# We can catalyse it for you thermally:
Catalytic transformations for thermal operations

Jakub Czartowski  Doctoral School of Exact and Natural Sciences, Jagiellonian
University, 30-348 Kraków, Poland  Faculty of Physics, Astronomy and Applied
Computer Science, Jagiellonian University, 30-348 Kraków, Poland  A. de
Oliveira Junior  Center for Macroscopic Quantum States bigQ, Department of
Physics, Technical University of Denmark, Fysikvej 307, 2800 Kgs. Lyngby,
Denmark

###### Abstract

What are the fundamental limits and advantages of using a catalyst to aid
thermodynamic transformations between quantum systems? In this work, we answer
this question by focusing on transformations between energy-incoherent states
under the most general energy-conserving interactions among the system, the
catalyst, and a thermal environment. The sole constraint is that the catalyst
must return unperturbed and uncorrelated with the other subsystems. More
precisely, we first characterise the set of states to which a given initial
state can thermodynamically evolve (the catalysable future) or from which it
can evolve (the catalysable past) with the help of a strict catalyst.
Secondly, we derive lower bounds on the dimensionality required for the
existence of catalysts under thermal process, along with bounds on the
catalyst’s state preparation. Finally, we quantify the catalytic advantage in
terms of the volume of the catalysable future and demonstrate its utility in
an exemplary task of generating entanglement using thermal resources.

††preprint: APS/123-QED

##  I Introduction

Classical thermodynamics is a theory of macroscopic systems in equilibrium,
whose description relies on quantities with fluctuations negligible compared
to their average values [1, 2]. As a byproduct, it gives a clear picture of
what state transformations are allowed in terms of a small number of
macroscopic quantities, such as work and entropy. Specifically, at constant
temperature and volume, transformations between equilibrium states are
governed by a simple function, the so-called Helmholtz free energy [3].
However, going beyond the original scenario of equilibrium thermodynamics has
profound consequences. For finite-dimensional quantum systems far from
equilibrium, state transformations are no longer governed by a single second
law but by an entire family of conditions known as the “second laws of quantum
thermodynamics” [4].

Under this paradigm, the existence of a family of generalised free energies,
which indicate what states can be reached from a given one (referred to as the
present state), naturally decomposes the space of states into three distinct
parts: the set of states to which the present state can evolve, known as the
future thermal cone; the set of states from which the present state can be
reached, or the past thermal cone; and the set of states that are neither in
the past nor the future thermal cone, constituting the incomparable region [5,
6] – three regions analogous to the structure of the light cone in special
relativity, which divides the space–time into future, past, and space-like
region (see Fig. 1). Consequently, the issue of thermodynamic transformations
in finite-size systems is characterised by the emergence of a finite-size
effect known as incomparability.

Interestingly, one can reduce this effect by introducing a catalyst – a system
that allows us to perform otherwise impossible transformations, but is not
modified itself, i.e., at the end of the process it is returned unchanged [7,
8]. Remarkably, if we allow the catalyst to become correlated with the main
system while keeping its local state intact, transformations are once again
described by a single function, the standard nonequilibrium free energy [9,
10, 11, 12].

Figure 1: Thermodynamic arrow of time. The second law of thermodynamics
introduces an ordering in the space of states such that, for a given initial
state, it divides the space into past (blue), incomparable (red), and future
(green) regions. By allowing a catalyst $\boldsymbol{c}$, the future and past
regions grow as the incomparable region shrinks.

Over the last two decades, catalytic transformations have become increasingly
relevant in the context of quantum information science [7, 8]. Their first
appearance focused on entanglement manipulation [13, 14, 15, 16, 17, 18, 19,
20, 7] which then broadened to include quantum thermodynamics [21, 22, 23, 24,
25, 12, 26, 27, 28, 29, 30, 31, 32, 33], and many other facets of quantum
theory [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 40, 41, 42, 43, 44, 45].
The only constraint of a catalytic process–returning the catalyst
unperturbed–leads to the classification of catalytic processes into two main
different types: _(i)_ allowing it to become correlated with the main system,
or _(ii)_ keeping it uncorrelated. As pointed out before, the former case of
correlated catalysis is powerful enough to close the gap between the future
and past thermal regions. The latter, referred to as ‘strict catalysis’, is
much closer to the original idea of returning the catalyst unperturbed. It is
thus natural to ask about limitations of strict catalysts – a problem we will
discuss in this manuscript.

Arguably, one of the fundamental problems in quantum thermodynamics concerns
quantification of the catalytic advantages in a given transformation. This
issue can be approached by investigating the behaviour of the thermal cones
when a catalyst is introduced. In this scenario, the catalyst cannot change
future into the past or vice-verse, nor can it shrink both regions.
Consequently, the growth of the past and future thermal cones occurs at the
expense of the incomparable region. However, determining the set of states
that can be achieved with the aid of a catalyst has been considered a highly
non-trivial problem. The first steps for a related problem were made in Ref.
[46, 47] within the realm of entanglement theory, where the authors focused on
pure state transformations governed by entanglement-assisted local operations
and classical communication (ELOCC). Therein, the authors consider bounds for
the entanglement of a catalyst necessary to catalyse transformation between a
given pair of otherwise incomparable states. Furthermore, they established a
lower bound for the dimension of the catalyst required for a specific ELOCC
transformation. As these results rely on the notion of majorisation, they
automatically extend to the resource theory of coherence, as well as to so-
called noisy operations, or incoherent thermal operations at infinite
temperature.

In this work, we extend our understanding of the interplay between quantum
thermodynamics and catalysis. Our main question can be framed as:

_What are the ultimate limits for the set of states achievable from a given
initial state, with the aid of catalyst, under the most general energy-
conserving interactions between the system and the thermal bath?_

We answer this question by providing for any given starting state
$\boldsymbol{p}$ a set of states $\mathcal{C}_{+}(\boldsymbol{p})$, called
_the catalysable future region of state $\boldsymbol{p}$_, into which it can
be transformed with the aid of a catalyst. The definition of catalysable past
region $\mathcal{C}_{-}(\boldsymbol{p})$ follows analogously by taking
$\boldsymbol{p}$ to be the target of a transformation to be catalysed. More
specifically, for a $d$-dimensional energy-incoherent state, we provide an
explicit construction of its catalysable past and future, along with a
characterisation of the extreme points of the catalysable future. These
results reveal fundamental limits on the advantages achievable with
uncorrelated catalysts [see Fig. 2 for a warm-up example, where we present the
future and past regions with and without catalyst for a three-and four-level
systems]. As these results are general and do not rely on any specific
assumptions regarding the catalytic state, our second focus is on the
dimensionality and populations of the catalyst. We derive lower bounds on the
dimensionality required for catalysts under thermal operations, together with
bounds on the catalysts population. Our results shed light on how to construct
a catalyst given two incomparable states. Surprisingly, our findings show that
the no-go result from [47] formulated for the resource theory of entanglement
and applicable for the infinite-temperature setting, stating that catalysis
has no effect under local operations and classical communication for a main
system of dimension three, ceases to apply in the context of thermodynamics at
finite temperatures.

Figure 2: Catalysable regions. For a three-level system with population given
by $\boldsymbol{p}=(0.42,0.51,0.07)$, represented by a black dot $\bullet$,
and energy spectrum $E_{1}=0,E_{2}=1$, and $E_{3}=2$ for $\beta=0.2$, we plot
its (a) catalysable future $C_{+}(\boldsymbol{p})$ (yellow) and (b) past
$C_{-}(\boldsymbol{p})$ (purple), together with their respective past and
future thermal cones. In (c) we show the catalysable future for the four-level
system with $\boldsymbol{p}=(0.05,0.28,0.40,0.27)$ and energy spectrum
$E_{1}=0,E_{2}=1$, $E_{3}=2$, and $E_{4}=3$.

Using the above framework, we also quantify the catalytic advantage in terms
of the volume of the catalysable future. In this context, we offer a detailed
discussion on its behaviour as a function of the ambient temperature. As an
application, we demonstrate the benefits of employing a catalyst in the
generation of entanglement. Recently Ref. [48] identified the set of states
that cannot become entangled under thermal operations. In this analysis, we
elucidate the catalytic advantages by identifying the set of states that, once
catalysed, enable entanglement generation under thermal operations.

The paper is organised as follows. In Section II, we introduce the resource-
theoretic approach to thermodynamics, review known results concerning the
conditions for state transformation under thermal operations, and establish
the notation of the paper. Section III, collecting our main results, begins
with the construction of the catalysable past and future regions, giving a
simple method for generating all the extreme points of the catalysable future,
thus providing the closest analogue to the Birkhoff theorem for the problem of
catalysis in thermal operations. Next, we state our second main result, which
establishes a lower bound on the dimensionality of the catalyst and introduces
a constraint on its population vector for any pair of incomparable states. In
Section IV we discuss a quantification of the catalytic advantage by volume of
the catalysable future and past regions. Then, in Section V, we provide an
application of our results to entanglement generation under thermal
operations. Finally, we conclude with an outlook in Section VI. The technical
derivation of our results can be found in the Appendices.

##  II Framework

In this work, our aim is to investigate how the structure of thermal cones is
modified when a catalyst is introduced. We approach this problem using the
resource-theoretic framework of thermodynamics [49, 50]. In this section, we
briefly review well-known results without going into details and set the scene
by introducing the relevant notation.

###  II.1 Thermal operations

We consider a composite system comprised of a finite-dimensional system and a
thermal environment at temperature $T=\beta^{-1}$, where we set $k_{B}=1$. The
main system is described by a Hamiltonian $H$ and is prepared in a state
$\rho$; while, the thermal environment, with Hamiltonian $H_{\textsf{E}}$, is
assumed to be in a thermal equilibrium state, $\gamma_{\textsf{E}}=e^{-\beta
H_{E}}/Z_{\textsf{E}}$, where $Z_{\textsf{E}}=\tr(e^{-\beta H_{\textsf{E}}})$
is the partition function.

The evolution of the composite system is modeled by considering the set of
_thermal operations_ (TOs) [51, 52, 50]. These are quantum channels defined
upon minimal assumptions, such as that the joint system is closed and evolves
via an energy-preserving unitary. This is captured by completely positive
trace-preserving (CPTP) maps that act on $\rho$ as

${\cal
E}(\rho)=\Tr_{\textsf{E}}\left[U\left(\rho\otimes\gamma_{E}\right)U^{\dagger}\right],$
(1)

where $U$ is a joint unitary that commutes with the total Hamiltonian of the
system and the bath $[U,H\otimes\mathbbm{1}_{E}+\mathbbm{1}\otimes H_{E}]=0$.

The fundamental question within the resource-theoretic approach is to identify
the set of states that a given state $\rho$ can be transformed to under
thermal operations. The reachability of states under TOs can be studied by
introducing the notion of thermal cones [6]. Given $\rho$, the set of states
achievable under a thermal operation is referred to as the future thermal cone
${\cal T}^{,\beta}_{+}(\rho)$. Conversely, the set of states that can evolve
into $\rho$ is called the past thermal cone ${\cal T}^{,\beta}_{-}(\rho)$. The
set of states that are neither in the past nor the future of $\rho$ is termed
the incomparable thermal region ${\cal T}^{,\beta}_{\emptyset}(\rho)$. The
general characterisation of thermal cones is not known beyond the simplest
qubit case [53, 54].

