Quantum Thermodynamics

Sai Vinjanampathya∗ and Janet Andersb†

aCentre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543.bDepartment of Physics and Astronomy, University of Exeter, Stocker Road, EX4 4QL, United Kingdom.

Quantum thermodynamics is an emerging research field aiming to extend standard thermodynamicsand non-equilibrium statistical physics to ensembles of sizes well below the thermodynamic limit, innon-equilibrium situations, and with the full inclusion of quantum effects. Fuelled by experimental ad-vances and the potential of future nanoscale applications this research effort is pursued by scientists withdifferent backgrounds, including statistical physics, many-body theory, mesoscopic physics and quantuminformation theory, who bring various tools and methods to the field. A multitude of theoretical questionsare being addressed ranging from issues of thermalisation of quantum systems and various definitionsof “work”, to the efficiency and power of quantum engines. This overview provides a perspective on aselection of these current trends accessible to postgraduate students and researchers alike.

Keywords: (quantum) information-thermodynamics link, (quantum) fluctuation theorems,thermalisation of quantum systems, single shot thermodynamics, quantum thermal machines

Contents

1 Introduction 2

2 Information and Thermodynamics 32.1 The first and second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . 32.2 Maxwell’s demon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Landauer’s erasure principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Erasure with quantum side information . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Measurement work due to coherences . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Classical and Quantum Non-Equilibrium Statistical Physics 73.1 Definitions of classical fluctuating work and heat . . . . . . . . . . . . . . . . . . . 73.2 Classical fluctuation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Fluctuation relations with feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Classical statistical physics experiments . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Definitions of quantum fluctuating work and heat . . . . . . . . . . . . . . . . . . . 123.6 Quantum fluctuation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.7 Quantum fluctuation experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Quantum Dynamics and Foundations of Thermodynamics 15

∗Email: sai@quantumlah.org†Email: janet@qipc.org

arX

iv:1

508.

0609

9v1

[qu

ant-

ph]

25

Aug

201

5

4.1 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Role of fixed points in thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Thermalisation of closed quantum systems . . . . . . . . . . . . . . . . . . . . . . . 18

5 Single Shot Thermodynamics 205.1 Work extraction in single shot thermodynamics . . . . . . . . . . . . . . . . . . . . 205.2 Work extraction and work of formation in thermodynamic resource theory . . . . . 215.3 Single shot second laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Single shot second laws with coherence . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Quantum Thermal Machines 256.1 Background and basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Carnot engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 Otto engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 Diesel engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.5 Quantumness of engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 Quantum refrigerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.7 Coherence and time in thermal machines . . . . . . . . . . . . . . . . . . . . . . . . 316.8 Coherence and work extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.9 Power and shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.10 Relationship to laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Discussion and Open Questions 33

1. Introduction

One of the biggest puzzles in quantum theory today is to show how the well-studied properties ofa few particles translate into a statistical theory from which new macroscopic quantum thermody-namic laws emerge. This challenge is addressed by the emerging field of quantum thermodynamicswhich has grown rapidly over the last decade. It is fuelled by recent equilibration experiments [1]in cold atomic and other physical systems, the introduction of new numerical methods [2], andthe development of novel theoretical tools from non-equilibrium statistical physics and quantuminformation theory [3–9]. With ultrafast experimental control of quantum systems and engineeringof small environments pushing the limits of conventional thermodynamics, the central goal of quan-tum thermodynamics is the extension of standard thermodynamics to include quantum effects andsmall ensemble sizes. Apart from the academic drive to clarify fundamental processes in nature,there is a strong industrial need for advancing our understanding of thermodynamic processes atthe nanoscale. Obtaining a detailed knowledge of how quantum fluctuations compete with thermalfluctuations is essential for us to be able to adapt existing technologies to operate at ever decreasingscales, and to uncover new technologies that may harness quantum thermodynamic features.

Various perspectives have emerged in quantum thermodynamics, due to the interdisciplinarynature of the field, and each contributes different insights. For example, the study of thermalisa-tion has been approached by quantum information theory from the standpoint of typicality andentanglement, and by many-body physics with a dynamical approach. Likewise, the recent studyof quantum thermal machines, originally studied from a quantum optics perspective [10], has sincereceived significant input from many-body physics, fluctuation relations, and linear response ap-proaches [11, 12]. These designs further contrast with the thermal machines based on quantuminformation theoretic approaches [13, 14]. The difference in perspectives on the same topics hasalso meant that there are ideas within quantum thermodynamics where consensus is yet to beestablished.

2

This article is aimed at non-expert readers and specialists in one subject who seek a brief overviewof quantum thermodynamics. The scope is to give an introduction to some of the different perspec-tives on current topics in a single paper, and guide the reader to a selection of useful references.Given the rapid progress in the field there are many aspects of quantum thermodynamics thatare not covered in this overview. Some topics have been intensely studied for several years anddedicated reviews are available, for example, in classical non-equilibrium thermodynamics [15],fluctuation relations [16], non-asymptotic quantum information theory [17], quantum engines [18],equilibration and thermalisation [19, 20] and a recent quantum thermodynamics review focussingon quantum information theory techniques [21]. We encourage researchers to take on board theinsights gained from different approaches and attempt to fit together pieces of the puzzle to createan overall united framework of quantum thermodynamics.

Section 2 discusses the standard laws of thermodynamics and introduces the link with informationprocessing tasks and Section 3 gives a brief overview of fluctuation relations in the classical andquantum regime. Section 4 then introduces quantum dynamical maps and discusses implications forthe foundations of thermodynamics. Properties of maps form the backbone of the thermodynamicresource theory approach to single shot thermodynamics discussed in Section 5. Section 6 discussesthe operation of thermal machines in the quantum regime. Finally, in Section 7 the current stateof the field is summarised and open questions are identified.

2. Information and Thermodynamics

This section defines averages of heat and work, introduces the first and second law of thermody-namics and discusses examples of the link between thermodynamics and information processingtasks, such as erasure.

2.1. The first and second law of thermodynamics

Thermodynamics is concerned with energy and changes of energy that are distinguished as heatand work. For a quantum system in state ρ and with Hamiltonian H at a given time the system’sinternal, or average, energy is identified with the expectation value U(ρ) = tr[ρH]. When a systemchanges in time, i.e. the pair of state and Hamiltonian [22] vary in time (ρ(t), H(t)) with t ∈ [0, τ ],the resulting average energy change

∆U = tr[ρ(τ)H(τ)]− tr[ρ(0)H(0)] (1)

is made up of two types of energy transfer - work and heat. Their intuitive meaning is that oftwo types of energetic resources, one fully controllable and useful, the other uncontrolled andwasteful [5, 22–30]. Since the time-variation of H is controlled by an experimenter the energychange associated with this time-variation is identified as work. The uncontrolled energy changeassociated with the reconfiguration of the system state in response to Hamiltonian changes andthe system’s coupling with the environment is identified as heat. The formal definitions of averageheat absorbed by the system and average work done on the system are then

〈Q〉 :=

∫ τ

0tr[ρ(t)H(t)] dt and 〈W 〉 :=

∫ τ

0tr[ρ(t) H(t)] dt. (2)

Here the brackets 〈·〉 indicate the ensemble average that is assumed in the above definition whenthe trace is performed. Work is extracted from the system when 〈Wext〉 := −〈W 〉 > 0, while heatis dissipated to the environment when 〈Qdis〉 := −〈Q〉 > 0.

3

The first law of thermodynamics states that the sum of average heat and work done on thesystem just makes up its average energy change,

〈Q〉+ 〈W 〉 =

∫ τ

0

d

dttr[ρ(t)H(t)] dt = tr[ρ(τ)H(τ)]− tr[ρ(0)H(0)] = ∆U. (3)

It is important to note that while the internal energy change only depends on the initial and finalstates and Hamiltonians of the evolution, heat and work are process dependent, i.e. it mattershow the system evolved in time from (ρ(0), H(0)) to (ρ(τ), H(τ)). Therefore heat and work for aninfinitesimal process will be denoted by 〈δQ〉 and 〈δW 〉 where the symbol δ indicates that heat andwork are (in general) not full differentials and do not correspond to observables [31], in contrastto the average energy with differential dU .

Choosing to split the energy change into two types of energy transfer is crucial to allow theformulation of the second law of thermodynamics. A fundamental law of physics, it sets limits onthe work extraction of heat engines and establishes the notion of irreversibility in physics. Clausiusobserved in 1865 that a new state function - the thermodynamic entropy Sth of a system - is helpfulto study the heat flow to the system when it interacts with baths at varying temperatures T [32].The thermodynamic entropy is defined through its change in a reversible thermodynamic process,

∆Sth :=

∫rev

〈δQ〉T

, (4)

where 〈δQ〉 is the heat absorbed by the system along the process and T is the temperature at whichthe heat is being exchanged between the system and the bath. Further observing that any cyclic

process obeys∮ 〈δQ〉

T ≤ 0 with equality for reversible processes Clausius formulated a version ofthe second law of thermodynamics for all thermodynamic processes, today known as the Clausius-inequality: ∫

〈δQ〉T≤ ∆Sth, becoming 〈Q〉 ≤ T ∆Sth for T = const. (5)

It states that the change in a system’s entropy must be equal or larger than the average heatabsorbed by the system during a process. In this form Clausius’ inequality establishes the existenceof an upper bound to the heat absorbed by the system and its validity is generally assumed toextend to the quantum regime [5, 22–30, 33]. Equivalently, by defining the free energy of a systemwith Hamiltonian H and in contact with a heat bath at temperature T as the state function

F (ρ) := U(ρ)− TSth(ρ), (6)

Clausius’ inequality becomes a statement of the upper bound on the work that can be extractedin a thermodynamic process,

〈Wext〉 = −〈W 〉 = −∆U + 〈Q〉 ≤ −∆U + T∆Sth = −∆F. (7)

While the actual heat absorbed/work extracted will depend on the specifics of the process thereexist optimal, thermodynamically reversible, processes that saturate the equality, see Eq. (4).However, modifications of the second law, and thus the optimal work that can be extracted, arisewhen the control of the working system is restricted to physically realistic, local scenarios [34].

For equilibrium states ρth = e−βH/tr[e−βH ] for Hamiltonian H and at inverse temperatures

β = 1kB T

the thermodynamic entropy Sth equals the information theory entropy Sth(ρth)kB

= S(ρth),

4

Figure 1. A gas in a box starts with a thermal distribution at temperature T . Maxwell’s demon inserts a wall and selectsfaster 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 into two boxes that are not in equilibrium. The demon attaches a bucket (or another suitable work storage system)

and allows the wall to move under the pressure of the gases. Some gas energy is extracted as work, Wext, raising the bucket.Finally, the gas is brought in contact with the environment at temperature T , equilibrating back to its initial state (∆U = 0)

while absorbing heat Qabs = Wext.

known as Shannon or von-Neumann entropy of a state ρ,

S(ρ) := −tr[ρ ln ρ]. (8)

Here we have normalised the information theory entropy with the Boltzmann constant kB. For theremainder of this article we will assume that the thermodynamic entropy is naturally extended tonon-equilibrium states by the information theoretic entropy which we will use from now on. Wenote that the suitability of this extension remains a debated issue [35, 36].

2.2. Maxwell’s demon

Maxwell’s demon is a creature that is able to observe the motion of individual particles and usethis information (by employing feedback protocols discussed in section 3.3) to convert heat intowork in a cyclic process. By separating slower, “colder” gas particles in a container from faster,“hotter” particles, and then allowing the hotter gas to expand while pushing a piston, see Fig. 1,the demon can extract work while returning to the initial mixed gas. For a single particle gas, thedemon’s extracted work in a cyclic process is

〈W demonext 〉 = kBT ln 2. (9)

Maxwell realised in 1867 that such a demon would appear to break the second law of thermody-namics as it converts heat completely into work in a cyclic process resulting in a positive extractedwork, 〈W demon

ext 〉 = 〈Q〉−∆U 6≤ T∆Sth−∆U = 0. The crux of this paradoxical situation is that thedemon acquires information about individual gas particles and uses this information to convertheat into useful work. One way of resolving the paradox was presented by Bennett [37] and invokesLandauer’s erasure principle [38], as described in the next section. A comprehensive recent reviewof Maxwell’s demon is [39]. Experimentally, the thermodynamic phenomenon of Maxwell’s demonremains highly relevant - for instance, cooling a gas can be achieved by mimicking the demonicaction [40]. An example of an experimental test of Maxwell’s demon is discussed in section 3.4.

5

2.3. Landauer’s erasure principle

In a seminal paper [38] Landauer investigated the thermodynamic cost of information processingtasks. He concluded that the erasure of one bit of information requires a minimum dissipation ofheat,

〈Qmindis 〉 = kB T ln 2, (10)

that the erased system dissipates to a surrounding environment, cf. Fig. 1, in equilibrium at tem-perature T . Erasure of one bit here refers to the change of a system being in one of two stateswith equal probability, 1

2 , to a definite known state, |0〉. Landauer’s principle has been testedexperimentally only very recently, see section 3.4.

The energetic cost of the dissipation is balanced by a minimum work that must be done on thesystem, 〈Wmin〉 = ∆U −〈Q〉 = 〈Qmin

dis 〉, to achieve the erasure at constant average energy, ∆U = 0.Bennett argued [37] that Maxwell’s demonic paradox can thus be resolved by taking into accountthat the demon’s memory holds bits of information that need to be erased to completely close thethermodynamic cycle and return to the initial conditions. The work that the demon extracted inthe first step, see previous section, then has to be spent to erase the information acquired, with nonet gain of work and in agreement with the second law.

While Landauer’s principle was originally formulated for the erasure of one bit of classicalinformation, it is straightforwardly extended to the erasure of a general mixed quantum state,ρ, which is transferred to the blank state |0〉. The minimum required heat dissipation is then〈Qmin

dis 〉 = kB T S(ρ) following from the second law, Eq. (5). A recent analysis of Landauer’s princi-ple [41] uses a framework of thermal operations, cf. section 5, to obtain corrections to Landauer’sbound when the effective size of the thermal bath is finite. They show that the dissipated heat isin general above Landauer’s bound and only converges to it for infinite-sized reservoirs.

2.4. Erasure with quantum side information

The heat dissipation during erasure is non-trivial when the initial state of the system is a mixedstate, ρS . In the quantum regime mixed states can always be seen as reduced states of globalstates, ρSM , of the system S and a memory M , with ρS = trM [ρSM ]. A recent paper [8] takesthis insight seriously and sets up an erasure scenario where an observer can operate on both, thesystem and the memory. During the global process the system’s local state is erased, ρS → |0〉,while the memory’s local state, ρM = trS [ρSM ], is not altered. In other words, this global processis locally indistinguishable from the erasure considered in the previous section. However, contraryto the previous case, erasure with side-information can be achieved while extracting a maximumamount of work

〈Wmaxext 〉 = −kB T S(ρSM |ρM ), (11)

where S(ρSM |ρM ) is the conditional von Neumann entropy, S(ρSM |ρM ) = S(ρSM )− S(ρM ). Cru-cially, the conditional entropy can be negative for some quantum correlated states (a subset of theset of entangled states) thus giving a positive extractable work. The possibility to extract workduring erasure is a purely quantum feature that relies on accessing the “side-information” of thequantum system stored in the memory [8]. The result contrasts strongly with Landauer’s principlevalid for both classical and quantum states when no side-information is available. The extractionof the maximum work does however require a maximally entangled state and the implementationof a finely tuned process.

