CONTEMPORARY PHYSICS, 2016 VOL. 57, NO. 4, 545–579 http://dx.doi.org/10.1080/00107514.2016.1201896

Quantum thermodynamics

Sai Vinjanampathya, Janet Andersb aCentre for Quantum Technologies, National University of Singapore, Singapore; bDepartment of Physics and Astronomy, University of Exeter, Exeter, UK

ABSTRACT ARTICLE HISTORY Quantum thermodynamics is an emerging research field aiming to extend standard thermodynamics Received 16 July 2015 and non-equilibrium statistical physics to ensembles of sizes well below the thermodynamic limit, in Accepted 13 May 2016 non-equilibrium situations and with the full inclusion of quantum effects. Fuelled by experimental KEYWORDS advances and the potential of future nanoscale applications, this research effort is pursued by (Quantum) information– scientists with different backgrounds, including statistical physics, many-body theory, mesoscopic thermodynamics link; physics and quantum information theory, who bring various tools and methods to the field. A (quantum) fluctuation multitude of theoretical questions are being addressed ranging from issues of thermalisation of theorems; thermalisation of quantum systems and various definitions of ‘work’ to the efficiency and power of quantum engines. quantum systems; single This overview provides a perspective on a selection of these current trends accessible to postgraduate shot thermodynamics; students and researchers alike. quantum thermal machines

1. Introduction field, and each contributes different insights. For exam- ple, the study of thermalisation has been approached One of the biggest puzzles in quantum theory today is to by quantum information theory from the standpoint of show how the well-studied properties of a few particles typicality and entanglement, and by many-body physics translate into a statistical theory from which new macro- with a dynamical approach. Likewise, the recent study scopic quantum thermodynamic laws emerge. This of quantum thermal machines (QTMs), originally app- challenge is addressed by the emerging field of quan- roached from a quantum optics perspective [10–12], has tum thermodynamics which has grown rapidly over the since received significant input from many-body physics, last decade. It is fuelled by recent equilibration experi- fluctuation relations and linear response approaches ments [1] in cold atomic and other physical systems, the [13,14]. These designs further contrast with studies on introduction of new numerical methods [2]andthedis- thermal machines based on quantum information theo- covery of fundamental theoretical relationships in non- retic approaches [15–20]. The difference in perspectives equilibrium statistical physics and quantum information on the same topics has also meant that there are ideas theory [3–9]. With ultrafast experimental control of qua- within quantum thermodynamics where consensus is yet ntum systems and engineering of small environments to be established. pushing the limits of conventional thermodynamics, the This article is aimed at non-expert readers and special- central goal of quantum thermodynamics is the exten- ists in one subject who seek a brief overview of quantum sion of standard thermodynamics to include quantum thermodynamics. The scope is to give an introduction to effects and small ensemble sizes. Apart from the academic some of the different perspectives on current topics in a drive to clarify fundamental processes in nature, it is single paper, and guide the reader to a selection of useful expected that industrial need for miniaturisation of tech- references. Given the rapid progress in the field, there nologies to the nanoscale will benefit from understanding are many aspects of quantum thermodynamics that are of quantum thermodynamic processes. Obtaining a deta- not covered in this overview. Some topics have been inte- iled knowledge of how quantum fluctuations compete nsely studied for several years and dedicated reviews are with thermal fluctuations is essential for us to be able to available, for example, classical non-equilibrium thermo- adapt existing technologies to operate at ever-decreasing dynamics [21], fluctuation relations [22], non-asymptotic scales, and to uncover new technologies that may harness quantum information theory [23], quantum engines [24], quantum thermodynamic features. equilibration and thermalisation [25,26]andarecent Various perspectives have emerged in quantum ther- quantum thermodynamics review focusing on quantum modynamics due to the interdisciplinary nature of the

CONTACT Sai Vinjanampathy [email protected]; Janet Anders [email protected] © 2016 Informa UK Limited, trading as Taylor & Francis Group 546 S. VINJANAMPATHY AND J. ANDERS

τ information theory techniques [27]. Other reviews of Q tr ρ(t) H(t) dt and interest discuss Maxwell’s demon and the physics of for- ⟨ ⟩:= 0 [˙ ] ! τ getting [28–30]andthermodynamicaspectsofinforma- ( ) ( ) W tr ρ t H t dt. (2) tion [31]. We encourage researchers to take on board the ⟨ ⟩:= [ ˙ ] !0 insights gained from different approaches and attempt to fit together the pieces of the puzzle to create an overall Here, the brackets indicate the ensemble average that ⟨·⟩ united framework of quantum thermodynamics. is assumed in the above definition when the trace is Section 2 discusses the standard laws of thermody- performed. Work is extracted from the system when namics and introduces the link with information W W > 0, while heat is dissipated to the ⟨ ext⟩:=−⟨ ⟩ processing tasks and Section 3 gives a brief overview of environment when Q Q > 0. ⟨ dis⟩:=−⟨ ⟩ fluctuation relations in the classical and quantum regimes. The first law of thermodynamics states that the sum of Section 4 then introduces quantum dynamical maps and average heat and work done on the system just makes up discusses implications for the foundations of thermody- its average energy change, namics. Properties of maps form the backbone of the thermodynamic resource theory approach to single shot τ d thermodynamics discussed in Section 5. Section 6 dis- Q W tr ρ(t) H(t) dt cusses the operation of thermal machines in the quantum ⟨ ⟩+⟨ ⟩= 0 dt [ ] ! (τ) (τ) ( ) ( ) regime. Finally, in Section 7,thecurrentstateofthefield tr ρ H tr ρ 0 H 0 #U. (3) = [ ]− [ ]= is summarised and open questions are identified. It is important to note that while the internal energy change only depends on the initial and final states and 2. Information and thermodynamics Hamiltonians of the evolution, heat and work are pro- This section defines averages of heat and work, intro- cess dependent, i.e. it matters how the system evolved in duces the first and second laws of thermodynamics and time from (ρ(0), H(0)) to (ρ(τ), H(τ)). Therefore, heat and discusses examples of the link between thermodynamics work for an infinitesimal process which will be denoted and information processing tasks, such as erasure. by δQ and δW where the symbol δ indicates that heat ⟨ ⟩ ⟨ ⟩ and work are (in general) not full differentials and do not correspond to observables [41], in contrast to the average 2.1. The first and second laws of thermodynamics energy with differential dU. Choosing to split the energy change into two types Thermodynamics is concerned with energy and changes of energy transfer is crucial to allow the formulation of energy that are distinguished as heat and work. For of the second law of thermodynamics. A fundamental ρ aquantumsysteminstate and with Hamiltonian H law of physics, it sets limits on the work extraction of at a given time, the system’s internal, or average, energy heat engines and establishes the notion of irreversibility is identified with the expectation value U(ρ) tr ρ H . = [ ] in physics. Clausius observed in 1865 that a new state When a system changes in time, i.e. the pair of state and function – the thermodynamic entropy Sth of a system – Hamiltonian [32]varyintime(ρ(t), H(t)) with t 0, τ , ∈[ ] is helpful to study the heat flow to the system when it the resulting average energy change interacts with baths at varying temperatures T [42]. The thermodynamic entropy is defined through its change in #U tr ρ(τ) H(τ) tr ρ(0) H(0) (1) = [ ]− [ ] a reversible thermodynamic process, is made up of two types of energy transfer – work and δQ heat. Their intuitive meaning is that of two types of #Sth ⟨ ⟩,(4) := T energetic resources, one fully controllable and useful, !rev the other uncontrolled and wasteful [5,32–40]. Since the time variation of H is controlled by an experimenter, where δQ is the heat absorbed by the system along the ⟨ ⟩ the energy change associated with this time variation process and T is the temperature at which the heat is is identified as work. The uncontrolled energy change being exchanged between the system and the bath. Fur- δQ associated with the reconfiguration of the system state ther observing that any cyclic process obeys ⟨ ⟩ 0 T ≤ in response to Hamiltonian changes and the system’s with equality for reversible processes, Clausius formu- " coupling with the environment are identified as heat.The lated a version of the second law of thermodynamics for all formal definitions of average heat absorbed by the system thermodynamic processes, today known as the Clausius and average work done on the system are then inequality: CONTEMPORARY PHYSICS 547

δQ ⟨ ⟩ #S ,becoming T ≤ th ! Q T #S for T =const. (5) ⟨ ⟩≤ th It states that the change in a system’s entropy must be equal or larger than the average heat absorbed by the system during a process divided by the temperature at which the heat is exchanged. In this form, Clausius’ inequality establishes the existence of an upper bound to the heat absorbed by the system and its validity is generally assumed to extend to the quantum regime [5,16,32–40]. Equivalently, by defining the free energy of Figure 1. A gas in a box starts with a thermal distribution at a system with Hamiltonian H and in contact with a heat temperature T. Maxwell’s demon inserts a wall and selects faster bath at temperature T as the state function particles to pass through a door in the wall to the left, while he lets slower particles pass to the right, until the gas is separated F(ρ) U(ρ) TS (ρ),(6) := − th into two boxes that are not in equilibrium. The demon attaches a bucket (or another suitable work storage system) and allows the Clausius’ inequality becomes a statement of the upper wall to move under the pressure of the gases. Some gas energy bound on the work that can be extracted in a thermody- is extracted as work, Wext,raisingthebucket.Finally,thegas is brought in contact with the environment at temperature T, namic process, equilibrating back to its initial state (#U 0) while absorbing heat Q W .AcomprehensivereviewofMaxwell’sdemon= Wext W #U Q abs ext ⟨ ⟩=−⟨ ⟩=− +⟨ ⟩ is [29]. = #U T#S #F. (7) ≤− + th =− While the actual heat absorbed/work extracted will dep- 2.2. Maxwell’s demon end on the specifics of the process, there exist optimal, Maxwell’s demon is a creature that is able to observe the thermodynamically reversible, processes that saturate the motion of individual particles and use this information equality, see Equation (4). However, modifications of (by employing feedback protocols discussed in Section the second law, and thus the optimal work that can be 3.3) to convert heat into work in a cyclic process using extracted, arise when the control of the working sys- only a single heat bath at temperature T.Byseparating tem is restricted to physically realistic, local scenarios slower, ‘colder’ gas particles in a container from faster, [43]. For equilibrium states ρ e βH /tr e βH for th = − [ − ] ‘hotter’ particles, and then allowing the hotter gas to Hamiltonian H and at inverse temperatures β 1 , = kB T expand while pushing a piston, see Figure 1,thedemon the thermodynamic entropy Sth equals the information can extract work while returning to the initial mixed gas. theory entropy, S,timestheBoltzmannconstantkB, i.e. For a single particle gas, the demon’s extracted work in a S (ρ ) k S(ρ ). The information theory entropy, th th = B th cyclic process is known as the Shannon or von Neumann entropy, for a ρ general state is defined as Wdemon k T ln 2. (9) ⟨ ext ⟩= B S(ρ) tr ρ ln ρ . (8) := − [ ] Maxwell realised in 1867 that such a demon would Many researchers in quantum thermodynamics assume appear to break the second law of thermodynamics, see that the thermodynamic entropy is naturally extended Equation (5), as it converts heat completely into work to non-equilibrium states by the information theoretic in a cyclic process, resulting in a positive extracted work, entropy. For example, this assumption is made when Wdemon Q #U T#S #U 0. The crux ⟨ ext ⟩=⟨ ⟩− ̸≤ th − = using the von Neumann entropy in connection with the of this paradoxical situation is that the demon acquires second law and the analysis of thermal processes, and information about individual gas particles and uses this in the calculation of efficiencies of QTMs. Evidence that information to convert heat into useful work. One way of this extension is appropriate has been provided via many resolving the paradox was presented by Bennett [46]and routes including Landauer’s original work, see [28] for invokes Landauer’s erasure principle [47], as described an introduction. The suitability of this extension and its in the next section and presented highly accessibly in limitations remain however debated issues [44,45]. For [28]. Other approaches consider the cost the demon has the remainder of this article, we will assume that the to pay upfront when identifying whether the particle von Neumann entropy S is the natural extension of the is slower or faster [30]. Experimentally, the thermody- thermodynamic entropy Sth. namic phenomenon of Maxwell’s demon remains highly 548 S. VINJANAMPATHY AND J. ANDERS relevant – for instance, cooling a gas can be achieved take infinitely long to implement. The time of erasure is by mimicking the demonic action [48]. An example of investigated in [51]anditisshownthatLandauer’slimit an experimental test of Maxwell’s demon is discussed in is achievable in finite time when allowing exponentially Section 3.4. suppressed erasure errors.

2.3. Landauer’s erasure principle 2.4. Erasure with quantum side information In a seminal paper [47], Landauer investigated the The heat dissipation during erasure is non-trivial when thermodynamic cost of information processing tasks. He the initial state of the system is a mixed state, ρS.In concluded that the erasure of one bit of information the quantum regime, mixed states can always be seen requires a minimum dissipation of heat, as reduced states of global states, ρSM,ofthesystem S and a memory M,withρ tr ρ .Aground- S = M[ SM] Qmin k T ln 2, (10) breaking paper [8] takes this insight seriously and sets ⟨ dis ⟩= B up an erasure scenario where an observer can operate that the erased system dissipates to a surrounding en- on both, the system and the memory. During the global process, the system’s local state is erased, ρ 0 ,while vironment, cf. Figure 1, in equilibrium at temperature S '→| ⟩ the memory’s local state, ρ tr ρ ,isnotaltered. T. The erasure of one bit, or ‘reset’, here refers to the M = S[ SM] change of a system being in one of two states with equal In other words, this global process is locally indistin- probability, I (i.e. 1 bit), to a definite known state, 0 (i.e. guishable from the erasure considered in the previous 2 | ⟩ 0bits).(Wenotethat,contrarytoeverydaylanguage,the section. However, contrary to the previous case, erasure technical term ‘information’ here refers to uncertainty with ‘side-information’ , i.e. using correlations of the rather than certainty. Erasure of information thus im- memory with the system, can be achieved while extracting plies increase of certainty.) Landauer’s principle has been a maximum amount of work tested experimentally only very recently, see Section 3.4. max W k TS(S M)ρ . (11) The energetic cost of the heat dissipation is balanced ⟨ ext ⟩=− B | SM by a minimum work that must be done on the system, min min W #U Q Q , to achieve the erasure at Here, S(S M)ρSM is the conditional von Neumann en- ⟨ ⟩= −⟨ ⟩=⟨ dis ⟩ | constant average energy, #U 0. Bennett argued [46] tropy between the system and memory, S(S M)ρSM = | = S(ρ ) S(ρ ).Crucially,theconditional entropy can that Maxwell’s demonic paradox can thus be resolved SM − M by taking into account that the demon’s memory holds be negative for some quantum correlated states (a sub- bits of information that need to be erased to completely set of the set of entangled states), thus giving a posi- close the thermodynamic cycle and return to the initial tive extractable work. This result contrasts strongly with conditions. The work that the demon extracted in the first Landauer’s principle valid for both classical and quantum step, see previous section, then has to be spent to erase states when no side-information is available. The possi- the information acquired, with no net gain of work and bility to extract work during erasure is a purely quan- in agreement with the second law. tum feature that relies on accessing the side-information While Landauer’s principle was originally formulated [8]. That is, to practically obtain positive work requires for the erasure of one bit of classical information, it knowledge of and access to an initial entangled state of is straightforwardly extended to the erasure of a gen- the system and the memory, and the implementation of eral mixed quantum state, ρ, which is transferred to the acarefullycontrolledprocessonthedegreesoffreedom blank state 0 . The minimum required heat dissipation of both parties. The entanglement between system and | ⟩ is then Qmin k TS(ρ) following from the second memory will be destroyed in the process and can be seen ⟨ dis ⟩= B law, Equation (5). A recent analysis of Landauer’s prin- as ‘fuel’ from which work is extracted. ciple [49] uses a framework of thermal operations, cf. Section 5, to obtain corrections to Landauer’s bound 2.5. Work from correlations when the effective size of the thermal bath is finite. They show that the dissipated heat is in general above Lan- The thermodynamic work and heat associated with cre- dauer’s bound and only converges to it for infinite-sized ating or destroying (quantum) correlations have been reservoirs. Probabilistic erasure is considered in [50]and studied intensely, e.g. [52–59], for a variety of settings, the trade-offbetween the probability of erasure and mini- including unitary and non-unitary processes. For exam- mal heat dissipation is demonstrated. In the simplest case, ple, the thermodynamic efficiency of an engine operating the erasure protocol achieving the Landauer’s bound will on pairs of correlated atoms can be quantified in terms of require an idealised quasi-static process which would quantum discord and it was shown to exceed the classical CONTEMPORARY PHYSICS 549 efficiency value [54]. In [60], the minimal heat dissipation Importantly, S(η) is larger than S(ρ) if and only if the for coupling a harmonic oscillator that starts initially in state ρ had coherences with respect to the energy eigen- local thermal equilibrium, and ends up correlated with a basis, while the entropy does not change under projection bath of harmonic oscillators in a global thermal equilib- processes for classical states. Thus, the extracted work rium state, is determined and it was shown that this heat here is due to the quantum coherences in the initial state. contribution resolves a previously reported second law We note that ‘decohering’ a state ρ is a physical imple- violation. mentation of the same state transfer, ρ η,during '→ Thermodynamic aspects of creating correlations are which coherences are washed out by the environment in also studied in [59] where the minimum work cost is an uncontrolled way and no work is extracted. This is established for unitarily evolving an initial, locally ther- a suboptimal process – to achieve the maximum work mal, state of N systemstoaglobalcorrelatedstate.Amax- an optimal implementation needs to be realised which imum temperature is derived at which entanglement can requires a carefully controlled protocol of interacting still be created, along with the minimal associated energy the system with heat and work sources [61]. It is worth cost. In turn, when wanting to extract work from many- noting that in contrast to Maxwell’s demon whose work body states that are initially globally correlated, while extraction is based on the knowledge of ‘microstates’ locally appearing thermal, the maximum extractable and appears to violate the second law when information work under global unitary evolutions is discussed in [58] erasure is not considered, gaining work from coherences for initial entangled and separable states, and those dia- is in accordance with the second law. Once the projection gonal in the energy basis. This contrasts with the non- process is completed, the final state has lost its coherence, unitary process of erasure with side-information, i.e. here the coherences have been used as ‘fuel’ to extract discussed in Section 2.4. work.