However, for states that are block-diagonal in the energy eigenbasis, also
known as _energy-incoherent states_ , a simple construction based on
thermomajorisation relation exists. For this class of states, the existence of
a thermal operation between two states $\rho$ and $\sigma$ is equivalent to
the existence of Gibbs preserving (GP) matrices acting on their eigenvalues
$\boldsymbol{p}=\operatorname{eig(\rho)}$ and
$\boldsymbol{q}=\operatorname{eig}(\sigma)$ [51, 55]. Furthermore, the
existence of a GP matrix connecting two states can be expressed by
thermomajorisation relations between $\boldsymbol{p}$ and $\boldsymbol{q}$
[52].

The framework of thermal operations naturally incorporates the phenomenon of
catalysis by simply assuming that the system $\rho$ is given by a joint and
uncorrelated system $\rho=\rho_{\textsf{S}}\otimes\omega_{\textsf{C}}$ with
Hamiltonian
$H=H_{\textsf{S}}\otimes\mathbbm{1}_{\textsf{C}}+\mathbbm{1}_{\textsf{S}}\otimes
H_{\textsf{C}}$111It is known in the literature that, without loss of
generality, one can always choose a trivial Hamiltonian for the catalyst,
$H_{\textsf{C}}\propto\mathbbm{1}$ [52]. However, since our results will not
rely on this and, furthermore, in practical settings one is unlikely to have a
trivial Hamiltonian, we will not use this assumption.. Thus, we consider
catalytic thermal operations (CTOs) to be transformations of the following
form

$\mathcal{E}(\rho\otimes\omega_{\textsf{C}})=\sigma\otimes\omega_{\textsf{C}}.$
(2)

When $\rho$ and $\sigma$ are energy-incoherent states, the necessary
conditions for the existence of a transformation as in Eq. (2) is captured by
a set of quantities called $\alpha$-free energies

$F_{\alpha}(\rho,\gamma)\geq
F_{\alpha}(\sigma,\gamma)\>\>,\>\>\forall\alpha\geq 0,$ (3)

where $F_{\alpha}(\rho):=[D_{\alpha}(\rho\|\sigma)-\log Z]/\beta$, with
$D_{\alpha}(\rho\|\sigma)$ being the quantum Rényi divergence [56, 57].
Nevertheless, a complete characterisation of the catalytic future thermal cone
has not yet been addressed.

###  II.2 Mathematical preliminaries

In our analysis we focus on $d$-dimensional states that are diagonal in the
energy eigenbasis and are equivalently described by probability vectors
corresponding to populations assigned to respective energy levels,

$\rho=\sum^{d}_{i=1}p_{i}\outerproduct{E_{i}}{E_{i}}\longmapsto\boldsymbol{p}=(p_{1},...,p_{d}).$
(4)

Thus, the states under consideration live in the $d$-dimensional probability
simplex,

$\Delta_{d}=\quantity{\boldsymbol{p}=(p_{1},...,p_{d})\in\mathbbm{R}_{\geq
0}^{d}:\sum_{i}p_{i}=1}.$ (5)

Unless stated otherwise, throughout this manuscript, we will work under the
assumption that the energy levels are non-degenerate, i.e., $i\neq
j\Rightarrow E_{i}\neq E_{j}$.

Furthermore, we define the thermal distribution $\boldsymbol{\gamma}$ and
slope vector $\boldsymbol{s}(\boldsymbol{p})$ associated with a probability
vector $\boldsymbol{p}$ as

$\displaystyle\boldsymbol{\gamma}:=\frac{1}{Z}(e^{-\beta E_{1}},...,e^{-\beta
E_{d}})\>\>\>\text{and}\>\>\>\boldsymbol{s}(\boldsymbol{p}):=\underbrace{\quantity(\frac{p_{1}}{\gamma_{1}},...,\frac{p_{d}}{\gamma_{d}})^{\downarrow}}_{\quantity(s_{1}(\boldsymbol{p}),\ldots,s_{d}(\boldsymbol{p}))},$
(6)

where the down arrow denotes the vector arranged in non-increasing order.
Next, we define the $\beta$-order of $\boldsymbol{p}$:

######  Definition 1 ($\beta$-ordering).

Given $\boldsymbol{p}$ and $\boldsymbol{\gamma}$, the $\beta$-ordering of
$\boldsymbol{p}$ is defined by a permutation
$\boldsymbol{\pi}_{\boldsymbol{p}}$ that satisfies

$s_{\boldsymbol{\pi}_{\boldsymbol{p}}(i)}(\boldsymbol{p})=\frac{p_{i}}{\gamma_{i}}.$
(7)

Thus, the $\beta$-ordered version of $\boldsymbol{p}$ is given by

$\boldsymbol{p}^{\,\beta}=\left(p_{\boldsymbol{\pi}_{\boldsymbol{p}}^{-1}(1)},\dots,p_{\boldsymbol{\pi}_{\boldsymbol{p}}^{-1}(d)}\right).$
(8)

Note that each permutation belonging to the symmetric group,
$\boldsymbol{\pi}\in\mathcal{S}_{d}$, defines a different $\beta$-ordering on
the energy levels of the Hamiltonian $H$

The above definition allows us to define a thermomajorisation curve:

######  Definition 2 (Thermomajorisation curve).

Given $\boldsymbol{p}$, a thermomajorisation curve
$f^{\beta}_{\boldsymbol{p}}:\\{0,1\\}\to\\{0,1\\}$ is a piecewise linear
function composed of segments connecting the point $(0,0)$ and the points
defined by consecutive subsums of the $\beta$-ordered form of the probability
$\boldsymbol{p}^{\beta}$ and the Gibbs state $\boldsymbol{\gamma}^{\beta}$,

$\left(\sum_{i=1}^{k}\gamma^{\,\beta}_{i},~{}\sum_{i=1}^{k}p^{\,\beta}_{i}\right):=\left(\sum_{i=1}^{k}\gamma_{\boldsymbol{\pi}^{-1}_{\boldsymbol{p}}(i)},~{}\sum_{i=1}^{k}p_{\boldsymbol{\pi}^{-1}_{\boldsymbol{p}}(i)}\right),$
(9)

for $k\in\\{1,\dots,d\\}$.

Finally, we define the notion of thermomajorisation in terms of the respective
thermomajorisation curves:

######  Definition 3 (Thermomajorisation).

Given two $d$-dimensional probability distributions $\boldsymbol{p}$ and
$\boldsymbol{q}$, and a fixed inverse temperature $\beta$, we say that
$\boldsymbol{p}$ _thermomajorises_ $\boldsymbol{q}$ and denote it as
$\boldsymbol{p}\succ_{\beta}\boldsymbol{q}$, if the thermomajorisation curve
$f^{\,\beta}_{\boldsymbol{p}}$ is above $f^{\,\beta}_{\boldsymbol{q}}$
everywhere, i.e.,

$\boldsymbol{p}\succ_{\beta}\boldsymbol{q}\iff\forall
x\in[0,1]:~{}f^{\,\beta}_{\boldsymbol{p}}(x)\geq
f^{\,\beta}_{\boldsymbol{q}}(x)\,.$ (10)

Let us make a few comments about the above definition. First, at the infinite-
temperature limit (or when $\beta=0$), the thermal distribution vector
$\boldsymbol{\gamma}$ becomes the uniform state

$\boldsymbol{\eta}=\frac{1}{d}(1,...,1).$ (11)

Consequently, the concept of $\beta$-ordering simplifies to arranging the
probability vectors in non-increasing order. Thus, thermomajorisation reduces
to the well-known concept of majorisation [58]. Second, thermomajorisation
does not introduce a total order, a given pair of states $\boldsymbol{p}$ and
$\boldsymbol{q}$ is said to be incomparable when neither $\boldsymbol{p}$
thermomajorises $\boldsymbol{q}$, nor $\boldsymbol{q}$ thermomajorises
$\boldsymbol{p}$. We denote incomparable states as
$\boldsymbol{p}\perp^{\beta}\boldsymbol{q}$.

The concept of incomparability can be studied by introducing the family of
tangent vectors, a concept stemming from the idea of tangent function. To gain
some intuition about their properties and interpretation, we first present its
construction for the simple case of $\beta=0$ and then generalise to
$\beta>0$.

For any vector $\boldsymbol{p}$, a tangent vector
$\boldsymbol{t}(\boldsymbol{p})\equiv\boldsymbol{t}$ is defined by imposing
that all its components, except the first and the last are equal,
$t_{i}=t_{j}$ for all $1<i<j<d$. Furthermore, we require that the majorisation
function $f_{\boldsymbol{t}}$ agrees with $f_{\boldsymbol{p}}$ at no more than
two consecutive elbows, i.e.,
$f_{\boldsymbol{t}}(i/d)=f_{\boldsymbol{p}}(i/d)$ and
$f_{\boldsymbol{t}}(x)>f_{\boldsymbol{p}}(x)$ for
$x\in\quantity(0,i/d)\cup(i/d,1)$ or
$f_{\boldsymbol{t}}(x)=f_{\boldsymbol{p}}(x)$ for $x\in[i/d,(i+1)/d]$ and
$f_{\boldsymbol{t}}(x)>f_{\boldsymbol{p}}(x)$ elsewhere. The two imposed
conditions follow the intuition of tangency and, by construction, satisfy the
majorisation relation $\boldsymbol{t}\succ\boldsymbol{p}$.

Figure 3:  Majorisation curves of the tangent vectors. For the initial state
$\boldsymbol{p}=(0.43,0.37,0.18,0.02)$, we plot the majorisation curve of
$\boldsymbol{p}$ (black curve) along with the tangent vectors
$\boldsymbol{t}^{(1)}$, $\boldsymbol{t}^{(2)}$, $\boldsymbol{t}^{(3)}$ and
$\boldsymbol{t}^{(4)}$.

Assuming equality between $f_{\boldsymbol{t}}(x)$ and $f_{\boldsymbol{p}}(x)$
on a single linear segment, $x\in[i/d,(i+1)/d]$, limits the tangent vectors to
a set of $d$ unique probability vectors $\boldsymbol{t}^{(n)}$, defined as
follows:

$\boldsymbol{t}^{(n)}=\left(t^{(n)}_{1},p_{n}^{\downarrow},...,p_{n}^{\downarrow},t^{(n)}_{d}\right),$
(12)

for $1\leq n\leq d$, where the first and last components are given by

$\displaystyle t_{1}^{(n)}$
$\displaystyle=\sum_{i=1}^{n-1}p_{i}^{\downarrow}-(n-2)p_{n}^{\downarrow},$
$\displaystyle t_{d}^{(n)}$
$\displaystyle=1-t^{(n)}_{1}-(d-2)p_{n}^{\downarrow}.$ (13)

An exemplary set of tangent vectors is depicted in Fig. 3.