6

2.5. Measurement work due to coherences

A recent paper [42] discusses a second fundamental information processing task that has a non-trivial thermodynamic contribution only in the quantum regime. Here projective processes areconsidered in which the coherences of a state ρ are removed with respect to a particular basisΠkk, i.e. ρ → η :=

∑k Πk ρΠk. Such a projective process can be interpreted as an unselective

measurement of an observable with eigenbasis Πkk, a measurement in which not the individualmeasurement outcomes are recorded but only the statistics of the outcomes is known. Just likeLandauer’s erasure this state transfer can be implemented in various physical ways with varyingwork and heat contributions that are bounded by optimal values. In particular, if the basis is theenergy eigenbasis of a Hamiltonian H =

∑k Ek Πk with eigenenergies Ek there exists a physical

way to implement the projection process such that a maximum amount of average work,

〈Wmaxext 〉 = kB T (S(η)− S(ρ)), (12)

can be extracted [42]. Importantly, S(η) > S(ρ) if and only if the state ρ had coherences withrespect to the energy eigenbasis Πkk. Thus the extracted work here is due to quantum coherencescontrasting with the work gained from quantum correlations discussed in the previous section.Again, the extraction of this maximum work requires a finely tuned process and is not achieved whenthe state transfer is implemented in a suboptimal process, such as “decohering” the state duringwhich the environment washes out coherences in an uncontrolled way and no work is extracted.

3. Classical and Quantum Non-Equilibrium Statistical Physics

In this section the statistical physics approach is outlined that rests on the definition of fluctuatingheat and work from which it derives the ensemble quantities of average heat and average workdiscussed in section 2. Fluctuation theorems and experiments are first described in the classicalregime before they are extended to the quantum regime.

3.1. Definitions of classical fluctuating work and heat

In classical statistical physics a single particle is assigned a point, x = (q, p), in phase space whilean ensemble of particles is described with a probability density function, P (x), in phase space.The Hamiltonian of the particle is denoted as H(x, λ) where x is the phase space point of theparticle for which the energy is evaluated and λ is an externally controlled force parameter that

can change in time. For example, a harmonic oscillator Hamiltonian H(x, λ) = p2

2m + mλ2q2

2 canbecome time-dependent through a protocol according to which the frequency, λ, of the potentialis varied in time, λ(t). This particular example will be discussed further in section 6 on quantumthermal machines.

Each particle’s state will evolve due to externally applied forces and due to the interaction withenvironments resulting in a trajectory xt = (qt, pt) in phase space, see Fig. 2. The fluctuating work(done on the system) and the fluctuating heat (absorbed by the system) for a single trajectory xtfollowed by a particle in the time window [0, τ ] is defined, analogously to (2), as the energy changedue to the externally controlled force parameter and the response of the system’s state,

Wτ,xt :=

∫ τ

0

∂H(xt, λ(t))

∂λ(t)λ(t) dt and Qτ,xt :=

∫ τ

0

∂H(xt, λ(t))

∂xtxt dt. (13)

These are stochastic variables that depend on the particle’s trajectory xt in phase space. Together

7

Figure 2. Phase space spanned by position and momentum coordinates, x = (q, p). A single trajectory, xt, starts at phase

space point x0 and evolves in time t to a final phase space point xτ . This single trajectory has an associated fluctuating work

Wτ,xt . An ensemble of initial phase space points described by an initial probability density functions, P0(x0), evolves to somefinal probability density function Pτ (xτ ). The ensemble of trajectories has an associated average work 〈Wτ 〉. Indicated with a

dashed line is the initial probability density shifted to be evaluated at the final phase space point xτ , P0(xτ ).

they give the energy change of the system, cf. (3), along the trajectory

Wτ,xt +Qτ,xt = H(xτ , λ(τ))−H(x0, λ(0)). (14)

For a closed system one has

Qclosedτ,xt =

∫ τ

0

(∂H(xt, λ(t))

∂qtqt +

∂H(xt, λ(t))

∂ptpt

)dt =

∫ τ

0(−ptqt + qtpt) dt = 0,

W closedτ,xt = H(xτ , λ(τ))−H(x0, λ(0)), (15)

since the (Lagrangian) equations of motion, ∂H∂q = −p and ∂H

∂p = q, apply to closed systems. This

means that a closed system has no heat exchange (a tautological statement) and that the change ofthe system’s energy during the protocol is identified entirely with work for each single trajectory.

The average work of the system refers to repeating the same experiment many times each timepreparing the same initial system state, with the same environmental conditions and applying thesame protocol. The average work is mathematically obtained as the integral over all trajectories,xt, explored by the system, 〈Wτ 〉 :=

∫Pt(xt)Wτ,xt D(xt), where Pt(xt) is the probability density

of a trajectory xt and D(xt) the phase space integral over all trajectories [6]. For a closed systemcombining this with Eqs. (13), (14) and (15), one obtains

〈W closedτ 〉 =

∫P0(x0) (H(xτ (x0), λ(τ))−H(x0, λ(0))) dx0,

=

∫P0(xτ )H(xτ , λ(τ))

∣∣∣∣dx0

dxτ

∣∣∣∣ dxτ −∫P0(x0)H(x0, λ(0)) dx0, (16)

= Uτ − U0,

i.e. the average work for a closed process is just the difference of the average energies. Here oneuses that each trajectory starting with x0 evolves deterministically to xτ at time τ ( dx0

dxτ= 1) and

the initial probability distribution, P0(x0), thus remains the same, P0(xτ ), just applied to the finaltrajectory points xτ (Liouville’s theorem), cf. Fig. 2. Here U0 =

∫P0(x)H(x, λ(0)) dx is the initial

average energy of the system, and similarly Uτ at time τ .In statistical physics experiments the average work is often established by measuring the fluc-

tuating work Wj for a trajectory observed in a particular run j of the experiment, repeating theexperiment N times, and taking the arithmetic mean of the results. In the limit N → ∞ this isequivalent to constructing (from a histogram of the outcomes Wj) the probability density distri-

8

bution for the work, P (W ), and then averaging with this function to obtain the average work

〈Wτ 〉 = limN→∞

1

N

N∑j=1

Wj =

∫P (W )W dW. (17)

3.2. Classical fluctuation relations

A central recent breakthrough in classical statistical mechanics is the extension of the secondlaw of thermodynamics, an inequality, to equalities valid for large classes of non-equilibrium pro-cesses. Detailed fluctuation relations show that the probability densities of stochastically fluctuatingquantities for a non-equilibrium process, such as entropy, work and heat, are linked to equilibriumproperties and corresponding quantities for the time-reversed process [4, 43, 44]. By integratingover the probability densities one obtains integral fluctuation relations, such as Jarzynski’s workrelation [3].

An important detailed fluctuation relation is Crooks relation for a system in contact with abath at inverse temperature β for which detailed balanced is valid [4]. The relation links the workdistribution PF (W ) associated with a forward protocol changing the force parameter λ(0)→ λ(τ),to the work distribution PB(−W ) associated with the time-reversed backwards protocol whereλ(τ)→ λ(0),

PF (W ) = PB(−W ) eβ(W−∆F ). (18)

The free energy difference ∆F = F (τ) − F (0) refers to the two thermal distributions with respectto the final, H(x, λ(τ)), and initial, H(x, λ(0)), Hamiltonian in the forward protocol at inversetemperature β. Here the free energies, defined in (6), can be expressed as F (0) = − 1

β lnZ(0) at

time 0 and similarly at time τ with classical partition functions Z(0) :=∫e−βH(x,λ(0)) dx and

similarly Z(τ). Rearranging and integrating over dW on both sides results in a well-known integralfluctuation relation, the Jarzynski equality [3],

〈e−βW 〉 =

∫PF (W ) e−βW dW = e−β∆F . (19)

Pre-dating Crooks relation, Jarzynski proved his equality [3] by considering a closed system

that starts with a thermal distribution, P0(x0) = e−βH(x0,λ(0))

Z(0) , for a given Hamiltonian H(x, λ(0))at inverse temperature β. The Hamiltonian is externally modified through its force parameter λwhich drives the system out of equilibrium and into evolution according to Hamiltonian dynamicsin a time-interval [0, τ ]. The averaged exponentiated work done on the system is then calculatedaccording to Eq. (15), and one obtains

〈e−βW closedτ 〉 =

∫P0(x0) e−βW

closedτ,xτ dx0 =

∫e−βH(x0,λ(0))

Z(0)e−β(H(xτ ,λ(τ))−H(x0,λ(0))) dx0, (20)

=1

Z(0)

∫e−βH(xτ ,λ(τ))

∣∣∣∣dx0

dxτ

∣∣∣∣ dxτ =Z(τ)

Z(0)= e−β∆F . (21)

The beauty of this equality is that for all closed but arbitrarily strongly non-equilibrium processesthe averaged exponentiated work along trajectories starting from an initially thermal distribution isdetermined entirely by equilibrium parameters contained in ∆F . As Jarzynski showed the equalitycan also be extended to open systems [45]. By applying Jensen’s inequality 〈e−βW 〉 ≥ e−β〈W 〉, the

9

Jarzynski equality turns into the standard second law of thermodynamics, cf. Eq. (7),

〈W 〉 ≥ ∆F. (22)

Thus Jarzynski’s relation strengthens the second law by including all moments of the non-equilibrium work resulting in an equality from which the inequality follows for the first moment.

Fluctuation theorems have been measured, for example, for a defect centre in diamond [46], fora torsion pendulum [47], and in an electronic system [48]. They have also been used to infer, fromthe measurable work in non-equilibrium pulling experiments, the desired equilibrium free-energysurface of bio-molecules [49, 50], which is not directly measurable.

3.3. Fluctuation relations with feedback

It is interesting to see how Maxwell demon’s feedback process can be captured in a generalisedJarzynski equation [6]. Here the system Hamiltonian is changed according to a protocol that dependson the phase space point the system is found in. The work fluctuation relation then becomes

〈e−β(W−∆F )〉 = γ, (23)

where γ characterises the feedback efficacy, which is related to the reversibility of the non-equilibrium process. Its maximum value is determined by the number of different protocol options.For example, for the two choices of a particle being in the left box or the right box, see Fig. 3, onehas γmax = 2. In this case the initial Hamiltonian H(x, λ(0)) will be changed to either H(x, λ1(τ))for the particle in the left box or H(x, λ2(τ)) for the particle in the right box. The average 〈·〉appearing in Eq. (23) includes an average over the two different protocols with the free energy dif-ference, ∆F , being itself a fluctuating function, here either ∆F = F (1)−F (0) or ∆F = F (2)−F (0)

depending on the measurement outcome in each particular run.Apart from measuring 〈e−β(W−∆F )〉 by performing the feedback protocol many times and record-

ing the work done on the system, γ can also be measured experimentally, see section 3.4. To do sothe system is initially prepared in a thermal state for one of the final Hamiltonians, say H(y, λ1(τ)),at inverse temperature β. The force parameter λ1(t) in the Hamiltonian is now changed backwards,λ1(τ) → λ(0)) without any feeback. The particle’s final phase space point yτ is recorded. Theprobability P (1)(yτ ) is established that the particle ended in the phase space volume that corre-sponds to the choice of H(y, λ1(τ)) in the forward protocol, in this example in the left box. Thesame is repeated for the other final Hamiltonians, i.e. H(y, λ2(τ)). The efficacy is now the addedprobability for each of the protocols, γ =

∑k P

(k)(yτ ).Applying Jensen’s inequality to Eq. (23) one now obtains 〈W 〉 ≥ 〈∆F 〉 − 1

β ln γ, which when

assuming that a cycle has been performed (〈∆F 〉 = 0) the maximal value of γ for two feedbackoptions has been reached (γ = 2) and the inequality is saturated, becomes

〈W demonext 〉 = −〈W 〉 = kBT ln 2, (24)

in agreement with Eq. (9).

3.4. Classical statistical physics experiments

A large number of statistical physics experiments are performed with colloidal particles, such aspolystyrene beads, that are suspended in a viscous fluid. The link between information theory andthermodynamics has been confirmed by groundbreaking colloidal experiments only very recently.

10

c

Figure 3. a. Dimeric polystyrene bead and electrodes for creating the spiral staircase. b. Schematic of a Brownian particle’s

dynamics climbing the stairs with the help of the demon. c. Experimental results for the feedback process 〈e−β(W−∆F )〉 anda time-reversed process γ, see Eq. (23), over the demon’s reaction time ε. (Figures taken from Nature Physics 6, 988 (2010).)

For example, the heat-to-work conversion achieved by a Maxwell demon intervening in the statis-tical dynamics of a system was investigated [51] with a dimeric polystyrene bead suspended in afluid and undergoing rotational Brownian motion, see Fig. 3a. An external electrical potential wasapplied so that the bead was effectively moving on a spiral staircase, see Fig. 3b. The demon’saction was realised by measuring if the particle had moved up and, depending on this, shifting theexternal ladder potential quickly, such that the particle’s potential energy was “saved” and theparticle would continue to climb up. The work done on the particle, W , was then measured forover 100000 trajectories to obtain an experimental value of 〈e−β(W−∆F )〉, required to be identicalto 1 by the standard Jarzynski equality, Eq. (20). Due the feedback operated by the demon thevalue did turn out larger than 1, see Fig. 3c, in agreement with Eq. (23). A separate experimentimplementing the time-reversed process without feedback was run to also determine the value of theefficacy parameter γ predicted to be the same as 〈e−β(W−∆F )〉. Fig. 3c show very good agreementbetween the two experimental results for varying reaction time of the demon, reaching the highestfeedback efficiency when the demon acted quickly on the knowledge of the particles position. Thetheoretical predictions of fluctuation relations that include Maxwell’s demon [6] have also beentested with a single electron box analogously to the original Szilard engine [52].

Another recent experiment measured the heat dissipated in an erasure process with a colloidalsilica bead that was optically trapped with tweezers in a double well potential [53]. The protocolemployed, see Fig. 4, ensured that a bead starting in the left well would move to the right well,while a starting position in the right well remained unchanged. The lowering and raising of theenergetic barrier between the wells was achieved by changing the trapping laser’s intensity. Thetilting of the potential, seen in Fig. 4c-e, was neatly realised by letting the fluid flow, resulting in aforce acting on the bead suspended in the fluid. The dissipated heat (negative absorbed heat) was

measured by following the trajectories xt of the bead and integrating, Qdis = −∫ τ

0∂U(xt,λ(t))

∂xtxt dt,

cf. Eq. (13), where U(xt, λ(t)) is the explicitly time-varying potential that together with a constantkinetic term makes up a time-varying Hamiltonian H(xt, λ(t)) = T (xt)+U(xt, λ(t)). The measuredheat distribution P (Qdis) and average heat 〈Qdis〉, defined analogously to Eq. (17), are shown onthe right in Fig. 4. The time taken to implement the protocol is denoted by τ . In the adiabaticlimit, i.e. the limit of long times in which the system equilibrates throughout its dynamics, it wasfound that the Landauer limit, kB T ln 2, for the minimum dissipated heat is indeed approached.

11

Figure 4. Left a-f: Erasure protocol for a particle trapped in a double well. Particles starting in either well will end up in theright 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 ), over varying cycle time τ .

