix. G with a completely different Since there is no interaction between the engine and the approach. Second, j is finite if the heat capacity of the baths in the initial and final states, the above defini- max bath is finite. Thus, if the bath is not at the first-order tions of work and heat contain no ambiguity. We note phase transition point, j is finite. Other quantities that although our setup and result are described in terms max (QL,QH, βL) are obviously finite. We thus conclude that of externally-operated heat engines, the same inequality if the operator norm of the Hamiltonian of the bath is holds for an autonomous evolution with a work storage, a finite and the baths are not at the first-order phase tran- catalyst, and a clock, which is frequently discussed in the sition point, Eqs. (6) and (8) are indeed stronger than the field of quantum thermodynamics [15, 71, 73–87] (details Carnot bound. Although, the difference from the Carnot are demonstrated in Appendix. A). bound might be small, it is indeed finite and the approach We now state our main result. In the above setup, the to the Carnot efficiency is fundamentally prohibited. efficiency η := W/QH is bounded by

b η ≤ ηC − D (6) (2vLRτ + C) III. OUTLINE OF THE PROOF with Our starting point to prove (6) is an expression of eﬃ- (QL)2 ciency with quantum relative entropy [88]. Since the von b := L H . (7) Neumann entropy S(ρ) := −Tr[ρ ln ρ] is invariant under 8β Q jmax unitary evolution, we rewrite the quantum relative en- Here, D is the spatial dimension, and jmax, vLR are pos- tropy between ρf and ρi as itive constants. Roughly speaking, jmax corresponds to the heat capacity of the bath per unit volume and vLR D(ρf ||ρi) := Tr[ρf (ln ρf − ln ρi)] = Tr[(ρi − ρf ) ln ρi]. (9) 4

A simple calculation yields the following inequality: X X Tr[(ρi − ρf ) ln ρi] H H H H L L L L = − β Tr[(ρi − ρf )H ] − β Tr[(ρi − ρf )H ] E E − D(ρf ||ρi ) ≤ − βHQH + βLQL, (10) Y Y where we used the cyclic property S(ρE) = S(ρE) and f i small t large t nonnegativity of relative entropy. Combining these two relations, we obtain a useful relation on efficiency: FIG. 3. Schematics of the Lieb-Robinson bound. The effect βH D(ρ ||ρ ) in the region Y cannot be detected in the region X far from η ≤1 − − f i . (11) βL βLQH Y unless we wait sufficiently long time. We remark that the quantum fluctuation theorem [69] leads to the expression of entropy production with rela- information corresponds to the heat capacity. jmax in tive entropy, which also implies the above relation. Our Eq. (6) is the maximum of the Fisher information nor- problem is thus how to evaluate the above relative en- malized by the volume. tropy in terms of microscopic description of the system. The evaluation is accomplished in two steps. In the first step, we use the fact that most of the emitted energy IV. STEP 1: BOUND ON ENERGY CHANGE OF BATH IN THE VICINITY OF ENGINE to the bath L remains in the region L close to the engine E (see Fig. 2). To state it rigorously, let L0 be a region A. Lieb-Robinson bound in L whose distance from the engine ΛE˜ is less than R. Then, the precise statement is that the energy increase in L0 is bounded below by QL/2 with R ≥ R∗, where R∗ Before going to the proof of the bound, we explain D is given in an explicit form. The quantity (2vLRτ + C) the celebrated Lieb-Robinson bound for spin or fermionic in Eq. (6) corresponds to the upper bound of the volume systems [46, 49]. (The case of bosonic systems is investi- of L0. gated in Refs. [50, 51].) The Lieb-Robinson bound pro- In the second step, we calculate the relative entropy vides a restriction on the commutator between two oper- D(ρf ||ρi). We here provide an intuitive idea of calculation ators acting on different space-time points (A schematic instead of a rigorous proof, which is shown in Sec. V. is seen in Fig. 3). This bound implies that two distant First, using the monotonicity of the relative entropy [88], points with short time difference have almost zero corre- L0 L0 lation. This is why some people say the Lieb-Robinson we have D(ρf ||ρi) & D(ρf ||ρi ). Second, we regard that 0 L0 bound as the non-relativistic counterpart of causality in the initial state of L is the canonical distribution: ρi ' L0 the relativistic theory. ρcan. We here introduce some quantities related to the lat- We now recall a property of relative entropy with a tice. As following Ref. [49], we suppose that every bonds canonical distribution. Let ρcan be a canonical distribu- 0 have their length 1 for simplicity. We then define a dis- tion with inverse temperature β and ρ be an arbitrary tance between two sites x and y denoted by d(x, y) as density matrix, whose expectation values of energy are 0 the length of the minimum path from x to y. We next respectively denoted by E and E . Then, the relative introduce the sphere with its center x and its radius r as entropy is evaluated as S(r; x) := {y|d(x, y) = r}. (13) 0 2 0 0 (E − E ) D(ρ ||ρcan) ≥ D(ρcan||ρcan) ≥ , (12) Using this, we define the dimension of the lattice D as 2Cmax the minimum number satisfying the following condition 0 where ρcan is a canonical distribution with inverse tem- for all x and r with a constant K: perature β0 whose expectation value of energy is E0, and |S(r; x)| ≤ KrD−1. (14) Cmax is the maximum heat capacity of the system with 0 inverse temperature between β and β . In the first in- Suppose a quantum spin or fermionic system on a dis- equality, we used the Jaynes’s maximum entropy princi- crete lattice. Its Hamiltonian is written as the sum of ple [89] that under a given energy expectation value E0 P the time-independent local Hamiltonians: H = hZ , the von Neumann entropy is maximized for the canonical Z where Z runs all finite subsets of sites and hZ is a local distribution. Hamiltonian whose support is Z. We assume that for The presented explanation is not a proof because the any x and y aforementioned approximation does not hold. In Sec. V, X −µd(x,y) we derive the lower bound of relative entropy by employ- khZ k ≤ λe (15) ing the Fisher information. Roughly speaking, the Fisher Z3x,y 5

max is satisfied with constants λ and µ, where kk represents where Hop := maxt kHop(t)k and c, µ, v are the same the operator norm [90]. The distance between two regions constants as those in the Lieb-Robinson bound. X and Y is defined as d(X,Y ) := minx∈X,y∈Y d(x, y). The Lieb-Robinson bound claims that any bounded op- We emphasize that this relation (18) is not obtained by erators on X and Y denoted by AX and BY satisfy a naive application of the Lieb-Robinson bound. This is because the Lieb-Robinson bound connects observables k[A (t),B ]k ≤ ckA kkB k|X||Y |e−µd(X,Y ) evt − 1 X Y X Y on two space-time points while Eq. (18) treats the effect (16) of an operation with finite time interval. To obtain an in- with finite constants c and v. The coefficient c depends equality for such a case, we need to complement carefully only on the Hamiltonian of the system and the metric of N − 1 new Hamiltonians between H and H + H (t). the lattice [91], and v is given by v := 4λ/c . A (t) is 0 0 op ~ X The lemma is proven in the Appendix. B. the Heisenberg picture of AX at time t. We define the Lieb-Robinson velocity as We then rigorously show that most of the energy emitted to the bath L remains near the engine. We fix a site v v := , (17) y in the engine and write l := maxx∈Λ˜ d(y, x) as the LR µ E maximum distance to any site in ΛE˜ from y. Then, a site which characterizes the maximum speed of the informa- x in L with d(x, y) = R is at least R − l far from the tion propagation in this system. The Lieb-Robinson ve- engine. We now divide the bath L into two regions: locity is the non-relativistic counterpart of the speed of light. • Region 1: sites x with d(x, y) ≤ R • Region 2: sites x with R + 1 ≤ d(x, y).