Note that the tangent vectors $\boldsymbol{t}^{(n)}$, which agrees with the
majorisation curve of $\boldsymbol{p}$ at two successive points, can be used
to construct all possible tangent vectors $\boldsymbol{t}$ that meet the
condition of agreement at least at a single point,
$f_{\boldsymbol{t}}(i/d)=f_{\boldsymbol{p}}(i/d)$. The fact that
$\boldsymbol{t}^{(n)}$ may be a quasi-probability distribution does not pose a
problem as this vector can always be projected back onto the probability
simplex.

The intuition behind tangent vectors can now finally be generalised to the
case of finite temperatures. This is achieved as follows:

######  Definition 4 (Thermal tangent vectors).

Given an energy-incoherent state $\boldsymbol{p}$ and a thermal state
$\boldsymbol{\gamma}$, consider distributions
$\boldsymbol{t}^{(n,\boldsymbol{\pi})}(\boldsymbol{p})\equiv
t^{(n,\boldsymbol{\pi})}$ in their $\beta$-ordered form, constructed for each
permutation $\boldsymbol{\pi}\in{\cal S}_{d}$ and $n\in\\{1,...,d\\}$,

$\left[\boldsymbol{t}^{(n,\boldsymbol{\pi})}\right]^{\,\beta}=\left(t^{(n,\boldsymbol{\pi})}_{\boldsymbol{\pi}(1)},s_{n}\gamma_{\boldsymbol{\pi}(2)},...,s_{n}\gamma_{\boldsymbol{\pi}(d-1)},t^{(n,\boldsymbol{\pi})}_{\boldsymbol{\pi}(d)}\right),$
(14)

with

$\displaystyle t^{(n,\boldsymbol{\pi})}_{\boldsymbol{\pi}(1)}$
$\displaystyle=\sum_{i=1}^{n}p^{\,\beta}_{i}-s_{n}\left(\sum_{i=1}^{n}\gamma^{\,\beta}_{i}-\gamma_{\boldsymbol{\pi}(1)}\right),$
(15a) $\displaystyle t^{(n,\boldsymbol{\pi})}_{\boldsymbol{\pi}(d)}$
$\displaystyle=1-t^{(n,\boldsymbol{\pi})}_{\boldsymbol{\pi}(1)}-s_{n}\sum_{i=2}^{d-1}\gamma_{\boldsymbol{\pi}(i)},$
(15b)

where $s_{n}:=\boldsymbol{s}_{n}(\boldsymbol{p}).$

Given that the previous definition naturally includes the case of $\beta=0$,
we will, from now on, simply refer to thermal tangent vectors as tangent
vectors.

Finally, notice that the same intuition as the case of $\beta=0$ holds for
$\beta\geq 0$: a tangent vector $\boldsymbol{t}^{(n,\pi)}$ is a probability
distribution with $\beta$-order given by of almost-constant slope, such that
$s[\boldsymbol{t}^{(n,\pi)}]_{i}=s_{n}(\boldsymbol{p})$ for $1<i<d$, with the
first and last slope adjusted such that
$\boldsymbol{t}^{(n,\pi)}\succ_{\beta}\boldsymbol{p}$ and there exists a point
$x$ at which $f_{\boldsymbol{t}^{(n,\pi)}}(x)=f_{\boldsymbol{p}}(x)$. For a
very large dimension $d$ such a state indeed looks like a tangent at point
$x$.

##  III Main results

In this section, we present our main results, which are divided into two
parts. First, we provide a condition that allows one to characterise the set
of states that can be catalysable for a given initial state. These results are
general and do not rely on any specific assumptions regarding the catalyst’s
structure, including its dimension, Hamiltonian, or state. Subsequently, the
second part of our results complements the first by establishing bounds on
dimensionality and the region of the state space within which a catalyst can
be found.

###  III.1 Catalysable past and future regions

We start by defining the catalysable sets:

######  Definition 5 (Catalysable region).

For a $d$-dimensional energy-incoherent state $\boldsymbol{p}$, we define the
catalysable future (past) region $\mathcal{C}_{\pm}(\boldsymbol{p})$ as
subsets of the incomparable region $\mathcal{T}_{\emptyset}(\boldsymbol{p})$
such that

$\displaystyle\mathcal{C}_{+}(\boldsymbol{p})$
$\displaystyle\equiv\quantity{\boldsymbol{q}\in\Delta_{d}:\boldsymbol{q}\perp_{\beta}\boldsymbol{p},\,\exists_{k\geq
2,\,\boldsymbol{c}\in\Delta_{k}}:\boldsymbol{p}\otimes\boldsymbol{c}\succ_{\beta}\boldsymbol{q}\otimes\boldsymbol{c}},$
(16) $\displaystyle\mathcal{C}_{-}(\boldsymbol{p})$
$\displaystyle\equiv\quantity{\boldsymbol{q}\in\Delta_{d}:\boldsymbol{q}\perp_{\beta}\boldsymbol{p},\,\exists_{k\geq
2,\,\boldsymbol{c}\in\Delta_{k}}:\boldsymbol{q}\otimes\boldsymbol{c}\succ_{\beta}\boldsymbol{p}\otimes\boldsymbol{c}}.$
(17)

In what follows, if $\boldsymbol{q}\in\mathcal{C}_{+}(\boldsymbol{p})$, we
will say that transformation $\boldsymbol{p}\rightarrow\boldsymbol{q}$ is
catalysable, or that $\boldsymbol{q}$ is catalysable, omitting explicit
relation to $\boldsymbol{p}$ whenever it is clear from the context.

The following result establishes a necessary condition for any state
$\boldsymbol{q}$ to belong to the catalysable future region of a state
$\boldsymbol{p}$:

######  Lemma 1 (Catalysability condition).

Consider a pair of incomparable states
$\boldsymbol{q}\perp_{\beta}\boldsymbol{p}$. The state $\boldsymbol{q}$ is
catalysable (with respect to $\boldsymbol{p}$), ie.
$\boldsymbol{q}\in\mathcal{C}_{+}(\boldsymbol{p})$ [or, equivalently,
$\boldsymbol{p}\in\mathcal{C}_{-}(\boldsymbol{q})$], only if it satisfies:

$s_{1}\quantity(\boldsymbol{p})>s_{1}\quantity(\boldsymbol{q})\quad\text{and}\quad
s_{d}\quantity(\boldsymbol{p})<s_{d}\quantity(\boldsymbol{q}).$ (18)

Figure 4: Catalysable regions for $d=3$. For a three-level system with a state
given by $\boldsymbol{p}=(0.34,0.59,0.07)$, represented by a black dot
$\bullet$, and a thermal state $\boldsymbol{\gamma}$, represented by a black
star $\bigstar$, with an energy spectrum $E_{1}=0,E_{2}=1$, and $E_{3}=2$, we
plot its catalysable thermal cone for $\beta$ ranging from (a) $\beta=0.2$ to
$\beta\to\infty$.

###### Proof.

Let us take a potential catalyst state $\boldsymbol{c}\in\Delta_{k}$. The
slope vectors will be given as

$\displaystyle\boldsymbol{s}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})=\biggl{(}$
$\displaystyle
s_{1}(\boldsymbol{p})s_{1}(\boldsymbol{c}),\ldots,s_{1}(\boldsymbol{p})s_{k}(\boldsymbol{c}),\ldots,$
(19) $\displaystyle
s_{d}(\boldsymbol{p})s_{1}(\boldsymbol{c}),\ldots,s_{d}(\boldsymbol{p})s_{k}(\boldsymbol{c})\biggr{)}^{\downarrow},$

and same for $\boldsymbol{q}\otimes\boldsymbol{c}$. In order for one state to
thermomajorise the other,
$\boldsymbol{p}\otimes\boldsymbol{c}\succ_{\beta}\boldsymbol{q}\otimes\boldsymbol{c}$,
we need

$\displaystyle
s_{1}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})>s_{1}\quantity(\boldsymbol{q}\otimes\boldsymbol{c}),$
$\displaystyle
s_{dk}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})<s_{dk}\quantity(\boldsymbol{q}\otimes\boldsymbol{c}).$
(20)

From the definition of $\beta$-ordering we know instantly that
$s_{i}(\boldsymbol{p})\geq s_{j}(\boldsymbol{p})$ whenever $i\leq j$,
similarly for $\boldsymbol{q}$ and $\boldsymbol{c}$. Consequently, we can
state that

$\displaystyle
s_{1}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})=s_{1}(\boldsymbol{p})s_{1}(\boldsymbol{c}),$
$\displaystyle
s_{dk}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})=s_{d}(\boldsymbol{p})s_{k}(\boldsymbol{c}),$
(21)

and, again, analogously for $\boldsymbol{q}$. Thus, we have

$\displaystyle
s_{1}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})>s_{1}\quantity(\boldsymbol{q}\otimes\boldsymbol{c})\Rightarrow
s_{1}\quantity(\boldsymbol{p})>s_{1}\quantity(\boldsymbol{q}),$ (22)
$\displaystyle
s_{dk}\quantity(\boldsymbol{p}\otimes\boldsymbol{c})<s_{dk}\quantity(\boldsymbol{q}\otimes\boldsymbol{c})\Rightarrow
s_{d}\quantity(\boldsymbol{p})<s_{d}\quantity(\boldsymbol{q}),$

which completes the proof. ∎

As a result, the catalysable past and future can be explicitly constructed by
using Lemma 1 together with the tangent vectors introduced in Sec. II.2.

######  Theorem 2 (Catalysable region characterisation).

Consider an energy-incoherent state $\boldsymbol{p}$ and its associated
tangent vectors $\boldsymbol{t}^{(n,\pi)}(\boldsymbol{p})$. We define two
auxiliary sets

$T_{i}(\boldsymbol{p})=\operatorname{conv}\quantity[\quantity{\boldsymbol{t}^{(i,\pi)}(\boldsymbol{p})}_{\pi\in\mathcal{S}_{d}}],$
(23)

where $\mathcal{S}_{d}$ is the set of all possible $d$-element permutations.
The catalysable future and past regions can then be reformulated as

$\displaystyle\mathcal{C}_{+}(\boldsymbol{p})$
$\displaystyle=\quantity[T_{1}(\boldsymbol{p})\cap
T_{d}(\boldsymbol{p})]\setminus\mathcal{T}_{+}(\boldsymbol{p}),$ (24)
$\displaystyle\mathcal{C}_{-}(\boldsymbol{p})$
$\displaystyle=\mathcal{T}_{\emptyset}(\boldsymbol{p})\setminus\quantity[T_{1}(\boldsymbol{p})\cup
T_{d}(\boldsymbol{p})].$ (25)

###### Proof.

The above follows from the fact that
$\boldsymbol{t}^{(i,\pi)}(\boldsymbol{p})$ by construction thermomajorises all
$\boldsymbol{q}$ with $\beta$-order $\boldsymbol{\pi}$ such that
$s_{i}(\boldsymbol{q})=s_{i}(\boldsymbol{p})$. In particular,
$\boldsymbol{t}^{(1,\pi)}(\boldsymbol{p})$ and
$\boldsymbol{t}^{(d,\pi)}(\boldsymbol{p})$ thermomajorise all $\boldsymbol{q}$
that satisfy first and second part of Lemma 1, respectively. As a consequence,
$\boldsymbol{q}$ thermomajorised by both
$\boldsymbol{t}^{(d,\pi)}(\boldsymbol{p})$ and
$\boldsymbol{t}^{(d,\pi)}(\boldsymbol{p})$ satisfies Lemma 1. The proof is
completed by noticing, that

$\bigcup_{\pi\in\mathcal{S}_{d}}\mathcal{T}_{+}[\boldsymbol{t}^{(i,\pi)}(\boldsymbol{p})]=\operatorname{conv}\quantity[\quantity{\boldsymbol{t}^{(i,\pi)}(\boldsymbol{p})}_{\pi\in\mathcal{S}_{d}}].$
(26)

∎

In what follows, when we appeal to the sets $T_{i}(\boldsymbol{p})$, we will
drop the argument whenever it is clear from the context.