3. Classical and quantum non-equilibrium statistical physics 2.6. Work from coherences In this section, the statistical physics approach is out- While erasure with side-information discussed in lined that rests on the definition of fluctuating heat and Section 2.4 and the results in Section 2.5 illustrate the work from which it derives the ensemble quantities of quantum thermodynamic aspect of correlations, a sec- average heat and average work discussed in Section 2. ond quantum thermodynamic feature arises due to the Fluctuation theorems and experiments are first described presence of coherences. A recent paper [61] identifies in the classical regime before they are extended to the projection processes as a route to analyse the thermo- quantum regime. dynamic role of coherences. Projection processes map an initial state, ρ, which has coherences in a particular 3.1. Definitions of classical fluctuating work and basis of interest, & with k 1, 2, 3, ...,toastate { k}k = heat in which these coherences are removed, i.e. ρ η '→ := k &k ρ&k. These processes can be interpreted as un- In classical statistical physics, a single particle is assigned selective measurements of an observable with eigenbasis a point, x (q, p), in phase space while an ensemble of # = & , a measurement in which not the individual mea- particles is described with a probability density function, { k}k surement outcomes are recorded but only the statistics of P(x), in phase space. The Hamiltonian of the particle is the outcomes is known. Just like in the case of Landauer’s denoted as H(x, λ) where x is the phase space point of erasure map, the state transfer achieved by the new math- the particle for which the energy is evaluated and λ is ematical map can be implemented in various physical an externally controlled force parameter that can change ways. Different physical implementations of the same in time. For example, a harmonic oscillator Hamilto- 2 λ2 2 map will have different work and heat contributions. nian H(x, λ) p m q can become time dependent = 2m + 2 It was shown that there exists a physical protocol to through a protocol according to which the frequency, λ, implement the projection process such that a non-trivial of the potential is varied in time, λ(t). This particular average work can be extracted from the initial state’s example will be discussed further in Section 6 on QTMs. coherences. For the example that the basis, & ,isthe { k}k Each particle’s state will evolve due to externally ap- energy eigenbasis of a Hamiltonian H E & with = k k k plied forces and due to the interaction with its environ- eigenenergies Ek,themaximum amount of average work ment, resulting in a trajectory x (q , p ) in phase # t = t t that can be extracted in a projection process is space, see Figure 2.Thefluctuating work (done on the system) and the fluctuating heat (absorbed by the system) Wmax k T (S(η) S(ρ)) 0. (12) for a single trajectory x followed by a particle in the ⟨ ext ⟩= B − ≥ t 550 S. VINJANAMPATHY AND J. ANDERS

with the same kind of environment and applying the same protocol. The average work is mathematically obtained as the integral over all trajectories, x , explored by the { t} system, Wτ P (x ) Wτ d (x ), where P (x ) ⟨ ⟩:= t t ,xt D t t t is the probability density of a trajectory x and d (x ) ( t D t the phase space integral over all trajectories [6]. For a closed system, the trajectories are fully determined by their initial phase space point alone: each trajectory starting with x0 evolves deterministically to xτ at time τ.Thus, Figure 2. Phase space spanned by position and momentum the probability of the trajectory is given by the initial coordinates, x (q, p). A single trajectory, xt,startsatphase probability P (x ) of starting in phase space point x , space point x and= evolves in time t to a final phase space point 0 0 0 0 ( ) dx0 d xt dx0, and the Jacobian determinant is dxτ 1. xτ . This single trajectory has an associated fluctuating work Wτ,xt . D → = In general, an ensemble of initial phase space points described Combining with Equations (13), (14)and(15), one by an initial probability density functions, P0(x0),evolvesto obtains some final probability density function Pτ (xτ ). The ensemble of trajectories has an associated average work Wτ .Forclosed closed ⟨ ⟩ Wτ P0(x0) H(xτ (x0), λ(τ)) dynamics, the final probability distribution P0(xτ ) (dashed line) ⟨ ⟩= has the same functional dependence as the initial probability ! $ distribution P0(x0) just evaluated at the final phase space H(x , λ(0)) dx , − 0 0 point xτ . % dx0 P0(xτ ) H(xτ , λ(τ)) dxτ = dxτ time window 0, τ are defined, analogously to (2), as ! ) ) [ ] ) ) P (x ) H(x , λ(0))) dx ), the energy change due to the externally controlled force − 0 0 0 ) 0) parameter and the response of the system’s state, ! Uτ U ,(16) = − 0 τ ∂H(x , λ(t)) t λ( ) Wτ,xt ˙ t dt and i.e. the average work for a closed process is just the dif- := 0 ∂λ(t) ! τ ference of the average energies. Because the evolution is ∂H(xt, λ(t)) Qτ,x xt dt. (13) closed, the distribution in phase space moves but keeps t := ∂x ˙ !0 t its volume, cf. Figure 2,whichisknownasLiouville’s ( ) These are stochastic variables that depend on the parti- theorem. The final probability distribution, P0 xτ ,has cle’s trajectory x in phase space. Together, they give the indeed the same functional form as the initial probability t ( ) energy change of the system, cf. (3), along the trajectory distribution, P0 x0 , just applied to the final trajectory points xτ . That is, the probability of finding xτ at the end is exactly the same as the probability of finding x0 Wτ,x Qτ,x H(xτ , λ(τ)) H(x0, λ(0)). (14) t + t = − initially. In contrast, the probability distribution would For a closed system,onehas change in time in open dynamics due to the system’s non-deterministic interaction with the environment. In τ (16), the initial average energy of the system is defined closed ∂H(xt, λ(t)) ∂H(xt, λ(t)) Qτ,xt qt pt dt as U0 P0(x) H(x, λ(0)) dx and similarly, Uτ is the = 0 ∂qt ˙ + ∂pt ˙ = ! τ $ % average energy at time τ. ( p q q p dt 0, = −˙t ˙t +˙t ˙t = In statistical physics experiments, the average work is !0 closed often established by measuring the fluctuating work Wj W H(x&τ , λ(τ)) H(x' , λ(0)),(15) τ,xt = − 0 for a trajectory observed in a particular run j of the expe- riment, repeating the experiment N times, and taking the since the (Hamiltonian) equations of motion, ∂H p ∂q =−˙ arithmetic mean of the results. In the limit N , and ∂H q,applytoclosedsystems.Thismeansthat →∞ ∂p =˙ this is equivalent to constructing (from a histogram of a closed system has no heat exchange (a tautological the outcomes Wj) the probability density distribution for statement) and that the change of the system’s energy the work, P(W), and then averaging with this function to during the protocol is identified entirely with work for obtain the average work each single trajectory. N The average work of the system refers to repeating the 1 Wτ lim Wj P(W) W dW. (17) same experiment many times, each time preparing the ⟨ ⟩=N N = →∞ j 1 ! same initial system state, bringing the system in contact *= CONTEMPORARY PHYSICS 551

3.2. Classical fluctuation relations thermal distribution. The average exponentiated work done on the system is then obtained similarly to (16)and A central recent breakthrough in classical statistical me- one obtains chanics is the extension of the second law of thermody- namics, an inequality, to equalities valid for large classes βWclosed βWclosed e− τ P (x ) e− τ,xτ dx of non-equilibrium processes. Detailed fluctuation rela- ⟨ ⟩= 0 0 0 ! tions show that the probability densities of stochastically βH(x0,λ(0)) e− β(H(xτ ,λ(τ)) H(x0,λ(0))) fluctuating quantities for a non-equilibrium process, such e− − dx0, = Z(0) as entropy, work and heat, are linked to equilibrium ! 1 βH(xτ ,λ(τ)) dx0 properties and corresponding quantities for the time- ( ) e− dxτ = Z 0 dxτ ! ) ) reversed process [4,62,63]. By integrating over the proba- (τ) ) ) Z β#F ) ) bility densities, one obtains integral fluctuation relations, − . (0) e ) ) (20) such as Jarzynski’s work relation [3]. = Z = An important detailed fluctuation relation is Crooks The beauty of this equality is that for all closed but arbi- relation for a system in contact with a bath at inverse trarily strongly non-equilibrium processes, the average temperature β 1/(k T) for which detailed balanced is = B exponentiated work is determined entirely by equilib- valid [4]. The relation links the work distribution PF(W) rium parameters contained in #F.AsJarzynskishowed, associated with a forward protocol changing the force pa- B the equality can also be generalised to open systems [64]. rameter λ(0) λ(τ), to the work distribution P ( W) βW β W → − By applying Jensen’s inequality e− e− ⟨ ⟩,the associated with the time-reversed backwards protocol ⟨ ⟩≥ Jarzynski equality turns into the standard second law of where λ(τ) λ(0), → thermodynamics, cf. Equation (7), F B β(W #F) P (W) P ( W) e − . (18) = − W #F. (21) ⟨ ⟩≥ Here, the forward and backward protocols each start with Thus, Jarzynski’s relation strengthens the second law by an initial distribution that is thermal at inverse temper- including all moments of the non-equilibrium work, res- ature β for the respective values of the force parameter. ulting in an equality from which the inequality follows The free energy difference #F F(τ) F(0) refers to = − for the first moment. the two thermal distributions with respect to the final, Fluctuation theorems have been measured, for exam- H(x, λ(τ)),andinitial,H(x, λ(0)),Hamiltonianinthe ple, for a defect centre in diamond [65], for a torsion pen- forward protocol at inverse temperature β.Here,the dulum [66] and in an electronic system [67]. They have free energies, defined in (6), can be expressed as F(0) = also been used to infer, from the measurable work in non- 1 ln Z(0) at time 0 and similarly at time τ with classical − β equilibrium pulling experiments, the desired equilibrium partition functions Z(0) e βH(x,λ(0)) dx and simi- := − free energy surface of biomolecules [68,69], which is not larly Z(τ). Rearranging and integrating over dW on both ( directly measurable otherwise. sides result in a well-known integral fluctuation relation, the Jarzynski equality [3], 3.3. Fluctuation relations with feedback βW F βW β#F e− P (W) e− dW e− . (19) ⟨ ⟩= = It is interesting to see how Maxwell demon’s feedback ! process discussed in Section 2.2 can be captured in a gen- Pre-dating Crooks relation, Jarzynski proved his eralised Jarzynski equation [6]. Again, one assumes that equality [3] by considering a closed system that starts the system undergoes closed dynamics with a βH(x ,λ(0)) e− 0 with a thermal distribution, P (x ) ( ) , for a time-dependent Hamiltonian; however, now the force 0 0 = Z 0 given Hamiltonian H(x, λ(0)) at inverse temperature β. parameter in the Hamiltonian is changed according to a The Hamiltonian is externally modified through its force protocol that depends on the phase space point the system parameter λ which drives the system out of equilibrium is found in. For example, for the two choices of a particle and into evolution according to Hamiltonian dynamics being in the left box or the right box, see Figure 3,the in a time interval 0, τ . Such experiments can be realised, initial Hamiltonian H(x, λ(0)) will be changed to either [ ] e.g. with colloidal particles see Section 3.4, where the H(x, λ1(τ)) for the particle in the left box or H(x, λ2(τ)) experiment is repeated many times, each time imple- for the particle in the right box. Calculating the average menting the same force protocol. Averages can then be exponentiated work for this situation, see Equation (20), calculated over many trajectories, each starting from an one now notices that the integration over trajectories initial phase space point that was sampled from an initial includes the two different evolutions, driven by one of 552 S. VINJANAMPATHY AND J. ANDERS the two Hamiltonians. The free energy difference, #F, other final Hamiltonians, i.e. H(y, λ2(τ)).Theefficacy previously corresponding to the Hamiltonian change, see is now the added probability for each of the protocols, (k) Equation (20), is now itself a fluctuating function. Either γ P (yτ ). = k it is #F F(1) F(0) or it is #F F(2) F(0),depending Applying Jensen’s inequality to Equation (22), one = − = − # on the measurement outcome in each particular run. As now obtains W #F 1 ln γ ,whichwhenassuming ⟨ ⟩≥⟨ ⟩− β aresult,thecorrespondingworkfluctuationrelationcan that a cycle has been performed, #F 0, the maximal ⟨ ⟩= be written as value of γ for two feedback options has been reached, γ 2, and the inequality is saturated, becomes = β(W #F) e− − γ ,(22) ⟨ ⟩= Wdemon W k T ln 2, (23) ⟨ ext ⟩=−⟨ ⟩= B where the average includes an average over the two dif- ⟨·⟩ ferent protocols. Here, γ 0characterisesthefeedback in agreement with Equation (9). A recent review pro- ≥ efficacy, which is related to the reversibility of the non- vides a detailed discussion of feedback in classical fluc- equilibrium process. When H(x, λ (τ)) H(x, λ (τ)), tuation theorems [31]. The efficiency of feedback control 1 = 2 i.e. no feedback is actually implemented, then γ becomes in quantum processes has also been analysed, see e.g. unity and the Jarzynski equality is recovered. The maxi- [70–72]. mum value of γ is determined by the number of different protocol options, e.g. for the two feedback choices dis- 3.4. Classical statistical physics experiments cussed above, one has γ max 2. = Apart from measuring e β(W #F) by performing A large number of statistical physics experiments are ⟨ − − ⟩ the feedback protocol many times and recording the work performed with colloidal particles, such as polystyrene done on the system, γ can also be measured experi- beads, that are suspended in a viscous fluid. The link mentally, see Section 3.4.Todoso,thesystemisini- between information theory and thermodynamics has tially prepared in a thermal state for one of the final been confirmed by groundbreaking colloidal experiments Hamiltonians, say H(y, λ1(τ)), at inverse temperature β. only very recently. For example, the heat-to-work con- The force parameter λ1(t) in the Hamiltonian is now version achieved by a Maxwell demon, see Sections 2.2 changed backwards, λ (τ) λ(0), without any feed- and 3.3,interveninginthestatisticaldynamicsofasys- 1 → back. The particle’s final phase space point yτ is recorded. tem was investigated [73] with a dimeric polystyrene (1) The probability P (yτ ) is established that the particle bead suspended in a fluid and undergoing rotational ended in the phase space volume that corresponds to Brownian motion, see Figure 3(a). An external electrical the choice of H(y, λ1(τ)) in the forward protocol, in this potential was applied so that the bead was effectively example in the left box. The same is repeated for the moving on a spiral staircase, see Figure 3(b). The demon’s

Figure 3. (a) Dimeric polystyrene bead and electrodes for creating the spiral staircase. (b) Schematic of a Brownian particle’s dynamics β(W #F) climbing the stairs with the help of the demon. (c) Experimental results of e− − for the feedback process, and of γ for a time-reversed (non-feedback) process, see Equation (22), over the demon’s reaction⟨ time ⟩ϵ.Figurestakenfrom[73]reproducedwith permission. CONTEMPORARY PHYSICS 553

Figure 4. Left (a)–(f) Erasure protocol for a particle trapped in a double well. Particles starting in either well will end up in the right-hand side well at the end of the protocol. Right (b) Measured heat probability distribution P(Qdis) over fluctuating heat value Qdis for a fixed protocol cycle time. Right (c) Average heat in units of kBT,i.e. Qdis /(kB T),overvaryingcycletimeτ in seconds. Figures taken from [75] reproduced with permission. ⟨ ⟩ action was realised by measuring if the particle had moved on the bead suspended in the fluid. The dissipated heat up and, depending on this, shifting the external ladder was measured by following the trajectories xt of the bead τ ∂ (xt ,λ(t)) potential quickly, such that the particle’s potential energy and integrating, Qdis U xt dt,cf.Equa- =− 0 ∂xt ˙ was ‘saved’ and the particle would continue to climb up. tion (13). Here, (x , λ(t)) is the explicitly time-varying U t ( The work done on the particle, W, was then measured potential that together with a constant kinetic term makes for over 100,000 trajectories to obtain an experimen- up a time-varying Hamiltonian H(x , λ(t)) T(x ) t = t + tal value of e β(W #F) , required to be identical to 1 (x , λ(t)). The measured heat distribution P(Q ) and ⟨ − − ⟩ U t dis by the standard Jarzynski equality, Equation (20). Due average heat Q ,definedanalogouslytoEquation(17), ⟨ dis⟩ the feedback operated by the demon, the value did turn are shown on the right in Figure 4.Thetimetakento out larger than 1, see Figure 3(c), in agreement with implement the protocol is denoted by τ. In the quasi- Equation (22). A separate experiment implementing the static limit, i.e. the limit of long times in which the system time-reversed process without feedback was run to also equilibrates throughout its dynamics, it was found that determine the value of the efficacy parameter γ predicted the Landauer limit, kB T ln 2, for the minimum dissipated to be the same as e β(W #F) .Figure3(c) shows very heat is indeed approached. ⟨ − − ⟩ good agreement between the two experimental results for Another beautiful high-precision experiment uses varying reaction times of the demon, reaching the highest electrical feedback to effectively trap a colloidal particle feedback efficiency when the demon acted quickly on the and implement an erasure protocol [76]. This experi- knowledge of the particle’s position. The theoretical pre- ment provides a direct comparison between the mea- dictions of fluctuation relations that include Maxwell’s sured dissipated heat for the erasure process as well as demon [6]havealsobeentestedwithasingleelectron anon-erasureprocessshowingthattheheatdissipation box analogously to the original Szilard engine [74]. is indeed a consequence of the erasure of information. Another recent experiment measured the heat dissi- Possible future implementations of erasure and work pated in an erasure process, see Section 2.3,withacol- extraction processes using a quantum optomechanical loidal silica bead that was optically trapped with tweezers system, consisting of a two-level system and a mechanical in a double well potential [75]. The protocol employed, oscillator, have also been proposed [77]. see Figure 4, ensured that a bead starting in the left well would move to the right well, while a starting position 3.5. Definitions of quantum fluctuating work and in the right well remained unchanged. The lowering and heat raising of the energetic barrier between the wells were achieved by changing the trapping laser’s intensity. The By its process character, it is clear that work is not an tilting of the potential, seen in Figure 4(c)–(e), was neatly observable [41], i.e. there is no operator, w,suchthat realised by letting the fluid flow, resulting in a force acting W tr w ρ . To quantise the Jarzynski equality, the = [ ] 554 S. VINJANAMPATHY AND J. ANDERS