B. Bound on change in a local operator far from 0 the time-dependent region The region 1 corresponds to the region L in Sec. III. The sets of lattices in the region 1 and 2 are labeled as Λ1 and Λ2, respectively. Correspondingly, we decompose HL, We now go back to the original problem. Let us de- the Hamiltonian on L, into three parts, H1, H2, and H12, compose the total Hamiltonian Htot(t) into the time- which are the sum of local Hamiltonians with its support independent part H0 := Htot(0) and the time-dependent on only Λ1,Λ2, and both Λ1 and Λ2, respectively. The part Hop(t) := Htot(t) − H0. We remark that Hop(t) total energy in L, Tr[HLρL], is decomposed into three acts on only ΛE˜ , and H0 keeps the initial state of the 1 L 2 L L parts: E1 := Tr[H ρ ], E2 := Tr[H ρ ], and E12 := bath L, ρcan, invariant. Thus, as long as we are inter- Tr[H12ρL]. We denote the difference of energy E (a = ested in quantities of the bath L, we can interpret the a 1, 2, 12) between ρi and ρf by ∆Ea. problem of the comparison of density matrices at two We set R larger than (D − 1)/µ + 1. Summing up different times, t = 0 and t = τ (i.e., ρi and ρf ), into the Eq. (18) with respect to AX for all local Hamiltonians comparison between those with two different operations, related to sites in the region 2, we find the following result H0 and H0 + Hop(t), at the equal-time t = τ. To solve (Calculation is shown in Appendix.C): By setting R as this problem, we here introduce a useful lemma which bounds the amount of information propagation from an R ≥ R∗ := 2v τ + C, (19) operation with finite time interval. LR with a constant Lemma: Consider a lattice system with single-body and two-body nearest-neighbor interaction Hamiltoni- max max D 2 KcHop Hsite 2|ΛE˜ |2 (D − 1)! ans. The full Hamiltonian H (t) is supposed to be de- C = ln + 2l + 1, (20) tot µ v~µDQL composed as Htot(t) = H0 + Hop(t), where H0 is time- independent and Hop(t) acts only on a finite region Y . L we find ∆E12 + ∆E2 ≤ Q /2, which implies that at least Let X be a finite region which does not have overlap energy of QL/2 remains in the region 1: with Y , and AX be a bounded operator on X. We here consider two different time evolutions: In both cases, the L ∆E1 > Q /2. (21) initial state is set to the same state ρi. In the first case, the state evolves to t = τ under the Hamiltonian Htot(t), Here, we defined while in the second case it evolves under H0. The density X X matrices under these two time evolutions are denoted by Hmax := kh k (22) ρtot(t) and ρ0(t), respectively. Then, the difference of the site Z 0 tot y Z3x,y expectation value of AX between ρ and ρ at t = τ is bounded as as the upper bound of the summation of all the energy 0 tot related to a single site (i.e., one-body Hamiltonian on |Tr[AX ρ (τ)] − Tr[AX ρ (τ)]| vτ the site and two-body Hamiltonians acting on this site c max −µd(Y,X) e − 1 and another site). The constants K and D are given in ≤ Hop kAX k|Y ||X|e − τ , (18) ~ v Eq. (14). 6

line of ρs ρs(1) region 1 hotter

region 2 ρ2 same temperature

ρ0 ρs

ρ1 FIG. 5. Schematic picture of the density matrix ρs and the Plane with same A Fisher information. The average energy in region 1 is raised, while region 2 is kept as before. FIG. 4. Schematic of the quantum information geometry. A single point corresponds to a single density matrix. The hor- izontal planes represent sets of states with the same expec- The Fisher information is a kind of capacity for A with tation value of A. The vertical line represents the trajectory respect to s. If we set A as the Hamiltonian of the sys- of ρs with different s. The Pythagorean theorem claims that tem, s as inverse temperature, and ρ1 as a canonical dis- the relative entropy from ρ1 to ρ2 is equal to the sum of the tribution with inverse temperature β, then the Fisher relative entropy from ρ1 to ρs(1) and that from ρs(1) to ρ2. information becomes the heat capacity and ρs becomes another canonical distribution with inverse temperature β − s. 1 L V. STEP 2: ENERGY CHANGE AND We set A in Eq. (24) to H , and ρ1 and ρ2 to ρi L QUANTUM RELATIVE ENTROPY and ρf . Although the Fisher information is not exactly equal to the heat capacity of the region 1, since the initial L1 We finally connect the difference of energy and relative state of the region 1, ρi , differs from the corresponding entropy. To do this, we first discuss general properties of canonical distribution only on its boundary, the Fisher relative entropy. information J(ρs) can be regarded as a local heat capac- Consider two density matrices ρ1 and ρ2 and an ob- ity of the region 1 (see Fig. 5). Thus, the asymptotic D servable A. Using the Pythagorean theorem in quantum behavior of J(ρs) is proportional to its volume R in information geometry [57], the relative entropy D(ρ2||ρ1) large R, which leads to define the renormalized Fisher is evaluated as information as

J(ρs) D(ρ2||ρ1) = D(ρ2||ρs(1)) + D(ρs(1)||ρ1) jmax := max . (28) R,s RD ≥ D(ρs(1)||ρ1). (23) Insulting Eq. (26) into Eq. (23) and using monotonicity Here, ρs is deﬁned as of relative entropy, we obtain eln ρ1+sA (QL)2 Z 1 1 − t ρs := ln ρ +sA , (24) L L Tr[e 1 ] D(ρf ||ρi) ≥ D(ρf ||ρi ) ≥ dt . (29) 4 0 J(ρs(t)) and s(t) is a real-valued function satisfying Rewriting the right-hand side of the above equation with jmax and using Eq. (19), we arrive at the desired inequal- Tr[Aρs(t)] = (1 − t)Tr[Aρ1] + tTr[Aρ2]. (25) ity (6).

