# Elementary Thermal Operations and the Universality of Thermodynamic
Constraints

Matteo Lostaglio11footnotemark: 1 Department of Physics, Imperial College
London, London SW7 2AZ, United Kingdom

###### Abstract

To what extent is the resource theory approach to thermodynamics relevant for
realistic experimental scenarios? We deconstruct this framework, showing that
each transition among incoherent states allowed in the theory can be obtained
by sequentially applying elementary thermal operations. These only couple the
bath to single transitions in the system and satisfy detailed balance. We show
that most of them can be realised within one of the simplest thermodynamic
models, a resonant Jaynes-Cummings interaction with a single mode bosonic
bath, and discuss the link to thermalisation models. No external control of
the Hamiltonian or ancillae are needed. We also present an extension of
Birkhoff’s theorem for bistochastic matrices to thermodynamics, and discuss
its consequences for the problem of simulating arbitrary thermal processes.
Finally, we show that the constraints found within thermal operations for
transitions among incoherent states are to a large extent _universal_ , i.e.
they apply to a broad class consisting of very different thermodynamic
frameworks.

Physicists agree that thermodynamics is one of the pillars of physics, but
disagree on its foundations and scope. The recent surge in interest in the
quantum information/single-shot approach to thermodynamics Janzing  _et al._
(2000); Dahlsten  _et al._ (2011); Brandão  _et al._ (2013); Åberg (2013);
Horodecki and Oppenheim (2013); Gour  _et al._ (2015); Brandão  _et al._
(2015); Masanes and Oppenheim (2014); Narasimhachar and Gour (2015); Renes
(2014); Åberg (2014); Lostaglio  _et al._ (2015a); Ng  _et al._ (2015);
Ćwikliński  _et al._ (2015); Lostaglio  _et al._ (2015b, c); Gemmer and Anders
(2015); Woods  _et al._ (2015); Alhambra  _et al._ (2015); Egloff  _et al._
(2015); Narasimhachar and Gour (2016); Wilming  _et al._ (2016); Goold  _et
al._ (2015) is no exception and has attracted some criticism.

The main aim of this paper is to present a “universal set of thermodynamic
gates”, i.e., a simple set of transformations that allows one to perform any
thermal processing achievable by thermal operations among incoherent states.
This set is given by elementary processes involving two-level subsystems and
satisfying detailed balance. Going beyond a previous result Perry  _et al._
(2015), we do not require any external control of the Hamiltonian or ancillae.

This has multiple consequences. First, we show that the gates in the universal
set can be approximated within the simplest Jaynes-Cummings model in the
rotating-wave approximation (RWA), paving the way for possible experimental
realisations. On the other hand, we address the connection to fluctuation
theorems. In fact, we show that even though every thermodynamic transition can
be realised by a process falling within the assumptions of Crooks theorem
Crooks (1998), there are also _indecomposable_ processes for which the theorem
cannot be applied. On the theoretical side, this leads to a generalisation of
a famous theorem by Birkhoff on bistochastic maps to thermodynamics, with
implications for the simulation of a generic thermal process and for the
geometry of thermalisation.

Another aim of this work is to clarify the scope of the resource theory
results. In this respect, we find that, at least for transformations among
incoherent states, the resource theory constraints actually apply to a wide
range of thermodynamic models. Moreover, we highlight connections between the
resource theory language and other approaches, such as collision models and
the master equation formalism, with the aim of developing a common language.
As an application, we discuss constraints that can be deduced from resource
theory for thermalisation models.

### Criticism of Thermal Operations

The resource theory approach to thermodynamics is based on the concept of
_Thermal Operations_ (TOs) Janzing  _et al._ (2000); Brandão  _et al._ (2013).
These are energy-preserving interactions of a system with a thermal bath. In
particular one assumes that given a system in an initial state $\rho$ and with
Hamiltonian $H_{S}$, this can be coupled to a bath at temperature $T_{B}$ with
_arbitrary_ Hamiltonian $H_{B}$ by _any_ unitary $U$ satisfying
$[U,H_{S}+H_{B}]=0$. So any Thermal Operation $\mathcal{T}$ can be written as
$\mathcal{T}(\rho)=\mathrm{Tr}_{B}\left[U(\rho\otimes e^{-\beta
H_{B}}/Z_{B})U^{\dagger}\right]$. Here $Z_{B}=\mathrm{Tr}\left[e^{-\beta
H_{B}}\right]$, $\beta=(kT_{B})^{-1}$; $H_{S}$ and $H_{B}$ should be
understood as the non-interacting Hamiltonians of system and bath, and $U$ as
the overall effect of an interaction Hamiltonian which is turned on at some
time $t_{i}$ and off at some later time $t_{f}$. The resource theory approach
to thermodynamics assumes that this is the set of transformations that can be
implemented for free, without any _external_ resource being consumed. In
particular, no classical work source is involved.222A battery may be needed to
turn on and off the couplings, or a field may be used to extract work. In the
resource theory approach these can be used, but they must be explicitly
included in the description as quantum-mechanical systems.

This framework has been criticised in at least two, somehow opposing, ways,
which we now present. TOs clearly contain a large set of transformations that
can be implemented only through a complete control of the system-bath
interaction $U$. This in general requires a fine control of the microscopic
degrees of freedom of the bath, on top of the assumption that any bath
Hamiltonian $H_{B}$ can be engineered; hence, achieving the full set of TOs
seems completely unrealistic from an experimental point of view Perry  _et
al._ (2015).

One may respond that we are interested in “fundamental” limitations, so that
by allowing such a large set of transformations we are able to embrace a large
variety of experimental situations; which is to say, the constraints derived
will still provide necessary conditions, however weak, to practically relevant
scenarios. But conversely TOs appear, in other ways, overly restrictive. In
particular, the constraint of energy conservation may seem a serious
limitation, as typically it will be violated as soon as interactions are
turned on. A deeper analysis of the connections to experimental
implementations is then crucial.

In this paper we aim to make a step towards overcoming the above criticism and
clarifying how to connect the resource theory approach to more experimentally-
friendly frameworks. Given a finite-dimensional system with Hamiltonian
$H_{S}=\sum_{i}\hbar\omega_{i}\left|i\right\rangle\\!\\!\left\langle
i\right|$, we will focus on transitions among incoherent states $\rho$ and
$\sigma$ i.e. $[\rho,H_{S}]=[\sigma,H_{S}]=0$. Let $\bm{p}$ and $\bm{q}$ be
the eigenvalues of $\rho$ and $\sigma$, representing the occupations of each
energy level. Denote by $\mathscr{T}$ the set of thermal stochastic processes,
i.e., all stochastic matrices $T$ with $T_{i|j}=\left\langle
i\right|\mathcal{T}(\left|j\right\rangle\\!\\!\left\langle
j\right|)\left|i\right\rangle$ for some TO $\mathcal{T}$ (we say that $T$ is
_induced_ by $\mathcal{T}$). These represent the random energy exchanges
between system and bath and do not carry a description of how superpositions
among different energy levels evolve. Since, by assumption, no such
superpositions are present in $\rho$ and $\sigma$, a TO $\mathcal{T}$ with
$\mathcal{T}(\rho)=\sigma$ exists if and only if there exists a thermal
process $T$ with $T\bm{p}=\bm{q}$ (see Appendix A.1).

### Elementary Thermal Operations

Among all stochastic processes describing the random heat exchanges between
system and bath, a subset stands out as particularly simple and physically
motivated. These are all stochastic processes $E$ satisfying two conditions:

  1. 1.

Only two energy levels of the system are involved.

  2. 2.

Detailed balance, $E_{i|j}=e^{-\beta\hbar(\omega_{i}-\omega_{j})}E_{j|i}$, is
satisfied.

This set of stochastic matrices will be called _elementary detailed balanced
processes_ (EDPs) and denoted by $\mathscr{E}$. One can check that
$\mathscr{E}\subset\mathscr{T}$, i.e., all EDPs can be induced by means of TOs
(Appendix B.2). We then call the set of TOs generating such simple stochastic
processes _Elementary Thermal Operations_ , or ETOs. Any subset of ETOs that
generates all EDPs will be called _complete_.

The heat exchanges described by $\mathscr{E}$ are much simpler than the
generic process in $\mathscr{T}$. In fact, each EDP is defined by the two
levels it acts on plus a single parameter. Of course, mathematical simplicity
is not by itself sufficient. Two natural questions arise: can we find a
complete set of ETOs that can be realised within a simple physical model? What
transitions between incoherent states can be achieved with a complete set?

### Physical realisations

#### The Jaynes-Cummings model approximates a complete set of ETOs

One of the core models describing the interaction of matter and radiation is
the Jaynes-Cummings model, and so we start investigating the previous
questions within it. For simplicity, consider a system whose relevant
transition frequencies are all well separated from each other. Assume one is
allowed to couple every transition of the system to a single-mode bosonic bath
in a thermal state through a resonant Jaynes-Cummings coupling in RWA
approximation, $H_{JC}=g(\sigma_{+}\otimes a+\sigma_{-}\otimes a^{\dagger})$,
for a time $t$. Here $a^{\dagger}$, $a$ are creation/annihilation operators on
the bath and $\sigma_{\pm}$ excite/de-excite a transition
$\omega:=\omega_{1}-\omega_{0}>0$ on the system. This model gives TOs. A
standard calculation provides the induced stochastic processes (Appendix C.1):

$J_{1|0}(s)=\sum_{n=1}^{\infty}\sin^{2}(s\sqrt{n})g_{n},\quad
J_{0|1}(s)=\sum_{n=1}^{\infty}\sin^{2}(s\sqrt{n})g_{n-1},$

where $g_{n}=e^{-\beta\hbar\omega n}/Z_{B}$ is the occupation of the $n$-th
energy level of the bath and $s=gt/\hbar$.333For any $s$, one can take $g$
small enough and $t$ large enough to avoid the short times/strong couplings
regime where the model breaks down. Also, $J_{0|0}=1-J_{1|0}$,
$J_{1|1}=1-J_{0|1}$. One has $J_{1|0}/J_{0|1}=e^{-\beta\hbar\omega}$, i.e. $J$
is an EDP. So the model generates ETOs defined by a single control, $s$. The
question then arises, to what extent can _all_ $E\in\mathscr{E}$ be achieved?

The independent parameter for a generic EDP on two levels can be chosen to be
the de-exciting probability $E_{0|1}$. If, for every $E_{0|1}\in[0,1]$, there
exists an $s$ such that $J_{0|1}(s)=E_{0|1}$, then all EDPs can be achieved.
In Appendix C.1 we show that in the Jaynes-Cummings model one can reach all
de-exciting probabilities in the interval $[0,J_{{\rm
max}}(\beta\hbar\omega)]$, where $J_{{\rm max}}(\beta\hbar\omega)$ lies in the
region presented in Figure 1. In particular when $\beta\rightarrow\infty$ (the
zero temperature limit) $J_{{\rm max}}\rightarrow 1$, so the Jaynes-Cummings
model gives a complete set of ETOs. But also the finite temperature scaling is
rather favourable: at room temperature $T_{B}=300K$ and frequencies of
$10^{13}$ Hz or above, and at millikelvin temperatures with frequencies of
around $10^{8}$ Hz, one has $J_{{\rm max}}(\beta\hbar\omega)>0.98$.

We conclude that, for a wide interval of reasonable parameters, almost all
EDPs can be realised within a simple subset of TOs corresponding to a
physically reasonable model, with no need to implement general unitaries or
engineering thermal states with arbitrary Hamiltonian. Of course, one still
requires considerable frequency control as we must be able to separately
couple to every transition frequency of the system. This is feasible only if
we are interested in the thermodynamics of small enough systems.