(τ) (τ) crucial step taken is to define the fluctuating quantum where p p are the marginals of the joint m := n n,m work for closed dynamics as a two-point correlation probability distribution. Assuming the initial state ρ(0) # (0) function. That is, a projective measurement of energy was diagonal in the initial energy basis, en n,the (τ) {| } needs to be performed at the beginning and end of the pm turn out to be just the probabilities for measur- (τ) (τ) (τ) process to gain knowledge of the system’s energetic ing the energies Em in the final state ρ , i.e. pm change. This enables the construction of a work dis- (τ) (τ) (τ) = em ρ em . Thus, the average work for the unitary tribution function (known as the two-point measure- | | -process is just. the internal energy difference, cf. (16), ment work distribution) and allows the formulation of the Tasaki–Crooks relation and the quantum Jarzynski’s closed (τ) (τ) (0) (0) Wτ tr H ρ tr H ρ #U. (27) equality [5,41,78,79]. ⟨ ⟩= [ ]− [ ]= To introduce the quantum fluctuating work and heat, consider a quantum system with initial state ρ(0) and ini- 3.6. Quantum fluctuation relations (0) (0) (0) (0) tial Hamiltonian H n En en en with eigen- ( ) = | ( )⟩⟨ | With the above definitions, in particular that of the work values E 0 and energy eigenstates e 0 .Aclosed system n # | n ⟩ distribution, the quantum Jarzynski equation can be undergoes dynamics due to its time-varying Hamiltonian readily formulated for a closed quantum system under- which generates a unitary transformation (here, stands τ ( ) going externally driven (with unitary V)non-equilibrium (τ) i H t Tdt/! for a time-ordered integral) V e− 0 and dynamics [5,41,78–80]. The average exponentiated (τ) (τ)= (T0) (τ)† ends in a final state ρ V ρ V ( and a final ρ(0) (τ) =(τ) (τ) (τ) work done on a system starting in initial state now Hamiltonian H m Em em em , with energies (τ) = (|τ) ⟩⟨ | becomes, cf. (20), E and energy eigenstates e . To obtain the fluctu- m # | m ⟩ closed (τ) (0) ating work, the energy is measured at the beginning of βWτ βW (τ) β(Em En ) (0) e− P(W) e− dW pn,m e− − . the process, giving e.g. En ,andthenagainattheend ⟨ ⟩= = (τ) ! *n,m of the process, giving e.g. in Em .Thedifferencebetween (28) the measured energies is now identified entirely with fluctuating work, When the initial state is a thermal state for the Hamilto- 0 (0) (0) (0) nian H at inverse temperature β, e.g. ρ n pn en (τ) ( ) = (0) | ⟩ closed 0 (0) (0) βEn Wm,n Em En ,(24) e− := − en ,withthermalprobabilitiespn Z(0#) and parti- ⟨ | (0) = tion function Z(0) e βEn ,oneobtainsthequan- = n − as the system is closed, the evolution is unitary and no tum Jarzynski equality heat dissipation occurs, cf. the classical case (15). The # ( ) work distribution is then peaked whenever the distribu- βE 0 closed n (τ) (0) τ βWτ e− (τ) β(Em En ) tion variable W coincides with Wm,n, e− ( ) pm n e− − ⟨ ⟩= Z 0 | *n,m (τ) (τ) (τ) (τ) (0) 1 βE Z β#F P(W) pn,m δ W (Em En ) . (25) − m − . = − − (0) e (0) e n,m = Z = Z = * + , *m (29) (τ) (0) (τ) Here, pn,m pn pm n is the joint probability distribution = | (0) Here, we have used that the conditional probabilities of finding the initial energy level En and the final energy (τ) (τ) (τ) (0) 2 (τ) sum to unity, n pm n n em V en 1and Em . This can be broken up into the probability of finding (τ) | = |⟨ | | ⟩| = (0) (0) (0) (0) (0) (τ) βEm the initial energy E , p e ρ e ,andthe Z m e#− is, like# in the classical case, the par- n n =⟨n | | n ⟩ = conditional probability for transferring from n at t 0 tition function of the hypothetical thermal configuration (τ) (τ) ( ) = # (τ) τ (τ) 0 2 for the final Hamiltonian, H . The free energy difference to m at t , pm n em V en . (τ) = | =|⟨ | | ⟩| 1 Z is, like in the classical case, #F ln ( ) . The average work for the unitarily driven non- =−β Z 0 equilibrium process can now be calculated as the average Similarly, the classical Crooks relation, cf. Equation over the work probability distribution, i.e. using (17), (18), can also be re-derived for the quantum regime and is known as the Tasaki–Crooks relation [5], Wclosed p(τ) δ(W (E(τ) E(0))) W dW ⟨ τ ⟩= n,m − m − n ! n,m F * P (W) β(W #F) ( ) (τ) (τ) ( ) − . 0 ( 0 ) B e (30) pn pm n Em En P ( W) = = | − − *n,m (τ) (τ) (0) (0) It is interesting to note that in the two-point mea- pm Em pn En ,(26) = − surement scheme, both, the quantum Jarzynski equality *m *n CONTEMPORARY PHYSICS 555 and the Tasaki–Crooks relation, show no difference to ρ(0) at inverse temperature β, for which four values were their classical counterparts, contrary to what one might realised. The experiment implemented the time-varying expect. A debated issue is that the energy measurements spin Hamiltonian, H(t), resulting in a unitary evolution, remove coherences with respect to the energy basis [41], i.e. V in Section 3.6, by applying a time-modulated radio and these do not show up in the work distribution, P(W). frequency field resonant with the 13Cnuclearspinina It has been suggested [61] that the non-trivial work and short time window τ. The (forward) work distribution heat that may be exchanged during the second measure- of the spin’s evolution, PF(W) see Figure 5(a), was then ment, see Section 2.6, implies that identifying the energy reconstructed through a series of one- and two-body op- change entirely with fluctuating work (24)isinconsistent. erations on the system and the ancilla, and measurement If the initial state is not thermal but has coherences, of the transverse magnetisation of the 1H nuclear spin. then also the first measurement will non-trivially affect Similarly, a backwards protocol was also implemented these initial coherences and lead to work and heat contri- and the backwards work distribution, PB(W),measured, butions. Work probability distributions and generalised cf. Sections 3.2 and 3.4. Jarzynski-type relations have been proposed to account The measured values of the Tasaki–Crooks ratio, for these coherences using path integral and quantum PF(W)/PB( W), are shown over the work value W − jump approaches [81,82]. in Figure 5(b) for four values of temperature. The trend followed by the data associated with each temperature 3.7. Quantum fluctuation experiments was in very good agreement with the expected linear While a number of interesting avenues to test fluctua- relation, confirming the predictions of the Tasaki–Crooks tion theorems at the quantum scale have been proposed relation. The cutting point between the two work distri- [83–86], the experimental reconstruction of the work butions was used to determine the value of #F experi- statistics for a quantum protocol has long remained mentally which is shown in Figure 5(c). Calculating the elusive. A new measurement approach has recently been average exponentiated work with the measured work devised that is based on well-known interferometric distribution, the data also showed good agreement with schemes of the estimation of phases in quantum sys- the quantum Jarzynski relation, Equation (29), see table tems, which bypasses the necessity of two direct pro- in Figure 5(d). jective measurements of the system state [87]. The first quantum fluctuation experiment that confirms the quan- 4. Quantum dynamics and foundations of tum Jarzynski relation, Equation (29), and the Tasaki– thermodynamics Crooks relation, Equation (30), with impressive accuracy has recently been implemented in a nuclear magnetic The information theoretic approach to thermodynamics resonance (NMR) system using such an interferomet- employs many standard tools of quantum information to ric scheme [88]. Another very recent experiment uses study mesoscopic systems. For instance, an abstract view trapped ions to measure the quantum Jarzynski equality of dynamics, minimal in the details of Hamiltonians, is [89]withthetwomeasurementmethodsusedtoderive often employed in quantum information. Such a view the theoretical result in Section 3.6. of dynamics as a map between quantum states serves The NMR experiment was carried out using liquid- to produce rules common to generic dynamics and has state NMR spectroscopy of the 1Hand13C nuclear spins served the study of computing and technologies well. In of a chloroform molecule sample. This system can be this section, we describe some of these techniques and regarded as a collection of identically prepared, non- theorems and highlight how they are used in quantum interacting, spin-1/2 pairs [90]. The 13Cnuclearspinis thermodynamics. the driven system, while the 1Hnuclearspinembodies an auxiliary degree of freedom, referred to as ‘ancilla’. 4.1. Completely positive maps Instead of performing two projective measurements on the system, it is possible to reconstruct the system’s work We begin by reviewing the description of generic quan- distribution with an interferometric scheme in which the tum maps [91–96]. The point of describing ancilla is instrumental [87]. The ancilla here interacts dynamics through a ‘map’ as opposed to a model of tem- with the system at the beginning and the end of the poral dynamics (i.e. a Hamiltonian) is deliberate. Maps process, and as a result, its state acquires a phase. This are not explicit functions of time (though they can be phase corresponds to the energy difference experienced parametrised by time, as we will see later), but are two- by the system and the ancilla is measured only once to ob- point functions. They accept initial states of the dynamics tain the energy difference of the system. The 13Cnuclear they model and output final states. Specifically, a com- spin was prepared initially in a pseudo-equilibrium state pletely positive trace-preserving (CPTP) map transforms 556 S. VINJANAMPATHY AND J. ANDERS

(b) (c)

(a)

(d)

Figure 5. (a) Experimental work distributions corresponding to the forward (backward) protocol, PF (W) (PB( W)), are shown as red squares (blue circles). (b) The Tasaki–Crooks ratio is plotted on a logarithmic scale for four values of the system’s− temperature. (c) Crosses indicate mean values and uncertainties for #F and β obtained from a linear fit to data compared with the theoretical prediction indicated by the red line. (d) Experimental values of the left- and right-hand side of the quantum Jarzynski relation measured for three (τ) temperatures together with their respective uncertainties, and theoretical predictions for ln Z showing good agreement. Figures taken Z(0) from [88], reproduced with permission. input density matrices ρ into physical output states ρ′. We can now consider the action of the transpose map The term ‘physical’ here refers to the requirement that on a subsystem. For example, for the two-qubit Bell state the output state is again a well-defined density matrix. φ ( 00 11 )/√2, one may apply transposition | +⟩= | ⟩+| ⟩ Amap, has to obey several rules to guarantee that its only on the second qubit. This map then gives outputs are physical states. These properties include: 1001 (1) trace preservation: ρ ,(ρ) has unit trace for ′ := 1 0000 all input states ρ of dimensionality d.Ifthisisvio- φ+ φ+ ⎛ ⎞ | ⟩⟨ |=2 0000 lated, the output states become unphysical in that ⎜ ⎟ ⎜ 1001⎟ Born’s rule cannot be applied directly anymore. ⎝ ⎠ (2) positivity: ,(ρ) has non-negative eigenvalues, in- 1000 1 0010 terpreted as probabilities. I T φ+ φ+ ⎛ ⎞ ,(32) ( ) (3) complete positivity: k 0, ... (I k ,) '→ ⊗ | ⟩⟨ | = 2 0100 (k d) ∀ ∈{ ∞}: ⊗ & ' ⎜ 0001⎟ (σ + ) has non-negative eigenvalues for all states ⎜ ⎟ (k d) ⎝ ⎠ σ + . This subsumes positivity. where the resulting global state turns out to have one The final property is the statement that if the CPTP map negative eigenvalue. Because of this property, transpo- is acting on a subsystem of dimension d which is part of sition is not a completely positive map. This example a larger (perhaps entangled) system of dimension k d, demonstrates that positivity, e.g. maintaining positive (k d) + whose state is σ + , then the resultant global state must eigenvalues of a single qubit state, does not imply com- also be a ‘physical’ state. plete positivity, e.g. maintaining positive eigenvalues of For instance, let us consider the transpose map T a two-qubit state when the map is only applied to one which, when applied to a given density matrix, transposes qubit. The fact the ‘partial’ transposition is not positive all its matrix elements in a fixed basis. When applied to when the initial state is entangled is used extensively as a physical states of dimension d,thismapalwaysoutputs criterion to detect entanglement [96]. physical states of dimension d. For example, for a general CPTP maps are related to dynamical equations, which qubit state ρ with Bloch vector r (x, y, z),thetranspo- ⃗ = sition map gives we discuss briefly. When the system is closed, and evolves under a unitary V,theevolutionofthedensitymatrixis given by ρ VρV †. A more general description of the ′ = 1 1 zx iy 1 1 zx iy transformations between states when the system is inter- ρ + − T(ρ) + + . = 2 x iy 1 z '→ = 2 x iy 1 z acting with an external environment is given by Lindblad- $ + − % $ − − (31)% type master equations. Such dynamics preserves trace Note that the eigenvalues of the matrix are invariant and positivity of the density matrix, while allowing the under the transposition map T. density matrix to vary otherwise. Such equations have CONTEMPORARY PHYSICS 557 the general form arbitrary environmental ancilla state, τE, a joint unitary and a partial trace. This assertion is written as dρ † 1 † 1 † i H, ρ AkρA A Akρ ρAkA . =−[ ]+ k − k − k † dt 2 2 ρ′ ,(ρ) tr V ρ τ V . (34) *k 5 6 = = E[ ⊗ E ] (33) Stinespring dilation allows for any generic quantum Here, Ak are Lindblad operators that describe the effect dynamics of the system, S,tobethoughtofinterms of the interaction between the system and the environ- of a global unitary V acting on the system and an en- ment on the system’s state. Notice that if an initial state vironment, E.Iftheinteractionsbetweenthesystemand ρ(t 0) is evolved to a final state ρ(t T), then the environment are known, then the time dependence = = the resulting transformation can be captured by a CPTP of the map may be specified. For instance, if the Hamil- map ,(ρ(t 0)) ρ(t T). Master equations are tonian governing the joint dynamics is H , then V = = = SE = typically derived with many assumptions, including that V (t) exp ( iH t). However, Stinespring dilation SE := − SE the system is weakly coupled to the environment and that allows the mathematical analysis of many properties of the environmental correlations decay sufficiently quickly maps for general sets of unitaries. We note that many such that the initial state of the system is uncorrelated to quantum thermodynamics papers assume τE to be the the environment. Lindblad-type master equations have Gibbs state of the environment and use this to study, been extensively employed to the study of quantum en- e.g., the influence of temperature on the dynamics of the gines, and this connection will be discussed in Section 6.2. quantum system which interacts with the environment. Some master equations are not in the Lindblad form, see For example, in [98], the authors considered thermali- [97] for example. We note that the structure of Equation sation of a quantum system, a topic we will discuss in (33) arises naturally if Markovianity, trace preservation Section 4.3. They prove a condition for the long time of the density matrix and positivity are considered. To states of a system to be independent of the initial state see that this should be the case, consider the general of the system. This condition relies on smooth min- and evolution step for the density matrix, which we write max-entropies which are defined below. without loss of generality as dρ (Mρ ρN)dt.From Another theorem used in [98]andotherstudies[99] =− + the Hermiticity of the density matrix, we have dρ† dρ, of quantum thermodynamics is the channel-state duality = implying N M†.Wecanhencebetemptedtowritean or Choi–Jamiołkowski isomorphism. It says that there = † evolution equation of the form dρ (Mρ ρM )dt. is a state ρ, isomorphic to every CPTP map ,, given =− + If we write M iH J, this form does not preserve by = + trace for the terms involving J. This problem is fixed by subtracting an appropriate term. If we write J F†F, ρ, I ,( ϕ ϕ ),(35) = := ⊗ | ⟩⟨ | then we can write a trace-preserving equation, namely: d dρ i H, ρ dt (2FρF† F†Fρ ρF†F)dt, where where ϕ (1/√d) i i is a maximally en- =−[ ] + − − | ⟩= i | ⟩⊗|⟩ ρ † tangled bipartite state. The main purpose of this the- 2F F ensures trace preservation. Writing the operator # F in an orthonormal basis and diagonalising the resulting orem is that it allows us to think of the CPTP map operator produces the general equation above. We note , as a state ρ, which is often useful to make state- that care has to be taken that the dynamical equation also ments about the amount of correlations generated by a preserves positivity, see [95] for a detailed derivation. given CPTP map. For example, see [100] for a discus- sion on entropic inequalities that employ this formal- ism. 4.1.1. Three theorems involving CPTP maps The final theorem is the operator sum representation, We now briefly mention three important theorems often which asserts that any map, which has the representation used in the quantum information theoretic study of ther- † modynamics. These theorems are the Stinespring dila- ρ′ ,(ρ) Kµ ρ K ,(36) = := µ tion theorem, the channel-state duality theorem and the µ * operator sum representation, respectively. The impor- † tance of these theorems lies in the alternative perspectives with K Kµ I, is completely positive (but not nec- µ = they afford an abstract dynamical map. We mention the essarily trace preserving). These operators, often known # theorems briefly, alongside a brief remark about the light as Sudarshan–Kraus operators, are related to the ancilla they shed on CPTP maps. state τE and V in Equation (34). We note that these The first of these, Stinespring dilation theorem asserts theorems are often implicitly assumed while discussing that every CPTP map , can be built up from three quantum thermodynamics from the standpoint of CPTP fundamental operations, namely: tensor product with an maps, see [86,99,101] for examples. 558 S. VINJANAMPATHY AND J. ANDERS