The density matrix ρs can be regarded as a propagated state from ρ1 in the direction of A (see Fig. 4). We propagate the state until the expectation value of A is VI. EXAMPLE: XX MODEL equal to that in ρ2. The role of t is re-labeling of s from 0 ≤ s ≤ s(1) to 0 ≤ t ≤ 1. The relative entropy We here demonstrate how the obtained bound (6) D(ρs(1)||ρ1) is calculated as [58] (see Appendix D for its works by taking a simple example, a one-dimensional derivation) XX spin chain. Suppose that the bath L consists of a one-dimensional discrete lattice with sites {1, 2, ··· ,L}, Z 1 2 1 − t where spins 1/2 are on the sites (see Fig. 6). The Hamil- D(ρs(1)||ρ1) = Tr[A(ρ2 − ρ1)] dt , (26) L 0 J(ρs(t)) tonian of the bath L, H , is given by where J(ρ ) is the Fisher information for the family L s L X x x y y {ρs} under the Bogoliubov-Kubo-Mori inner product [57] H = − J(σj σj+1 + σj σj+1), (30) given by j=1

∂Tr[Aρ ] where J > 0 is an interaction constant, σ is the Pauli J(ρ ) := s . (27) s ∂s operator, and we identify the site L + 1 to the site 1. We 7

Combining them, the simpliﬁed version of our main inequality (6) reads

L ln 2(QL)2(1 + e−β J )2 1 η ≤ η − ~ . (36) H C 256(βL)2QHJ 3 τ E We can easily see the finiteness of the coefficient of 1/τ. L L 1 2 VII. TRANSIENT CASE L-1 3 In this section, we consider the case of transient processes with multiple baths. We again assume that there FIG. 6. Schematic of the analyzed XX model. The bath L is is no interaction between the system and the baths at a periodic one-dimensional chain with length L. The engine the initial and the final state, and the initial states of the is a single site and it interacts with the bath L only through heat baths are in canonical distributions. We shall eval- the site 1. uate the entropy production of the total systems given by consider that the engine is on a single site and it interacts X ˆ j j E hσî := βjTr[Hj(ρf − ρi )] + ∆S(ρ ). (37) only with the site 1. We in particular treat a slow process j (i.e., large R). We first see quantities in the Lieb-Robinson bound. In Here, ρj represents the reduced density matrix of ρ to the E E E this setup, c should satisfy c ≥ 1, and λ and µ should j-th bath, and ∆S(ρ ) := S(ρf )−S(ρi ) is the difference satisfy two constraints: of the von Neumann entropy of the engine. In a manner similar to Sec. III, we obtain J ≤ λ, (31) D(ρ ||ρ ) J f i ≤ λe−µ, (32) = − S(ρ ) − Tr[ρ ln ρ ] 2 i f i X =Tr[ρE ln ρE] + Tr[ρj · (−βjHj)] − Tr[ρE ln ρE] which are in the case of d(x, y) = 0 and d(x, y) = 1, i i i f i j respectively. Thus, we set c = 1, λ = J, and µ = ln 2, X j j j which implies v = 4J/~ and vLR = 4J/~ ln 2. In addition, − Tr[ρf · (−β H )] max j K = 1, D = 1, l = 1, and Hsite = J are satisfied in this setup. We then find X j j j E E E E = βjTr[H (ρf − ρi )] + S(ρf ) − S(ρi ) + D(ρf ||ρi ), 8J j R∗ = τ + C (33) (38) ~ ln 2 j with where H is the Hamiltonian of the j-th bath. The sub- additivity of the von Neumann entropy [88] suggests max 2 3 Hop C = ln + 3. (34) E E E X j j ln 2 ln 2 QH D(ρf ||ρi) − D(ρf ||ρi ) =S(ρf ) − S(ρf ) − Tr[ρf ln ρi ] j X X Let us make the operation slower and slower. In this ≥ − S(ρj) − Tr[ρj ln ρj] max H f f i situation, Hop decreases with the increase of τ while Q j j is fixed, which implies that C will become negative. If X j j ∗ the above condition is satisfied, we have R ≥ 8Jτ/ ln 2. = D(ρf ||ρi ), (39) We next see the Fisher information. The initial state j L −βHL −βHL of the bath L is written as ρi = e /Tr[e ]. This which directly implies density matrix is solvable through the Jordan-Wigner X transformation, which maps this system to a free fermion hσî ≥ D(ρj||ρj) (40) system. By approximating the Fisher information by the f i j 1 variance of H with ρ0 and neglecting oscillating terms with high frequency, jmax is estimated above as (Deriva- with the aid of Eq. (38). The right-hand side can be tion is shown in Appendix.E) evaluated in a manner similar to that in the previous section. 8βLJ 3 Let us consider an isothermal process in 0 ≤ t ≤ τ with j ≤ . (35) max (1 + e−βLJ )2 heat emission Q to the bath as an example. In this case, 8 the conventional second law is equivalent to the principle We remark that we consider only the contribution from of maximum work: the bath L to the entropy production for a simple expression of the efficiency. We can evaluate that from the bath W ≤ −∆F (41) H in a similar manner, and by taking its contribution into account we obtain the following stronger bound: where the extracted work W is given in Eq. (3) and the difference of the Helmholtz free energy ∆F is defined as (QL)2 η ≤ ηC − L H D 8β Q jmax(2vLRτ + C) 1 ∆F := Tr[HE(τ)ρE] − Tr[HE(0)ρE] + (S(ρE) − S(ρE)). QH f i β i f − L 0 0 0 D , (44) (42) 8β jmax(2vLRτ + C ) Using a relation W + ∆F = hσî, we arrive at the ex- 0 0 0 tended version of the principle of maximum work with where jmax, vLR and C are defined for the bath H in a finite speed operation: similar manner to jmax, vLR, and C. We also remark on our setup of lattice systems. We re- Q2 strict our attention to lattice systems, which allows rigor- W ≤ −∆F − . (43) 8j (2v τ + C)D ous evaluation of the speed of energy spread by the Lieb- max LR Robinson bound. Although some baths are not described as lattice systems, we consider that the restriction to lattice systems is only for rigorous evaluation and our essen- VIII. DISCUSSION tial idea still holds for non-lattice baths. In fact, if one succeed in extending the Lieb-Robinson bound to non- In this paper, we have shown a novel no-go theorem lattice systems, our result is directly extended to such that non-Markovian quantum heat engines never attain non-lattice systems. the Carnot efficiency at finite power. Our result relies on Our result still provides meaningful information in the non-quick energy relaxation of baths, which is rig- thermodynamic limit. We suppose that the time inter- orously characterized by the Lieb-Robinson bound. The val of an operation τ is scaled by the size of the engine most basic result in this paper is Eq. (18), which is appli- O(V 1/D), where V is the volume of the engine. Since cable to any observable AX , not only to conserved quan- both QH and QL are of order O(V ) and C is of order tities. This relation simply exhibits that two different O(l) = O(V 1/D), we find that the second term of the operations far from a region X are hard to distinguish left-hand side of Eq. (6) is O(1) and Eq. (6) is still a through observation on X within a short time interval. meaningful inequality in the thermodynamic limit. In When AX is set to a conserved quantity, this relation this case, the obtained bound corresponds to the bound serves as a bound on relaxation speed. This technique of efficiency in terms of exergy, which is a bound for the will be helpful to clarify relaxation and thermalization case with a finite size bath and is less than the Carnot phenomena in a rigorous way. efficiency. It is worth noting that there exists a completely different stream to investigate a relation on efficiency and power: the problem on efficiency at maximum power. ACKNOWLEDGMENTS This problem treats the efficiency of a heat engine when its power is maximum. A phenomenological treatment We appreciate Keiji Saito, who we think almost as a within the linear response regime (i.e., endoreversible co-author, for fruitful discussion and many helpful com- thermodynamics [21]) shows that the efficiency at max- ments. We also appreciate Eiki Iyoda and Kaoru Ya- imum power is given by the celebrated Chambadal- mamoto for careful reading and useful comments. NS is Novikov-Curzon-Ahlborn efficiency [92–94]. Some other grateful to Kazuya Kaneko for his readable seminar on universal properties are discovered in the first and second the Lieb-Robinson bound. NS was supported by Grant- order expansion [95–97], which are again obtained in the in-Aid for JSPS Fellows JP17J00393. phenomenological irreversible thermodynamics. These stream and the investigation of a trade-off relation between efficiency and speed of operation are irrelevant to each other: Since a trade-off relation is an upper bound, Appendix A: Expression with work storage it does not characterize efficiency at maximum power. In contrast, efficiency at maximum power consider only In the present paper, we treat the work storage im- an engine at maximum power, and thus it does not give plicitely. However, recently, the explicit treatment of the a trade-off relation between efficiency and speed which work storage is studied actively [15, 71, 73–87]. It is note- covers, of course, engines not at the maximum power. worthy that our results are valid perfectly under the ex- In particular, the information on efficiency at maximum plicit treatments of the work storage. In this Appendix, power does not exclude the compatibility of the Carnot we introduce the setup for the explicit treatment, and efficiency and finite power. show that our results are indeed applicable to it. 9