Figure 1: _Realising EDPs in the simplest Jaynes-Cummings model:_ to achieve
every elementary detailed balanced process (EDP), one needs to be able to
obtain all de-exciting probabilities $E_{i|j}\in[0,1]$,
$\omega=\omega_{j}-\omega_{i}>0$. In Appendix B.2 we show that one can realise
every transition probability within $[0,J_{{\rm max}}(\beta\hbar\omega)]$,
where $J_{{\rm max}}(\beta\hbar\omega)$ lies somewhere within the shaded area.
Hence, far from the high temperature limit, almost all EDPs can be achieved
within the Jaynes-Cummings model.

#### Thermalisation models are Markovian ETO

To bridge the gap between resource theories and applications, we investigate
the connection to thermalisation models and the master equation formalism,
widely used in quantum thermodynamics Kosloff (2013).

Consider a _collision model_ of thermalisation of a system in the presence of
a large thermal bath Ziman  _et al._ (2005). A thermal particle from the bath
approaches, interacts with a two-level subsystem and scatters away; the
interaction conserves energy and the particle is lost in the bath Scarani  _et
al._ (2002). Each “collision” is then an ETO. Reasoning as in Scarani  _et
al._ (2002), one finds that the combined effect of many identical weak
interactions on each two-level subsystem, initially described by a population
$\bm{p}(0)$, leads to an exponential relaxation to the thermal distribution
$\bm{g}$ (Appendix C.2):

$\bm{p}(t)=e^{-t/\xi}\bm{p}(0)+N(1-e^{-t/\xi})\bm{g},$ (1)

where $\xi>0$ and $N$ is the normalisation of $\bm{p}(0)$.

Alternatively, from the master equation point of view, we can consider Davies
maps, which are routinely used as a simple model of a system weakly
interacting with a large thermal bath Davies (1974), or in the low-density
limit Dümcke (1985). On two-level systems, one can check that they induce the
evolution of Eq. (1).

Now, what is the connection with the resource theory approach? This can be
clarified as follows. _Markovian ETOs_ are defined here as the subset of ETOs
whose induced processes $E\in\mathscr{E}$ are embeddable stochastic matrices,
i.e. there is some generator $L$ and $t\geq 0$ such that $E=e^{Lt}$ Davies
(2010). Then

######  Proposition 1.

Markovian ETOs induce the same evolution (1) of collision models and Davies
maps.

###### Proof.

For $E$ to be a valid stochastic matrix, $L$ must satisfy $L_{i|j}\geq 0$ for
$i\neq j$ and $\sum_{i}L_{i|j}=0$ for all $j$ Davies (2007). Detailed balance
is equivalent to $L_{i|j}=e^{-\beta\hbar(\omega_{i}-\omega_{j})}L_{j|i}$. A
direct calculation then gives Eq. (1). ∎

### Elementary Thermal Operations are universal for incoherent processes

#### Sequential application of ETOs is universal

The Jaynes-Cummings model approximates a complete set of ETOs. But what
thermodynamic transformations can we achieve with a complete set of ETOs? We
show that we can realise _every_ incoherent transition possible under Thermal
Operations:

######  Theorem 2.

Let $[\rho,H_{S}]=0$. Then all transitions $\rho\rightarrow\sigma$ that can be
achieved through TOs can be obtained by a finite sequence of ETOs drawn from a
complete set and energy-preserving unitaries on the system alone.

The proof can be found in Appendix D.1. If $\bm{p}$ and $\bm{q}$ are the
eigenvalues of $\rho$ and $\sigma$, the proof consists in showing that the
thermal process $T\in\mathscr{T}$ such that $T\bm{p}=\bm{q}$ can always be
chosen to be a sequence of EDPs.

From the previous analysis of the Jaynes-Cummings model we conclude that, by
resonant interactions between two level subspaces and single-mode bosonic
baths and by energy-preserving unitaries on the system alone, we can realise
every incoherent thermal processing that could be achieved by arbitrary TOs.

#### Crooks theorem and indecomposable thermal processes

Theorem 2 ensures that, if there exists a thermal process $T\in\mathscr{T}$
such that $T\bm{p}=\bm{q}$, then we can achieve the same final state through a
sequence of steps, each satisfying the detailed balance condition. In other
words, we can always choose the thermodynamic process to satisfy Crooks
theorem Crooks (1998). More subtly, however, this does _not_ imply that all
thermal processes $T\in\mathscr{T}$ can be decomposed as a sequence of EDPs
(see Fig. 2). An explicit example of such _indecomposable_ process can be
constructed for trivial Hamiltonians, see Marshall  _et al._ (2010), Chapter
2.

#### Thermal Birkhoff’s theorem and simulation of thermal processes

The existence of indecomposable maps implies that not all thermal processes
can be simulated as a sequence of EDPs. What is needed then to simulate
generic thermal processes? The answer to this problem for bistochastic
processes is a classic theorem by Birkhoff: every bistochastic matrix can be
decomposed as a convex mixture of permutations Marshall  _et al._ (2010). Is
there a thermodynamic analogue?

Recall that every permutation is made up of a sequence of transpositions.
Thermodynamically, however, transpositions are not allowed processes due to
the unavoidable dissipation of heat in the bath. Given any two energy levels
$0$ and $1$, with $\omega_{1}-\omega_{0}=\omega>0$, we can construct a process
$E$ moving all the population from the upper to the lower level, $E_{0|1}=1$;
but detailed balance limits the probability of the inverse process to
$E_{1|0}=e^{-\beta\hbar\omega}$. Stochastic processes involving only two
levels and with the previous transition probabilities, as well as the
identity, are the thermodynamic analogue of transpositions. As such, they will
be called _thermo-transpositions_. By the same logic, sequences of thermo-
transpositions will be called _thermo-permutations_. Similarly to
permutations, these constitute a finite set. We then have the following
extension of Birkhoff’s theorem:

Figure 2: _Indecomposable thermal processes:_ If there exists a thermal
process $T\bm{p}=\bm{q}$, then there exists also $T^{\prime}\in\mathscr{T}$
that achieves the same thermodynamic transition and is composed of elementary
steps satisfying Crooks theorem. However, $T$ may not be decomposable in this
way.

######  Theorem 3.

$\mathscr{T}$ coincides with the set of $T$ that can be written as a convex
combination of thermo-permutations $\\{P^{i}\\}$:

$T=\sum_{i}\lambda_{i}P^{i},$ (2)

where $\lambda_{i}\geq 0$ and $\sum_{i}\lambda_{i}=1$.

For the proof, see Appendix D.2. In terms of ETOs,

######  Corollary 4.

Every thermal process $T\in\mathscr{T}$ can be simulated by finite sequences
of ETOs drawn from a complete set and classical randomness.

As already discussed, a complete set of ETOs can be approximated within the
Jaynes-Cummings model. So this simple physical model is not only sufficient to
(approximately) obtain all final states permitted by thermal processes; more
strongly it also allows, when coupled to classical randomness, to
(approximately) simulate arbitrary thermal processes.

An intermediate lemma in proving the previous theorem shows that the set of
thermal processes $\mathscr{T}$ coincides with the set $\mathscr{G}$ of
stochastic maps that preserve the Gibbs distribution $\bm{g}$:

######  Lemma 5.

$\mathscr{G}=\mathscr{T}$.

As an application, we show how this makes the proof of the core result
Horodecki and Oppenheim (2013) in the resource theory of thermodynamics
straightforward (Appendix D.3). Moreover, we discuss consequences for the
optimisation of coherence preservation under Thermal Operations (Appendix
D.4).

#### Geometric characterisation of the thermal cones

To characterise the set of states accessible through Thermal Operations from
an incoherent $\rho$ with spectrum $\bm{p}$, the central non-trivial problem
is to obtain $\mathscr{T}_{\bm{p}}$, the _thermal cone_ of $\bm{p}$. This is
defined as $\mathscr{T}_{\bm{p}}:=\\{T\bm{p}:T\in\mathscr{T}\\}$, i.e. the set
of final populations $\bm{q}$ that can be achieved from $\bm{p}$ by means of
thermal processes. The thermal cone has the following geometric
characterisation (see Fig. 3):

######  Corollary 6.

$\mathscr{T}_{\bm{p}}$ is the convex hull of the states that can be reached
from $\bm{p}$ by means of thermo-permutations.

Figure 3: _Geometric characterisation of thermal cone:_ the set
$\mathscr{T}_{\bm{p}}$ of states accessible from $\bm{p}$ through thermal
processes (orange cone) is the convex combination of those that can be
obtained by means of a finite set of thermo-permutations $P^{i}$ (red dots at
the bottom and $\bm{p}$ itself, $P^{0}$ being the trivial thermo-permutation).

#### Comparison to previous work

A comparison with the recent work Perry  _et al._ (2015) is useful (also see
the related work Åberg (2013)). In Ref Perry  _et al._ (2015) the following
operations are allowed: (1) Partially thermalise any two levels of the system,
as in Eq. (1); (2) Externally control in an arbitrary manner the energy levels
of the system Hamiltonian, while paying the required work; (3) Bring in an
extra qubit system in a thermal state with arbitrary Hamiltonian; (4) Apply
steps (1) and (2) on system plus ancilla.

One can readily check from the previous discussion that Partial Level
Thermalisations coincide with the thermal processes induced by Markovian ETOs;
from Eq. (1) this implies a maximum probability of de-exciting a transition
$\omega$ equal to $(1+e^{-\beta\hbar\omega})^{-1}$. However within the Jaynes-
Cummings model described above this limit is overcome for a large set of
parameters (in Appendix F we show this for $\beta\hbar\omega\in(0.1,6.4)$).
The advantage of Perry  _et al._ (2015) – that only requires _Markovian_ ETO –
is offset by the need for arbitrary external control of each energy level in
the system Hamiltonian, as well as the need to generate and control ancillary
qubit thermal states with arbitrary Hamiltonians. Moreover, both here and in
Perry  _et al._ (2015) a high degree of control is assumed anyway, because one
must be able to selectively couple any transition in the system to the
environment. When such control is present, the greater freedom of ETOs as
compared to Markovian ETOs seems justified; in fact, it is achieved in the
simplest model for such controlled situations – a Jaynes-Cummings interaction
in the RWA approximation.

### Equivalent thermodynamic frameworks

#### Are Thermal Operations too restrictive?

When we perform a Thermal Operation $\mathcal{T}$ on a system, its population
undergoes a thermal process $T\in\mathscr{T}$. More generally, a classical
framework can be defined as a set of stochastic maps $\mathscr{F}$ that can be
induced on the energy occupations without paying any work. We proved that
$\mathscr{E}\subset\mathscr{T}$ is already sufficient to achieve all the final
states that would be possible using the full set $\mathscr{T}$. Conversely,
would it be possible to allow for a wider thermodynamic framework in which a
set of maps $\mathscr{F}\supset\mathscr{T}$ can be induced at no cost? A
negative answer is given by the following result:

######  Theorem 7.

Let $\mathscr{F}\supseteq\mathscr{E}$ be a classical thermodynamic framework
that forbids work extraction from a single heat bath. Then
$\mathscr{F}\subseteq\mathscr{T}$.

This shows that the stochastic maps induced by TOs are hardly a too
restrictive set. In fact, they lie precisely at the boundary of processes that
would generate a perpetuum mobile.

This may seem puzzling in the light of the criticism presented in the
introduction. The reason is in part a misunderstanding about the role of TOs.
These are often compared to frameworks in which a classical field can be
applied to the system – as it is often the case in thermodynamic protocols.
Compared to such frameworks, TOs seem unduly limited. The crucial point is,
however, that TOs claim to provide a framework for a careful accounting of
thermodynamic resources, _not_ to set the limits of what can be achieved in
the lab. For example, if a resource state $\sigma$ is available, representing
a large enough coherent field, _every_ quantum operation $\mathcal{Q}$ can be
approximated on a system $\rho$ using TOs $\mathcal{T}$ via
$\mathcal{Q}(\rho):=\mathrm{Tr}_{2}\left[\mathcal{T}(\rho\otimes\sigma)\right]$.
The caveat is that, because $\sigma$ cannot be prepared freely within TOs, an
appropriate accounting of its deterioration is necessary Åberg (2014);
Korzekwa  _et al._ (2016). This includes, but is not limited to, an analysis
of the energy flows to and from $\sigma$.