4.1.2. Entropy inequalities and majorisation β-ordered σ ,thenthestateρ can be transformed to σ in We state key results from quantum information theory the resource theoretic setting discussed in Section 5. relating to entropy often employed in studying thermo- dynamics. Specifically, we discuss convexity, subadditiv- 4.1.3. Smooth entropies ity, contractivity and majorisation. We begin by defining Smooth entropies are related to von Neumann entropy quantum relative entropy as defined in (8)andtheconditionalentropyusedinSection 2.4. Smooth entropies are central in the single shot and S ρ σ tr ρ ln ρ ρ ln σ . (37) resource theoretic approaches to quantum thermody- [ ∥ ]:= [ − ] namics, presented in Section 5. The negative of the first term is the von Neumann entropy Let us consider a bipartite system, whose parties are of the state ρ,seeSection2.1. labelled A and B. The smooth entropies are defined in The relative entropy is jointly convex in both its inputs. terms of min- and max-entropies conditioned on B, given This means that if ρ p ρ and σ p σ , by = m m m = m m m with pm probabilities then λ # # Smin(A B)ρ sup sup λ R 2− IA σB ρAB , | := σB { ∈ : ⊗ ≥ } S ρ σ pkS ρk σk . (38) (41) ∥ ≤ [ ∥ ] k 2 7 8 * Smax(A B)ρ sup ln F(ρAB, IA σB) . (42) | := σB ⊗ Furthermore, for any tripartite state ρABC defined over 7 8 three physical systems labelled A, B and C,theentropies Here, the supremum is over all states σB in the space of various marginals are bounded by the inequality of subsystem B,andF(x, y) √x√y ,with g := ∥ ∥1 ∥ ∥1 := tr g†g is the fidelity. Note that we retain standard nota- S(ρ ) S(ρ ) S(ρ ) S(ρ ). (39) [ ] ABC + B ≤ AB + BC tion9 here, employed, for instance, in standard textbooks on quantum information theory [112]. The smoothed ε This inequality, known as the strong subadditivity in- versions of min-entropy, Smin, are defined as the maxi- equality, is equivalent to the joint convexity of quantum mum over the min-entropy Smin of all states σAB which relative entropy and is used, for instance in [102], to are in a radius (measured in terms of purified distance ε discuss area laws in quantum systems. [113]) of ε from ρAB. Likewise, Smax is defined as the The quantum relative entropy is monotonic, or con- minimum Smax over all states in the radius ε.Thiscanbe tractive,undertheapplicationofCPTPmaps.Forany written as (see Chapter 4 in [23]) two density matrices ρ and σ , this contractivity relation [103] is written as min Sα(A)ρ,ifα < 1 ϵ ρ ˜ Sα(A)ρ ⎧ ˜ (43) := max Sα(A)ρ,ifα > 1, 0 ϵ < 1, S ρ σ S ,(ρ) ,(σ) . (40) ⎨⎪ ρ ˜ ≤ [ ∥ ]≥ [ ∥ ] ˜ ⎪ Contractivityiscentraltocertainextensionsofthesecond where ρ ε ρ⎩are an ε-ball of states close to ρ.Wenote ˜ ≈ law to entropy production inequalities, see [100,104–106] that there is more than one smoothing procedure, and and is also obeyed by trace distance. There has been recent care needs to be taken to not confuse them [98]. interest in improving these inequality laws in relation to irreversibility of quantum dynamics [107–110]. 4.2. Role of fixed points in thermodynamics Finally, majorisation is a quasi-ordering relationship between two vectors. Consider two vectors x and y,which Fixed points of CPTP maps play an important role in ⃗ ⃗ are n-dimensional. Often in the context of quantum ther- quantum thermodynamics. This role is completely anal- modynamics, these vectors will be the eigenvalues of ogous to the role played by equilibrium states in equi- density matrices. The vector x is said to majorise y, writ- librium thermodynamics. In equilibrium thermodynam- ⃗ ⃗ ten as x y, if all partial sums of the ordered vector ics, the equilibrium state is defined by the laws of ther- ⃗ ≻⃗ k n k n obey the inequality ≤ x↓ ≤ y↓.Here,x is the modynamics. If two bodies are placed in contact with i i ≥ i i ⃗↓ vector x,sortedindescendingorder.Arelatedideaisthe each other, and can exchange energy, they will eventually ⃗ # # notion of thermo-majorisation [111]. Consider p(E, g) to equilibrate to a unique state, see Section 4.3.Onthe be the probability of a state ρ to be in a state g with other hand, if the system of interest is not in thermal energy E. At a temperature β,suchastateisβ-ordered equilibrium, notions such as equilibrium, temperature βEk βEk 1 if e p(Ek, gk) e − p(Ek 1, gk 1), for k 1, 2, ... . and thermodynamic entropy are ill-defined. In the non- ≥ − − ={ } If two states ρ and σ are such that β-ordered ρ majorises equilibrium setting, [114] proposed a thermodynamic CONTEMPORARY PHYSICS 559 framework where the equilibrium state was replaced by use the CP formalism described here to derive fluctuation the fixed point of the map that generates the dynamics. relations discussed in Section (3.3). If the steady state of a particular dynamics is not the thermal state, since there is entropy produced in the steady state, ‘housekeeping heat’ was introduced as a way 4.3. Thermalisation of closed quantum systems of taking into account the deviation from equilibrium. One of the fundamental problems in physics involves This programme, called ‘steady-state thermodynamics’ understanding the route from microscopic to macro- [114–116], for a classical system described by Langevin scopic dynamics. Most microscopic descriptions of equation is described in [116]andtheconnectionto physical reality, for instance, are based on laws that are fluctuation theorems is elaborated. time-reversal invariant [124]. Such symmetries play a For a quantum system driven by a Lindblad equation strong role in the construction of dynamical descrip- given in Equation (33), the fixed point is defined as the tions at the microscopic level, but many problems persist steadystateoftheevolution.Suchasteadystateisfixedby when the derivation of thermodynamic laws is attempted the Hamiltonian and the Lindblad operators that describe from quantum mechanical laws. In this subsection, we the dynamics. If a system whose steady state is an equi- highlight some examples of the problems considered and librium state is taken out of equilibrium, the system will introduce the reader to some important ideas. We refer to relax back to equilibrium. This relaxation mechanism will three excellent reviews [125–127] for a detailed overview be accompanied by a certain entropy production, which of the field. ceases once the system stops evolving (this can be an An information theoretic example of the foundational asymptotic process). In the non-equilibrium setting, let problems in deriving thermodynamics involves the log- the given dynamics be described by a CPTP map ,.Such ical paradox in reconciling the foundations of thermo- a map is a sufficient description of the dynamical pro- dynamics with those of quantum mechanics. Suppose cesses of interest to us. Every such map, , is guaranteed we want to describe the universe, consisting of system to have at least one fixed point [117], which we refer to S and environment E. From the stand point of quantum as ρ⋆. If we compare the relative entropy S ρ ρ⋆ before [ ∥ ] mechanics, then we would assign this universe a pure and after the application of the map ,, then this quan- state, commensurate with the (tautologically) isolated tity reduces monotonically with the application of the nature of the universe. This wave function is constrained CPTP map ,. This is because of contractivity inequality, by the constant energy of the universe. On the other hand, given by Equation (40). Since the entropy production if we thought of the universe as a closed system, from the ceases only when the state reaches the fixed point ρ⋆,we standpoint of statistical mechanics, we assume the axiom can compare an arbitrary initial state ρ with the fixed of ‘equal a priori probability’ to hold. By this, we mean point to study the deviation from steady-state dynamics. that each configuration commensurate with the energy Using this substitution, the contractivity inequality can constraint must be equally probable. Hence, we expect be rewritten as a difference of von Neumann entropies, the universe to be in a maximally mixed state in the namely given energy shell. This can be thought of as a Bayesian [128] approach to states in a given energy shell. This is in S , ρ S ρ tr , ρ ln ρ⋆ ρ ln ρ⋆ ς. [ [ ]]− [ ]≥− [ ] − := − (44) contradiction with the assumption made about isolated 7 8 quantum systems, namely: that states of such isolated This result [104,118] states that the entropy production systems are pure states. In [7], the authors resolved this by is bounded by the term on the right-hand side, a factor pointing out that entanglement is the key to understand- which depends on the initial density matrix and the map. ing the resolution to this paradox. They replace the axiom If N maps ,i are applied in succession to an initial density of equal a priori probability with a provable statement matrix ρ, the total entropy change can be shown easily known as the ‘general canonical principle’. This principle [105] to be bounded by can be stated as follows: if the state of the universe is a pure state φ ,thenthereducedstateofthesystem N | ⟩ S , ρ S ρ ςk. (45) [ [ ]]− [ ]≥− ρ tr φ φ (46) k 1 S E *= := [| ⟩⟨ |] Here, , is the concatenation of the maps , , is very close (in trace distance, see Section 4)tothestate := N ◦ ,N 1 ... ,1.Wereferthereaderto[70,105,119–121] 4S for almost all choices of the pure state φ .Thephrase − ◦ ◦ | ⟩ for a discussion of the relationship between steady states ‘almost all’ can be quantified into the notion of typicality, and laws of thermodynamics and [122] for a discussion see [7]. The state 4S is the canonical state, defined as on limit cycles for quantum engines. In [123], the authors 4 tr ( ), the partial reduction of the equiprobable S := E E 560 S. VINJANAMPATHY AND J. ANDERS state corresponding to the maximally mixed state in a to thermalise if this represents a model of thermalisation. E shell (or subspace) corresponding to a general restriction. But from the axioms of the micro-canonical ensemble, An example of such a restriction is the constant energy the expectation value is expected to be proportional to restriction, and hence the ‘shell’ corresponds to the sub- k Bkk,inkeepingwithequal‘aprioriprobability’about space of states with the same energy. Since this applies to the mean energy E .Thisisasourceofmysterysince # 0 any restriction, and not just the energy shell restriction the long time average c 2B somehow needs to k | k| kk we motivated our discussion with, the principle is known ‘forget’ about the initial state of the system φ , given # | ⟩ as the general canonical principle. by coefficients ck. The intuition here is that almost any pure state of This can happen in three ways. The first mechanism is the universe is consistent with the system of interest to demand that Bkk be non-zero only in the narrow band being close to the canonical state, and hence the axiom discussed above and come outside the sum. This is called of equal a priori probability is modified to note that the ‘eigenstate thermalisation hypothesis’ (ETH) [132,133], system cannot tell the difference between the universe see [26] for more details on ETH. The second method being pure or mixed for a sufficiently large universe. We to reconcile the long time average with the axiom of the emphasise that though almost all pure states have a lot of micro-canonical ensemble is to demand that the coeffi- entanglement across any cut (i.e. across any number of cients ck can be constant and non-zero for a subset of parties you choose in each subsystem and any number of indices k.Thethirdmechanismfortheindependenceof subsystems you divide the universe into), not all states the long time average of expectation values of observ- do. A simple example of this is product states of the ables, and hence the observation of thermalisation in a form ψ ψ ... ψ .Entanglementbetweenthe closed quantum system is to demand that the coefficients | 1⟩⊗| 2⟩ | N ⟩ subsystems is understood to play an important role in ck be uncorrelated to Bkk. This causes ck to uniformly this kinematic (see below) description of thermalisation, sample Bkk the average given simply by the mean value, leading to the effect that sufficiently small subsystems in agreement with the micro-canonical ensemble. ETH are unable to distinguish highly entangled states from was numerically verified for hardcore bosons in [133] maximally mixed states of the universe, see [15,129,130] and we refer the interested reader to a review of the field for related works. in [25], see [134,135] for a geometric approach to the This issue of initial states is only one of the many topics discussion of thermodynamics and thermalisation. of investigation in this modern study of equilibration and An integrable system is defined as a system where the thermalisation [131]. When thermalisation of a quantum number of conserved local quantities grows extensively. system is considered, it is generally expected that the Here, the notions of local and global are with respect to system will thermalise for any initial condition of the the constituent subsystems. We write local to differen- system (i.e. for ‘almost all’ initial states of the system, tiate quantities such as eigenstates of the Hamiltonian, the system will end up in a thermal state) and that many which commute with the Hamiltonian but are usually (but perhaps not all) macroscopic observables will ‘ther- global. Consider the eigenvalues of such an integrable malise’ (i.e. their expectation values will saturate to values system, which are then perturbed with a non-integrable predicted from thermal states, apart from infrequent re- Hamiltonian perturbation. Since equilibration happens currences and fluctuations). Let us consider this issue of by a system exploring all possible transitions allowed the thermalisation of macroscopic observables in a closed between its various states, any restriction to the set of all quantum system. Let the initial state of the system be allowed transitions will have an effect on equilibration. given by φ c 6 , where the 6 are the energy Since there are a number of conserved local quantities | ⟩= k k| k⟩ | k⟩ eigenstates of the Hamiltonian, with energies assumed implied by the integrability of a quantum system, such a # non-degenerate. The corresponding coefficients ck are quantum system is not expected to thermalise. (We note non-zero only in a small band around a given energy that this is one of the many definitions of thermalisation E0. If we consider an observable B, its expectation value and it has been criticised as being inadequate [136], see in the time evolving state is given by also Section 9 of [26] for a detailed criticism of various definitions of thermalisation). On the other hand, the iHt iHt 2 φ e Be− φ B ck Bkk absence of integrability is no guarantee for thermalisation ⟨ | | ⟩:=⟨ ⟩= | | + [137]. Furthermore, if an integrable quantum system is *k i Em Ek t perturbed so as to break integrability slightly, then the c∗ cke [ − ] Bkm,(47) m various invariants are assumed to approximately hold. km * Hence, the intuition is that there should be some gap where B 6 B 6 . The long time average of this (in the perturbation parameter) between the breaking of km := ⟨ m| | k⟩ expectation value is given by the first term and is expected integrability and the onset of thermalisation. This phe- CONTEMPORARY PHYSICS 561

Figure 6. Energy levels are indicated by the black lines, populations of levels are indicated as red columns. (a) LTs of the Hamiltonian change only the energy levels while the populations of the levels remain the same. The energy levels can be chosen such that the initial probabilities correspond to thermal probabilities for a given temperature. (b) ‘Thermalisation’ of the population with respect to the energy levels resulting in the Gibbs state. (c) An isothermal reversible transformation (ITR) changing both the Hamiltonian’s energy levels and the state population to remain thermal throughout the process. (Figures taken from Nat. Commun. 4, 1925 (2013).) (d) No (0) (0) (1) level transfer is assumed during LTs with the energy change from level En corresponding to initial Hamiltonian H H to En corresponding to changed Hamiltonian H(1) associated with single shot work. Level transfers during thermalisation are assumed= to end (1) (1) (1) with thermal probabilities in each of the energy eigenstates Em1 of the same Hamiltonian H , with the energy difference between En (1) and Em1 identified as single shot heat. An ITR is a sequence of infinitesimal small steps of LTs and thermalisations, here, resulting in the system in an energy E(1) after N 1steps. mN − nomenon is called pre-thermalisation, and owing to the thermalisation steps where the Hamiltonian is held fixed invariants, such systems which are perturbed out of in- and the system equilibrates with a bath at temperature tegrability settle into a quasi-steady state [138]. This is a T to a thermal state [142]. Let the initial Hamiltonian (0) (0) (0) (0) result of the interplay between thermalisation and inte- be H k,gk Ek Ek , gk Ek , gk where Ek are the = | (⟩⟨) | grability [139] and was experimentally observed in [140]. energy eigenvalues and E 0 , g the corresponding de- # | k k⟩ generate energy eigenstates, labelled with the degeneracy (0) 5. Single shot thermodynamics index gk. For the system starting with energy En ,the ( ) ( ) energy change during LTs, E 0 E 1 see Figure 6(d), is n → n The single shot regime refers to operating on a single entirely associated with single shot work while the energy quantum system, which can be a highly correlated system (1) (2) change, Em Em see Figure 6(d), for thermalisation of many subsystems, rather than on an infinite ensemble 1 → 2 steps is associated with single shot heat [142]. See also of identical and independently distributed copies of a the definition of average work and average heat in Equa- quantum system, often referred to as the i.i.d. regime tion (2) associated with these processes for ensembles [96]. For long, the issue of whether information theory [32]. Through a sequence of LTs and thermalisations, see and hence physics can be formulated reasonably in the Figure 6(a)–(c), it is found that the single shot random single shot regime has been an important open question work yield of a system with diagonal distribution ρ and which has only been addressed in the last decade, see e.g. Hamiltonian H is [142] [141]. An active programme of applying quantum infor- rn Wyield kB T ln . (48) mation theory techniques, in particular the properties of = tn CPTP maps discussed in Section 4, to thermodynamics Here, rn and tn are energy-level populations of ρ and the is the extension of the laws and protocols applicable to βH thermal state, τ e− , respectively, which are both thermodynamically large systems to ensembles of finite := Z size that in addition may host quantum properties. Finite diagonal in the Hamiltonian’s eigenbasis. The particular size systems typically deviate from the assumptions made index n appearing here refers to the initial energy state ( ) E 0 , g that the system happens to start with in a ran- in the context of equilibrium thermodynamics and there | n n⟩ has been a large interest in identifying limits of work dom single shot, see Figure 6(d). Averaging the single extraction from such small quantum systems. shot work in Equation (48)withtheprobabilityrn of obtaining it, gives the average work yield A(ρ, H) = F(ρ) F(τ) as expected from Equation (7), where the 5.1. Work extraction in single shot thermodynamics − free energy F is defined in Equation (6). One single shot thermodynamics approach to analyse Deterministic work extraction is rarely possible in the how much work may be extracted from a system in a single shot setting and proofs typically allow a non-zero classical, diagonal distribution ρ with Hamiltonian H in- probability ϵ of failing to extract work in the construction troduces level transformations (LTs) of the Hamiltonian’s of single shot work. The ϵ-deterministic work content energy eigenvalues while leaving the state unchanged and Aϵ(ρ, H) is found to be [142] 562 S. VINJANAMPATHY AND J. ANDERS

Figure 7. Left (a) Work in classical thermodynamics is identified with lifting a bucket against gravity, thus raising its potential energy. (b) In the quantum thermodynamic resource theory, a work bit, i.e. a quantum version of a bucket, is introduced where the jump from one energetic level to another defines the extracted work during the process. In the text, the lower level is denoted as 0 ,i.e.theground | ⟩W state with energy 0 and the excited state is w W with energy w.(FiguretakenfromNat. Commun. 4 2059 (2013).) Right: Global energy levels of combined system, bath and work storage| ⟩ system have a high degeneracy indicated by the sets of levels all in the same energy shell, E.Theresourcetheoryapproachtoworkextraction[111] allows a global system starting in a particular energy state made up of energies E ES EB 0, indicated in red, to be moved to a final state made up of energies E ES′ EB′ w, indicated in blue, in the same energy= shell,+E. Work+ extraction now becomes an optimisation problem to squeeze the population= + of+ the initial state into the final state and find the maximum possible w [149].

recent contributions include references [9,111,142,144– ϵ ϵ Z7∗(ρ,H) A (ρ, H) F (ρ, H) F(τ) kB T ln , 148]. Resource theories in quantum information identify ≈ − := − Z a set of restrictive operations that can act on ‘valuable (49) resource states’. For a given initial state, these restrictive with the free energy difference defined through a ratio of a operations then define a set of states that are reachable. For example, applying stochastic local operations assisted subspace partition function Z7∗ over the full thermal par- βE by classical communication (SLOCC) on an initial prod- tition function Z.Here,Z7 (ρ, H) e− is ∗ := (E,g) 7∗ the partition function for the minimal∈ subspace uct state of two parties will produce a restricted, separable # 7 (ρ, H) inf 7 E, g ρ E, g > 1 ϵ set of two-party states. The same SLOCC operations ap- ∗ := { : (E,g) 7⟨ | | ⟩ − } defined such that the state ρ’s probability∈ in that subspace plied to a two-party Bell state will result in the entire # makes up the total required probability of 1 ϵ. state space for two parties, thus the Bell state is a ‘valu- − As an example consider an initial state ρ with probabil- able resource’. In the thermodynamic resource theory, ities (5/15, 4/15, 1/15, 2/15, 3/15, 0) for non-degenerate the valuable resource states are non-equilibrium states energies (0E,1E,2E,3E,4E,5E) for E>0andassume while the restrictive operations include thermal states of that the allowed error is ϵ 1/10. The thermal partition auxiliary systems. β0 β=1E β2E β3E β4E The thermodynamic resource theory setting involves function is Z e− e− e− e− e− β5E = + + + + + three components: the system of interest, S,abathB e . The subspace partition function is then Z7 − ∗ = e β0 e β1E e β3E e β4E including the thermal terms and a weight (or work storage system or battery) W. − + − + − + − for the most populated energies until the corresponding Additional auxiliary systems may also be included [146] state probability is at least 1 1/10, i.e. p to explicitly model the energy exchange that causes the − 0E,1E,3E,4E = 14/15 1 1/10. time dependence of the system Hamiltonian, cf. Section 3, ≥ − The proof of Equation (49) employs a variation of and to model catalytic participants in thermal operations, Crook’s relation, see Section 3.2,andreliesonthenotion see Section 5.3. In the simplest case, the Hamiltonian at of smooth entropies, see Section 4.1.Contrastingwiththe the start and the end of the process is assumed to be the same and the sum of the three local terms, H H H ensemble situation that allows probabilistic fluctuations = S+ B+ P(W) around the average work, see e.g. Equation (17), the HW . Thermal operations are those transformations of the single shot work Aϵ(ρ, H) discussed here is interpreted system that can be generated by a global unitary, V,that acts on system, bath and work storage system initially in a as the amount of ordered energy, i.e. energy with no ini ini product state ρS τB EW W EW ,wheretheworkstor- fluctuations, that can be extracted from a distribution ⊗ ⊗| ⟩ ⟨ | ini age system starts in one of its energy eigenstates, EW , in a single shot setting. βH | ⟩ e− B and the bath in a thermal state, τB ,whichis = ZB considered a ‘free resource’. The non-trivial resource for 5.2. Work extraction and work of formation in the process is the non-equilibrium state of the system, ρS. thermodynamic resource theory Perfect energy conservation is imposed by requiring that An alternative single shot approach is the resource the- the unitary may only induce transitions within energy shells of total energy E E E E ,seeFigure7(b). ory approach to quantum thermodynamics [143], where = S + B + W CONTEMPORARY PHYSICS 563

