CONTEMPORARY PHYSICS, 2016

VOL. 57, NO. 4, 545–579

http://dx.doi.org/10.1080/00107514.2016.1201896

Quantum thermodynamics
Sai Vinjanampathya , Janet Andersb
a Centre for Quantum Technologies, National University of Singapore, Singapore; b Department of Physics and Astronomy, University of Exeter,
Exeter, UK

ABSTRACT ARTICLE HISTORY

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

1. Introduction field, and each contributes different insights. For exam-

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

CONTACT Sai Vinjanampathy sai@quantumlah.org; Janet Anders janet@qipc.org

© 2016 Informa UK Limited, trading as Taylor & Francis Group
546 S. VINJANAMPATHY AND J. ANDERS

! τ
information theory techniques [27]. Other reviews of ⟨Q⟩ := tr[ρ̇ (t) H (t) ] dt and
interest discuss Maxwell’s demon and the physics of for- !0 τ
getting [28–30] and thermodynamic aspects of informa-
⟨W⟩ := tr[ρ (t) Ḣ (t) ] dt. (2)
tion [31]. We encourage researchers to take on board the 0
insights gained from different approaches and attempt to
fit together the pieces of the puzzle to create an overall Here, the brackets ⟨·⟩ indicate the ensemble average that
united framework of quantum thermodynamics. is assumed in the above definition when the trace is
Section 2 discusses the standard laws of thermody- performed. Work is extracted from the system when
namics and introduces the link with information ⟨Wext ⟩ := −⟨W⟩ > 0, while heat is dissipated to the
processing tasks and Section 3 gives a brief overview of environment when ⟨Qdis ⟩ := −⟨Q⟩ > 0.
fluctuation relations in the classical and quantum regimes. The first law of thermodynamics states that the sum of
Section 4 then introduces quantum dynamical maps and average heat and work done on the system just makes up
discusses implications for the foundations of thermody- its average energy change,
namics. Properties of maps form the backbone of the
thermodynamic resource theory approach to single shot !
thermodynamics discussed in Section 5. Section 6 dis-
τd
⟨Q⟩ + ⟨W⟩ = tr[ρ (t) H (t) ] dt
cusses the operation of thermal machines in the quantum 0 dt
regime. Finally, in Section 7, the current state of the field = tr[ρ (τ ) H (τ ) ] − tr[ρ (0) H (0) ] = #U. (3)
is summarised and open questions are identified.
It is important to note that while the internal energy
change only depends on the initial and final states and
2. Information and thermodynamics
Hamiltonians of the evolution, heat and work are pro-
This section defines averages of heat and work, intro- cess dependent, i.e. it matters how the system evolved in
duces the first and second laws of thermodynamics and time from (ρ (0) , H (0) ) to (ρ (τ ) , H (τ ) ). Therefore, heat and
discusses examples of the link between thermodynamics work for an infinitesimal process which will be denoted
and information processing tasks, such as erasure. by ⟨δQ⟩ and ⟨δW⟩ where the symbol δ indicates that heat
and work are (in general) not full differentials and do not
correspond to observables [41], in contrast to the average
2.1. The first and second laws of thermodynamics energy with differential dU.
Choosing to split the energy change into two types
Thermodynamics is concerned with energy and changes of energy transfer is crucial to allow the formulation
of energy that are distinguished as heat and work. For of the second law of thermodynamics. A fundamental
a quantum system in state ρ and with Hamiltonian H law of physics, it sets limits on the work extraction of
at a given time, the system’s internal, or average, energy heat engines and establishes the notion of irreversibility
is identified with the expectation value U(ρ) = tr[ρ H]. in physics. Clausius observed in 1865 that a new state
When a system changes in time, i.e. the pair of state and function – the thermodynamic entropy Sth of a system –
Hamiltonian [32] vary in time (ρ (t) , H (t) ) with t ∈ [0, τ ], is helpful to study the heat flow to the system when it
the resulting average energy change interacts with baths at varying temperatures T [42]. The
thermodynamic entropy is defined through its change in
#U = tr[ρ (τ ) H (τ ) ] − tr[ρ (0) H (0) ] (1) a reversible thermodynamic process,

is made up of two types of energy transfer – work and !

⟨δQ⟩
heat. Their intuitive meaning is that of two types of #Sth := , (4)
energetic resources, one fully controllable and useful, rev T
the other uncontrolled and wasteful [5,32–40]. Since the
time variation of H is controlled by an experimenter, where ⟨δQ⟩ is the heat absorbed by the system along the
the energy change associated with this time variation process and T is the temperature at which the heat is
is identified as work. The uncontrolled energy change being exchanged between the system and the"bath. Fur-
associated with the reconfiguration of the system state ther observing that any cyclic process obeys ⟨δQ⟩T ≤ 0
in response to Hamiltonian changes and the system’s with equality for reversible processes, Clausius formu-
coupling with the environment are identified as heat. The lated a version of the second law of thermodynamics for all
formal definitions of average heat absorbed by the system thermodynamic processes, today known as the Clausius
and average work done on the system are then inequality:
 CONTEMPORARY PHYSICS 547

!
⟨δQ⟩
≤ #Sth , becoming
T
⟨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 heat absorbed by
the system during a process divided by the temperature
at which the heat is exchanged. In this form, Clausius’
inequality establishes the existence of an upper bound
to the heat absorbed by the system and its validity is
generally assumed to extend to the quantum regime
[5,16,32–40]. Equivalently, by defining the free energy of Figure 1. A gas in a box starts with a thermal distribution at
a system with Hamiltonian H and in contact with a heat temperature T . Maxwell’s demon inserts a wall and selects faster
bath at temperature T as the state function particles to pass through a door in the wall to the left, while he
lets slower particles pass to the right, until the gas is separated
F(ρ) := U(ρ) − TSth (ρ), (6) into two boxes that are not in equilibrium. The demon attaches a
bucket (or another suitable work storage system) and allows the
Clausius’ inequality becomes a statement of the upper wall to move under the pressure of the gases. Some gas energy
bound on the work that can be extracted in a thermody- is extracted as work, Wext , raising the bucket. Finally, the gas
is brought in contact with the environment at temperature T ,
namic process,
equilibrating back to its initial state (#U = 0) while absorbing
⟨Wext ⟩ = −⟨W⟩ = −#U + ⟨Q⟩ heat Qabs = Wext . A comprehensive review of Maxwell’s demon
is [29].
≤ −#U + T#Sth = −#F. (7)

While the actual heat absorbed/work extracted will dep- 2.2. Maxwell’s demon
end on the specifics of the process, there exist optimal, Maxwell’s demon is a creature that is able to observe the
thermodynamically reversible, processes that saturate the motion of individual particles and use this information
equality, see Equation (4). However, modifications of (by employing feedback protocols discussed in Section
the second law, and thus the optimal work that can be 3.3) to convert heat into work in a cyclic process using
extracted, arise when the control of the working sys- only a single heat bath at temperature T. By separating
tem is restricted to physically realistic, local scenarios slower, ‘colder’ gas particles in a container from faster,
[43]. For equilibrium states ρth = e−βH /tr[e−βH ] for ‘hotter’ particles, and then allowing the hotter gas to
Hamiltonian H and at inverse temperatures β = kB1T , expand while pushing a piston, see Figure 1, the demon
the thermodynamic entropy Sth equals the information can extract work while returning to the initial mixed gas.
theory entropy, S, times the Boltzmann constant kB , i.e. For a single particle gas, the demon’s extracted work in a
Sth (ρth ) = kB S(ρth ). The information theory entropy, cyclic process is
known as the Shannon or von Neumann entropy, for a
general state ρ is defined as demon
⟨Wext ⟩ = kB T ln 2. (9)
S(ρ) := −tr[ρ ln ρ]. (8)
Maxwell realised in 1867 that such a demon would
Many researchers in quantum thermodynamics assume appear to break the second law of thermodynamics, see
that the thermodynamic entropy is naturally extended Equation (5), as it converts heat completely into work
to non-equilibrium states by the information theoretic in a cyclic process, resulting in a positive extracted work,
entropy. For example, this assumption is made when demon ⟩ = ⟨Q⟩ − #U ̸ ≤ T#S − #U = 0. The crux
⟨Wext th
using the von Neumann entropy in connection with the of this paradoxical situation is that the demon acquires
second law and the analysis of thermal processes, and information about individual gas particles and uses this
in the calculation of efficiencies of QTMs. Evidence that information to convert heat into useful work. One way of
this extension is appropriate has been provided via many resolving the paradox was presented by Bennett [46] and
routes including Landauer’s original work, see [28] for invokes Landauer’s erasure principle [47], as described
an introduction. The suitability of this extension and its in the next section and presented highly accessibly in
limitations remain however debated issues [44,45]. For [28]. Other approaches consider the cost the demon has
the remainder of this article, we will assume that the to pay upfront when identifying whether the particle
von Neumann entropy S is the natural extension of the is slower or faster [30]. Experimentally, the thermody-
thermodynamic entropy Sth . namic phenomenon of Maxwell’s demon remains highly
548 S. VINJANAMPATHY AND J. ANDERS

relevant – for instance, cooling a gas can be achieved take infinitely long to implement. The time of erasure is
by mimicking the demonic action [48]. An example of investigated in [51] and it is shown that Landauer’s limit
an experimental test of Maxwell’s demon is discussed in is achievable in finite time when allowing exponentially
Section 3.4. suppressed erasure errors.

2.3. Landauer’s erasure principle 2.4. Erasure with quantum side information
In a seminal paper [47], Landauer investigated the The heat dissipation during erasure is non-trivial when
thermodynamic cost of information processing tasks. He the initial state of the system is a mixed state, ρS . In
concluded that the erasure of one bit of information the quantum regime, mixed states can always be seen
requires a minimum dissipation of heat, as reduced states of global states, ρSM , of the system
S and a memory M, with ρS = trM [ρSM ]. A ground-
min breaking paper [8] takes this insight seriously and sets
⟨Qdis ⟩ = kB T ln 2, (10)
up an erasure scenario where an observer can operate
that the erased system dissipates to a surrounding en- on both, the system and the memory. During the global
vironment, cf. Figure 1, in equilibrium at temperature process, the system’s local state is erased, ρS ' → |0⟩, while
T. The erasure of one bit, or ‘reset’, here refers to the the memory’s local state, ρM = trS [ρSM ], is not altered.
change of a system being in one of two states with equal In other words, this global process is locally indistin-
probability, 2I (i.e. 1 bit), to a definite known state, |0⟩ (i.e. guishable from the erasure considered in the previous
0 bits). (We note that, contrary to everyday language, the section. However, contrary to the previous case, erasure
technical term ‘information’ here refers to uncertainty with ‘side-information’ , i.e. using correlations of the
rather than certainty. Erasure of information thus im- memory with the system, can be achieved while extracting
plies increase of certainty.) Landauer’s principle has been a maximum amount of work
tested experimentally only very recently, see Section 3.4.
max
The energetic cost of the heat dissipation is balanced ⟨Wext ⟩ = −kB T S(S|M)ρSM . (11)
by a minimum work that must be done on the system,
⟨W min ⟩ = #U − ⟨Q⟩ = ⟨Qdis min ⟩, to achieve the erasure at Here, S(S|M)ρSM is the conditional von Neumann en-
constant average energy, #U = 0. Bennett argued [46] tropy between the system and memory, S(S|M)ρSM =
that Maxwell’s demonic paradox can thus be resolved S(ρSM ) − S(ρM ). Crucially, the conditional entropy can
by taking into account that the demon’s memory holds be negative for some quantum correlated states (a sub-
bits of information that need to be erased to completely set of the set of entangled states), thus giving a posi-
close the thermodynamic cycle and return to the initial tive extractable work. This result contrasts strongly with
conditions. The work that the demon extracted in the first Landauer’s principle valid for both classical and quantum
step, see previous section, then has to be spent to erase states when no side-information is available. The possi-
the information acquired, with no net gain of work and bility to extract work during erasure is a purely quan-
in agreement with the second law. tum feature that relies on accessing the side-information
While Landauer’s principle was originally formulated [8]. That is, to practically obtain positive work requires
for the erasure of one bit of classical information, it knowledge of and access to an initial entangled state of
is straightforwardly extended to the erasure of a gen- the system and the memory, and the implementation of
eral mixed quantum state, ρ, which is transferred to the a carefully controlled process on the degrees of freedom
blank state |0⟩. The minimum required heat dissipation of both parties. The entanglement between system and
min ⟩ = k T S(ρ) following from the second
is then ⟨Qdis memory will be destroyed in the process and can be seen
B
law, Equation (5). A recent analysis of Landauer’s prin- as ‘fuel’ from which work is extracted.
ciple [49] uses a framework of thermal operations, cf.
Section 5, to obtain corrections to Landauer’s bound
2.5. Work from correlations
when the effective size of the thermal bath is finite. They
show that the dissipated heat is in general above Lan- The thermodynamic work and heat associated with cre-
dauer’s bound and only converges to it for infinite-sized ating or destroying (quantum) correlations have been
reservoirs. Probabilistic erasure is considered in [50] and studied intensely, e.g. [52–59], for a variety of settings,
the trade-off between the probability of erasure and mini- including unitary and non-unitary processes. For exam-
mal heat dissipation is demonstrated. In the simplest case, ple, the thermodynamic efficiency of an engine operating
the erasure protocol achieving the Landauer’s bound will on pairs of correlated atoms can be quantified in terms of
require an idealised quasi-static process which would quantum discord and it was shown to exceed the classical
 CONTEMPORARY PHYSICS 549

efficiency value [54]. In [60], the minimal heat dissipation Importantly, S(η) is larger than S(ρ) if and only if the
for coupling a harmonic oscillator that starts initially in state ρ had coherences with respect to the energy eigen-
local thermal equilibrium, and ends up correlated with a basis, while the entropy does not change under projection
bath of harmonic oscillators in a global thermal equilib- processes for classical states. Thus, the extracted work
rium state, is determined and it was shown that this heat here is due to the quantum coherences in the initial state.
contribution resolves a previously reported second law We note that ‘decohering’ a state ρ is a physical imple-
violation. mentation of the same state transfer, ρ ' → η, during
Thermodynamic aspects of creating correlations are which coherences are washed out by the environment in
also studied in [59] where the minimum work cost is an uncontrolled way and no work is extracted. This is
established for unitarily evolving an initial, locally ther- a suboptimal process – to achieve the maximum work
mal, state of N systems to a global correlated state. A max- an optimal implementation needs to be realised which
imum temperature is derived at which entanglement can requires a carefully controlled protocol of interacting
still be created, along with the minimal associated energy the system with heat and work sources [61]. It is worth
cost. In turn, when wanting to extract work from many- noting that in contrast to Maxwell’s demon whose work
body states that are initially globally correlated, while extraction is based on the knowledge of ‘microstates’
locally appearing thermal, the maximum extractable and appears to violate the second law when information
work under global unitary evolutions is discussed in [58] erasure is not considered, gaining work from coherences
for initial entangled and separable states, and those dia- is in accordance with the second law. Once the projection
gonal in the energy basis. This contrasts with the non- process is completed, the final state has lost its coherence,
unitary process of erasure with side-information, i.e. here the coherences have been used as ‘fuel’ to extract
discussed in Section 2.4. work.