(Figures taken from Nature 483, 187 (2012).)

3.5. Definitions of quantum fluctuating work and heat

By its process character it is clear that work is not an observable [31], i.e. there is no operator, w,such that W = tr[w ρ]. To quantise the Jarzynski equality the crucial step taken is to define thefluctuating quantum work for closed dynamics as a two-point correlation function [5, 31, 54, 55],i.e. a projective measurement of energy needs to be performed at the beginning and end of theprocess to gain knowledge of the system’s energetic change. This enables the construction of a work-distribution function and allows the formulation of the Tasaki-Crooks relation and the quantumJarzynski equality.

To introduce the quantum fluctuating work and heat, consider a quantum system with ini-

tial state ρ(0) and initial Hamiltonian H(0) =∑

nE(0)n |e(0)

n 〉〈e(0)n | with eigenvalues E

(0)n . A closed

system undergoes dynamics due to its time-varying Hamiltonian which generates a unitary1

V (τ) = T e−i∫ τ0H(t) dt/~ and ends in a final state ρ(τ) = V (τ) ρ(0) V (τ)† and a final Hamiltonian

H(τ) =∑

mE(τ)m |e(τ)

m 〉〈e(τ)m |, with energies E

(τ)m . To obtain the fluctuating work the energy is mea-

sured at the beginning of the process, giving e.g. E(0)n , and then again at the end of the process,

giving e.g. in E(τ)m . The difference between the measured energies is now identified entirely with

fluctuating work,

W closedm,n := E(τ)

m − E(0)n , (25)

as the system is closed, the evolution is unitary and no heat dissipation occurs, cf. the classicalcase (15). The work distribution is then peaked whenever the distribution variable W coincideswith W τ

m,n,

P (W ) =∑n,m

p(τ)n,m δ(W − (E(τ)

m − E(0)n )). (26)

Here p(τ)n,m = p

(0)n p

(τ)m|n is the joint probability distribution of finding the initial energy level E

(0)n

1Here T stands for a time-ordered integral.

12

and the final energy E(τ)m . This can be broken up into the probability of finding the initial energy

E(0)n , p

(0)n = 〈e(0)

n |ρ(0)|e(0)n 〉, and the conditional probability for transferring from n at t = 0 to m at

t = τ , p(τ)m|n = |〈e(τ)

m |V (τ)|e(0)n 〉|2.

The average work for the unitarily-driven non-equilibrium process can now be calculated as theaverage over the work probability distribution, i.e. using (17),

〈W closedτ 〉 =

∫ ∑n,m

p(τ)n,m δ(W − (E(τ)

m − E(0)n ))W dW

=∑n,m

p(0)n p

(τ)m|n (E(τ)

m − E(0)n ) =

∑m

p(τ)m E(τ)

m −∑n

p(0)n E(0)

n , (27)

where p(τ)m :=

∑n p

(τ)n,m = 〈e(τ)

m |ρ(τ)|e(τ)m 〉 are the final state marginals of the joint probability

distribution, i.e. just the probabilities for measuring the energies E(τ)m in the final state ρ(τ). Thus

the average work for the unitary process is just the internal energy difference, cf. (16),

〈W closedτ 〉 = tr[H(τ) ρ(τ)]− tr[H(0) ρ(0)] = ∆U. (28)

3.6. Quantum fluctuation relations

With the above definitions, in particular that of the work distribution, the quantum Jarzynskiequation can be readily formulated for a closed quantum system undergoing externally driven(with V ) non-equilibrium dynamics [5, 31, 54–56]. The average exponentiated work done on asystem starting in initial state ρ(0) now becomes, cf. (20),

〈e−βW closedτ 〉 =

∫P (W ) e−βW dW =

∑n,m

p(τ)n,m e

−β(E(τ)m −E(0)

n ). (29)

When the initial state is a thermal state for the Hamiltonian H0 at inverse temperature β, e.g.

ρ(0) =∑

n p(0)n |e(0)

n 〉〈e(0)n |, with thermal probabilities, p

(0)n = e−βE

(0)n

Z(0) and partition function Z(0) =∑n e−βE(0)

n , one gets

〈e−βW closedτ 〉 =

1

Z(0)

∑n,m

p(τ)m|n e

−βE(τ)m =

1

Z(0)

∑m

e−βE(τ)m =

Z(τ)

Z(0)= e−β∆F , (30)

where we have used that the conditional probabilities sum to unity,∑

n p(τ)m|n =∑

n |〈e(τ)m |V (τ)|e(0)

n 〉|2 = 1 and Z(τ) =∑

m e−βE(τ)m is, like in the classical case, the partition func-

tion of the hypothetical thermal configuration for the final Hamiltonian, H(τ). The free energy

difference is, like in the classical case, ∆F = − 1β ln Z(τ)

Z(0) . The classical Crooks relation, cf. Eq. (18),

can also be re-derived for the quantum regime [5] and is known as the Tasaki-Crooks relation,

PF (W )

PB(−W )= eβ(W−∆F ). (31)

It is interesting to note that both the quantum Jarzynski equality and the Tasaki-Crooks relationshow no difference to their classical counterparts, contrary to what one might expect. A debatedissue is that the energy measurements remove coherences with respect to the energy basis [31], and

13

Mag

netiz

atio

n

Time (ms)

0.5

0.0

-0.50 20 40 60 80 100

2

1

0-6 -4 -2 0 2 4 6

6

2

1

0-6 -4 -2 0 2 4

Characteristic func. Work distribution(a)

(b)

Mag

netiz

atio

n

0.5

0.0

-0.50 20 40 60 80 100

(c)

(d)

Time (ms)

(b)M

agne

tizat

ion

Time (ms)

0.5

0.0

-0.50 20 40 60 80 100

2

1

0-6 -4 -2 0 2 4 6

6

2

1

0-6 -4 -2 0 2 4

Characteristic func. Work distribution(a)

(b)

Mag

netiz

atio

n

0.5

0.0

-0.50 20 40 60 80 100

(c)

(d)

Time (ms)

(b)(a)

-4 -3 -2 -1 0

1.0

0.8

0.6

0.4

0.2

0.00.0 0.1 0.2 0.3 0.4 0.541 2 3

101(a) (b)

100

10-1

(c)

(c) (d)

(e)

(b) (c)

(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 comparedwith 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 temperatures together with their respective uncertainties, and theoretical predictions for

ln Z(τ)

Z(0) showing good agreement. (Figures taken from Phys. Rev. Lett. 113, 140601 (2014).)

these do not show up in the work distribution, P (W ). It has been suggested [42] that the non-trivialwork that may be extracted during this process, see section 2.5, affects the identification of energychange with fluctuating work (25), however, this remains an open question.

3.7. Quantum fluctuation experiments

While a number of interesting avenues to test fluctuation theorems at the quantum scale have beenproposed [57–59], the experimental reconstruction of the work statistics for a quantum protocolhas long remained elusive. A new measurement approach has recently been devised that is basedon well-known interferometric schemes of the estimation of phases in quantum systems, whichbypasses the necessity of two direct projective measurements of the system state [60]. The firstquantum fluctuation experiment that confirms the quantum Jarzynski relation, Eq. (30), and theTasaki-Crooks relation, Eq. (31), with impressive accuracy has recently been implemented in aNuclear Magnetic Resonance (NMR) system using such an interferometric scheme [61]. Another,very recent experiment uses trapped ions to measure the quantum Jarzynski equality [62] with thetwo measurement method used to derive the theoretical result in section 3.6.

The NMR experiment was carried out using liquid-state NMR-spectroscopy of the 1H and 13Cnuclear spins of a chloroform-molecule sample. This system can be regarded as a collection ofidentically prepared, non-interacting, spin-1/2 pairs [63]. The 13C nuclear spin is the driven system,while the 1H nuclear spin embodies an ancilla that is instrumental for the reconstruction of thework distribution in the interferometric scheme [60]. The 13C nuclear spin was prepared initially ina pseudo-equilibrium state ρ(0) at inverse temperature β, for which four values were realised. Theexperiment implemented the time-varying spin Hamiltonian, H(t), resulting in a unitary evolution,i.e. V in section 3.6, by applying a time-modulated radio frequency field resonant with the 13Cnuclear spin in a short time window τ . The (forward) work distribution of the spin’s evolution,PF (W ) see Fig. 5(a), was then reconstructed through a series of one- and two-body operations onthe system and the ancilla, and measurement of the transverse magnetization of the 1H nuclearspin. Similarly a backwards protocol was also implemented and the backwards work distribution,PB(W ), measured, cf. sections 3.2 and 3.4.

The measured values of the Tasaki-Crooks ratio, PF (W )/PB(−W ), are shown over the work

14

value W in Fig. 5(b) for four values of temperature. The trend followed by the data associatedwith each temperature was in very good agreement with the expected linear relation confirming thepredictions of the Tasaki-Crooks relation. The cutting point between the two work distributionswas used to determine the value of ∆F experimentally which is shown in Fig. 5(c). Calculatingthe average exponentiated work with the measured work distribution, the data also showed goodagreement with the quantum Jarzynski relation, Eq. (30), see table in Fig. 5(d).

4. Quantum Dynamics and Foundations of Thermodynamics

In this section, we describe important quantum information theoretic techniques used in quantumthermodynamics. We employ these techniques to discuss the role played by steady states in quantumthermodynamics and thermalisation of closed quantum systems.

4.1. Completely positive maps

We begin by reviewing the description of generic quantum maps [64–69]. A completely positivetrace preserving (CPTP) map transforms input density matrices into output density matrices. Itis commonly denoted by Φ and has the following properties

(1) trace preservation: ρ′ := Φ(ρ) has unit trace for all input states ρ of dimensionality d.(2) positivity: Φ(ρ) has non-negative eigenvalues, interpreted as probabilities.

(3) complete positivity: Φ⊗ I(k)(Y(k+d)) ≥ 0, ∀k ∈ 0, . . .∞. This subsumes positivity.

The final property is the statement that if the given CPTP map is acting on a subsystem of alarger entangled system, whose state is Y(k+d), then the resultant state is also a “physical” state.

CPTP maps are related to dynamical equations, which we discuss briefly. When the system isclosed, and evolves under a unitary V , the evolution of the density matrix is given by ρ′ = V ρV †.A more general description of the transformations between states when the system is interactingwith an external environment is given by Lindblad type master equations. Such dynamics preservestrace and positivity of the density matrix, while allowing the purity, tr[ρ2] of the density matrix tovary. Such equations have the general form

dρ

dt= −i[H, ρ] +

∑k

[AkρA

†k −

1

2A†kAkρ−

1

2ρAkA

†k

]. (32)

Here Ak are Lindblad operators that describe the effect of the interaction between the system andthe environment on the system’s state. Master equations are typically derived with many assump-tions, including that the system is weakly coupled to the environment, that the environmentalcorrelations decay exponentially and that the initial state of the system is uncorrelated to theenvironment. Lindblad type master equations are central to the study of quantum engines, andwill be discussed in section 6.2. Some master equations are not in the Lindblad form, see [70] forexample.

Three theorems

We now briefly mention three important theorems often used in the quantum information theo-retic study of thermodynamics. The first of these, Stinespring dilation asserts that every CPTPmap Φ can built up from three fundamental operations, namely tensor product with an arbitraryenvironmental ancilla state, τE , a joint unitary and a partial trace. This assertion is written as

ρ′ = Φ(ρ) = trE [V ρ⊗ τE V †]. (33)

15

If the interactions between the system, S, and the environment, E, are known, then the timedependance of the map may be specified. For instance, if the Hamiltonian governing the jointdynamics is HSE , then V = VSE(t) := exp (iHSEt). In [71], the authors considered thermalisationof a quantum system, a topic we will discuss in section 4.3. Stinespring dilation allows for anygeneric quantum dynamics to be thought of in terms of a system and an environment, and theunitary then represents the interactions between the system and the environment. They prove acondition for the long time states of a system to be independent of the initial state of the system.This condition relies on smooth min- and max-entropies. These smooth entropies are central in thesingle shot and resource theoretic approaches to quantum thermodynamics, presented in section 5.They are defined in terms of min- and max-entropies, given by

Smin(A|B)ρ := supσB

supλ ∈ R : 2−λIA ⊗ σB ≥ ρAB, (34)

Smax(A|B)ρ := supσB

ln [F (ρAB, IA ⊗ σB)]2 . (35)

Here the supremum is over all states in the space of subsystem B and F (x, y) := ‖√x√y‖1, with

‖x‖1 := tr[√x†x] is the fidelity. The smoothed versions of min-entropy, Sεmin is defined as the

maximum over the min-entropy Smin of all states σAB which are in a radius (measured in termsof purified distance [72]) of ε from ρAB. Likewise, Sεmax is defined as the minimum Smax over allstates in the radius ε. We note that there is more than one smoothing procedure, and care needsto be taken to not confuse them (see reference 11 in [71]).

Another theorem used in [71] and other studies [73] of quantum thermodynamics is the channel-state duality or Choi-Jamio lkowski isomorphism. It says that there is a state ρΦ isomorphic toevery CPTP map Φ, given by

ρΦ := I⊗ Φ(|ϕ〉〈ϕ|), (36)

where |ϕ〉 = 1√d

∑di |ii〉 is a maximally entangled bipartite state.

The final theorem is the operator sum representation, which asserts that any map, which has therepresentation

ρ′ = Φ(ρ) :=∑µ

Kµ ρK†µ, (37)

with∑

K†µKµ = I is completely positive (but not necessarily trace preserving). These operators,often known as Sudarshan-Kraus operators, are related to the ancilla state τE and V in Eq. (33). Wenote that these theorems are often implicitly assumed while discussing quantum thermodynamicsfrom the standpoint of CPTP maps, see [73–75] for examples.

Entropy and majorisation

We end this section with an enumeration of results from quantum information relating to entropyoften employed in studying thermodynamics. Specifically, we discuss convexity, subadditivity, con-tractivity and majorisation. We begin by defining quantum relative entropy as

S[ρ‖σ] := tr[ρ ln ρ− ρ lnσ]. (38)

The negative of the first term is the von Neumann entropy of the state ρ, see section 2.1. Relativeentropy is jointly convex in both its inputs. This means that if ρ =

∑m pmρm and σ =

∑m pmσm,

16

then

S [ρ‖σ] ≤∑k

pkS[ρk‖σk]. (39)

Furthermore, for any tripartite state ρABC , the entropies of various marginals are bounded by theinequality

S(ρABC) + S(ρB) ≤ S(ρAB) + S(ρBC). (40)

This inequality, known as the strong subadditivity inequality is equivalent to the joint convexity ofquantum relative entropy and is used, for instance in [76], to discuss area laws in quantum systems.Quantum relative entropy is monotonous (or contractive) under the application of CPTP maps.For any two density matrices ρ and σ, this contractivity relation [77] is written as

S[ρ‖σ] ≥ S[Φ(ρ)‖Φ(σ)]. (41)

Contractivity is central to certain extensions of the second law to entropy production inequalities,see [78–80] and is also obeyed by trace distance.

