Elementary Thermal Operations
Matteo Lostaglio
1
,
´
Alvaro M. Alhambra
2
, and Christopher Perry
3
1
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
2
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
3
QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
February 5, 2018
To what extent do thermodynamic resource
theories capture physically relevant constraints?
Inspired by quantum computation, we define
a set of elementary thermodynamic gates that
only act on 2 energy levels of a system at a
time. We show that this theory is well re-
produced by a Jaynes-Cummings interaction in
rotating wave approximation and draw a con-
nection to standard descriptions of thermalisa-
tion. We then prove that elementary thermal
operations present tighter constraints on the al-
lowed transformations than thermal operations.
Mathematically, this illustrates the failure at fi-
nite temperature of fundamental theorems by
Birkhoff and Muirhead-Hardy-Littlewood-Polya
concerning stochastic maps. Physically, this im-
plies that stronger constraints than those im-
posed by single-shot quantities can be given if we
tailor a thermodynamic resource theory to the
relevant experimental scenario. We provide new
tools to do so, including necessary and sufficient
conditions for a given change of the population
to be possible. As an example, we describe the
resource theory of the Jaynes-Cummings model.
Finally, we initiate an investigation into how our
resource theories can be applied to Heat Bath
Algorithmic Cooling protocols.
1 Introduction
In recent years, ideas first appearing in the field of phys-
ical chemistry [1–4] have been used in conjunction with
information theory to analyse thermodynamics in the
quantum and nano-scale regimes. Often referred to as
the resource theory approach to quantum thermody-
namics, it has encompassed 2nd law-like statements in-
vestigating constraints on the allowed state transforma-
tions, e.g. [5–16], considerations on the 3rd law [17–19]
and the construction of fluctuation theorems [20–23].
For a more thorough review of results within the field
and beyond, see [24, 25] and references therein.
Many of these results are based upon the definition of
a restricted set of quantum operations known as ther-
mal operations (TO). These aim to capture all processes
that can be realised without an external source of work
or coherence and thus derive fundamental limitations
and bounds on thermodynamic processes and transfor-
mations. More specifically, they consist of all maps on
a system with Hamiltonian H
S
and in a state ρ that
can be written as
E(ρ) = Tr
X
U
ρ ⊗
e
−βH
B
Tr [e
−βH
B
]
U
†
. (1)
Here β = 1/(kT
B
) is a fixed inverse temperature
of the environment, H
B
is an arbitrary bath Hamil-
tonian, U is an energy-preserving unitary satisfying
[U, H
S
+ H
B
] = 0, and typically (but not necessarily)
X = B, i.e. the partial trace is over the bath de-
grees of freedom. Note that if [U, H
S
+ H
B
] 6= 0, then
U requires work to be performed, as can be seen from
an argument involving the notion of passivity (see Ap-
pendix A).
Questions can be raised about the connection of this
formalism to relevant experimental scenarios. It is im-
portant to stress that giving the definition of thermal
operations in the “Stinespring form” [26] of Eq. (1) does
not imply that an agent needs full control of system-
bath interactions and arbitrary baths to implement
them; furthermore, there are instances in which a large
number of unitaries lead to the same transformation on
the system [6]. Nevertheless, it is not clear what realis-
tic constraints are captured by Eq. (1) [27]. In this pa-
per we provide answers to this question by considering
subsets of thermal operations which have clear experi-
mental realisations and contrasting them with the full
set.
In quantum computing, it is common to study which
operations can be implemented by sequences of ele-
mentary gates. Most famously, every n-qubit gate can
be implemented using sequences of one and two-qubit
gates [28]. From this perspective it is natural to ask
if every transition induced by thermal operations can
be implemented through sequences of two-level thermal
Accepted in Quantum 2018-01-14, click title to verify 1
arXiv:1607.00394v3 [quant-ph] 2 Feb 2018
Figure 1: Decomposing TO: is it possible to realise any transi-
tion allowed by TO by sequences of two-level processes?
operations, which we call elementary thermal operations
(ETOs), see Fig. 1.
We analyse the above question by comparing the pop-
ulation dynamics that can be induced by thermal op-
erations and sequences of ETOs. On the one hand,
we show that the dynamics induced by an ETO can
be satisfactorily captured within a very simple physi-
cal model, a Jaynes-Cumming interaction with a single
bosonic mode in rotating-wave approximation (RWA);
moreover, ETOs are connected to collision models. On
the other hand, we show that ETOs constitute a gen-
uinely new resource theory – specifically, one that is
more restricted in the allowed transformations and eas-
ier to relate to physical models. The presence of a gap
between TOs and ETOs already at the classical level
may come as a surprise; in fact, it is based on the failure
at finite temperature of famous theorems by Muirhead
[29], Hardy, Littlewood and P´olya [30] and Birkhoff [31]
for doubly-stochastic matrices [32].
Furthermore, we provide necessary and sufficient con-
ditions to compute the population dynamics that are
allowed in the theory of ETOs. We then use these re-
sults to investigate what further limitations, compared
to the thermal operation framework, arise in determin-
istic work processes. We show that the work of forma-
tion under ETOs can be infinite, and we point out a gap
between ETOs and TOs in deterministic work extrac-
tion. This shows how restrictions in control translate
into the impossibility to achieve some of the traditional
results of single-shot thermodynamics. In addition, we
discuss the role of coherence within the resource the-
ory, showing that ETOs are more limited than thermal
operations also with regards to processing of coherence.
The present investigation can be understood as part
of a general effort to develop tools to study thermody-
namic transformations under restricted control [18, 33–
36]. We show that, while at infinite temperature ther-
modynamic theories are largely universal, i.e. the spe-
cific constraints at hand do not matter in terms of the
thermodynamic laws arising, this is no longer the case at
finite temperature. This implies that every restriction
will lead to a different resource theory; luckily, how-
ever, some of these can be solved using the same tools
we develop here for ETOs. For example, we can give
necessary and sufficient conditions for the population
dynamics achievable in the resource theory of 2-level
Jaynes-Cummings interactions in RWA, which is a the-
ory strictly contained in ETOs. This is important since,
for large enough systems and/or specific choices of phys-
ical parameters, we expect the two theories to depart
from each other. We also show how to solve a rather
general set of thermodynamic resource theories satis-
fying two assumptions: (a) partial level thermalisation
involving 2 levels at a time are (strictly) included and
(b) the set of extremal maps in the set of allowed op-
erations is known. Finally, we discuss an application
of this framework to an algorithmic cooling protocol,
showing that a particular ETO can be used to speed up
recently proposed protocols.
2 Elementary thermal operations
2.1 Definition and first properties
A thermal operation will be called elementary if it
acts non-trivially only on a two-dimensional subspace
spanned by two eigenstates of the system Hamiltonian
H
S
. We will denote by E
i
= ~ω
i
the eigenvalues of H
S
.
Given a d-dimensional system in state ρ, denote by
p
i
the occupation probability of energy level E
i
, and let
p = (p
0
, ..., p
d−1
). Note that every thermal operation E
induces a corresponding population dynamics G on p:
E(ρ) = σ ⇒ Gp = q, (2)
where q is the vector of occupation probabilities q
i
of
σ and G
k
0
|k
= hk
0
|E(|kihk|) |k
0
i. G is simply a matrix
of transition probabilities G
k
0
|k
from energies {E
k
} to
{E
k
0
}. Since G
k
0
|k
≥ 0 and
P
k
0
G
k
0
|k
= 1 (because
E is completely positive and trace-preserving), G is a
stochastic matrix. Moreover, one can directly show
from Eq. (1) that G satisfies Gg = g, where g is the
thermal Gibbs distribution of H
S
at fixed inverse tem-
perature β:
g =
e
−βE
0
Z
S
, . . . ,
e
−βE
d−1
Z
S
, Z
S
=
d−1
X
i=0
e
−βE
i
. (3)
Such matrices G are referred to as Gibbs-stochastic. In
fact, the converse is true: every Gibbs-stochastic matrix
can be realised as the population dynamics of (the limit
of some sequence of) thermal operations [8, 37].
Elementary thermal operations give rise to 2-level
Gibbs-stochastic matrices G. These are a set of ma-
trices exhibiting a particularly simple structure:
1. Only two energy levels of the system (i, j) are in-
volved, i.e., G
k|k
= 1 for every k 6= i, j.
Accepted in Quantum 2018-01-14, click title to verify 2
2. Detailed balance holds, i.e. G
i|j
= e
−β(E
i
−E
j
)
G
j|i
.
Property 1 follows immediately from the definition of
ETOs. Property 2 follows since Gg = g is equivalent to
detailed balance in the case of 2 × 2 matrices.
Note that any population dynamics G induced by an
ETO is characterised by the two energy levels i and j
on which it acts and a single parameter describing the
“strength” of the interaction. This can be chosen to be
G
j|i
∈ [0, 1] if E
i
≥ E
j
. So,
G =
1 −G
j|i
e
−β~ω
ij
G
j|i
G
j|i
e
−β~ω
ij
1 −G
j|i
⊕ 1
\(i,j)
,
where 1
\(i,j)
denotes the identity matrix on every en-
ergy level different from i, j and ω
ij
:= ω
i
− ω
j
is the
transition frequency. It is worth stressing that the set
of allowed operations we propose does not require the
ability to control the energy levels of the system Hamil-
tonian. We highlight that changing the energy levels
is difficult in practice for many experimental scenarios,
but the ETOs do not require this ability.
Every 2-level Gibbs stochastic transition matrix can
be realised by means of ETOs, without the need for
arbitrary bath Hamiltonians:
Lemma 1. Every population dynamics satisfying prop-
erties 1 and 2 can be realised by an ETO involving a
single-mode bosonic bath.
See Appendix B for the simple proof. ETOs hence
have a much simpler structure than the generic pro-
cesses induced by thermal operations and can be im-
plemented using a simple bath. Due to Lemma 1, and
since from now on we will only focus on the population
dynamics, with some abuse of language we will refer to
the set of 2-level Gibbs-stochastic matrices (stochastic
matrices satisfying properties 1 and 2) as ETOs. Note
that this identification is restricted to the allowed popu-
lation dynamics. ETOs and Gibbs-preserving maps on
two levels do not coincide as sets of quantum channels
(see Sec. 4.6).
Before we turn to the question of achieving ETOs
within simple physical models, let us introduce an ETO
of central importance, the β-swap, and describe how the
question posed in Fig. 1 fits within the general theory
of stochastic processes.
2.2 β-swaps and partial level thermalisations
A direct swap of population between two energy levels
cannot be achieved by thermal operations, since the as-
sociated transition matrix is not Gibbs-stochastic (un-
less E
i
= E
j
or β = 0). Nevertheless, there is a trans-
formation that is the thermodynamic analogue of such
a swap: the β-swap. It corresponds to an ETO β
(i,j)
among levels i and j (E
i
≥ E
j
) for which G
j|i
= 1:
β
(i,j)
=
1 −e
−β~ω
ij
1
e
−β~ω
ij
0
⊕ 1
\(i,j)
. (4)
An important fact regarding β-swaps is that if G is any
ETO acting on energy levels i and j, we have for some
λ ∈ [0, 1]
G = (1 − λ)1 + λβ
(i,j)
(5)
i.e. every ETO can be written as a convex combination
of the identity matrix and a β-swap. This decomposi-
tion will be useful later.