3. Classical and quantum non-equilibrium

statistical physics
2.6. Work from coherences
In this section, the statistical physics approach is out-
While erasure with side-information discussed in
lined that rests on the definition of fluctuating heat and
Section 2.4 and the results in Section 2.5 illustrate the
work from which it derives the ensemble quantities of
quantum thermodynamic aspect of correlations, a sec-
average heat and average work discussed in Section 2.
ond quantum thermodynamic feature arises due to the
Fluctuation theorems and experiments are first described
presence of coherences. A recent paper [61] identifies
in the classical regime before they are extended to the
projection processes as a route to analyse the thermo-
quantum regime.
dynamic role of coherences. Projection processes map
an initial state, ρ, which has coherences in a particular
3.1. Definitions of classical fluctuating work and
basis of interest, {&k }k with k = 1, 2, 3, . . ., to a state
heat
in
# which these coherences are removed, i.e. ρ ' → η :=
k &k ρ &k . These processes can be interpreted as un- In classical statistical physics, a single particle is assigned
selective measurements of an observable with eigenbasis a point, x = (q, p), in phase space while an ensemble of
{&k }k , a measurement in which not the individual mea- particles is described with a probability density function,
surement outcomes are recorded but only the statistics of P(x), in phase space. The Hamiltonian of the particle is
the outcomes is known. Just like in the case of Landauer’s denoted as H(x, λ) where x is the phase space point of
erasure map, the state transfer achieved by the new math- the particle for which the energy is evaluated and λ is
ematical map can be implemented in various physical an externally controlled force parameter that can change
ways. Different physical implementations of the same in time. For example, a harmonic oscillator Hamilto-
map will have different work and heat contributions. p2 mλ2 q2
nian H(x, λ) = 2m + 2 can become time dependent
It was shown that there exists a physical protocol to through a protocol according to which the frequency, λ,
implement the projection process such that a non-trivial of the potential is varied in time, λ(t). This particular
average work can be extracted from the initial state’s example will be discussed further in Section 6 on QTMs.
coherences. For the example that the basis,#{&k }k , is the Each particle’s state will evolve due to externally ap-
energy eigenbasis of a Hamiltonian H = k Ek &k with plied forces and due to the interaction with its environ-
eigenenergies Ek , the maximum amount of average work ment, resulting in a trajectory xt = (qt , pt ) in phase
that can be extracted in a projection process is space, see Figure 2. The fluctuating work (done on the
system) and the fluctuating heat (absorbed by the system)
max
⟨Wext ⟩ = kB T (S(η) − S(ρ)) ≥ 0. (12) for a single trajectory xt followed by a particle in the
550 S. VINJANAMPATHY AND J. ANDERS

with the same kind of environment and applying the same

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

3.2. Classical fluctuation relations thermal distribution. The average exponentiated work
done on the system is then obtained similarly to (16) and
A central recent breakthrough in classical statistical me-
one obtains
chanics is the extension of the second law of thermody-
!
namics, an inequality, to equalities valid for large classes −βWτclosed closed
of non-equilibrium processes. Detailed fluctuation rela- ⟨e ⟩ = P0 (x0 ) e−βWτ ,xτ dx0
tions show that the probability densities of stochastically ! −βH(x0 ,λ(0))
e
fluctuating quantities for a non-equilibrium process, such = e−β(H(xτ ,λ(τ ))−H(x0 ,λ(0))) dx0 ,
Z (0) ) )
as entropy, work and heat, are linked to equilibrium !
1 ) dx0 )
properties and corresponding quantities for the time- = (0) e−βH(xτ ,λ(τ )) )) ) dxτ
Z dxτ )
reversed process [4,62,63]. By integrating over the proba-
bility densities, one obtains integral fluctuation relations, Z (τ )
= (0) = e−β#F . (20)
such as Jarzynski’s work relation [3]. Z
An important detailed fluctuation relation is Crooks
The beauty of this equality is that for all closed but arbi-
relation for a system in contact with a bath at inverse
trarily strongly non-equilibrium processes, the average
temperature β = 1/(kB T) for which detailed balanced is
exponentiated work is determined entirely by equilib-
valid [4]. The relation links the work distribution P F (W)
rium parameters contained in #F. As Jarzynski showed,
associated with a forward protocol changing the force pa-
the equality can also be generalised to open systems [64].
rameter λ(0) → λ(τ ), to the work distribution P B ( − W)
By applying Jensen’s inequality ⟨e−βW ⟩ ≥ e−β⟨W⟩ , the
associated with the time-reversed backwards protocol
Jarzynski equality turns into the standard second law of
where λ(τ ) → λ(0),
thermodynamics, cf. Equation (7),
P F (W) = P B ( − W) eβ(W−#F) . (18)
⟨W⟩ ≥ #F. (21)
Here, the forward and backward protocols each start with
Thus, Jarzynski’s relation strengthens the second law by
an initial distribution that is thermal at inverse temper-
including all moments of the non-equilibrium work, res-
ature β for the respective values of the force parameter.
ulting in an equality from which the inequality follows
The free energy difference #F = F (τ ) − F (0) refers to
for the first moment.
the two thermal distributions with respect to the final,
Fluctuation theorems have been measured, for exam-
H(x, λ(τ )), and initial, H(x, λ(0)), Hamiltonian in the
ple, for a defect centre in diamond [65], for a torsion pen-
forward protocol at inverse temperature β. Here, the
dulum [66] and in an electronic system [67]. They have
free energies, defined in (6), can be expressed as F (0) =
also been used to infer, from the measurable work in non-
− β1 ln Z (0) at time 0 and similarly at time τ with classical
( equilibrium pulling experiments, the desired equilibrium
partition functions Z (0) := e−βH(x,λ(0)) dx and simi- free energy surface of biomolecules [68,69], which is not
larly Z (τ ) . Rearranging and integrating over dW on both directly measurable otherwise.
sides result in a well-known integral fluctuation relation,
the Jarzynski equality [3],
! 3.3. Fluctuation relations with feedback
⟨e−βW ⟩ = P F (W) e−βW dW = e−β#F . (19) It is interesting to see how Maxwell demon’s feedback
process discussed in Section 2.2 can be captured in a gen-
Pre-dating Crooks relation, Jarzynski proved his eralised Jarzynski equation [6]. Again, one assumes that
equality [3] by considering a closed system that starts the system undergoes closed dynamics with a
−βH(x0 ,λ(0))
with a thermal distribution, P0 (x0 ) = e Z (0) , for a time-dependent Hamiltonian; however, now the force
given Hamiltonian H(x, λ(0)) at inverse temperature β. parameter in the Hamiltonian is changed according to a
The Hamiltonian is externally modified through its force protocol that depends on the phase space point the system
parameter λ which drives the system out of equilibrium is found in. For example, for the two choices of a particle
and into evolution according to Hamiltonian dynamics being in the left box or the right box, see Figure 3, the
in a time interval [0, τ ]. Such experiments can be realised, initial Hamiltonian H(x, λ(0)) will be changed to either
e.g. with colloidal particles see Section 3.4, where the H(x, λ1 (τ )) for the particle in the left box or H(x, λ2 (τ ))
experiment is repeated many times, each time imple- for the particle in the right box. Calculating the average
menting the same force protocol. Averages can then be exponentiated work for this situation, see Equation (20),
calculated over many trajectories, each starting from an one now notices that the integration over trajectories
initial phase space point that was sampled from an initial includes the two different evolutions, driven by one of
552 S. VINJANAMPATHY AND J. ANDERS

the two Hamiltonians. The free energy difference, #F, other final Hamiltonians, i.e. H(y, λ2 (τ )). The efficacy
previously corresponding to the Hamiltonian change, see is now
#the added probability for each of the protocols,
Equation (20), is now itself a fluctuating function. Either γ = k P (k) (yτ ).
it is #F = F (1) −F (0) or it is #F = F (2) −F (0) , depending Applying Jensen’s inequality to Equation (22), one
on the measurement outcome in each particular run. As now obtains ⟨W⟩ ≥ ⟨#F⟩− β1 ln γ , which when assuming
a result, the corresponding work fluctuation relation can that a cycle has been performed, ⟨#F⟩ = 0, the maximal
be written as value of γ for two feedback options has been reached,
γ = 2, and the inequality is saturated, becomes
⟨e−β(W−#F) ⟩ = γ , (22)
demon
⟨Wext ⟩ = −⟨W⟩ = kB T ln 2, (23)
where the average ⟨·⟩ includes an average over the two dif-
ferent protocols. Here, γ ≥ 0 characterises the feedback in agreement with Equation (9). A recent review pro-
efficacy, which is related to the reversibility of the non- vides a detailed discussion of feedback in classical fluc-
equilibrium process. When H(x, λ1 (τ )) = H(x, λ2 (τ )), tuation theorems [31]. The efficiency of feedback control
i.e. no feedback is actually implemented, then γ becomes in quantum processes has also been analysed, see e.g.
unity and the Jarzynski equality is recovered. The maxi- [70–72].
mum value of γ is determined by the number of different
protocol options, e.g. for the two feedback choices dis-
3.4. Classical statistical physics experiments
cussed above, one has γ max = 2.
Apart from measuring ⟨e−β(W−#F) ⟩ by performing A large number of statistical physics experiments are
the feedback protocol many times and recording the work performed with colloidal particles, such as polystyrene
done on the system, γ can also be measured experi- beads, that are suspended in a viscous fluid. The link
mentally, see Section 3.4. To do so, the system is ini- between information theory and thermodynamics has
tially prepared in a thermal state for one of the final been confirmed by groundbreaking colloidal experiments
Hamiltonians, say H(y, λ1 (τ )), at inverse temperature β. only very recently. For example, the heat-to-work con-
The force parameter λ1 (t) in the Hamiltonian is now version achieved by a Maxwell demon, see Sections 2.2
changed backwards, λ1 (τ ) → λ(0), without any feed- and 3.3, intervening in the statistical dynamics of a sys-
back. The particle’s final phase space point yτ is recorded. tem was investigated [73] with a dimeric polystyrene
The probability P (1) (yτ ) is established that the particle bead suspended in a fluid and undergoing rotational
ended in the phase space volume that corresponds to Brownian motion, see Figure 3(a). An external electrical
the choice of H(y, λ1 (τ )) in the forward protocol, in this potential was applied so that the bead was effectively
example in the left box. The same is repeated for the moving on a spiral staircase, see Figure 3(b). The demon’s

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

Figure 4. Left (a)–(f) Erasure protocol for a particle trapped in a double well. Particles starting in either well will end up in the right-hand
side well at the end of the protocol. Right (b) Measured heat probability distribution P(Qdis ) over fluctuating heat value Qdis for a fixed
protocol cycle time. Right (c) Average heat in units of kB T , i.e. ⟨Qdis ⟩/(kB T ), over varying cycle time τ in seconds. Figures taken from [75]
reproduced with permission.