#### Equivalence of thermodynamic frameworks

The discussion above naturally leads to the concept of _equivalent_
thermodynamic frameworks. Two classical frameworks, each represented by a set
of stochastic matrices $\mathscr{F}$ and $\mathscr{F}^{\prime}$, are
equivalent if they induce the same transitions; i.e. for any $\bm{p}$ and
$\bm{q}$ there exists $F\in\mathscr{F}$ such that $F\bm{p}=\bm{q}$ if and only
if there exists $F^{\prime}\in\mathscr{F}^{\prime}$ such that
$F^{\prime}\bm{p}=\bm{q}$.

A priori the specific choice of $\mathscr{F}$ may significantly modify what
transformations are possible within the thermodynamic theory. If this was the
case, the constraints found in the resource theory approach would have a
narrow scope. Contrary to this, we find as a corollary of Theorems 2 and 7 and
that a large class of classical thermodynamic framework are _equivalent_ :

######  Corollary 8.

All classical thermodynamic frameworks $\mathscr{F}\supseteq\mathscr{E}$ that
forbid work extraction from a single heat bath are equivalent. In particular,
there exists $F\in\mathscr{F}$ such that $F\bm{p}=\bm{q}$ if and only if
$\bm{p}\succ_{T}\bm{q}$.

Figure 4: _Thermodynamic universality and thermal processes:_ thermo-
majorisation, the constraint characterising achievable transitions with
thermal processes $\mathscr{T}$, also characterises all frameworks
$\mathscr{F}$ containing elementary thermal processes and not allowing for a
perpetuum mobile (Corollary 8). In fact, the largest of these sets coincides
with $\mathscr{T}$ (Theorem 7).

For a definition of the thermo-majorisation constraint, $\succ_{T}$, see
Appendix A.2. Corollary 8 says that even if two physicists use very different,
but consistent, thermodynamic frameworks, the resource theory results will
apply to them both – as far as they agree on allowing EDPs (see Fig. 4).
Theorem 8 is also a result on the _universality_ of thermo-majorisation within
a large class of thermodynamic models, for transitions among classical states.
Somehow inverting the logic of resource theories, it suggests a focus on the
analysis of classes of universal constraints, instead of the detailed
definition of the framework.

### Conclusions

We presented a direct link between resource theory results and experimental
implementations, showing that to induce any thermal process one only needs a
“universal set of thermodynamic gates”, which in turn can be well approximated
within a simple Jaynes-Cummings model. Thermodynamic equivalence, moreover,
enlarged the scope and applicability of the results. In Appendix D.1 we
discuss how to explicitly find the set of transformations of Theorem 2 and in
Appendix G we discuss extensions to the fully quantum case.

However, we wish to conclude by showing that the framework is flexible enough
to provide results beyond the resource theory domain. Consider for example
thermalisation models, which describe situations in which one has little or no
control on the system. Drawing on the idea of analysing classes of
constraints, can we find a stronger requirement than thermo-majorisation,
relevant for these situations? In this spirit, we introduce the following
definition: we say that $\bm{q}$ is a _thermalisation_ of $\bm{p}$ if and only
if $\bm{p}\succ_{T}\bm{q}$ and the $\beta$-ordering of $\bm{p}$ coincides with
that of $\bm{q}$ (recall that a sequence $\bm{p}$ is said to be
$\beta$-ordered if for all elements $p_{i}/g_{i}\geq p_{i+1}/g_{i+1}$). This
constraint is immediately seen to be relevant for thermalisation models. Let
$\bm{p}$ be the energy occupations of the initial, incoherent, state. Assume
thermalisation can be described through a sequence of collisions as explained
before. Then the final occupation $\bm{q}$ is given by a sequence of thermal
processes on $\bm{p}$ induced by Markovian ETOs. Hence,
$\bm{p}\succ_{T}\bm{q}$. Moreover each of these processes, being a Partial
Level Thermalisation, cannot change the $\beta$-ordering Perry  _et al._
(2015). So $\bm{q}$ is a thermalisation of $\bm{p}$. The converse is also
true: if $\bm{q}$ is a thermalisation of $\bm{p}$, then there exists a
sequence of Markovian ETOs transforming $\bm{p}$ into $\bm{q}$ (this is a
simple corollary of Theorem 12 in Perry  _et al._ (2015)). In other words, the
constraint that $\bm{q}$ is a thermalisation of $\bm{p}$ is the strongest
statement we can make, unless further information about the thermalisation
model is provided.

A final comment on the applicability of the results of the present
investigation beyond small-scale thermodynamics should be made. Relevant
systems for quantum thermodynamics may be mesoscopic – and _any_ control of
the microscopic degrees of freedom of the system unrealistic. However,
stochastic processes satisfying detailed balance can be realised at very
different scales (see, _e.g._ , England (2013)). In fact, thermo-majorisation
has already been applied in very different situations from the thermodynamics
of single quantum systems, such as diffusion of a diluted, ideal solution and
chemical reactions of ideal gas mixtures Ruch and Mead (1976). Some
thermodynamic constraints may then be universal not only in the sense of being
applicable within a wide range of models, but also throughout a wide range of
scales. It is after all universality that makes thermodynamics so useful and
its study so appealing.

Acknowledgments. I would like to warmly thank K. Korzekwa for insightful
discussions and for suggesting that a result in the spirit of Eq. (2) may
hold. I also thank D. Jennings and T. Rudolph for their continuous support. I
thank the people above and A. Milne, R. Uzdin, M. Müller, A. Levy for useful
comments on a previous draft. Finally, I would like to thank the above-
mentioned people and N. Yunger Halpern, M. Plenio, S. F. Huelga, M.
Christandl, R. Kosloff, P. Salamon for discussions at various stages of my PhD
on the merits and limitations of Thermal Operations. I am supported in part by
EPSRC and COST Action MP1209.

## Appendix

_About notation:_ We will denote probabilities with Roman bold letters
$\bm{x}$, $\bm{y}$,…, leaving the Greek letters $\rho,\sigma,...$ to denote
quantum states, described by trace one, non-negative operators.

###  .1 Preliminary facts

####  .1.1 Thermal Operations, thermal processes $\mathscr{T}$

Let $H_{S}$ be the Hamiltonian of the system. Then

######  Definition 1 (Thermal Operations (TO) Janzing  _et al._ (2000);
Brandão  _et al._ (2013)).

Thermal Operations in the presence of an environment at temperature $T_{B}$
are the set of quantum maps $\mathcal{T}$ that can be written as

$\mathcal{T}(\rho)=\mathrm{Tr}_{B}\left[U\left(\rho\otimes\frac{e^{-\beta
H_{B}}}{\mathrm{Tr}\left[e^{-\beta H_{B}}\right]}\right)U^{\dagger}\right],$
(3)

where

  1. 1.

$H_{B}$ is an arbitrary bath Hamiltonian,

  2. 2.

$U$ is an arbitrary energy-preserving unitary, i.e. $[U,H_{S}+H_{B}]=0$,

  3. 3.

${\rm Tr}_{B}$ represents a partial trace over the bath degrees of freedom,

  4. 4.

$\beta=(kT_{B})^{-1}$, with $k$ Boltzmann’s constant.

$\mathcal{T}$ is trace-preserving,
$\mathrm{Tr}\left[\mathcal{T}(\rho)\right]=\mathrm{Tr}\left[\rho\right]$, and
maps the energy eigenbasis into itself. So $\mathcal{T}$ induces a stochastic
matrix on the set of probability distributions representing the occupation of
different energy levels through the transition probabilities

$T_{i|j}=\left\langle
i\right|\mathcal{T}(\left|j\right\rangle\\!\\!\left\langle
j\right|)\left|i\right\rangle,$ (4)

if $\left|i\right\rangle$ are the eigenstates of $H_{S}$.444More generally,
one has projectors $\Pi_{i}$ on the energy eigenspace $\hbar\omega_{i}$. This
set of stochastic matrices will be denoted by $\mathscr{T}$:

######  Definition 2 (Thermal processes $\mathscr{T}$).

The set of stochastic maps that can be realised as in Eq. (4), where
$\mathcal{T}$ is a Thermal Operation, will be denoted by $\mathscr{T}$ and
called thermal processes.

We have the following reduction result for TO among incoherent states:

######  Proposition 9.

Let $\rho$ and $\sigma$ quantum states with $[\rho,H_{S}]=[\sigma,H_{S}]=0$.
Denote by $\bm{p}$ and $\bm{q}$ the eigenvalues of $\rho$ and $\sigma$,
respectively. Then there exists a TO $\mathcal{T}$ such that
$\mathcal{T}(\rho)=\sigma$ if and only if there exists a thermal process
$T\in\mathscr{T}$ such that $T\bm{p}=\bm{q}$.

###### Proof.

Unitaries $U_{S}$ on the system satisfying $[U_{S},H_{S}]=0$ are TOs. Let
$\mathcal{U}_{S,1}(\cdot)=U_{S,1}(\cdot)U^{\dagger}_{S,1}$ and
$\mathcal{U}_{S,2}(\cdot)=U_{S,2}(\cdot)U^{\dagger}_{S,2}$, where $U_{S,1}$,
$U_{S,2}$ are unitaries diagonalising the energy blocks of $\rho$ and
$\sigma$, respectively. Then if $T\in\mathscr{T}$ is such that
$T\bm{p}=\bm{q}$, let $\mathcal{T}$ be a TO that acts as $T$ in the basis of
the eigenstates of $\rho$. Then
$\tilde{\mathcal{T}}=\mathcal{U}^{\dagger}_{S,2}\circ\mathcal{T}\circ\mathcal{U}_{S,1}$
is a TO and satisfies $\tilde{\mathcal{T}}(\rho)=\sigma$. The converse also
follows easily. ∎

So we can focus on the existence of a thermal process between two arbitrary
probability distributions. This leads to the following considerations.

####  .1.2 Thermo-majorisation

Let $\bm{p}$, $\bm{q}$ denote $n$-dimensional probability distributions. Let
$\bm{g}=(g_{1},...,g_{n})$ be the thermal Gibbs distribution, i.e.

$g_{i}=e^{-\beta\hbar\omega_{i}}/Z,$ (5)

where {$\hbar\omega_{i}$} are the eigenstates of the Hamiltonian $H_{S}$ of
the system, $\beta=1/kT_{B}$ is the inverse temperature of the environment,
$Z=\sum_{i}e^{-\beta\hbar\omega_{i}}$ is the partition function of $H_{S}$. We
now review a relation between probability vectors that generalises the notion
of majorisation Bhatia (2013), which found widespread applications in quantum
information, especially in the context of entanglement theory Nielsen and
Chuang (2010). In the same way in which majorisation measures how “mixed” a
probability distribution is (i.e. how close to the uniform distribution),
thermo-majorisation measures how “thermal” a distribution is (i.e. how close
to $\bm{g}$):

######  Definition 3 (Thermo-majorisation Horodecki and Oppenheim (2013)).

$\bm{p}\succ_{T}\bm{q}$ if and if and only if the piecewise linear curve
obtained by joining the points
$\left(\sum_{i=1}^{k}g_{\pi(i)},\sum_{i=1}^{k}p_{\pi(i)}\right)$ lies all over
the curve obtained joining the points
$\left(\sum_{i=1}^{k}g_{\pi^{\prime}(i)},\sum_{i=1}^{k}q_{\pi^{\prime}(i)}\right)$,
where $k=1,..,n$. Here $\pi$ (respectively, $\pi^{\prime}$) is the permutation
ensuring that $p_{\pi(i)}/g_{\pi(i)}$ (respectively, $q_{\pi(i)}/g_{\pi(i)}$)
are sorted in decreasing order.