Another decomposition of any ETO can be obtained
by combining β-swaps and partial level thermalisations
(PLTs). PLTs partially swap the state of two levels i, j
(E
i
≥ E
j
) for the corresponding thermal probabilities:
(q
i
, q
j
) = (1 −λ)(p
i
, p
j
) + λNg
(i,j)
, λ ∈ [0, 1], (6)
and p
k
= q
k
for k 6= i, j. Here N = p
i
+ p
j
,
g
(i,j)
=
1
1+e
−β~ω
ij
,
e
−β~ω
ij
1+e
−β~ω
ij
. PLTs take the form
T
λ
=
1 −λe
−β~ω
ij
λ
λe
−β~ω
ij
1 −λ
⊕ 1
\(i,j)
. (7)
for λ ∈ [0, 1/(1 +e
−β~ω
ij
)]. If we take λ to be such that
λ = 1/(1 + e
−β~ω
ij
), the final state is proportional to a
thermal state on the corresponding energy levels and T
λ
is hence a full thermalisation on such levels. Given any
ETO G, one can directly check that either G is a PLT
or there exists a PLT T
λ
and a β-swap β
(i,j)
such that
G = T
λ
β
(i,j)
.
2.3 The infinite temperature limit
A central question, introduced in Fig. 1, is whether ev-
ery population dynamics induced by thermal operations
can be realised by either
1. Sequences of ETOs.
2. Sequences of ETOs and convex combinations
thereof.
A fruitful perspective on these problems comes from the
fact that, in the infinite temperature limit β → 0 (or
within each energy subspace), thermal operations in-
duce population dynamics G given by doubly-stochastic
matrices. These are stochastic matrices G satisfying
G1 = 1. In this limit (studied in Ref. [38]) both of the
above questions are answered by two fundamental theo-
rems, one due to Birkhoff [31] and the other to Muirhead
[29] and Hardy, Littlewood and P´olya [30].
Accepted in Quantum 2018-01-14, click title to verify 3
Specifically, from Eq. (5), one can recognise that
ETOs in the infinite temperature limit are convex com-
binations of the identity and a swap, since β-swaps are
swaps when β → 0. These matrices are simply gen-
eral 2-level doubly-stochastic matrices, also known as
T -transforms. They play a crucial role due to the fol-
lowing theorem [32]:
Theorem 2 (Muirhead [29] and Hardy, Littlewood
and P´olya [30]). Given p and q, there exists a doubly-
stochastic matrix G such that Gp = q if and only if
there exists a finite sequence of T -transforms G
(i)
such
that
q = G
(k)
···G
(1)
p.
At infinite temperature, this theorem answers the
first question above, since it shows that all transforma-
tions induced by thermal operations can also be realised
by finite sequences of ETOs.
If we also allow convex combinations, we can take
advantage of a fundamental theorem by Birkhoff to
strengthen the above result:
Theorem 3 (Birkhoff [31]). Every doubly-stochastic
matrix can be written as a convex combination of per-
mutations.
Since permutations are sequences of swaps, Birkhoff’s
theorem implies that, if we allow convex combinations,
then not only do ETOs induce the same transformations
p → q as thermal operations at infinite temperature,
but they can in fact simulate every transition matrix
generated by a thermal operation, i.e. every doubly-
stochastic matrix.
1
2.4 Failure of finite temperature extensions:
ETO 6= TO
We can now see that asking if ETOs coincide with ther-
mal operations (in terms of the achievable transforma-
tions or in terms of the set of transition matrices) corre-
sponds to asking: do the two central theorems 2 and 3
in the theory of doubly stochastic matrices extend from
infinite to finite temperatures? We will show that, in
both the weaker and stronger sense,
elementary thermal operations 6= thermal operations,
and hence both Theorems 2 and 3 fail at finite temper-
atures, i.e. for Gibbs-stochastic matrices. The conse-
quence is that ETOs define a genuinely new resource
theory of thermodynamics. We leave the proof of these
facts to Section 4 of this manuscript.
1
This is strictly stronger. In fact, the set of matrices that can
be realised through sequences of T -transforms is strictly smaller
than the set of all doubly-stochastic matrices, see [32, Chapter 2].
Knowing that ETOs provide a new resource theory
based on a simple set of allowed operations, two natural
questions arise:
1. Can we realise every ETO within simple physical
models?
2. What transformations can be realised by combin-
ing ETOs?
We begin by tackling the first question and show that, in
contrast to thermal operations, ETOs are easily related
to simple physical models.
3 Physical models for elementary ther-
mal operations
While ETOs only require interactions with a single
mode bosonic bath (Lemma 1), we are interested here
in considering physically realistic models. This may re-
quire additional restrictions, e.g. on the dimension of
the bath or on the allowed system-bath interactions.
The results of Ref. [18] imply that not all ETOs can
be achieved with a finite-size bath. Here we consider
instead some natural limitations on the allowed system-
bath interactions. We begin this study by analysing if
the Jaynes-Cummings (JC) model reproduces the set of
ETOs to a satisfactory extent. For many practical pur-
poses, which is what interests us in providing a physical
model, we shall see that the answer is affirmative.
3.1 Jaynes-Cummings model
One of the core descriptions of the interaction between
matter and radiation is the Jaynes-Cummings (JC)
model [39, 40]. Hence, we wish to know if it can provide
a natural model for ETOs.
For simplicity, consider a system with Hamiltonian
H
S
=
P
d−1
k=0
E
k
|E
k
ihE
k
| whose relevant transition fre-
quencies are all well separated from each other and non-
degenerate, i.e.
E
j
− E
i
6= E
j
0
− E
i
0
if (j, i) 6= (j
0
, i
0
). (8)
Assume one is allowed to couple every transition ω
ij
of
the system to a single-mode bosonic bath with Hamilto-
nian H
B
=
P
∞
n=0
n~ω
ij
|nihn|. The interaction is given
by a resonant JC Hamiltonian H
JC
in rotating wave
approximation,
H
JC
= g(σ
+
⊗ a + σ
−
⊗ a
†
).
Here a
†
, a are creation/annihilation operators on the
bath and σ
+
= |E
j
ihE
i
|, σ
−
= σ
†
+
excite/de-excite the
relevant transition.
Accepted in Quantum 2018-01-14, click title to verify 4
Without loss of generality let us denote the two
levels of the system involved by |E
0
i and |E
1
i and
set ~ω := E
1
− E
0
> 0. One can compute ([41], Ap-
pendix E.2)
e
−
it
~
H
JC
=
∞
X
n=1
1
X
k,k
0
=0
u
(n)
k,k
0
(s) |E
k
ihE
k
0
| ⊗ |n − kihn − k
0
|
+ |E
0
ihE
0
| ⊗ |0ih0|,
where s = gt/~ and, for each n, u
(n)
k,k
0
(s) are matrix
elements of the unitary
U
(n)
(s) =
cos(s
√
n) −i sin(s
√
n)
−i sin(s
√
n) cos(s
√
n)
.
Take the system and bath to be in the initial state
ρ ⊗ e
−βH
B
/Z
B
, where Z
B
= (1 − e
−β~ω
)
−1
. One then
has that the dynamics of the system in the interaction
picture is described by
2
ρ 7→ Tr
B
e
−
it
~
H
JC
ρ ⊗
e
−βH
B
Z
B
e
it
~
H
JC
.
Since [H
JC
, H
S
+ H
B
] = 0, this is an ETO. The trans-
formation induced on the vector p of occupation proba-
bilities is a stochastic process G whose transition prob-
abilities are:
G
1|0
(s) =
∞
X
n=1
sin
2
(s
√
n)
e
−β~ωn
Z
B
, G
0|0
= 1 − G
1|0
,
G
0|1
(s) =
∞
X
n=1
sin
2
(s
√
n)
e
−β~ω(n−1)
Z
B
, G
1|1
= 1−G
0|1
. (9)
Of course, G
k|k
= 1 for every k 6= 0, 1. As described
before, G can be parametrised by G
0|1
∈ [0, 1]. The
question is if this parameter interval can be covered
in the JC model. All values between 0 and the max-
imum achievable G
0|1
(s) (denoted by G
max
(β~ω)) can
be achieved, since G
0|1
(s) is a continuous function in s
and it takes the value 0 at s = 0.
First, let us investigate the low temperature limit.
For every s we have G
0|1
(s) ≥ (1 − e
−β~ω
) sin
2
(s).
Hence, in the zero temperature limit β → ∞,
G
max
(β~ω) → 1. In this limit every ETO can be re-
produced within the JC model with arbitrary accuracy.
Now consider any finite temperature. Setting
¯
β =
β~ω one can prove (see Appendix C)
G
max
(
¯
β) ≤
(
1
16
8e
−
¯
β
− e
2
¯
β
+ e
3
¯
β
+ 8
, for
¯
β ∈ [0,
log(4)
3
],
e
−4
¯
β
− e
−3
¯
β
+ 1, for
¯
β ≥
log(4)
3
.
(10)
2
To realise any given s, one should take g small enough and
t large enough to avoid the short times/strong couplings regime
where the model breaks down.
Figure 2: Realising ETOs in the simplest Jaynes-Cummings
model: to achieve every ETO one needs to be able to ob-
tain all de-exciting probabilities G
j|i
∈ [0, 1], ω = ω
j
− ω
i
>
0. The JC models realises every transition probability within
[0, G
max
(β~ω)], where G
max
(β~ω) lies somewhere within the
shaded area. Hence, far from the high temperature limit, al-
most all ETOs can be achieved within the JC model. Here the
blue line corresponds to the upper bound given in Eq. (10),
while the red line is the lower bound is obtained by truncating
the infinite series in Eq. (9).
On the other hand, a rough lower bound on G
max
(
¯
β)
(good when temperatures are not too high) can be ob-
tained by truncating the sum in Eq. (9) at some finite
m
max
, and setting every other term to zero. We take
m
max
= 12. Then numerics suggest to choose s = 98.92.
This, together with the previous upper bounds, provides
a region where G
max
(
¯
β) must lie as a function of
¯
β, see
Fig. 2.
The finite temperature scaling is rather favourable:
at room temperature T
B
= 300K and frequencies
ω/(2π) larger than 10
13
Hz, and at millikelvin tem-
peratures with frequencies larger than 10
8
Hz, one has
G
max
(β~ω) > 0.989. Hence Fig. 2 shows that, even
though not every ETO can be realised within the JC
in RWA, a vast majority of them can be achieved, espe-
cially at lower temperatures. This establishes the con-
nection at the level of single operations,
ETOs ↔ Jaynes-Cummings model.
Of course, to implement a sequence of ETOs one still
requires considerable frequency control as we must be
able to separately couple to every transition frequency
of the system. This is feasible to the extent in which we
are interested in the thermodynamics of small enough
systems. In Section 6, we give some suggestions relating
to protocols exploiting this set of operations.
Accepted in Quantum 2018-01-14, click title to verify 5
3.2 Collision models
We now turn to investigating the connection between
the framework of ETOs and thermalisation models,
widely used in quantum thermodynamics [42].
Assume again for simplicity that Eq. (8) holds. Con-
sider a collision model of thermalisation of a system in
the presence of a large thermal bath [43]. If we describe
the bath as a collection of qubits with various energy
gaps, in this simple model a thermal particle from the
bath approaches, interacts with a two-level subspace of
the system and scatters away; the interaction conserves
energy and the particle is lost in the bath [44]. Each
“collision” is then an ETO. Reasoning as in Ref. [44],
one finds that the combined effect of many identical
weak interactions on each two-level subsystem, initially
described by a population p(0), leads to an exponential
relaxation to the thermal distribution g (Appendix D):
p(t) = e
−t/ξ
p(0) + N(1 − e
−t/ξ
)g, (11)
where ξ ≥ 0 and N is the normalisation of p(0). This
relates ETOs to the collision model.