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

(τ ) # (τ )
crucial step taken is to define the fluctuating quantum where pm := n pn,m are the marginals of the joint
work for closed dynamics as a two-point correlation probability distribution. Assuming the initial state ρ (0)
function. That is, a projective measurement of energy was diagonal in the initial energy basis, {|en(0) }n , the
needs to be performed at the beginning and end of the (τ )
pm turn out to be just the probabilities for measur-
process to gain knowledge of the system’s energetic -ing the energies
(τ ) (τ )
. Em in the final state ρ , i.e. pm =
(τ )
change. This enables the construction of a work dis- (τ ) (τ )
em |ρ (τ ) |em . Thus, the average work for the unitary
tribution function (known as the two-point measure- process is just the internal energy difference, cf. (16),
ment work distribution) and allows the formulation of
the Tasaki–Crooks relation and the quantum Jarzynski’s ⟨Wτclosed ⟩ = tr[H (τ ) ρ (τ ) ] − tr[H (0) ρ (0) ] = #U. (27)
equality [5,41,78,79].
To introduce the quantum fluctuating work and heat,
consider a quantum system with initial state ρ (0) and ini- 3.6. Quantum fluctuation relations
# (0) (0) (0)
tial Hamiltonian H (0) = n En |en ⟩⟨en | with eigen-
(0) (0) With the above definitions, in particular that of the work
values En and energy eigenstates |en ⟩. A closed system distribution, the quantum Jarzynski equation can be
undergoes dynamics due to its time-varying Hamiltonian readily formulated for a closed quantum system under-
which generates a unitary transformation (here, ( τ (t)T stands going externally driven (with unitary V ) non-equilibrium
for a time-ordered integral) V (τ ) = T e−i 0 H dt/! and dynamics [5,41,78–80]. The average exponentiated
ends in a final state ρ# (τ ) = V (τ ) ρ (0) V (τ ) † and a final
(τ ) (τ ) (τ )
work done on a system starting in initial state ρ (0) now
Hamiltonian H (τ ) = m Em |em ⟩⟨em |, with energies becomes, cf. (20),
(τ ) (τ )
Em and energy eigenstates |em ⟩. To obtain the fluctu- !
ating work, the energy is measured at the beginning of closed * (τ ) (0)
⟨e−βWτ ⟩ = P(W) e−βW dW = (τ ) −β(Em −En )
pn,m e .
the process, giving e.g. En(0) , and then again at the end n,m
(τ )
of the process, giving e.g. in Em . The difference between (28)
the measured energies is now identified entirely with
fluctuating work, When the initial state is a thermal state for the
# Hamilto-
nian H 0 at inverse temperature β, e.g. ρ (0) = n pn(0) |en(0) ⟩
closed (τ ) (0)
Wm,n := Em − En(0) , (24) −βEn
⟨en(0) |, with thermal probabilities pn(0) = e Z (0) and parti-
# (0)
tion function Z (0) = n e−βEn , one obtains the quan-
as the system is closed, the evolution is unitary and no tum Jarzynski equality
heat dissipation occurs, cf. the classical case (15). The
work distribution is then peaked whenever the distribu- * e−βEn(0) (τ )
−βWτclosed
(τ ) (0)
tion variable W coincides with Wm,n
τ ,
⟨e ⟩= pm|n e−β(Em −En )
n,m
Z (0)

* + ,
P(W) = (τ )
pn,m (τ )
δ W − (Em − En(0) ) . (25) 1 * −βEm(τ ) Z (τ )
= e = = e−β#F .
n,m Z (0) m Z (0)
(29)
(τ ) (0) (τ )
Here, pn,m = pn pm|n is the joint probability distribution
Here, we have used that the conditional probabilities
of finding the initial energy level En(0) and the final energy # (τ ) # (τ ) (τ ) (0) 2
(τ ) sum to unity, n pm|n = n |⟨em |V |en ⟩| = 1 and
Em . This can be broken up into the probability of finding # −βE(τ )
(0) (0) (0) (0)
the initial energy En , pn = ⟨en |ρ (0) |en ⟩, and the Z = me
(τ ) m is, like in the classical case, the par-
conditional probability for transferring from n at t = 0 tition function of the hypothetical thermal configuration
(τ )
to m at t = τ , pm|n = |⟨em(τ ) (τ ) (0) 2
|V |en ⟩| . for the final Hamiltonian, H (τ ) . The free energy difference
is, like in the classical case, #F = − β1 ln ZZ (0) .
(τ )
The average work for the unitarily driven non-
equilibrium process can now be calculated as the average Similarly, the classical Crooks relation, cf. Equation
over the work !probability distribution, i.e. using (17), (18), can also be re-derived for the quantum regime and
* is known as the Tasaki–Crooks relation [5],
⟨Wτclosed ⟩ = (τ )
pn,m δ(W − (Em (τ )
− En(0) )) W dW
*
n,m P F (W)
(τ ) = eβ(W−#F) . (30)
= pn(0) pm|n (Em
(τ )
− En(0) ) P B ( − W)
n,m
* *
= pm(τ ) (τ )
Em − pn(0) En(0) , (26) It is interesting to note that in the two-point mea-
m n surement scheme, both, the quantum Jarzynski equality
 CONTEMPORARY PHYSICS 555

and the Tasaki–Crooks relation, show no difference to ρ (0) at inverse temperature β, for which four values were
their classical counterparts, contrary to what one might realised. The experiment implemented the time-varying
expect. A debated issue is that the energy measurements spin Hamiltonian, H (t) , resulting in a unitary evolution,
remove coherences with respect to the energy basis [41], i.e. V in Section 3.6, by applying a time-modulated radio
and these do not show up in the work distribution, P(W). frequency field resonant with the 13 C nuclear spin in a
It has been suggested [61] that the non-trivial work and short time window τ . The (forward) work distribution
heat that may be exchanged during the second measure- of the spin’s evolution, P F (W) see Figure 5(a), was then
ment, see Section 2.6, implies that identifying the energy reconstructed through a series of one- and two-body op-
change entirely with fluctuating work (24) is inconsistent. erations on the system and the ancilla, and measurement
If the initial state is not thermal but has coherences, of the transverse magnetisation of the 1 H nuclear spin.
then also the first measurement will non-trivially affect Similarly, a backwards protocol was also implemented
these initial coherences and lead to work and heat contri- and the backwards work distribution, P B (W), measured,
butions. Work probability distributions and generalised cf. Sections 3.2 and 3.4.
Jarzynski-type relations have been proposed to account The measured values of the Tasaki–Crooks ratio,
for these coherences using path integral and quantum P F (W)/P B ( − W), are shown over the work value W
jump approaches [81,82]. in Figure 5(b) for four values of temperature. The trend
followed by the data associated with each temperature
3.7. Quantum fluctuation experiments was in very good agreement with the expected linear
While a number of interesting avenues to test fluctua- relation, confirming the predictions of the Tasaki–Crooks
tion theorems at the quantum scale have been proposed relation. The cutting point between the two work distri-
[83–86], the experimental reconstruction of the work butions was used to determine the value of #F experi-
statistics for a quantum protocol has long remained mentally which is shown in Figure 5(c). Calculating the
elusive. A new measurement approach has recently been average exponentiated work with the measured work
devised that is based on well-known interferometric distribution, the data also showed good agreement with
schemes of the estimation of phases in quantum sys- the quantum Jarzynski relation, Equation (29), see table
tems, which bypasses the necessity of two direct pro- in Figure 5(d).
jective measurements of the system state [87]. The first
quantum fluctuation experiment that confirms the quan- 4. Quantum dynamics and foundations of
tum Jarzynski relation, Equation (29), and the Tasaki– thermodynamics
Crooks relation, Equation (30), with impressive accuracy
has recently been implemented in a nuclear magnetic The information theoretic approach to thermodynamics
resonance (NMR) system using such an interferomet- employs many standard tools of quantum information to
ric scheme [88]. Another very recent experiment uses study mesoscopic systems. For instance, an abstract view
trapped ions to measure the quantum Jarzynski equality of dynamics, minimal in the details of Hamiltonians, is
[89] with the two measurement methods used to derive often employed in quantum information. Such a view
the theoretical result in Section 3.6. of dynamics as a map between quantum states serves
The NMR experiment was carried out using liquid- to produce rules common to generic dynamics and has
state NMR spectroscopy of the 1 H and 13 C nuclear spins served the study of computing and technologies well. In
of a chloroform molecule sample. This system can be this section, we describe some of these techniques and
regarded as a collection of identically prepared, non- theorems and highlight how they are used in quantum
interacting, spin-1/2 pairs [90]. The 13 C nuclear spin is thermodynamics.
the driven system, while the 1 H nuclear spin embodies
an auxiliary degree of freedom, referred to as ‘ancilla’.
4.1. Completely positive maps
Instead of performing two projective measurements on
the system, it is possible to reconstruct the system’s work We begin by reviewing the description of generic quan-
distribution with an interferometric scheme in which the tum maps [91–96]. The point of describing
ancilla is instrumental [87]. The ancilla here interacts dynamics through a ‘map’ as opposed to a model of tem-
with the system at the beginning and the end of the poral dynamics (i.e. a Hamiltonian) is deliberate. Maps
process, and as a result, its state acquires a phase. This are not explicit functions of time (though they can be
phase corresponds to the energy difference experienced parametrised by time, as we will see later), but are two-
by the system and the ancilla is measured only once to ob- point functions. They accept initial states of the dynamics
tain the energy difference of the system. The 13 C nuclear they model and output final states. Specifically, a com-
spin was prepared initially in a pseudo-equilibrium state pletely positive trace-preserving (CPTP) map transforms
556 S. VINJANAMPATHY AND J. ANDERS

(b) (c)

(a)

(d)

Figure 5. (a) Experimental work distributions corresponding to the forward (backward) protocol, P F (W) (P B ( − W)), are shown as red
squares (blue circles). (b) The Tasaki–Crooks ratio is plotted on a logarithmic scale for four values of the system’s temperature. (c)
Crosses indicate mean values and uncertainties for #F and β obtained from a linear fit to data compared with the theoretical prediction
indicated by the red line. (d) Experimental values of the left- and right-hand side of the quantum Jarzynski relation measured for three
temperatures together with their respective uncertainties, and theoretical predictions for ln ZZ (0) showing good agreement. Figures taken
(τ )

from [88], reproduced with permission.

input density matrices ρ into physical output states ρ ′ . We can now consider the action of the transpose map
The term ‘physical’ here refers to the requirement that on a subsystem. For example,
√ for the two-qubit Bell state
the output state is again a well-defined density matrix. |φ + ⟩ = (|00⟩ + |11⟩)/ 2, one may apply transposition
A map , has to obey several rules to guarantee that its only on the second qubit. This map then gives
outputs are physical states. These properties include: ⎛ ⎞
1001
(1) trace preservation: ρ ′ := ,(ρ) has unit trace for 1 ⎜0 0 0 0⎟
all input states ρ of dimensionality d. If this is vio- |φ + ⟩⟨φ + | = ⎜ ⎟
2 ⎝0 0 0 0⎠
lated, the output states become unphysical in that
1001
Born’s rule cannot be applied directly anymore. ⎛ ⎞
(2) positivity: ,(ρ) has non-negative eigenvalues, in- 1000
& ' 1 ⎜0 0 1 0⎟
terpreted as probabilities. ' → I ⊗ T |φ + ⟩⟨φ + | = ⎜ ⎟, (32)
(3) complete positivity: ∀k ∈ {0, . . . ∞} : (I(k) ⊗ ,) 2 ⎝0 1 0 0⎠
(σ (k+d) ) has non-negative eigenvalues for all states 0001
σ (k+d) . This subsumes positivity.
where the resulting global state turns out to have one
The final property is the statement that if the CPTP map negative eigenvalue. Because of this property, transpo-
is acting on a subsystem of dimension d which is part of sition is not a completely positive map. This example
a larger (perhaps entangled) system of dimension k + d, demonstrates that positivity, e.g. maintaining positive
whose state is σ (k+d) , then the resultant global state must eigenvalues of a single qubit state, does not imply com-
also be a ‘physical’ state. plete positivity, e.g. maintaining positive eigenvalues of
For instance, let us consider the transpose map T a two-qubit state when the map is only applied to one
which, when applied to a given density matrix, transposes qubit. The fact the ‘partial’ transposition is not positive
all its matrix elements in a fixed basis. When applied to when the initial state is entangled is used extensively as a
physical states of dimension d, this map always outputs
criterion to detect entanglement [96].
physical states of dimension d. For example, for a general
CPTP maps are related to dynamical equations, which
qubit state ρ with Bloch vector ⃗r = (x, y, z), the transpo-
sition map gives we discuss briefly. When the system is closed, 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 inter-
1 1 + z x − iy 1 1 + z x + iy
ρ= '→ T(ρ) = . acting with an external environment is given by Lindblad-
2 x + iy 1 − z 2 x − iy 1 − z
(31) type master equations. Such dynamics preserves trace
Note that the eigenvalues of the matrix are invariant and positivity of the density matrix, while allowing the
under the transposition map T. density matrix to vary otherwise. Such equations have
 CONTEMPORARY PHYSICS 557