the following unitary time evolution: H Z τ 0 Utot0 := T exp[− Htot(t)dt], (A3) E interact W 0 where T represents the time-ordered product. We require L that Utot0 satisfies the law of energy conservation: 0 [Utot0 ,Htot(0)] = 0. (A4) FIG. 7. Schematic of a system with work storage W, which If you want, we can also require some extra conditions interacts only with E, not H or L. The work storage stores which make the extracted energy in work storage “work.” extracted work as in the usable form of internal energy. This extra conditions change depending on the “definition of work” that we choose. However, as we will show in the end of this section, our results will be valid whichever We firstly rewrite our setup. The engine E and the definition we choose. two heat baths H and L are the same as the main text. Let us show the definition of work and heat. We de- Namely, we prepare a quantum lattice system, and refer note the density matrix of the total system at time t to the subsets of the sites ΛE,ΛH,ΛL as the engine, the by ρtot0 (t), and write ρi := ρtot0 (0) and ρf := ρtot0 (τ). hot bath, and the cold bath, respectively. ΛE is attached The partial trace of ρ to X (X = E, H, L, W) is de- to both ΛH and ΛL, and ΛH is not attached to ΛL. We noted by ρX . Using the canonical distribution of baths X X also define Λ˜ as a set of sites in ΛE and its nearest- X −β H X E X expressed as ρcan := e /Z (X =H,L) with neighbor ones. We then add the work storage W to the X X ZX := Tr[e−β H ], we set the initial state as a product setup (see Fig. 7). The work storage W is not necessarily state ρ = ρE ⊗ ρH ⊗ ρL ⊗ ρW, where ρE is arbitrary. a lattice system. Λ is also attached to W, and neither i i can can i i E ρW is the initial state of W, which is discussed below. Λ nor Λ is attached to W. i H L The cyclicity of the process requires that the final state Next, we explain the dynamics. We refer to the Hamil- and the initial state of the engine have the same energy tonian of the total system including the work storage as E E E E 0 0 expectation value: Tr[H (0)ρf ] = Tr[H (0)ρi ]. The ex- Htot(t). We assume that Htot(t) is written as follows: tracted work W , the heat released from the bath H and 0 EW W that absorbed by the bath L are respectively written as Htot(t) = Htot(t) + H (t) + H . (A1) W W W W W := Tr[H (τ)ρcan(τ)] − Tr[H ρcan(0)], (A5) Here, Htot is the Hamiltonian of E, H, and L. As in the H H H H Q := Tr[H (ρi − ρf )], (A6) main text, we assume that Htot is the sum of one-body L L L L Hamiltonians and nearest-neighbor two-body interaction Q := Tr[H (ρf − ρi )]. (A7) W EW Hamiltonians. H is the Hamiltonian of W, and H (t) Here, the amount of the extracted work is given as is the interaction between W and E. The support of EW the difference of the expectation value of energy in the H (t) is only on ΛE and W. In a cyclic process in 0 ≤ work storage. By adding extra conditions, our defini- t ≤ τ, the Hamiltonian is changed with time as satisfying EW EW tion is equal to each of the work definitions used in Refs. Htot(0) = Htot(τ) and H (0) = H (τ). Same as the [15, 71, 73–87]. For example, if we add the condition main part, the Hamiltonian Htot(t) is time-dependent that ρW (τ) and ρW (0) are energy pure states, our defi- only on Λ . Hence, the Hamiltonian H0 (t) is time- can can E˜ tot nition of work is equal to the single-shot work extraction dependent only on Λ and W , and can be decomposed E˜ which is treated in [71, 73, 76–78, 81]. If we add the into seven parts as W condition that the von Neumann entropy of ρcan(τ) and ρW (0) are the same, our condition is equal to the average H (t) =HE(t) + HEH(t) + HEL(t) + HEW(t) can tot work extraction which is used in [15, 74, 75, 79, 80, 82– + HH + HL + HW, (A2) 87]. Regardless of which of these conditions we add, our inequality is still valid, because our inequality is a where HX (X = E, H, L) acts only on X, and HEX necessary condition for the possibility of the transforma- (X = H, L) is the sum of all interaction Hamiltonians tion without these extra conditions. (Note that a neces- between a site in ΛE and that in ΛX . The Hamiltonians sary condition for the possibility of transformation with of the baths and the work storage are set to be time- looser conditions is also necessary for the possibility of independent because we consider the situation that the the transformation with tighter conditions.) Therefore, external operation acts only on the engine and the bath is our results are valid for any definition of work. not driven as explained above. We also suppose that the engine is initially separated from baths and work storage: HEH(0) = HEL(0) = HEW(0) = 0. This setup corre- Appendix B: Derivation of Lemma sponds to the situation that the initial state of the engine and that of the baths (i.e., canonical distributions) are We here derive the Lemma shown in Sec. IV B. To com- 0 prepared independently. The Hamiltonian Htot(t) causes pare the two operations H0 and H0 + Hop(t), we divide 10 the time interval τ into N ingredients ∆t = τ/N and H0+Hop(t) label tn := n∆t. Correspondingly, we introduce N + 1 ρ(0) ρN(τ) different Hamiltonians Hn(t)(n = 0, 1, ··· ,N) as ( H + H (t) : 0 ≤ t ≤ t Hn(t) := 0 op n (B1) H : t < t ≤ τ. 0 n H +H (t) t H ρ(0) 0 op n 0 ρn(τ) We also define ρn(t) as the density matrix of the com- ρ(0) ρn-1(τ) posite system at time t evolving under the Hamiltonian H +H (t) H 0 op tn-1 0 Hn(t) (see Fig. 8). We finally take N → ∞ limit with fixed τ. n We first evaluate the change in AX between ρ (τ) and ρn−1(τ), which corresponds to the contribution to the H0 0 change in AX from Hop(t) in tn−1 ≤ t < tn. Defining ρ(0) ρ (τ) the time evolution operator with the Hamiltonian H0 for time interval τ − tn as FIG. 8. Schematic image of the proof of Eq. (18). We com- N 0 i pare the expectation value of AX between ρ (τ) and ρ (τ) n Un := exp − (τ − tn) · H0 , (B2) by introducing N − 1 novel Hamiltonians H (t) and density ~ matrices ρn(t) and piling up the difference of that between ρn(τ) and ρn−1(τ). we obtain