Finally, majorisation is a quasi-ordering relationship between two vectors. Consider two vectors~x and ~y, which are n-dimensional. Often in the context of quantum thermodynamics, these vectorswill be the eigenvalues of density matrices. The vector ~x is said to majorise ~y, written as ~x ~y if all

partial sums of the ordered vector obey the inequality∑k≤n

i x↓i ≥∑k≤n

i y↓i . Here ~x↓ is the vector~x, sorted in descending order. A related idea is the notion of thermo-majorisation [81]. Considerp(E, g) to be the probability of a state ρ to be in a state g with energy E. At a temperature β,such a state is β-ordered if eβEkp(Ek, gk) ≥ eβEk−1p(Ek−1, gk−1), for k = 1, 2, . . .. If two states ρand σ are such that β-ordered ρ majorises β-ordered σ, then the state ρ can be transformed to σin the resource theoretic setting discussed in section 5.

4.2. Role of fixed points in thermodynamics

Fixed points of CPTP maps play an important role in quantum thermodynamics. This role iscompletely analogous to the role played by equilibrium states in equilibrium thermodynamics. Inequilibrium thermodynamics, the equilibrium state is defined by the laws of thermodynamics. Iftwo bodies are placed in contact with each other, and can exchange energy, they will eventuallyequilibrate to a unique state, see section 4.3. On the other hand, if the system of interest is notin thermal equilibrium, notions such as equilibrium, temperature and thermodynamic entropy areill-defined. In the non-equilibrium setting, [82] proposed a thermodynamic framework where theequilibrium state was replaced by the fixed point of the map that generates the dynamics. If thesteady state of a particular dynamics is not the thermal state, since there is entropy produced in thesteady state, “housekeeping heat” was introduced as a way of taking into account the deviation fromequilibrium. This program, called “steady-state thermodynamics” [82–84], for a classical systemdescribed by Langevin equation is described in [84] and the connection to fluctuation theorems iselaborated.

For a quantum system driven by a Lindblad equation given in Eq. (32), the fixed point is definedas the steady state of the evolution, described by a specific Hamiltonian and a set of Lindbladoperators. If a system whose steady state is an equilibrium state is taken out of equilibrium, thesystem will relax back to the equilibrium. This relaxation mechanism will be accompanied by acertain entropy production, which ceases once the system stops evolving (this can be an asymptoticprocess). In the non-equilibrium setting, let the given dynamics be described by a CPTP map Φ.Such a map is a sufficient description of the dynamical processes of interest to us. Every such map

17

Φ is guaranteed to have at least one fixed point [85], which we refer to as ρ?. If the initial stateof the system is ρ, the passage of time is accompanied by the repeated application of the mapΦ. Since the fixed point is defined as ρ? := limn→∞Φn[ρ], we are assured that the state of thesystem will either “fall into” the fixed point or become close to the fixed point, in the case of alimit cycle. If we compare the relative entropy S[ρ‖ρ?], then this quantity reduces monotonicallywith the application of the CPTP map Φ. This is because monotonicity of relative entropy insiststhat S[ρ1‖ρ2] ≥ S[Φ[ρ1]‖Φ[ρ2]], ∀ρ1, ρ2. Since the entropy production ceases only when the statereaches the fixed point ρ?, we can compare an arbitrary initial state ρ with the fixed point to studythe deviation from steady state dynamics. Using this substitution, the monotonicity inequality canbe rewritten as a difference of von Neumann entropies, namely

S[Φ[ρ]]− S[ρ] ≥ −tr [Φ[ρ] ln ρ? − ρ ln ρ?] := −ς. (42)

This result [78, 86] states that the entropy production is bounded by the term on the right handside, a factor which depends on the initial density matrix and the map. If N maps Φi are appliedin succession to an initial density matrix ρ, the total entropy change can be shown easily [79] tobe bounded by

S[Φ[ρ]]− S[ρ] ≥ −∑k

ςk. (43)

Here Φ is the concatenation of the maps Φ := ΦN ΦN−1 . . . Φ1. We refer the reader to thefollowing papers [79, 87–90] for a discussion of the relationship between steady states and laws ofthermodynamics.

4.3. Thermalisation of closed quantum systems

One of the fundamental problems in physics involves understanding the route from microscopic tomacroscopic dynamics. Most microscopic descriptions of physical reality, for instance, are based onlaws that are time-reversal invariant [91]. Such symmetries play a strong role in the construction ofdynamical descriptions at the microscopic level however, many problems persist when the deriva-tion of thermodynamic laws is attempted from quantum mechanical laws. In this subsection, wehighlight some examples of the problems considered and introduce the reader to some importantideas. We refer to three excellent reviews [92], [93] and [94] for a detailed overview of the field.

An information theoretic example of the foundational problems in deriving thermodynamics in-volves the logical paradox in reconciling the foundations of thermodynamics with those of quantummechanics. Suppose we want to describe the universe, consisting of system S and environment E.From the stand point of quantum mechanics, then we would assign this universe a pure state,commensurate with the (tautologically) isolated nature of the universe. This wave-function is con-strained by the constant energy of the universe. On the other hand, if we thought of the universeas a closed system, from the standpoint of statistical mechanics, we assume the axiom of “equala priori probability” to hold. By this, we mean that each configuration commensurate with theenergy constraint must be equally probable. Hence, we expect the universe to be in a maximallymixed state in the given energy shell. This can be thought of as a Bayesian [95] approach to statesin a given energy shell. This is in contradiction with the assumption of pure wave-functions madeby quantum mechanics. In [7], the authors resolved this by pointing out that entanglement is thekey to understanding the resolution to this paradox. They replace the axiom of equal a prioriprobability with a provable statement known as the “general canonical principle”. This principlecan be stated as follows: If the state of the universe is a pure state |φ〉, then the reduced state of

18

the system

ρS := trE [|φ〉〈φ|] (44)

is very close (in trace distance, see section 4) to the state ΩS for almost all choices of the pure state|φ〉. The phrase “almost all” can be quantified into the notion of typicality, see [96]. The state ΩS

is the canonical state, defined as ΩS := trE(E), the partial reduction of the equiprobable state Ecorresponding to the maximally mixed state in a shell corresponding to a general restriction. Sincethis applies to any restriction, and not just the energy shell restriction we motivated our discussionwith, the principle is known as the general canonical principle.

The intuition here is that almost any pure state of the universe is consistent with the systemof interest being close to the canonical state, and hence the axiom of equal a priori probabilityis modified to note that the system cannot tell the difference between the universe being pure ormixed for a sufficiently large universe. Entanglement between the subsystems is understood to playan important role in this kinematic description of thermalisation since the states of the universe areentangled, leading to the effect that sufficiently small subsystems are unable to distinguish highlyentangled states from maximally mixed states of the universe, see [97] and [98] for related works.

This issue of initial states is only one of the many topics of investigation in this modern studyof equilibration and thermalisation. When thermalisation of a quantum system is considered, itis generally expected that the system will thermalise for any initial condition of the system (i.e.,for “almost all” initial states of the system, the system will end up in a thermal state) and thatmany (but perhaps not all) macroscopic observables will “thermalise” (i.e., their expectation val-ues will saturate to values predicted from thermal states, apart from infrequent recurrences andfluctuations). Let us consider this issue of the thermalisation of macroscopic observables in a closedquantum system. Let the initial state of the system be given by |φ〉 =

∑k ck|Ψk〉, where the |Ψk〉

are the eigenstates of the Hamiltonian such that the corresponding coefficients ck are non-zero onlyin a small band around a given energy E0. If we consider an observable B, its expectation value isgiven by

〈B〉 =∑k

|ck|2Bkk +∑km

c∗mckei[Em−Ek]tBkm. (45)

Here Bkm := 〈Ψm|B|Ψk〉, and the energies are assumed to be non-degenerate. The long time averageof this expectation value, which is expected to thermalise is given by the first term. But from theaxioms of the micro-canonical ensemble, the expectation value is expected to be proportional to∑

k Bkk, in keeping with equal “a priori probability” about the mean energy E0. This is a sourceof mystery, since the long time average

∑k |ck|2Bkk somehow needs to “forget” about the initial

state of the system |φ〉. This can happen in three ways. The first mechanism is to demand thatBkk be constant and come outside the sum. This is called “eigenstate thermalisation hypothesis”(ETH)[99, 100], see [20] for a technically precise definition of ETH.

The second method to reconcile the long time average with the axiom of the micro-canonicalensemble is to demand that the coefficients ck can be constant and non-zero for a subset of indicesk. The third mechanism for the independence of the long time average of expectation values ofobservables, and hence the observation of thermalisation in a closed quantum system is to demandthat the coefficients ck be uncorrelated to Bkk. This causes ck to uniformly sample Bkk, the aver-age given simply by the mean value, in agreement with the micro-canonical ensemble. ETH wasnumerically verified for hardcore bosons in [100] and we refer the interested reader to a review ofthe field in [19], see [101, 102] for a geometric approach to the discussion of thermodynamics andthermalisation.

Consider the eigenvalues of an integrable system, which are then perturbed with a non-integrableHamiltonian perturbation. Since equilibration happens by a system exploring all possible transi-

19

Figure 6. a: Level transformations of the Hamiltonian, indicated by the black lines, while population of the level remains

the same. 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 thermalthroughout the process. (Figures taken from Nat. Commun. 4, 1925 (2013).) d: No level transfer is assumed during level

transformations with the energy change from level E(0)n corresponding to initial Hamiltonian H = H(0) to E

(1)n corresponding

to changed Hamiltonian H(1) associated with single shot work. Level transfers during thermalisation are assumed to end with

thermal probabilities in each of the energy eigenstates E(1)m1 of the same Hamiltonian H(1), with the energy difference between

E(1)n and E

(1)m1 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)mN after N − 1 steps.

tions allowed between its various states, any restriction to the set of all allowed transitions willhave an effect on equilibration. Since there are a number of conserved local quantities implied bythe integrability of a quantum system, such a quantum system is not expected to thermalise (Wenote that this is one of the many definitions of thermalisation, and has been criticised as beinginadequate [103]. See Section 9 of [20] for a detailed criticism of various definitions of thermali-sation). On the other hand, the absence of integrability is no guarantee for thermalisation [104].Furthermore, if an integrable quantum system is perturbed so as to break integrability slightly,then the various invariants are assumed to approximately hold. Hence, the intuition is that thereshould be some gap (in the perturbation parameter) between the breaking of integrability and theonset of thermalisation. This phenomenon is called prethermalisation, and owing to the invariants,such systems which are perturbed out of integrability settle into a quasi-steady state [105]. Thisis a result of the interplay between thermalisation and integrability [106] and was experimentallyobserved in [107].

5. Single Shot Thermodynamics

The single shot regime refers to operating on a single quantum system rather than an ensembleof infinite identical and independent copies of the quantum system often referred to as the i.i.d.regime [69]. For long, the issue of whether information theory and hence physics can be formulatedreasonably in the single shot regime has been an important open question which has only beenaddressed in the last decade, see e.g. [108]. An active programme of applying quantum informationtheory techniques, in particular the properties of CPTP maps discussed in section 4, to thermody-namics is the extension of the laws and protocols applicable to thermodynamically large systems toensembles of finite size that in addition may host quantum properties. Finite size systems typicallydeviate from the assumptions made in the context of equilibrium thermodynamics and there hasbeen a large interest in identifying limits of work extraction from such small quantum systems.

5.1. Work extraction in single shot thermodynamics

One single shot thermodynamics approach to analyse how much work may be extracted from asystem in a classical, diagonal distribution ρ with Hamiltonian H introduces level transformations(LTs) of the Hamiltonian’s energy eigenvalues while leaving the state unchanged and thermalisation

20

steps where the Hamiltonian is held fixed and the system equilibrates with a bath at temperature

T to a thermal state [109]. The energy change during LTs, E(0)n → E

(1)n see Fig. 6d, is entirely

associated with single shot work while the energy change, E(1)m1 → E

(2)m2 see Fig. 6d, for thermalisation

steps is associated with single shot heat [109]. See also the definition of average work and averageheat in Eq. (2) associated with these processes for ensembles [22]. Through a sequence of LTs andthermalisations, see Fig. 6a-c, it is found that the single shot random work yield of a system withdiagonal distribution ρ and Hamiltonian H is [109]

Wyield = kB T lnrntn, (46)

where rkk and tkk are the eigenvalues of ρ and the thermal state, τ := e−βH

Z , respectively.Since both are assumed diagonal in the Hamiltonian’s eigenbasis, |Ek, gk〉k, the density matrices’eigenvalues are labelled by the energy basis index. The particular n = (En, gn) appearing here

refers to the initial energy level E(0)n that the system happens to start with in a random single

shot, see Fig. 6d.Deterministic work extraction is rarely possible in the single shot setting and proofs typically

allow a non-zero probability ε of failing to extract work in the construction of single shot work.The ε-deterministic work content Aε(ρ,H) is found to be [109]

Aε(ρ,H) ≈ F ε(ρ,H)− F (τ) := −kB T lnZΛ∗(ρ,H)

Z, (47)

with the free energy difference defined through a ratio of two partition functions. The proofof Eq. (47) employs a variation of Crook’s relation, see section 3.2, and relies on the notionof smooth entropies, see section 4.1. Here ZΛ∗(ρ,H) :=

∑(E,g)∈Λ∗ e−βE is a single shot par-

tition function determined by a subspace Λ∗ spanned by energy eigenstates, |E, g〉 denotingenergy and degeneracy, of the Hamiltonian H [109]. It is defined as the minimal subspace,Λ∗(ρ,H) := infΛ :

∑(E,g)∈Λ〈E, g|ρ|E, g〉 > 1 − ε, such that the state ρ’s probability in that

subspace makes up the total required probability of 1− ε. Contrasting with the ensemble situationthat allows probabilistic fluctuations P (W ) around the average work, see e.g. Eq. (17), the singleshot work Aε(ρ,H) discussed here is interpreted as the amount of ordered energy, i.e. energy withno fluctuations, that can be extracted from a distribution in a single-shot setting.

5.2. Work extraction and work of formation in thermodynamic resource theory

An alternative single shot approach is the resource theory approach to quantum thermodynamics[9, 81, 109–114]. Resource theories in quantum information identify a set of restrictive operationsthat can act on “valuable resource states”. For a given initial state these restrictive operations thendefine a set of states that are reachable. For example, applying stochastic local operations assistedby classical communication (SLOCC) on an initial product state of two parties will produce arestricted, separable set of two-party states. The same SLOCC operations applied to a two partyBell state will result in the entire state space for two parties, thus the Bell state is a “valuableresource”. In the thermodynamic resource theory the valuable resource states are non-equilibriumstates while the restrictive operations include thermal states of auxiliary systems.

The thermodynamic resource theory setting involves three components: the system of interest, S,a bath B, and a weight (or work storage system or battery) W . Additional auxiliary systems mayalso be included [112] to explicitly model the energy exchange that causes the time-dependence ofthe system Hamiltonian, cf. section 3, and to model catalytic participants in thermal operations, seesection 5.3. In the simplest case, the Hamiltonian at the start and the end of the process is assumed

21

Figure 7. Left a: Work in classical thermodynamics is identified with lifting a bucket against gravity thus raising its potentialenergy. Left b: In the quantum thermodynamic resource theory a work bit (=wit) is introduced where the jump from one

energetic level to another defines the extracted work during the process. (Figures taken from Nature Comm. 4 2059 (2013).)