the general form arbitrary environmental ancilla state, τE , a joint unitary

dρ *5 1 1
6 and a partial trace. This assertion is written as
= −i[H, ρ]+ Ak ρA†k − A†k Ak ρ − ρAk A†k .
dt 2 2 ρ ′ = ,(ρ) = trE [V ρ ⊗ τE V † ]. (34)
k
(33)
Stinespring dilation allows for any generic quantum
Here, Ak are Lindblad operators that describe the effect dynamics of the system, S, to be thought of in terms
of the interaction between the system and the environ- of a global unitary V acting on the system and an en-
ment on the system’s state. Notice that if an initial state vironment, E. If the interactions between the system and
ρ(t = 0) is evolved to a final state ρ(t = T), then the environment are known, then the time dependence
the resulting transformation can be captured by a CPTP of the map may be specified. For instance, if the Hamil-
map ,(ρ(t = 0)) = ρ(t = T). Master equations are tonian governing the joint dynamics is HSE , then V =
typically derived with many assumptions, including that VSE (t) := exp ( − iHSE t). However, Stinespring dilation
the system is weakly coupled to the environment and that allows the mathematical analysis of many properties of
the environmental correlations decay sufficiently quickly maps for general sets of unitaries. We note that many
such that the initial state of the system is uncorrelated to quantum thermodynamics papers assume τE to be the
the environment. Lindblad-type master equations have Gibbs state of the environment and use this to study,
been extensively employed to the study of quantum en- e.g., the influence of temperature on the dynamics of the
gines, and this connection will be discussed in Section 6.2. quantum system which interacts with the environment.
Some master equations are not in the Lindblad form, see For example, in [98], the authors considered thermali-
[97] for example. We note that the structure of Equation sation of a quantum system, a topic we will discuss in
(33) arises naturally if Markovianity, trace preservation Section 4.3. They prove a condition for the long time
of the density matrix and positivity are considered. To states of a system to be independent of the initial state
see that this should be the case, consider the general of the system. This condition relies on smooth min- and
evolution step for the density matrix, which we write max-entropies which are defined below.
without loss of generality as dρ = −(Mρ + ρN)dt. From Another theorem used in [98] and other studies [99]
the Hermiticity of the density matrix, we have dρ † = dρ, of quantum thermodynamics is the channel-state duality
implying N = M † . We can hence be tempted to write an or Choi–Jamiołkowski isomorphism. It says that there
evolution equation of the form dρ = −(Mρ + ρM † )dt. is a state ρ, isomorphic to every CPTP map ,, given
If we write M = iH + J, this form does not preserve by
trace for the terms involving J. This problem is fixed by
subtracting an appropriate term. If we write J = F † F, ρ, := I ⊗ ,(|ϕ⟩⟨ϕ|), (35)
then we can write a trace-preserving equation, namely: √ #
dρ = −i[H, ρ]dt + (2FρF † − F † Fρ − ρF † F)dt, where where |ϕ⟩ = (1/ d) di |i⟩ ⊗ |i⟩ is a maximally en-
2FρF † ensures trace preservation. Writing the operator tangled bipartite state. The main purpose of this the-
F in an orthonormal basis and diagonalising the resulting orem is that it allows us to think of the CPTP map
operator produces the general equation above. We note , as a state ρ, which is often useful to make state-
that care has to be taken that the dynamical equation also ments about the amount of correlations generated by a
preserves positivity, see [95] for a detailed derivation. given CPTP map. For example, see [100] for a discus-
sion on entropic inequalities that employ this formal-
ism.
4.1.1. Three theorems involving CPTP maps The final theorem is the operator sum representation,
We now briefly mention three important theorems often which asserts that any map, which has the representation
used in the quantum information theoretic study of ther-
*
modynamics. These theorems are the Stinespring dila- ρ ′ = ,(ρ) := Kµ ρ K†µ , (36)
tion theorem, the channel-state duality theorem and the µ
operator sum representation, respectively. The impor- #
tance of these theorems lies in the alternative perspectives with K†µ Kµ = I, is completely positive (but not nec-
they afford an abstract dynamical map. We mention the essarily trace preserving). These operators, often known
theorems briefly, alongside a brief remark about the light as Sudarshan–Kraus operators, are related to the ancilla
they shed on CPTP maps. state τE and V in Equation (34). We note that these
The first of these, Stinespring dilation theorem asserts theorems are often implicitly assumed while discussing
that every CPTP map , can be built up from three quantum thermodynamics from the standpoint of CPTP
fundamental operations, namely: tensor product with an maps, see [86,99,101] for examples.
558 S. VINJANAMPATHY AND J. ANDERS

4.1.2. Entropy inequalities and majorisation β-ordered σ , then the state ρ can be transformed to σ in
We state key results from quantum information theory the resource theoretic setting discussed in Section 5.
relating to entropy often employed in studying thermo-
dynamics. Specifically, we discuss convexity, subadditiv- 4.1.3. Smooth entropies
ity, contractivity and majorisation. We begin by defining Smooth entropies are related to von Neumann entropy
quantum relative entropy as defined in (8) and the conditional entropy used in Section
2.4. Smooth entropies are central in the single shot and
S[ρ∥σ ] := tr[ρ ln ρ − ρ ln σ ]. (37) resource theoretic approaches to quantum thermody-
namics, presented in Section 5.
The negative of the first term is the von Neumann entropy Let us consider a bipartite system, whose parties are
of the state ρ, see Section 2.1. labelled A and B. The smooth entropies are defined in
The relative entropy is# jointly convex in both#its inputs. terms of min- and max-entropies conditioned on B, given
This means that if ρ = m pm ρm and σ = m pm σm , by
with pm probabilities then
Smin (A|B)ρ := sup sup{λ ∈ R : 2−λ IA ⊗ σB ≥ ρAB },
7 8 * σB
S ρ∥σ ≤ pk S[ρk ∥σk ]. (38) (41)
k 7 82
Smax (A|B)ρ := sup ln F(ρAB , IA ⊗ σB ) . (42)
σB
Furthermore, for any tripartite state ρABC defined over
three physical systems labelled A, B and C, the entropies Here, the supremum is over all states σB in the space
√ √
of various marginals are bounded by the inequality subsystem B, and F(x, y) := ∥ x y∥1 , with ∥g∥1 :=
of 9
tr[ g † g] is the fidelity. Note that we retain standard nota-
S(ρABC ) + S(ρB ) ≤ S(ρAB ) + S(ρBC ). (39) tion here, employed, for instance, in standard textbooks
on quantum information theory [112]. The smoothed
This inequality, known as the strong subadditivity in- versions of min-entropy, Smin ε , are defined as the maxi-
equality, is equivalent to the joint convexity of quantum mum over the min-entropy Smin of all states σAB which
relative entropy and is used, for instance in [102], to are in a radius (measured in terms of purified distance
discuss area laws in quantum systems. [113]) of ε from ρAB . Likewise, Smax ε is defined as the
The quantum relative entropy is monotonic, or con- minimum Smax over all states in the radius ε. This can be
tractive, under the application of CPTP maps. For any written as (see Chapter 4 in [23])
two density matrices ρ and σ , this contractivity relation ⎧
[103] is written as ⎪
⎨min Sα (A)ρ̃ , if α < 1
ϵ ρ̃
Sα (A)ρ := (43)
⎪
⎩ max Sα (A)ρ̃ , if α > 1, 0 ≤ ϵ < 1,
S[ρ∥σ ] ≥ S[,(ρ)∥,(σ )]. (40) ρ̃

Contractivity is central to certain extensions of the second where ρ̃ ≈ε ρ are an ε-ball of states close to ρ. We note
law to entropy production inequalities, see [100,104–106] that there is more than one smoothing procedure, and
and is also obeyed by trace distance. There has been recent care needs to be taken to not confuse them [98].
interest in improving these inequality laws in relation to
irreversibility of quantum dynamics [107–110].
4.2. Role of fixed points in thermodynamics
Finally, majorisation is a quasi-ordering relationship
between two vectors. Consider two vectors x⃗ and y⃗, which Fixed points of CPTP maps play an important role in
are n-dimensional. Often in the context of quantum ther- quantum thermodynamics. This role is completely anal-
modynamics, these vectors will be the eigenvalues of ogous to the role played by equilibrium states in equi-
density matrices. The vector x⃗ is said to majorise y⃗, writ- librium thermodynamics. In equilibrium thermodynam-
ten as x⃗ ≻ y⃗, if all partial sums of the ordered vector ics, the equilibrium state is defined by the laws of ther-
# ↓ # ↓
obey the inequality k≤n i xi ≥ k≤n i yi . Here, x⃗ ↓ is the modynamics. If two bodies are placed in contact with
vector x⃗ , sorted in descending order. A related idea is the each other, and can exchange energy, they will eventually
notion of thermo-majorisation [111]. Consider p(E, g) to equilibrate to a unique state, see Section 4.3. On the
be the probability of a state ρ to be in a state g with other hand, if the system of interest is not in thermal
energy E. At a temperature β, such a state is β-ordered equilibrium, notions such as equilibrium, temperature
if eβEk p(Ek , gk ) ≥ eβEk−1 p(Ek−1 , gk−1 ), for k = {1, 2, . . .}. and thermodynamic entropy are ill-defined. In the non-
If two states ρ and σ are such that β-ordered ρ majorises equilibrium setting, [114] proposed a thermodynamic
 CONTEMPORARY PHYSICS 559

framework where the equilibrium state was replaced by use the CP formalism described here to derive fluctuation
the fixed point of the map that generates the dynamics. relations discussed in Section (3.3).
If the steady state of a particular dynamics is not the
thermal state, since there is entropy produced in the
steady state, ‘housekeeping heat’ was introduced as a way 4.3. Thermalisation of closed quantum systems
of taking into account the deviation from equilibrium. One of the fundamental problems in physics involves
This programme, called ‘steady-state thermodynamics’ understanding the route from microscopic to macro-
[114–116], for a classical system described by Langevin scopic dynamics. Most microscopic descriptions of
equation is described in [116] and the connection to physical reality, for instance, are based on laws that are
fluctuation theorems is elaborated. time-reversal invariant [124]. Such symmetries play a
For a quantum system driven by a Lindblad equation strong role in the construction of dynamical descrip-
given in Equation (33), the fixed point is defined as the tions at the microscopic level, but many problems persist
steady state of the evolution. Such a steady state is fixed by when the derivation of thermodynamic laws is attempted
the Hamiltonian and the Lindblad operators that describe from quantum mechanical laws. In this subsection, we
the dynamics. If a system whose steady state is an equi- highlight some examples of the problems considered and
librium state is taken out of equilibrium, the system will introduce the reader to some important ideas. We refer to
relax back to equilibrium. This relaxation mechanism will three excellent reviews [125–127] for a detailed overview
be accompanied by a certain entropy production, which of the field.
ceases once the system stops evolving (this can be an An information theoretic example of the foundational
asymptotic process). In the non-equilibrium setting, let problems in deriving thermodynamics involves the log-
the given dynamics be described by a CPTP map ,. Such ical paradox in reconciling the foundations of thermo-
a map is a sufficient description of the dynamical pro- dynamics with those of quantum mechanics. Suppose
cesses of interest to us. Every such map, , is guaranteed we want to describe the universe, consisting of system
to have at least one fixed point [117], which we refer to S and environment E. From the stand point of quantum
as ρ⋆ . If we compare the relative entropy S[ρ∥ρ⋆ ] before mechanics, then we would assign this universe a pure
and after the application of the map ,, then this quan- state, commensurate with the (tautologically) isolated
tity reduces monotonically with the application of the nature of the universe. This wave function is constrained
CPTP map ,. This is because of contractivity inequality, by the constant energy of the universe. On the other hand,
given by Equation (40). Since the entropy production if we thought of the universe as a closed system, from the
ceases only when the state reaches the fixed point ρ⋆ , we standpoint of statistical mechanics, we assume the axiom
can compare an arbitrary initial state ρ with the fixed of ‘equal a priori probability’ to hold. By this, we mean
point to study the deviation from steady-state dynamics. that each configuration commensurate with the energy
Using this substitution, the contractivity inequality can constraint must be equally probable. Hence, we expect
be rewritten as a difference of von Neumann entropies, the universe to be in a maximally mixed state in the
namely given energy shell. This can be thought of as a Bayesian
7 8 [128] approach to states in a given energy shell. This is in
S[,[ρ]] − S[ρ] ≥ −tr ,[ρ] ln ρ⋆ − ρ ln ρ⋆ := −ς.
(44) contradiction with the assumption made about isolated
quantum systems, namely: that states of such isolated
This result [104,118] states that the entropy production systems are pure states. In [7], the authors resolved this by
is bounded by the term on the right-hand side, a factor pointing out that entanglement is the key to understand-
which depends on the initial density matrix and the map. ing the resolution to this paradox. They replace the axiom
If N maps ,i are applied in succession to an initial density of equal a priori probability with a provable statement
matrix ρ, the total entropy change can be shown easily known as the ‘general canonical principle’. This principle
[105] to be bounded by can be stated as follows: if the state of the universe is a
N
pure state |φ⟩, then the reduced state of the system
*
S[,[ρ]] − S[ρ] ≥ − ςk . (45)
ρS := trE [|φ⟩⟨φ|] (46)
k=1