This is equivalent to requiring that V must commute with Similar to the asymmetry between distillable entangle- the sum of the three local Hamiltonians. ment and entanglement of formation, it is also possible A key result in this setting is the identification of a to derive a work of formation to create a diagonal non- maximal work [111] that can be extracted from a single equilibrium state ρS which is in general smaller than the system starting in a diagonal non-equilibrium state ρS to maximum extractable work derived above [111], the work storage system under thermal operations. The V min formation 1 ϵ desired thermal transformation, ρS τB 0 W 0 wϵ inf ln min λ ρ λτS , ⊗ ⊗| ⟩ ⟨ | −→ = β ρϵ { : S ≤ } σ w w , enables the lifting of the work storage & ' S SB ⊗| ⟩W ⟨ | & ' system by an energy w>0 from the ground state Eini (51) | W ⟩= 0 W to another single energy level w W ,seeFigure7(a). | ⟩ | ⟩ where ρϵ are states close to ρ , ρϵ ρ ϵ with Here,thesystemandbathstartinaproductstateandmay S S || S − S|| ≤ || · || the trace norm. end in a correlated state σSB.Theperfecttransformation can be relaxed by requesting the transformation to be successful with probability 1 ϵ,seeSection5.1.The − 5.3. Single shot second laws maximum extractable work in this setting is then [111, 149] Beyond optimising work extraction, a recent paper iden- tifies the set of all states that can be reached in a single shot max 1 w ln t( ) hρ (E , g , ϵ) by a broader class of thermal operations in the resource ϵ =−β ES,gS S S S E*S,gS theory setting [146]. This broader class of operations, min(ρ ) (τ ) ,iscalledcatalytic thermal operations and involves, in Fϵ S F S ,(50)EC =: − addition to a system S (which here may include the work where τS is the thermal state of the system at the bath’s storage system) and a heat bath, B,acatalystC [151]. The e βES inverse temperature β with eigenvalues t( ) − role of the catalyst is to participate in the operation while ES,gS ZS βE = and partition function Z e− S .Here,E and starting and being returned uncorrelated to system and S = ES,gS S g refer to the energy and degeneracy index of the eigen- bath, and in the same state σC, S # states ES, gS of the Hamiltonian HS. hρS (ES, gS, ϵ) is a | ⟩ (ρ ) σ tr V (ρ τ σ ) V † . (52) binary function that leads to either including or exclud- EC S ⊗ C := B[ S ⊗ B ⊗ C ] ing a particular t(ES, gS) in the summation, see [111,149] for details. Comparing with the result in Equation (49) The unitary now must commute with the sum of the three derived with the single shot approach discussed above, Hamiltonians that the system, bath and catalyst start and / end with, V, HS HB HC 0. Note that the map C one notices that the ratio Z7∗ Z is just the sum given [ + + ]= E on the system state depends upon the catalyst state σ . in (50), i.e. Z7 /Z t(E ,g ) hρ (ES, gS, ϵ) and the C ∗ = ES,gS S S S ρ expressions for the extractable work coincide. For system states S diagonal in the energy basis of HS,it # For non-zero ϵ, the derivation can no longer rely on is shown [146]thatastatetransferundercatalyticthermal the picture of lifting the work storage system from its operations is possible only under a family of necessary and ground state to a pure excited state. A more general sufficient conditions, mixed work storage system state is used in [150]and ρ ρ′ (ρ ) a general link between the uncertainty arising from the S → S = EC S ϵ-probabilities and the fluctuations in work as discussed α 0 Fα(ρS, τS) Fα(ρ′ , τS),(53) ⇔∀≥ : ≥ S in Section 3.5 is derived. This approach recovers the results, Equations (49)and(50), as the lower bounds on where τS is the thermal state of the system. τS is the the extractable work. When extending the single-level fixed point of the map C and the proof makes use of E transitions of the work storage system to multiple levels the contractivity of map, discussed in Section 4.1.Here, [149], it becomes apparent that this resource theory work Fα(ρS, τS) kB T Sα(ρS τS) ln ZS is a family of free := || − cannot directly be compared to lifting a weight to a certain energies defined through the α-relative Renyi entropies, sgn(α&) α 1 α ' height or above. Allowing the weight to rise to an energy Sα(ρ τ ) ln r t , applicable for states S|| S := α 1 n n n− level w or a range of higher energy levels turns the diagonal in the same− basis. The r and t are the eigen- | ⟩W # n n problem into a new optimisation and that would have a values of ρS and τS corresponding to energy eigenstates different associated ‘work value’, w , contrary to physical E , g of the system, cf. Equation (48). An extension of ′ | n n⟩ intuition. Due to its restriction to a particular energetic these laws to non-diagonal system states and including transition, the above resource theory work may very well changes of the Hamiltonian is also discussed [146]. be a useful concept for resonance processes where such a The resulting continuous family of second laws can restriction is crucial [149]. be understood as limiting state transfer in the single 564 S. VINJANAMPATHY AND J. ANDERS shot regime in a more restrictive way than the limita- tion enforced by the standard second law of thermody- namics, Equation (5), valid for macroscopic ensembles [146], see Section 2.Indeed,thelatterisincludedinthe family of second laws: for limα 1 where Sα(ρS τS) rn → || → S1(ρS τS) rn ln [23], one recovers from Equa- || = n tn tion (53)thestandardsecondlaw, #

rn rn′ kB T rn ln ln ZS kB T rn′ ln ln ZS , > tn − ?≥ > tn − ? *n *n (54) Figure 8. For a fixed Hamiltonian HS and temperature T,the manifold of time symmetric states, indicated in grey, including rn ln rn rn′ ln rn′ (rn′ rn) ln tn, − ≥− − the thermal state, γ τS, as well as any diagonal state, D(ρ) n n n ≡ ≡ * * * HS (ρS), is a submanifold of the space of all system states (blue (55) Doval). Under thermal operations, an initial state ρ ρ moves ≡ S #S β#U β Q ,(56)towards the thermal state in two independent ‘directions’. The ≥ = ⟨ ⟩ thermodynamic purity p measured by Fα and the asymmetry a with no work contribution, W 0, as no explicit measured by Aα must both decrease. (Figure taken from Nat. ⟨ ⟩= Commun. 6, 6383 (2015).) source of work was separated here (although this can be done, see Section 5.2). Moreover, in the limit of large ensembles and for systems that are not highly correlated, the contractivity of the relative entropy with respect to the map ,seeSection4.1,anditscommutingwith .It the Renyi free energies reduce to the standard Helmholtz E D free energy, Fα(ρ , τ ) F (ρ , τ ) for all α, providing is noted that (57) provides necessary conditions only – it S S ≈ 1 S S a single shot explanation of why Equation (5)istheonly is not known if the conditions on the Aα(ρS′ ) are sufficient second law relation for macroscopic ensembles. for a thermal operation to exist that turns ρS into ρS′ . Allowing the catalyst in Equation (52) to be returned The conditions (57) mean that thermal operations on after the operation not in the identical state, but a state a single copy of a quantum state with coherence must close in trace distance, leads to the rather unphysical puz- decrease their coherence, just like free energy must de- zle of thermal embezzling: a large set of transformations crease for diagonal non-equilibrium states. The two fam- are allowed without the usual second law restrictions. ilies of second laws (53)and(57)canbeinterpreted This puzzle has been tamed by recent results showing that geometrically as moving states in state space under ther- when physical constraints, such as fixing the dimension mal operations ‘closer’(the Renyi divergences Sα are not and the energy-level structure of the catalyst, are incorpo- proper distance measures [23]) to the thermal state in rated, the allowed state transformations are significantly two independent ‘directions’, in thermodynamic purity restricted [152]. and time asymmetry [148], see Figure 8.Thedecreaseof coherence implies a quantum aspect to irreversibility in the resource theory setting [153]. Again, when taking the 5.4. Single shot second laws with coherence single shot results to the macroscopic limit, ρ ρ N , → ⊗ Recently, it was discovered that thermal operations, conditions (57)becometrivialandthestandardsecond (ρ ) tr V (ρ τ ) V † ,onaninitialstateρ that law (5)isrecoveredastheonlyconstraint. E S := B[ S ⊗ B ] S has coherences (i.e. non-zero off-diagonals) with respect to the energy eigenbasis of the Hamiltonian HS,require 6. Quantum thermal machines a second family of inequalities to be satisfied [148], in Until now, we have discussed fundamental issues in addition to the free energy relations for diagonal states quantum thermodynamics. Thermodynamics is a field (53). The derived necessary conditions are whose focus from the very beginning has been both fun- ρ ρ′ (ρ ) α 0 Aα(ρ ) Aα(ρ′ ), damental as well as applied. Applications of thermody- S → S = E S ⇒∀≥ : S ≥ S (57) namics are mostly in designing thermal machines, which extract useful work from thermal baths. Examples of this where Aα(ρ ) Sα(ρ (ρ )) and (ρ ) is the are engines and refrigerators. In Section 2, we discussed S = S||DHS S DHS S operation that removes all coherences between energy work that can be extracted from correlations and from eigenspaces, i.e. (ρ) η in Section 2.6.TheAα are coherences in the energy eigenbasis. These coherences DH ≡ coherence measures, i.e. they are coherence monotones and correlations hence affect cyclical and non-cyclical for thermal operations. The proof of (57)reliesagainon machine operations and contribute towards the quantum CONTEMPORARY PHYSICS 565 features of thermal machine design. Here, we will present From standard thermodynamics, it is well known that some of the recent progress in the design of quantum the most efficient engines also output zero power since thermal machines that apply these aforementioned prin- they are quasi-static. Hence, for the operation of a real ciples, which generalise classical machines. We will focus engine, other objective functions such as power have to on two main types of QTMs. The first are cyclical ma- be optimised. The seminal paper by Curzon and Ahlborn chines which are quantum generalisations of engines and [163]consideredtheissueoftheefficiencyofaclassical refrigerators. The second type are non-cyclical machines, Carnot engine operating between two temperatures, cold which highlight important aspects of QTMs such as work TC and hot TH , at maximum power, which they found to extraction, power generation and correlations between be subsystems. TC ηCA 1 . (59) = − T 6.1. Quantum thermal machines @ H The study of the efficient conversion of various forms of This limit is also reproduced with quantum working flu- energy to mechanical energy has been a topic of interest ids, as discussed in Section 6.8.Wenotethattheafore- for more than a century. QTMs, defined as quantum mentioned formula for the efficiency at maximum power machines that convert heat to useful forms of work, have is sensitive to the constraints on the system, and changing been a topic of intense study. Engines, heat-driven re- the constraints changes this formula. For instance, see frigerators, power-driven refrigerators and several other [164,165] for results generalising the Curzon–Ahlborn kinds of thermal machines have been studied in the quan- efficiency to stochastic thermodynamics and to quantum tum regime, initially motivated from areas such as quan- systems with other constraints. tum optics [10,11]. QTMs can be classified in various The rest of this section is a brief description of how ways. Dynamically, QTMs can be classified as those that engines are designed in the quantum regime. We consider operate in discrete strokes, and those that operate as three examples, namely: Carnot engines with spins as continuous devices, where the steady state of the con- the working fluid, harmonic oscillator Otto engines and tinuous time device operates as a QTM. Furthermore, Diesel engines operating with a particle in a box as the such continuous engines can be further classified as those working fluid. These examples were chosen to illustrate that employ linear response techniques [14]andthose the design of a variety of engines operating with a variety that employ dynamical equation techniques [24]. These of quantum working fluids. models of quantum engines are in contrast with biological motors that extract work from fluctuations and have been studied extensively [154]. 6.2. Carnot engine Engines use a working fluid and two (or more) reser- Classically, the Carnot engine consists of two sets of voirs to transform heat to work. These reservoirs model alternating adiabatic strokes and isothermal strokes. The the bath that inputs energy into the working fluid (hot quantum analogue of the Carnot engine consists of a bath) and another that accepts energy from the working working fluid, which can be a particle in a box [166], fluid (cold bath). Two main classes of quantum systems qubits [18,159], multiple-level atoms [157]orharmonic have been studied as working fluids, namely: discrete oscillators [155,156]. We emphasise that for all such en- quantum systems and continuous variable quantum sys- gines, the efficiency of the engine is strictly bounded by tems [155–158]. This study of continuous variable sys- the Carnot efficiency [167]. For the engine consisting of tems complements various models of finite level-systems non-interacting qubits considered in [159], the Carnot [159,160]andhybridmodels[161,162] studied as quan- cycle consists of tum engines. Both continuous variable models of engines and finite dimensional heat engines have been theoreti- (1) Adiabatic Expansion: An expansion wherein the cally shown to be operable at theoretical maximal effi- spin is uncoupled from any heat baths and its ciencies. Such efficiencies are only well defined once the frequency is changed from ω1 to ω2 > ω1 adia- engine operation is specified. batically. Work is done by the spin in this step due The efficiency of any engine is given by the ratio of the to a change in the internal energy. Since the spin is net work done by the system to the heat that flows into uncoupled, its von Neumann entropy is conserved the system, namely in this expansion stroke. (2) Cold Isotherm: The spin is coupled to a cold bath at Wnet η ⟨ ⟩. (58) inverse temperature βC. This transfers heat from = Q ⟨ in⟩ the engine to the cold bath. 566 S. VINJANAMPATHY AND J. ANDERS

HO (3) Adiabatic Compression:Acompressionstroke (2) Hot Isochore: At frequency ω2 ,theharmonic where the spin is uncoupled from all heat baths oscillator working fluid is coupled to a bath whose and its frequency is changed from ω3 to ω4 adi- inverse temperature is βH,andisallowedtorelax abatically. Work is done on the medium in this to the new thermal state. step. (3) Adiabatic Expansion:Inthisstroke,themediumis (4) Hot Isotherm: The spin is coupled to a hot bath at uncoupled from any heat baths and its frequency HO HO inverse temperature βH < βC. This transfers heat is changed from ω2 to ω1 . HO to the engine from the hot bath. (4) Cold Isochore: At frequency ω1 ,theharmonic oscillator working fluid is coupled to a bath whose The inverse temperature β of the thermal state of this ′ inverse temperature is β and it is allowed to relax working fluid is given at any point (ω,S) on the cycle, by C back to its initial thermal state. the magnetisation relation S tanh (β ω/2)/2. The =− ′ dynamics is described by a Lindblad equation, which will In the two strokes where the frequencies change, work then be used to derive the behaviour of heat currents. The is exchanged. In the two thermalizing strokes on the other engine cycle is described in Figure 9. The Hamiltonian hand, only heat is exchanged. Hence the efficiency is describing the dynamics is given by H(t) ω(t)σ /2. easily calculated to be = 3 The Heisenberg equation of motion is written in terms of D(σ3), the Lindblad operators describing coupling to δW δW L η 1 3 . the baths at inverse temperatures βH and βC, respectively. Otto ⟨ ⟩ +⟨ ⟩ (62) =− δQ 2 This equation can be used to calculate the expectation ⟨ ⟩ value of the Hamiltonian H(t) ,whichleadsto ⟨ ⟩ The experimental implementation of such a quantum d H(t) 1 dω Otto engine was considered in [168,171] and is presented ⟨ ⟩ σ ω (σ ) = ⟨ 3⟩+ ⟨LD 3 ⟩ in Figure 10. The Paul or quadrupole trap uses rapidly dt 2 $ dt % 1 dω d σ3 oscillating electromagnetic fields to confine ions using σ3 ω ⟨ ⟩ . (60) = 2 dt ⟨ ⟩+ dt an effectively repulsive field. The ion, trapped in the $ % modified trap, presented in Figure 10 is initially cooled As described in Section 2,thedefinitionofwork δW in all spatial directions. The engine is coupled to hot ⟨ ⟩= σ δω/2 and heat δQ ωδ σ /2 emerges naturally and cold reservoirs composed of blue and red detuned ⟨ 3⟩ ⟨ ⟩= ⟨ 3⟩ from the above discussion, and is identified with the laser beams. The tapered design of the trap translates to time derivative of the first law. For this quantum Carnot an axial force that the ion experiences. A change in the engine, the maximal efficiency which is that of the temperature of the radial state of the ion, and hence the reversible engine [159] is given by the standard formula width of its spatial distribution, leads to a modification of namely the axial component of the repelling force. Thus, heating and cooling the ion move it back and forth along the TC trap axis, as induced in the right-hand side of the figure. ηCarnot 1 . (61) = − TH The frequency of the oscillator is controlled by the trap parameters. Energy is stored in the axial mode and can 6.3. Otto engine be transferred to other systems and used. Since we have considered an Otto engine operating be- As a counterpoint to the Lindblad study of Carnot tween two thermal baths, we should not be surprised that engines implemented via qubits, let us consider the har- the standard formula for efficiency still applies. monic oscillator implementation of Otto engines [12,158, Making one of the components of the engine genuinely 168–170]whosetime-dependentfrequencyofthehar- non-thermal (and quantum) makes the analysis more monic oscillator working fluid is ωHO(t).Theinitialstate interesting. In [171], the authors show that the stan- of the oscillator is a thermal state at inverse temperature dard Carnot efficiency can be overcome by squeezing β , and the oscillator frequency is ωHO.Thefourstrokes C 1 the thermal baths. In [172], the authors derive a ‘gen- of the Otto cycle are given by: eralised’ Carnot efficiency that correctly accounts for the (1) Adiabatic Compression:Ancompression,where first and second laws. Using this analysis, they demon- the medium is uncoupled from all heat baths and strate that though the engine efficiency exceeds the stan- HO HO HO its frequency is changed from ω1 to ω2 > ω1 . dard Carnot formula, the ‘generalized’ efficiency is not Since the oscillator is uncoupled, its von Neumann exceeded, in keeping with the laws of thermodynamics entropy is conserved during this stroke, though it [173]. We note that the cost of squeezing the bath has not no longer is a thermal state. been accounted for, similar to how the cost of preparing a CONTEMPORARY PHYSICS 567

Figure 9. The left figure is a reversible Carnot cycle, operating in the limit ω 0, depicted in the space of the normalised magnetic field ω and the magnetisation S. The horizontal lines represent adiabats wherein˙ → the engine is uncoupled from the heat baths and the magnetic field is changed between two values. The lines connecting the two horizontal strokes constitute changing the magnetisation by changing ω while the qubits are connected to a heat bath at constant temperature. We note that in the figure, S1,2 < 0represent two values of the magnetisation, with S1 < S2.Figuretakenfrom[159]. The right figure is a particle in a box with engine strokes being defined in terms of expansion and contraction of the box. This defines a quantum isobaric process, by defining force. This definition is used to define a Diesel cycle in the text. We note that in the figure force is denoted by F, while it’s denoted by F in the text. Figure taken from [175]. Figures reproduced with permission. cold bath is unaccounted for in classical thermodynamics be employed to construct a Diesel cycle, presented in (it is assumed that the baths are free resources). Figure 9, and whose four strokes are given by 6.4. Diesel engine (1) Isobaric Expansion:Theexpansionofthewallsof Finally, we discuss Diesel cycles designed with a particle the particle in a box happen at constant pressure. in a box as the working fluid [175]. The classical Diesel The width of the walls goes from L2 to L3 >L1. engine is composed of isobaric strokes, where the pres- (2) Adiabatic Expansion: An adiabatic expansion pro- sure is held constant. We will discuss how the notion cess, wherein the length goes from L3 to L1 >L3. of ‘pressure’ is generalised to the quantum setting and Entropy is conserved in this stroke. how this defines a ‘quantum isobaric’ stroke. To define (3) Isochoric Compression:Thecompressionhappens pressure, we begin by defining force. After defining av- at constant volume L1 where the force on the box erage work and heat as before, generalised forces can be is reduced. defined by analogy with classical thermodynamics using (4) Adiabatic Compression:Thecompressionprocess the relation takes the box from volume L1 to L2