Finally, the extremal points of the above set can be derived directly from its
thermomajorisation curve:

######  Corollary 3 (Extreme points of the catalysable future).

Consider an energy-incoherent state $\boldsymbol{p}$ and an extreme point
$\boldsymbol{v}^{\pi}$ of its catalysable future region
$\mathcal{C}_{+}(\boldsymbol{p})$, corresponding to the $\beta$-order
$\boldsymbol{\pi}$. The elbows of its thermomajorisation curve
$f_{\boldsymbol{v}^{\boldsymbol{\pi}}}$ are determined by

$f_{\boldsymbol{v}^{\pi}}\quantity(\Gamma_{i})=\min\quantity[f_{\boldsymbol{t}^{(1,\boldsymbol{\pi})}(\boldsymbol{p})}\quantity(\Gamma_{i}),f_{\boldsymbol{t}^{(d,\boldsymbol{\pi})}(\boldsymbol{p})}\quantity(\Gamma_{i})],$
(27)

where we have $\Gamma_{i}=\sum_{j=1}^{i}\gamma_{\pi(j)}$.

###### Proof.

It follows directly from translating Theorem 2 to thermomajorisation curves. ∎

It is interesting to note that the full catalysable future of any given state
will have much fewer extreme points than the corresponding future thermal
cone.

######  Lemma 4.

Consider a state $\boldsymbol{p}\in\Delta_{d}$. The number of extreme points
of the full catalysable future thermal cone
$\mathcal{CT}_{+}(\boldsymbol{p})\equiv\mathcal{T}_{+}(\boldsymbol{p})\cup\mathcal{C}_{+}(\boldsymbol{p})$
of any given state $\boldsymbol{p}\in\Delta_{d}$ is upper-bounded by

$\absolutevalue{\operatorname{Ext}(\mathcal{CT}_{+}(\boldsymbol{p}))}\leq\left\lceil\frac{d}{2}\right\rceil\binom{d}{\left\lceil\frac{d}{2}\right\rceil}\ll
d!,$ (28)

where
$d!\geq\absolutevalue{\operatorname{Ext}(\mathcal{T}_{+}(\boldsymbol{p}))}$ is
the upper bound on the number of vertices of the future thermal cone of
$\boldsymbol{p}$.

The proper proof of this geometric insight is given in the Appendix C.
However, an intuition for the upper bound can be found by considering a ball
$\mathcal{B}(R,\boldsymbol{c})$ of radius $R$ centred at the centre
$\boldsymbol{c}$ of the simplex $\Delta_{d}$. First, for some $R=R_{d-1}$ the
ball becomes inscribed, with a single common point with each hyperface of the
simplex. Next, we have $R=R_{d-2}$, when the ball touches each hyperedge at
exactly one point. For $R_{d-1}<R<R_{d-2}$, we find a $d-1$-dimensional ball
on each of the hyperfaces. Replacing the ball with an inscribed inverted
simplex yields the desired result.

The analogy drawn between special relativity and Gibbs-preserving matrices
introduces a causal structure into the probability simplex $\Delta_{d}$,
suggesting a “light cone” for each point within $\Delta_{d}$ that divides the
space into past, incomparable, and future regions. This analogy implies that,
much like in special relativity, a specific division of space-time exists.
This division separates space-time into future, past, and space-like regions.
This allows for the identification of the generating event, which is referred
to as the present. Furthermore, there exists a one-to-one correspondence
between events and the space-time divisions they induce. This concept is
mirrored in the thermal cones for $\beta>0$: given a specific configuration of
the incomparable region along with future and past thermal cones, one can
precisely determine the current state of the system [see black dot $\bullet$
in Fig.4a-d]. This is in sharp contrast to the situation with $\beta=0$, where
each division into past, future, and incomparable has a $d!$-fold symmetry,
making it impossible to deduce the present state of the system based solely on
this division without additional details such as the permutation that arranges
the probabilities in non-decreasing order. Interestingly, in certain
temperature regimes, the structure of the catalysable future mimics this
feature, where the past and future regions meet again [see Fig. 4a-d].

###  III.2 Dimensionality bounds for thermal catalyst

Our second main result concerns general bounds on the catalyst dimension for
thermal operations, extending the bounds described in Ref. [47] for local
operations and classical communication. Notably, unlike the entanglement
scenario, catalysis is possible even when the main system is a three-level
system.

######  Theorem 5 (Dimensionality criteria).

For two incomparable energy-incoherent states
$\boldsymbol{p}\perp_{\beta}\boldsymbol{q}$, we define a maximal interval
$\mathcal{L}$ as

$\mathcal{L}=\quantity{x\in(0,1)\mid
f_{\boldsymbol{p}}(x)-f_{\boldsymbol{q}}(x)<0},$ (29)

and denote $m=\min(\mathcal{L})$ and $n=\max(\mathcal{L})$. Consider a
catalyst $\boldsymbol{r}\in\Delta_{k}$ with Hamiltonian $H_{c}$. The
transformation can be catalysed by $\boldsymbol{r}$, that is,
$\boldsymbol{p}\otimes\boldsymbol{r}\succ_{\beta}\boldsymbol{q}\otimes\boldsymbol{r}$,
only if its dimension satisfies the required conditions:

$\displaystyle k$ $\displaystyle>k^{*}=\frac{\log b}{\log a}+1,$ (30)

with the coefficients $a,b$ defined as

$\displaystyle a$
$\displaystyle=\min\quantity(\frac{s_{1}\quantity(\boldsymbol{p})}{f^{\prime}_{\boldsymbol{p}}(m_{-})},\frac{f^{\prime}_{\boldsymbol{p}}(n_{+})}{s_{d}\quantity(\boldsymbol{p})})>\max_{v\in(0,1)}\quantity(\frac{f^{\prime}_{\boldsymbol{r}}(v_{-})}{f^{\prime}_{\boldsymbol{r}}(v_{+})}),$
(31) $\displaystyle b$
$\displaystyle=\max_{l\in\mathcal{L}}\quantity(\frac{f^{\prime}_{\boldsymbol{p}}(l_{-})}{f^{\prime}_{\boldsymbol{p}}(l_{+})})<\frac{s_{1}\quantity(\boldsymbol{r})}{s_{k}\quantity(\boldsymbol{r})},$
(32)

where for brevity we use $f^{\prime}(x)\equiv\derivative*{f(x)}{x}$ and
$f^{\prime}(x_{i})\equiv\evaluated{f^{\prime}(x)}_{x=x_{i}}$. Furthermore, we
denote the left and right derivatives as
$f^{\prime}(x_{\pm})=\lim_{y\rightarrow x_{\pm}}f^{\prime}(y)$.

The full proof of the above theorem is given in Appendix B. Therein, we
present a heuristic proof in which we treat (thermo)majorisation curves as
continuous objects, and an exact proof based on the notion of an embedding map
[52], where thermomajorisation $\succ_{\beta}$ is equivalent to majorisation
$\succ$ in the (embedded) higher-dimensional space. Both approaches agree in
the obtained extension of the results presented in [47].

We may further simplify the expressions above to remove optimisation over an
interval in favour of a finite set of points.

######  Lemma 6 (Interval to point optimisation).

For a given state $\boldsymbol{p}$, we define an auxiliary set of indices
$\mathcal{L}^{\prime}\subset\quantity{1,\ldots,d}$ as

$\mathcal{L}^{\prime}=\quantity{l\in\quantity{1,\ldots,d}\mid\Gamma_{l}=\sum_{i=1}^{l}\gamma_{i}^{\beta},f_{\boldsymbol{p}}\quantity(\Gamma_{l})-f_{\boldsymbol{q}}(\Gamma_{l})<0},$
(33)

where $\boldsymbol{\gamma}^{\beta}$ is assumed to be ordered according to the
$\beta$-order of $\boldsymbol{p}$. Then we find that

$\max_{l^{\prime}\in\mathcal{L}^{\prime}}\frac{s_{l^{\prime}}\quantity(\boldsymbol{p})}{s_{l^{\prime}+1}\quantity(\boldsymbol{p})}=\max_{l\in\mathcal{L}}\quantity(\frac{f^{\prime}_{\boldsymbol{p}}(l_{-})}{f^{\prime}_{\boldsymbol{p}}(l_{+})})=b.$
(34)

###### Proof.

The ratio between left and right derivative is equal to one whenever the
function is differentiable. Therefore, the only points where we might find
nontrivial values are for the elbows of the thermomajorisation curve
$f_{\boldsymbol{p}}$, described entirely by the set $\mathcal{L}^{\prime}$. ∎

Theorem 5 has notable implications for the catalysability of low-dimensional
systems. Specifically, we derive the following two corollaries:

######  Observation 7 (Thermal non-catalysability for $d=2$).

Consider two states described by population vectors
$\boldsymbol{p},\,\boldsymbol{q}\in\Delta_{2}$ such that
$\boldsymbol{p}\perp_{\beta}\boldsymbol{q}$. There exists no state
$\boldsymbol{r}\in\Delta_{k}$ which can catalyse the transformation.

###### Proof.

A heuristic argument can be given based on the embedding lattice introduced in
[6] – any two states $\boldsymbol{p},\boldsymbol{q}$ of dimension $d=2$, for
$\beta>0$ and different $\beta$-orders, have an effective dimension
$\tilde{d}=\absolutevalue{\quantity{\sum_{i\in I}\gamma_{i}|I\in
2^{\quantity{1,2}}}}-1=3$, where by $\absolutevalue{\cdot}$ we denote the size
of the set, and they fall within the scope of Property 2 from [47].