Here, , is the concatenation of the maps , := ,N ◦ is very close (in trace distance, see Section 4) to the state
,N−1 ◦ . . . ◦ ,1 . We refer the reader to [70,105,119–121] 4S for almost all choices of the pure state |φ⟩. The phrase
for a discussion of the relationship between steady states ‘almost all’ can be quantified into the notion of typicality,
and laws of thermodynamics and [122] for a discussion see [7]. The state 4S is the canonical state, defined as
on limit cycles for quantum engines. In [123], the authors 4S := trE (E), the partial reduction of the equiprobable
560 S. VINJANAMPATHY AND J. ANDERS

state E corresponding to the maximally mixed state in a to thermalise if this represents a model of thermalisation.
shell (or subspace) corresponding to a general restriction. But from the axioms of the micro-canonical ensemble,
An example of such a restriction is the constant energy the
# expectation value is expected to be proportional to
restriction, and hence the ‘shell’ corresponds to the sub- k Bkk , in keeping with equal ‘a priori probability’ about
space of states with the same energy. Since this applies to the mean energy E0 . This # is a source of mystery since
any restriction, and not just the energy shell restriction the long time average k |ck |2 Bkk somehow needs to
we motivated our discussion with, the principle is known ‘forget’ about the initial state of the system |φ⟩, given
as the general canonical principle. by coefficients ck .
The intuition here is that almost any pure state of This can happen in three ways. The first mechanism is
the universe is consistent with the system of interest to demand that Bkk be non-zero only in the narrow band
being close to the canonical state, and hence the axiom discussed above and come outside the sum. This is called
of equal a priori probability is modified to note that the ‘eigenstate thermalisation hypothesis’ (ETH) [132,133],
system cannot tell the difference between the universe see [26] for more details on ETH. The second method
being pure or mixed for a sufficiently large universe. We to reconcile the long time average with the axiom of the
emphasise that though almost all pure states have a lot of micro-canonical ensemble is to demand that the coeffi-
entanglement across any cut (i.e. across any number of cients ck can be constant and non-zero for a subset of
parties you choose in each subsystem and any number of indices k. The third mechanism for the independence of
subsystems you divide the universe into), not all states the long time average of expectation values of observ-
do. A simple example of this is product states of the ables, and hence the observation of thermalisation in a
form |ψ1 ⟩ ⊗ |ψ2 ⟩ . . . |ψN ⟩. Entanglement between the closed quantum system is to demand that the coefficients
subsystems is understood to play an important role in ck be uncorrelated to Bkk . This causes ck to uniformly
this kinematic (see below) description of thermalisation, sample Bkk the average given simply by the mean value,
leading to the effect that sufficiently small subsystems in agreement with the micro-canonical ensemble. ETH
are unable to distinguish highly entangled states from was numerically verified for hardcore bosons in [133]
maximally mixed states of the universe, see [15,129,130] and we refer the interested reader to a review of the field
for related works. in [25], see [134,135] for a geometric approach to the
This issue of initial states is only one of the many topics discussion of thermodynamics and thermalisation.
of investigation in this modern study of equilibration and An integrable system is defined as a system where the
thermalisation [131]. When thermalisation of a quantum number of conserved local quantities grows extensively.
system is considered, it is generally expected that the Here, the notions of local and global are with respect to
system will thermalise for any initial condition of the the constituent subsystems. We write local to differen-
system (i.e. for ‘almost all’ initial states of the system, tiate quantities such as eigenstates of the Hamiltonian,
the system will end up in a thermal state) and that many which commute with the Hamiltonian but are usually
(but perhaps not all) macroscopic observables will ‘ther- global. Consider the eigenvalues of such an integrable
malise’ (i.e. their expectation values will saturate to values system, which are then perturbed with a non-integrable
predicted from thermal states, apart from infrequent re- Hamiltonian perturbation. Since equilibration happens
currences and fluctuations). Let us consider this issue of by a system exploring all possible transitions allowed
the thermalisation of macroscopic observables in a closed between its various states, any restriction to the set of all
quantum system. # Let the initial state of the system be allowed transitions will have an effect on equilibration.
given by |φ⟩ = k ck |6k ⟩, where the |6k ⟩ are the energy Since there are a number of conserved local quantities
eigenstates of the Hamiltonian, with energies assumed implied by the integrability of a quantum system, such a
non-degenerate. The corresponding coefficients ck are quantum system is not expected to thermalise. (We note
non-zero only in a small band around a given energy that this is one of the many definitions of thermalisation
E0 . If we consider an observable B, its expectation value and it has been criticised as being inadequate [136], see
in the time evolving state is given by also Section 9 of [26] for a detailed criticism of various
definitions of thermalisation). On the other hand, the
*
⟨φ|eiHt Be−iHt |φ⟩ := ⟨B⟩ = |ck |2 Bkk + absence of integrability is no guarantee for thermalisation
k [137]. Furthermore, if an integrable quantum system is
* perturbed so as to break integrability slightly, then the
∗
cm ck ei[Em −Ek ]t Bkm , (47)
various invariants are assumed to approximately hold.
km
Hence, the intuition is that there should be some gap
where Bkm := ⟨6m |B|6k ⟩. The long time average of this (in the perturbation parameter) between the breaking of
expectation value is given by the first term and is expected integrability and the onset of thermalisation. This phe-
 CONTEMPORARY PHYSICS 561

Figure 6. Energy levels are indicated by the black lines, populations of levels are indicated as red columns. (a) LTs of the Hamiltonian
change only the energy levels while the populations of the levels remain the same. The energy levels can be chosen such that the
initial probabilities correspond to thermal probabilities for a given temperature. (b) ‘Thermalisation’ of the population with respect to
the energy levels resulting in the Gibbs state. (c) An isothermal reversible transformation (ITR) changing both the Hamiltonian’s energy
levels and the state population to remain thermal throughout the process. (Figures taken from Nat. Commun. 4, 1925 (2013).) (d) No
(0) (1)
level transfer is assumed during LTs with the energy change from level En corresponding to initial Hamiltonian H = H (0) to En
corresponding to changed Hamiltonian H associated with single shot work. Level transfers during thermalisation are assumed to end
(1)
(1) (1)
with thermal probabilities in each of the energy eigenstates Em1 of the same Hamiltonian H (1) , with the energy difference between En
(1)
and Em1 identified as single shot heat. An ITR is a sequence of infinitesimal small steps of LTs and thermalisations, here, resulting in the
(1)
system in an energy EmN after N − 1 steps.

nomenon is called pre-thermalisation, and owing to the thermalisation steps where the Hamiltonian is held fixed
invariants, such systems which are perturbed out of in- and the system equilibrates with a bath at temperature
tegrability settle into a quasi-steady state [138]. This is a T to a thermal
# state [142]. Let the initial Hamiltonian
(0) (0) (0) (0)
result of the interplay between thermalisation and inte- be H = k,gk Ek |Ek , gk ⟩⟨Ek , gk | where Ek are the
grability [139] and was experimentally observed in [140]. energy eigenvalues and |Ek(0) , gk ⟩ the corresponding de-
generate energy eigenstates, labelled with the degeneracy
(0)
5. Single shot thermodynamics index gk . For the system starting with energy En , the
(0) (1)
energy change during LTs, En → En see Figure 6(d), is
The single shot regime refers to operating on a single entirely associated with single shot work while the energy
quantum system, which can be a highly correlated system (1) (2)
change, Em 1 → Em2 see Figure 6(d), for thermalisation
of many subsystems, rather than on an infinite ensemble
steps is associated with single shot heat [142]. See also
of identical and independently distributed copies of a
the definition of average work and average heat in Equa-
quantum system, often referred to as the i.i.d. regime
tion (2) associated with these processes for ensembles
[96]. For long, the issue of whether information theory
[32]. Through a sequence of LTs and thermalisations, see
and hence physics can be formulated reasonably in the
Figure 6(a)–(c), it is found that the single shot random
single shot regime has been an important open question
work yield of a system with diagonal distribution ρ and
which has only been addressed in the last decade, see e.g.
Hamiltonian H is [142]
[141]. An active programme of applying quantum infor- rn
Wyield = kB T ln . (48)
mation theory techniques, in particular the properties of tn
CPTP maps discussed in Section 4, to thermodynamics
is the extension of the laws and protocols applicable to Here, rn and tn are energy-level populations of ρ and the
−βH
thermodynamically large systems to ensembles of finite thermal state, τ := e Z , respectively, which are both
size that in addition may host quantum properties. Finite diagonal in the Hamiltonian’s eigenbasis. The particular
size systems typically deviate from the assumptions made index n appearing here refers to the initial energy state
in the context of equilibrium thermodynamics and there |En(0) , gn ⟩ that the system happens to start with in a ran-
has been a large interest in identifying limits of work dom single shot, see Figure 6(d). Averaging the single
extraction from such small quantum systems. shot work in Equation (48) with the probability rn of
obtaining it, gives the average work yield A(ρ, H) =
F(ρ) − F(τ ) as expected from Equation (7), where the
5.1. Work extraction in single shot thermodynamics
free energy F is defined in Equation (6).
One single shot thermodynamics approach to analyse Deterministic work extraction is rarely possible in the
how much work may be extracted from a system in a single shot setting and proofs typically allow a non-zero
classical, diagonal distribution ρ with Hamiltonian H in- probability ϵ of failing to extract work in the construction
troduces level transformations (LTs) of the Hamiltonian’s of single shot work. The ϵ-deterministic work content
energy eigenvalues while leaving the state unchanged and Aϵ (ρ, H) is found to be [142]
562 S. VINJANAMPATHY AND J. ANDERS

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

recent contributions include references [9,111,142,144–

Z7∗ (ρ,H)
Aϵ (ρ, H) ≈ F ϵ (ρ, H) − F(τ ) := −kB T ln , 148]. Resource theories in quantum information identify
Z a set of restrictive operations that can act on ‘valuable
(49)
resource states’. For a given initial state, these restrictive
with the free energy difference defined through a ratio of a operations then define a set of states that are reachable.
subspace partition function Z7∗ over the For example, applying stochastic local operations assisted
#full thermal par- by classical communication (SLOCC) on an initial prod-
tition function Z. Here, Z7∗ (ρ, H) := (E,g)∈7∗ e−βE is
the partition function# for the minimal subspace uct state of two parties will produce a restricted, separable
7∗ (ρ, H) := inf {7 : set of two-party states. The same SLOCC operations ap-
(E,g)∈7 ⟨E, g|ρ|E, g⟩ > 1 − ϵ}
defined such that the state ρ’s probability in that subspace plied to a two-party Bell state will result in the entire
makes up the total required probability of 1 − ϵ. state space for two parties, thus the Bell state is a ‘valu-
As an example consider an initial state ρ with probabil- able resource’. In the thermodynamic resource theory,
ities (5/15, 4/15, 1/15, 2/15, 3/15, 0) for non-degenerate the valuable resource states are non-equilibrium states
energies (0E, 1E, 2E, 3E, 4E, 5E) for E > 0 and assume while the restrictive operations include thermal states of
that the allowed error is ϵ = 1/10. The thermal partition auxiliary systems.
function is Z = e−β0 + e−β1E + e−β2E + e−β3E + e−β4E + The thermodynamic resource theory setting involves
e−β5E . The subspace partition function is then Z7∗ = three components: the system of interest, S, a bath B
e−β0 +e−β1E +e−β3E +e−β4E including the thermal terms and a weight (or work storage system or battery) W.
for the most populated energies until the corresponding Additional auxiliary systems may also be included [146]
state probability is at least 1 − 1/10, i.e. p0E,1E,3E,4E = to explicitly model the energy exchange that causes the
14/15 ≥ 1 − 1/10. time dependence of the system Hamiltonian, cf. Section 3,
The proof of Equation (49) employs a variation of and to model catalytic participants in thermal operations,
Crook’s relation, see Section 3.2, and relies on the notion see Section 5.3. In the simplest case, the Hamiltonian at
of smooth entropies, see Section 4.1. Contrasting with the the start and the end of the process is assumed to be the
ensemble situation that allows probabilistic fluctuations same and the sum of the three local terms, H = HS +HB +
P(W) around the average work, see e.g. Equation (17), the HW . Thermal operations are those transformations of the
single shot work Aϵ (ρ, H) discussed here is interpreted system that can be generated by a global unitary, V , that
as the amount of ordered energy, i.e. energy with no acts on system, bath and work storage system initially in a
product state ρS ⊗τB ⊗|EW ini ⟩ ⟨E ini |, where the work stor-
fluctuations, that can be extracted from a distribution W W
age system starts in one of its energy eigenstates, |EW ini ⟩,
in a single shot setting. −βH
and the bath in a thermal state, τB = e ZB , which is
B

considered a ‘free resource’. The non-trivial resource for

5.2. Work extraction and work of formation in the process is the non-equilibrium state of the system, ρS .
thermodynamic resource theory Perfect energy conservation is imposed by requiring that
An alternative single shot approach is the resource the- the unitary may only induce transitions within energy
ory approach to quantum thermodynamics [143], where shells of total energy E = ES + EB + EW , see Figure 7(b).
 CONTEMPORARY PHYSICS 563

