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 [14] 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. [516], considerations on the 3rd law [1719]
and the construction of fluctuation theorems [2023].
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
d1
). 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
d1
Z
S
, Z
S
=
d1
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 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 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
d1
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(n1)
Z
B
, G
1|1
= 1G
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
d1
), 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
d1
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 r1. 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 r1. 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
pq
, is the highest value of w such that:
p |0ih0|
ETO
q |1ih1|. (16)
If W
ETO
pq
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 [79]. The main difference
with other notions of work, such as those involving ap-
plying arbitrary unitaries to systems and the notion of
passive states