n Tr[AX ρ (τ)]

−i/ ·(H0+Hop(tn−1))∆t N =Tr[AX Une ~ ρ (tn−1)

i/~·(H0+Hop(tn−1))∆t † 2 · e Un] + O(∆t ) limit N → ∞, we ﬁnally obtain the desired relation n−1 i N † =Tr[AX ρ (τ)] + Tr[AX Un [ρ (tn−1),Hop(tn−1)]Un]∆t ~ + O(∆t2). (B3) 0 N N n−1 |Tr[AX ρ (τ)] − Tr[AX ρ (τ)]| In the fourth line, we used ρ (tn−1) = ρ (tn−1) and Z τ c ≤ dt kH (t)kkA k|Λ ||X|e−µd(E˜,B) ev(τ−t) − 1 i op X E˜ ρn−1(t ) + [ρn−1(t ),H ]∆t = ρn−1(t ) + O(∆t2). 0 ~ n−1 n−1 0 n vτ ~ c ˜ e − 1 (B4) ≤ HmaxkA k|Λ ||X|e−µd(E,B) − τ , (B7) op X E˜ v The equation (B3) suggests ~

n−1 n |Tr[AX ρ (τ)] − Tr[AX ρ (τ)]|

1 † N 2 = |Tr[[Hop(tn−1),UnAX Un]ρ (tn−1)]|∆t + O(∆t ) max ~ where we deﬁned Hop := maxt kHop(t)k. This is the 1 † 2 desired bound. ≤ k[Hop(tn−1),UnAX Un]k∆t + O(∆t ), (B5) ~ where we used the cyclic property of the trace in the second line and a relation kρk = 1 in the third line. By not- † ing that Hop(tn−1) is an operator on ΛE˜ , and UnAX Un is an operator on X which evolves during time interval τ − tn, the Lieb-Robinson bound (16) implies

1 † 2 k[Hop(tn−1),UnAX Un]k∆t + O(∆t ) ~ Appendix C: Derivation of Eq. (21) c −µd(E˜,X) v(N−n)∆t ≤ kHop(tn−1)kkAX k|ΛE˜ ||X|e e − 1 ∆t ~ + O(∆t2), (B6) We here evaluate the upper bound for energy increase where c, µ, ν take the same values as in the Lieb- outside of the region 1. We set R larger than (D−1)/µ+ Robinson bound (16). Combining Eqs. (B5) and (B6), 1. We set AX in Eq. (18) as a local Hamiltonian and and summing them from n = 1 to N and taking the sum up this inequality for all local Hamiltonians whose 11 support has an overlap with the region 2. We then obtain Using both the above relation and the following relation

s(1) ∆E12 + ∆E2 Z ∞ dsTr[Aρs] = µ(s(1)), (D5) X evτ − 1 0 ≤ kvrD−1e−µr − τ v r=R we arrive at Eq. (26): Z ∞ ≤kevτ drrD−1e−µr Z 1 2 1 − t R−1 (Tr[Aρs(1)] − Tr[Aρ1]) dt D−1 0 J(ρs(t)) e−µ(R−1) X 1 (D − 1)! =kevτ (R − 1)D−1−i Z s(1) µ µi (D − 1 − i)! i=0 = ds(Tr[Aρs(1)] − Tr[Aρs]) 0 D−1 e−µ(R−1) X 2i (D − 1)! ≤kevτ (R − 1)D−1−i =D(ρs(1)||ρ1). (D6) µ µi (D − 1 − i)! i=0 D−1 e−µ(R−1) X 2i (D − 1)! ≤kevτ (R − 1)D−1−i Appendix E: Derivation of Eq. (35) µ µi (D − 1 − i)! i=−∞ In this Appendix, we evaluate the normalized Fisher e−µ(R−1)/22D−1(D − 1)! vτ information j in the XX model discussed in Sec. VI. =ke D , (C1) max µ First, using the Jordan-Wigner transformation, this system is mapped to a free fermion system with the where we deﬁned Hamiltonian Kc k := HmaxHmax2|Λ |eµl. (C2) L op site E˜ X J v~ H = − (c† c + c†c ), (E1) 2 j+1 j j j+1 In the second line, we used the fact that the number j=1 of sites whose distance from the site y is r is less than † D−1 Kr . In the last line, we used the Taylor expansion where cj and cj are the creation and annihilation opera- of the exponential function. The constants K and D are tor of a fermion at the site j. This Hamiltonian is known L to be diagonalized as [98] given in Eq. (14). The condition ∆E12 + ∆E2 ≤ Q /2 directly implies the desired condition for R. L−1 X † H = − J cos φq · cqcq, (E2) q=0 Appendix D: Derivation of Eq. (26)

where we defined the wave number φq := 2πq/L (q = We here demonstrate the derivation of Eq. (26). By 0, 1, ··· ,L − 1) and fermion operators inserting the definition of ρs(1), the relative entropy L D(ρs(1)||ρ1) is calculated as 1 X c† := √ eiφq jc†, (E3) q L j D(ρs(1)||ρ1) = s(1)Tr[ρs(1)A] − µ(s(1)), (D1) j=1 1 L ln ρ +sA X −iφq j where we defined µ(s) := ln Tr[e 1 ]. The differenti- cq := √ e cj. (E4) ation of Eq. (25) with respect to t is written as L j=1