For the more formal proof, we start by assuming that $\boldsymbol{p}$ and
$\boldsymbol{q}$ have different $\beta$-orders. From their incomparability we
find that they have to satisfy

$\displaystyle q_{2}\geq p_{1}\frac{\gamma_{2}}{\gamma_{1}},$ $\displaystyle
q_{2}+q_{1}\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}}\leq p_{1},$ (35)

By simple manipulation, the second inequality is turned into

$\displaystyle 1-q_{1}+q_{1}\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}}$
$\displaystyle\leq 1-p_{2}$ (36) $\displaystyle
q_{1}\frac{\gamma_{2}}{\gamma_{1}}$ $\displaystyle\geq p_{2}.$ (37)

Together with the first inequality we have

$\displaystyle
s_{1}(\boldsymbol{q})=\frac{q_{2}}{\gamma_{2}}\geq\frac{p_{1}}{\gamma_{1}}=s_{1}(\boldsymbol{p})$
$\displaystyle
s_{2}(\boldsymbol{q})=\frac{q_{1}}{\gamma_{1}}\geq\frac{p_{2}}{\gamma_{2}}=s_{2}(\boldsymbol{p}).$
(38)

which contradicts the necessary condition expressed in Eq. (18) for
catalysability. ∎

The more interesting case occurs in dimension $d=3$:

######  Observation 8 (Thermal catalysability in $d=3$).

There exist pairs of states described by population vectors
$\boldsymbol{p},\,\boldsymbol{q}\in\Delta_{3}$ such that
$\boldsymbol{q}\in\mathcal{C}_{+}(\boldsymbol{p})$ and a catalyst
$\boldsymbol{r}\in\Delta_{k}$ such that
$\boldsymbol{p}\otimes\boldsymbol{r}\succ_{\beta}\boldsymbol{q}\otimes\boldsymbol{r}$.

###### Proof.

An explicit example can be given by considering a pair of incomparable vectors
$\boldsymbol{p}=\quantity(0.42,0.51,0.07)$ and
$\boldsymbol{q}=\quantity(0.52,0.13,0.05)$, both with energy spectrum
$E_{0}=0$, $E_{1}=1$ and $E_{2}=2$ and inverse temperature $\beta=0.2$. Then,
a 2-dimensional catalyst, described by a state $\boldsymbol{r}=(0.55,0.45)$
with trivial Hamiltonian $H_{\boldsymbol{c}}=0$, catalyses the transformation,
since
$\boldsymbol{p}\otimes\boldsymbol{r}\succ_{\beta}\boldsymbol{q}\otimes\boldsymbol{r}$.
More generally, it is simple to see that as soon as $\beta>0$, we find that
for a generic state $\boldsymbol{p}$, the catalysable future is non-empty
$\mathcal{C}_{+}(\boldsymbol{p})\neq\emptyset$. ∎

Furthermore, using Theorem 5 we can formulate certain bounds on qubit
catalysts, limiting the corresponding probability vector $\boldsymbol{r}$ to a
specific interval.

######  Corollary 9 (Trivial qubit catalysts).

Consider a pair of incomparable states
$\boldsymbol{p},\,\boldsymbol{q}\in\Delta_{d}$ such that
$\boldsymbol{q}\in\mathcal{C}_{+}(\boldsymbol{p})$. A qubit catalyst in a
state $\boldsymbol{r}=(1-t,t)$ with $t\leq 1/2$ and described by a trivial
Hamiltonian $H_{c}=0$ can catalyse the transformation,
$\boldsymbol{p}\otimes\boldsymbol{r}\succ\boldsymbol{q}\otimes\boldsymbol{r}$
only if

$\frac{1}{1+a}\leq t\leq\frac{1}{1+b}$ (39)

with $a,b$ defined as in Theorem 5.

In contrast to [47], the provided bound on qubit catalyst is expressed in
terms of coefficients $a,b$ which are implicitly functions of both the
underlying probability vector $\boldsymbol{p}$ and the underlying energy level
structure given by the Gibbs state $\boldsymbol{\gamma}$. Moreover, similar
bounds can be given for qubit catalysts with nontrivial energy level
structure.

######  Corollary 10 (Nontrivial qubit catalysts).

Consider a pair of incomparable states
$\boldsymbol{p},\,\boldsymbol{q}\in\Delta_{d}$ such that
$\boldsymbol{q}\in\mathcal{C}_{+}(\boldsymbol{p})$ and a qubit catalyst in a
state $\boldsymbol{r}=(1-t,t)$ with Hamiltonian such that the Gibbs state is
given by
$\boldsymbol{\gamma}_{\boldsymbol{r}}=\quantity(1-\gamma_{\boldsymbol{r}},\,\gamma_{\boldsymbol{r}})$.
For $t\leq\gamma_{\boldsymbol{r}}$ it can catalyse the process only if

$\frac{\gamma_{\boldsymbol{r}}}{a(1-\gamma_{\boldsymbol{r}})+\gamma_{\boldsymbol{r}}}\leq
t\leq\frac{\gamma_{\boldsymbol{r}}}{b(1-\gamma_{\boldsymbol{r}})+\gamma_{\boldsymbol{r}}};$
(40)

for $t\geq\gamma_{\boldsymbol{r}}$ it can catalyse the transformation only if

$\frac{1-\gamma_{\boldsymbol{r}}}{b\gamma_{\boldsymbol{r}}+1-\gamma_{\boldsymbol{r}}}\leq
1-t\leq\frac{1-\gamma_{\boldsymbol{r}}}{a\gamma_{\boldsymbol{r}}+1-\gamma_{\boldsymbol{r}}},$
(41)

where in the above we take $a,\,b$ from Theorem 5.

It is worth noting that the two bounds above are symmetric with simultaneous
replacement of $t\leftrightarrow 1-t$ and
$\gamma_{\boldsymbol{r}}\leftrightarrow 1-\gamma_{\boldsymbol{r}}$ and thus,
in a certain sense, the allowable region for qubit catalysts retains the
symmetry with respect to the Gibbs state.

Figure 5: Isovolumetric curves for $d=3$. For a three-level system described
by an equidistant energy spectrum with $E_{1}=0$, $E_{2}=1$, and $E_{3}=2$, we
plot the volume of the catalysable future for several regimes of $\beta$. The
lines indicate states with equal volumes, and the colors represent their
magnitudes as a fraction of the maximal volume,
$\mathcal{V}_{+}/\quantity(\max_{\Delta_{3}}\mathcal{V}_{+})$. The yellow star
depicts the thermal state, and the dashed line indicates the boundaries
between the six possible $\beta$-orderings, dividing the simplex into six
chambers. Note that at the boundaries $\mathcal{V}_{+}$ goes to zero, while
maxima are located at the outer edges of each chamber.

##  IV Quantifying catalytic advantages

The characterisation of the catalysable regions naturally enables the
quantification of the catalytic advantage through their respective relative
volumes. Given a state $\boldsymbol{p}$, we define

$\mathcal{V}_{i}(\boldsymbol{p})=\frac{\operatorname{Vol}[\mathcal{C}_{i}(\boldsymbol{p})]}{\operatorname{Vol}(\Delta_{d})}\quad\text{for}\quad
i\in\\{+,-\\},$ (42)

as the volumes of the catalysable past and future, where
$\operatorname{Vol}(\cdot)$ denotes the Euclidean metric volume. Both volumes
also help us to understand the behaviour of the catalytic set for a given
choice of state and temperature.

We begin our analysis of the volumes of catalysable thermal cones by
presenting general properties that can be directly read from our results. The
starting point is a straightforward result that follows directly from Theorem
2, namely

######  Corollary 11 (Zero volume for non-full rank states).

The catalysable past thermal cone of a non-full rank state has zero volume.

###### Proof.

Without loss of generality, consider a non-full rank state
$\boldsymbol{p}=(p_{1},...,p_{d-1},0)$. Applying Eq. (14) yields
$t^{(d,\boldsymbol{\pi})}_{i}(\boldsymbol{p})=\delta_{\boldsymbol{\pi}(1),i}$,
for all $\boldsymbol{\pi}\in\mathcal{S}_{d}$. Consequently, the incomparable
region is given by all points in the interior of the probability simplex,
except those that are in the future of $\boldsymbol{p}$. Then, all the points
of the past will be located at the edge. Therefore according to Theorem 2
there is no catalysable past and its volume is zero. ∎

The above result also enables us to state a general behaviour for sharp states
$\boldsymbol{v}_{k}$ with $(\boldsymbol{v}_{k})_{j}=\delta_{jk}$:

######  Corollary 12 (Zero volume for sharp states).

The catalysable future thermal cone of a sharp state $\boldsymbol{v}_{k}$ has
zero volume.

###### Proof.

We proceed by showing that the extreme points of the catalysable set coincide
with the extreme points of the future thermal cone of the sharp state. First,
we consider the slope vector $\boldsymbol{s}(\boldsymbol{v}_{k})$ of a sharp
state $\boldsymbol{v}_{k}$. We readily determine that
$s_{j}(\boldsymbol{v}_{k})=\frac{\delta_{j1}}{\gamma_{k}}$. From this, it is
straightforward to conclude that the corresponding $d$-th tangent vectors are
represented by sharp states, i.e.,
$\boldsymbol{t}^{(d,\boldsymbol{\pi})}(\boldsymbol{v}_{k})=\boldsymbol{v}_{\pi(1)}$.
The components of the first ones are expressed as
$t^{(1,\boldsymbol{\pi})}_{i}(\boldsymbol{v}_{k})=\gamma_{i}/\gamma_{k}$ for
$i<d$. Second, upon defining a projection operator $\hat{P}$ that acts on a
probability vector, we ensure that the thermomajorisation function at each
elbow is defined to be

$f_{\hat{P}\boldsymbol{t}}(\Gamma_{i})=\max\quantity[1,f_{\hat{P}\boldsymbol{t}}(\Gamma_{i})]\quad,\quad\forall\,\Gamma_{i}=\sum_{j=1}^{i}\gamma_{\pi(j)}.$
(43)

Using the above projection onto the simplex $\Delta_{d}$ one finds that either
$\hat{P}\boldsymbol{t}^{(1,\boldsymbol{\pi})}(\boldsymbol{v}_{k})=\boldsymbol{v}_{k}$
or $\hat{P}\boldsymbol{t}^{(1,\boldsymbol{\pi})}(\boldsymbol{v}_{k})$ is an
extreme point of the future thermal cone
$\mathcal{T}_{+}(\boldsymbol{v}_{k})$. Thus, we find that
$\mathcal{T}_{+}(\boldsymbol{v}_{k})=T_{1}(\boldsymbol{v}_{k})$ and, in
consequence, $T_{1}(\boldsymbol{v}_{k})\cap
T_{d}(\boldsymbol{v}_{k})\setminus\mathcal{T}_{+}(\boldsymbol{v}_{k})=\emptyset$,
which completes the proof. ∎

Note that there is an interplay between the volumes of the catalysable future
and past, and the volume of the incomparable thermal region. Understanding the
general behaviour of their volumes is complex as it depends on the state under
consideration. Nevertheless, a significant insight can be obtained by
resorting to the fact that the state of non-full rank with the smallest future
thermal cone is known [6]. For this state, the following result can be proven:

######  Corollary 13 (Non-catalysable state).

The non-full rank state with the smallest future thermal cone

$\boldsymbol{g}=\frac{1}{Z}_{\boldsymbol{g}}\left(e^{-\beta
E_{1}},...,e^{-\beta E_{d-1}},0\right)\quad\text{where}\quad
Z_{\boldsymbol{g}}=\sum_{i=1}^{d-1}e^{-\beta E_{i}},$ (44)

cannot be catalysed, i.e., the volume of its catalysable future is zero.