It is worth noticing that this is a particular instance of the notion of
$d-$majorisation (Marshall  _et al._ (2010), Chapter 14). Majorisation
(denoted by $\succ$) is recovered by taking $\bm{g}$ to be the uniform
distribution. Definition 3 gives a geometrical interpretation to thermo-
majorisation by means of curves called thermo-majorisation curves, a
generalisation of the so-called Lorenz curves. The thermal state $\bm{g}$ is
described by a straight line connecting the origin $(0,0)$ to the point
$(1,1)$ (an equivalent definition uses the unnormalised Gibbs-state, so that
the final point is $(Z,1)$). Any other curve will lie above this one. There
exists $\bm{p}$, $\bm{q}$ that are incomparable, in the sense that neither
$\bm{p}\succ_{T}\bm{q}$ nor $\bm{p}\succ_{T}\bm{q}$. These are the
distributions whose thermo-majorisation curves cross.

The interest in this condition lies in the fact that it characterises the
constraints present in various thermodynamic models. In particular we have
defined the set of thermal processes $\mathscr{T}$. Another interesting set of
transformations is that of Gibbs-preserving quantum maps:

######  Definition 4 (Gibbs-preserving stochastic processes $\mathscr{G}$).

For a system with thermal distribution $\bm{g}$, the set of stochastic
matrices $G$ such that $G\bm{g}=\bm{g}$ is denoted by $\mathscr{G}$.

One can check $\mathscr{T}\subseteq\mathscr{G}$. Moreover, these two sets
induce the same transitions and are characterised by thermo-majorisation:

######  Lemma 10.

Let $\bm{p}$ and $\bm{q}$ be two probability distributions. Then the following
are equivalent:

  1. 1.

There exists $G\in\mathscr{G}$ such that $G(\bm{p})=\bm{q}$.

  2. 2.

There exists $T\in\mathscr{T}$ such that $T(\bm{p})=\bm{q}$.

  3. 3.

$\bm{p}\succ_{T}\bm{q}$

###### Proof.

For $1\Leftrightarrow 3$ see Ruch  _et al._ (1980). For $2\Leftrightarrow 3$
see Janzing  _et al._ (2000); Horodecki and Oppenheim (2013). Later we will
give an alternative, direct proof of this equivalence, showing that actually
$\mathscr{T}=\mathscr{G}$ (see Lemma 24) ∎

Essentially this generalises the classic result that $\bm{p}\succ\bm{q}$ if
and only if there exists a bistochastic matrix $M$ such that $M\bm{p}=\bm{q}$
(bistochastic means stochastic and preserving the uniform distribution) Bhatia
(2013).

For technical purposes it will be useful to use an equivalent definition of
thermo-majorisation:

######  Lemma 11.

$\bm{p}\succ_{T}\bm{q}$ if and only if

$\sum_{j=1}^{n}g_{j}\left|\frac{q_{j}}{g_{j}}-a\right|\leq\sum_{j=1}^{n}g_{j}\left|\frac{p_{j}}{g_{j}}-a\right|\quad\forall
a\geq 0.$ (6)

The original proof can be found, in a more general context, in Ruch  _et al._
(1978, 1980). Note that, if $\bm{g}=(1/n,...,1/n)$ is the uniform
distribution, then the condition of Lemma 11 reduces to a characterisation of
majorisation, $\bm{p}\succ\bm{q}$ (Bhatia (2013), Chapter 2). Lemma 11 can
also be derived from the above mentioned result on majorisation using the
techniques introduced in the next section.

####  .1.3 Embedding

Let us now introduce the concept of embedding map, first introduced in Brandão
_et al._ (2015). The definition is given for thermal states whose energies are
rational numbers, i.e. for which there exists a vector of integers
$\bm{d}=(d_{1},...,d_{n})$ with

$\bm{g}=\left(\frac{d_{1}}{D},...,\frac{d_{n}}{D}\right),\quad
D=\sum_{i=1}^{n}d_{i}.$ (7)

This can be assumed without loss of generality because real numbers can be
approximated to any precision by rationals and, at any rate, the case in which
energies are irrationals can formally be dealt with Brandão  _et al._ (2015);
Lostaglio  _et al._ (2015c). From now on we will then assume that $\bm{g}$ can
be written as in Eq. (7):

######  Definition 5.

Let $\bm{g}$ be a thermal state with rational entries and let $\bm{d}$ be
defined as above. Then the embedding map is an application
$\Gamma_{\bm{d}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{D}$ such that

$\Gamma_{\bm{d}}(\bm{p})=\oplus_{i=1}^{n}p_{i}\bm{\eta}^{i},$ (8)

where $\bm{\eta}^{i}$ is a $d_{i}$-dimensional uniform distribution and
$(p,p^{\prime})\oplus(q,q^{\prime}):=(p,p^{\prime},q,q^{\prime})$.

Note that, by definition, $\Gamma_{\bm{d}}(\bm{g})=\bm{\eta}$, where
$\bm{\eta}$ is a $D$-dimensional uniform distribution. We are now in the
position of proving a useful technical lemma:

######  Lemma 12.

$\bm{p}\succ_{T}\bm{q}$ if and only if
$\Gamma_{\bm{d}}(\bm{p})\succ\Gamma_{\bm{d}}(\bm{q})$.

###### Proof.

Using Lemma 11 for a trivial Hamiltonian in a $D$-dimensional space (so that
$\bm{\eta}$ plays the role of $\bm{g}$ in $\mathbb{R}^{D}$), one has
$\Gamma_{\bm{d}}(\bm{p})\succ\Gamma_{\bm{d}}(\bm{q})$ if and only if

$\sum_{j=1}^{D}\left|D\Gamma(\bm{q})_{j}-a\right|\leq\sum_{j=1}^{D}\left|D\Gamma(\bm{p})_{j}-a\right|\,\,\forall
a\geq 0.$ (9)

Taking into account the structure of the embedding map, both sides of (9) can
be split into two distinct sums: one over $i$, running from $1$ to $n$ and,
for every $i$, a sum over $j$ from $j_{i-1}$ to $j_{i}$, where
$j_{i}=\sum_{k=0}^{i}d_{k}$ (here $d_{0}:=0$ by convention). Then,

$\sum_{j=1}^{D}\left|D\Gamma(\bm{p})_{j}-a\right|=\sum_{i=1}^{n}\sum_{j=j_{i-1}}^{j_{i}}\left|D\frac{p_{i}}{d_{i}}-a\right|$
$=\sum_{i=1}^{n}d_{i}\left|D\frac{p_{i}}{d_{i}}-a\right|=\sum_{i=1}^{n}Dg_{i}\left|\frac{p_{i}}{g_{i}}-a\right|,$

where in the last line we used Eq. (7). From Eq. (9) and the previous equation
one can see that $\Gamma_{\bm{d}}(\bm{x})\succ\Gamma_{\bm{d}}(\bm{y})$ if and
only if

$\sum_{j=1}^{n}g_{i}\left|\frac{q_{i}}{g_{i}}-a\right|\leq\sum_{j=1}^{n}g_{i}\left|\frac{p_{i}}{g_{i}}-a\right|\quad\forall
a\geq 0.$

Using again Lemma 11, this time for the Hamiltonian of the system, i.e. for
the $n$-dimensional probability distribution $\bm{g}$, we obtain the result we
wanted. ∎

#### T-transformations

In the context of bistochastic maps, i.e. stochastic matrices that preserve
the uniform distribution, a special role is played by the so-called
$T$-transformations or $T$-transforms (Marshall  _et al._ (2010), Chapter 2):

######  Definition 6 ($T$-transformation).

A $T$-transform is stochastic map $T$ with

$T=\lambda\mathbb{I}+(1-\lambda)Q,\quad\lambda\in[0,1],$ (10)

where $Q$ is a transposition of any two elements of the probability
distribution and $\mathbb{I}$ is the identity matrix.

One then has the following Lemma (B.1 of Marshall  _et al._ (2010)), due to
Muirhead, Hardy, Littlewood, Pólya:

######  Lemma 13.

$\bm{p}\succ\bm{q}$ if and only if $\bm{p}$ can be achieved from $\bm{q}$ by
only applying a sequence of T-transformations. The number of
$T$-transformation required is bounded by the dimension of the probability
vectors.

###### Proof.

A proof can be found in Marshall  _et al._ (2010) (Lemma B.1) as well as Gour
_et al._ (2015) (Lemma 19) and Bhatia (2013). ∎

###  .2 Elementary Thermal Operations (ETOs)

In this section we introduce the concept of Elementary Thermal Operations. We
discuss their appealing physical properties and show that most of them can be
realised if we can resonantly couple any transition of the system to a single-
mode bosonic bath by a standard Jaynes-Cummings coupling. We also study the
Markovian limit of ETO, showing that they generate the same dynamics of Davies
maps.

####  .2.1 Definition

In the context of thermodynamics, a central role is played by the detailed
balance condition Crooks (1998, 1999):

######  Definition 7 (Detailed Balance).

A stochastic map $M$ is called detailed balanced with respect to the thermal
state $\bm{g}$ if

$\frac{M_{i|j}}{M_{j|i}}=e^{-\beta\hbar(\omega_{i}-\omega_{j})}=\frac{g_{i}}{g_{j}}$
(11)

Every detailed balanced map is in $\mathscr{G}$, i.e. $M\bm{g}=\bm{g}$:

$(M\bm{g})_{j}=\sum_{i}M_{j|i}g_{i}=\sum_{i}M_{i|j}g_{j}=g_{j}$ (12)

However the opposite implication is not true (see Halpern  _et al._ (2015),
Appendix A). Also, $\mathscr{T}$ contains transformations that do not satisfy
detailed balance (Horodecki and Oppenheim (2013), Supplementary Note 6). A
particularly simple set of detailed balanced stochastic maps is defined as
follows:

######  Definition 8 (Elementary Detailed balanced process (EDP)).

An EDP is a stochastic map $E$ satisfying two properties:

  1. 1.

The transformation only couples two energy levels, i.e. there are two energies
$\hbar\omega_{i}$, $\hbar\omega_{j}$ of $H_{S}$, with
$\omega_{i}\neq\omega_{j}$, such that $E_{k|m}=0$ if $k,m\not\in\\{i,j\\}$.

  2. 2.

$E$ is detailed balanced.

The set of all EDP will be denoted by $\mathscr{E}$. Elementary Thermal
Operations are defined as Thermal Operations that induce such elementary
stochastic processes:

######  Definition 9 (Elementary Thermal Operation (ETO)).

An ETO is a Thermal Operation $\mathcal{T}$ whose induced stochastic map $T$
is an EDP.

More explicitly, every $E\in\mathscr{E}$ has the especially simple structure

$E=\left[\begin{matrix}\ddots&\vdots&\vdots&\vdots&\iddots\\\
\dots&E_{i|i}&\dots&E_{i|j}&\dots\\\ \dots&\dots&\dots&\dots&\dots\\\
\dots&E_{j|i}&\dots&E_{j|j}&\dots\\\
\iddots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right]$

where $E_{i|j}=E_{j|i}e^{-\beta\hbar(\omega_{i}-\omega_{j})}$ and dots are all
zeros for off-diagonal elements and $1$ for diagonal elements. Due to
stochasticity, if $H_{S}$ is known then each EDP is fully determined by the
labels $i$, $j$ and a single transition probability. This can be chosen to be
$E_{i|j}\in[0,1]$, if $\omega_{j}>\omega_{i}$.

####  .2.2 Proof $\mathscr{E}\subset\mathscr{T}$

######  Lemma 14.

$\mathscr{E}\subset\mathscr{T}$, i.e., for any EDP there exists a TO inducing
it. Conversely, not all thermal processes are elementary.