This is equivalent to requiring that V must commute with Similar to the asymmetry between distillable entangle-
the sum of the three local Hamiltonians. ment and entanglement of formation, it is also possible
A key result in this setting is the identification of a to derive a work of formation to create a diagonal non-
maximal work [111] that can be extracted from a single equilibrium state ρS which is in general smaller than the
system starting in a diagonal non-equilibrium state ρS to maximum extractable work derived above [111],
the work storage system under thermal operations. The & '
V min formation 1 & '
desired thermal transformation, ρS ⊗ τB ⊗ |0⟩W ⟨0| −→ wϵ = infϵ ln min{λ : ρSϵ ≤ λτS } ,
σSB ⊗ |w⟩W ⟨w|, enables the lifting of the work storage β ρS
system by an energy w > 0 from the ground state |EW ini ⟩ = (51)
|0⟩W to another single energy level |w⟩W , see Figure 7(a).
Here, the system and bath start in a product state and may where ρSϵ are states close to ρS , ||ρSϵ − ρS || ≤ ϵ with || · ||
end in a correlated state σSB . The perfect transformation the trace norm.
can be relaxed by requesting the transformation to be
successful with probability 1 − ϵ, see Section 5.1. The 5.3. Single shot second laws
maximum extractable work in this setting is then [111,
149] Beyond optimising work extraction, a recent paper iden-
* tifies the set of all states that can be reached in a single shot
1
wϵmax = − ln t(ES ,gS ) hρS (ES , gS , ϵ) by a broader class of thermal operations in the resource
β
ES ,gS theory setting [146]. This broader class of operations,
=: Fϵmin (ρS ) − F(τS ), (50) EC , is called catalytic thermal operations and involves, in
addition to a system S (which here may include the work
where τS is the thermal state of the system at the bath’s storage system) and a heat bath, B, a catalyst C [151]. The
−βES
inverse temperature β with eigenvalues t(ES ,gS ) = e ZS role of the catalyst is to participate in the operation while
# starting and being returned uncorrelated to system and
and partition function ZS = ES ,gS e−βES . Here, ES and
gS refer to the energy and degeneracy index of the eigen- bath, and in the same state σC ,
states |ES , gS ⟩ of the Hamiltonian HS . hρS (ES , gS , ϵ) is a
binary function that leads to either including or exclud- EC (ρS ) ⊗ σC := trB [V (ρS ⊗ τB ⊗ σC ) V † ]. (52)
ing a particular t(ES , gS ) in the summation, see [111,149]
for details. Comparing with the result in Equation (49) The unitary now must commute with the sum of the three
derived with the single shot approach discussed above, Hamiltonians that the system, bath and catalyst start and
one notices that the # ratio Z7∗ /Z is just the sum given end with, [V , HS + HB + HC ] = 0. Note that the map EC
in (50), i.e. Z7∗ /Z = ES ,gS t(ES ,gS ) hρS (ES , gS , ϵ) and the on the system state depends upon the catalyst state σC .
expressions for the extractable work coincide. For system states ρS diagonal in the energy basis of HS , it
For non-zero ϵ, the derivation can no longer rely on is shown [146] that a state transfer under catalytic thermal
the picture of lifting the work storage system from its operations is possible only under a family of necessary and
ground state to a pure excited state. A more general sufficient conditions,
mixed work storage system state is used in [150] and
a general link between the uncertainty arising from the ρS → ρS′ = EC (ρS )
ϵ-probabilities and the fluctuations in work as discussed ⇔ ∀α ≥ 0 : Fα (ρS , τS ) ≥ Fα (ρS′ , τS ), (53)
in Section 3.5 is derived. This approach recovers the
results, Equations (49) and (50), as the lower bounds on where τS is the thermal state of the system. τS is the
the extractable work. When extending the single-level fixed point of the map EC and the proof makes use of
transitions of the work storage system to multiple levels the contractivity of &map, discussed in Section
' 4.1. Here,
[149], it becomes apparent that this resource theory work Fα (ρS , τS ) := kB T Sα (ρS ||τS ) − ln ZS is a family of free
cannot directly be compared to lifting a weight to a certain energies defined through the α-relative Renyi entropies,
sgn(α) #
height or above. Allowing the weight to rise to an energy Sα (ρS ||τS ) := α−1 ln n rnα tn1−α , applicable for states
level |w⟩W or a range of higher energy levels turns the diagonal in the same basis. The rn and tn are the eigen-
problem into a new optimisation and that would have a values of ρS and τS corresponding to energy eigenstates
different associated ‘work value’, w ′ , contrary to physical |En , gn ⟩ of the system, cf. Equation (48). An extension of
intuition. Due to its restriction to a particular energetic these laws to non-diagonal system states and including
transition, the above resource theory work may very well changes of the Hamiltonian is also discussed [146].
be a useful concept for resonance processes where such a The resulting continuous family of second laws can
restriction is crucial [149]. be understood as limiting state transfer in the single
564 S. VINJANAMPATHY AND J. ANDERS

shot regime in a more restrictive way than the limita-

tion enforced by the standard second law of thermody-
namics, Equation (5), valid for macroscopic ensembles
[146], see Section 2. Indeed, the latter is included in the
family of second # laws: for limα→1 where Sα (ρS ||τS ) →
S1 (ρS ||τS ) = n rn ln rtnn [23], one recovers from Equa-
tion (53) the standard second law,
> ? > ?
* rn *
′ rn′
kB T rn ln − ln ZS ≥ kB T rn ln − ln ZS ,
n
tn n
tn
(54) Figure 8. For a fixed Hamiltonian HS and temperature T , the
* * * manifold of time symmetric states, indicated in grey, including
rn ln rn − rn′ ln rn′ ≥− (rn′ − rn ) ln tn ,
the thermal state, γ ≡ τS , as well as any diagonal state, D(ρ) ≡
n n n
DHS (ρS ), is a submanifold of the space of all system states (blue
(55) oval). Under thermal operations, an initial state ρ ≡ ρS moves
#S ≥ β#U = β⟨Q⟩, (56) towards the thermal state in two independent ‘directions’. The
thermodynamic purity p measured by Fα and the asymmetry a
with no work contribution, ⟨W⟩ = 0, as no explicit measured by Aα must both decrease. (Figure taken from Nat.
Commun. 6, 6383 (2015).)
source of work was separated here (although this can
be done, see Section 5.2). Moreover, in the limit of large
ensembles and for systems that are not highly correlated, the contractivity of the relative entropy with respect to
the Renyi free energies reduce to the standard Helmholtz the map E, see Section 4.1, and its commuting with D. It
free energy, Fα (ρS , τS ) ≈ F1 (ρS , τS ) for all α, providing is noted that (57) provides necessary conditions only – it
a single shot explanation of why Equation (5) is the only is not known if the conditions on the Aα (ρS′ ) are sufficient
second law relation for macroscopic ensembles. for a thermal operation to exist that turns ρS into ρS′ .
Allowing the catalyst in Equation (52) to be returned The conditions (57) mean that thermal operations on
after the operation not in the identical state, but a state a single copy of a quantum state with coherence must
close in trace distance, leads to the rather unphysical puz- decrease their coherence, just like free energy must de-
zle of thermal embezzling: a large set of transformations crease for diagonal non-equilibrium states. The two fam-
are allowed without the usual second law restrictions. ilies of second laws (53) and (57) can be interpreted
This puzzle has been tamed by recent results showing that geometrically as moving states in state space under ther-
when physical constraints, such as fixing the dimension mal operations ‘closer’(the Renyi divergences Sα are not
and the energy-level structure of the catalyst, are incorpo- proper distance measures [23]) to the thermal state in
rated, the allowed state transformations are significantly two independent ‘directions’, in thermodynamic purity
restricted [152]. and time asymmetry [148], see Figure 8. The decrease of
coherence implies a quantum aspect to irreversibility in
the resource theory setting [153]. Again, when taking the
5.4. Single shot second laws with coherence single shot results to the macroscopic limit, ρ → ρ ⊗N ,
Recently, it was discovered that thermal operations, conditions (57) become trivial and the standard second
E(ρS ) := trB [V (ρS ⊗ τB ) V † ], on an initial state ρS that law (5) is recovered as the only constraint.
has coherences (i.e. non-zero off-diagonals) with respect
to the energy eigenbasis of the Hamiltonian HS , require 6. Quantum thermal machines
a second family of inequalities to be satisfied [148], in
Until now, we have discussed fundamental issues in
addition to the free energy relations for diagonal states
quantum thermodynamics. Thermodynamics is a field
(53). The derived necessary conditions are
whose focus from the very beginning has been both fun-
ρS → ρS′ = E(ρS ) ⇒ ∀α ≥ 0 : Aα (ρS ) ≥ Aα (ρS′ ), damental as well as applied. Applications of thermody-
(57) namics are mostly in designing thermal machines, which
extract useful work from thermal baths. Examples of this
where Aα (ρS ) = Sα (ρS ||DHS (ρS )) and DHS (ρS ) is the are engines and refrigerators. In Section 2, we discussed
operation that removes all coherences between energy work that can be extracted from correlations and from
eigenspaces, i.e. DH (ρ) ≡ η in Section 2.6. The Aα are coherences in the energy eigenbasis. These coherences
coherence measures, i.e. they are coherence monotones and correlations hence affect cyclical and non-cyclical
for thermal operations. The proof of (57) relies again on machine operations and contribute towards the quantum
 CONTEMPORARY PHYSICS 565

features of thermal machine design. Here, we will present From standard thermodynamics, it is well known that
some of the recent progress in the design of quantum the most efficient engines also output zero power since
thermal machines that apply these aforementioned prin- they are quasi-static. Hence, for the operation of a real
ciples, which generalise classical machines. We will focus engine, other objective functions such as power have to
on two main types of QTMs. The first are cyclical ma- be optimised. The seminal paper by Curzon and Ahlborn
chines which are quantum generalisations of engines and [163] considered the issue of the efficiency of a classical
refrigerators. The second type are non-cyclical machines, Carnot engine operating between two temperatures, cold
which highlight important aspects of QTMs such as work TC and hot TH , at maximum power, which they found to
extraction, power generation and correlations between be
subsystems. @
TC
ηCA = 1 − . (59)
TH
6.1. Quantum thermal machines
The study of the efficient conversion of various forms of This limit is also reproduced with quantum working flu-
energy to mechanical energy has been a topic of interest ids, as discussed in Section 6.8. We note that the afore-
for more than a century. QTMs, defined as quantum mentioned formula for the efficiency at maximum power
machines that convert heat to useful forms of work, have is sensitive to the constraints on the system, and changing
been a topic of intense study. Engines, heat-driven re- the constraints changes this formula. For instance, see
frigerators, power-driven refrigerators and several other [164,165] for results generalising the Curzon–Ahlborn
kinds of thermal machines have been studied in the quan- efficiency to stochastic thermodynamics and to quantum
tum regime, initially motivated from areas such as quan- systems with other constraints.
tum optics [10,11]. QTMs can be classified in various The rest of this section is a brief description of how
ways. Dynamically, QTMs can be classified as those that engines are designed in the quantum regime. We consider
operate in discrete strokes, and those that operate as three examples, namely: Carnot engines with spins as
continuous devices, where the steady state of the con- the working fluid, harmonic oscillator Otto engines and
tinuous time device operates as a QTM. Furthermore, Diesel engines operating with a particle in a box as the
such continuous engines can be further classified as those working fluid. These examples were chosen to illustrate
that employ linear response techniques [14] and those the design of a variety of engines operating with a variety
that employ dynamical equation techniques [24]. These of quantum working fluids.
models of quantum engines are in contrast with biological
motors that extract work from fluctuations and have been
studied extensively [154]. 6.2. Carnot engine
Engines use a working fluid and two (or more) reser-
Classically, the Carnot engine consists of two sets of
voirs to transform heat to work. These reservoirs model
alternating adiabatic strokes and isothermal strokes. The
the bath that inputs energy into the working fluid (hot
quantum analogue of the Carnot engine consists of a
bath) and another that accepts energy from the working
working fluid, which can be a particle in a box [166],
fluid (cold bath). Two main classes of quantum systems
qubits [18,159], multiple-level atoms [157] or harmonic
have been studied as working fluids, namely: discrete
oscillators [155,156]. We emphasise that for all such en-
quantum systems and continuous variable quantum sys-
gines, the efficiency of the engine is strictly bounded by
tems [155–158]. This study of continuous variable sys-
the Carnot efficiency [167]. For the engine consisting of
tems complements various models of finite level-systems
non-interacting qubits considered in [159], the Carnot
[159,160] and hybrid models [161,162] studied as quan-
cycle consists of
tum engines. Both continuous variable models of engines
and finite dimensional heat engines have been theoreti- (1) Adiabatic Expansion: An expansion wherein the
cally shown to be operable at theoretical maximal effi- spin is uncoupled from any heat baths and its
ciencies. Such efficiencies are only well defined once the frequency is changed from ω1 to ω2 > ω1 adia-
engine operation is specified. batically. Work is done by the spin in this step due
The efficiency of any engine is given by the ratio of the to a change in the internal energy. Since the spin is
net work done by the system to the heat that flows into uncoupled, its von Neumann entropy is conserved
the system, namely in this expansion stroke.
(2) Cold Isotherm: The spin is coupled to a cold bath at
⟨Wnet ⟩ inverse temperature βC . This transfers heat from
η= . (58)
⟨Qin ⟩ the engine to the cold bath.
566 S. VINJANAMPATHY AND J. ANDERS