Right: Global energy levels of combined system, bath and work storage system have a high degeneracy indicated by the setsof levels all in the same energy shell, E. The resource theory approach to work extraction [81] 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 = E′S + E′B + w, indicated in blue, in the same energy shell, E. Work extraction now becomes an optimisationproblem to squeeze the population of the initial state into the final state and find the maximum possible w [115].

to be the same and the sum of the three local terms, H = HS + HB + HW . Thermal operationsare those transformations of the system that can be generated by a global unitary, V , that acts onsystem, bath and work storage system initially in a product state ρS ⊗ τB ⊗ |Eini

W 〉W 〈EiniW |, where

work storage system in one of its energy eigenstates, |EiniW 〉 and the “free resource” is the bath in

a thermal state, τB = e−βHBZB

. Perfect energy conservation is imposed by requiring that the unitarymay only induce transitions within an energy shell of total energy E = ES +EB +EW , see Fig. 7b.This is equivalent to requiring that V must commute with the sum of the three local Hamiltonians.

A key result in this setting is the identification of a maximal work [81] that can be extractedfrom a single system starting in a diagonal non-equilibrium state ρS to the work storage systemunder thermal operations. The desired thermal transformation enables the lifting of a work storagesystem by an energy w > 0 from the ground state |Eini

W 〉 = |0〉W to another single energy level|w〉W , see Fig. 7a,

ρSB ⊗ |0〉W 〈0|V−→ σSB ⊗ |w〉W 〈w|, (48)

where the system and bath start in a fixed state ρSB and end in some state σSB. The perfecttransformation can be relaxed by requesting the transformation to be successful with probability1− ε, see section 5.1. The maximum extractable work in this setting is [81, 115]

wmaxε = − 1

βln∑ES ,gS

t(ES ,gS) hρS(ES , gS , ε) =: Fminε (ρS)− F (τS), (49)

where τS is the thermal state of the system at the bath’s inverse temperature β with eigenvaluest(ES ,gS) = e−βES

ZSand partition function ZS =

∑ES ,gS

e−βES . Here ES and gS refer to the energy

and degeneracy index of the eigenstates |ES , gS〉 of the Hamiltonian HS . hρS(ES , gS , ε) is a binaryfunction that leads to either including or excluding a particular t(ES ,gS) in the summation, see [81,

115] for details. Recalling kB T = 1β and identifying that ZΛ∗/Z =

∑ES ,gS

t(ES ,gS) hρS(ES , gS , ε),

expression (49) becomes identical to the result (47) derived with the single shot approach discussedabove. When extending the single level transitions of the work storage system to multiple levels[115] it becomes apparent that this resource theory work cannot directly be compared to lifting aweight to a certain height or above. Allowing the weight to rise to an energy level |w〉 or a rangeof higher energy levels turns the problem into a new optimisation and that would have a differentassociated “work value”, w′, contrary to physical intuition. Due to its restriction to a particular

22

energetic transition the above resource theory work may very well be a useful concept for resonanceprocesses [115] where such a restriction is crucial.

Similar to the asymmetry between distillable entanglement and entanglement of formation it isalso possible to derive a work of formation to create a diagonal non-equilibrium state ρS which isin general smaller than the maximum extractable work derived above [81],

wmin (formation)ε =

1

βinfρεS

(ln minλ : ρεS ≤ λτS) , (50)

where ρεS are states close to ρS , ||ρεS − ρS || ≤ ε with || · || the trace norm.

5.3. Single shot second laws

Beyond optimising work extraction, a recent paper identifies the set of all states that can be reachedin a single shot by a broader class of thermal operations in the resource theory setting [112]. Thisbroader class of operations, EC , is called catalytic thermal operations and involves, in addition to asystem S (which here may include the work storage system) and a heat bath, B, a catalyst C. Therole of the catalyst is to participate in the operation while starting and being returned uncorrelatedto system and bath, and in the same state σC ,

EC(ρS)⊗ σC := trB[V (ρS ⊗ τB ⊗ σC)V †], (51)

where the unitary now must commute with each of the three Hamiltonians that the system, bathand catalyst start and end with, [V,HS + HB + HC ] = 0. For system states ρS diagonal in theenergy-basis of HS it is shown [112] that a state transfer under catalytic thermal operations ispossible only under a family of necessary and sufficient conditions,

ρS → ρ′S = EC(ρS) ⇔ ∀α ≥ 0 : Fα(ρS , τS) ≥ Fα(ρ′S , τS), (52)

where τS is the thermal state of the system. τS is the fixed point of the map EC and theproof makes use of the contractivity of map, discussed in section 4.1. Here Fα(ρS , τS) :=kB T (Sα(ρS ||τS)− lnZS) is a family of free energies defined through the α-relative Renyi en-

tropies, Sα(ρS ||τS) :=sgn(α)α−1 ln

∑n r

αn t

1−αn , applicable for states diagonal in the same basis. The

rn and tn are the eigenvalues of ρS and τS corresponding to energy eigenstates |En, gn〉 of the sys-tem, cf. Eq. (46). An extension of these laws to non-diagonal system states and including changesof the Hamiltonian is also discussed [112].

The resulting continuous family of second laws can be understood as limiting state transfer in thesingle shot regime in a more restrictive way than the limitation enforced by the standard second lawof thermodynamics, Eq. (5), valid for macroscopic ensembles [112], see section 2. Indeed, the latteris included in the family of second laws: for limα→1 where Sα(ρS ||τS) → S1(ρS ||τS) =

∑n rn ln rn

tn[17] one recovers from Eq. (52) the standard second law,

kB T

(∑n

rn lnrntn− lnZS

)≥ kB T

(∑n

r′n lnr′ntn− lnZS

), (53)∑

n

rn ln rn −∑n

r′n ln r′n ≥ −∑n

(r′n − rn) ln tn, (54)

∆S ≥ β∆U = β〈Q〉, (55)

with no work contribution, 〈W 〉 = 0, as no explicit source of work was separated here (althoughthis can be done, see section 5.2). Moreover, in the i.i.d. limit of large ensembles, ρ → ρ⊗N with

23

Figure 8. For a fixed Hamiltonian HS and temperature T , the manifold of time-symmetric states, indicated in grey, including

the thermal state, γ ≡ τS , as well as any diagonal state, D(ρ) ≡ DHS (ρS), is a submanifold of the space of all system states(blue oval). Under thermal operations an initial state ρ ≡ ρS moves towards the thermal state in two independent “directions”.

The thermodynamic purity p measured by Fα, and the asymmetry a measured by Aα, must both decrease. (Figure taken from

Nat. Commun. 6, 6383 (2015).)

N → ∞, the second law family, ∀α ≥ 0 : Fα(ρ⊗NS , τ⊗NS ) ≥ Fα(ρ′⊗NS , τ⊗NS ), reduces to just thesingle constraint, F1(ρS , τS) ≥ F1(ρ′S , τS), providing a single-shot derivation of why Eq. (5) appliesto macroscopic ensembles.

Allowing the catalyst in Eq. (51) to be returned after the operation not in the identical state, buta state close in trace distance, leads to the unphysical puzzle of thermal embezzling. It has beenshown that when physical constraints, such as fixing the dimension and the energy level structureof the catalyst, are incorporated the allowed state transformations are significantly restricted [116].

5.4. Single shot second laws with coherence

Recently it was discovered that thermal operations, E(ρS) := trB[V (ρS⊗τB)V †], on an initial stateρS that has coherences with respect to its Hamiltonian HS , require a second family of inequalitiesto be satisfied [114], in addition to the free energy relations for diagonal states (52). These necessaryconditions are

ρS → ρ′S = E(ρS) ⇒ ∀α ≥ 0 : Aα(ρS) ≥ Aα(ρ′S), (56)

where Aα(ρS) = Sα(ρS ||DHS(ρS)) and DHS(ρS) is the operation that removes all coherences be-tween energy eigenspaces, i.e. DH(ρ) ≡ η in section 2.5. The Aα are coherence measures, i.e. theyare coherence monotones for thermal operations. Again the proof relies on the contractivity of themap E , see section 4.1, and its commuting with D. The result (56) means that thermal operationson a single copy of a quantum state with coherence must decrease its coherence, just like free energymust decrease for diagonal non-equilibrium states. This decrease of coherence implies a quantumaspect to irreversibility in the resource theory setting [117]. The two families of second laws (52)and (56) can be interpreted geometrically as moving states in state space under thermal opera-tions “closer”2 to the thermal state in two independent “directions”, in thermodynamic purity andtime-asymmetry [114], see Fig. 8. When taking the single shot results to the macroscopic i.i.d.limit, conditions (56) become trivial since limN→∞Aα(ρ⊗NS ) = 0 for initial and final state, and thestandard second law (5) is the only constraint.

2The Renyi divergences Sα are not proper distance measures [17].

24

6. Quantum Thermal Machines

Until now, we have discussed how researchers in quantum thermodynamics have dealt with a varietyof fundamental issues. Thermodynamics is a field whose focus from the very beginning has beenboth fundamental as well as applied. We will present some of the recent progress in the design ofquantum thermal machines, which generalise well-known classical thermal machines.

6.1. Background and basics

The study of the efficient conversion of various forms of energy to mechanical energy has been atopic of interest for more than a century. Quantum thermal machines (QTMs), defined as quantummachines that mediate between different forms of energy have been a topic of intense study. Engines,heat driven refrigerators, power driven refrigerators and several other kinds of thermal machineshave been studied in the quantum regime, initially motivated from areas such as quantum optics[118, 119]. QTMs can be classified in various ways. Dynamically, QTMs can be classified as thosethat operate in discrete strokes, and those that operate as continuous devices, where the steadystate of the continuous time device operates as a QTM. Furthermore, such continuous engines canbe further classified as those that employ linear response techniques [12] and those that employdynamical equation techniques [18]. These models of quantum engines are in contrast with biologicalmotors, that extract work from fluctuations and have been studied extensively [120].

Engines use a working fluid and two (or more) reservoirs to transform heat to work. These reser-voirs model the bath that inputs energy into the working fluid (hot bath) and another that acceptsenergy from the working fluid (cold bath). Two main classes of quantum systems have been stud-ied as working fluids, namely discrete quantum systems and continuous variable quantum systems.Following [121], various authors have studied oscillators as models of heat engines [122–124]. Thiscompliments various models of finite level systems [125, 126] studied as quantum engines. Bothcontinuous variable models of engines and finite dimensional heat engines have been theoreticallyshown to be operable at theoretical maximal efficiencies. Such efficiencies are only well defined oncethe engine operation is specified.

The efficiency of any engine is given by the ratio of the net work done by of the system to theheat that flows into the system, namely

η =〈Wnet〉〈Qin〉

. (57)

From standard thermodynamics, it is well known that the most efficient adiabatic engines alsooutput zero power. Hence, for the operation of a real engine, other objective functions such aspower have to be optimised. The seminal paper by Curzon and Ahlborn [127] considered the issueof the efficiency of a classical Carnot engine operating between two temperatures, cold TC and hotTH , at maximum power, which they found to be

ηCA = 1−√TC

TH. (58)

This limit is also reproduced with quantum working fluids, as discussed in section 6.8. We notethat the aforementioned formula for the efficiency at maximum power is sensitive to the constraintson the system, and changing the constraints changes this formula. For instance, see [128, 129] forresults generalising the Curzon-Ahlborn efficiency to stochastic thermodynamics and to quantumsystems with other constraints.

The rest of this section is a brief description of how engines are designed in the quantum regime.We consider three examples, namely Carnot engines with spins as the working fluid, harmonic

25

oscillator Otto engines and Diesel engines operating with a particle in a box as the working fluid.

6.2. Carnot engine

Classically, the Carnot engine consists of two sets of alternating adiabatic strokes and isothermalstrokes. The quantum analogue of the Carnot engine consists of a working fluid, which can be aparticle in a box [130], qubits [14, 125], multiple level atoms [123] or harmonic oscillators [121, 122].For the engine consisting of non-interacting qubits considered in [125], the Carnot cycle consists of

(1) Adiabatic Expansion: An expansion wherein the spin is uncoupled from any heat baths andits frequency is changed from ω1 to ω2 > ω1 adiabatically. Work is done by the spin in thisstep due to a change in the internal energy. Since the spin is uncoupled, its von Neumannentropy is conserved in this expansion stroke.

(2) Cold Isotherm: The spin is coupled to a cold bath at inverse temperature βC . This transfersheat from the engine to the cold bath.

(3) Adiabatic Compression: An compression wherein again, the spin is uncoupled from all heatbaths and its frequency is changed from ω2 to ω1 adiabatically. Work is done on the mediumin this step.

(4) Hot Isotherm: The spin is coupled to a hot bath at inverse temperature βH < βC . Thistransfers heat to the engine from the hot bath.

The inverse temperature β′ of the thermal state of this working fluid is given at any point (ω,S)on the cycle, by the magnetisation relation S = − tanh(β′ω/2)/2. The dynamics is described by aLindblad equation, which will then be used to derive the behaviour of heat currents. The enginecycle is described in Fig. 9. The Hamiltonian describing the dynamics is given by H(t) = ω(t)σ3/2.The Heisenberg equation of motion is written in terms of LD(σ3), the Lindblad operators describingcoupling to the baths at inverse temperatures βH and βC respectively. This equation can be usedto calculate the expectation value of the Hamiltonian 〈H(t)〉, which leads to

d〈H(t)〉dt

=1

2

(dω

dt〈σ3〉+ ω〈LD(σ3)〉

)=

1

2

(dω

dt〈σ3〉+ ω

d〈σ3〉dt

). (59)

As described in section 2, the definition of work 〈δW 〉 = 〈σ3〉δω/2 and heat 〈δQ〉 = ωδ〈σ3〉/2emerge naturally from the above discussion, and are identified with the time derivative of the firstlaw. For the quantum Carnot engine, the maximal efficiency, which is that of the reversible engine[125], is given by

ηCarnot = 1− TC

TH. (60)

6.3. Otto engine

As a counterpoint to the Lindblad study of Carnot engines implemented via qubits, let’s considerthe harmonic oscillator implementation of Otto engines [10, 124, 131–133]. The frequency of theworking fluid is time dependant ωHO(t), and undergoes four strokes. The initial state of the oscil-lator is a thermal state at inverse temperature βC, and the oscillator frequency is ωHO

1 . The fourstrokes of the Otto cycle are given by:

(1) Isentropic Compression: An compression, where the medium is uncoupled from all heat bathsand its frequency is changed from ωHO

1 to ωHO2 > ωHO

1 . Since the oscillator is uncoupled, itsvon Neumann entropy is conserved during this stroke, though it no longer is a thermal state.

26

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 theheat 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 < 0 represent two values of the magnetisation, with S1 < S2.(Figure taken from Jour. Chem. Phys. 96,3054 (1992).) 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 its denoted by F in the text. (Figure taken from Phys. Rev. E79, 041129 (2009).)

(2) Hot Isochore: At frequency ωHO2 , the harmonic oscillator working fluid is coupled to a bath

whose inverse temperature is βH, and is allowed to relax to the new thermal state.(3) Isentropic Expansion: In this stroke, the medium is uncoupled from any heat baths and its

frequency is changed from ωHO2 to ωHO

1 .(4) Cold Isochore: At frequency ωHO

1 , the harmonic oscillator working fluid is coupled to a bathwhose inverse temperature is βC and it is allowed to relax back to its initial thermal state.