Alternatively, from the master equation point of
view [45], we can consider Davies quantum dynami-
cal semigroups, which are routinely used as a simple
model of a system weakly interacting with a large ther-
mal bath [46], or in the low-density limit [47]. On 2-
level systems, a full characterisation of their action ex-
ists [48] and it coincides with Eq. (11). Hence, the colli-
sion model of ETOs reproduces Davies maps on qubits.
Comparing Eq. (11) with Eq. (6), we see that the
models described effectively realise a subclass of ETO
dynamics, i.e. partial level thermalisations (defined
in Sec. 2.2). These can be understood as Markovian
ETOs, defined as the subset of ETOs whose induced
transition matrices G are embeddable stochastic ma-
trices, i.e. there is some generator L and t ≥ 0 such
that G = e
Lt
[49]. For G to be a valid stochas-
tic matrix, L must satisfy L
i|j
≥ 0 for i 6= j and
P
i
L
i|j
= 0 for all j [50]. Detailed balance is equiv-
alent to L
i|j
= e
−β~(ω
i
−ω
j
)
L
j|i
, where (i, j) are the two
levels involved. Computing e
Lt
p then gives Eq. (11).
In conclusion, Markovian ETOs can be related to two
different physical models: collision models and Davies
maps. For a summary of the relations between ETOs
and physical models, see Fig. 3.
4 The resource theory of elementary
thermal operations
We now describe in more detail the resource theory of
ETOs and provide necessary and sufficient conditions
for a transformation to be possible under this set. In
Figure 3: ETOs and physical models: The set of ETOs is closely
reproduced by the Jaynes-Cummings model for a wide set of
physically relevant parameters and with growing precision as
temperatures lowers. The Markovian subset of ETOs, on the
other hand, is connected to simple thermalisation models.
particular, we will present explicit results for qutrit sys-
tems, since these can be readily visualised, and discuss
a typical thermodynamic task, work extraction. Let us
begin with
Definition 1 (Resource theory of elementary thermal
operations). The resource theory of ETOs allows all
transformations that can realised by sequential applica-
tion of ETOs and convex combinations thereof. When
such operations exist transforming p into q, we denote
this by p
ETO
→ q.
Allowing convex combination simply corresponds to
the ability to condition the choice of our protocol on
some random variable (like a coin toss). It is difficult
to envisage an experimental setup in which this should
be forbidden. If no sequence of ETOs (no matter how
long) nor convex combinations thereof can transform p
into q, we will write p
ETO
9 q. We will use a similar nota-
tion if a transformation is possible by means of thermal
operations: p
TO
→ q (for any other set A, p
A
→ q).
4.1 Counterexample: ETOs do not have the
same power as thermal operations
As anticipated in Sec. 2.4, the resource theory of ETOs
is distinct from thermal operations, in that we are able
to show that there are p and q such that p
TO
→ q but
p
ETO
9 q.
3
To do this, let us recall the definition of a mono-
tone in a resource theory: it is a quantity that never
decreases (or never increases) under the allowed oper-
ations. We shall show that there is a monotone in the
resource theory of ETO that is not a monotone for ther-
mal operations, namely the size of the support of p.
3
A result that can be proven to be equivalent to the one we
present in this subsection was claimed in Ref. [51]. However,
to the best of our knowledge, its proof was never published [32,
Chapter 14B].
Accepted in Quantum 2018-01-14, click title to verify 6
The second concept we need to introduce to build
the counterexample is an important tool in the study
of thermal operations: thermo-majorisation. Given
an initial state p with corresponding energy levels
E = (E
0
, ..., E
d−1
), the associated thermo-majorisation
curve is constructed as follows:
1. Reorder p and E so that p
i
e
βE
i
is in non-increasing
order. Such an ordering is called the β-order of p.
2. Plot the ordered points
n
P
k
i=0
e
−βE
i
,
P
k
i=0
p
i
o
d−1
k=0
together with the point (0, 0) and connect them
piecewise linearly to form a concave curve - the
thermo-majorisation curve of p.
Given two probability distributions p and q associated
with the same energy levels, we say that p thermo-
majorizes q if the thermo-majorisation curve of p is
never below that of q (examples are given in Fig. 4). Re-
calling that a Gibbs-stochastic matrix G with Gp = q
exists if and only if p thermo-majorises q [52], and the
fact that every such G can be realised through TOs, it
follows [8]
p
TO
→ q ⇔ p thermo-majorises q. (12)
This implies that under TOs the height of the thermo-
majorisation curve at each value on the x-axis cannot
increase; furthermore, the decrease of these heights is
not only necessary but also sufficient for p
TO
→ q.
As ETOs form a subset of thermal operations, the
heights of a thermo-majorisation curve are also mono-
tones in the theory of ETOs. However, we shall now
show that additional monotones exist:
Lemma 4. Given a probability distribution p, let
|supp (p) | = {number of i : p
i
> 0}. Then p
ETO
→ q im-
plies
|supp (p) | ≤ |supp (q) |, (13)
i.e. the size of the support is a monotone in the theory
of ETOs.
Proof. First note that given two probability distribu-
tions p
1
and p
2
and λ ∈ [0, 1]:
|supp (λp
1
+ (1 − λ) p
2
) | ≥ max {|supp (p
1
) |, |supp (p
2
) |},
so taking convex combinations of ETOs can only in-
crease the size of the support. Hence, we can assume
q = Mp, with M a sequence of ETOs. Furthermore,
when combined with the decomposition of an ETO
given in Eq. (5), this implies that M can be written as
a convex combination of sequences of β-swaps. Then,
q will have support larger or equal to the support of
the distributions obtained by applying such sequences
Figure 4: Thermo-majorisation curves: We present the curves
for three probability distributions p = (0, 0.4, 0.6), q = (1, 0, 0)
and r = (0.15, 0.05, 0.8), each associated with the same Hamil-
tonian H satisfying e
−βE
0
= 1.5, e
−βE
1
= 0.7 and e
−βE
2
=
0.3. Here, p
TO
→ q, as the thermo-majorisation curve of p is
never below that of q. However, since the thermo-majorisation
curve of r crosses those of p and q, we have r
TO
9 p, q and
p, q
TO
9 r. The states p and q form the basis of Corollary 5.
Note that the size of the support of p is 2, while for q it is 1.
to p. We thus need to consider only the effect of β-
swaps β
(i,j)
. As the matrix associated with β-swaps,
given in Eq. (4), has full rank, it cannot decrease the
size of the support.
These considerations lead us to the following
Corollary 5. There exists p and q such that p
TO
→ q
but p
ETO
9 q.
Proof. Consider the three-level Hamiltonian with en-
ergy levels such that e
−βE
1
+ e
−βE
2
≤ e
−βE
0
together
with the initial state p such that p
0
= 0 and the final
state q such that q
0
= 1. For example, p = (0, 0.4, 0.6),
q = (1, 0, 0) and e
−βE
0
= 1.5, e
−βE
1
= 0.7, e
−βE
2
=
0.3, as presented in Fig. 4. As the support of q in
smaller than p, p
ETO
9 q. However, as shown in Fig. 4, p
thermo-majorizes q and hence p
TO
→ q.
To make the counterexample robust to the un-
avoidable experimental imprecisions, one would like a
stronger statement: there are transformations allowed
by thermal operations that cannot be approximated ar-
bitrarily well in the resource theory of ETOs. Specifi-
cally, we want to exclude that there exists a sequence
of distributions q
with q
-close to q and such that
p
ETO
→ q
for every > 0. This stronger result follows
from the fact that set of q that can be achieved in the
resource theory of ETOs is a closed set; in turn, this is a
by-product of the considerations of the next section, in
Accepted in Quantum 2018-01-14, click title to verify 7
which we develop necessary and sufficient conditions for
a transformation to be possible in the resource theory
of ETOs.
4.2 Necessary and sufficient conditions
Given some initial state p, we will now focus on finding
all possible states that can be reached from it in the
theory of ETOs. In other words, we want to study
Definition 2 (ETO cone). Given an initial state p, let
C
ETO
(p) be the set of all q such that p
ETO
→ q.
The strategy will be to characterise the extremal
points of the above convex set. We can show that the
ETO cone can be completely explored by convex com-
binations of a finite number of β-swaps β
(i,j)
:
Theorem 6 (Extremal points of ETO cone). All ex-
tremal points q of C
ETO
(p) can be written as
q = β
(i
n
,j
n
)
···β
(i
1
,j
1
)
p, (14)
where n ≤ d!.
Proof. Eq. (5) allows us to distinguish the identity and
β-swaps as extremal ETOs. The first step of the proof
will be to show that if q is extremal, then it can
be obtained through some sequence of β-swaps, as in
Eq. (14). By assumption, since q ∈ C
ETO
(p), we can
write q =
P
i
a
i
M
i
p, where each M
i
is a sequence of
ETOs and {a
i
} is a probability distribution. Each se-
quence of ETOs M
i
can contain both extremal and non-
extremal ETOs. Any non-extremal ETO can be decom-
posed using Eq. (5), obtaining
q =
X
x
b
x
β
x
p, (15)
where each β
x
denotes a sequence of β-swaps and {b
x
}
is some probability distribution, with b
x
6= 0 for all x.
We must then have β
x
p = q for every x, otherwise q is
not extremal. This completes the first part of the proof.
The second part of the proof derives the bound n ≤ d!
on the length n of the required sequence of β-swaps. By
contradiction, assume that the length of the shortest
sequence of β-swaps β
min
such that β
min
p = q is r > d!.
For k = 1, ..., r, define q
k
= β
(i
k
,j
k
)
···β
(i
1
,j
1
)
p to be the
intermediate states obtained by applying the elements
of the sequence β
min
. Without loss of generality, we can
assume that q
k
6= q
k
0
for every k, k
0
, otherwise there
would be a sequence of length r
0
< r mapping p into q,
in contradiction with our hypothesis.
For any q
k
, define the β-ordering of q
k
as a vector of
indexes π
k
, obtained by a permutation of (0, ..., d − 1).
Since there are only d! distinct β-orderings, after n > d!
β-swaps we necessarily have k
1
, k
2
with k
1
< k
2
and
1. q
k
1
6= q
k
2
and π
k
1
= π
k
2
,
2. q
k
2
can be obtained from q
k
1
through a thermal
operation.
Note that the second property holds trivially, since se-
quences of ETOs are also sequences of TOs and the lat-
ter are closed under composition. From the two proper-
ties above, applying Theorem 12 of Ref. [33], we deduce
that there exists a sequence of PLTs transforming q
k
1
into q
k
2
. As in the first part of the proof, since PLTs
are non-extremal, we can decompose them through β-
swaps using Eq. (5). Hence, q
k
2
=
P
y
c
y
β
y
q
k
1
, where
as before β
y
denotes a sequence of β-swaps. This allows
us to substitute part of the sequence β
min
, obtaining an
expression with the following structure:
q = β
min
p = β
x
2
X
y
c
y
β
y
!
β
x
1
p,
where β
x
1
p = q
k
1
, β
x
2
q
k
2
= q and {c
y
} is some prob-
ability distribution with c
y
6= 0 for all y. Note that the
sum of the lengths of β
x
1
and β
x
2
is at most r−1. Since
q is extremal we must have, for every y,
q = β
x
2
β
y
β
x
1
p.