###### Proof.

Given the assumption that if $i>j\Rightarrow E_{i}<E_{j}$, we conclude that
$\gamma_{i}\geq\gamma_{j}$. From this, we observe that that either
$\boldsymbol{t}^{(d,\boldsymbol{\pi})}(\boldsymbol{g})=\boldsymbol{g}$ or
$\boldsymbol{t}^{(d,\boldsymbol{\pi})}(\boldsymbol{g})$ is an extreme point of
the future thermal cone $\mathcal{T}_{+}(\boldsymbol{g})$. Therefore, we find
that $\mathcal{T}_{+}(\boldsymbol{g})=T_{d}(\boldsymbol{g})$, leading to the
conclusion that $T_{1}(\boldsymbol{g})\cap
T_{d}(\boldsymbol{g})\setminus\mathcal{T}_{+}(\boldsymbol{g})=\emptyset$. ∎

The analysis and discussion of the volumes has proceeded without explicit
computation. Several algorithms are known for calculating the volumes of
convex polytopes [59, 60]. More specifically, it is observed that the
catalysable future is given by the intersection of two simplices, which
reduces the complexity of volume calculation thanks to the reduction of the
number of vertices (see Lemma 6) in comparison to the full thermal future
cone.

One might ask about the states in $\Delta_{d}$ that possess the highest
volumes of catalysable future. Answering this question generally is highly
non-trivial. The challenge arises because the volumes depend significantly on
the initial state, and computing them for higher dimensions is not
straightforward. However, by resorting to low-dimensional systems, we can
still gain some insights into how catalysability behaves as a function of
inverse temperature. In this regard, we illustrate the profile of the
catalysable future’s volume as a function of $\beta$ in Fig. 5 for a three-
level system. It is observed that states with the highest volumes are
concentrated near the edges of the probability simplex, and the volume of the
catalysable future tends to be higher for states that are not of full rank.
Interestingly, in the extreme cases of $\beta=0$ and $\beta\to\infty$,
catalysis has no effect.

##  V Entanglement generation under thermal operations with catalyst

Figure 6: Thermally non-entanglable states with (and without) catalyst. Two-
qubit states that cannot become entangled under thermal operations are
illustrated for two scenarios: without a catalyst (shown in the left panel in
blue) and with a catalyst (shown in the right panel in red), for (a) $\beta=0$
and (b) $\beta=0.5$

As a direct application of our results, we demonstrate the catalytic
advantages of generating entanglement from separable bipartite states under
thermal operations. Throughout this section, we will focus on a composite
system consisting in two 2-level systems with identical energy levels
$H=\operatorname{diag}(0,\delta)$. For this reason, the total Hamiltonian of
the system, $H_{AB}=H\otimes\mathbbm{1}+\mathbbm{1}\otimes H$ has a degenerate
energy subspace corresponding to $E_{01}=E_{10}=\delta$, and without loss of
generality we may set $\delta=1$. Within this subspace any unitary operation
remains Gibbs-preserving, which allows for generation of entanglement while
simultaneously respecting the laws of thermodynamics.

Recently, the necessary and sufficient conditions for producing bipartite
qubit entanglement from an initially separable states via thermal operations
were addressed in Ref. [48]. States that cannot get entangled under thermal
operations are said to belong to the _thermally non-entanglable set_ ,
mathematically denoted by $\mathbb{TN}$. Conversely, states that can get
entangled under thermal operations lie in the _entanglable set_ $\mathbb{TE}$.
By utilising the results on the catalysable future region $\mathcal{C}_{+}$,
we may now answer the natural question whether borrowing catalyst allow more
states to get entangled. Defining the _catalytically non-entanglable set_
$\mathbb{CN}$ as the set of states that cannot be entangled even with the aid
of catalysis, the question can be reframed – _is the set $\mathbb{CN}$ smaller
than $\mathbb{TN}$, or in terms of volumes,
$\mathcal{V}(\mathbb{CN})/\mathcal{V}(\mathbb{TN})\overset{?}{<}1$ for a given
$\beta$?_

To begin with, we focus on the infinite temperature case $\beta=0$.
Remarkably, in this regime, the thermally non-entanglable set $\mathbb{TN}$
reaches its maximum volume and can be analytically constructed. In order to
investigate the catalytically non-entanglable set $\mathbb{CN}$, we start by
using Theorem 2, where we construct the catalysable future for two qubits and
verify whether it lies within the thermally non-entanglable set. Next, by
employing a numerical bisection-based algorithm inspired by vacuum-packaging,
we generate an approximation of the boundary $\partial\mathbb{CN}$. This
allows us to estimate the ratio of the volumes
$\mathcal{V}(\mathbb{CN})/\mathcal{V}(\mathbb{TN})\approx 0.88$. In Fig. 6a,
we show the thermally non-entanglable set without (left panel) and with (right
panel) a catalyst.

It is interesting to note that for $\beta>0$, although the set $\mathbb{TN}$
remains convex as verified by numerical approximations, the set $\mathbb{CN}$
becomes non-convex (see Fig. 6b for an example considering $\beta=0.5$). In
fact, by reverse-engineering the numerical approximations, we find that
$\mathbb{CN}$ is a set sum of a certain convex set $\mathbb{CN}_{0}$ and a
future of a certain distinguished subspace-thermalised state,
$\mathcal{\boldsymbol{p}_{*}}$. First, concerning the distinguished state, we
have

######  Proposition 14 (Partially thermalised state & non-entanglability).

The future thermal cone $\mathcal{T}_{+}(\boldsymbol{p}^{*})$ of the partially
thermalised state

$\boldsymbol{p}^{*}=\frac{1}{4+2\cosh(\beta)}\quantity(e^{\beta},\,1,\,1,\,2+e^{-\beta})$
(45)

is fully contained in the catalytically non-entanglable set,
$\mathcal{T}_{+}(\boldsymbol{p}^{*})\subset\mathbb{CN}$.

We give the entire proof of this fact in Appendix D. Furthermore, based on
numerics we put forward the following

######  Conjecture 15 (Non-entangable decomposition).

The catalytically non-entanglable set can be decomposed into

$\mathbb{CT}=\mathbb{CT}_{0}\cup\mathcal{T}_{+}(\boldsymbol{p}^{*}),$ (46)

where

$\mathbb{CT}_{0}=\mathbb{CT}\setminus\quantity[\mathcal{T}_{+}(\boldsymbol{p}^{*})\setminus\mathcal{T}_{+}(\boldsymbol{p}^{**})],$
(47)

with
$\boldsymbol{p}^{**}=\frac{1}{4+2\cosh(\beta)}\quantity(e^{\beta},\,3,\,1,\,e^{-\beta})$.

The conjecture is based on numerical approximations with high precision, and
we believe it to hold true. Furthermore, it can be partially supported by
noting that for a ball $\mathcal{B}(\boldsymbol{p}^{*},\,\delta)$ of radius
$\delta$ around the state $\boldsymbol{p}^{*}$ any state
$\boldsymbol{q}\in\quantity[\mathcal{B}(\boldsymbol{p}^{*},\,\delta)\setminus\mathcal{T}_{+}(\boldsymbol{p}^{*})]$
does not belong to $\mathbb{CN}$ by simple arguments of either not belonging
to $\mathbb{TN}$ or by the fact that
$\mathcal{C}_{+}\quantity(\boldsymbol{q})\setminus\mathbb{TN}\neq\emptyset$.

Using the introduced numerical approximations we derived the dependence of the
volumes $\mathcal{V}(\mathbb{TN})$ and $\mathcal{V}(\mathbb{CN})$ for both
thermall and catalytically non-entanglable sets and their ratio as a function
of $\beta$, demonstrated in Fig 7.

Figure 7: Thermally non-entanglable volume with(out) catalyst. Numerical
estimates for the volumes $\mathcal{V}(\mathbb{CN})$ (blue line),
$\mathcal{V}(\mathbb{TN})$ (orange line) and their ratio (green line), with
the axis on the left for volumes and on the right for the ration. The
character of the curves shows that, even though the volumes of both thermally
and catalytically non-entanglable sets both go to zero together with $\beta$
going to infinity, the ratio of the volumes also diminishes, pointing to an
increasing catalytic advantage in entanglement generation.

##  VI Summary and Discussion

In this paper, we explore the fundamental limits on state achievability with
the aid of a strict catalyst under a thermal process. We propose a novel
approach to address this question by constructing the sets of states that can
be achieved from a given energy-incoherent state under catalytic thermal
operations, as well as the sets of states that can, through catalytic thermal
operations, achieve the initial state. These regions naturally highlight the
advantages of using a strict catalyst in general thermodynamic processes,
whose only assumption is energy conservation. Our construction enables us to
characterise the catalysable past and future for main systems of arbitrary
dimensions. This analytical approach is based solely on thermomajorisation
relations. Furthermore, we established general bounds on the catalyst
dimension for thermal operations and demonstrated that catalysis is effective
under thermal operations even for systems of dimensionality as small as $d=3$.
The catalytic advantages were quantified by the volume of the catalysable
future, with a detailed discussion on its behaviour as a function of
temperature. Finally, we applied our findings to a practical question: the
generation of entanglement under thermal operations. We showed that thermal
processes, which were once incapable of generating entanglement, allows its
generation when a strict catalyst is employed.

To address this problem, we overcome several challenges. Firstly, there is no
efficient method for characterizing the set of allowed state transformations
under catalytic thermal operations. A comprehensive analysis for inexact
catalysis (where the catalyst is allowed to be returned with some minor error)
was performed in [25]; while, in Ref. [30], the authors solved the
characterisation problem for three-dimensional systems and for a subset of
initial states of generic dimensions in the context of catalysis in elementary
thermal operations. Secondly, keeping track of the the set of achievable state
after “decoupling” the catalyst without specifying the precise form of the
catalyst state required by the transformation is a highly difficult
mathematical problem. Approaching from the perspective of lattice theory, this
problem can be framed by asking _(i)_ how to characterise the higher-
dimensional lattice for composite systems (main system plus catalyst) and
_(ii)_ how to project back to the native lattice of the main system while
preserving the catalytic effects in the native lattice of the initial state.
For small-dimensional systems (such as $d=3$), the entire convex polytope
$\boldsymbol{p}\otimes\boldsymbol{c}$ can be constructed, and intersections
between such a polytope and plane equations enforcing the catalyst recovery
condition can be numerically identified. However, this method does not scale
well and depends heavily on the initial state of the system and the catalyst.
In contrast, our approach requires only constructing a set of quasi-
probability vectors or determining the extremal points through
thermomajorisation relations.

There are many possibilities for extending our results. For instance, they can
be leveraged to study catalysis within the framework of elementary thermal
operations. Note that the characterisation of the catalysable past and future
relies entirely on the concept of tangent vectors, which tightly
thermomajorise $\boldsymbol{p}$. On the other hand, the extreme points of set
of states achievable under elementary thermal operations are known to be
$\beta$-swaps that tightly thermomajorise the initial distribution.
Consequently, one could explore how to adapt the technical aspects of our
findings to this specific context. In a similar spirit, we could relax the
condition requiring the catalyst to be returned unperturbed and allow for some
small error. A natural quantifier for this error is the trace distance, and
the problem could be reframed by asking for the set of states achievable with
a given catalyst up to this specified amount of error [25].