Figure 10. Experimental proposal of an Otto engine consisting of a single trapped ion, currently being built by K. Singer’s group [174]. On the left, the energy frequency diagram corresponding to the Otto cycle implemented on the radial degree of freedom of the ion. The inset on the left-hand side is the geometry of the Paul trap. On the right-hand side, a representation of the four strokes, namely (1) adiabatic compression, (2) hot isochore, (3) adiabatic expansion and (4) cold isochore, see text for details. Figures taken from [171] reproduced with permission. is that when thermal machines operate between two heat speed limit [182]. The quantum speed limit places a limit baths with well-defined temperatures, the efficiency of on the power of quantum engines, and has been used to such a machine is also limited by the classical efficiencies, improve engine performance [180]. Another approach i.e. the Carnot and Curzon–Ahlborn efficiencies. The to time in quantum thermodynamics employs so-called Carnot efficiency, for instance, is derived independent of adiabatic shortcuts to improve engine performance in the details of the working fluid and depends only on the both the classical [169] and quantum regime [170]. laws of thermodynamics. It is expected that these bounds An adiabatic quantum system that starts in an eigen- must also hold in the quantum setting. state of the Hamiltonian remains in the instantaneous Once the thermal machines are committed to operate eigenstate of the Hamiltonian when the parameters of between two temperatures and in equilibrium, not much the Hamiltonian are swept slowly enough (distinguish can be done to change the efficiency of these machines this from the thermodynamic definition of adiabatic- (see [177] for a recent example). Hence, the deviation ity, which relates to having zero heat exchange). This from classical performance of these thermal machines is typically slow dynamics can be sped up both in the clas- only seen when either the baths are made non-thermal or sical and quantum contexts by the addition of exter- the working fluid of the thermal machine is not allowed nal control fields that are bound to be transitionless. to equilibrate, operating in a non-equilibrium setting. Finally, we note that more foundational aspects of quan- To characterise such non-equilibrium thermal machines, tum mechanics, such as non-commutativity of operators, several authors have studied the role of quantum corre- have been shown to have a detrimental effect on engine lations [178] and the role they play in work extraction. performance. This is known as quantum friction and Using non-thermal heat baths [171,179]hasbeen is also a genuinely quantum effect that impacts engine another strategy to see the effect of quantum states on performance [183]. Besides issues such as non-thermal thermal machines. An example of such a study focuses on baths and non-commutativity, we refer the reader to Otto engines operated with squeezing thermal baths we [184] for an information theoretic perspective on engine discussed in the previous subsection. In all such design. examples, non-classical resources are employed to imp To answer the question about the quantumness of rove engine performance, from power generation [170, engines, recent studies have focused on trying to study 180] to efficiency [168,171,172]. Finally, we point out that alternative non-quantum models (to compare and con- time has an important role to play in engine performance trast with quantum engines) or produce genuine quan- since it relates to power. As detailed in Section 6.7,both tum effects to demonstrate the quantumness of engines. coherence and time play important roles in the design of On the difference between quantum mechanics and quantum machines [58,59,181]. As an example, consider stochastic formulation of thermodynamics, in [185], the that the evolution of an initial state to a final state cannot authors study quantumness of engines and show that happen faster than a fundamental bound that depends on there is an equivalence between different types of en- the Hamiltonian and states involved. The minimum time gines, namely: two-stroke, four-stroke and continuous associated with this bound is often called the quantum engines. Furthermore, the authors define and discuss CONTEMPORARY PHYSICS 569 quantum signatures in thermal machines by comparing COP can be larger than one, and in [194], the authors these machines with equivalent stochastic machines, and show the relationship of the COP to the cooling rate, they demonstrate that quantum engines operating with which is simply the rate at which heat is transported from coherence can in general output more power than their the system. stochastic counterparts. Finally, we note an example of a The other example of quantum refrigerators takes a QTM [186] operating in the continuous time regime that more quantum information theoretic point of view. In produces entanglement. The working fluid considered by [18], the authors were inspired by algorithmic cooling the authors consists of two qubits, which are coupled to and studied the smallest refrigerators possible. One of two different baths and to each other. The steady state of the models consists of a qubit coupled to a qutrit, see this thermal machine far from thermal equilibrium (in Figure 11. The intuition for cooling the qubit comes the presence of a heat current) exhibits the interesting from a computational model of entropy reduction. This property of creating bipartite entanglement in the work- procedure, called algorithmic cooling [17], is a procedure ing fluid. Such an engine can be said to have a ‘genuinely to cool quantum systems which uses ideas of entropy quantum’ output. shunting to ‘move’ entropy around a large system in a way that lowers the entropy of a subsystem. Consider n copies of a quantum system which is not in its maximal entropy state. Joint unitary operations on the n copies can 6.6. Quantum refrigerators allow for the distillation of a small number m of systems Refrigerators are engines operating in a regime where which are colder (in this context, of lower entropy) than before, while leaving the remaining (n m) systems in a the heat flow is reversed. Like engines, the role that quan- − tised energy states, coherence and correlations play in the state that is hotter than before (this is required by unitar- operation of a quantum refrigerator have been fields of ity). This is the intuition behind cooling of the qubit in extensive study [24,187–192]. Figure 11. The refrigerator in [18]isthesmallest(in We present two examples of refrigerator cycles consid- Hilbert space dimensionality) refrigerator possible. See ered in the literature. The first of these [193]isbuiltona [187] for a discussion of the relationship of power to three-level atom, coupled to two baths, as shown in Fig- COP and [187] for a discussion of correlations in the ure 11. The refrigerator is driven by coherent radiation. design of quantum absorption refrigerators. Finally, we This induces transitions between level 2 and level 3. The note that there is a connection between models of cooling population in level 3 then relaxes to level 1 by rejecting based on study of heat flows of continuous models, such heat to the hot bath at temperature TH.Thesystemthen as sideband cooling, and quantum information theoretic transitions from level 1 to level 2 by absorbing energy models of cooling based on control theory [195–197]. from a cold bath. Like before, the dynamics can be written in terms of the Heisenberg equation for an observable A. 6.7. Coherence and time in thermal machines Let refer to the Lindblad operators corresponding to LC,H the cold and hot baths, respectively. Since this is a thermal In this subsection, we want to briefly discuss the role machine operated continuously in time, the steady-state coherence and time play in QTMs. The fundamental energy transport is the measure of heat transported by the starting point of finite time classical [198] and quan- machine. Assuming the Hamiltonian H H V(t),the = 0 + tum thermodynamics is the fact that the most efficient energy transport can be written as macroscopic processes are also the ones that are quasi- static and hence slow. Hence, they are impractical from d H ∂V(t) ⟨ ⟩ (H) (H) . (66) the standpoint of power generation. The role of quantum = ∂ +⟨LH ⟩+⟨LC ⟩ dt A t B coherences was illustrated by a series of studies involv- ing commutativity of parts of the Hamiltonian, as ex- Using this, the authors in [193] investigated the refrig- plained below. Consider a quantum Otto engine, whose erator efficiency. This efficiency is expressed in terms of energy balance equation is given in terms of the Hamil- the coefficient of performance (COP), which is defined tonian part H H and dissipative Lindblad oper- ext + int as the ratio of the cooling energy to the work input into ator [199]. There are two sources of heat that were L the system. For two thermal baths at temperatures TC studied in this model, and they are related to coher- and TH, respectively, the universal reversible limit is the ence and time. The first of these sources of heat are Carnot COP, given by attributed to energy transfer from the hot bath to the cold bath via the system. The second source of heat comes TC COP . (67) from the finite time driving of the adiabatic strokes of = T T H − C the engine. Such a finite time driving means that the 570 S. VINJANAMPATHY AND J. ANDERS

Figure 11. Two designs of refrigerators, the details of which are discussed in the text. The left figure consists of a three-level atom coupled to two baths and interacting with a field, figure taken from [193]. The right figure consists of the smallest refrigerator, composed of a qubit and a qutrit. Here, cooling of the qubit is achieved through joint transformations between the system and the reservoir, wherein the system’s entropy is shunted to the reservoir. Figure taken from [18]. Figures reproduced with permission. system causes irreversible heating which is understood as A non-cyclical example of the use of quantum correla- follows. Suppose the initial state of the system is an eigen- tions to improve machine performance involves quan- state of H .If H , H 0, then the quantum state, tum batteries. These are quantum work storage devices ext [ ext int] ̸= even in the absence of Lindblad terms, does not ‘follow’ and their key issues relate to their capacity and the speed the instantaneous eigenstate of the external Hamiltonian. at which work can be deposited in them. This question of The precession of the quantum state of the system about work deposition can be studied in two steps. Firstly, it is this instantaneous eigenstate, that is induced by the non- desirable to understand the limits on the work extractable commutativity of the external and internal Hamiltonians, from a quantum system under unitary transformations, is a source of a friction like heat [183]. V. Here, consider states to be written in an ordered eigen- Finally, we present another model of a non-thermal basis as ρ r↑ r↑ r↑ and H ϵ↓ ϵ↓ ϵ↓ . = k k | k ⟩⟨ k | = k k | k ⟩⟨ k | bath consisting of three-level atoms. In [179], the authors Note that r↑ ,theeigenstatesofρ, are a priori unrelated | k ⟩ # # extract work from coherences using a technique similar to ϵ↑ ,theeigenstatesofH.Thearrowsshowtheorder- | k ⟩ to lasing without inversion. The photon Carnot engine ing of eigenvalues, namely: r↑ r↑ ...and likewise for 0 ≤ 1 proposed by the authors to study the role of coherences the Hamiltonian. The optimal unitary that extracts the follows by considering a cavity with a single mode ra- most work from ρ is given by the unitary that maps ρ diation field enclosed in it. The radiation interacts with to the state π r↑ ϵ↓ ϵ↓ [32–34,200]. Such states, = k | k ⟩⟨ k | phased three-level atoms at two temperatures during the where the eigenvalues are ordered inversely with respect # isothermal parts of the Carnot engine cycle. The three- to the eigenvectors of the Hamiltonian, are known as level atoms are phased by making them interact with an ‘passive states’ [33,34]andthecorrespondingmaximum external microwave source. The efficiency of this engine work extracted from the unitary transformation, was calculated to be η η π cos (,),(68) Wmax tr H(ρ π) . (69) = Carnot − ⟨ ext ⟩= [ − ] where the phased three-level atoms have an off-diagonal is called ‘ergotropy’. Note, that the thermal states τβ = coherence term , that is used to improve the Carnot e βH /tr e βH for any inverse temperature β are passive. − [ − ] efficiency. This engine is depicted in Figure 12.Thus,by Ground states of a given Hamiltonian, for instance, are employing the coherence dynamics of three-level atoms, examples of states passive with respect to the Hamilto- the authors demonstrate enhancement in work output. nian and correspond to β . Thermal states maximise =∞ We end this section by noting that quantum features the entropy for a given energy and minimise the energy are not necessarily beneficial to a QTM’s performance. for a given entropy. Passive states which maximise the For example, a study of entanglement dynamics in three- energy for a given entropy were sought in [201], and are qubit refrigerators [191]demonstratesthatthereisonly useful in bounding the cost of thermodynamic processes. very little entanglement found when the machine is op- Passive states are important to understand the function- erated near the Carnot limit. ing of quantum engines, see [202] for example. The role entangling operations play in extracting work 6.8. Correlations, work extraction and power using unitarys transformations was explored within the Finally, let us discuss the aspects of work extraction com- simultaneous extraction of work from n quantum bat- mon to both cyclical and non-cyclical thermal machines. teries [203]. They considered n copies of a given state CONTEMPORARY PHYSICS 571

Figure 12. An optical Carnot engine, discussed in the text. The engine is driven by the radiation in the cavity being in equilibrium with atoms with two separate temperatures during the hot and cold parts of the Carnot cycle. The phased three-level atoms have an off-diagonal coherence term , that is used to improve the Carnot efficiency. Figure taken from [179], reproduced with permission.

σ and a Hamiltonian H for each system. As discussed systems). To remedy this, both classically and quantum before, σ and H imply that there exists a passive state π. mechanically, engines which optimise power have been Two strategies for extracting work exist, the first of these considered. These studies find the efficiency at maximum being to simply process each quantum system separately. power, see Equation (59), first derived in the classical The second strategy is to do joint entangling operations, context by Curzon and Ahlborn [163] but also valid in the n which transform the initial state σ ⊗ to a final state quantum case [155]. See [170] for a discussion of short- which has the same entropy as n copies of σ .Theauthors cuts to adiabaticity and their role in optimising power considered a Gibbs state τβ where the temperature was in Otto engines and [205] for a calculation of efficiency fixed by matching the entropy of this state with σ . They at maximum power of an absorption heat pump, which considered such a state since this state provided a upper also proved the weak dependence of efficiency and power bound on the amount of work that could be extracted by with dimensionality. In [168,206], the authors consider n transforming σ ⊗ intoapassivestate.Sincethefinalstate the power of Otto cycles in various regimes, and derive n τβ⊗ has the same entropy as n copies of the initial state, the steady-state efficiency n a transformation between σ ⊗ and the neighbourhood n √β /β of τβ⊗ would be the desired transformation that extracts 1 H C ηSS − . (70) maximum work. Such a transformation was shown to be = 2 √β /β + H C entangling, using typicality arguments [112]. However, the authors in [204] pointed out that there is always a 6.9. Relationship to laws of thermodynamics protocol to extract all the work from non-passive states using non-entangling operations, although in compari- We end the discussion of QTMs with a small report on son to entangling unitaries, more non-entangling oper- the role that the laws of thermodynamics, discussed in ations are needed. This suggests a relationship between Section 2, play in constraining design. As was noted in power and entanglement in work extraction, an assertion the context of engines, the first law emerges naturally as demonstrated for quantum batteries in [180]. A different a partitioning of the energy of the system in terms of role can be played by correlations in storing and extract- heat and work. See [101] for a partitioning of internal ing work, as discussed in Section 2. This role relates energy change for CPTP processes. In the context of to storing work in correlations. It has been shown by QTMs, for instance, this can be seen in Equation (60). several authors [58,59,191] that measures of correlations The second law manifests itself usually by inspecting the like quantum mutual information and entanglement can entropy production of the universe [193]. This is given affect the performance of thermal machines. for a typical continuous regime QTM, with a system and A final noteworthy point in the consideration of time two baths as explicitly in the context of QTMs relates to maximum dS Q˙ H Q˙ C power engines. Firstly, since the optimal performance of ⟨ ⟩ ⟨ ⟩. (71) dt =− TH − TC engines corresponds to quasi-static transformations, the power is always negligible and the time of operation of Analysis of the dynamics shows that the second law en- one cycle is infinite (or simply much longer compared forces the rule that the net entropy production, given to the internal dynamical time scales of the quantum by the equation above, is non-negative. See [187,207] 572 S. VINJANAMPATHY AND J. ANDERS to study the relationship between weak coupling and is starting to emerge. This difference in perspectives has the second law. Information theoretic approaches often also meant that there are ideas within quantum thermo- use monotonicity condition of relative entropy to track dynamics where consensus is yet to be established. entropy production, see section 4. Finally, we note the One example of such disagreement is the definition of work in [190]relatingtothesecondlawand[208] for heat work in the quantum regime. Various notions of work engine fluctuation relation and experimental proposal, have been introduced in the field, including the average all in the context of SWAP engines. In [209], the authors work defined for an ensemble of experimental runs in discuss how in a network of quantum systems coupled to Equation (2), classical and quantum fluctuating work reservoirs at different temperatures, the second law is vi- defined for a single experimental run in Equations (15) olated locally, though its always valid globally, in line with and (24), optimal single shot work given in Equation intuition (see Ref. [100] for a discussion on the violation (48) and optimal thermodynamic resource theory work of entropic inequalities, see [210] for a discussion on the given in Equation (50). It is reassuring to see that the role of non-linear couplings in refrigerator design). The latter two, quite separate single shot approaches, result role of the second law, in all these examples, tends to be to in the same optimal work value. Despite advances in modify the existing figures of merit, like efficiency [171], unifying these work concepts, our understanding of work and hence show a path towards understanding nanoscale in the quantum regime remains patchy. For example, QTMs. resource theory work, Equation (50), is a work associated The third law states that the entropy of any quantum with an optimised thermal operations process –tomove system goes to a constant as temperature approaches a work storage system almost deterministically as high zero. This constant is commonly assumed to be zero. This as possible – while the fluctuating work concept, Equa- means, that in the context of the equation above, the heat tions (15)and(24), is applicable to general closed and α 1 current corresponding to the cold bath, Q T + open dynamical processes [64,217]. Thus, they appear ⟨ ˙ C⟩∝ C as T 0withα > 0. Since we are discussing the to refer to different types of work, a situation that may C → limit of the cold bath temperature going to zero, we be compared to different entanglement measures each expect that the third law will inform the operation of of importance for a different quantum communication quantum refrigerators and heat engines, where the cold and computational task. For example, it has been sug- bath temperature is quite low. Furthermore, the third law gested that the resource theory work is a suitable measure affects the ability for us to cool a quantum system as the to quantify resonance processes [149]. A second issue is system approaches absolute zero. This is because there is the link between the work definition in the ensemble a trade-offbetween the entropy generated by the system sense, see (2), and the single shot work. Beyond taking coupling to an environment that is used to cool the system mathematical limits of Renyi entropies, these definitions and the fact that approaching absolute zero means that have to be understood within the context of each other the system becomes a pure state (assuming the ground operationally, i.e. how can one measure each of these state is unique). This trade-offpractically limits COP. quantities and in what sense do these quantities converge This was studied in [193], see also [24] for a discussion experimentally? Indeed, while traditional classical ther- on another formulation of the third law and [211] for modynamics is a manifestly practical theory, made for a recent review of the laws of thermodynamics in the steam engines and fridges, quantum thermodynamics has quantum regime. In [212], the authors discuss periodi- only made a small number of experimentally checkable cally driven quantum systems and investigate the laws of predictions of new thermodynamic effects yet. thermodynamics for a model of quantum refrigerator, see Another point of discussion is the kinematic versus [213–215]. Finally, in [216], the authors study the third dynamical approach to thermalisation and equilibration. law in the context of refrigerators and show that the rate The kinematic approach arises from typicality discus- of cooling is determined entirely by the cold reservoir sions in quantum information where states in Hilbert and its interaction to the system, and is insensitive to space are discussed from the standpoint of a property underlying the particle statistics. of their marginals. In the case of thermalisation, this property is the closeness to the canonical Gibbs state. A Hamiltonian and its eigenstates never appear in the kine- 7. Discussion and open questions matic description. The situation is quite different in the We have presented an overview of a selection of current case of the dynamical approach to thermalisation implied approaches to quantum thermodynamics pursued with by the ETH, where the eigenstates of the Hamiltonian are various techniques and interpreted from different per- of crucial importance. A connection between these two spectives. Substantial insight has been gained from these techniques to study thermalisation is thus desirable. This advances and their combination, and a unified language difference in perspectives between quantum information CONTEMPORARY PHYSICS 573 theoretic and kinematic approaches also needs to be rec- Notes on contributors onciled in the study of pre-thermalisation. Exceptional states such as ‘rare states’ need to be fully understood Sai Vinjanampathy is a theoretical physicist working at the Centre for from a kinematic standpoint, see [218] for a discussion Quantum Technologies at the National on rare states and thermalisation. University of Singapore. Starting July 1, In the context of QTMs, the interplay between statis- 2016, he assumed a faculty position at the tics and engine performance needs more study [158,219]. Indian Institute of Technology-Bombay, This is crucial since typical work fluids of QTMs consti- in Mumbai, India. He received his tute several particles. Though some authors have consid- PhD from Louisiana State University in 2010. Besides quantum thermodynamics, ered non-thermal baths in contact with a working fluid, his current work focuses on quantum almost all such studies involved unitary transformations metrology and quantum control theory. on thermal states. There is a need for the study of thermal machines, wherein at least one of the baths is non-thermal in a way that is different from unitary transformations Janet Anders received her PhD in on thermal baths. The role of entanglement and other quantum information theory from the correlations would clearly become very important in such National University of Singapore in a regime. Finally, we note that various experimental im- 2008. After a short postdoc, followed by plementations of quantum engines are expected in the an independent Royal Society Dorothy near future [220]. Hodgkin research fellowship at Uni- versity College London, she moved To close, quantum thermodynamics is a rapidly evolv- in 2013 to the University of Exeter ing research field that promises to change our under- where she now heads the quantum standing of the foundations of physics while enabling non-equilibrium group. Anders’ interests the discovery of novel thermodynamic techniques and are in quantum thermodynamics and applications at the nanoscale. This overview provided an quantum information theory, with a focus on the concept of work in the quantum regime, levitating nanosphere introduction to a number of current trends and perspec- experiments, non-equilibrium fluctuation relations, thermal tives in quantum thermodynamics and concluded with entanglement in many-body systems and measurement-based three particular discussions where there is still disagree- quantum computation. ment and flux. Resolving these and other riddles will no doubt deepen our understanding of the interplay between quantum mechanics and thermodynamics. References [1] S. Trotzky, Y.A. Chen, A. Flesch, I.P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Probing the Acknowledgements relaxation towards equilibrium in an isolated strongly correlated one-dimensional bose gas,Nat.Phys.8(2012), We are grateful to A. Acin, A. Auffeves, G. De Chiara, pp. 325–330. V. Dunjko, J. Eisert, R. Fazio, A. Fisher, I. Ford, P. Gonzalo [2] U. Schollwöck, The density-matrix renormalization Manzano, M. Horodecki, K. Hovhannisyan, M. Huber, group in the age of matrix product states,Ann.Phys.326 K. Jacobs, D. Jennings, R. Kosloff, M.A. Martin-Delgado, (2011), pp. 96–192. M. Olshanii, M. Paternostro, M. Perarnau-Llobet, M. Plenio, [3] C. Jarzynski, Nonequilibrium equality for free energy D. Poletti, S. Salek, P. Skrzypczyk, R. Uzdin and M. Wilde for differences, Phys. Rev. Lett. 78 (1997), pp. 2690–2693. their input on the overview. [4] G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60 (1999), p. 2721. [5] H. Tasaki, Jarzynski relations for quantum systems and Disclosure statement some applications, preprint (2000), Available at arXiv cond-mat/0009244. No potential conflict of interest was reported by the authors. [6] T. Sagawa and M. Ueda, Generalized jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104 (2010), 090602. Funding [7] S. Popescu, A.J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics,Nat.Phys.2 The Centre for Quantum Technologies is a Research Centre (2006), pp. 754–758. of Excellence funded by the Ministry of Education and the [8] L. Del Rio, J. Åberg, R. Renner, O. Dahlsten, and National Research Foundation of Singapore. JA acknowledges V. Vedral, The thermodynamic meaning of negative support by the Royal Society and EPSRC [EP/M009165/1]. This entropy,Nature474(2011), pp. 61–63. work was supported by the European COST network MP1209 [9] F.G. Brandao, M. Horodecki, J. Oppenheim, J.M. Renes, ‘Thermodynamics in the quantum regime’. and R.W. Spekkens, Resource theory of quantum states 574 S. VINJANAMPATHY AND J. ANDERS