Note however that there is a y such that β
y
is the iden-
tity, i.e. a sequence of length zero. It follows that there
is a sequence of β-swaps mapping p to q and of length at
most r−1. This is in contradiction with our hypothesis.
Hence, it must be that n ≤ d!.
The above theorem gives, in particular, necessary and
sufficient conditions for determining whether p
ETO
→ q.
One can proceed as follows:
1. Construct the extremal points of C
ETO
(p) using
Theorem 6.
2. p
ETO
→ q if and only if q ∈ C
ETO
(p). Determining the
latter, given that the extremal points of C
ETO
(p)
are known, can be solved by a linear program.
We illustrate the above general results by comparing
the ETO cone with the TO cone
4
for d = 3, see Fig. 5.
We also note that Theorem 6 implies that C
ETO
(p) is
a simplex with a finite number of vertices. As such, it
is a closed set (as noted at the end of Section 4.1).
We complete the study of the main aspects of the re-
source theory of ETOs by addressing some thermody-
namically relevant questions within the new framework:
what is the maximum amount of work that can be ex-
tracted from a state? Conversely, how much work is
4
For completeness, we show how to construct the TO cone in
Appendix E
Accepted in Quantum 2018-01-14, click title to verify 8
(1,0,0) (0,1,0)
(0,0,1)
(1,0,0) (0,1,0)
(0,0,1)
Initial State, p
C
TO
(p)
C
ETO
(p)
Gibbs State, g
Figure 5: Comparison of the ETO and TO cones: Here we
plot C
ETO
(p) and C
TO
(p) for a 3-level system in state p and
with associated energy levels E on the 3 outcome probability
simplex. Note the region of states accessible using TO but not
by ETO. Here p = (0.7, 0.2, 0.1), βE = (0, 0.2, 0.5).
necessary to prepare a state? We will show that there
is a gap between what can be achieved with TO and
ETOs. As a by-product, this shows that there will also
be gaps between the predictions of the resource theory
of thermal operations and physical models in which only
ETOs are allowed.
4.3 Work in the theory of ETOs
Under thermal operations, if it is not possible to convert
p into q then an additional resource, work, may have to
be supplied to achieve the conversion. Similarly, if p can
be converted into q, it may be possible to extract work.
Within the resource theory approach, the deterministic
work can be regarded as the minimal amount of work
that must be supplied to guarantee a transformation or
the maximal amount of work that can be guaranteed
to be extracted in converting one state into another.
In this section, we investigate these concepts for ETOs
and highlight the differences with the theory of thermal
operations.
Definition 3 (Deterministic Work under ETOs).
Given two probability distributions p and q with Hamil-
tonian H
S
and a 2-level system (called a wit for short),
with Hamiltonian H
W
which has energy levels E
W
0
= 0
and E
W
1
= w, the work of transition under ETO,
W
ETO
p→q
, is the highest value of w such that:
p ⊗ |0ih0|
ETO
→ q ⊗ |1ih1|. (16)
If W
ETO
p→q
is positive, work is extracted in converting p
into q, while if it is negative, work must be supplied to
perform the transformation.
This notion of deterministic work, which is stored in
the extra 2-level wit system, is very common within the
resource theory framework [7–9]. The main difference
with other notions of work, such as those involving ap-
plying arbitrary unitaries to systems and the notion of
passive states [53] are: (a) only energy-preserving uni-
taries are allowed, (b) the unitaries can be applied to
both system and heat bath rather than just system and
(c) as the work is stored in a 2-level system rather than
a (often implicit) weight-like device, it is a single-shot
quantity, achieved in every run of an extraction proto-
col, rather than an average yield.
As a small caveat, note that the zero error statement
of Eq. (16) is unobservable, and we can instead ask the
more physically relevant question as to whether a given
w can be achieved up to an arbitrarily small, but poten-
tially nonzero, error. I.e., is there a sequence of states
(q ⊗ |1ih1|)
-close to q ⊗ |1ih1| and such that
p ⊗ |0ih0|
ETO
→ (q ⊗ |1ih1|)
, (17)
for every > 0? If this is the case, we will write
W
ETO
p→q
≥ w.
Let us denote the set of joint energy levels of system
and wit by
˜
E
i
2d−1
i=0
. These neatly divide into two sets.
Firstly, let A
0
=
˜
E
i
d−1
i=0
where
˜
E
i
= E
i
. These are
associated with states for which the wit is in the state
|0i. Secondly, let A
w
=
˜
E
i
2d−1
i=d
where
˜
E
i
= E
i−d
+w.
These are associated with states for which the wit is in
the state |1i.
A crucial tool for analysing deterministic work under
ETOs is given by the following Lemma:
Lemma 7. Given distributions p and q, if W
ETO
p→q
≥ w,
then there is a wit with energy gap w and an injective
map Φ between elements of
˜
E
i
∈ A
0
such that p
i
> 0
and A
w
such that Φ
˜
E
i
≤
˜
E
i
.
Proof. Let us fix w. Suppose that with respect to this
energy gap we have:
p ⊗ |0ih0|
ETO
→ q ⊗ |1ih1|,
and consider the convex combination of sequences of
ETOs that implements this transformation. The initial
state has support only on energy levels contained in A
0
while the final state has support only on energy levels
contained in A
w
. Hence, every sequence of ETOs within
the convex combination must take p ⊗ |0ih0| to a state
of the form q
0
⊗|1ih1|. In what follows, we will consider
one such sequence and use it to construct the required
injection. We denote this sequence by π and note that:
q
0
⊗ |1ih1| = π (p ⊗ |0ih0|)
=
G
(i
n
,j
n
)
. . . G
(i
1
,j
1
)
(p ⊗ |0ih0|)
Accepted in Quantum 2018-01-14, click title to verify 9
where G
(i,j)
denotes an ETO acting non-trivially on
energy levels
˜
E
i
and
˜
E
j
.
We now characterise the scenarios where an ETO act-
ing on energy levels
˜
E
i
and
˜
E
j
maps the associated oc-
cupation probabilities (r
i
, r
j
) such that r
i
> 0 to
r
0
i
, r
0
j
with r
0
i
= 0. The occupation probabilities are related
via:
r
0
i
r
0
j
=
G
i|i
G
i|j
G
j|i
G
j|j
r
i
r
j
, (18)
which implies that we need both G
i|i
= 0 and r
j
= 0.
If G
i|i
= 0, then to achieve Gibbs-stochasticity we need
that G
i|j
e
−β
˜
E
j
= e
−β
˜
E
i
which, together with G
i|j
≤ 1,
implies that it must hold that
˜
E
i
≥
˜
E
j
.
In light of this, we now modify our protocol π as
follows:
1. Without loss of generality assume that
˜
E
i
≥
˜
E
j
.
First, remove all of the G
(i,j)
such that G
i|i
> 0.
Note that the resulting protocol still maps
p ⊗ |0ih0| to a state of the form
ˆ
q ⊗ |1ih1| as we have
not removed any of the ETOs that empty popula-
tion from a given energy level and in particular all
population must eventually leave A
0
.
2. Next, remove all of the G
(i,j)
that are applied to
intermediate states in the protocol such that r
j
>
0. Again, the resulting protocol still maps p⊗|0ih0|
to a state of the form
ˆ
q ⊗ |1ih1| as we have not
removed any of the ETOs that empty population
from a given energy level.
3. Finally, remove all G
(i,j)
that are applied to inter-
mediate states in the protocol such that r
i
= r
j
= 0
(as these operations now act trivially).
We call the resulting sequence π
0
and note that:
π
0
(p ⊗ |0ih0|) =
ˆ
q ⊗ |1ih1|,
for some state
ˆ
q which potentially differs from both q
and q
0
. That the resulting state has this form, follows
from the fact that the sequence of ETOs in π
0
still con-
tains the operations which empty the energy levels in
A
0
.
The effect of each ETO in π
0
is to swap the occupa-
tion probability of an occupied energy level
˜
E
i
with an
unoccupied energy level
˜
E
j
. In addition, each pair of
energy levels is such that
˜
E
i
≥
˜
E
j
. By combining each
of these swaps to construct a permutation, we hence
find an injection Φ between elements of A
0
with p
i
> 0
and A
w
such that Φ
˜
E
i
≤
˜
E
i
.
As an example to illustrate the construction of
such an injection, consider the two states p =
(0, p
1
, p
2
) and q = (p
1
, p
2
, 0) with Hamiltonian
H
S
= ~ω(|1ih1| + 2 |2ih2|). Suppose we wish to extract
w = ~ω in transforming p into q. It is easy to see that
this can be done by performing a sequence of two β-
swaps. First, a β-swap is performed between the energy
levels
˜
E
1
and
˜
E
3
. Next, a β-swap is performed between
the energy levels
˜
E
2
and
˜
E
4
. The injection Φ is thus
given by Φ
˜
E
1
=
˜
E
3
, Φ
˜
E
2
=
˜
E
4
.
4.4 Unbounded work of formation
Note that finding such an injection is a necessary,
but not sufficient condition for being able to achieve
W
ETO
p→q
≥ w. Indeed, while an injection satisfying the
required properties will always exist for large negative
values of w, there are state transformations for which
supplying no finite amount of work enables the transfor-
mation, i.e. W
ETO
p→q
= −∞. This is in stark contrast to
the case of thermal operations where, provided the tar-
get state is block-diagonal in the energy basis, providing
enough work will always make the transition possible,
i.e. W
ETO
p→q
> −∞.
Prime examples of this occur for a special case of de-
terministic work processes: the work of formation. If
g denotes the probability distribution which is thermal
with respect to H
S
at the fixed temperature β, then
the work of formation of p under ETO is the minimum
amount of work required to deterministically create p
from the thermal state: W
ETO
form
(p) = W
ETO
g→p
. If p does
not have full support, then Lemma 4 implies the trans-
formation is impossible regardless of the value of work
chosen. Further examples that do not rely on argu-
ments based on the size of the support of p can also be
constructed (see Appendix F).
4.5 Gap in work extraction
The work of distillation of p under ETOs is the maxi-
mum amount of work that can be deterministically ex-
tracted from p and is given by W
ETO
distil
(p) = W
ETO
p→g
.
A canonical toy example of work distillation is given
by Szilard engines. In its most basic form, this con-
sists of a 2-level system with degenerate Hamiltonian
H
S
∝ 1 and initially in state p = (1, 0). For such a
setup, W
TO
distil
(p) = β
−1
ln 2. However, under ETOs, for
any value of w > 0 it is not possible to find the injection
required for work extraction as detailed in Lemma 7 and
hence W
ETO
distil
(p) = 0. This indicates that the resource
theory of ETOs cannot exploit purity to extract work
to the same extent as thermal operations.
This discrepancy between work distillation under
TOs and ETOs is in fact completely general, as cap-
tured in the following theorem:
Theorem 8. Given a probability distribution p and as-
sociated Hamiltonian H
S
, if W
TO
distil
(p) > 0, then there
Accepted in Quantum 2018-01-14, click title to verify 10
exists a positive constant C bounded away from zero
such that:
W
ETO
distil
(p) + C ≤ W
TO
distil
(p) .
Proof. Without loss of generality, let p be ordered such
that p
i
> 0 for i ∈ {0, . . . , m − 1} and p
i
= 0 for
i ∈ {m, . . . , d − 1}. From the results of Ref. [8] it follows
that W
TO
distil
(p) satisfies:
e
−βW
TO
distil
(p)
Z
S
=
m−1
X
i=0
e
−βE
i
. (19)
For W
TO
distil
(p) > 0 to hold, we must have m < d.