Since the two strokes where the frequencies change, work is exchanged, whereas the two ther-malising strokes involve heat exchanges, the efficiency is easily calculated to be

ηOtto = −〈δW 〉1 + 〈δW 〉3〈δQ〉2

. (61)

The experimental implementation of such a quantum Otto engine was considered in [134] and ispresented in Fig. 10. The Paul or quadrupole trap uses rapidly oscillating electromagnetic fields toconfine ions using an effectively repulsive field. The ion, trapped in the modified trap, presented inFig. 10 is initially cooled in all spatial directions. The engine is coupled to a hot and cold reservoirscomposed of blue and red detuned laser beams. The tapered design of the trap translates to anaxial force that the ion experiences. A change in the temperature of the radial state of the ion,and hence the width of its spatial distribution, leads to a modification of the axial component ofthe repelling force. Thus heating and cooling the ion moves it back and forth along the trap axis,as induced in the right hand side of the figure. The frequency of the oscillator is controlled by thetrap parameters. Energy is stored in the axial mode and can be transferred to other systems andused, see [134] for details.

6.4. Diesel engine

Finally, we discuss Diesel cycles designed with a particle in a box as the working fluid [136]. Theclassical Diesel engine is composed of isobaric strokes, where the pressure is held constant. We will

27

Figure 10. Experimental proposal of an Otto engine consisting of a single trapped ion, currently being built by K. Singer’s

group [135]. 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 representationof the four strokes, namely (1) isentropic compression, (2) hot isochore, (3) isentropic expansion and (4) cold isochore, see text

for details. (Figure taken from Phys. Rev. Lett. 112, 030602 (2014).)

discuss how the notion of “pressure” is generalised to the quantum setting and how this defines a“quantum isobaric” stroke. To define pressure, we begin by defining force. After defining averagework and heat as before, generalised forces can be defined analogues to classical thermodynamicsusing the relation

Yn =dW

dyn, (62)

where yn, Yn form the n-th conjugate pair in the definition of work dW =∑

n Yndyn. For eigenstateswith energies En distributed according to a probability distribution Pn, this yields a definition offorce, namely

F =∑n

PndEndL

. (63)

If the system is in equilibrium with a heat bath at inverse temperature β, with the correspondingFree energy being F = − log(Z)/β, then the force is given by the usual formula F = −[dF/dL]β,evaluated at constant temperature β−1. This force is calculated to be F = (βL)−1. Hence to executean “isobaric process”, wherein the pressure is held constant across a stroke, the temperature mustvary as β = (FL)−1. This can be employed to construct a Diesel cycle, presented in Fig. 9, andwhose four strokes are given by

(1) Isobaric Expansion: The expansion of the walls of the particle in a box happen at constantpressure. The width of the walls goes from L2 to L3 > L1.

(2) Adiabatic Expansion:An adiabatic expansion expansion process, wherein the length goes fromL3 to L1 > L3. Entropy is conserved in this stroke.

(3) Isochoric Compression: The compression happens at constant volume L1 where the force onthe box is reduced.

(4) Adiabatic Compression: The compression process takes the box from volume L1 to L2 < L1.This is done by isolating the quantum system, conserving the entropy in the process.

Like before, the efficiency is calculated by considering the ratio of the net work done by the systemto the heat into the system. Given that the system returns to its original state at the beginningof each cycle, this efficiency is given by the ratio of the difference between heat in and heat out of

28

the system, to the heat into the system. A straightforward calculation of this efficiency yields

ηDiesel = 1− 1

3(r2

E + rErC + r2C). (64)

Here rE = L3/L1 is the expansion ratio and rC = L2/L1 is the compression ratio.

6.5. Quantumness of engines

Since the design of quantum engines reflects their classical counterparts, one might seek to knowwhat, if anything, is genuinely quantum about the design of engines? There are two importantpoints here. The first point is that the efficiencies calculated by Carnot, and by Curzon and Ahlbornare generic and only rely on a working fluid operating between two reservoirs whose temperaturesare given. Hence these efficiencies are recovered by analogous quantum devices. The second pointis that there is much that is still genuinely quantum about these engine designs. As explained in6.7, both coherence and time play important roles in the design of quantum engines. The quantumspeed limit places a limit on the power of quantum engines, and adiabatic shortcuts are techniquesthat have improved engine performance in the quantum regime. Quantum friction, caused bythe non-commutativity of propagators, is also a genuinely quantum effect that impacts engineperformance. The role of entanglement and correlations have been explored in the case of workextraction, a study that is connected to the design of engines. Note that some authors [134, 137]have proposed protocols like adding coherence or squeezing thermal baths as a way to surpass theCarnot efficiency. A correct accounting of the energy that goes into increasing the efficiency has tobe undertaken to see if this increase in efficiency still remains.

In [138], the authors study quantumness of engines and show that there is an equivalence betweendifferent types of engines, namely two-stroke, four-stroke and continuous engines. Furthermore, theauthors define and discuss quantum signatures in thermal machines by comparing these machineswith equivalent stochastic machines, and they demonstrate that quantum engines operating withcoherence can in general output more power than their stochastic counterparts.

6.6. Quantum refrigerators

Quantum refrigerators act in a regime where they move heat from cold to hot sources by consumingwork. This is different but related to the engine’s operating regime. Refrigerators are hence enginesoperating in a regime where the heat flow is reversed. Like engines, the role that quantised energystates, coherence and correlations play in the operation of a quantum refrigerator have been fieldsof extensive study [18, 139–143].

We present two examples of refrigerator cycles considered in the literature. The first of these[144] is built on a three level atom, coupled to two baths, as shown in Figure 11. The refrigeratoris driven by the coherent radiation. This induces transitions between levels 2 and 3. The level 3then relaxes to 1 by by rejecting heat to the hot bath Th. The system then transitions from 1 to2 by absorbing energy from a cold bath. The coupling to the field is given by ε and the couplingto the baths is given by λ in the figure. Like before, the dynamics can be written in terms of theHeisenberg equation for an observable A. Let Lc,h refer to the Lindblad operators correspondingto the cold and hot baths respectively. Assuming the Hamiltonian H = H0 + V (t), the energytransport can be investigated 〈H〉, namely

d〈H〉dt

=

⟨∂V (t)

∂t

⟩+ 〈Lh(H)〉+ 〈Lc(H)〉 (65)

Using this, the authors in [144] investigated the refrigerator efficiency. This efficiency is expressed

29

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 Jour. Appl. Phys. 87, 8093 (2000).). The right

figure consists of the smallest refrigerator, composed of a qubit and a qutrit. (Figure taken from Phys. Rev. Lett. 105, 130401(2010).)

in terms of the coefficient of performance (COP), which is defined as the ratio of the cooling energyto the work input into the system. For two thermal baths at temperatures Tc and Th respectively,the universal reversible limit is the Carnot COP, given by

COP =Tc

Th − Tc. (66)

COP can be larger than one, and in [145], the authors show the relationship of COP to coolingrate, which is simply the rate at which heat is transported form the system.

Let us consider another example of quantum refrigerators which has a more quantum informationtheoretic point of view. In [14], the authors were inspired by algorithmic cooling and studied thesmallest refrigerators possible. One of the three models discussed is presented in Fig. 11. Algorithmiccooling [13] is a procedure to cool quantum systems which uses ideas from information theory.Consider n copies of a quantum system which is not in its maximal entropy state. Joint unitaryoperations on the n copies can allow for the distillation of a small number m of systems which arecolder (in this context, of lower entropy) than before, whilst leaving the remaining (n−m) systemsin a state that is hotter than before (this is required by unitarity).

The refrigerator in [14] is the smallest (in Hilbert space dimensionality) refrigerator possible.Before we discuss this refrigerator, consider a slightly bigger refrigerator built from two qubits andone qutrit. The qubit is the system which is to be cooled. The bare Hamiltonian is the usual sumof projectors onto energy eigenstates of each system. The interaction Hamiltonian for this system

is given by Hint = H(1)int +H

(2)int , where

H(1)int = g(|02〉〈11|+ h.c.)⊗ I(3) (67)

H(2)int = hI(1) ⊗ (|01〉〈10|+ h.c.). (68)

Particles 1 and 3 are qubits and particle 2 is a qutrit. The energy levels of each particle are tunedin such a way as to make |020〉 and |101〉 degenerate. Though these interaction Hamiltoniansdo not induce the cooling transition |020〉 → |101〉 in first order, they do induce this transitionin second order. The cooling is achieved by biassing the two qubits by placing them in contactwith the thermal reservoirs at different temperatures. Now, instead of inducing transitions fromtwo different qubits, one placed in contact with a hot reservoir and one placed in contact with acold reservoir, the proposal in Fig. 11 places different transitions of the qutrit at different rates,corresponding to the temperatures indicated. This can be done physically, for instance, if the qutrithas spatial extent. The entropy shunting transformations inspired by algorithmic cooling are clearly

30

seen in the interaction Hamiltonian above. See [139] for a discussion of the relationship of power toCOP and [139] for a discussion of correlations in the design of quantum absorption refrigerators.

6.7. Coherence and time in thermal machines

In this subsection, we want to briefly discuss the role coherence and time play in quantum thermalmachines. The fundamental starting point of finite time classical [146] and quantum thermodynam-ics is the fact that the most efficient macroscopic processes are also the ones that are adiabaticallyslow. Hence they are impractical from the standpoint of power generation. The role of quantumcoherences was illustrated by a series of studies involving commutativity. Consider a quantumsystem, whose energy balance equation, as discussed before, is given in terms of the unitary anddissipative Lindblad operators L = LH + LD by

dU

dt= 〈L(H)〉+

⟨∂H

∂t

⟩. (69)

If the HamiltonianH = Hext+Hint is the sum of an external time dependent partHext = ω(t)∑

iHi

and an internal time independent part, then one can calculate the second term, the power P, as

P =

⟨∂H

∂t

⟩=

dω

dt

∑i

〈Hi〉. (70)

Likewise, the first term, corresponding to the heat, is calculated as

dQdt

= 〈L(Hext +Hint)〉 (71)

Two important observations can be made from the equations above [147]. Firstly, since the heatgenerated is non-zero if 〈L(Hext + Hint)〉 6= 0, not all of the change in the energy is converted topower. Thus the dissipative Lindblad processes limit the ultimate power that can be generated byquantum thermal machines. For the second observation lets assume that initial state of the systemis an eigenstate of Hext. If [Hext, Hint] 6= 0, then the quantum state, even in the absence of Lindbladterms, does not “follow” the instantaneous eigenstate of the external Hamiltonian. This precession,that is induced by the non-commutativity of the external and internal Hamiltonians is a source ofa friction like term [148].

In [137], the authors extract work from coherences by using a technique similar to lasing withoutinversion. The photon Carnot engine proposed by the authors to study the role of coherencesfollows by considering a cavity with a single mode radiation field enclosed in it. The radiationinteracts with phased three level atoms at two temperatures during the isothermal parts of theCarnot engine cycle. The three level atoms are phased by making them interact with an externalmicrowave source. The efficiency of this engine was calculated to be

η = ηCarnot − π cos(Φ), (72)

where the phased three level atoms have an off-diagonal coherence term Φ that is used to improvethe Carnot efficiency. This engine is depicted in Fig. 12. Thus by employing the coherence dynamicsof three level atoms, the authors demonstrate enhancement in work output. See [143] for a studyof entanglement dynamics in three-qubit refrigerators, wherein the authors demonstrate that thereis little entanglement found when the machine is operated near the Carnot limit.

31

Figure 12. An optical Carnot engine, discussed in the text. The engine is driven by the radiation in the cavity being inequilibrium with atoms with two separate temperatures during the hot and cold part of the Carnot cycle. The phased three

level atoms have an off-diagonal coherence term Φ that is used to improve the Carnot efficiency.(Figure taken from Science299, 862 (2003).)

6.8. Coherence and work extraction

Another example of the use of coherence involves quantum batteries. These are quantum workstorage devices and their key issues relate to their capacity and the speed at which they can be“charged” and “discharged”. Important for this discussion is the extractable work from a quantum

system under unitary transformations, V, for states, ρ =∑

k r↑k|r↑k〉〈r

↑k|, that have coherences with

respect to a Hamiltonian’s eigenbasis, where H =∑

k ε↓k|ε↓k〉〈ε

↓k|. Here, the arrows show the ordering

of eigenvalues, namely r↑0 ≤ r↑1 . . . and likewise for the Hamiltonian. When searching for the optimal

unitary that extracts the most work from ρ one finds [22, 24, 149] that it is the unitary that maps ρ

to the state π =∑r↑k|ε

↓k〉〈ε

↓k|. Such states, where the eigenvalues are ordered inversely with respect

to the eigenvectors of the Hamiltonian, are known as “passive states” [23, 24] and the correspondingmaximum work extracted from the unitary transformation,

〈Wmaxext 〉 = tr[H(ρ− π)]. (73)

is called “ergotropy”. Note, that the thermal states τβ = e−βH/tr[e−βH ] for any inverse temperatureβ are passive.

The role entanglement plays in extracting work using unitary transformations was exploredwithin the simultaneous extraction of work from n quantum batteries [150]. They considered ncopies of a given state σ and a Hamiltonian H for each system. As discussed before, σ and Himply that there exists a passive state π. Two strategies for extracting work exist, the first ofthese being to simply process each quantum system separately. The second strategy is to do jointentangling operations, which transform the initial state σ(⊗n) to a final state which has the same

entropy as n copies of σ. Since the final state τ(⊗n)β has the same entropy as n copies of the

initial state, a transformation between σ(⊗n) and the neighbourhood of τ(⊗n)β would be the desired

transformation that extracts maximum work. Such a transformation was shown to be entangling,using typicality arguments [96]. However, the authors in [151] pointed out that there is always aprotocol to extract all the work from non-passive states by using more operations in comparisonto entangling unitaries. This suggests a relationship between power and entanglement in workextraction, an assertion demonstrated for quantum batteries in [152].

32

6.9. Power and shortcuts

A final noteworthy point in the consideration of time explicitly in the context of quantum thermalmachines relates to maximum power engines. Firstly, since the optimal performance of enginescorresponds to adiabatic transformations, the power is always negligible and the time of operationof one cycle is infinite (or simply much longer compared to the internal dynamical timescales ofthe quantum systems). To remedy this, both classically and quantum mechanically, engines whichoptimise power have been considered. These studies find the efficiency at maximum power, seeEq. (58), first derived in the classical context by Curzon and Ahlborn [127] but also valid inthe quantum case [121]. See [133] for a discussion of shortcuts to adiabaticity and their role inoptimising power in Otto engines and [153] for a calculation of efficiency at maximum power ofan absorption heat pump, which also proved the weak dependance of efficiency and power withdimensionality. In [131, 154], the authors consider the power of Otto cycles in various regimes, andderive the steady-state efficiency

ηSS =1−

√βH/βC

2 +√βH/βC

. (74)

6.10. Relationship to laws of thermodynamics

We end the discussion of QTMs with a small report on the role that the laws of thermodynamics,discussed in section 2, play in constraining design. As was noted in the context of engines, the firstlaw emerges naturally as a partitioning of the energy of the system in terms of heat and work. See[75] for a partitioning of internal energy change for CPTP processes. In the context of QTMs, forinstance, this can be seen in Eq. (59). The second law manifests itself usually by inspecting theentropy production of the universe [144]. This is given for a typical QTM, with a system and twobaths as

dS

dt= −〈QH〉

TH− 〈QC〉

TC. (75)

Analysis of the dynamics shows that the second law enforces the rule that the net entropy produc-tion, given by the equation above, is non-negative. See [139, 155] to study the relationship betweenweak coupling and the second law. Information theoretic approaches often use monotonicity con-dition of relative entropy to track entropy production, see section 4. See [142] for a discussion ofsecond law in the context of SWAP engines. The third law states that the entropy of any quantumsystem goes to a constant as temperature approaches zero. This constant is commonly assumedto be zero. This means, that in the context of the equation above, the heat current correspondingto the cold bath, 〈QC〉 ∝ Tα+1

C as TC → 0 with α > 0. This was studied in [144], see also [18] fora discussion on another formulation of the third law and [156] for a recent review of the laws ofthermodynamics in the quantum regime. Finally, in [157], the authors discuss periodically drivenquantum systems and investigate the laws of thermodynamics for a model of quantum refrigerator,see [158] for a review.