(3) Adiabatic Compression: A compression stroke (2) Hot Isochore: At frequency ω2HO , the harmonic
where the spin is uncoupled from all heat baths oscillator working fluid is coupled to a bath whose
and its frequency is changed from ω3 to ω4 adi- inverse temperature is βH , and is allowed to relax
abatically. Work is done on the medium in this to the new thermal state.
step. (3) Adiabatic Expansion: In this stroke, the medium is
(4) Hot Isotherm: The spin is coupled to a hot bath at uncoupled from any heat baths and its frequency
inverse temperature βH < βC . This transfers heat is changed from ω2HO to ω1HO .
to the engine from the hot bath. (4) Cold Isochore: At frequency ω1HO , the harmonic
oscillator working fluid is coupled to a bath whose
The inverse temperature β ′ of the thermal state of this
inverse temperature is βC and it is allowed to relax
working fluid is given at any point (ω, S) on the cycle, by
back to its initial thermal state.
the magnetisation relation S = − tanh (β ′ ω/2)/2. The
dynamics is described by a Lindblad equation, which will In the two strokes where the frequencies change, work
then be used to derive the behaviour of heat currents. The is exchanged. In the two thermalizing strokes on the other
engine cycle is described in Figure 9. The Hamiltonian hand, only heat is exchanged. Hence the efficiency is
describing the dynamics is given by H(t) = ω(t)σ3 /2. easily calculated to be
The Heisenberg equation of motion is written in terms
of LD (σ3 ), the Lindblad operators describing coupling to ⟨δW⟩1 + ⟨δW⟩3
the baths at inverse temperatures βH and βC , respectively. ηOtto = − . (62)
⟨δQ⟩2
This equation can be used to calculate the expectation
value of the Hamiltonian ⟨H(t)⟩, which leads to
The experimental implementation of such a quantum
$ %
d⟨H(t)⟩ 1 dω Otto engine was considered in [168,171] and is presented
= ⟨σ3 ⟩ + ω⟨LD (σ3 )⟩
dt 2 dt in Figure 10. The Paul or quadrupole trap uses rapidly
$ % oscillating electromagnetic fields to confine ions using
1 dω d⟨σ3 ⟩
= ⟨σ3 ⟩ + ω . (60) an effectively repulsive field. The ion, trapped in the
2 dt dt
modified trap, presented in Figure 10 is initially cooled
As described in Section 2, the definition of work ⟨δW⟩ = in all spatial directions. The engine is coupled to hot
⟨σ3 ⟩δω/2 and heat ⟨δQ⟩ = ωδ⟨σ3 ⟩/2 emerges naturally and cold reservoirs composed of blue and red detuned
from the above discussion, and is identified with the laser beams. The tapered design of the trap translates to
time derivative of the first law. For this quantum Carnot an axial force that the ion experiences. A change in the
engine, the maximal efficiency which is that of the temperature of the radial state of the ion, and hence the
reversible engine [159] is given by the standard formula width of its spatial distribution, leads to a modification of
namely the axial component of the repelling force. Thus, heating
and cooling the ion move it back and forth along the
TC trap axis, as induced in the right-hand side of the figure.
ηCarnot = 1 − . (61)
TH The frequency of the oscillator is controlled by the trap
parameters. Energy is stored in the axial mode and can
6.3. Otto engine be transferred to other systems and used.
Since we have considered an Otto engine operating be-
As a counterpoint to the Lindblad study of Carnot
tween two thermal baths, we should not be surprised that
engines implemented via qubits, let us consider the har-
the standard formula for efficiency still applies.
monic oscillator implementation of Otto engines [12,158,
Making one of the components of the engine genuinely
168–170] whose time-dependent frequency of the har-
non-thermal (and quantum) makes the analysis more
monic oscillator working fluid is ωHO (t). The initial state
interesting. In [171], the authors show that the stan-
of the oscillator is a thermal state at inverse temperature
dard Carnot efficiency can be overcome by squeezing
βC , and the oscillator frequency is ω1HO . The four strokes
the thermal baths. In [172], the authors derive a ‘gen-
of the Otto cycle are given by:
eralised’ Carnot efficiency that correctly accounts for the
(1) Adiabatic Compression: An compression, where first and second laws. Using this analysis, they demon-
the medium is uncoupled from all heat baths and strate that though the engine efficiency exceeds the stan-
its frequency is changed from ω1HO to ω2HO > ω1HO . dard Carnot formula, the ‘generalized’ efficiency is not
Since the oscillator is uncoupled, its von Neumann exceeded, in keeping with the laws of thermodynamics
entropy is conserved during this stroke, though it [173]. We note that the cost of squeezing the bath has not
no longer is a thermal state. been accounted for, similar to how the cost of preparing a
 CONTEMPORARY PHYSICS 567

Figure 9. The left figure is a reversible Carnot cycle, operating in the limit ω̇ → 0, depicted in the space of the normalised magnetic
field ω and the magnetisation S. The horizontal lines represent adiabats wherein the engine is uncoupled from the heat baths and the
magnetic field is changed between two values. The lines connecting the two horizontal strokes constitute changing the magnetisation
by changing ω while the qubits are connected to a heat bath at constant temperature. We note that in the figure, S1,2 < 0 represent
two values of the magnetisation, with S1 < S2 . Figure taken from [159]. The right figure is a particle in a box with engine strokes being
defined in terms of expansion and contraction of the box. This defines a quantum isobaric process, by defining force. This definition is
used to define a Diesel cycle in the text. We note that in the figure force is denoted by F, while it’s denoted by F in the text. Figure taken
from [175]. Figures reproduced with permission.

cold bath is unaccounted for in classical thermodynamics be employed to construct a Diesel cycle, presented in
(it is assumed that the baths are free resources). Figure 9, and whose four strokes are given by
6.4. Diesel engine (1) Isobaric Expansion: The expansion of the walls of
Finally, we discuss Diesel cycles designed with a particle the particle in a box happen at constant pressure.
in a box as the working fluid [175]. The classical Diesel The width of the walls goes from L2 to L3 > L1 .
engine is composed of isobaric strokes, where the pres- (2) Adiabatic Expansion: An adiabatic expansion pro-
sure is held constant. We will discuss how the notion cess, wherein the length goes from L3 to L1 > L3 .
of ‘pressure’ is generalised to the quantum setting and Entropy is conserved in this stroke.
how this defines a ‘quantum isobaric’ stroke. To define (3) Isochoric Compression: The compression happens
pressure, we begin by defining force. After defining av- at constant volume L1 where the force on the box
erage work and heat as before, generalised forces can be is reduced.
defined by analogy with classical thermodynamics using (4) Adiabatic Compression: The compression process
the relation takes the box from volume L1 to L2 < L1 . This is
⟨δW⟩ done by isolating the quantum system, conserving
Yn = , (63) the entropy in the process.
δyn
Like before, the efficiency is calculated by considering
where yn , Yn form the
# nth conjugate pair in the definition the ratio of the net work done by the system to the
of work ⟨δW⟩ = n Yn δyn . Since work is defined for heat into the system. A straightforward calculation of this
such continuous machines as ⟨δW⟩ = tr[ρ δH] [176], efficiency yields
for eigenstates with energies En distributed according to
a probability distribution Pn , the force F is given by 1
* δEn ηDiesel = 1 − (rE2 + rE rC + rC2 ). (65)
3
F= Pn . (64)
δL
n Here, rE = L3 /L1 is the expansion ratio and rC = L2 /L1
is the compression ratio.
If the system is in equilibrium with a heat bath at
inverse temperature β, with the corresponding free en-
ergy being F = − log (Z)/β, then the force is given by
6.5. Quantumness of engines
the usual formula F = −[dF/dL]β , evaluated at con-
stant temperature β −1 . This force is calculated to be Since the design of quantum engines is often analogous
F = (βL)−1 . Hence, to execute an ‘isobaric process’, to their classical counterparts, a central question is: What
wherein the pressure is held constant across a stroke, are genuinely quantum ingredients for engines? There
the temperature must vary as β = (FL)−1 . This can are two important points to consider here: the first point
568 S. VINJANAMPATHY AND J. ANDERS

Figure 10. Experimental proposal of an Otto engine consisting of a single trapped ion, currently being built by K. Singer’s group [174].
On the left, the energy frequency diagram corresponding to the Otto cycle implemented on the radial degree of freedom of the ion.
The inset on the left-hand side is the geometry of the Paul trap. On the right-hand side, a representation of the four strokes, namely
(1) adiabatic compression, (2) hot isochore, (3) adiabatic expansion and (4) cold isochore, see text for details. Figures taken from [171]
reproduced with permission.

is that when thermal machines operate between two heat speed limit [182]. The quantum speed limit places a limit
baths with well-defined temperatures, the efficiency of on the power of quantum engines, and has been used to
such a machine is also limited by the classical efficiencies, improve engine performance [180]. Another approach
i.e. the Carnot and Curzon–Ahlborn efficiencies. The to time in quantum thermodynamics employs so-called
Carnot efficiency, for instance, is derived independent of adiabatic shortcuts to improve engine performance in
the details of the working fluid and depends only on the both the classical [169] and quantum regime [170].
laws of thermodynamics. It is expected that these bounds An adiabatic quantum system that starts in an eigen-
must also hold in the quantum setting. state of the Hamiltonian remains in the instantaneous
Once the thermal machines are committed to operate eigenstate of the Hamiltonian when the parameters of
between two temperatures and in equilibrium, not much the Hamiltonian are swept slowly enough (distinguish
can be done to change the efficiency of these machines this from the thermodynamic definition of adiabatic-
(see [177] for a recent example). Hence, the deviation ity, which relates to having zero heat exchange). This
from classical performance of these thermal machines is typically slow dynamics can be sped up both in the clas-
only seen when either the baths are made non-thermal or sical and quantum contexts by the addition of exter-
the working fluid of the thermal machine is not allowed nal control fields that are bound to be transitionless.
to equilibrate, operating in a non-equilibrium setting. Finally, we note that more foundational aspects of quan-
To characterise such non-equilibrium thermal machines, tum mechanics, such as non-commutativity of operators,
several authors have studied the role of quantum corre- have been shown to have a detrimental effect on engine
lations [178] and the role they play in work extraction. performance. This is known as quantum friction and
Using non-thermal heat baths [171,179] has been is also a genuinely quantum effect that impacts engine
another strategy to see the effect of quantum states on performance [183]. Besides issues such as non-thermal
thermal machines. An example of such a study focuses on baths and non-commutativity, we refer the reader to
Otto engines operated with squeezing thermal baths we [184] for an information theoretic perspective on engine
discussed in the previous subsection. In all such design.
examples, non-classical resources are employed to imp To answer the question about the quantumness of
rove engine performance, from power generation [170, engines, recent studies have focused on trying to study
180] to efficiency [168,171,172]. Finally, we point out that alternative non-quantum models (to compare and con-
time has an important role to play in engine performance trast with quantum engines) or produce genuine quan-
since it relates to power. As detailed in Section 6.7, both tum effects to demonstrate the quantumness of engines.
coherence and time play important roles in the design of On the difference between quantum mechanics and
quantum machines [58,59,181]. As an example, consider stochastic formulation of thermodynamics, in [185], the
that the evolution of an initial state to a final state cannot authors study quantumness of engines and show that
happen faster than a fundamental bound that depends on there is an equivalence between different types of en-
the Hamiltonian and states involved. The minimum time gines, namely: two-stroke, four-stroke and continuous
associated with this bound is often called the quantum engines. Furthermore, the authors define and discuss
 CONTEMPORARY PHYSICS 569

quantum signatures in thermal machines by comparing COP can be larger than one, and in [194], the authors
these machines with equivalent stochastic machines, and show the relationship of the COP to the cooling rate,
they demonstrate that quantum engines operating with which is simply the rate at which heat is transported from
coherence can in general output more power than their the system.
stochastic counterparts. Finally, we note an example of a The other example of quantum refrigerators takes a
QTM [186] operating in the continuous time regime that more quantum information theoretic point of view. In
produces entanglement. The working fluid considered by [18], the authors were inspired by algorithmic cooling
the authors consists of two qubits, which are coupled to and studied the smallest refrigerators possible. One of
two different baths and to each other. The steady state of the models consists of a qubit coupled to a qutrit, see
this thermal machine far from thermal equilibrium (in Figure 11. The intuition for cooling the qubit comes
the presence of a heat current) exhibits the interesting from a computational model of entropy reduction. This
property of creating bipartite entanglement in the work- procedure, called algorithmic cooling [17], is a procedure
ing fluid. Such an engine can be said to have a ‘genuinely to cool quantum systems which uses ideas of entropy
quantum’ output. shunting to ‘move’ entropy around a large system in a
way that lowers the entropy of a subsystem. Consider n
copies of a quantum system which is not in its maximal
entropy state. Joint unitary operations on the n copies can
6.6. Quantum refrigerators allow for the distillation of a small number m of systems
Refrigerators are engines operating in a regime where which are colder (in this context, of lower entropy) than
the heat flow is reversed. Like engines, the role that quan- before, while leaving the remaining (n − m) systems in a
tised energy states, coherence and correlations play in the state that is hotter than before (this is required by unitar-
operation of a quantum refrigerator have been fields of ity). This is the intuition behind cooling of the qubit in
extensive study [24,187–192]. Figure 11. The refrigerator in [18] is the smallest (in
We present two examples of refrigerator cycles consid- Hilbert space dimensionality) refrigerator possible. See
ered in the literature. The first of these [193] is built on a [187] for a discussion of the relationship of power to
three-level atom, coupled to two baths, as shown in Fig- COP and [187] for a discussion of correlations in the
ure 11. The refrigerator is driven by coherent radiation. design of quantum absorption refrigerators. Finally, we
This induces transitions between level 2 and level 3. The note that there is a connection between models of cooling
population in level 3 then relaxes to level 1 by rejecting based on study of heat flows of continuous models, such
heat to the hot bath at temperature TH . The system then as sideband cooling, and quantum information theoretic
transitions from level 1 to level 2 by absorbing energy models of cooling based on control theory [195–197].
from a cold bath. Like before, the dynamics can be written
in terms of the Heisenberg equation for an observable A.
Let LC,H refer to the Lindblad operators corresponding to 6.7. Coherence and time in thermal machines
the cold and hot baths, respectively. Since this is a thermal In this subsection, we want to briefly discuss the role
machine operated continuously in time, the steady-state coherence and time play in QTMs. The fundamental
energy transport is the measure of heat transported by the starting point of finite time classical [198] and quan-
machine. Assuming the Hamiltonian H = H0 +V (t), the tum thermodynamics is the fact that the most efficient
energy transport can be written as macroscopic processes are also the ones that are quasi-
A B static and hence slow. Hence, they are impractical from
d⟨H⟩ ∂V (t) the standpoint of power generation. The role of quantum
= + ⟨LH (H)⟩ + ⟨LC (H)⟩. (66)
dt ∂t coherences was illustrated by a series of studies involv-
ing commutativity of parts of the Hamiltonian, as ex-
Using this, the authors in [193] investigated the refrig- plained below. Consider a quantum Otto engine, whose
erator efficiency. This efficiency is expressed in terms of energy balance equation is given in terms of the Hamil-
the coefficient of performance (COP), which is defined tonian part Hext + Hint and dissipative Lindblad oper-
as the ratio of the cooling energy to the work input into ator L [199]. There are two sources of heat that were
the system. For two thermal baths at temperatures TC studied in this model, and they are related to coher-
and TH , respectively, the universal reversible limit is the ence and time. The first of these sources of heat are
Carnot COP, given by attributed to energy transfer from the hot bath to the cold
bath via the system. The second source of heat comes
TC from the finite time driving of the adiabatic strokes of
COP = . (67)
T H − TC the engine. Such a finite time driving means that the
570 S. VINJANAMPATHY AND J. ANDERS