From Lemma 7, to distil work W
ETO
distil
(p) we
must be able to find an injection Φ such that
Φ(E
i
) := E
0
i
+ W
ETO
distil
(p) ≤ E
i
for i ∈ {0, . . . , m − 1}.
We thus have:
m−1
X
i=0
e
−βE
i
≤
m−1
X
i=0
e
−βE
0
i
e
−βW
ETO
distil
(p)
.
From Eq. (19),
W
TO
distil
(p) ≥ W
ETO
distil
(p) −
1
β
ln
m−1
X
i=0
e
−βE
0
i
Z
S
.
Set C := −
1
β
ln
P
m−1
i=0
e
−βE
0
i
Z
S
. The result follows from
the fact that C > 0, since {E
0
i
}
d−1
i=0
is a relabeling of the
energies {E
i
}
d−1
i=0
and m < d.
4.6 Quantum coherence and ETOs
So far we have focused on population dynamics. We
could do this because under any ETO (in fact, any ther-
mal operation) the evolution of population and coher-
ence among energy eigenstates decouples (see Eq. (2)).
However it is natural to wonder how the two theories
compare in relation to their ability to process quantum
coherence.
In this subsection we consider the problem in its full
generality: ETOs and thermal operations are two differ-
ent sets of quantum channels (and not simply stochastic
maps, as in the rest of this paper). How do they com-
pare in terms of the coherent evolutions that they can
induce?
First of all, it is important to clarify that ETOs do
not coincide with the set of quantum operations that
preserve the Gibbs state and act non-trivially only on
two level subsystems (i.e., with the set of 2-level Gibbs-
preserving operations). The simplest way to see this
is that 2-level Gibbs-preserving channels can generate
coherence from an initially incoherent state, while ther-
mal operations, and hence ETOs, do not [54]. The two
sets only coincide classically.
To tackle the issue of quantum coherence, we need to
introduce the concept of modes of coherence. Given a
quantum state ρ, one can decompose it into parts that
transform in the same way under time-translations [55].
If H
S
has eigenvalues ~ω
i
, then
ρ =
X
Ω
ρ
(Ω)
, e
−iH
S
t
ρ
(Ω)
e
iH
S
t
= e
−i~Ωt
ρ
(Ω)
, (20)
where the set of {Ω} is the set of all transition fre-
quencies within H
S
(also called the Bohr spectrum of
H
S
, this is the set of all distinct differences ω
i
− ω
j
).
Each Ω defines a so-called mode of coherence. If E is a
thermal operation, coherence transformations can take
place only within each mode, in the sense that
E(ρ) = σ ⇒ E(ρ
(Ω)
) = σ
(Ω)
, for each Ω, (21)
which generalises Eq. (2) from Ω = 0 to every Ω in the
Bohr spectrum. Quantum maps satisfying Eq. (21) are
called time-translation covariant.
Quantum coherence can be transported within each
mode under thermal operations, sometimes perfectly
[56]. On the other hand, we now show that coherence
transport cannot take place under ETOs. Denoting
ρ
xy
= hx|ρ |yi,
Lemma 9. Let σ = E(ρ), with E an elementary ther-
mal operation acting on energy levels ω
x
0
and ω
y
0
, with
ω
x
0
> ω
y
0
. Then
|σ
x
0
y
0
| ≤
q
G
x
0
|x
0
G
y
0
|y
0
|ρ
x
0
y
0
|. (22)
Proof. Thermal operations are time-translation covari-
ant maps [12]. ETOs are a subset of them, so they
are as well. Theorem 2 of Ref. [57] then immediately
implies that,
|σ
x
0
y
0
| ≤
X
x,y:
ω
x
−ω
y
=ω
x
0
−ω
y
0
q
G
x
0
|x
G
y
0
|y
|ρ
xy
|, (23)
where the sum is restricted to all indexes such that ω
x
−
ω
y
= ω
x
0
−ω
y
0
and, as before, G
k
0
|k
= hk
0
|E(|kihk|) |k
0
i.
Consider now any term in the above sum in which
ω
x
0
6= ω
x
. Since ω
y
0
− ω
y
= ω
x
0
− ω
x
, it follows that
ω
y
6= ω
y
0
(so there are no terms in which ω
x
0
6= ω
x
but
ω
y
0
= ω
y
). From ω
x
0
6= ω
y
0
one gets ω
x
6= ω
y
. Further-
more,
ω
x
0
− ω
y
= ω
x
− ω
y
+ ω
y
0
− ω
y
(24)
ω
y
0
− ω
x
= −(ω
x
− ω
y
) + ω
x
0
− ω
x
. (25)
Since ω
y
0
− ω
y
= ω
x
0
− ω
x
, it follows that one could
have ω
x
0
= ω
y
or ω
y
0
= ω
x
, but not both. This shows,
for every term with ω
x
0
6= ω
x
or ω
y
0
6= ω
y
, that at
most two of the four levels ω
x
0
, ω
x
, ω
y
0
, ω
y
have the
Accepted in Quantum 2018-01-14, click title to verify 11
same energy. Hence, three distinct energy levels are
involved in the corresponding pre-factor
p
G
x
0
|x
G
y
0
|y
.
ETOs, on the other hand, can only act non-trivially
on two energy levels at any single time. Hence, either
G
x
0
|x
= 0 or G
y
0
|y
= 0, so every such term is zero.
Eq. (23) follows.
The previous lemma implies that there cannot be any
transport of coherence within a mode. For example,
given a qutrit system with H
S
= ~ω(|1ih1| + 2 |2ih2|)
(the ground state having zero energy), one cannot
move coherence from |0ih1| into |1ih2| using ETOs, even
though these are part of the same mode Ω = ω. This
shows that ETOs are more limited than thermal opera-
tions, not only from the point of view of population dy-
namics, but also in terms of coherence processing. An-
other interesting consequence of the previous lemma is
that β-swaps destroy all quantum coherence in the lev-
els on which they act, since they have either G
x
0
|x
0
= 0
or G
y
0
|y
0
= 0.
A superset of ETOs that can be easily characterised is
that of enhanced ETOs, which are defined as enhanced
thermal operations acting on 2 energy levels. Recall
that an enhanced thermal operation is a quantum chan-
nel E such that [58]
1. E(e
−βH
S
) = e
−βH
S
, (Gibbs-preservation)
2. Eq. (21) is satisfied, (Time-translation covari-
ance)
There are enhanced ETOs achieving the bound of the
previous lemma. Their Kraus decomposition is given as
(taking ω
x
0
> ω
y
0
)
K
1
=
q
e
−β~ω
x
0
y
0
G
y
0
|x
0
x
0
y
0
, K
−1
=
p
G
y
0
|x
0
y
0
x
0
,
K
0
=
q
1 − G
y
0
|x
0
e
−β~ω
x
0
y
0
y
0
y
0
+
p
1 − G
y
0
|x
0
x
0
x
0
.
In fact, one can deduce from Ref. [58] that ETOs
achieving the bound exist as well.
It is worth noticing that the parallel we drew
in Section 3.2 between ETOs, collision models and
Davies maps extends to the quantum case. Taking
G
k
0
|k
= T/n (where T > 0 denotes the total time
and n the number of collisions) one obtains in the
n → ∞ limit of infinitely many weak collisions the stan-
dard [48] exponential damping of each coherence term,
|σ
x
0
y
0
| := ρ
x
0
y
0
(T ) ≤ e
−ξT
|ρ
x
0
y
0
|, ξ > 0.
5 Thermodynamic resource theories at
finite temperature
So far, we have introduced and analysed ETOs as a
suitable set of allowed transformations, distinct from
TOs and related to physically relevant models. We now
broaden the discussion to more general thermodynamic
resource theories. We discuss how, in the infinite tem-
perature limit, many different thermodynamic theories
“flow” toward a universal model, characterised by ma-
jorisation theory. On the one hand, we can reinterpret
the results of the previous sections as showing that this
universality is lost at finite temperature. On the other
hand, we show that some of the techniques developed in
this manuscript can be applied more generally and tai-
lored very specifically to the experimental constraints
at hand.
5.1 Lack of universality at finite temperature
What set of operations can be performed at no work
cost, if we require compatibility with the second law
of thermodynamics? Each (state-independent) choice
defines a thermodynamic resource theory.
5
As mentioned before, thermal operations realise the
full set of Gibbs-preserving maps on the energy levels’
populations. In terms of allowed population dynam-
ics, this is arguably the largest thermodynamic resource
theory, since any other stochastic process would allow
the extraction of work from a single heat bath. This
follows from the fact that if a map M with Mg = x
and x 6= g is allowed (g being the thermal Gibbs state
of Eq. (3)), then we can create many copies of x from
many copies of g; Theorem 1 of Ref. [6] then implies that
we could raise a weight using a single bath, in disagree-
ment with the second law of thermodynamics. A much
more restricted theory is one in which only partial level
thermalisations (PLTs) on pairs of levels are allowed
(see Eq. (11)). At infinite temperature, the classical
result of Theorem 2 immediately implies the following
universality result:
Lemma 10. In the limit T → ∞, every thermodynamic
theory that allows for PLTs on pairs of levels and does
not extract work from a single heat bath is governed by
the same laws at the classical level. In other words, if
A and B are two such theories and p, q are any two
states
p
A
→ q ⇔ p
B
→ q
Furthermore, well-known results on doubly-stochastic
maps ([32], Chapter 2) imply that at infinite temper-
ature a transformation is possible if and only if p ma-
jorises q. This remarkable degree of independence of the
thermodynamic constraints from the specific choice of
5
A framework not included in this discussion is, for example,
catalysis [9]. Catalysts allow one to perform some non Gibbs-
stochastic maps on some non thermal states, as can be deduced
from the fact that they present weaker constraints than thermo-
majorisation [9].
Accepted in Quantum 2018-01-14, click title to verify 12
the set of allowed free operations is lost at finite temper-
ature, as shown by the counterexample of Sec. 4. Also,
universality does not hold for transformations involv-
ing quantum coherence since, as mentioned in Sec. 4.6,
Gibbs-preserving quantum operations and thermal op-
erations allow different sets of transformations. This
lack of universality implies that to obtain tighter con-
straints on the allowed transformations we need to de-
velop models that are more closely connected to the
relevant experimental conditions.
For example, we can observe that the construction of
Theorem 6 holds for any restriction of the set of ETOs,
provided that PLTs are allowed. In the next section we
explicitly consider the example of the Jaynes-Cummings
model in RWA, and then consider a generalisation that
characterises a large class of thermodynamic theories
given the knowledge of the set of extremal maps.
5.2 The resource theory of Jaynes-Cummings
operations
We have demonstrated that the theory of ETOs com-
bines appealing theoretical properties with the ability
to reproduce, within a large set of parameters, 2-level
Jaynes-Cummings interactions. However, for high di-
mensional systems, due to the need of applying long se-
quences of operations, the differences between the ETOs
and the Jaynes-Cummings model will become impor-
tant. Hence, we now go a step further and define a re-
stricted resource theory, contained within ETOs, that
coincides exactly with the Jaynes-Cummings model.