out of thermal equilibrium, Phys. Rev. Lett. 111 (2013), [27] J. Goold, M. Huber, A. Riera, L. del Rio, and 250404. P. Skrzypczyk, The role of quantum information in [10] H. Scovil and E. Schulz-DuBois, Three-level masers as thermodynamics–a topical review, J. Phys. A: Math. heat engines, Phys. Rev. Lett. 2 (1959), pp. 262–263. Theor. 49 (2016), 143001. [11] J. Geusic, E. Schulz-DuBios, and H. Scovil, Quantum [28] M.B. Plenio and V. Vitelli, The physics of forgetting: equivalent of the carnot cycle, Phys. Rev. 156 (1967), Landauer’s erasure principle and information theory, pp. 343–351. Contemp. Phys. 42 (2001), pp. 25–60. [12] M.O. Scully, Quantum afterburner: Improving the [29] K. Maruyama, F. Nori, and V. Vedral, Colloquium: The efficiency of an ideal heat engine, Phys. Rev. Lett. 88 physics of maxwell’s demon and information, Rev. Mod. (2002), 050602. Phys. 81 (2009), pp. 1–23. [13] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den [30] I.J. Ford, Maxwell’s demon and the management of Broeck, Quantum-dot carnot engine at maximum power, ignorance in stochastic thermodynamics, Contemp. Phys. Phys. Rev. E 81 (2010), 041106. (2016), pp. 1–22. [14] F. Mazza, R. Bosisio, G. Benenti, V. Giovannetti, [31] J.M. Parrondo, J.M. Horowitz, and T. Sagawa, R. Fazio, and F. Taddei, Thermoelectric efficiency of three- Thermodynamics of information,Nat.Phys.11(2015), terminal quantum thermal machines,NewJ.Phys.16 pp. 131–139. (2014), 085001. [32] J. Anders and V. Giovannetti, Thermodynamics of [15] J. Gemmer and G. Mahler, Distribution of local entropy discrete quantum processes,NewJ.Phys.15(2013), in the hilbert space of bi-partite quantum systems: Origin 033022. of jaynes’ principle, Eur. Phys. J. B-Condens. Matter [33] W. Pusz and S. Woronowicz, Passive states and kms states Complex Syst. 31 (2003), pp. 249–257. for general quantum systems, Commun. Math. Phys. 58 [16] J. Gemmer, M. Michel, and G. Mahler, Quantum (1978), pp. 273–290. Thermodynamcis-emergence of Thermodynamic Behav- [34] A. Lenard, Thermodynamical proof of the gibbs formula ior within Composite Quantum Systems, Lecture Notes for elementary quantum systems, J. Stat. Phys. 19 (1978), in Physics, 2nd ed., Springer LNP, Berlin, 2005. pp. 575–586. [17] L.J. Schulman and U.V. Vazirani, Molecular scale [35] R. Alicki, The quantum open system as a model of the heat engines and scalable quantum computation,in heat engine, J. Phys. A: Math. General 12 (1979), pp. Proceedings of the Thirty-first Annual ACM Symposium L103–L107. on Theory of Computing, ACM Press, El Paso, TX, 1998, [36] R. Kosloffand M.A. Ratner, Beyond linear response: pp. 322–329. Line shapes for coupled spins or oscillators via direct [18] N. Linden, S. Popescu, and P. Skrzypczyk, How small can calculation of dissipated power, J. Chem. Phys. 80 (1984), thermal machines be? the smallest possible refrigerator, pp. 2352–2362. Phys. Rev. Lett. 105 (2010), 130401. [37] T.D. Kieu, The second law, maxwell’s demon, and work [19] K. Jacobs, Quantum measurement and the first law of derivable from quantum heat engines, Phys. Rev. Lett. 93 thermodynamics: The energy cost of measurement is the (2004), 140403. work value of the acquired information, Phys. Rev. E 86 [38] A. Allahverdyan and T.M. Nieuwenhuizen, Fluctuations (2012), 040106. of work from quantum subensembles: The case against [20] K. Jacobs, Quantum Measurement Theory and Its quantum work-fluctuation theorems, Phys. Rev. E 71 Applications, Cambridge University Press, Cambridge, (2005), 066102. 2014. [39] M. Esposito and S. Mukamel, Fluctuation theorems for [21] S.R. De Groot and P. Mazur, Non-equilibrium quantum master equations, Phys. Rev. E 73 (2006), Thermodynamics, Courier Corporation, New York, 046129. 1962. [40] M.J. Henrich, G. Mahler, and M. Michel, Driven [22] M. Campisi, P. Hänggi, and P. Talkner, Colloquium: spin systems as quantum thermodynamic machines: Quantum fluctuation relations: Foundations and Fundamental limits, Phys. Rev. E 75 (2007), 051118. applications, Rev. Mod. Phys. 83 (2011), pp. 771–791. [41] P.Talkner,E.Lutz,andP.Hänggi, Fluctuation theorems: [23] M. Tomamichel, A framework for non-asymptotic Work is not an observable, Phys. Rev. E 75 (2007), 050102. quantum information theory, Ph.D. thesis, Diss., [42] I. Ford, Statistical Physics: An Entropic Approach, John Eidgenössische Technische Hochschule ETH Zürich, Wiley & Sons, Chichester, 2013. Nr. 20213, 2012. [43] H. Wilming, R. Gallego, and J. Eisert, Second law of [24] R. Kosloffand A. Levy, Quantum heat engines and thermodynamics under control restrictions, Phys. Rev. E refrigerators: Continuous devices, Ann. Rev. Phys. Chem. 93 (2016), p. 042126. Available at http://link.aps.org/doi/ 65 (2014), pp. 365–393. 10.1103/PhysRevE.93.042126. [25] V. Dunjko and M. Olshanii, Thermalization from the [44] J. Deutsch, Thermodynamic entropy of a many-body perspective of eigenstate thermalization hypothesis,Ann. energy eigenstate,NewJ.Phys.12(2010), 075021. Rev. Cold Atoms Mol. 1 (2012), pp. 443–471. [45] J. Gemmer and R. Steinigeweg, Entropy increase in k-step [26] C. Gogolin and J. Eisert, Equilibration, thermalisation, markovian and consistent dynamics of closed quantum and the emergence of statistical mechanics in closed systems, Phys. Rev. E 89 (2014), 042113. quantum systems, Rep. Prog. Phys. 79 (2016), p. 056001. [46] C.H. Bennett, The thermodynamics of computation—a Available at http://stacks.iop.org/0034-4885/79/i=5/ review, Int. J. Theor. Phys. 21 (1982), pp. 905–940. a=056001. CONTEMPORARY PHYSICS 575

[47] R. Landauer, Irreversibility and heat generation in the [67] O.P. Saira, Y. Yoon, T. Tanttu, M. Möttönen, D. Averin, computing process, IBM J. Res. Dev. 5 (1961), pp. 183– and J. Pekola, Test of the jarzynski and crooks fluctuation 191. relations in an electronic system, Phys. Rev. Lett. 109 [48] M.G. Raizen, Demons, entropy and the quest for absolute (2012), 180601. zero,Sci.Am.304(2011), pp. 54–59. [68] J. Liphardt, S. Dumont, S.B. Smith, I. Tinoco, [49] D. Reeb and M.M. Wolf, An improved landauer principle and C. Bustamante, Equilibrium information from with finite-size corrections, New J. Phys. 16 (2014), nonequilibrium measurements in an experimental test of 103011. jarzynski’s equality,Science296(2002), pp. 1832–1835. [50] M.H. Mohammady, M. Mohseni, and Y. Omar, [69] D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, Minimising the heat dissipation of quantum information and C. Bustamante, Verification of the crooks fluctuation erasure, New J. Phys. 18 (2016), 015011. theorem and recovery of rna folding free energies,Nature [51] C. Browne, A.J. Garner, O.C. Dahlsten, and V. Vedral, 437 (2005), pp. 231–234. Guaranteed energy-efficient bit reset in finite time, Phys. [70] T. Sagawa and M. Ueda, Second law of thermodynamics Rev. Lett. 113 (2014), 100603. with discrete quantum feedback control, Phys. Rev. Lett. [52] J. Oppenheim, M. Horodecki, P. Horodecki, and 100 (2008), 080403. R. Horodecki, Thermodynamical approach to quanti- [71] Y. Morikuni and H. Tasaki, Quantum jarzynski-sagawa- fying quantum correlations, Phys. Rev. Lett. 89 (2002), ueda relations, J. Stat. Phys. 143 (2011), pp. 1–10. 180402. [72] J.M. Horowitz and K. Jacobs, Quantum effects improve [53] W.H. Zurek, Quantum discord and maxwell’s demons, the energy efficiency of feedback control, Phys. Rev. E 89 Phys. Rev. A 67 (2003), p. 012320. (2014), 042134. [54] R. Dillenschneider and E. Lutz, Energetics of quantum [73] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and correlations,EPL(Europhys.Lett.)88(2009), 50003. M. Sano, Experimental demonstration of information- [55] O.C. Dahlsten, R. Renner, E. Rieper, and V. Vedral, to-energy conversion and validation of the generalized Inadequacy of von neumann entropy for characterizing jarzynski equality,Nat.Phys.6(2010), pp. 988–992. extractable work, New J. Phys. 13 (2011), 053015. [74] J.V. Koski, V.F. Maisi, T. Sagawa, and J.P. Pekola, [56] S. Jevtic, D. Jennings, and T. Rudolph, Maximally and Experimental observation of the role of mutual minimally correlated states attainable within a closed information in the nonequilibrium dynamics of a maxwell evolving system, Phys. Rev. Lett. 108 (2012), 110403. demon, Phys. Rev. Lett. 113 (2014), 030601. [57] K. Funo, Y. Watanabe, and M. Ueda, Thermodynamic [75] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, work gain from entanglement, Phys. Rev. A 88 (2013), R. Dillenschneider, and E. Lutz, Experimental verifi- 052319. cation of landauer/’s principle linking information and [58] M. Perarnau-Llobet, K.V. Hovhannisyan, M. Huber, thermodynamics,Nature483(2012), pp. 187–189. P. Skrzypczyk, N. Brunner, and A. Acín, Extractable [76] Y. Jun, M. Gavrilov, and J. Bechhoefer, High-precision work from correlations, Phys. Rev. X 5 (2015), test of landauer’s principle in a feedback trap, Phys. Rev. 041011. Lett. 113 (2014), 190601. [59] M. Huber, M. Perarnau-Llobet, K.V. Hovhannisyan, [77] C. Elouard, M. Richard, and A. Auffèves, Reversible work P. Skrzypczyk, C. Klöckl, N. Brunner, and A. Acín, extraction in a hybrid opto-mechanical system, New J. Thermodynamic cost of creating correlations, New J. Phys. 17 (2015), 055018. Phys. 17 (2015), 065008. [78] J. Kurchan, A quantum fluctuation theorem, preprint [60] S. Hilt, S. Shabbir, J. Anders, and E. Lutz, Landauer’s (2000). Available at arXiv cond-mat/0007360. principle in the quantum regime, Phys. Rev. E 83 (2011), [79] S. Mukamel, Quantum extension of the jarzynski relation: 030102. Analogy with stochastic dephasing, Phys. Rev. Lett. 90 [61] P. Kammerlander and J. Anders, Coherence and (2003), 170604. measurement in quantum thermodynamics,Sci.Rep.6 [80] M. Esposito, U. Harbola, and S. Mukamel, Nonequi- (2016), 22174. librium fluctuations, fluctuation theorems, and counting [62] D.J. Evans, E. Cohen, and G. Morriss, Probability of statistics in quantum systems, Rev. Mod. Phys. 81 (2009), second law violations in shearing steady states, Phys. Rev. pp. 1665–1702. Lett. 71 (1993), pp. 2401–2404. [81] P. Solinas and S. Gasparinetti, Full distribution of work [63] R. Kawai, J. Parrondo, and C. Van den Broeck, done on a quantum system for arbitrary initial states, Dissipation: The phase-space perspective, Phys. Rev. Lett. Phys. Rev. E 92 (2015), 042150. 98 (2007), 080602. [82] C. Elouard, A. Auffeves, and M. Clusel, Stochastic [64] C. Jarzynski, Nonequilibrium work theorem for a system thermodynamics in the quantum regime, preprint (2015), strongly coupled to a thermal environment, J. Stat. Mech.: Available at arXiv:1507.00312. Theory Exp. 2004 (2004), p. P09005. [83] G. Huber, F. Schmidt-Kaler, S. Deffner, and E. Lutz, [65] S. Schuler, T. Speck, C. Tietz, J. Wrachtrup, and U. Employing trapped cold ions to verify the quantum Seifert, Experimental test of the fluctuation theorem for a jarzynski equality, Phys. Rev. Lett. 101 (2008), 070403. driven two-level system with time-dependent rates, Phys. [84] M. Heyl and S. Kehrein, Crooks relation in optical Rev. Lett. 94 (2005), 180602. spectra: Universality in work distributions for weak local [66] F. Douarche, S. Joubaud, N.B. Garnier, A. Petrosyan, quenches, Phys. Rev. Lett. 108 (2012), 190601. and S. Ciliberto, Work fluctuation theorems for harmonic [85] J.P. Pekola, P. Solinas, A. Shnirman, and D. Averin, oscillators, Phys. Rev. Lett. 97 (2006), 140603. Calorimetric measurement of work in a quantum system, New J. Phys. 15 (2013), 115006. 576 S. VINJANAMPATHY AND J. ANDERS