Figure 11. Two designs of refrigerators, the details of which are discussed in the text. The left figure consists of a three-level atom
coupled to two baths and interacting with a field, figure taken from [193]. The right figure consists of the smallest refrigerator, composed
of a qubit and a qutrit. Here, cooling of the qubit is achieved through joint transformations between the system and the reservoir,
wherein the system’s entropy is shunted to the reservoir. Figure taken from [18]. Figures reproduced with permission.

system causes irreversible heating which is understood as A non-cyclical example of the use of quantum correla-
follows. Suppose the initial state of the system is an eigen- tions to improve machine performance involves quan-
state of Hext . If [Hext , Hint ] ̸ = 0, then the quantum state, tum batteries. These are quantum work storage devices
even in the absence of Lindblad terms, does not ‘follow’ and their key issues relate to their capacity and the speed
the instantaneous eigenstate of the external Hamiltonian. at which work can be deposited in them. This question of
The precession of the quantum state of the system about work deposition can be studied in two steps. Firstly, it is
this instantaneous eigenstate, that is induced by the non- desirable to understand the limits on the work extractable
commutativity of the external and internal Hamiltonians, from a quantum system under unitary transformations,
is a source of a friction like heat [183]. V. Here, consider states to be written in an ordered eigen-
# ↑ ↑ ↑ # ↓ ↓ ↓
Finally, we present another model of a non-thermal basis as ρ = k rk |rk ⟩⟨rk | and H = k ϵk |ϵk ⟩⟨ϵk |.
bath consisting of three-level atoms. In [179], the authors ↑
Note that |rk ⟩, the eigenstates of ρ, are a priori unrelated
extract work from coherences using a technique similar ↑
to |ϵk ⟩, the eigenstates of H. The arrows show the order-
to lasing without inversion. The photon Carnot engine ↑ ↑
ing of eigenvalues, namely: r0 ≤ r1 . . . and likewise for
proposed by the authors to study the role of coherences the Hamiltonian. The optimal unitary that extracts the
follows by considering a cavity with a single mode ra- most work from ρ is given by the unitary that maps ρ
diation field enclosed in it. The radiation interacts with # ↑ ↓ ↓
to the state π = rk |ϵk ⟩⟨ϵk | [32–34,200]. Such states,
phased three-level atoms at two temperatures during the where the eigenvalues are ordered inversely with respect
isothermal parts of the Carnot engine cycle. The three- to the eigenvectors of the Hamiltonian, are known as
level atoms are phased by making them interact with an ‘passive states’ [33,34] and the corresponding maximum
external microwave source. The efficiency of this engine work extracted from the unitary transformation,
was calculated to be
max
η = ηCarnot − π cos (,), (68) ⟨Wext ⟩ = tr[H(ρ − π)]. (69)

where the phased three-level atoms have an off-diagonal is called ‘ergotropy’. Note, that the thermal states τβ =
coherence term , that is used to improve the Carnot e−βH /tr[e−βH ] for any inverse temperature β are passive.
efficiency. This engine is depicted in Figure 12. Thus, by Ground states of a given Hamiltonian, for instance, are
employing the coherence dynamics of three-level atoms, examples of states passive with respect to the Hamilto-
the authors demonstrate enhancement in work output. nian and correspond to β = ∞. Thermal states maximise
We end this section by noting that quantum features the entropy for a given energy and minimise the energy
are not necessarily beneficial to a QTM’s performance. for a given entropy. Passive states which maximise the
For example, a study of entanglement dynamics in three- energy for a given entropy were sought in [201], and are
qubit refrigerators [191] demonstrates that there is only useful in bounding the cost of thermodynamic processes.
very little entanglement found when the machine is op- Passive states are important to understand the function-
erated near the Carnot limit. ing of quantum engines, see [202] for example.
The role entangling operations play in extracting work
6.8. Correlations, work extraction and power
using unitarys transformations was explored within the
Finally, let us discuss the aspects of work extraction com- simultaneous extraction of work from n quantum bat-
mon to both cyclical and non-cyclical thermal machines. teries [203]. They considered n copies of a given state
 CONTEMPORARY PHYSICS 571

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

σ and a Hamiltonian H for each system. As discussed systems). To remedy this, both classically and quantum
before, σ and H imply that there exists a passive state π. mechanically, engines which optimise power have been
Two strategies for extracting work exist, the first of these considered. These studies find the efficiency at maximum
being to simply process each quantum system separately. power, see Equation (59), first derived in the classical
The second strategy is to do joint entangling operations, context by Curzon and Ahlborn [163] but also valid in the
which transform the initial state σ ⊗n to a final state quantum case [155]. See [170] for a discussion of short-
which has the same entropy as n copies of σ . The authors cuts to adiabaticity and their role in optimising power
considered a Gibbs state τβ where the temperature was in Otto engines and [205] for a calculation of efficiency
fixed by matching the entropy of this state with σ . They at maximum power of an absorption heat pump, which
considered such a state since this state provided a upper also proved the weak dependence of efficiency and power
bound on the amount of work that could be extracted by with dimensionality. In [168,206], the authors consider
transforming σ ⊗n into a passive state. Since the final state the power of Otto cycles in various regimes, and derive
τβ⊗n has the same entropy as n copies of the initial state, the steady-state efficiency
a transformation between σ ⊗n and the neighbourhood √
of τβ⊗n would be the desired transformation that extracts 1 − βH /βC
ηSS = √ . (70)
maximum work. Such a transformation was shown to be 2 + βH /βC
entangling, using typicality arguments [112]. However,
the authors in [204] pointed out that there is always a
6.9. Relationship to laws of thermodynamics
protocol to extract all the work from non-passive states
using non-entangling operations, although in compari- We end the discussion of QTMs with a small report on
son to entangling unitaries, more non-entangling oper- the role that the laws of thermodynamics, discussed in
ations are needed. This suggests a relationship between Section 2, play in constraining design. As was noted in
power and entanglement in work extraction, an assertion the context of engines, the first law emerges naturally as
demonstrated for quantum batteries in [180]. A different a partitioning of the energy of the system in terms of
role can be played by correlations in storing and extract- heat and work. See [101] for a partitioning of internal
ing work, as discussed in Section 2. This role relates energy change for CPTP processes. In the context of
to storing work in correlations. It has been shown by QTMs, for instance, this can be seen in Equation (60).
several authors [58,59,191] that measures of correlations The second law manifests itself usually by inspecting the
like quantum mutual information and entanglement can entropy production of the universe [193]. This is given
affect the performance of thermal machines. for a typical continuous regime QTM, with a system and
A final noteworthy point in the consideration of time two baths as
explicitly in the context of QTMs relates to maximum dS ⟨Q̇H ⟩ ⟨Q̇C ⟩
power engines. Firstly, since the optimal performance of =− − . (71)
dt TH TC
engines corresponds to quasi-static transformations, the
power is always negligible and the time of operation of Analysis of the dynamics shows that the second law en-
one cycle is infinite (or simply much longer compared forces the rule that the net entropy production, given
to the internal dynamical time scales of the quantum by the equation above, is non-negative. See [187,207]
572 S. VINJANAMPATHY AND J. ANDERS

to study the relationship between weak coupling and is starting to emerge. This difference in perspectives has
the second law. Information theoretic approaches often also meant that there are ideas within quantum thermo-
use monotonicity condition of relative entropy to track dynamics where consensus is yet to be established.
entropy production, see section 4. Finally, we note the One example of such disagreement is the definition of
work in [190] relating to the second law and [208] for heat work in the quantum regime. Various notions of work
engine fluctuation relation and experimental proposal, have been introduced in the field, including the average
all in the context of SWAP engines. In [209], the authors work defined for an ensemble of experimental runs in
discuss how in a network of quantum systems coupled to Equation (2), classical and quantum fluctuating work
reservoirs at different temperatures, the second law is vi- defined for a single experimental run in Equations (15)
olated locally, though its always valid globally, in line with and (24), optimal single shot work given in Equation
intuition (see Ref. [100] for a discussion on the violation (48) and optimal thermodynamic resource theory work
of entropic inequalities, see [210] for a discussion on the given in Equation (50). It is reassuring to see that the
role of non-linear couplings in refrigerator design). The latter two, quite separate single shot approaches, result
role of the second law, in all these examples, tends to be to in the same optimal work value. Despite advances in
modify the existing figures of merit, like efficiency [171], unifying these work concepts, our understanding of work
and hence show a path towards understanding nanoscale in the quantum regime remains patchy. For example,
QTMs. resource theory work, Equation (50), is a work associated
The third law states that the entropy of any quantum with an optimised thermal operations process – to move
system goes to a constant as temperature approaches a work storage system almost deterministically as high
zero. This constant is commonly assumed to be zero. This as possible – while the fluctuating work concept, Equa-
means, that in the context of the equation above, the heat tions (15) and (24), is applicable to general closed and
current corresponding to the cold bath, ⟨Q̇C ⟩ ∝ TCα+1 open dynamical processes [64,217]. Thus, they appear
as TC → 0 with α > 0. Since we are discussing the to refer to different types of work, a situation that may
limit of the cold bath temperature going to zero, we be compared to different entanglement measures each
expect that the third law will inform the operation of of importance for a different quantum communication
quantum refrigerators and heat engines, where the cold and computational task. For example, it has been sug-
bath temperature is quite low. Furthermore, the third law gested that the resource theory work is a suitable measure
affects the ability for us to cool a quantum system as the to quantify resonance processes [149]. A second issue is
system approaches absolute zero. This is because there is the link between the work definition in the ensemble
a trade-off between the entropy generated by the system sense, see (2), and the single shot work. Beyond taking
coupling to an environment that is used to cool the system mathematical limits of Renyi entropies, these definitions
and the fact that approaching absolute zero means that have to be understood within the context of each other
the system becomes a pure state (assuming the ground operationally, i.e. how can one measure each of these
state is unique). This trade-off practically limits COP. quantities and in what sense do these quantities converge
This was studied in [193], see also [24] for a discussion experimentally? Indeed, while traditional classical ther-
on another formulation of the third law and [211] for modynamics is a manifestly practical theory, made for
a recent review of the laws of thermodynamics in the steam engines and fridges, quantum thermodynamics has
quantum regime. In [212], the authors discuss periodi- only made a small number of experimentally checkable
cally driven quantum systems and investigate the laws of predictions of new thermodynamic effects yet.
thermodynamics for a model of quantum refrigerator, see Another point of discussion is the kinematic versus
[213–215]. Finally, in [216], the authors study the third dynamical approach to thermalisation and equilibration.
law in the context of refrigerators and show that the rate The kinematic approach arises from typicality discus-
of cooling is determined entirely by the cold reservoir sions in quantum information where states in Hilbert
and its interaction to the system, and is insensitive to space are discussed from the standpoint of a property
underlying the particle statistics. of their marginals. In the case of thermalisation, this
property is the closeness to the canonical Gibbs state. A
Hamiltonian and its eigenstates never appear in the kine-
7. Discussion and open questions matic description. The situation is quite different in the
We have presented an overview of a selection of current case of the dynamical approach to thermalisation implied
approaches to quantum thermodynamics pursued with by the ETH, where the eigenstates of the Hamiltonian are
various techniques and interpreted from different per- of crucial importance. A connection between these two
spectives. Substantial insight has been gained from these techniques to study thermalisation is thus desirable. This
advances and their combination, and a unified language difference in perspectives between quantum information
 CONTEMPORARY PHYSICS 573

theoretic and kinematic approaches also needs to be rec- Notes on contributors

onciled in the study of pre-thermalisation. Exceptional
states such as ‘rare states’ need to be fully understood Sai Vinjanampathy is a theoretical
physicist working at the Centre for
from a kinematic standpoint, see [218] for a discussion
Quantum Technologies at the National
on rare states and thermalisation. University of Singapore. Starting July 1,
In the context of QTMs, the interplay between statis- 2016, he assumed a faculty position at the
tics and engine performance needs more study [158,219]. Indian Institute of Technology-Bombay,
This is crucial since typical work fluids of QTMs consti- in Mumbai, India. He received his
tute several particles. Though some authors have consid- PhD from Louisiana State University in
2010. Besides quantum thermodynamics,
ered non-thermal baths in contact with a working fluid, his current work focuses on quantum
almost all such studies involved unitary transformations metrology and quantum control theory.
on thermal states. There is a need for the study of thermal
machines, wherein at least one of the baths is non-thermal
in a way that is different from unitary transformations
Janet Anders received her PhD in
on thermal baths. The role of entanglement and other quantum information theory from the
correlations would clearly become very important in such National University of Singapore in
a regime. Finally, we note that various experimental im- 2008. After a short postdoc, followed by
plementations of quantum engines are expected in the an independent Royal Society Dorothy
near future [220]. Hodgkin research fellowship at Uni-
versity College London, she moved
To close, quantum thermodynamics is a rapidly evolv- in 2013 to the University of Exeter
ing research field that promises to change our under- where she now heads the quantum
standing of the foundations of physics while enabling non-equilibrium group. Anders’ interests
the discovery of novel thermodynamic techniques and are in quantum thermodynamics and
applications at the nanoscale. This overview provided an quantum information theory, with a focus on the concept
of work in the quantum regime, levitating nanosphere
introduction to a number of current trends and perspec-
experiments, non-equilibrium fluctuation relations, thermal
tives in quantum thermodynamics and concluded with entanglement in many-body systems and measurement-based
three particular discussions where there is still disagree- quantum computation.
ment and flux. Resolving these and other riddles will no
doubt deepen our understanding of the interplay between
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