More precisely, we are assuming that one can per-
form the operations described in Section 3.1, i.e. all
stochastic matrices
J
(i,j)
(s) =
1 − G
j|i
(s)e
−β~ω
ji
G
j|i
(s)
G
j|i
(s)e
−β~ω
ij
1 − G
j|i
(s)
, (26)
acting on any pair of levels i, j, E
i
≥ E
j
, and satisfy-
ing Eq. (9) (recall that s is the normalised interaction
time, s = gt/~). Sequential composition and convex
combinations are allowed. With obvious notation,
Definition 4 (JC cone). Given an initial state p, let
C
JC
(p) be given by all q such that p
JC
→ q.
As in ETOs, any operation among two given levels is
described by a single parameter, G
j|i
(s). If s is limited
by some s
(i,j)
max
(possibly dependent on the levels i, j in-
volved), then G
j|i
(s) can be numerically optimised to
find its maximum within
h
0, s
(i,j)
max
i
. If the interaction
time s is not limited, one needs to find bounds indepen-
dent of s, for example the upper and lower bounds given
at the end of Sec. 3.1. In every case, we are left with
a value (or interval) for the greatest relaxation proba-
bility G
max
(β~ω
ij
) allowed in the model for any given
transition. The only assumption we will make is that
the theory allows to perform PLTs between any pair of
energy levels, i.e. G
max
(β~ω
ij
) > 1/(1 + e
−β~ω
ij
) for all
i,j.
6
We now proceed in constructing C
JC
(p). This can be
done immediately from Theorem 6, since it holds un-
changed within this restricted theory, with the caveat of
substituting β-swaps with the relevant extremal maps:
J
(i,j)
=
1 − G
max
(β~ω
ij
)e
−β~ω
ij
G
max
(β~ω
ij
)
G
max
(β~ω
ij
)e
−β~ω
ij
1 − G
max
(β~ω
ij
)
.
If we only know that G
max
(β~ω
ij
) ∈ [G
<
ij
, G
>
ij
], as
in our Jaynes-Cummings example if we do not en-
force any limitations on s, we can still build an
outer cone C
>
JC
(p) and an inner cone C
<
JC
(p) satisfy-
ing C
>
JC
(p) ⊃ C
JC
(p) ⊃ C
<
JC
(p), simply using the lower
or upper bounds of Fig. 2.
This allow us to construct cones that faithfully repro-
duce the specific experimental constraints at hand in the
Jaynes-Cummings model described above. Some results
are presented in Figs. 6 and 7. Fig. 6 considers opti-
cal frequencies and room temperature; this is a regime
in which, from the considerations of Sec. 3, we expect
ETOs to provide good constraints for the JC model. In
fact, we can see that the allowed transitions are almost
indistinguishable, i.e. C
JC
(p) ≈ C
ETO
(p), while there is
a clear distinction between C
ETO
(p) and C
TO
(p). Fig. 7,
in contrast, considers microwave frequencies and a tem-
perature of 0.1K, a parameter regime in which we do
not expect ETOs to be a provide a faithful model. In
fact, the correspondent cones are rather different, high-
lighting the lack of thermodynamic universality and the
usefulness of the techniques developed here to build re-
source theories that fit the specific experimental condi-
tions.
5.3 Restricted thermodynamic theories
The previous considerations can be extended to more
general restricted thermodynamic theories. As an ex-
ample, Ref. [36] has identified the set of extremal Gibbs-
preserving maps involving 3-levels at any time. Then,
one would like to construct the set of achievable states
for the 3-dimensional generalisation of ETOs, in which
we allow all extremal 2-level and 3-level maps in the set
of Gibbs-stochastic maps. More generally, one could
consider theories in which every extremal map on k
6
The reason for the strict inequality is a technical one. The
second part of the proof of Theorem 6 required that we could
decompose PLTs into non-trivial convex combinations of extremal
maps and identity transformations. If the extremal map were a
PLT (the inequality was not strict), this would not be possible.
Accepted in Quantum 2018-01-14, click title to verify 13
(1,0,0)
Initial State, p
C
TO
(p )
C
E TO
(p )
C
>
JC
(p )
C
<
JC
(p )
Gi bbs S tate, g
Figure 6: Comparison of the ETO and JC cones for T = 300K:
Here we plot C
ETO
(p), C
TO
(p) and upper and lower bounds
on C
JC
(p) for a 3-level system in state p and with associated
energy levels E on the 3 outcome probability simplex. Here
p = (0.85, 0.03, 0.12), E/~ = (0, 69 THz, 130 THz).
(1,0,0) (0,1,0)
(0,0,1)
Initial State, p
C
TO
(p)
C
ETO
(p)
C
>
JC
(p)
C
<
JC
(p)
Gibbs State, g
Figure 7: Comparison of the ETO and JC cones for T = 0.1K:
Here we plot C
ETO
(p), C
TO
(p) and upper and lower bounds
on C
JC
(p) for a 3-level system in state p and with associated
energy levels E on the 3 outcome probability simplex. Here
p = (0.1, 0.3, 0.6), E/~ = (0, 6.2 GHz, 11.7 GHz) .
energy levels is allowed, with k = 2 corresponding to
ETOs and k = d thermal operations (see Ref. [36]). An-
other appealing restriction is given by ETOs augmented
by a global PLT involving all energy levels at once or,
in a more refined version, PLTs on any set of k > 2
levels. Ultimately, the relevant set of restrictions will
depend on the experimental setup. However, all of the
above theories can be solved from the knowledge of the
set of extremal maps using the following generalisation
of Theorem 6:
Theorem 11. Let A denote the allowed operations in
a convex thermodynamic resource theory. Suppose A
contains, for every i, j, E
i
> E
j
a map
A
(i,j)
=
1 − G
max
j|i
e
−β~ω
ij
G
max
j|i
G
max
j|i
e
−β~ω
ij
1 − G
max
j|i
!
, (27)
for some G
max
j|i
> 1/(1 + e
−β~ω
ij
) (in other words, 2-
level PLTs are strictly contained).
If A
i
are the extremal maps in A, then p
A
→ q if and
only if q is in the convex hull of the following points:
{A
i
n
···A
i
1
p},
where n ≤ d!.
Proof. The proof is a straightforward generalisation of
that of Theorem 6. Firstly, we can prove that any ex-
tremal points of the set {r|p
A
→ r} must be achiev-
able through a sequence of extremal maps in A, as in
Eq. (11). Secondly, if by contradiction we need n > d!
extremal maps, then some β-ordering will be repeated
at least twice in the sequence; but then we can substi-
tute part of the sequence with 2-level PLTs. Since by
assumption these 2-level PLTs can be written as a non-
trivial convex combination of extremal points in A, the
rest of the proof of Theorem 6 holds.
6 Applications in cooling algorithms
We now explore one potential application of the frame-
work presented in the previous sections, regarding the
cooling of quantum states. The generation of highly
pure, “cold”, quantum states is one of the necessary re-
sources in the implementation of quantum technologies
[59]. These are needed, for instance, as ancilla qubits
in quantum error correction, or at the initial stage of
many quantum algorithms.
A potential method for generating these pure states
is via algorithmic cooling, which is a generic name for
a set of schemes in which unitary operations are ap-
plied to a set of qubits in order to reduce the entropy of
individual ones. It was found that such entropy extrac-
tion processes can be enhanced with the introduction
of interactions with a heat bath [60]. These so called
Heat Bath Algorithmic Cooling (HBAC) protocols, can
thus be thought of as sets of thermodynamical opera-
tions. For a review of HBAC in the context of NMR
experiments, see Ref. [61].
In such a scenario, we have a set of n qubits with iden-
tical individual Hamiltonians H
S
i
=
∆
2
σ
Z
⊗1
\S
i
, where
σ
Z
= |1ih1| − |0ih0| is the Pauli-Z operator, 1
\S
i
is the
identity on the complementary subspace to qubit i and
i = 0, .., n − 1. Here ∆ denotes the energy gap. The aim
is to perform operations that increase the polarisation
of a target qubit, say i = 0, as much as possible, tak-
ing advantage of the auxiliary qubits. Given a density
Accepted in Quantum 2018-01-14, click title to verify 14
●
β-swap
■
JC model
◆
PLT
●
●
●
● ●
■
■
■
■
■
◆
◆
◆
◆
◆
1 2 3 4 5
0.00
0.02
0.04
0.06
k
ϵ
Figure 8: Comparison of 2 qubit HBAC protocols. Here we
show how the polarisation = hσ
Z
i of a qubit increases with
the number of iterations k in the HBAC protocol described in
Section 6.1. We compare our β-swap protocol to the original
PLT protocol presented in Ref. [62]. One sees that the new
protocol reaches the asymptotic polarisation value more rapidly
than the original. Also the JC model performs better, even
though less remarkably so.
matrix ρ, the polarisation of the target qubit is defined
as
= Tr
ρ(σ
Z
⊗ 1
\S
0
)
.
Note that in terms of , the population of qubit i = 0
can be written as
p =
1
2
(1 + , 1 − ) .
Given that these protocols involve thermodynamical
operations on qubits, one can ask whether having access
to the set of ETOs provides an enhancement to the per-
formance of existing algorithms. We initiate the study
of this question by analysing the recent cooling scheme
introduced in Ref. [62].
In this work, the authors use protocols containing
PLTs to boost the performance of HBAC. Building on
that idea, we now present a modified HBAC based on
β-swaps, and explore how these improve the perfor-
mance of the protocols. In the low polarisation limit
we show that the asymptotic cooling of the HBAC pro-
tocol proposed in Ref. [62] can be achieved in a single
step using β-swaps. Furthermore, we give a preliminary
discussion of the cooling achievable within a Jaynes-
Cummings model, that also improves on the HBAC
based on PLTs although less significantly (see Fig. 8).
While this provides another illustration of the applica-
tions of our framework, we leave a more in-depth study
to further work.
Figure 9: HBAC protocol for 2 qubits. In this figure we present
a generalisation of the protocol given in [62, Fig. 4]. While
they consider a thermalisation step for the ETO, we will instead
utilise a β-swap. The box contains one iteration of the protocol.
6.1 Enhanced HBAC with two-qubit interactions
The scheme presented in Ref. [62] involves a set of n
qubits and an iterative procedure in which thermalisa-
tion of joint energy eigenstates is performed. We begin
by explaining the protocol for n = 2, which is illustrated
in Fig. 9.
The scheme begins with two qubits: S
0
(tar-
get) and S
1
(auxiliary), in their thermal states
g
b
=
1
2
(1 +
b
, 1 −
b
), where
b
denotes the polarisa-
tion of the Gibbs state and is related to the energy gap
of the Hamiltonian via:
b
= tanh
β
∆
2
.
The protocol in Ref. [62] then runs as follows:
1. Realise a population inversion on qubit S
1
via a
Pauli-X gate. The joint state of the two qubits is
now athermal.
2. Apply a fully thermalising PLT between the joint
energy levels |0i
S
0
|0i
S
1
and |1i
S
0
|1i
S
1
. This is an
ETO for which matrix acting on those two levels
can be written as
T
(00,11)
therm
=
1
1 + e
−β2∆
1 1
e
−β2∆
e
−β2∆
,
This can be seen from Eq. (7), since
T
(00,11)
therm
:= T
λ=1/(1+e
−β2∆
)
with ~ω = 2∆ and
(i, j) = (00, 11).
3. Reset qubit S
1
to the initial thermal state via an-
other fully thermalising PLT. In terms of the en-
ergy levels, this corresponds to applying two full
thermalisations, T
(00,01)
therm
and T
(10,11)
therm
.
4. Repeat until the polarisation of qubit S
0
reaches a
stationary value.