###### Proof.

To prove this, let $\left|\phi_{0}\right\rangle$,
$\left|\phi_{1}\right\rangle$ be two eigenstates of $H_{S}$ with energies
$\hbar\omega_{0}$ and $\hbar\omega_{1}$, with $\omega_{1}>\omega_{0}$. Let $E$
be an EDP involving them, fully characterised by the transition probability
$E_{0|1}\in[0,1]$. We now show that $E\in\mathscr{T}$, by explicitly
constructing a TO inducing it.

Consider a single-mode thermal bath with Hamiltonian
$H_{B}=\sum_{n=0}^{\infty}n\hbar\omega\left|n\right\rangle\\!\\!\left\langle
n\right|$, with $\omega=\omega_{1}-\omega_{0}$. Let the bath be in the state
$\gamma_{B}$, with $\gamma_{B}=e^{-\beta H_{B}}/Z_{B}$, where
$Z_{B}=(1-e^{-\beta\hbar\omega})^{-1}$. Take now the energy-preserving unitary
(appearing in Åberg (2013))

$U=\left|\phi_{0}\right\rangle\\!\\!\left\langle\phi_{0}\right|\otimes\left|0\right\rangle\\!\\!\left\langle
0\right|+\sum_{n=1}^{\infty}\sum_{k,k^{\prime}=0}^{1}v_{kk^{\prime}}\left|\phi_{k}\right\rangle\\!\\!\left\langle\phi_{k^{\prime}}\right|\otimes\left|n-k\right\rangle\\!\\!\left\langle
n-k^{\prime}\right|,$

where $V$ (whose elements are $v_{kk^{\prime}}$) is a $2$ by $2$ unitary.
Hence, by Definition 1, the transformation

$\rho\mapsto\mathrm{Tr}_{B}\left[U(\rho\otimes\gamma_{B})U^{\dagger}\right]:=\mathcal{T}(\rho),$

is a TO $\mathcal{T}$. One can readily compute the induced transition
probabilities of $\mathcal{T}$, as defined in Eq. (4). These give

$T_{1|0}=e^{-\beta\hbar\omega}|v_{10}|^{2},\quad T_{0|1}=|v_{01}|^{2},$

the others being determined by the condition of stochasticity (that follow
from the unitarity of $V$). We can take $V$ of the form

$V=\left[\begin{matrix}\cos(x)&-i\sin(x)\\\
-i\sin(x)&\cos(x)\end{matrix}\right].$

Then for any $E_{0|1}\in[0,1]$ we can choose $x$ such that
$\sin^{2}(x)=E_{0|1}$, so that $T_{0|1}=E_{0|1}$. Hence we realised every EDP
through a TO, which implies $\mathscr{E}\subseteq\mathscr{T}$. The inclusion
is strict because, as mentioned just after Definition 7, there exists TOs
whose induced stochastic process does not satisfy the detailed balance
condition. ∎

We also introduce

######  Definition 10 (Complete set of ETOs).

A set of ETOs is called complete if all $E\in\mathscr{E}$ can be induced by
them.

The previous result has shown that such sets exist.

####  .2.3 Further properties of EDPs, thermo-transpositions

Detailed balance is enjoyed by the commonly used operations modelling
reversible systems coupled with a heat bath (see Davies (1974); Dümcke (1985);
Crooks (1999); Breuer and Petruccione (2002); Roga  _et al._ (2010) and
references therein). Detailed balance is sometimes known under the name of
“microscopic reversibility” and it is used by Crooks to derive his famous
theorem Crooks (1999), and more recently in single-shot generalisations
Halpern  _et al._ (2015). Hence this property not only makes the set of
transformations more tightly connected to practical realisations, but it is
also of great theoretical interest because it ensures that fluctuation
theorems are satisfied by processes built composing them. For completeness, we
show the following:

######  Lemma 15.

Let $E$ be an EDP on two given energy levels. Then,
$\lim_{n\rightarrow\infty}E^{n}$ maps the subsystem it acts on to a
distribution proportional to the thermal state.

###### Proof.

Let the relevant two-level subsystem, with energies $\hbar\omega_{0}$,
$\hbar\omega_{1}$, be described by the (possibly unnormalised) distribution
$\bm{q}(0):=(q_{0}(0),q_{1}(0))$. Without loss of generality, take
$\omega_{0}=0$ and $\omega_{1}=\omega$. Let $\bm{q}(n):=E^{n}\bm{q}(0)$. One
can easily compute $q_{0}{(1)}=q_{0}{(0)}(1-\lambda Z)+\lambda N$, where
$Z=1+e^{-\beta\hbar\omega}$, $N=q_{0}(0)+q_{1}(0)$ and $\lambda:=E_{0|1}$. By
recursion,

$q_{0}{(n)}=q_{0}{(0)}(1-\lambda Z)^{n}+\frac{N}{Z}\left(1-(1-\lambda
Z)^{n}\right)$

Then $q_{0}(\infty):=\lim_{n\rightarrow\infty}q_{0}(n)=N/Z$, independently of
the initial distribution. Because the map is stochastic,

$\bm{q}(\infty)=N\bm{g}$

where $\bm{g}=(1/Z,1-1/Z)$ is a thermal distribution within the two-level
subsystem. ∎

Transpositions are stochastic maps acting only on two levels and having
transition probabilities $E_{0|1}=1$, $E_{1|0}=1$. Such processes are not
thermal processes. The closest we can get to such process while satisfying the
condition that the thermal state is a fixed point is $E_{0|1}=1$,
$E_{1|0}=e^{-\beta\hbar\omega}$, where $\omega$ is the transition under
consideration. We then define

######  Definition 11 (Thermo-transposition).

A thermo-transposition is a stochastic map $\tilde{E}$ such that either

  1. 1.

$\tilde{E}=\mathbb{I}$

  2. 2.

Or there are indexes $i$,$j$ with $\omega_{j}>\omega_{i}$ and

$\tilde{E}_{i|j}=1,\quad\tilde{E}_{j|i}=e^{-\beta\hbar\omega},\quad\tilde{E}_{k|m}=\delta_{km},\;\forall(k,m)\neq(i,j).$

As expected by analogy to the theory of bistochastic processes, EDP are the
thermodynamic analogue of T-transformations. The following property of EDP
makes clear that they are the thermodynamic analogue of $T$-transformations:

######  Lemma 16.

Any $E\in\mathscr{E}$ can be decomposed as a convex combination of the
identity matrix and a thermo-transposition $\tilde{E}$:

$E=(1-\lambda)\mathbb{I}+\lambda\tilde{E},\quad\lambda\in[0,1].$ (13)

###### Proof.

By direct inspection, choosing $\lambda=E_{0|1}$. ∎

This should be compared with Eq. (10). We now analyse if all EDPs can be
induced by Thermal Operations corresponding to some simple physical toy model
(the transformations used to prove Lemma 14 are not good for this purpose, as
they involve rather non-standard couplings; see Appendix E.3 of Åberg (2014)).

###  .3 Physical realisations

####  .3.1 The Jaynes-Cummings model

One of the simplest toy models of open system dynamics is the Jaynes-Cummings
model Jaynes and Cummings (1963); Shore and Knight (1993), describing the
interaction of a two-level system with a single mode of the electromagnetic
field, for example in an optical cavity. If the field is taken to be in a
thermal state, can we achieve all EDPs within this model? In other words, do
we obtain a complete set of ETOs?

######  Lemma 17.

Let $H_{S}$ be a non-degenerate Hamiltonian with no equal spacings. Then, the
Jaynes-Cummings interactions in RWA approximation with a single-mode bosonic
bath are ETOs. In fact, in the low-temperature limit they are a complete set
of ETOs.

###### Proof.

Let $\left|\phi_{k}\right\rangle$, $k=0,1$ be two eigenstates in $H_{S}$, with
energy $\hbar\omega_{k}$, $\omega_{1}>\omega_{0}$ and
$\omega_{1}-\omega_{0}=\omega$. Consider the Jaynes-Cummings Hamiltonian
$H_{JC}$ resonantly interacting these two levels with a single mode bosonic
bath of frequency $\omega$ in RWA approximation, i.e.

$H_{JC}=g(\sigma_{+}\otimes a+\sigma_{-}\otimes a^{\dagger}),$

where $[a,a^{{\dagger}}]=\mathbb{I}_{B}$ and $g$ is a coupling constant.
Notice that we work in interaction picture. Here
$H_{B}=\hbar\omega\,a^{\dagger}a$ is the bath Hamiltonian,
$\sigma_{+}=\left|\phi_{1}\right\rangle\\!\\!\left\langle\phi_{0}\right|$ and
$\sigma_{-}=\sigma_{+}^{\dagger}$. One can compute (Åberg (2014), Appendix
E.2)

$\displaystyle e^{-\frac{it}{\hbar}H_{JC}}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\sum_{k,k^{\prime}=0}^{1}u^{(n)}_{k,k^{\prime}}(s)\left|\phi_{k}\right\rangle\\!\\!\left\langle\phi_{k^{\prime}}\right|\otimes\left|n-k\right\rangle\\!\\!\left\langle
n-k^{\prime}\right|$ $\displaystyle+$
$\displaystyle\left|\phi_{0}\right\rangle\\!\\!\left\langle\phi_{0}\right|\otimes\left|0\right\rangle\\!\\!\left\langle
0\right|,$

where $s=gt/\hbar$ and, for each $n$, $u^{(n)}_{k,k^{\prime}}(s)$ are matrix
elements of the unitary

$U^{(n)}(s)=\left[\begin{matrix}\cos(s\sqrt{n})&-i\sin(s\sqrt{n})\\\
-i\sin(s\sqrt{n})&\cos(s\sqrt{n}).\end{matrix}\right]$

Take system and bath to be in the initial state $\rho\otimes e^{-\beta
H_{B}}/Z_{B}$, where $Z_{B}=(1-e^{-\beta\hbar\omega})^{-1}$. One then has that
the dynamics of the system is described as

$\rho\mapsto\mathrm{Tr}_{B}\left[e^{-\frac{it}{\hbar}H_{JC}}(\rho\otimes
e^{-\beta H_{B}}/Z_{B})e^{\frac{it}{\hbar}H_{JC}}\right],$

so it is a Thermal Operation. This induces a stochastic process $J$ whose
transition probabilities are:

$J_{1|0}(s)=\sum_{n=1}^{\infty}\sin^{2}(s\sqrt{n})\frac{e^{-\beta\hbar\omega
n}}{Z_{B}},\quad J_{0|0}=1-J_{1|0},$
$J_{0|1}(s)=\sum_{n=1}^{\infty}\sin^{2}(s\sqrt{n})\frac{e^{-\beta\hbar\omega(n-1)}}{Z_{B}},\quad
J_{1|1}=1-J_{0|1}.$ (14)

So $J_{1|0}/J_{0|1}=e^{-\beta\hbar\omega}$, i.e. the transformation is
detailed balanced. This shows that every transformation within the resonant
Jaynes-Cummings model in RWA approximation gives an ETO.