Finally, our results imply that one can potentially (for a fixed finite
dimension) replace an infinity family of second laws by a finite set of
conditions. As it can be checked, within the catalysable region the second
laws are fulfilled and as soon as we get out of it, they are broken. So, an
ambitious question is whether our results can be reformulated (or adapted), so
that given $\boldsymbol{p}$ and $\boldsymbol{q}$, one can check a finite
number of conditions to verify whether $\boldsymbol{p}$ can catalytically
thermomajorises $\boldsymbol{q}$. Note that, our results shows that for every
$\boldsymbol{p}$, we have a method to calculate the extended future thermal
cone. For each $\beta$-order, this cone is defined by an extreme point of the
catalysable set. While membership in this region does not guarantee the
existence of a catalyst, non-membership guarantees its non-existence.

###### Acknowledgements.

We thank Karol Życzkowski for asking a stimulating question at the beginning
of the authors’ PhD, which led to this work, and for his fruitful comments on
the first version of this manuscript. We also thank Kamil Korzekwa for
valuable discussions that significantly contributed to the execution of this
project, as well as for his insightful comments during the final stages of
preparation of this manuscript. JCz acknowledges financial support by NCN
PRELUDIUM BIS no. DEC-2019/35/O/ST2/01049 and ID.UJ Research Support
minigrant. AOJ acknowledges financial support from VILLUM FONDEN through a
research grant (40864).

##  Appendix A Reframing entanglement catalysis bounds for thermal operations

We start by revisiting the results presented in Ref. [47] and adapt them to
the context of thermal operations at the infinite temperature limit. First, we
establish a simple necessary condition: For a pair of vectors $\boldsymbol{p}$
and $\boldsymbol{q}$, a catalyst $\boldsymbol{r}$ such that
$\boldsymbol{p}\otimes\boldsymbol{r}\succ\boldsymbol{q}\otimes\boldsymbol{r}$
exists only if $p^{\downarrow}_{1}<q^{\downarrow}_{1}$ and
$p^{\downarrow}_{d}>q^{\downarrow}_{d}$. Based on this assumption, we then
restate the main theorem from the aforementioned work.

######  Theorem 16 (Theorem 1 of Ref. [47]).

Consider two states described by population vectors
$\boldsymbol{p},\,\boldsymbol{q}\in\Delta_{d}$ which are incomparable,
$\boldsymbol{p}\perp\boldsymbol{q}$ (ie. neither
$\boldsymbol{p}\prec\boldsymbol{q}$ nor $\boldsymbol{q}\prec\boldsymbol{p}$)
and $p^{\downarrow}_{1}>q^{\downarrow}_{1}$ and
$p^{\downarrow}_{d}<q^{\downarrow}_{d}$. We define a set of indices
$\mathcal{L}$ as:

$\mathcal{L}=\quantity{l\in\quantity{1,2,...,d}\mid\sum_{i=1}^{l}p^{\downarrow}_{i}-q^{\downarrow}_{i}<0}$
(48)

and take $m=\min(\mathcal{L})$ and $n=\max(\mathcal{L})$. The catalyst
$\boldsymbol{r}\in\Delta_{k}$ can catalyse the transformation, ie.
$\boldsymbol{p}\otimes\boldsymbol{r}\succ\boldsymbol{q}\otimes\boldsymbol{r}$,
if and only if its dimensionality satisfies

$\displaystyle k$ $\displaystyle>k^{*}=\frac{\log b}{\log a}+1$ (49)

with the coefficients $a,b$ defined as

$\displaystyle a$
$\displaystyle=\min\quantity(\frac{p^{\downarrow}_{1}}{p^{\downarrow}_{m}},\frac{p^{\downarrow}_{n+1}}{p^{\downarrow}_{d}})>\max_{v\in\quantity{1,2,\ldots,k-1}}\quantity(\frac{r^{\downarrow}_{v}}{r^{\downarrow}_{v+1}})$
(50) $\displaystyle b$
$\displaystyle=\max_{l\in\mathcal{L}}\quantity(\frac{p^{\downarrow}_{l}}{p^{\downarrow}_{l+1}})<\frac{r^{\downarrow}_{1}}{r^{\downarrow}_{k}}$
(51)

and provide bounds for the entries of the catalyst state $\boldsymbol{r}$.

Note that the bound depends explicitly only on the entries of the initial
state $\boldsymbol{p}$, with the target state $\boldsymbol{q}$ entering the
picture indirectly via the set $\mathcal{L}$, which defines the range of the
entries taken into consideration for optimisation purposes. In a more recent
work bounds using both $\boldsymbol{p}$ and $\boldsymbol{q}$ entries
explicitly have been proposed [61]; however, the bounds introduced therein are
necessarily weaker than the ones introduced in [47], and therefore redundant.

We also recall the resulting limitations of the catalytic processes for
$\beta=0$:

######  Observation 17.

For incomparable states of dimension $d\leq 3$ catalysis with incoherent
catalyst is not possible for noisy operations.

The proof is given in Ref. [62]. Moreover, in Ref. [47], we find two further
corollaries of interest.

######  Corollary 18.

Whenever $p^{\downarrow}_{1}=p^{\downarrow}_{m}$ or
$p^{\downarrow}_{n+1}=p^{\downarrow}_{d}$, catalysis is impossible.

###### Proof.

In either case we have $a=1$, and hence by Theorem 16 one finds that
$k^{*}\rightarrow\infty$. ∎

Finally, it is important to note that the results immediately define an
admissible region for an incoherent qubit catalyst.

######  Corollary 19.

Consider a pair of incomparable states
$\boldsymbol{p},\,\boldsymbol{q}\in\Delta_{d}$ such that
$\boldsymbol{p}\perp\boldsymbol{q}$, $p_{1}^{\downarrow}<q_{1}^{\downarrow}$
and $p_{d}^{\downarrow}>q_{d}^{\downarrow}$. A qubit catalyst in a state
$\boldsymbol{r}=(1-t,t)$ with $t\leq 1/2$ can catalyse the process,
$\boldsymbol{p}\otimes\boldsymbol{r}\succ\boldsymbol{q}\otimes\boldsymbol{r}$
only if

$\frac{1}{1+a}\leq t\leq\frac{1}{1+b}$ (52)

with $a,b$ defined as in Theorem 16.

It has to be stressed, however, that it is only a necessary condition. As has
been already noticed in [47], not only the actual catalysis happens for a
smaller sub-interval, but may also happen in several disjoint sub-intervals.

##  Appendix B Proof of Theorem 5

Below we provide two approaches to extend the Theorem 16 to the thermal
operations. First, we focus on a heuristic approach by treating the
majorisation and thermomajorisation curves as continuous functions, which
allows us to reformulate the expressions in terms of slopes instead of
probabilities. The second, more formal approach, takes advantage of the
embedding scheme introduced in [52], which relates thermomajorisation relation
with a majorisation relation in an extended space of potentially infinite
dimension.

###  B.1 Heuristic approach via derivatives

Let us consider a state $\boldsymbol{p}$ and the corresponding majorisation
curve $f_{\boldsymbol{p}}(x)$. In what follows, for brevity of notation, we
will use primed notation $f^{\prime}(x)\equiv\derivative*{f(x)}{x}$ for
derivative of a function and
$f^{\prime}(x_{i})\equiv\evaluated{f^{\prime}(x)}_{x=x_{i}}$. For $\beta=0$,
we find a simple relation between probabilities and derivatives of the
majorisation curve, namely

$d\cdot
p_{i}^{\downarrow}=\evaluated{f^{\prime}_{\boldsymbol{p}}(x)}_{\frac{i-1}{d}<x<\frac{i}{d}}\equiv
f^{\prime}_{\boldsymbol{p}}(x_{i}),$ (53)

where we define $x_{i}\in\quantity[(i-1)/d,i/d]$ as an arbitrary point from
the interval between the consecutive elbows of the majorisation curve. Using
the relation given by Eq. (53) in Eqs. (50) - (51), we find that the
coefficients $a$ and $b$ can be expressed in terms of derivatives as follows:

$\displaystyle a$
$\displaystyle=\min\quantity(\frac{f^{\prime}_{\boldsymbol{p}}(x_{1})}{f^{\prime}_{\boldsymbol{p}}(x_{m})},\frac{f^{\prime}_{\boldsymbol{p}}(x_{n+1})}{f^{\prime}_{\boldsymbol{p}}(x_{d})})>\max_{v\in\quantity{1,2,\ldots,k-1}}\quantity(\frac{f^{\prime}_{\boldsymbol{r}}(x_{\nu})}{f^{\prime}_{\boldsymbol{r}}(x_{\nu+1})}),$
(54) $\displaystyle b$
$\displaystyle=\max_{l\in\mathcal{L}}\frac{f^{\prime}_{\boldsymbol{p}}(x_{l})}{f^{\prime}_{\boldsymbol{p}}(x_{l+1})}<\frac{f^{\prime}_{\boldsymbol{r}}(x_{1})}{f^{\prime}_{\boldsymbol{r}}(x_{k})}.$
(55)

Now, the expressions above remain valid when we replace the subset
$\mathcal{L}$, over which we maximize, with an open interval
$\mathcal{L}=(m^{\prime},n^{\prime})=\quantity{x:f^{\prime}_{\boldsymbol{p}}(x)-f^{\prime}_{\boldsymbol{q}}(x)<0}$;
the same substitution can be applied to the right-hand side (54). However, to
maintain the correct interpretation, we must substitute the ratios of
subsequent points with the ratios of the left and right derivatives,
specifically focusing on the left-hand side of Eq. (55). As result, leads to
the following

$b=\max_{x\in\mathcal{L}}\frac{f^{\prime}_{\boldsymbol{p}}(x_{-})}{f^{\prime}_{\boldsymbol{p}}(x_{+})}.$
(56)

where we used shorthand notation for left and right derivatives
$f^{\prime}(x_{\pm})\equiv\lim_{y\rightarrow x_{\pm}}f^{\prime}(x)$. Note
that, whenever a function is differentiable at point $x$, the ratio between
left and right derivative is equal to one. Therefore, any ratio value
differing from one can be considered indicative of non-differentiability. This
is a characteristic observed exclusively at the elbows of majorisation
curves—a property we will later exploit to further simplify our calculations.

Finally, we return to the discussion of thermomajorisation curves by
reintroducing $\beta>0$, which establishes the connection between the
derivatives and probabilities.

$p_{i}^{\beta}=\evaluated{\gamma_{i}^{\beta}f^{\prime}_{\boldsymbol{p}}(x)}_{\Gamma_{i-1}<x<\Gamma_{i}},$
(57)

where we used the subsums of the Gibbs distribution
$\Gamma_{i}=\sum_{j=1}^{i}\gamma_{j}^{\beta}$, which correspond to the
locations of the elbows on the thermomajorisation curve
$f_{\boldsymbol{p}}(x)$. While all inequalities remain valid, they now depend
not only on the probability vectors $\boldsymbol{p}$ and $\boldsymbol{r}$ but
also on the respective underlying Gibbs distributions $\boldsymbol{\gamma}$
and $\boldsymbol{\gamma}_{\boldsymbol{r}}$.