For small initial polarisation
b
, it was shown that
after k iterations of the above protocol the polarisation
of the target is given by [62]
n=2
k
=
3 −
1
2
k−1
b
. (28)
Accepted in Quantum 2018-01-14, click title to verify 15
Hence, the polarisation triples in the limit of large k.
More generally, for any
b
, the polarisation after each
run obeys the recursion relation
n=2
k+1
=
sech(β∆)
2
sinh
3
2
β∆
sech
β
∆
2
+
n=2
k
,
from this the asymptotic value
n=2
∞
= tanh
3
2
β∆
can
be derived.
Here, we shall modify the above algorithm, substitut-
ing the fully thermalising PLT of step 2 for a β-swap
between the two energy levels:
β
(00,11)
=
1 − e
−2β∆
1
e
−2β∆
0
.
The first iteration of this algorithm creates
g
⊗2
b
=
1
4 cosh
2
β
∆
2
e
β∆
, 1, 1, e
−β∆
σ
X
−→
1
4 cosh
2
β
∆
2
1, e
β∆
, e
−β∆
, 1
β-swap
−→
1
4 cosh
2
β
∆
2
2 − e
−2β∆
, e
β∆
, e
−β∆
, e
−2β∆
PLT
−→
1
4 cosh
2
β
∆
2
2 − e
−2β∆
+ e
β∆
, e
−β∆
+ e
−2β∆
⊗
1
1 + e
−β∆
1, e
−β∆
More generally, one obtains the following recursion
relation for the polarisation after each run
˜
n=2
k+1
=
˜
n=2
k
e
2β∆
− 1
+ e
3β∆
− 1
e
2β∆
+ e
3β∆
.
This also converges to tanh
3
2
β∆
in the limit of large
k. A comparison of the rate at which the two protocols
converge to this asymptotic value is given in Fig. 8.
If one focuses on the low polarisation regime, it can
be shown from the above derivation that for the β-swap
protocol
˜
n=2
k=1
= 3
b
.
That is, the asymptotic value in the low polarisation
regime is achieved after a single run of the algorithm.
This is in stark contrast to the algorithm using a ther-
malisation step where this polarisation is only achieved
in the limit of large k. From the discussion in Sec. 3
one can infer that this enhancement is due to the fact
that β-swaps are non-markovian processes.
6.2 Implementation via Jaynes-Cumming model
The previous results indicate that considering more gen-
eral thermal operations than thermalisations in cooling
protocols can improve performance dramatically. Of
Figure 10: HBAC protocol for 3 qubits. In this figure we present
a generalisation of the protocol given in [62, Fig. 6]. While
they consider a thermalisation step for the ETO, we will instead
utilise a β-swap. Alg
n=2
implements the algorithm from Fig. 9.
The box contains one iteration of the protocol.
course, a more detailed investigation is necessary in or-
der to assess the feasibility of the proposal within spe-
cific platforms and to fairly compare with other HBAC
proposals. While this goes beyond the scope of the
present investigation, a preliminary discussion of this
topic constitutes a useful illustration of how the theo-
retical framework presented here can be used in practice
to explore new interesting thermodynamic protocols.
For low initial polarisation (i.e. high temperatures,
the regime in which our protocol at face value improves
considerably over the proposal of Ref. [62]), the Jaynes-
Cummings model can only realise poor approximations
of β-swaps. Nevertheless, improvements can still be
seen. As an example, assume one implements step 2 of
the cooling protocol through a Jaynes-Cummings model
with β~∆ = 0.05 and a suitable coupling g that may
depend on the relevant experimental parameters. One
then has G
1|0
(t) following Eq. (9). G
1|0
reaches a first
local maximum quite quickly (at around
¯
t = 1.5t
th
with
t
th
being the time necessary to implement a full ther-
malisation). At that point, one has G
1|0
(
¯
t) ≈ 0.656.
Despite this being very far from the probability required
by a β-swap (i.e., G
1|0
≈ 1) one can see the improve-
ments over the PLT protocol, see Fig. 8. In fact, a
more detailed analysis shows that after a single step
the improvement on the HBAC protocol is linear in
G
1|0
(
¯
t)−G
1|0
(t
th
), i.e. in the excess de-excitation prob-
ability as compared to the full thermalisation process.
We only mention in passing that a variety of Jaynes-
Cummings protocols can be obtained by imposing dif-
ferent time constraints, and this will depend on the de-
tails of the experimental proposal.
6.3 Many qubit case
The above algorithms can be extended to act on an arbi-
trary number of auxiliary qubits S
1
, .., S
n−1
, enhancing
the final asymptotic polarisation for the target qubit
S
0
. It becomes an iterative algorithm, in the sense that
each step on n qubits requires an implementation of
the algorithm for n −1 qubits as detailed in Fig. 10 for
n = 3.
Initially all n qubits start in the thermal state. The
Accepted in Quantum 2018-01-14, click title to verify 16
●
β-swap
■
PLT
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
● ● ● ● ● ●
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
0 5 10 15 20
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
k
ϵ
Figure 11: Comparison of 3 qubit protocols. Here we show how
the polarisation increases with number of iterations when the
ETO is taken to be either a β-swap or a full thermalisation. One
sees that the new protocol reaches the asymptotic polarisation
value more rapidly than the original.
steps are the following
1. Apply the n − 1 algorithm to all but the target
qubit.
2. Apply a Pauli-X gate to each of these n −1 qubits.
3. Apply an ETO to the lowest and maximally excited
state of the n qubits. That is, |0i
S
0
..... |0i
S
n−1
and
|1i
S
0
..... |1i
S
n−1
.
4. Reset all the qubits but the target one in the ther-
mal state.
5. Repeat until asymptotic value is reached.
It is shown in Ref. [62] that if the ETO is chosen to be
a full thermalisation, then for general n the asymptotic
polarisation is given by
n
∞
= tanh
2
n−1
−
1
2
β∆
. (29)
What happens if the ETO is instead chosen to be a
β-swap? It can be shown that in general the asymptotic
value of Eq. (29) stays the same, but the convergence
to it is still enhanced. We give a numerical example of
this for the case of n = 3 in Fig. 11. However, for small
initial polarisation (and in contrast to the 2-qubit case),
one run of the n qubit algorithm utilizing a β-swap is
not sufficient to achieve the asymptotic polarisation.
7 Discussion and conclusions
In this paper we have introduced the thermodynamic re-
source theory of elementary thermal operations (ETOs)
and shown that for certain parameter regimes that they
can be well reproduced by a simple Jaynes-Cummings
interaction. While we proved that they are not as
powerful as the much studied resource theory of ther-
mal operations (TOs), both in terms of population dy-
namics and coherence manipulations, this link to the
Jaynes-Cummings model makes ETOs more experimen-
tally palatable than the complete set of TO. In doing so,
we have show that natural analogues of powerful theo-
rems from the theory of doubly stochastic matrices do
not apply to finite temperature thermodynamics; phys-
ically this reflects a dependence of the thermodynamic
constraints on the specific experimental restrictions at
hand, in contrast to the large degree of universality ob-
served in the infinite temperature limit. To overcome
this we provide necessary and sufficient conditions for a
given population dynamics to be achievable under ETO,
but also some general tools to study more general re-
stricted thermodynamic theories, including the resource
theory of the Jaynes-Cummings model.
This set of results shows how natural limitations in
the control assumed in a thermodynamic resource the-
ory translate into stricter constraints than the standard
thermodynamic single-shot quantities. For example, we
have proven that for any p for which the maximum ex-
tractable work under thermal operations is positive one
has
W
JC
distil
(p) ≤ W
ETO
distil
(p) < W
TO
distil
(p) = F
0
(p) +
1
β
log Z
S
.
Hence, the min-free energy F
0
(p) =
−β
−1
log
P
i:p
i
>0
e
−βE
i
is not the thermodynami-
cally relevant quantity in the theory of ETOs or
the Jaynes-Cummings model. Similarly, we have
seen that there are incoherent states that cannot
be created under ETOs from a thermal state by
adding work. Hence, for some p, W
ETO
form
(p) = +∞,
even though W
ETO
form
(p) = F
∞
(p) + kT log Z
S
< +∞
for every p; this shows that the max-free energy
F
∞
(p) = β
−1
log max
i
p
i
e
βE
i
is also not the ther-
modynamically relevant quantity in the restricted
scenario.
It is worth comparing our framework with other ap-
proaches to providing experimentally feasible realisa-
tions of thermal operations. In Ref. [33] it was shown
that any population dynamics achievable using TO can
also be produced by a protocol which first appends
a single thermal qubit to the system and then ap-
plies sequences of partial levels thermalisations (PLTs)
while slowly adjusting the energy levels of the combined
system-qubit Hamiltonian (in a way judiciously chosen
so as to not cost any work) before discarding the qubit
in its original thermal state. While such a set of op-
erations is more powerful than that considered here, in
addition to the ability to selectively couple any pair of
Accepted in Quantum 2018-01-14, click title to verify 17
energy levels in the system to the environment (as is also
the case for ETOs) it also requires the ability to pre-
cisely manipulate energy levels and thus adds an extra
level of complexity to their implementation. This ad-
vantage suggests that NMR experiments may provide a
platform to implement ETOs.
Since our necessary and sufficient conditions can be
directly applied also to the Jaynes-Cummings model,
even in parameter regimes where this is not well repro-
duced by ETOs, we can readily answer physically mo-
tivated questions. For example, in a cooling scenario
one can ask: what is the maximum ground state pop-
ulation achievable by applying sequences and convex
combinations of 2-level Jaynes-Cummings interactions
in RWA, when the bath has temperature T
B
? In Fig. 7
the answer corresponds to the ground state population
of the extremal point closest to (1, 0, 0). In the exam-
ple of Fig. 7 the two qutrit transitions are ω
10
= 6.2
GHz, ω
21
= 5.5 GHz and the bath has temperature
T
B
= 0.1K; if the initial state is p = (0.1, 0.3, 0.6), we
can bound the maximum final ground state population
attainable through arbitrary Jaynes-Cummings interac-
tions p
max
g
by 0.7607 < p
max
g
< 0.7757. Note that this
is much higher than the population that one would get
by a trivial strategy consisting of simply thermalising
the system with the bath at temperature T
B
= 0.1K,
which gives p
th
g
≈ 0.4921. A generic sequence of ETOs
can get to p
max
g
= 0.8061, while a general thermal opera-
tion achieves p
max
g
= 0.895, illustrating that the cooling
power of the different sets is distinct, in accordance to
the lack of universality discussed in Sec. 5.1. Another
crucial aspect is that, unlike other resource theory re-
sults, we know exactly the sequence of ETOs or Jaynes-
Cummings interactions needed to get to the final state,
so that we can devise explicit protocols.
Finally, there remain many open theoretical ques-
tions regarding ETOs. Firstly, it would be interesting
to investigate whether a simple (and similarly experi-
mentally motivated) operation can be added to them
which allows the complete power of TOs to be realised.
One could also investigate more general work extrac-
tion paradigms than the deterministic one consider here
in Section 4.3, such as those including a weight system
[10, 23].
Thermodynamic resource theories have insofar been
looking at fundamental constraints, but it is also nec-
essary to close the gap with experimental implementa-
tions. We hope our contribution moves in this direction.
While we have connected theory and physical models
and gave tools to compute what protocols are possi-
ble, it remains to be seen how these results may help
us devise efficient thermodynamical protocols for the
quantum and nano scale. The improvements observed
in the HBAC protocol gives some preliminary evidence
in this direction.