ds dTr[Aρs] By setting the vacuum |0i as the state with no fermion = Tr[Aρs(1)] − Tr[Aρ1], (D2) dt ds (i.e., cj |0i = 0 for all j), the energy eigenstates are ob- † tained by operating some of the operators {cq} to the which leads to the transformation of a differential form vacuum as c† c† ··· c† |0i, whose energy eigenvalue is q1 q2 qm with t to that with s as Pm − i=1 J cos φqi . We set the region 1 as {L − l + 1,L − l + −1 1 dtJ (ρs(t)) = ds. (D3) Pl+1 † Tr[Aρs(1)] − Tr[Aρ1] 2, ··· , L, 1, 2, ··· , l + 1}, and A = J/2 j=−l+1(cj+1cj + † 1 † † cjcj+1) =: H , where we denote c−a := cL−a. We now We define the inverse function of s(t) (given in Eq. (25)) use some approximations. First, since R is large, which as t(s), which satisfies implies small s(1), we find that J(ρs) (0 ≤ s ≤ s(1)) is close to J(ρ0) and therefore approximate the former by Tr[Aρs] − Tr[Aρs(1)] the latter. Second, although H and H1 are not commute, 1 − t(s) = − . (D4) 1 Tr[Aρs(1)] − Tr[Aρ1] the commutator is small compared to H and H . This 12

1 fact suggests that J(ρ0) is approximated as the variance calculate Tr[H ρ0] as of H1:

1 2 1 2 L−1 J(ρ0) ' Tr[(H ) ρ0] − Tr[H ρ0] . (E5) 1 1 X cos φq Tr[H ρ0] = 2lJ · L (E6) L 1 + e−β J cos φq −βHL −βHL q=0 Note that ρ0 = e /Tr[e ] is the canonical distribution with inverse temperature βL. † † 1 2 Expressing ci and ci in terms of cq and cq, we can and Tr[(H ) ρ0] as

2 L−1 ! 2 l L−1 0 1 X cos φq J X X cos((j − j )(φq − φq0 ))(1 + cos(φq + φq0 )) Tr[(H1)2ρ ] = 2lJ · + . (E7) 0 L 2 L L L 1 + e−β J cos φq L −β J cos φq −β J cos φq0 q=0 j,j0=−l q,q0=0 (1 + e )(1 + e )

−βLJ cos φ In large L limit, φq and φq0 become continuous and run quickly compared to its denominator (1+e q )(1+ L from 0 to 2π. Consider the situation that φq + φq0 is e−β J cos φq0 ), and the summation turns to be negligible. 0 ﬁxed and φq − φq0 runs from −2π to 2π. If |j − j | > We thus ﬁnd L 0 β J, the numerator cos((j − j )(φq − φq0 )) oscillates

2 l L−1 0 J X X cos((j − j )(φq − φq0 ))(1 + cos(φq + φq0 )) Tr[(H1)2ρ ] − Tr[H1ρ ]2 ' 0 0 2 L L L (1 + e−β J cos φq )(1 + e−β J cos φq0 ) j,j0=−l q,q0=0 |j−j0|≤βLJ

2 L−1 J X 1 + cos(φq + φq0 ) ≤4lβLJ 2 L L L −β J cos φq −β J cos φq0 q,q0=0 (1 + e )(1 + e ) 2J 2 ≤4lβLJ , (E8) (1 + e−βLJ )2

Appendix F: Case of general Hamiltonian 13 H region 1 In this Appendix, we consider the case with general exponentially-decaying local Hamiltonians (i.e., those region 2 H123 satisfying Eq. (15)) not restricted to nearest-neighbor interactions. In this case, we divide the bath L into three regions (see Fig. 9):

region 3 • Region 1: sites y with dist(x, y) ≤ R • Region 2: sites y with R < dist(x, y) ≤ 2R

FIG. 9. Schematic of the division of the bath into the region • Region 3: sites y with 2R + 1 ≤ dist(x, y). 1,2, and 3, and examples of local Hamiltonians in H13 and H123. Correspondingly, we decompose the Hamiltonian on L into seven parts, H1, H2, H3, H12, H23, H13, and H123, which are the sum of local Hamiltonians with its support on only Λ1,Λ2,Λ3, among Λ1 and Λ2, among Λ2 and Λ3, among Λ1 and Λ3, and among Λ1 and Λ2 and Λ3, respectively. We also decompose the total energy into seven parts, E1,E2,E3,E12,E13,E23,E123, where the former three energies are in the region 1∼3, and the which implies the desired Eq. (35). middle three energies are many-body interactions with 13 sites among two regions (1,2), (1,3), and (2,3), and the that the interaction in the bath is at most Nint-body last energy is many-body interactions with sites among interaction. three regions (1,2,3). We next evaluate the change in energy of many-body We first evaluate the change in energy inside the com- interactions among sites in both region 1 and 3 (i.e., E13+ 23 E123). To obtain this, we first evaluate the sum of the posite region of 2 and 3, ∆Ein := |∆E2|+|∆E3|+|∆E23|, from t = 0 to τ. By following argument similar to that in operator norm of H13 and H123 as follows: Sec. IV B, the maximum of the change in energy in these X X X regions is evaluated as kH13k + kH123k ≤ khZ k x∈Λ1 y∈Λ3 Z3x,y R ∞ ∞ vτ X D−1 X D−1 −µr 23 X 0 D−1 −µr e − 1 ≤K x λ Kr e ∆Ein ≤ k vr e − τ . (F1) v x=1 r=R+1 r=R+1 ∞ Z R+1 X ≤K xD−1dx · λ KrD−1e−µr 0 Here, the coefficient k is set to 0 r=R+1 ∞ (R + 1)D X =K · λ KrD−1e−µr. Kc D k0 := HmaxHmax|Λ |N eµl (F2) r=R+1 v op site E˜ int ~ (F3) corresponding to Eq. (C2), where we defined Nint such Setting R ≥ (D − 1)/µ, we obtain

12 ∆Eout :=|∆E3| + |∆E13| + |∆E23| + |∆E123| 23 ≤∆Ein + |∆E13| + |∆E123| ∞ ∞ X (R + 1)D X ≤ k0evτ rD−1e−µr + K2 λ rD−1e−µr D r=R+1 r=2R+1 ∞ ∞ (R + 1)D X (R + 1)D X ≤ k0evτ rD−1e−µr + K2 λ rD−1e−µr D D r=R+1 r=R+1 (R + 1)D Z ∞ ≤ k0evτ + K2λ drrD−1e−µr D R D−1 1 e−µR X 1 (D − 1)! ≤ k0evτ + K2λ RD−1−i D µ µi (D − 1 − i)! i=0 2D−1 1 e−µR X 2i (2D − 1)! ≤ k0evτ + K2λ (R + 1)2D−1−i D µ µi (2D − 1 − i)! i=−∞ 22D−1(2D − 1)! =(k0evτ + K2λ) e−µ(R−1)/2. (F4) Dµ2D