To investigate the low temperature limit notice that we only need to show that
in such limit $J_{0|1}(s)$ can achieve all values in $[0,1]$. For every $s$ we
have

$J_{0|1}(s)\geq(1-e^{-\beta\hbar\omega})\sin^{2}(s)$

At $s=0$, in fact, equality holds. By continuity of $J_{0|1}(s)$ and the
previous inequality, we see that all values in $[0,1]$ can be achieved,
varying $s$, in the limit $\beta\rightarrow\infty$. Hence, at zero temperature
all EDPs can be induced within the Jaynes-Cummings model. Hence, varying $s$,
we get a complete set of ETOs. ∎

One may be tempted to hope that all EDPs can be achieved within the Jaynes-
Cummings model at all temperatures. Analytically one immediately notices that
$J_{0|1}(s)<1$ for all times $s$. However this does not exclude that there is
a sequence of $s_{i}$ such that $J_{0|1}(s_{i})$ gets arbitrarily close to
$1$, i.e. that we can approximate EDPs arbitrarily well within the Jaynes-
Cummings model. As the dynamics in $s$ is very chaotic, this cannot be
excluded numerically. What we need is a bound on $J_{0|1}$ independent of $s$.
In the following lemma we prove such bound:

######  Lemma 18.

For the Jaynes-Cummings model, set $\bar{\beta}=\beta\hbar\omega$. Then for
every $s$

$\displaystyle J_{0|1}(s)$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\left(8e^{-\bar{\beta}}-e^{2\bar{\beta}}+e^{3\bar{\beta}}+8\right),\;\textrm{for
}\bar{\beta}\in[0,\log(4)/3]$ $\displaystyle J_{0|1}(s)$ $\displaystyle\leq$
$\displaystyle
e^{-4\bar{\beta}}-e^{-3\bar{\beta}}+1\quad\quad\quad\quad\quad\;\;\;\textrm{for
}\bar{\beta}\geq\log(4)/3.$

The usefulness of these bounds is that they prove that for any fixed
temperature $\beta\in(0,1)$ one will have a constant $C_{\beta}\in(0,1)$ such
that $J_{0|1}<1-C_{\beta}$ for all $s$. So, the Jaynes-Cummings model cannot
achieve all EDPs nor arbitrary approximations of them:

######  Corollary 19.

For any fixed $\beta\in(0,1)$, the Jaynes-Cummings cannot induce arbitrary
close approximations of every EDP.

###### Proof of Lemma 18.

Consider the bound on $J_{0|1}(s)$ constructed following these steps:

  1. 1.

Split the series in Eq. (14) into a finite sum $F$ up to $m$, plus a residue
$R$. In the residue, bound each factor $\sin^{2}(s\sqrt{n})$ with $1$. Then
one has $R=e^{-m\beta\omega}$.

  2. 2.

In $F$, bound every oscillating term with irrational frequency as
$\sin^{2}(s\sqrt{n})\leq 1$.

This provides a bound of $J_{0|1}(s)$ with a periodic function of $s$, which
can then be simply analysed. For $m=4$, the function has one global maximum at
$s=\pi/2$ when $\bar{\beta}\geq\log(4)/3$. On the other hand, when
$\bar{\beta}<\log(4)/3$ it has two global maxima at
$s=\pm\arccos(-e^{3\bar{\beta}}/4)/2$. The values achieved at the maxima give
the bounds of Lemma 18. ∎

On the other hand, a rough lower bound (good when temperatures are not too
high) of the achievable $J_{0|1}$ can be obtained by truncating the sum in Eq.
(14) at some finite $m_{{\rm max}}$, and setting every other term to zero. We
take $m_{{\rm max}}=12$. Then numerics suggest to choose $s=98.92$. This,
together with the upper bounds of Lemma 18, provides a region where the curve
of maximum achievable probability $J_{0|1}$ must lie as a function of
$\bar{\beta}$ (Fig. 1 in the main text). For any $\bar{\beta}$, all values
from $0$ to this limit can be achieved, because $J_{0|1}(s)$ is a continuous
function in $s$ and it takes the value $0$ at $s=0$. Hence Figure 1 shows
that, even though not every EDPs can be realised within the Jaynes-Cummings
model, a vast majority of them can be achieved, especially at lower
temperatures. In other words, we realised within a physically reasonable model
an approximation to a complete set of ETOs.

####  .3.2 Thermalisation models: proof of Eq. (1)

Following the first part of the proof of Lemma 15, we obtain that the effect
of $n$ interactions on the occupation of the lower energy state is

$q_{0}{(n)}=q_{0}{(0)}(1-\lambda Z)^{n}+\frac{N}{Z}(1-(1-\lambda Z)^{n}),$
(15)

where $\lambda=E_{0|1}$, $Z=1+e^{-\beta\hbar\omega}$, $N=q_{0}(0)+q_{1}(0)$.
We can define $n=t/t_{{\rm int}}$, with $t_{{\rm int}}$ the duration of a
single interaction. Then we will take the limit $t_{{\rm int}}\rightarrow 0$,
$\lambda\rightarrow 0$, while keeping $Z\lambda/t_{{\rm
int}}\rightarrow\xi\geq 0$ finite ($Z$ is a constant that we absorbed in the
definition of $\xi$). Then one can check that $(1-\lambda Z)^{n}\rightarrow
e^{-t/\xi}$ and hence

$\bm{q}(t)=e^{-t/\xi}\bm{q}(0)+N(1-e^{-t/\xi})\bm{g}.$ (16)

Here $\xi\geq 0$ is a free parameter. So the state decays exponentially to a
Gibbs distribution (whenever $\xi>0$).

####  .3.3 Thermalisation models: comparison to Davies maps

An alternative, but equivalent, point of view is given by Davies maps. These
maps emerge as the Markovian description of a system weakly coupled to an
infinite thermal bath Davies (1974). Let us begin with a definition Roga  _et
al._ (2010):

######  Definition 12.

A Davies map $\mathcal{M}$ is a map such that

  1. 1.

$\mathcal{M}=\exp(\mathcal{L}t)$. Here
$\mathcal{L}=\mathcal{H}+\mathcal{L^{\prime}}$, with
$\mathcal{H}(\rho)=[\rho,H]$, $H$ hermitian operator and
$\mathcal{L}^{\prime}$ a general Lindbladian.

  2. 2.

$[\mathcal{H},\mathcal{L^{\prime}}]=0$ (covariance).

  3. 3.

$\mathrm{Tr}\left[\mathcal{L^{\prime}}(Be^{-\beta
H})A\right]=\mathrm{Tr}\left[B\mathcal{L^{\prime}}(e^{-\beta H}A)\right]$ for
all $A$, $B$ (quantum detailed balance).

The action of the most general qubit Davies map on a two-level system was
studied in Roga  _et al._ (2010), Section 3.

Notice that by taking $A=\left|i\right\rangle\\!\\!\left\langle i\right|$,
$B=\left|j\right\rangle\\!\\!\left\langle j\right|$ in the third property we
can show that the induced stochastic map (defined through Eq. (4)) satisfies
detailed balance in the sense of Definition 7. The covariance condition
ensures that every Davies map is time-translation symmetric, in the sense
discussed in Appendix G.

As expected, acting with Davies maps on a state whose population is initially
described by $\bm{q}(0)$, one gets back again Eq. (1). Every Davies map
induces an ETO. In particular, as shown in the main text, it gives Markovian
ETOs.

###  .4 Elementary Thermal Operations are universal for incoherent processes

####  .4.1 Proof of Theorem 2

We can now prove one of the main results, showing the sufficiency of ETOs. TOs
commute with dephasing in energy, so we get $[\sigma,H_{S}]=0$. Hence, using
Proposition 9, Theorem 2 can be rewritten as follows:

######  Theorem 20.

A thermal process $T\in\mathscr{T}$ such that $T(\bm{p})=\bm{q}$ exists if and
only if $\bm{q}$ can be obtained from $\bm{p}$ through a finite sequence of
EDPs.

###### Proof.

From Lemma 10 we need to equivalently prove $\bm{p}\succ_{T}\bm{q}$ if and
only if $\bm{q}$ can be obtained from $\bm{p}$ through a sequence of EDPs.
From Lemma 12, $\bm{p}\succ_{T}\bm{q}$ if and only if
$\Gamma_{\bm{d}}(\bm{p})\succ\Gamma_{\bm{d}}(\bm{q})$ (recall that
$\bm{p}\in\mathbb{R}^{n}$ and
$\Gamma_{\bm{d}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{D}$). Now, due to Lemma
13, we know that $\Gamma_{\bm{d}}(\bm{q})$ can be obtained from
$\Gamma_{\bm{d}}(\bm{p})$ through a sequence of $T$-transformations $T^{i}$:

$\Gamma_{\bm{d}}(\bm{q})=T^{1}\dots T^{D}\Gamma_{\bm{d}}(\bm{p}),$ (17)

perhaps some of them trivial. Here we used the fact that the number of
$T$-transformations required is finite and bounded by the dimension of the
probability distribution.

Next define $E^{i}=\Gamma_{\bm{d}}^{-1}\circ T^{i}\circ\Gamma_{\bm{d}}$. We
will prove the following:

  1. 1.

$\bm{q}=E^{1}\dots E^{D}(\bm{p})$

  2. 2.

Each $E^{i}$ is an EDP.

The first point is immediately checked by substituting
$T^{i}=\Gamma_{\bm{d}}\circ E^{i}\circ\Gamma_{\bm{d}}^{-1}$ in Eq. (17).
Regarding the second point note that, from Def. 6,

$E^{i}=\lambda\mathbb{I}+(1-\lambda)\tilde{E}^{i}\quad\lambda\in[0,1],$ (18)

where $\mathbb{I}$ is the identity matrix in $\mathbb{R}^{n}$,
$\tilde{E}^{i}=\Gamma_{\bm{d}}^{-1}\circ Q^{i}\circ\Gamma_{\bm{d}}$ and
$Q^{i}$ is a transposition in $\mathbb{R}^{D}$. Because the convex combination
of an EDP and the identity is readily seen to be just another EDP, we only
need to show that $\tilde{E}_{i}$ is an EDP.

Consider the action of $\tilde{E}^{i}$ on a probability distribution $\bm{p}$.
Defining $\bm{q}=\tilde{E}^{i}(\bm{p})$, we get

$q_{k}=\left(1-\frac{1}{d_{k}}\right)p_{k}+\frac{1}{d_{j}}p_{j},\quad
q_{j}=\frac{1}{d_{k}}p_{k}+\left(1-\frac{1}{d_{j}}\right)p_{j},$

where we used the definition of embedding map, Def. 5. One can immediately see
that the choice of $i$ fixes the two indexes $k,j$ (but many choices of $i$ in
general correspond to the same $k$,$j$). The previous expression immediately
shows that the transformation involves only two energy levels. Moreover, using
$d_{i}=g_{i}D$ (see Eq. (7)) and the previous equation, we obtain for the
transition probabilities of $\tilde{E}^{i}$,

$\tilde{E}^{i}_{k|j}/\tilde{E}^{i}_{j|k}=g_{k}/g_{j}=e^{-\beta\hbar(\omega_{k}-\omega_{j})},\quad\forall\,i.$
(19)

Hence, $\tilde{E}^{i}$ is detailed balanced. By construction, the transition
is achieved using a finite number of EDPs. ∎

In other words, EDPs are the analogue in thermodynamics of T-transforms, and
Theorem 2 is the generalisation to thermodynamics of Lemma 13 by Muirhead,
Hardy, Littlewood, Pólya.

#### Explicit construction of EDPs sequence in Theorem 2

Because the proof of Lemma 13 is constructive, i.e. it explicitly gives a
prescription to build each $T^{i}$, the above embedding gives a way to
explicitly construct the maps $\tilde{E}^{i}=\Gamma_{\bm{d}}^{-1}\circ
T^{i}\circ\Gamma_{\bm{d}}$. Suppose one wants to construct a sequence of
transformations (18) achieving the transition from $\bm{p}$ to $\bm{q}$. By
assumption a thermal process transforming $\bm{p}$ into $\bm{q}$ exists, so
$\bm{p}\succ_{T}\bm{q}$ from Lemma 10. But then from Lemma 12 we know that
$\Gamma_{\bm{d}}(\bm{p})\succ\Gamma_{\bm{d}}(\bm{q})$, so we can run the
construction of Marshall  _et al._ (2010) (Lemma B.1), also reported in Gour
_et al._ (2015) (Lemma 19). This construction gives the sequence of
$T$-transformations mapping $\Gamma_{\bm{d}}(\bm{p})$ into
$\Gamma_{\bm{d}}(\bm{q})$. Sort both $\bm{p}$ and $\bm{q}$ in such a way that
$p_{i}/g_{i}$ and $q_{i}/g_{i}$ are non-increasing.555In case for some $i,j$
$p_{i}/g_{i}=p_{j}/g_{j}$, $i$ will precede $j$ if $p_{i}\geq p_{j}$. This is
called $\beta$-ordering in Horodecki and Oppenheim (2013) and will be denoted
by $\bm{p}^{\downarrow\beta}$, $\bm{q}^{\downarrow\beta}$. One can then take
$T^{i}=\lambda\mathbb{I}+(1-\lambda)Q^{i}$, where

  1. 1.