Acknowledgments
We would like to thank D. Jennings, A. Milne, A. Levy,
T. Rudolph and R. Uzdin for comments on a previous
draft and M. Christandl, R. Kosloff, N. Yunger Halpern,
M. Horodecki, S. F. Huelga, P. Mazurek, M. Plenio,
P. Salamon and A. Smith for useful discussions. ML
is particularly grateful to M. M¨uller for pointing out
a mistake in a previous draft, which led to many de-
velopments, and to P. Mazurek for interesting discus-
sions about generalisations of Theorem 6 to k-level op-
erations. The result that ETO ⊂ TO (which follows
from Corollary 5) has been obtained independently by
different methods by P. Mazurek and M. Horodecki
in [36]. ML acknowledges financial support from the
Spanish MINECO (Severo Ochoa SEV-2015-0522 and
project QIBEQI FIS2016-80773-P), Fundacio Cellex,
and Generalitat de Catalunya (CERCA Programme
and SGR 875). CP acknowledges financial support from
the European Research Council (ERC Grant Agree-
ment no 337603), the Danish Council for Independent
Research (Sapere Aude) and VILLUM FONDEN via
the QMATH Centre of Excellence (Grant No. 10059).
AMA acknowledges support from the FQXi. This work
was supported by EPSRC and in part by COST Action
MP1209.
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A Energy-preservation and complete
passivity
In this appendix, we briefly analyse why the con-
dition [U, H
S
+ H
B
] = 0 is necessary for the (state-
independent) set of allowed operations in Eq. (1) to re-
quire no work to be performed. To do so, we will use
the notions of passivity and complete passivity [63, 64].
It is useful to rewrite the restriction on U in
an equivalent way. Let γ
β
(H) :=
e
−βH
Tr[e
−βH
]
. Then
[U, H
S
+ H
B
] = 0 is equivalent to
U (γ
β
(H
S
) ⊗ γ
β
(H
B
)) U
†
= γ
β
(H
S
) ⊗ γ
β
(H
B
). (30)
Note that in the present manuscript we are assuming
that the Hamiltonian of system and bath are the same
at the end of the protocol as they are at the beginning
(for generalisations, see Ref. [6, 8]). If U does not sat-
isfy the previous equation, then it would be possible to
transform the completely passive state γ
β
(H
S
)⊗γ
β
(H
B
)
into one that is not. By applying such U on several
copies of γ
β
(H
S
) ⊗ γ
β
(H
B
), we would then get a state
σ that is not passive, i.e. a state from which it is pos-
sible to extract work. Specifically, there exists a time-
dependent Hamiltonian H(t) and τ > 0 such that
W =
Z
τ
0
Tr
σ(t)
dH(t)
dt
dt < 0, (31)
where σ(0) = σ, H(0) = H(τ ) = 0 and σ(t) is given by
the Schr¨odinger equation for H
S
+H
B
+H(t) [64]. This
tells us that an experimenter turning on the interaction
H(t) would be able to extract work from σ. Since no
work could be extracted in this way from any number of
copies of γ
β
(H
S
) ⊗ γ
β
(H
B
), by consistency the unitary
U must cost some work to perform, in contradiction to
our assumption.
B Proof of Lemma 1
Lemma 1. Every population dynamics satisfying prop-
erties 1 and 2 can be realised by an ETO involving a
single-mode bosonic bath.
Proof. Let |E
0
i, |E
1
i be two eigenstates of H
S
with
energies ~ω
0
and ~ω
1
, with ω
1
> ω
0
. Intro-
duce a single-mode thermal bath with Hamiltonian
H
B
=
P
∞
n=0
n~ω |nihn|, with ω = ω
1
−ω
0
. Let the bath
be in the state γ
β
(H
B
), with γ
β
(H
B
) = e
−βH
B
/Z
B
,
Accepted in Quantum 2018-01-14, click title to verify 21
where Z
B
= (1 − e
−β~ω
)
−1
. Take now the energy-
preserving unitary (appearing in [7])
U = |E
0
ihE
0
|⊗|0ih0|+
∞
X
n=1
1
X
k,k
0
=0
v
kk
0
|E
k
ihE
k
0
|⊗|n − ki
n − k
0
,
where V (whose elements are v
kk
0
) is a 2 by 2 unitary. Hence,
by Eq. (1), the transformation
E(ρ) = Tr
B
U(ρ ⊗ γ
B
)U
†
,
is an ETO. One can readily compute the induced population
dynamics G (defined as in Eq. (2))
G
1|0
= e
−β~ω
|v
10
|
2
, G
0|1
= |v
01
|
2
,
and G
0|0
= 1 − G
1|0
, G
1|1
= 1 − G
0|1
. Since V is unitary
|v
10
|
2
= |v
01
|
2
, thus G
1|0
= e
−β~ω
G
0|1
. Given G satisfying
properties 1 and 2, we can take V of the form
V =
cos(x) −i sin(x)
−i sin(x) cos(x)
.
and choose x such that sin
2
(x) = G
0|1
.
C Upper bound on G
max
(
¯
β)
Consider the bound on G
0|1
(s) constructed following
these steps:
1. Split the series in Eq. (9) into a finite sum F up
to m, plus a residue R: G
0|1
(s) = F + R, with
F =
P
m
n=1
sin
2
(s
√
n)
e
−β~ω(n−1)
Z
B
. In the residue,
bound each factor sin
2
(s
√
n) with 1. Then one has
R = e
−mβω
.
2. In F , bound every oscillating term with irrational
frequency as sin
2
(s
√
n) ≤ 1.
This provides a bound on G
0|1
(s) with a periodic func-
tion of s, which can then be simply analysed. For
m = 4, the function has one global maximum at
s = π/2 when
¯
β ≥ log(4)/3. On the other hand,
when
¯
β < log(4)/3, it has two global maxima at
s = ±arccos(−e
3
¯
β
/4)/2. The values achieved at the
maxima give the bounds provided in the main text.
D Collision models
Let the relevant two-level subsystem, with energies ~ω
0
,
~ω
1
, be described by the (possibly unnormalised) dis-
tribution p(0) := (p
0
(0), p
1
(0)). Without loss of gener-
ality, take ω
0
= 0 and ω
1
= ω. Let p(n) := G
n
p(0),
where G is the 2-level Gibbs-stochastic matrix induced
by an individual collision. One can easily compute
p
0
(1) = p
0
(0)(1 − λZ) + λN , where Z = 1 + e
−β~ω
,
N = p
0
(0) + p
1
(0) and λ := G
0|1
. By recursion,
p
0
(n) = p
0
(0)(1 − λZ)
n
+
N
Z
(1 − (1 − λZ)
n
) .
Then p
0
(∞) := lim
n→∞
p
0
(n) = N/Z, independently of
the initial distribution. Because the map is stochastic,
p(∞) = Ng
where g = (1/Z, 1 − 1/Z) is a thermal distribution
within the two-level subsystem.
Reasoning as in Ref. [44], we can define n = t/t
int
,
with t
int
the duration of a single interaction. We then
take the limit of short interactions t
int
→ 0, λ → 0,
while keeping Zλ/t
int
→ ξ ≥ 0 finite (Z is a constant
that we absorbed in the definition of ξ). One can check
that (1 − λZ)
n
→ e
−t/ξ
and hence
p(t) = e
−t/ξ
p(0) + N(1 − e
−t/ξ
)g. (32)
Here ξ ≥ 0 is a free parameter. Hence the subsystem
decays exponentially to a distribution proportional to
the Gibbs distribution (whenever ξ > 0).
E The TO Cone
In this appendix we describe how to construct C
TO
(p)
given a state p.
Given a state p and Hamiltonian H
S
, let
c
p
: [0, Z
S
] → [0, 1] be the function defined by the
thermo-majorisation curve of p i.e. c
p
(x) is the height
of the thermo-majorisation curve of p at x.
Lemma 12. Given p, consider the following distribu-
tions p
π
∈ C
TO
(p) constructed for each permutation
π ∈ S
d
. For i ∈ {0, . . . , d − 1}:
1. Let x
π
i
=
P
i
j=0
e
−βE
π
−1
(j)
and y
π
i
= c
p
(x
π
i
).
2. Define p
π
i
:= y
π
π(i)
− y
π
π(i)−1
, with y
−1
:= 0.
Then all extremal points in C
TO
(p) have the form p
π
for some π. This is in particular implies that C
TO
(p)
has at most d! extremal points.
Note that each π ∈ S
d
defines a different β-ordering
on the energy levels of H
S
. For each possible β-order,
p
π
defines a thermo-majorization curve c
π
p
which coin-
cides with the thermo-majorization curve of p at the
points (x
π
i
, y
π
i
). In general, one has c
p
(x) ≥ c
p
π
(x)
for every x in [0, Z
S
] so p
π
∈ C
TO
(p). Intuitively, p
π
is a distribution with β-ordering defined by π that is
“just thermo-majorised” by p. More precisely, there
are no other distributions q with the same β-order as
p
π
whose thermo-majorisation curve lies under that of
p and above that of p
π
.
Accepted in Quantum 2018-01-14, click title to verify 22
Proof. We show that all extremal points of C
TO
(p)
must have the above form. To see this, consider a
state q ∈ C
TO
(p) and suppose it were an extremal
point. Let π
q
denote the β-order of q and suppose
that q 6= p
π
q
. Now q is thermo-majorised not only
by p, but also by p
π
q
. This follows from the fact
that q and p
π
q
have the same β-ordering and, fur-
thermore, c
q
(x
π
q
i
) ≤ c
p
(x
π
q
i
) = c
p
π
q
(x
π
q
i
). Using Theo-
rem 12 of Ref. [33], we have that there exists a se-
quence of PLTs transforming p
π
q
into q. Using Eq. (5),
we can then write q = c
0
p
π
+
P
x
c
x
β
x
p
π
q
where β
x
denotes a non-trivial sequence of β-swaps and c
0
> 0.
As q 6= p
π
q
, at least one other c
x
is non-zero. This
contradicts q being extremal.
With this lemma in place, C
TO
(p) can be found by
constructing the states p
π
for each π ∈ S
d
and taking
the convex hull.
F Infinite work of formation: a coun-
terexample not based on rank
Here we show that the phenomena of infinite work of
formation under ETO is not limited to the formation of
states without full rank.
First note that as the initial state in formation pro-
cesses is g ⊗ |0ih0|, without loss of generality we can
begin by applying β-swaps between elements of A
0
and
A
w
according to the injection Φ (here Φ is actually a
bijection as g has full support). To see this, note that
applying β-swaps between elements of A
0
at any stage
of the protocol will either do nothing (if both elements
are occupied) or leave us unable to move all popula-
tion out of A
0
(if one of the elements is unoccupied)
as we will no longer be able to find a suitable injective
map between A
0
and A
w
. Similarly, applying β-swaps
between elements of A
w
can either be postponed until
we have applied all β-swaps associated with Φ (if both
elements are occupied) or leave us unable to move all
population out of A
0
(if one of the elements is unoccu-
pied). Applying β-swaps according to Φ to g ⊗ |0ih0|
creates a state of the form g
π
⊗|1ih1| where g
π
is a per-
mutation of g. Which permutations can be achieved is
determined by the value of w. From this we can con-
clude that any p such that g
π
ETO
9 p for any choice of
π we have W
ETO
form
(p) = −∞. Examples of this include
any state such that p
i
e
βE
i
>
e
β
(
E
d−1
−E
0
)
Z
S
, for some i.
Accepted in Quantum 2018-01-14, click title to verify 23