In the third line, we used |∆E13| + |∆E123| ≤ kH13k + haves as 2v/µ · τ with a constant term. Corresponding ∗ kH123k and Eq. (F3). In the fourth line, we used to Eq. (6), the efficiency is bounded in terms of R as (R + 1)D/D ≥ 1. In the fifth line, we used the fact that rD−1e−µr is monotonically decreasing for r ≥ (D − 1)/µ. L 12 Hence, the condition ∆E1 +∆E2 +∆E12 ≥ Q −∆Eout ≥ QL/2 is satisfied for R ≥ R∗ with

2 K2λ k0Dµ2D+1QL R∗ := ln evτ + − ln + 1. µ k0 22D(2D − 1)! (F5) (QL)2 vτ 2 0 ∗ η ≤ ηC − L H ∗ D . (F6) For suﬃciently large τ such that e K λ/k , R be- 8β Q jmax(R ) 14

Baths 2. Quantum entropy production and trade-oﬀ Interact relation

Engine Interact We here introduce the transition probability of a forward and its time-reversal process as ∆t 0 0 0 2 Pa,j ,j →a0,j0 ,j0 := | ha , jH, jL|∆t U |a, jH, jLi0 | (G1) Interact H L H L ˜∆t ˜ ˜ † ˜0 ˜0 ˜0 2 P ˜0 ˜0 ˜0 ˜ ˜ := | ha,˜ jH, jL|0 U |a , jH, jLi∆t | , a ,jH,jL→a,˜ jH,jL t=0 t=Δt t=2Δt (G2) where |a˜i represents the time-reversal state of |ai. Time- FIG. 10. Schematic picture of quantum Markov processes. reversal symmetry of unitary evolution implies We prepare inﬁnitely many baths in canonical distribution. ∆t ∆t We attach the engine to one of the baths in small time interval P 0 0 0 = P˜ . (G3) a,jH,jL→a ,j ,j a˜0,j˜0 ,j˜0 →a,˜ j˜ ,j˜ ∆t, and detach the bath and attach a new bath to the system. H L H L H L It is known that the stochastic entropy production can be expressed as [99]

∆t 0 X X X Appendix G: Case of Markovian limit σˆ :=s(p∆t(a )) − s(p0(a)) + β Q i H L ∆t p0(a)pcan(jH)pcan(jL)Pa,j ,j →a0,j0 ,j0 1. Problem and Setup = ln H L H L , 0 H 0 L 0 ˜∆t p∆t(a )pcan(jH)pcan(jL)P ˜0 ˜0 ˜0 ˜ ˜ a ,jH,jL→a,˜ jH,jL Our main result (6) contains the operator norm of the (G4) local Hamiltonian of the bath L in its right-hand side in X X X where we defined the heat release Q := εj − ε 0 , the X jX the form of the Lieb-Robinson velocity vLR. Because the X X X −β εj operator norm of the local Hamiltonian in baths diverges canonical distribution pcan(jX ) := e X /ZX with the −βX εX and so does vLR in Markovian limit, our inequality falls partition function Z := P e jX , and the stochas- X jX down into the conventional second law in this limit. To tic entropy s(p(a)) := − ln p(a), which reproduces the avoid this insufficiency, in this Appendix we demonstrate Shannon entropy by taking its average. With these defi- a completely different approach to exclude the possibility nitions, both the fluctuation theorem, hexp −σˆ∆ti = 1, of coexistence of finite power and the Carnot efficiency and the second law of thermodynamics, σˆ∆t ≥ 0, are in quantum Markov processes. satisfied. We again treat an engine with two heat baths L and We here require a physically plausible property that H with inverse temperatures βH and βL. It is straight- in sufficiently small time interval ∆t energy cannot be forward to extend our analysis to the case with three transported directly from a bath to the other bath. In ∆t or more baths and with particle baths. Differently from other words, P 0 0 0 turns to be zero if both a,jH,jL→a ,jH,jL the main part on non-Markovian engines, we here do not 0 0 jH 6= jH and jL 6= jL hold. We then safely define the assume that the system is on a lattice and the interac- transition rate induced by each bath as tion is short-range. The initial state of the total sys- E H L ∆t X ∆t tem is again given by ρ (0) = ρ ⊗ ρ ⊗ ρ . The P 0 0 := P 0 0 (G5) tot i can can a,jH→a ,jH a,jH,jL→a ,jH,jL spectrum decomposition of the initial state of the engine jL E P reads ρ = p0(a) |φai hφa|. By denoting the energy ∆t X ∆t i a P 0 0 := P 0 0 . (G6) X a,jL→a ,jL a,jH,jL→a ,jH,jL eigenstates of the bath X (X =H, L) by |εj i, the basis X jH {|a, j , j i} := {|φ i |εH i |εL i} diagonalizes ρ (0). H L 0 a jH jL tot Taking ∆t → 0 limit and defining P 0 0 := a,jX →a ,jX Suppose that the total system evolves in a small ∆t lim∆t→0 Pa,j →a0,j0 /∆t, we have an expression of en- time interval from t = 0 to t = ∆t (see Fig. 10). X X ∆t The density matrix of the total system at t = ∆t is tropy production rateσ ˙ := lim∆t→0 σˆ /∆t and heat † flux as given by ρtot(∆t) = Uρtot(0)U with the time evolution operator U. The spectrum decomposition of σ˙ :=σ ˙ H +σ ˙ L (G7) the reduced density matrix reads TrH,L[ρtot(∆t)] = X P X X p0(a)pcan(jX ) a p∆t(a) |ψai hψa|, where the label of a in |ψai is set σ˙ X := p0(a)p (jX )Pa,j →a0,j0 ln can X X p (a0)pX (j0 ) as satisfying lim∆t→0 hψa|φai = 1. Suppose that we per- 0 0 0 can X a,jX ,a ,jX form a projection measurement on ρtot(∆t) with a basis (G8) H L {|a, jH, jLi}∆t := {|ψai |ε i |ε i}, which does not affect jH jL X X X J := p (a)p (j )P 0 0 (E 0 − E ), the evolution of the system because the engine is diago- Q 0 can X a,jX →a ,jX a a 0 0 nalized with this basis and the baths are not used in the a,jX ,a ,jX subsequent evolution of the engine as explained later. (G9) 15