$Q^{i}$ is the transposition between the index $j_{ex}$ and $j_{df}$. Here
$j_{ex}$ (using the notation of Gour  _et al._ (2015)) is the largest index
$j$ such that
$\Gamma_{\bm{d}}(\bm{p}^{\downarrow\beta})_{j}>\Gamma_{\bm{d}}(\bm{q}^{\downarrow\beta})_{j}$
and $j_{df}$ is the smallest index $j$ such that
$\Gamma_{\bm{d}}(\bm{p}^{\downarrow\beta})_{j}<\Gamma_{\bm{d}}(\bm{q}^{\downarrow\beta})_{j}$.
The existence of both these indexes can be proven as shown in Marshall  _et
al._ (2010), Lemma B.1.

  2. 2.

Defining

$\delta=\min\\{\Gamma_{\bm{d}}(\bm{p}^{\downarrow\beta})_{j_{ex}}-\Gamma_{\bm{d}}(\bm{q}^{\downarrow\beta}){j_{ex}},\Gamma_{\bm{d}}(\bm{q}^{\downarrow\beta})_{j_{df}}-\Gamma_{\bm{d}}(\bm{p}^{\downarrow\beta})_{j_{df}}\\},$

one transfers $\delta$ from $j_{ex}$ to $j_{df}$, i.e.

$\Gamma_{\bm{d}}(\bm{x}^{\downarrow\beta})_{j_{ex}}\mapsto\Gamma_{\bm{d}}(\bm{x}^{\downarrow\beta})_{j_{ex}}-\delta,\quad\Gamma_{\bm{d}}(\bm{x}^{\downarrow\beta})_{j_{df}}\mapsto\Gamma_{\bm{d}}(\bm{x}^{\downarrow\beta})_{j_{df}}+\delta,$

by taking
$\lambda=1-\delta/(\Gamma_{\bm{d}}(\bm{p}^{\downarrow\beta})_{j_{ex}}-\Gamma_{\bm{d}}(\bm{p}^{\downarrow\beta})_{j_{df}})$.

As proved in Marshall  _et al._ (2010) (Lemma B.1), iterating the reasoning
above one finds a finite sequence of $T-$transforms mapping
$\Gamma_{\bm{d}}(\bm{p})$ into $\Gamma_{\bm{d}}(\bm{q})$. As should be clear
from the proof of the previous theorem, the corresponding sequence of maps
(18), where $\tilde{E}^{i}=\Gamma_{\bm{d}}^{-1}\circ
T^{i}\circ\Gamma_{\bm{d}}$, transforms $\bm{p}$ into $\bm{q}$.

An interesting line of research is to go beyond this construction and find
shorter sequences of transformations; one obvious reason is that there are
many $T^{i}$ that give an EDP among the same two levels $(k,l)$. However,
composing EDPs among the same levels gives yet another EDP, so all of these
can be grouped into a single step. This can cut massively the number of
required elementary steps. Another idea may be to require that the target
state is obtained only within a given level of precision (which is physically
reasonable).

####  .4.2 Proof of Theorem 3, Lemma 5, Corollary 4

A second result showing the theoretical importance of EDPs, and in particular
thermo-permutations (Def. 11), is the following. It is a famous theorem of
Birkhoff that bistochastic maps are the convex hull of permutations. This
generalises to thermal processes (the set $\mathscr{T}$ of Def. 2) as follows
(Theorem 3 in the main text):

######  Theorem 21.

The set $\mathscr{T}$ coincides with the convex hull of thermo-permutations,
i.e. with the set of $T$ that can be written as

$T=\sum_{i}\lambda_{i}P^{i},\quad\lambda_{i}\geq
0,\;\;\;\sum_{i}\lambda_{i}=1,$ (20)

where each $P^{i}$ is a thermo-permutation.

The result above is an immediate consequence of two Lemmas, the second of
which has some interest by itself as we will discuss. Recall the definition of
Gibbs-preserving processes $\mathscr{G}$ given in Def. 4.

######  Lemma 22.

The set $\mathscr{G}$ coincides with the maps that can be decomposed as in Eq.
(20).

###### Proof.

Let $G\in\mathscr{G}$. Then the map $\Gamma_{\bm{d}}\circ
G\circ\Gamma^{-1}_{\bm{d}}$ is bistochastic. In fact, it is a composition of
stochastic maps and it is immediate to prove that the uniform distribution is
preserved using the properties of $\Gamma_{\bm{d}}$ and the fact that $G$ is
Gibbs-preserving. So, using Birkhoff theorem this map can be decomposed as a
convex combination of permutations ${M^{k}}$. Hence, defining
$D^{k}=\Gamma^{-1}_{\bm{d}}\circ M^{k}\circ\Gamma_{\bm{d}}$ we obtain

$G=\sum_{k}x_{k}D^{k},\quad x_{k}\geq 0,\quad\sum_{k}x_{k}=1.$ (21)

Each $D^{k}$ is a convex combination of thermo-permutations. In fact,
$M^{k}=Q^{k}_{1}\dots Q^{k}_{n_{k}}$, where $Q^{k}_{j}$ are transpositions. So
$D^{k}=E^{k}_{1}\dots E^{k}_{n_{k}}$, where
$E^{k}_{j}=\Gamma^{-1}_{\bm{d}}\circ Q^{k}_{j}\circ\Gamma_{\bm{d}}$. As we
have seen in the second part of the proof of Theorem 2, $E^{k}_{j}$ are EDPs.
So

$G=\sum_{k}x_{k}E^{k}_{1}\dots E^{k}_{n_{k}}.$ (22)

Moreover, by Lemma 16,
$E^{k}_{j}=(1-\lambda^{k}_{j})\mathbb{I}+\lambda^{k}_{j}\tilde{E}^{k}_{j}$,
where $\tilde{E}^{k}_{j}$ is a thermo-transposition. Hence, after substitution
we have an expression of each $D^{k}$ as a convex combination of thermo-
permutations. After rearranging we obtain a final expression
$G=\sum_{i}\lambda_{i}P^{i}$, for some probability distribution
$\\{\lambda_{i}\\}$ and thermo-permutations $P^{i}$.

Conversely, let $G=\sum_{i}\lambda_{i}P^{i}$ for thermo-permutations $P^{i}$.
Each $P^{i}$ is a composition of thermo-transpositions. Each thermo-
transposition is an element of $\mathscr{G}$, as it is their composition.
$\mathscr{G}$ is clearly a convex set, so $G\in\mathscr{G}$. ∎

As an intermediate result, we proved Corollary 4:

######  Corollary 23.

Every thermal process can be simulated by finite sequences of ETOs from a
complete set and classical randomness.

###### Proof.

Every thermal process is Gibbs-preserving, so it can be decomposed as in Eq.
(22). This gives the statement. ∎

It is known that $\mathscr{G}$ allows the same transitions as $\mathscr{T}$
Faist  _et al._ (2015) (Lemma 10). However $\mathscr{T}$ and $\mathscr{G}$ may
well be different sets of maps. For example, the set of unital and that of
noisy channels Horodecki  _et al._ (2003) are distinct sets, even if the same
set of final states can be reached from an arbitrary initial state by either
set of maps. However we prove here that $\mathscr{G}$ and $\mathscr{T}$
coincide as sets of stochastic maps:

######  Lemma 24.

The set of Gibbs-preserving stochastic maps coincides with the set of thermal
processes, i.e. $\mathscr{T}=\mathscr{G}$.

###### Proof.

Every thermal process is Gibbs-preserving by direct inspection. Conversely,
let $G\in\mathscr{G}$. From Lemma 22, it can be decomposed as in Eq. (20).
Each $P_{i}$ is a thermo-permutation. Hence, it can be achieved by composing
thermo-transpositions. Thermo-transpositions are EDPs, and EDPs are thermal
processes (Lemma 14). As the set of thermal processes is closed under
composition, thermo-permutations are thermal processes. So $G$ is a convex
combination of thermal processes. However $\mathscr{T}$ is a convex set,
because TO are a convex set (see Lostaglio  _et al._ (2015b), Appendix C). So
we conclude that $G$ is a thermal process. Hence, the two sets coincide. ∎

Note that this result does not hold true in the quantum case Faist  _et al._
(2015). This is for the simple reason that TOs cannot generate quantum
coherence in the energy basis, whereas Gibbs-preserving quantum maps can (see
Appendix G). One can now prove Theorem 21 simply applying the previous two
Lemmas one after the other.

####  .4.3 A simple proof of the central result on Thermal Operations

Lemma 24 (Lemma 5 in the main text) greatly simplifies the proof of the
central result of the theory of Thermal Operations, Lemma 10 (the result of
Horodecki and Oppenheim (2013)). This is important because it clarifies the
main reasoning, reduces the number of assumptions and makes it easier to build
over the known result. Previous proofs of this central lemma are significantly
harder and seem to require extra assumptions about the bath (see (i)-(iv) of
Supplementary Note 1 of Horodecki and Oppenheim (2013)). Here, the proof
follows directly as an immediate corollary of Lemma 24:

######  Theorem 25 (Horodecki and Oppenheim (2013)).

Let $\rho$, $\sigma$ be block-diagonal states in the energy eigenbasis, with
eigenvalues $\bm{p}$ and $\bm{q}$, respectively. Then a Thermal Operation
$\mathcal{T}$ such that $T\rho=\sigma$ exists if and only if
$\bm{p}\succ_{T}\bm{q}$.

###### Alternative proof.

First, from Proposition 9, this is the same as proving that
$\bm{p}\succ_{T}\bm{q}$ is equivalent to the existence of $T\in\mathscr{T}$
such that $T\bm{p}=\bm{q}$. From Lemma 24, this can be reformulated as the
existence of a map $G$ such that $G(\bm{p})=\bm{q}$ and $G(\bm{g})=\bm{g}$.
Using the embedding map this is equivalent to the existence of a bistochastic
map $M$ such that $M(\Gamma_{\bm{d}}(\bm{p}))=\Gamma_{\bm{d}}(\bm{q})$ (to see
this, define $M=\Gamma_{\bm{d}}\circ G\circ\Gamma^{-1}_{\bm{d}}$). But the
answer to this problem is a classic result in the theory of majorisation
Bhatia (2013): it is equivalent to
$\Gamma_{\bm{d}}(\bm{p})\succ\Gamma_{\bm{d}}(\bm{q})$. However, from Lemma 12,
this is equivalent to $\bm{p}\succ_{T}\bm{q}$.

∎

####  .4.4 Lemma 5 and coherence preservation under Thermal Operations

Lemma 5 is a useful tool to investigate optimal coherence-preserving
processes. Given an arbitrary quantum state $\rho$ with energy population
given by $\bm{p}$ and some target population $\bm{q}$, it is natural to wonder
what is the Thermal Operation $\mathcal{T}$ that maps $\bm{p}$ into $\bm{q}$
while preserving the maximum amount of the initial coherence of $\rho$. This
corresponds to the question: given some heat exchange with the bath, what is
the minimal degradation of its coherent properties that the system must
undergo?

This problem can be split into two steps:

  1. 1.

Find all $T\in\mathscr{T}$ such that there exists $\mathcal{T}$ satisfying Eq.
(4) and $T\bm{p}=\bm{q}$.

  2. 2.

For each $T$, find a Thermal Operation $\mathcal{T}$ that induces $T$ and
optimally preserves coherence, according to some given measure

One of the problems that one immediately faces is the lack of a simple
characterisation of $\mathscr{T}$. Lemma 24 provides it: the first step
becomes finding all $G$ such that $G\bm{p}=\bm{q}$ and $G\bm{g}=\bm{g}$. For
the second step, one may use the inequalities presented in Lostaglio  _et al._
(2015b).