7. Discussion and Open Questions

We have presented an overview of a selection of current approaches to quantum thermodynamicspursued with various techniques and interpreted from different perspectives. Substantial insighthas been gained from these advances and their combination, and a unified language is startingto emerge. This difference in perspective has also meant that there are ideas within quantum

33

thermodynamics where consensus is yet to be established.One example of such disagreement is the definition of work in the quantum regime. Various

notions of work have been introduced in the field, including the average work defined for an en-semble of experimental runs in Eq. (2), classical and quantum fluctuating work defined for a singleexperimental run in Eq. (15) and Eq. (25), optimal single shot work given in Eq. (46) and optimalthermodynamic resource theory work given in Eq. (49). It is reassuring to see that the latter two,quite separate single shot approaches, result in the same optimal work value. Despite advances inunifying these work concepts our understanding of work in the quantum regime remains patchy.For example, resource theory work Eq. (49) is a work associated with an optimised thermal op-erations process - to move a work storage system almost deterministically as high as possible -while the fluctuating work concept, Eq. (15) and Eq. (25), is applicable to general closed and opendynamical processes [45, 159]. Thus they appear to refer to different types of work, a situation thatmay be compared to different entanglement measures each of importance for a different quantumcommunication and computational tasks. For example, it has been suggested that the resourcetheory work is a suitable measure to quantify resonance processes [115]. A second issue is thelink between the work definition in the ensemble sense, see (2), and the single shot work. Beyondtaking mathematical limits of Renyi entropies these definitions have to be understood within thecontext of each other operationally, i.e. how can one measure each of these quantities and in whatsense do these quantities converge experimentally? Indeed, while old-fashioned thermodynamics isa manifestly practical theory, made for steam engines and fridges, quantum thermodynamics hasonly made a small number of experimentally checkable predictions of new thermodynamic effectsyet.

Another point of discussion is the kinematic versus dynamical approach to thermalisation andequilibration. The kinematic approach arises from typicality discussions in quantum informationwhere states in Hilbert space are discussed from the standpoint of a property of their marginals. Inthe case of thermalisation, this property is the closeness to the canonical Gibbs state. A Hamilto-nian and its eigenstates never appear in the kinematic description. The situation is quite differentin the case of the dynamical approach to thermalisation implied by the eigenstate thermalisationhypothesis (ETH), where the eigenstates of the Hamiltonian are of crucial importance. A con-nection between these two techniques to study thermalisation is thus desirable. This differencein perspectives between quantum information theoretic and kinematic approaches also needs tobe reconciled in the study of pre-thermalisation. Exceptional states such as “rare states” needto be fully understood from a kinematic standpoint, see [160] for a discussion on rare states andthermalisation.

In the context of quantum thermal machines, the interplay of statistics and engine performancehas not been completely understood. This is important, since quantum fluids typically constitutemore than one subsystem, where statistics becomes important. The relationship of correlationsand entanglement to engine performance is an area where a lot of progress is expected in the nearfuture. Proper inclusion of non-Markovian baths in the discussion of engines is another area underinvestigation. A systematic theory of multiparameter optimisation is still unavailable for engines.Finally, we note that various experimental implementations of quantum engines are expected inthe near future.

To close, quantum thermodynamics is a rapidly evolving research field that promises to changeour understanding of the foundations of physics while enabling the discovery of novel thermody-namic techniques and applications at the nanoscale. We have provided an overview of a numberof current trends and perspectives on quantum thermodynamics and ended with three particulardiscussions where there is still disagreement and flux. Reshaping these discussions and resolvingthem will no doubt deepen our understanding of the interplay between quantum mechanics andthermodynamics.

34

Acknowledgements

We are grateful to G. De Chiara, V. Dunjko, J. Eisert, S. Fazio, I. Ford, M. Horodecki, D. Jennings,R. Kosloff, M. Olshanii, M. Paternostro, D. Poletti, P. Skrzypczyk and R. Uzdin for their inputfor the review. Centre for Quantum Technologies is a Research Centre of Excellence funded bythe Ministry of Education and the National Research Foundation of Singapore. JA acknowledgessupport by the Royal Society and EPSRC (EP/M009165/1). This work was supported by theEuropean COST network MP1209 “Thermodynamics in the quantum regime”.

References

[1] S. Trotzky, Y.A. Chen, A. Flesch, I.P. McCulloch, U. Schollwock, J. Eisert, and I. Bloch, Probingthe relaxation towards equilibrium in an isolated strongly correlated one-dimensional bose gas, NaturePhysics 8 (2012), pp. 325–330.

[2] M. Banuls, R. Orus, J. Latorre, A. Perez, and P. Ruiz-Femenia, Simulation of many-qubit quantumcomputation with matrix product states, Physical Review A 73 (2006), p. 022344.

[3] C. Jarzynski, Nonequilibrium equality for free energy differences, Physical Review Letters 78 (1997),p. 2690.

[4] G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for freeenergy differences, Physical Review E 60 (1999), p. 2721.

[5] H. Tasaki, Jarzynski relations for quantum systems and some applications, arXiv preprint cond-mat/0009244 (2000).

[6] T. Sagawa and M. Ueda, Generalized jarzynski equality under nonequilibrium feedback control, Physicalreview letters 104 (2010), p. 090602.

[7] S. Popescu, A.J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics,Nature Physics 2 (2006), pp. 754–758.

[8] L. Del Rio, J. Aberg, R. Renner, O. Dahlsten, and V. Vedral, The thermodynamic meaning of negativeentropy, Nature 474 (2011), pp. 61–63.

[9] F.G. Brandao, M. Horodecki, J. Oppenheim, J.M. Renes, and R.W. Spekkens, Resource theory ofquantum states out of thermal equilibrium, Physical review letters 111 (2013), p. 250404.

[10] M.O. Scully, Quantum afterburner: Improving the efficiency of an ideal heat engine, Physical reviewletters 88 (2002), p. 050602.

[11] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Quantum-dot carnot engine at maxi-mum power, Physical Review E 81 (2010), p. 041106.

[12] F. Mazza, R. Bosisio, G. Benenti, V. Giovannetti, R. Fazio, and F. Taddei, Thermoelectric efficiencyof three-terminal quantum thermal machines, New Journal of Physics 16 (2014), p. 085001.

[13] L.J. Schulman and U.V. Vazirani, Molecular scale heat engines and scalable quantum computation, inProceedings of the thirty-first annual ACM symposium on Theory of computing, 1999, pp. 322–329.

[14] N. Linden, S. Popescu, and P. Skrzypczyk, How small can thermal machines be? the smallest possiblerefrigerator, Physical review letters 105 (2010), p. 130401.

[15] S.R. De Groot and P. Mazur, Non-equilibrium thermodynamics, Courier Corporation, 2013.[16] M. Campisi, P. Hanggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and

applications, Reviews of Modern Physics 83 (2011), p. 771.[17] M. Tomamichel, A framework for non-asymptotic quantum information theory, Ph.D. thesis, Diss.,

Eidgenossische Technische Hochschule ETH Zurich, Nr. 20213, 2012.[18] R. Kosloff and A. Levy, Quantum heat engines and refrigerators: Continuous devices, Annual Review

of Physical Chemistry 65 (2014), pp. 365–393.[19] V. Dunjko and M. Olshanii, Thermalization from the perspective of eigenstate thermalization hypoth-

esis, Annual Review of Cold Atoms and Molecules 1 (2012), p. 443.[20] C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in

closed quantum systems, arXiv preprint arXiv:1503.07538 (2015).[21] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypzyk, The role of quantum information in

thermodynamics—a topical review, arXiv preprint arXiv:1505.07835 (2015).

35

[22] J. Anders and V. Giovannetti, Thermodynamics of discrete quantum processes, New Journal of Physics15 (2013), p. 033022.

[23] W. Pusz and S. Woronowicz, Passive states and kms states for general quantum systems, Communi-cations in Mathematical Physics 58 (1978), pp. 273–290.

[24] A. Lenard, Thermodynamical proof of the gibbs formula for elementary quantum systems, Journal ofStatistical Physics 19 (1978), pp. 575–586.

[25] R. Alicki, The quantum open system as a model of the heat engine, Journal of Physics A: Mathematicaland General 12 (1979), p. L103.

[26] R. Kosloff and M.A. Ratner, Beyond linear response: Line shapes for coupled spins or oscillators viadirect calculation of dissipated power, The Journal of chemical physics 80 (1984), pp. 2352–2362.

[27] T.D. Kieu, The second law, maxwell’s demon, and work derivable from quantum heat engines, Physicalreview letters 93 (2004), p. 140403.

[28] A. Allahverdyan and T.M. Nieuwenhuizen, Fluctuations of work from quantum subensembles: The caseagainst quantum work-fluctuation theorems, Physical Review E 71 (2005), p. 066102.

[29] M. Esposito and S. Mukamel, Fluctuation theorems for quantum master equations, Physical Review E73 (2006), p. 046129.

[30] M.J. Henrich, G. Mahler, and M. Michel, Driven spin systems as quantum thermodynamic machines:Fundamental limits, Physical Review E 75 (2007), p. 051118.

[31] P. Talkner, E. Lutz, and P. Hanggi, Fluctuation theorems: Work is not an observable, Physical ReviewE 75 (2007), p. 050102.

[32] I. Ford, Statistical physics: An entropic approach, John Wiley & Sons, 2013.[33] J. Gemmer, M. Michel, and G. Mahler, Quantum thermodynamcis-emergence of thermodynamic be-

havior within composite quantum systems, Lecture Notes in Physics 2nd ed.(Springer, 2009) (2004).[34] H. Wilming, R. Gallego, and J. Eisert, Weak thermal contact is not universal for work extraction,

arXiv preprint arXiv:1411.3754 (2014).[35] J. Deutsch, Thermodynamic entropy of a many-body energy eigenstate, New Journal of Physics 12

(2010), p. 075021.[36] J. Gemmer and R. Steinigeweg, Entropy increase in k-step markovian and consistent dynamics of

closed quantum systems, Physical Review E 89 (2014), p. 042113.[37] C.H. Bennett, The thermodynamics of computationa review, International Journal of Theoretical

Physics 21 (1982), pp. 905–940.[38] R. Landauer, Irreversibility and heat generation in the computing process, IBM journal of research and

development 5 (1961), pp. 183–191.[39] K. Maruyama, F. Nori, and V. Vedral, Colloquium: The physics of maxwells demon and information,

Reviews of Modern Physics 81 (2009), p. 1.[40] M.G. Raizen, Demons, entropy and the quest for absolute zero, Scientific American 304 (2011), pp.

54–59.[41] D. Reeb and M.M. Wolf, An improved landauer principle with finite-size corrections, New Journal of

Physics 16 (2014), p. 103011.[42] P. Kammerlander and J. Anders, Quantum measurement and its role in thermodynamics, arXiv

preprint arXiv:1502.02673 (2015).[43] D.J. Evans, E. Cohen, and G. Morriss, Probability of second law violations in shearing steady states,

Physical Review Letters 71 (1993), p. 2401.[44] R. Kawai, J. Parrondo, and C. Van den Broeck, Dissipation: the phase-space perspective, Physical

review letters 98 (2007), p. 080602.[45] C. Jarzynski, Nonequilibrium work theorem for a system strongly coupled to a thermal environment,

Journal of Statistical Mechanics: Theory and Experiment 2004 (2004), p. P09005.[46] S. Schuler, T. Speck, C. Tietz, J. Wrachtrup, and U. Seifert, Experimental test of the fluctuation

theorem for a driven two-level system with time-dependent rates, Physical review letters 94 (2005), p.180602.

[47] F. Douarche, S. Joubaud, N.B. Garnier, A. Petrosyan, and S. Ciliberto, Work fluctuation theoremsfor harmonic oscillators, Physical review letters 97 (2006), p. 140603.

[48] O.P. Saira, Y. Yoon, T. Tanttu, M. Mottonen, D. Averin, and J. Pekola, Test of the jarzynski andcrooks fluctuation relations in an electronic system, Physical review letters 109 (2012), p. 180601.

[49] J. Liphardt, S. Dumont, S.B. Smith, I. Tinoco, and C. Bustamante, Equilibrium information from

36

nonequilibrium measurements in an experimental test of jarzynski’s equality, Science 296 (2002), pp.1832–1835.

[50] D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, and C. Bustamante, Verification of the crooksfluctuation theorem and recovery of rna folding free energies, Nature 437 (2005), pp. 231–234.

[51] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental demonstration ofinformation-to-energy conversion and validation of the generalized jarzynski equality, Nature Physics6 (2010), pp. 988–992.

[52] J.V. Koski, V.F. Maisi, T. Sagawa, and J.P. Pekola, Experimental observation of the role of mutualinformation in the nonequilibrium dynamics of a maxwell demon, Physical review letters 113 (2014),p. 030601.

[53] A. Berut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimentalverification of landauer/’s principle linking information and thermodynamics, Nature 483 (2012), pp.187–189.

[54] J. Kurchan, A quantum fluctuation theorem, arXiv preprint cond-mat/0007360 (2000).[55] S. Mukamel, Quantum extension of the jarzynski relation: Analogy with stochastic dephasing, Physical

review letters 90 (2003), p. 170604.[56] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and

counting statistics in quantum systems, Reviews of modern physics 81 (2009), p. 1665.[57] G. Huber, F. Schmidt-Kaler, S. Deffner, and E. Lutz, Employing trapped cold ions to verify the quantum

jarzynski equality, Physical review letters 101 (2008), p. 070403.[58] M. Heyl and S. Kehrein, Crooks relation in optical spectra: universality in work distributions for weak

local quenches., Physical review letters 108 (2012), pp. 190601–190601.[59] J.P. Pekola, P. Solinas, A. Shnirman, and D. Averin, Calorimetric measurement of work in a quantum

system, New Journal of Physics 15 (2013), p. 115006.[60] L. Mazzola, G. De Chiara, and M. Paternostro, Measuring the characteristic function of the work

distribution, Physical review letters 110 (2013), p. 230602.[61] T.B. Batalhao, A.M. Souza, L. Mazzola, R. Auccaise, R.S. Sarthour, I.S. Oliveira, J. Goold, G.

De Chiara, M. Paternostro, and R.M. Serra, Experimental reconstruction of work distribution andstudy of fluctuation relations in a closed quantum system, Physical review letters 113 (2014), p. 140601.