H Here,σ ˙ X (X =H, L) is a part of entropy production η := W/Q satisfy rate contributed from the bath X, and Ea := hφa| H |φai W is the energy expectation value of a state |φai, which ¯ L X X ≤ Θβ η(ηC − η), (G14) satisfies Ea0 − Ea = εj − ε 0 in ∆t → 0 limit due to the τ X jX law of energy conservation. which clearly shows that a finite power heat engine never In line with Ref. [41], we can derive the following in- attains the Carnot efficiency. equality for a quantum Markov engine: We remark that our result on Markovian engines is applicable to broader class of engines than the result for stationary thermoelectric transport shown in Ref. [101]. X X p The result in Ref. [101] is applicable only to stationary |JQ | ≤ Θ(0)σ. ˙ (G10) X=H,L systems described by the Lindblad equation. In contrast, our result also covers transient systems and stationary Markovian systems not described by the Lindblad equa- Here, Θ(0) is defined as tion (e.g., systems without the rotating wave approximation or without the weak-coupling approximation). We note that the Markovness of the system requires that the   interaction between the bath and the system is weak with 9 X X 2 X Θ(0) :=  (∆Ea0 ) Aa,j ;a0,j0  (G11) respect to the bath, while it is not necessarily weak with 8 X X 0 0 respect to the system. Our result is applicable to the X=H,L a,jX ,a ,jX latter one. with energy fluctuation and activity: 3. Derivation of Eq. (G10)

∆Ea0 :=Ea0 − Tr[Hρ(0)], (G12) In this Appendix, we derive Eq. (G10) in line with X X Ref. [41]. We first derive A 0 0 :=p0(a)p (jX )Pa,j →a0,j0 a,jX ;a ,jX can X X 0 X + p0(a )p (jX )Pa0,j0 →a,j . (G13) X p can X X |JQ | ≤ ΘX σ˙ X , (G15) from which we can easily prove Eq. (G10). We here in- The derivation of Eq. (G10) is shown in the next subsec- troduce quantities tion. X,± X ˜ 0 0 A 0 0 :=p0(a)pcan(jX )Pa,jX →a ,j Since the bath equilibrates extremely quick compared a,jX ;a ,jX X to the time scale of the engine, we argue that through- 0 X 0 ˜ ± p0(a )pcan(jX )Pa˜0,˜j0 →a,˜ ˜j , (G16) out a process in 0 ≤ t ≤ τ the heat bath is regarded X X as always in canonical distribution, which implies that 0 0 which consist of a probability flux a, jX → a , jX and Eq. (G10) holds at any time t by replacing p0 and Θ(0) 0 ˜0 ˜ its time-reversal onea ˜ , jX → a,˜ jX . We note a relation with pt and Θ(t). The same assumption is seen in the corresponding to the time-reversal invariance of escape Born-Markov approximation, which is used in a stan- rate in classical Markov jump processes as dard derivation of the Lindblad equation [100]. Accept- ing this plausible requirement, we obtain the trade-off X X ˜ 0 0 Pa˜0,˜j0 →a,˜ ˜j = Pa ,j →a,jX relation between power and efficiency for a cyclic pro- X X X a,jX a,jX 0 0 0 0 cess during 0 ≤ t ≤ τ with two heat baths with inverse (a,jX )6=(a ,j ) (a,jX )6=(a ,j ) H L H L X X temperatures β and β (β < β ). The integration of (G17) Eq. (G10) from t = 0 to τ with the Schwarz inequality for any a0, j0 , which follows from the following time- H L 2 X leads to (Q + Q ) ≤ τΘ¯ ∆S, where we defined time- reversal symmetry ¯ R τ averaging of Θ as Θ := 1/τ 0 dtΘ(t) and entropy increase as ∆S := βLQL − βHQH. Here, the change in the P ∆t = P˜∆t . (G18) a,jX →a,jX a,˜ ˜jX →a,˜ ˜jX entropy of the engine is zero due to cyclicity. Then, a L H 2 thermodynamic relation η(ηC − η) = W ∆S/{β (Q ) } Using Eq. (G17), the normalization condition and the suggests that the work W := QH − QL and efficiency Schwarz inequality, we obtain Eq. (G15): 16

X 2 X X 0 X 0 |J | = Ea0 p0(a)p (jX )Pa,j →a0,j0 − p0(a )p (j )Pa0,j0 →a,j Q can X X can X X X 0 0 a.jX ,a ,jX 0 0 (a,jX )6=(a ,jX ) 2

X ˜X,− = Ea0 A 0 0 a,jX ;a ,jX 0 0 a.jX ,a ,jX 0 0 (a,jX )6=(a ,jX ) 2

X ˜X,− = ∆Ea0 A 0 0 a,jX ;a ,jX 0 0 a.jX ,a ,jX 0 0 (a,jX )6=(a ,jX ) 2

˜X,− q Aa,j ;a0,j0 X ˜X,+ X X = ∆Ea0 A 0 0 · a,jX ;a ,j q X ˜X,+ 0 0 A 0 0 a.jX ,a ,jX a,jX ;a ,j 0 0 X (a,jX )6=(a ,jX )     2 ˜X,−    Aa,j ;a0,j0   X 2 ˜X,+   X X X  ≤ ∆Ea0 Aa,j ;a0,j0  X X   ˜X,+   0 0   0 0 Aa,j ;a0,j0  a.jX ,a ,jX a.jX ,a ,jX X X 0 0 0 0 (a,jX )6=(a ,jX ) (a,jX )6=(a ,jX )  

X  X 2 X,+  9 X X p0(a)pcan(jX )  ˜  0 0 ≤ ∆Ea0 Aa,j ;a0,j0 p0(a)pcan(jX )Pa,jX →a ,j ln 0  X X  8 X p (a0)pX (j )  0 0  0 0 0 can X a.jX ,a ,jX a.jX ,a ,jX 0 0 0 0 (a,jX )6=(a ,jX ) (a,jX )6=(a ,jX )  

X  X 2 X  9 X X p0(a)pcan(jX )  0  0 0 = ∆Ea0 Aa,j ;a0,j p0(a)pcan(jX )Pa,jX →a ,j ln  X X  8 X p (a0)pX (j0 )  0 0  0 0 0 can X a.jX ,a ,jX a.jX ,a ,jX 0 0 0 0 (a,jX )6=(a ,jX ) (a,jX )6=(a ,jX )

=ΘX σ˙ X , (G19)

where we deﬁned By applying the Schwarz inequality again, Eq. (G15) directly implies the desired inequality (G10): X 2 X ΘX := (∆Ea0 ) Aa,j ;a0,j0 . (G20) X X X X X p 0 0 |J | ≤ Θ σ˙ a.jX ,a ,jX Q X X 0 0 (a,jX )6=(a ,jX ) X X v u ! ! In the second and seventh lines, we used Eq. (G17). In u X X ≤t ΘX σ˙ X the ﬁfth line, we used the Schwarz inequality. In the X X sixth line, we used an inequality a ln a/b + b − a ≥ 8(a − p b)2/9(a + b) and the time-reversal symmetry of unitary = Θ(0)σ. ˙ (G21) evolution (G3).

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