[86] R. Dorner, S.R. Clark, L. Heaney, R. Fazio, J. Goold, Quantum Computing, Thermodynamics and Statistical and V. Vedral, Extracting quantum work statistics and Physics Vol. 8, M. Nakahara and S. Tanaka, ed., fluctuation theorems by single-qubit interferometry, Phys. Singapore, 2012, p. 127. Rev. Lett. 110 (2013), 230601. [106] J.M. Horowitz and T. Sagawa, Equivalent definitions of [87] L. Mazzola, G. De Chiara, and M. Paternostro, the quantum nonadiabatic entropy production, J. Stat. Measuring the characteristic function of the work Phys. 156 (2014), pp. 55–65. distribution, Phys. Rev. Lett. 110 (2013), 230602. [107] D. Petz, Sufficient subalgebras and the relative entropy of [88] T.B. Batalhão, A.M. Souza, L. Mazzola, R. Auccaise, states of a von neumann algebra, Commun. Math. Phys. R.S. Sarthour, I.S. Oliveira, J. Goold, G. De 105 (1986), pp. 123–131. Chiara, M. Paternostro, and R.M. Serra, Experimental [108] D. Petz, Sufficiency of channels over von neumann reconstruction of work distribution and study of algebras, Q. J. Math. 39 (1988), pp. 97–108. fluctuation relations in a closed quantum system, Phys. [109] D. Petz, Monotonicity of quantum relative entropy Rev. Lett. 113 (2014), 140601. revisited, Rev. Math. Phys. 15 (2003), pp. 79–91. [89] S. An, J.N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.Q. [110] M.M. Wilde, Recoverability in quantum information Yin, H. Quan, and K. Kim, Experimental test of the theory, Proc. R. Soc. A 471 (2015), 20150338. quantum jarzynski equality with a trapped-ion system, [111] M. Horodecki and J. Oppenheim, Fundamental Nat. Phys. 11 (2015), pp. 193–199. limitations for quantum and nanoscale thermodynamics, [90] I. Oliveira, R. Sarthour Jr, T. Bonagamba, E. Azevedo, Nat. Commun. 4 (2013). and J.C. Freitas, NMR Quantum Information Processing, [112] M.M. Wilde, Quantum Information Theory,Cambridge Elsevier, Amsterdam, 2007. University Press, Cambridge, 2013. [91] A. Kossakowski, On quantum statistical mechanics of [113] M. Tomamichel, R. Colbeck, and R. Renner, Duality non-hamiltonian systems, Rep. Math. Phys. 3 (1972), between smooth min-and max-entropies,IEEETrans.Inf. pp. 247–274. Theory 56 (2010), pp. 4674–4681. [92] V. Gorini, A. Kossakowski, and E.C.G. Sudarshan, [114] Y. Oono and M. Paniconi, Steady state thermodynamics, Completely positive dynamical semigroups of n-level Prog. Theor. Phys. Suppl. 130 (1998), pp. 29–44. systems, J. Math. Phys. 17 (1976), pp. 821–825. [115] F. Schlögl, On thermodynamics near a steady state,Z.für [93] G. Lindblad, On the generators of quantum dynamical Phys. 248 (1971), pp. 446–458. semigroups, Commun. Math. Phys. 48 (1976), pp. 119– [116] S.i. Sasa, and H. Tasaki, Steady state thermodynamics, 130. J. Stat. Phys. 125 (2006), pp. 125–224. [94] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and [117] A. Granas and J. Dugundji, Fixed Point Theory, Springer E. Sudarshan, Properties of quantum markovian master Science & Business Media, 2013. equations, Rep. Math. Phys. 13 (1978), pp. 149–173. [118] S. Yukawa, The second law of steady state thermody- [95] H.P. Breuer and F. Petruccione, The Theory of Open namics for nonequilibrium quantum dynamics, preprint Quantum Systems, Oxford University Press, New York, (2001). Available at arXiv cond-mat/0108421. 2002. [119] W.H. Zurek and J.P. Paz, Decoherence, chaos, and the [96] M.A. Nielsen and I.L. Chuang, Quantum Computation second law, Phys. Rev. Lett. 72 (1994), pp. 2508–2511. and Quantum Information, Cambridge University Press, [120] M. Esposito and C. Van den Broeck, Three faces of the Cambridge, 2010. second law. i. master equation formulation, Phys. Rev. E [97] A.G. Redfield, On the theory of relaxation processes,IBM 82 (2010), 011143. J. Res. Dev. 1 (1957), pp. 19–31. [121] M. Esposito and C. Van den Broeck, Second law and [98] A. Hutter and S. Wehner, Almost all quantum states have landauer principle far from equilibrium, EPL (Europhys. low entropy rates for any coupling to the environment, Lett.) 95 (2011), 40004. Phys. Rev. Lett. 108 (2012), 070501. [122] T. Feldmann and R. Kosloff, Characteristics of the limit [99] P. Faist, J. Oppenheim, and R. Renner, Gibbs-preserving cycle of a reciprocating quantum heat engine, Phys. Rev. maps outperform thermal operations in the quantum E70(2004), 046110. regime,NewJ.Phys.17(2015), 043003. [123] G. Manzano, J.M. Horowitz, and J.M.R. Parrondo, [100] S. Vinjanampathy and K. Modi, Entropy bounds for Nonequilibrium potential and fluctuation theorems for quantum processes with initial correlations, Phys. Rev. quantum maps, Phys. Rev. E 92 (2015), 032129. A92(2015), 052310. [124] U.FanoandA.R.P.Rau, Symmetries in Quantum Physics, [101] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Academic Press, San Diego, CA, 1996. Quantum thermodynamics of general quantum processes, [125] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics Phys. Rev. E 91 (2015), 032119. with ultracold gases, Rev. Mod. Phys. 80 (2008), pp. 885– [102] M.M. Wolf, F. Verstraete, M.B. Hastings, and J.I. Cirac, 964. Area laws in quantum systems: Mutual information and [126] M.R. von Spakovsky and J. Gemmer, Some trends in correlations, Phys. Rev. Lett. 100 (2008), 070502. quantum thermodynamics, Entropy 16 (2014), pp. 3434– [103] V. Vedral, The role of relative entropy in quantum 3470. information theory, Rev. Mod. Phys. 74 (2002), pp. 197– [127] A. Polkovnikov, K. Sengupta, A. Silva, and 234. M. Vengalattore, Colloquium: Nonequilibrium dynamics [104] H. Spohn, Entropy production for quantum dynamical of closed interacting quantum systems, Rev. Mod. Phys. semigroups, J. Math. Phys. 19 (1978), pp. 1227–1230. 83 (2011), pp. 863–883. [105] T. Sagawa, Second Law-like Inequalities with Quantum [128] E.T. Jaynes, Probability Theory: The Logic of Science, Relative Entropy: An Introduction,inLectures on Cambridge University Press, Cambridge, 2003. CONTEMPORARY PHYSICS 577

[129] S. Goldstein, J.L. Lebowitz, R. Tumulka, and N. Zanghí, [150] S. Salek and K. Wiesner, Fluctuations in single-shot Canonical typicality, Phys. Rev. Lett. 96 (2006), 050403. ϵ-deterministic work extraction, preprint (2015). [130] S. Dubey, L. Silvestri, J. Finn, S. Vinjanampathy, and Available at arXiv:1504.05111. K. Jacobs, Approach to typicality in many-body quantum [151] D. Jonathan and M.B. Plenio, Entanglement-assisted systems, Phys. Rev. E 85 (2012), 011141. local manipulation of pure quantum states, Phys. Rev. [131] N. Linden, S. Popescu, A.J. Short, and A. Winter, Lett. 83 (1999), pp. 3566–3569. Quantum mechanical evolution towards thermal [152] N.H.Y. Ng, L. Manˇcinska, C. Cirstoiu, J. Eisert, and equilibrium, Phys. Rev. E 79 (2009), 061103. S. Wehner, Limits to catalysis in quantum thermodynam- [132] M. Srednicki, Chaos and quantum thermalization, Phys. ics,NewJ.Phys.17(2015), 085004. Rev. E 50 (1994), pp. 888–901. [153] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, [133] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization Quantum coherence, time-translation symmetry, and and its mechanism for generic isolated quantum systems, thermodynamics, Phys. Rev. X 5 (2015), 021001. Nature 452 (2008), pp. 854–858. [154] P.HänggiandF.Marchesoni, Artificial brownian motors: [134] D.A. Sivak and G.E. Crooks, Thermodynamic metrics Controlling transport on the nanoscale, Rev. Mod. Phys. and optimal paths, Phys. Rev. Lett. 108 (2012), 190602. 81 (2009), pp. 387–442. [135] M. Olshanii, Geometry of quantum observables and [155] R. Kosloff, A quantum mechanical open system as a thermodynamics of small systems, Phys. Rev. Lett. 114 model of a heat engine, J. Chem. Phys. 80 (1984), (2015), 060401. pp. 1625–1631. [136] S. Weigert, The problem of quantum integrability, Phys. [156] B. Lin and J. Chen, Performance analysis of an irreversible D: Nonlinear Phenom. 56 (1992), pp. 107–119. quantum heat engine working with harmonic oscillators, [137] M. Rigol, Breakdown of thermalization in finite one- Phys. Rev. E 67 (2003), 046105. dimensional systems, Phys. Rev. Lett. 103 (2009), 100403. [157] H. Quan, Y.x. Liu, C. Sun, and F. Nori, Quantum [138] R. Barnett, A. Polkovnikov, and M. Vengalattore, thermodynamic cycles and quantum heat engines, Phys. Prethermalization in quenched spinor condensates, Phys. Rev. E 76 (2007), 031105. Rev. A 84 (2011), 023606. [158] Y. Zheng and D. Poletti, Work and efficiency of quantum [139] M. Kollar, F.A. Wolf, and M. Eckstein, Generalized gibbs otto cycles in power-law trapping potentials, Phys. Rev. E ensemble prediction of prethermalization plateaus and 90 (2014), 012145. their relation to nonthermal steady states in integrable [159] E. Geva and R. Kosloff, A quantum-mechanical heat systems, Phys. Rev. B 84 (2011), 054304. engine operating in finite time. a model consisting of spin- [140] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, 1/2 systems as the working fluid, J. Chem. Phys. 96 (1992), M. Schreitl, I. Mazets, D.A. Smith, E. Demler, and pp. 3054–3067. J. Schmiedmayer, Relaxation and prethermalization in [160] T. Zhang, W.T. Liu, P.X. Chen, and C.Z. Li, Four-level an isolated quantum system,Science337(2012), pp. entangled quantum heat engines, Phys. Rev. A 75 (2007), 1318–1322. 062102. [141] R. Renner, Security of quantum key distribution, Int. J. [161] M. Youssef, G. Mahler, and A.S. Obada, Quantum optical Quantum Inf. 6 (2008), pp. 1–127. thermodynamic machines: Lasing as relaxation, Phys. [142] J. Åberg, Truly work-like work extraction via a single-shot Rev. E 80 (2009), 061129. analysis,Nat.Commun.4(2013), 1925. [162] M. Youssef, G. Mahler, and A.S. Obada, Quantum [143] E. Ruch and A. Mead, The principle of increasing mixing heat engine: A fully quantized model, Phys. E: Low- character and some of its consequences, Theor. Chim. dimensional Syst. Nanostruct. 42 (2010), pp. 454–460. Acta 41 (1976), pp. 95–117. [163] F. Curzon and B. Ahlborn, Efficiency of a carnot engine [144] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, at maximum power output,Am.J.Phys.43(1975), Thermodynamic cost of reliability and low temperatures: pp. 22–24. Tightening landauer’s principle and the second law,Int. [164] R. Uzdin and R. Kosloff, Universal features in the J. Theor. Phys. 39 (2000), pp. 2717–2753. efficiency at maximal work of hot quantum otto engines, [145] F.G. Brandao and M.B. Plenio, Entanglement theory and EPL (Europhys. Lett.) 108 (2014), 40001. the second law of thermodynamics,Nat.Phys.4(2008), [165] T.SchmiedlandU.Seifert, Efficiency at maximum power: pp. 873–877. An analytically solvable model for stochastic heat engines, [146] F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and EPL (Europhys. Lett.) 81 (2008), 20003. S. Wehner, The second laws of quantum thermodynamics, [166] C.M. Bender, D.C. Brody, and B.K. Meister, Quantum Proc. Nat. Acad. Sci. 112 (2015), pp. 3275–3279. mechanical carnot engine, J. Phys. A: Math. Gen. 33 [147] P. Skrzypczyk, A.J. Short, and S. Popescu, Work (2000), pp. 4427–4436. extraction and thermodynamics for individual quantum [167] B. Gardas and S. Deffner, Thermodynamic universality systems,Nat.Commun.5(2014), 4185. of quantum carnot engines, Phys. Rev. E 92 (2015), [148] M. Lostaglio, D. Jennings, and T. Rudolph, Description of 042126. quantum coherence in thermodynamic processes requires [168] O. Abah, J. Rossnagel, G. Jacob, S. Deffner, F. Schmidt- constraints beyond free energy,Nat.Commun.6(2015), Kaler, K. Singer, and E. Lutz, Single-ion heat engine at 6383. maximum power, Phys. Rev. Lett. 109 (2012), 203006. [149] J. Gemmer and J. Anders, From single-shot towards [169] J. Deng, Q.h. Wang, Z. Liu, P. Hänggi, and J. Gong, general work extraction in a quantum thermodynamic Boosting work characteristics and overall heat-engine framework, New J. Phys. 17 (2015), 085006. performance via shortcuts to adiabaticity: Quantum and classical systems, Phys. Rev. E 88 (2013), 062122. 578 S. VINJANAMPATHY AND J. ANDERS

[170] A. Del Campo, J. Goold, and M. Paternostro, More bang [190] R. Uzdin and R. Kosloff, The multilevel four-stroke swap for your buck: Super-adiabatic quantum engines,Sci.Rep. engine and its environment,NewJ.Phys.16(2014), 4(2014), 6208. 095003. [171] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and [191] N. Brunner, M. Huber, N. Linden, S. Popescu, E. Lutz, Nanoscale heat engine beyond the carnot limit, R. Silva, and P. Skrzypczyk, Entanglement enhances Phys. Rev. Lett. 112 (2014), 030602. cooling in microscopic quantum refrigerators, Phys. Rev. [172] O. Abah and E. Lutz, Efficiency of heat engines coupled E89(2014), 032115. to nonequilibrium reservoirs,EPL(Europhys.Lett.)106 [192] J.B. Brask and N. Brunner, Small quantum absorption (2014), 20001. refrigerator in the transient regime: Time scales, enhanced [173] W. Niedenzu, D. Gelbwaser-Klimovsky, A.G. Kofman, cooling, and entanglement, Phys. Rev. E 92 (2015), and G. Kurizki, , Efficiency bounds for quantum engines 062101. Available at http://link.aps.org/doi/10.1103/ powered by non-thermal baths, preprint (2015). Available PhysRevE.92.062101. at arXiv:1508.06519. [193] R. Kosloff, E. Geva, and J.M. Gordon, Quantum [174] Special issue on quantum thermodynamics, New J. refrigerators in quest of the absolute zero, J. Appl. Phys. Phys. (2015). Available at http:// 87 (2000), pp. 8093–8097. iopscience.iop.org/1367-2630/focus/ [194] J.P. Palao, R. Kosloff, and J.M. Gordon, Quantum Focus%20on%20Quantum%20Thermodynamics. thermodynamic cooling cycle, Phys. Rev. E 64 (2001), [175] H. Quan, Quantum thermodynamic cycles and quantum 056130. heat engines. ii, Phys. Rev. E 79 (2009), 041129. [195] X. Wang, S. Vinjanampathy, F.W. Strauch, and [176] R. Alicki, The quantum open system as a model of the K. Jacobs, Ultraefficient cooling of resonators: Beating heat engine, J. Phys. A: Math. Gen. 12 (1979), p. L103. sideband cooling with quantum control, Phys. Rev. Lett. [177] W. Niedenzu, D. Gelbwaser-Klimovsky, and G. Kurizki, 107 (2011), 177204. Performance limits of multilevel and multipartite [196] X. Wang, S. Vinjanampathy, F.W. Strauch, and quantum heat machines, Phys. Rev. E 92 (2015), 042123. K. Jacobs, Absolute dynamical limit to cooling weakly [178] R. Gallego, A. Riera, and J. Eisert, Thermal machines coupled quantum systems, Phys. Rev. Lett. 110 (2013), beyond the weak coupling regime,NewJ.Phys.16(2014), 157207. 125009. [197] J.M. Horowitz and K. Jacobs, Energy cost of controlling [179] M.O.Scully,M.S.Zubairy,G.S.Agarwal,andH.Walther, mesoscopic quantum systems, Phys. Rev. Lett. 115 (2015), Extracting work from a single heat bath via vanishing 13501. quantum coherence,Science299(2003), pp. 862–864. [198] B. Andresen, Current trends in finite-time thermodynam- [180] F.C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, ics, Angew. Chem. Int. Ed. 50 (2011), pp. 2690–2704. Quantacell: Powerful charging of quantum batteries,New [199] T. Feldmann and R. Kosloff, Quantum four-stroke heat J. Phys. 17 (2015), 075015. engine: Thermodynamic observables in a model with [181] E. Geva and R. Kosloff, On the classical limit of quantum intrinsic friction, Phys. Rev. E 68 (2003), 016101. thermodynamics in finite time, J. Chem. Phys. 97 (1992), [200] A.E. Allahverdyan, R. Balian, and T.M. Nieuwenhuizen, pp. 4398–4412. Maximal work extraction from finite quantum systems, [182] L. Mandelstam and I. Tamm, The uncertainty relation EPL (Europhys. Lett.) 67 (2004), pp. 565–571. between energy and time in nonrelativistic quantum [201] M. Perarnau-Llobet, K.V. Hovhannisyan, M. Huber, mechanics, J. Phys. (USSR) 9 (1945), p. 1. P. Skrzypczyk, J. Tura, and A. Acín, Most energetic [183] R. Kosloffand T. Feldmann, Discrete four-stroke passive states, Phys. Rev. E 92 (2015), 042147. quantum heat engine exploring the origin of friction, Phys. [202] D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, Rev. E 65 (2002), 055102. Work and energy gain of heat-pumped quantized [184] J.D. de la Cruz and M. Martin-Delgado, Quantum- amplifiers,EPL(Europhys.Lett.)103(2013), 60005. information engines with many-body states attaining [203] R. Alicki and M. Fannes, Entanglement boost for optimal extractable work with quantum control, Phys. extractable work from ensembles of quantum batteries, Rev. A 89 (2014), 032327. Phys. Rev. E 87 (2013), 042123. [185] R. Uzdin, A. Levy, and R. Kosloff, Equivalence of [204] K.V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, and quantum heat machines, and quantum-thermodynamic A. Acín, Entanglement generation is not necessary for signatures, Phys. Rev. X 5 (2015), 031044. optimal work extraction, Phys. Rev. Lett. 111 (2013), [186] J.B. Brask, G. Haack, N. Brunner, and M. Huber, 240401. Autonomous quantum thermal machine for generating [205] L.A. Correa, Multistage quantum absorption heat pumps, steady-state entanglement,NewJ.Phys.17(2015), Phys. Rev. E 89 (2014), 042128. 113029. [206] Y. Rezek and R. Kosloff, Irreversible performance of a [187] L.A. Correa, J.P. Palao, G. Adesso, and D. Alonso, quantum harmonic heat engine,NewJ.Phys.8(2006), Performance bound for quantum absorption refrigerators, 83. Phys. Rev. E 87 (2013), 042131. [207] E. Geva and R. Kosloff, The quantum heat engine and [188] L.A. Correa, J.P. Palao, D. Alonso, and G. Adesso, heat pump: An irreversible thermodynamic analysis of Quantum-enhanced absorption refrigerators,Sci.Rep.4 the three-level amplifier, J. Chem. Phys. 104 (1996), (2014), 3949. pp. 7681–7699. [189] F. Ticozzi and L. Viola, Quantum resources for [208] M. Campisi, J. Pekola, and R. Fazio, Nonequilibrium purification and cooling: fundamental limits and fluctuations in quantum heat engines: Theory, example, opportunities,Sci.Rep.4(2014), 5192. CONTEMPORARY PHYSICS 579

and possible solid state experiments,NewJ.Phys.17 [215] R. Alicki and D. Gelbwaser-Klimovsky, Non-equilibrium (2015), 035012. quantum heat machines,NewJ.Phys.17(2015), 115012. [209] A. Levy and R. Kosloff, The local approach to quantum [216] A. Levy, R. Alicki, and R. Kosloff, Quantum refrigerators transport may violate the second law of thermodynamics, and the third law of thermodynamics, Phys. Rev. E 85 EPL (Europhys. Lett.) 107 (2014), 20004. (2012), 061126. [210] E.A. Martinez and J.P. Paz, Dynamics and thermodynam- [217] M. Campisi, P. Talkner, and P. Hänggi, Fluctuation ics of linear quantum open systems, Phys. Rev. Lett. 110 theorem for arbitrary open quantum systems, Phys. Rev. (2013), 130406. Lett. 102 (2009), 210401. [211] R. Kosloff, Quantum thermodynamics: A dynamical [218] G. Biroli, C. Kollath, and A.M. Läuchli, Effect of rare viewpoint, Entropy 15 (2013), pp. 2100–2128. fluctuations on the thermalization of isolated quantum [212] M. Koláˇr, D. Gelbwaser-Klimovsky, R. Alicki, and systems, Phys. Rev. Lett. 105 (2010), 250401. G. Kurizki, Quantum bath refrigeration towards absolute [219] Z. Gong, S. Deffner, and H.T. Quan, Interference of zero: Challenging the unattainability principle, Phys. Rev. identical particles and the quantum work distribution, Lett. 109 (2012), 090601. Phys. Rev. E 90 (2014), 062121. [213] R. Alicki, D. Gelbwaser-Klimovsky, and G. Kurizki, [220] J. Roßnagel, S.T. Dawkins, K.N. Tolazzi, O. Abah, Periodically driven quantum open systems: Tutorial, E. Lutz, F. Schmidt-Kaler, and K. Singer, Asingle-atom preprint (2012). Available at arXiv:1205.4552. heat engine,Science352(2016), pp. 325–329. [214] D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, Minimal universal quantum heat machine, Phys. Rev. E87(2013), 012140.