[62] S. An, J.N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.Q. Yin, H. Quan, and K. Kim, Experimentaltest of the quantum jarzynski equality with a trapped-ion system, Nature Physics (2014).

[63] I. Oliveira, R. Sarthour Jr, T. Bonagamba, E. Azevedo, and J.C. Freitas, NMR quantum informationprocessing, Elsevier, 2011.

[64] A. Kossakowski, On quantum statistical mechanics of non-hamiltonian systems, Reports on Mathe-matical Physics 3 (1972), pp. 247–274.

[65] V. Gorini, A. Kossakowski, and E.C.G. Sudarshan, Completely positive dynamical semigroups of n-levelsystems, Journal of Mathematical Physics 17 (1976), pp. 821–825.

[66] G. Lindblad, On the generators of quantum dynamical semigroups, Communications in MathematicalPhysics 48 (1976), pp. 119–130.

[67] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. Sudarshan, Properties of quantum markovianmaster equations, Reports on Mathematical Physics 13 (1978), pp. 149–173.

[68] H.P. Breuer and F. Petruccione, The theory of open quantum systems, Oxford university press, 2002.[69] M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge university

press, 2010.[70] A.G. Redfield, On the theory of relaxation processes, IBM Journal of Research and Development 1

(1957), pp. 19–31.[71] A. Hutter and S. Wehner, Almost all quantum states have low entropy rates for any coupling to the

environment, Physical review letters 108 (2012), p. 070501.[72] M. Tomamichel, R. Colbeck, and R. Renner, Duality between smooth min-and max-entropies, Infor-

mation Theory, IEEE Transactions on 56 (2010), pp. 4674–4681.[73] P. Faist, J. Oppenheim, and R. Renner, Gibbs-preserving maps outperform thermal operations in the

quantum regime, New Journal of Physics 17 (2015), p. 043003.[74] R. Dorner, S.R. Clark, L. Heaney, R. Fazio, J. Goold, and V. Vedral, Extracting quantum work statis-

tics and fluctuation theorems by single-qubit interferometry, Phys. Rev. Lett. 110 (2013), p. 230601,Available at http://link.aps.org/doi/10.1103/PhysRevLett.110.230601.

37

[75] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantum thermodynamics of general quantumprocesses, Physical Review E 91 (2015), p. 032119.

[76] M.M. Wolf, F. Verstraete, M.B. Hastings, and J.I. Cirac, Area laws in quantum systems: mutualinformation and correlations, Physical review letters 100 (2008), p. 070502.

[77] V. Vedral, The role of relative entropy in quantum information theory, Reviews of Modern Physics 74(2002), p. 197.

[78] H. Spohn, Entropy production for quantum dynamical semigroups, Journal of Mathematical Physics19 (1978), pp. 1227–1230.

[79] T. Sagawa, Second law-like inequalities with quantum relative entropy: An introduction, Lectures onQuantum Computing, Thermodynamics and Statistical Physics 8 (2012), p. 127.

[80] S. Vinjanampathy and K. Modi, Second law for quantum operations, arXiv preprint arXiv:1405.6140(2014).

[81] M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynam-ics, Nature communications 4 (2013).

[82] Y. Oono and M. Paniconi, Steady state thermodynamics, Progress of Theoretical Physics Supplement130 (1998), pp. 29–44.

[83] F. Schlogl, On thermodynamics near a steady state, Zeitschrift fur Physik 248 (1971), pp. 446–458.[84] S.i. Sasa and H. Tasaki, Steady state thermodynamics, Journal of statistical physics 125 (2006), pp.

125–224.[85] A. Granas and J. Dugundji, Fixed point theory, Springer Science & Business Media, 2013.[86] S. Yukawa, The second law of steady state thermodynamics for nonequilibrium quantum dynamics,

arXiv preprint cond-mat/0108421 (2001).[87] W.H. Zurek and J.P. Paz, Decoherence, chaos, and the second law, Physical Review Letters 72 (1994),

p. 2508.[88] T. Sagawa and M. Ueda, Second law of thermodynamics with discrete quantum feedback control, Phys-

ical review letters 100 (2008), p. 080403.[89] M. Esposito and C. Van den Broeck, Three faces of the second law. i. master equation formulation,

Physical Review E 82 (2010), p. 011143.[90] M. Esposito and C. Van den Broeck, Second law and landauer principle far from equilibrium, EPL

(Europhysics Letters) 95 (2011), p. 40004.[91] U. Fano and A.R.P. Rau, Symmetries in quantum physics, Academic Press, 1996.[92] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80

(2008), pp. 885–964, Available at http://link.aps.org/doi/10.1103/RevModPhys.80.885.[93] M.R. von Spakovsky and J. Gemmer, Some trends in quantum thermodynamics, Entropy 16 (2014),

pp. 3434–3470.[94] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics

of closed interacting quantum systems, Reviews of Modern Physics 83 (2011), p. 863.[95] E.T. Jaynes, Probability theory: the logic of science, Cambridge university press, 2003.[96] M.M. Wilde, Quantum information theory, Cambridge University Press, 2013.[97] S. Goldstein, J.L. Lebowitz, R. Tumulka, and N. Zanghı, Canonical typicality, Physical review letters

96 (2006), p. 050403.[98] J. Gemmer and G. Mahler, Distribution of local entropy in the hilbert space of bi-partite quantum

systems: origin of jaynes’ principle, The European Physical Journal B-Condensed Matter and ComplexSystems 31 (2003), pp. 249–257.

[99] M. Srednicki, Chaos and quantum thermalization, Physical Review E 50 (1994), p. 888.[100] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum

systems, Nature 452 (2008), pp. 854–858.[101] D.A. Sivak and G.E. Crooks, Thermodynamic metrics and optimal paths, Physical review letters 108

(2012), p. 190602.[102] M. Olshanii, Geometry of quantum observables and thermodynamics of small systems, Physical review

letters 114 (2015), p. 060401.[103] S. Weigert, The problem of quantum integrability, Physica D: Nonlinear Phenomena 56 (1992), pp.

107–119.[104] M. Rigol, Breakdown of thermalization in finite one-dimensional systems, Physical review letters 103

(2009), p. 100403.

38

[105] R. Barnett, A. Polkovnikov, and M. Vengalattore, Prethermalization in quenched spinor condensates,Physical Review A 84 (2011), p. 023606.

[106] M. Kollar, F.A. Wolf, and M. Eckstein, Generalized gibbs ensemble prediction of prethermalizationplateaus and their relation to nonthermal steady states in integrable systems, Physical Review B 84(2011), p. 054304.

[107] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D.A. Smith, E. Dem-ler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science337 (2012), pp. 1318–1322.

[108] R. Renner, Security of quantum key distribution, International Journal of Quantum Information 6(2008), pp. 1–127.

[109] J. Aberg, Truly work-like work extraction via a single-shot analysis, Nature communications 4 (2013).[110] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Thermodynamic cost of reliability and low

temperatures: tightening landauer’s principle and the second law, International Journal of TheoreticalPhysics 39 (2000), pp. 2717–2753.

[111] F.G. Brandao and M.B. Plenio, Entanglement theory and the second law of thermodynamics, NaturePhysics 4 (2008), pp. 873–877.

[112] F. Brandao, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum ther-modynamics, Proceedings of the National Academy of Sciences 112 (2015), pp. 3275–3279.

[113] P. Skrzypczyk, A.J. Short, and S. Popescu, Work extraction and thermodynamics for individual quan-tum systems, Nature communications 5 (2014).

[114] M. Lostaglio, D. Jennings, and T. Rudolph, Description of quantum coherence in thermodynamicprocesses requires constraints beyond free energy, Nature communications 6 (2015).

[115] J. Gemmer and J. Anders, From single-shot towards general work extraction in a quantum thermody-namic framework, arXiv preprint arXiv:1504.05061 (2015).

[116] N.H.Y. Ng, L. Mancinska, C. Cirstoiu, J. Eisert, and S. Wehner, Limits to catalysis in quantumthermodynamics, New Journal of Physics 17 (2015), p. 085004.

[117] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Quantum coherence, time-translation sym-metry, and thermodynamics, Physical Review X 5 (2015), p. 021001.

[118] H. Scovil and E. Schulz-DuBois, Three-level masers as heat engines, Physical Review Letters 2 (1959),p. 262.

[119] J. Geusic, E. Schulz-DuBios, and H. Scovil, Quantum equivalent of the carnot cycle, Physical Review156 (1967), p. 343.

[120] P. Hanggi and F. Marchesoni, Artificial brownian motors: Controlling transport on the nanoscale,Reviews of Modern Physics 81 (2009), p. 387.

[121] R. Kosloff, A quantum mechanical open system as a model of a heat engine, The Journal of chemicalphysics 80 (1984), pp. 1625–1631.

[122] B. Lin and J. Chen, Performance analysis of an irreversible quantum heat engine working with har-monic oscillators, Physical Review E 67 (2003), p. 046105.

[123] H. Quan, Y.x. Liu, C. Sun, and F. Nori, Quantum thermodynamic cycles and quantum heat engines,Physical Review E 76 (2007), p. 031105.

[124] Y. Zheng and D. Poletti, Work and efficiency of quantum otto cycles in power-law trapping potentials,Physical Review E 90 (2014), p. 012145.

[125] E. Geva and R. Kosloff, A quantum-mechanical heat engine operating in finite time. a model consistingof spin-1/2 systems as the working fluid, The Journal of chemical physics 96 (1992), pp. 3054–3067.

[126] T. Zhang, W.T. Liu, P.X. Chen, and C.Z. Li, Four-level entangled quantum heat engines, PhysicalReview A 75 (2007), p. 062102.

[127] F. Curzon and B. Ahlborn, Efficiency of a carnot engine at maximum power output, American Journalof Physics 43 (1975), pp. 22–24.

[128] R. Uzdin and R. Kosloff, Universal features in the efficiency at maximal work of hot quantum ottoengines, EPL (Europhysics Letters) 108 (2014), p. 40001.

[129] T. Schmiedl and U. Seifert, Efficiency at maximum power: An analytically solvable model for stochasticheat engines, EPL (Europhysics Letters) 81 (2008), p. 20003.

[130] C.M. Bender, D.C. Brody, and B.K. Meister, Quantum mechanical carnot engine, Journal of PhysicsA: Mathematical and General 33 (2000), p. 4427.

[131] O. Abah, J. Rossnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, Single-ion

39

heat engine at maximum power, Physical review letters 109 (2012), p. 203006.[132] J. Deng, Q.h. Wang, Z. Liu, P. Hanggi, and J. Gong, Boosting work characteristics and overall heat-

engine performance via shortcuts to adiabaticity: Quantum and classical systems, Physical Review E88 (2013), p. 062122.

[133] A. Del Campo, J. Goold, and M. Paternostro, More bang for your buck: Super-adiabatic quantumengines, Scientific reports 4 (2014).

[134] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, Nanoscale heat engine beyond thecarnot limit, Physical review letters 112 (2014), p. 030602.

[135] Special issue on quantum thermodynamics, New Journal of Physics (2015), Available athttp://iopscience.iop.org/1367-2630/focus/Focus%20on%20Quantum%20Thermodynamics.

[136] H. Quan, Quantum thermodynamic cycles and quantum heat engines. ii., Physical Review E 79 (2009),p. 041129.

[137] M.O. Scully, M.S. Zubairy, G.S. Agarwal, and H. Walther, Extracting work from a single heat bath viavanishing quantum coherence, Science 299 (2003), pp. 862–864.

[138] R. Uzdin, A. Levy, and R. Kosloff, Quantum equivalence and quantum signatures in heat engines,arXiv preprint arXiv:1502.06592 (2015).

[139] L.A. Correa, J.P. Palao, G. Adesso, and D. Alonso, Performance bound for quantum absorption re-frigerators, Physical Review E 87 (2013), p. 042131.

[140] L.A. Correa, J.P. Palao, D. Alonso, and G. Adesso, Quantum-enhanced absorption refrigerators, Sci-entific reports 4 (2014).

[141] F. Ticozzi and L. Viola, Quantum resources for purification and cooling: fundamental limits and op-portunities, Scientific reports 4 (2014).

[142] R. Uzdin and R. Kosloff, The multilevel four-stroke swap engine and its environment, New Journal ofPhysics 16 (2014), p. 095003.

[143] N. Brunner, M. Huber, N. Linden, S. Popescu, R. Silva, and P. Skrzypczyk, Entanglement enhancescooling in microscopic quantum refrigerators, Physical Review E 89 (2014), p. 032115.

[144] R. Kosloff, E. Geva, and J.M. Gordon, Quantum refrigerators in quest of the absolute zero, Journal ofApplied Physics 87 (2000), pp. 8093–8097.

[145] J.P. Palao, R. Kosloff, and J.M. Gordon, Quantum thermodynamic cooling cycle, Physical Review E64 (2001), p. 056130.

[146] B. Andresen, Current trends in finite-time thermodynamics, Angewandte Chemie International Edition50 (2011), pp. 2690–2704.

[147] T. Feldmann and R. Kosloff, Quantum four-stroke heat engine: Thermodynamic observables in a modelwith intrinsic friction, Physical Review E 68 (2003), p. 016101.

[148] R. Kosloff and T. Feldmann, Discrete four-stroke quantum heat engine exploring the origin of friction,Physical Review E 65 (2002), p. 055102.

[149] A.E. Allahverdyan, R. Balian, and T.M. Nieuwenhuizen, Maximal work extraction from finite quantumsystems, EPL (Europhysics Letters) 67 (2004), p. 565.

[150] R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries,Phys. Rev. E 87 (2013), p. 042123, Available at http://link.aps.org/doi/10.1103/PhysRevE.87.042123.

[151] K.V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, and A. Acın, Entanglement generation isnot necessary for optimal work extraction, Phys. Rev. Lett. 111 (2013), p. 240401, Available athttp://link.aps.org/doi/10.1103/PhysRevLett.111.240401.

[152] F.C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantacell: Powerful charging of quantumbatteries, arXiv preprint arXiv:1503.07005 (2015).

[153] L.A. Correa, Multistage quantum absorption heat pumps, Physical Review E 89 (2014), p. 042128.[154] Y. Rezek and R. Kosloff, Irreversible performance of a quantum harmonic heat engine, New Journal

of Physics 8 (2006), p. 83.[155] E. Geva and R. Kosloff, The quantum heat engine and heat pump: An irreversible thermodynamic

analysis of the three-level amplifier, The Journal of chemical physics 104 (1996), pp. 7681–7699.[156] R. Kosloff, Quantum thermodynamics: a dynamical viewpoint, Entropy 15 (2013), pp. 2100–2128.[157] M. Kolar, D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, Quantum bath refrigeration towards

absolute zero: Challenging the unattainability principle, Physical review letters 109 (2012), p. 090601.[158] R. Alicki, D. Gelbwaser-Klimovsky, and G. Kurizki, Periodically driven quantum open systems: Tuto-

rial, arXiv preprint arXiv:1205.4552 (2012).

40

[159] M. Campisi, P. Talkner, and P. Hanggi, Fluctuation theorem for arbitrary open quantum systems,Physical review letters 102 (2009), p. 210401.

[160] G. Biroli, C. Kollath, and A.M. Lauchli, Effect of rare fluctuations on the thermalization of isolatedquantum systems, Physical review letters 105 (2010), p. 250401.

41