###  .5 Comparison to previous work

It is useful to make a connection with a work related to the present one Perry
_et al._ (2015). There, a set of transformations called “Partial Level
Thermalisations” are allowed, but seen to be insufficient to perform every
transition. In agreement with that we have the following connection with EDPs:

######  Lemma 26.

Partial Level Thermalisations involving two energy levels $\hbar\omega_{i}$
and $\hbar\omega_{j}$, i.e. the transformation:

$(x_{i},x_{j})\mapsto(1-\epsilon)(x_{i},x_{j})+\epsilon N\bm{g}$ (23)

with all other $x_{k}$ unchanged are a strict subset of EDPs. Here
$N=x_{i}+x_{j}$, $\omega=\omega_{j}-\omega_{i}>0$,
$\bm{g}=\left(\frac{1}{1+e^{-\beta\hbar\omega}},\frac{e^{-\beta\hbar\omega}}{1+e^{-\beta\omega}}\right)$.

###### Proof.

One can easily verify that the transformation in Eq. (23) is characterised by
the following transition probabilities:

$p_{i|j}=\frac{\epsilon}{1+e^{-\beta\hbar\omega}},\quad
p_{j|i}=\frac{\epsilon}{1+e^{\beta\hbar\omega}},$ (24)

and $p_{i|i}=1-p_{j|i}$, $p_{j|j}=1-p_{i|j}$. This and the fact that
$p_{j|i}/p_{i|j}=e^{-\beta\hbar\omega}$ shows that every Partial Level
Thermalisation is an EDP. The converse is not true because $p_{0|1}\leq
1/(1+e^{-\beta\hbar\omega})<1$, for all $\beta\in[0,\infty)$, whereas EDP
require one to be able to achieve all $p_{0|1}\in[0,1]$. ∎

As mentioned in the main text, Partial Level Thermalisations correspond to
embeddable EDP. In fact, for a wide range of parameters, there are Jaynes-
Cummings interactions that cannot be realised by means of Partial Level
Thermalisations:

######  Lemma 27.

There exist resonant Jaynes-Cummings interactions with a single-mode bosonic
bath that cannot be realised by means of Partial Level Thermalisations.

###### Proof.

Let $\omega_{j}>\omega_{i}$. From the proof of Lemma 26, the maximum
probability of de-exciting the level is $p_{{\rm
max}}(\beta\hbar\omega)=(1+e^{-\beta\hbar\omega})^{-1}$. However, suppose we
resonantly couple the two energy levels to a single-mode bosonic bath via a
Jaynes-Cummings coupling in RWA approximation. Following the calculations in
the proof of Lemma 17, we find a de-exciting probability equal to
$J_{i|j}(s)$. In Appendix B.2 we obtained a lower bound on the maximum
achievable de-exciting probability within the Jaynes-Cummings model, $J_{{\rm
max}}(\beta\hbar\omega)$. This can be compared with the maximum achievable
transition probability through Partial Level Thermalisations This allows us to
check that at least for $\beta\hbar\omega\in(0.1,6.4)$

$J_{{\rm max}}(\beta\hbar\omega)>p_{{\rm max}}(\beta\hbar\omega).$ (25)

∎

###  .6 Equivalent thermodynamic frameworks

A classical framework $\mathscr{F}$ is defined as a set of stochastic
processes that can be induced on diagonal states without paying any work.
Whatever our choice is, it should lead to a consistent thermodynamic framework
in which work cannot be extracted from a single heat bath at a given
temperature. In other words, we clearly do not want our theory to allow for
the construction of a perpetuum mobile. This is made precise by the following:

######  Definition 13 (Perpetuum mobile).

A perpetuum mobile (of second kind) is a machine extracting work from a single
heat bath at given temperature. Specifically, if $\bm{g}$ is the thermal
distribution, it allows to approximate arbitrarily well the transition

$\bm{g}^{\otimes a}\rightarrow\bm{w}^{\otimes b}$ (26)

for some $a\in\mathbb{N}\backslash\\{0\\}$, $b\in\mathbb{N}\backslash\\{0\\}$
and $\bm{w}=(0,1)$. The Hamiltonian of the final state is taken to be
$H_{w}=w\left|w\right\rangle\\!\\!\left\langle w\right|$.

Eq. (26) can be understood as a transformation of heat into work, in the form
of $b$ weight systems in a pure excited state of energy $w$.

####  .6.1 Proof of Theorem 8

First, let us rewrite the theorem in a more precise way:

######  Theorem 28.

Let $\mathscr{F}$ a set of stochastic matrices such that
$\mathscr{F}\supseteq\mathscr{E}$ and such that it is impossible to obtain
arbitrary approximations of the transformation in Eq. (26). Then,
$\mathscr{F}\subseteq\mathscr{T}$.

###### Proof.

From Lemma 24, we only need to prove the above statement for $\mathscr{G}$
instead of $\mathscr{T}$.

By contradiction, assume $\mathscr{G}\subset\mathscr{F}$. Let
$M\in\mathscr{F}$, $M\not\in\mathscr{G}$. Then there exists $\bm{x}\neq\bm{g}$
such that $M(\bm{g})=\bm{x}$. Take many copies of $\bm{g}$ (which can be
thought of as a larger thermal bath composed of many thermal particles).
Applying the map $M$ to each system independently we obtain by assumption

$M^{\otimes a}(\bm{g}^{\otimes a})=\bm{x}^{\otimes a}$ (27)

Consider now the transition

$\bm{x}^{\otimes a}\rightarrow\bm{w}^{\otimes b}$ (28)

Taking $a\rightarrow\infty$, there exists a thermal process $T$ that achieves
with arbitrary precision a number $b\rightarrow\infty$ of pure excited states
with any finite energy $w$ at a rate $R$ given by Theorem 1 of Brandão  _et
al._ (2013):

$R=\frac{S(\bm{x}||\bm{g})}{w}>0,$ (29)

where $S(\bm{x}||\bm{g})=\sum_{i}p_{i}\log(p_{i}/g_{i})$. $R>0$ because for
the relative entropy $S(\bm{x}||\bm{g})=0$ if and only if $\bm{x}=\bm{g}$.
Equivalently, for every $\epsilon>0$ there exists a thermal process
$T\in\mathscr{T}$ such that $T\bm{x}^{\otimes
a}\stackrel{{\scriptstyle\epsilon}}{{\approx}}\bm{w}^{\otimes b}$. However,
from Theorem 2, the thermodynamic transition (28) can be realised as a
sequence of EDPs $E^{1}$,…,$E^{N}$. Hence, using only operations in
$\mathscr{F}$, we achieved the perpetuum mobile transition of Eq. (26) with
$b=aS(\bm{x}||\bm{g})/w$. This is a contradiction, so it must be
$\mathscr{F}\subseteq\mathscr{G}$, that gives
$\mathscr{F}\subseteq\mathscr{T}$ ∎

####  .6.2 Proof of Corollary 8

###### Proof.

This can be proven by showing that all thermodynamic frameworks with the
mentioned properties are equivalent to the $\mathscr{T}$. In fact

  1. 1.

Equivalence of frameworks is an equivalence relation, so this proves the
equivalence of any two frameworks.

  2. 2.

Because thermal processes are characterised by thermo-majorisation (Lemma 10
or Theorem 25), this proves the second part of the claim as well.

Let $\mathscr{F}$ be a thermodynamic framework. Let us fix $\bm{p}$ and
$\bm{q}$. If there exists a $T\in\mathscr{T}$ such that $T(\bm{p})=\bm{q}$,
then there exists a sequence of EDPs mapping $\bm{p}$ into $\bm{q}$ (Theorem
20). Because $\mathscr{E}\subseteq\mathscr{F}$, we conclude that the
transformation $\bm{p}\rightarrow\bm{q}$ can be achieved using only maps in
$\mathscr{F}$.

Conversely, assume that one can map $\bm{p}$ into $\bm{q}$ using only
transformations in $\mathscr{F}$. Because $\mathscr{F}$ does not allow for a
perpetuum mobile, it is contained in the set $\mathscr{T}$ of thermal
processes (Theorem 28). Hence, one can transform $\bm{p}$ into $\bm{q}$ by
means of thermal processes. We conclude that $\mathscr{F}$ is equivalent to
$\mathscr{T}$. ∎

###  .7 Fully quantum considerations

An obvious question is how to generalise the results of the present
investigation to the fully quantum case. Natural questions are in particular:
to what extent all ETOs can be realised within a natural physical model?
Assuming that all ETOs can be realised, is it possible to induce any
transition allowed by means of Thermal Operations?

It is interesting also to discuss thermodynamic equivalence in the fully
quantum case. Theorem 7 does not holds, because Gibbs-preserving quantum
channels are a strict superset of Thermal Operations and do not allow for a
perpetuum mobile. Moreover, also Corollary 8 requires further discussion. In
fact, the framework of Gibbs-preserving quantum channels is inequivalent to
that of Thermal Operations.

The reason for these differences has to do with the physics more than to some
technical difficulty. In fact, whereas work is a universal resource in
thermodynamics classically, it is not such quantum-mechanically Lostaglio  _et
al._ (2015a). A second resource in thermodynamics is quantum coherence among
energy eigenspaces, and the inequivalence of various thermodynamic models
arises from the different role coherence plays in them.

Broadly speaking, there are two general classes of models:

  1. M1

Those in which coherence cannot be freely created or exchanged with work, an
example being Thermal Operations. In these models thermodynamics can be seen
as a particular instance of coherence theory (Lostaglio  _et al._ (2015a),
Theorem 1, Lostaglio  _et al._ (2015b); Marvian and Spekkens (2016)). This
better describes scenarios in which a quantum system interacts with a
dissipative environment and no classical external field is applied.

  2. M2

Those in which coherence can be freely exchanged with work, an example being
Gibbs-preserving quantum operations Faist  _et al._ (2015). These models
implicitly assume the possibility of using an external classical field
Lostaglio  _et al._ (2015a) that allows to freely transform coherence into
work and convert it back, but not to do work on the system.

In the models M1, the crucial property is _time-translation symmetry_ Marvian
and Spekkens (2014). If $H_{S}$ describes the Hamiltonian part of the
evolution, the quantum map $\mathcal{E}$ is required to satisfy for all $\rho$
and $t$

$\mathcal{E}\circ\mathcal{U}_{t}(\rho)=\mathcal{U}_{t}\circ\mathcal{E}(\rho),$
(30)

where $\mathcal{U}_{t}(\rho)=e^{-iHt}\rho e^{iHt}$. For processes described by
a master equation, this is the same as requiring that the dissipative and
unitary part of the generator of the dynamics commute. In fact, this is a
property of standard thermalisation models, such as Davies maps Davies (1974);
Roga  _et al._ (2010). Time-translation symmetry makes coherence a resource,
as it is unavoidably dissipated in time, and “decouples” its evolution from
that of population Marvian and Spekkens (2014).

On the other hand, models M2 are characterised by the _lack_ of time-
translation symmetry, even though the condition that the Gibbs state is a
fixed point still ensures that external work cannot be used to push a state
farther away from equilibrium. Davies maps are not recovered in the Markovian
limit, so these models appear to describe more controlled situations where one
can drive the system with a classical field (but without pushing the system
more out of equilibrium). It would be useful to understand if such situation
can be realised in a simple physical model.

Hence, the lack of thermodynamic equivalence (Corollary 8) in the quantum
regime then reflects the different roles coherence can play in the model, but
this is ultimately a question of the physical scenario we wish to describe. A
promising line of research is to find whether quantum thermodynamic models are
equivalent once we specify one of the two alternatives M1 or M2.

These considerations could also offer a new approach to the problem of
analysing coherence constraints in thermodynamics. For example, one could
compare decomposable and indecomposable thermodynamic processes in terms of
their ability to preserve coherence. We leave this to future investigations.

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