Now, let us focus on the derivation of lower bound on catalyst dimensionality
$k_{*}$ from the derivatives’ perspective. In the original approach the
starting point was to note the following chain of bounds,

$b<\frac{r^{\downarrow}_{1}}{r^{\downarrow}_{k}}=\prod_{i=1}^{k-1}\frac{r^{\downarrow}_{i}}{r^{\downarrow}_{i+1}}<\quantity[\max_{v\in\quantity{1,2,\ldots,k-1}}\quantity(\frac{r^{\downarrow}_{v}}{r^{\downarrow}_{v+1}})]^{k-1}<a^{k-1}.$
(58)

Unfortunately, the heuristic approach does not allow us to extend this result
directly, as we shifted our perspective from piecewise-linear curves defined
solely by their elbows to continuous functions. However, by taking the
logarithm of Eq. (55), one can observe that it involves the ratio between the
first and last slope. This allows for the computation of the left-derivative
at $x_{1}=0$ and the right-derivative at $x_{k}=1$,

$\log(\frac{f^{\prime}_{\boldsymbol{r}}(0_{+})}{f^{\prime}_{\boldsymbol{r}}(1_{-})})=\log[f^{\prime}_{\boldsymbol{r}}(0_{+})]-\log[f^{\prime}_{\boldsymbol{r}}(1_{-})].$
(59)

The above can also be reformulated as the integral of a derivative,

$\displaystyle\log(\frac{f^{\prime}_{\boldsymbol{r}}(0_{+})}{f^{\prime}_{\boldsymbol{r}}(1_{-})})$
$\displaystyle=\lim_{\delta\rightarrow
0}\int_{\delta}^{1-\delta}\quantity[-\log(f^{\prime}_{\boldsymbol{r}}(x))]^{\prime}\differential{x}.$
(60)

Next, we observe that the logarithmic terms appearing in Eq. (60) can be
explicitly expressed as

$\displaystyle\log(f^{\prime}_{\boldsymbol{r}}(x))=$
$\displaystyle\sum_{i=1}^{k}\log(s_{i}(\boldsymbol{r}))\quantity(\Theta\quantity(x-\Gamma_{i-1})-\Theta\quantity(x-\Gamma_{i})),$
(61) $\displaystyle\quantity[\log(f^{\prime}_{\boldsymbol{r}}(x))]^{\prime}=$
$\displaystyle\sum_{i=1}^{k}\log(s_{i}(\boldsymbol{r}))\quantity(\delta\quantity(x-\Gamma_{i-1})-\delta\quantity(x-\Gamma_{i}))$
$\displaystyle=$
$\displaystyle\log(s_{1}(\boldsymbol{r}))\delta(x)-\log(s_{d}(\boldsymbol{r}))\delta(x-1)$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{k-1}\delta(x-\Gamma_{i})\quantity[\log(s_{i+1}(\boldsymbol{r}))-\log(s_{i}(\boldsymbol{r}))].$
(62)

In Eqs. (61) and (62), $\Theta(x)$ represents the Heaviside step function, and
$\delta(x)$ denotes the Dirac delta function, respectively.

Thus, using Eqs.(61) and (62), one can compute the integral given in Eq.(60)
as follows:

$\displaystyle\lim_{\delta\rightarrow
0}\int_{\delta}^{1-\delta}\\!\\!\\!\\!\\!\\!\\!\\!\quantity[-\log(f^{\prime}_{\boldsymbol{r}}(x))]^{\prime}\differential{x}$
$\displaystyle=\sum_{i=1}^{k-1}\log(\frac{s_{i}(\boldsymbol{r})}{s_{i+1}(\boldsymbol{r})})$
$\displaystyle\leq(k-1)\max_{i}\log(\frac{s_{i}(\boldsymbol{r})}{s_{i+1}(\boldsymbol{r})}).$
(63)

where we recall that $\boldsymbol{s}(\boldsymbol{r})$ is the slope vector
corresponding to the population of the catalyst $\boldsymbol{r}$. Finally,
using (54), we retrieve the bound on the dimensionality of the catalyst state
$\boldsymbol{r}$,

$k\geq\frac{\log b}{\log a}+1=k_{*},$ (64)

within the heuristic approach.

###  B.2 Embedding map approach

We start by recalling the embedding map [52] (see Ref. [50] for a detailed
discussion). This allows us to draw a connection between thermomajorisation
and majorisation in a space of larger dimension.

######  Definition 6 (Embedding map).

Consider a thermal distribution $\boldsymbol{\gamma}$ with rational entries,
$\gamma_{i}=D_{i}/D$ and $D_{i},D\in\mathbb{N}$, the embedding map $\Xi$ sends
a $d$-dimensional probability distribution $\boldsymbol{p}$ to a
$D$-dimensional probability distribution
$\hat{\boldsymbol{p}}:=\Xi(\boldsymbol{p})$ as follow:

$\hat{\boldsymbol{p}}=\Biggl{[}\underbrace{\frac{p_{1}}{D_{1}},\ldots,\frac{p_{1}}{D_{1}}}_{D_{1}\>\text{times}},\underbrace{\frac{p_{2}}{D_{2}},\ldots,\frac{p_{2}}{D_{2}}}_{D_{2}\>\text{times}},\ldots,\underbrace{\frac{p_{d}}{D_{d}},\ldots,\frac{p_{d}}{D_{d}}}_{D_{d}\>\text{times}}\Biggr{]}.$
(65)

A generic thermal distribution $\boldsymbol{\gamma}$ will not fall within the
scope of Definition 6. Nevertheless, since the rational numbers $\mathbb{Q}$
are dense subset of the real numbers $\mathbb{R}$. Thanks to this the limit
$\lim_{D\rightarrow\infty}\frac{\lceil
D\gamma_{i}\rceil}{D}=\lim_{D\rightarrow\infty}\frac{\lfloor
D\gamma_{i}\rfloor}{D}=\gamma_{i}$ exists. With this chain of approximations
in mind, we proceed to work within the limit of $D\rightarrow\infty$.

It follows from the above definition that the majorisation between embedded
vectors coincides exactly with the notion of thermo-majorisation

$\hat{\boldsymbol{p}}\succ\hat{\boldsymbol{q}}\Longleftrightarrow\boldsymbol{p}\succ_{\beta}\boldsymbol{q}$
(66)

Now, we introduce a function $\varphi$ that maps elements of the embedded
distribution to the corresponding elements of the original one

$\hat{p}^{\downarrow}_{j}=\frac{p_{\varphi(j)}}{D_{\varphi(j)}},$ (67)

which, as a consequence of non-increasing ordering, yields

$j>k\Rightarrow\frac{p_{\varphi(j)}}{D_{\varphi(j)}}\leq\frac{p_{\varphi(k)}}{D_{\varphi(k)}}.$
(68)

The function $\phi$ is used in order to convert the ordering of the embedded
vector $\hat{\boldsymbol{p}}^{\downarrow}$ to the $\beta$-ordering of
$\boldsymbol{p}^{\beta}$. Suppose, for example, that $p^{\beta}_{1}=p_{3}$.
Knowing this, we find that $\phi(j)=3$ for all $1\leq j\leq D_{3}$. More
generally, it is defined to be, we find that $\phi(j)=\pi^{-1}(i)$ for
$\sum_{n=1}^{i}D_{\pi^{-1}(n)}+1\leq j\leq\sum_{n=1}^{i}D_{\pi^{-1}(n)}$.

Focusing on the expression (51) for $b$ and applying it to the embedded vector
$\hat{\boldsymbol{p}}$, we substitute

$\displaystyle\frac{\hat{p}^{\downarrow}_{l}}{\hat{p}^{\downarrow}_{l+1}}$
$\displaystyle=\frac{p_{\varphi(l)}}{D_{\varphi(l)}}\frac{D_{\varphi(l+1)}}{p_{\varphi(l+1)}}$
(69)
$\displaystyle=\frac{p_{\varphi(l)}}{D_{\varphi(l)}/D}\frac{D_{\varphi(l+1)}/D}{p_{\varphi(l+1)}}$
(70)
$\displaystyle\overset{D\rightarrow\infty}{\longrightarrow}\frac{p_{\varphi(l)}}{\gamma_{\varphi(l)}}\frac{\gamma_{\varphi(l+1)}}{p_{\varphi(l+1)}}=\frac{s_{\pi(\varphi(l))}(\boldsymbol{p})}{s_{\pi(\varphi(l+1))}(\boldsymbol{p})}$
(71)

where in the last expression $\pi$ is the permutation responsible for the
beta-ordering of $\boldsymbol{p}^{\beta}$. Treating all the remaining ratios
in (50), (51), we retrieve the full form of Theorem 5.

##  Appendix C Upper-bounding number of intersections between d-point
simplices

We start by considering a regular $d$-point simplex $\Delta_{d}$ spanned by
vertices $\boldsymbol{v}_{i}$ for $i=\\{1,\ldots,d\\}$, with the central point
$\boldsymbol{c}=\frac{1}{d}\sum_{i}\boldsymbol{v}_{i}$. In what follows, we
will consider the rescaled versions of the simplex defined as

$t\Delta_{d}=\operatorname{conv}\quantity(\quantity{t\boldsymbol{v}_{i}+(1-t)\boldsymbol{c}}_{i=1}^{d}).$
(72)

First, consider the intersection $\Delta_{d}\cap t\Delta_{d}$ for $t\leq 0$
which, in fact, defines an inverted or dual simplex. This will be denoted as
$t_{d}\nabla_{d}$ with $t_{d}=-t$, a choice whose rationale will soon become
apparent. For $t_{d}\in[0,1/(d-1)]$, we observe that there is no intersection
between the boundaries,
$\partial\Delta_{d}\cap\partial(t_{d}\nabla_{d})=\emptyset$. The first
intersection occurs at $t=-1/(d-1)$, at which point there is only a single
extreme point per $d$-dimensional face of $\Delta_{d}$, since

$\frac{1}{d-1}\nabla_{d}=\operatorname{conv}\quantity(\quantity{\sum_{j=1}^{d}\frac{1-\delta_{j}}{d-1}\boldsymbol{v}_{j}}_{i=1}^{d}).$
(73)

For $t_{d}\geq 1/(d-1)$, the intersection points of $-t\nabla_{d}$ with each
of the $d$ faces of $\Delta_{d}$—which are effectively $(d-1)$-dimensional
simplices—result in a $(d-1)$-dimensional inverted simplex, denoted as
$t_{d-1}\nabla_{d-1}$. Here, we define
$t_{d-1}=\frac{d-1}{d}\min(0,t_{d}-\frac{1}{d-1})$. Consequently, there are
exactly $(d-1)$ intersections per face of $\Delta_{d}$, continuing until
$t_{d-1}=1/(d-2)$. The analysis then continues for the $(d-2)$-dimensional
faces of $\Delta_{d}$.

Based on the reasoning above, we can identify only two types of situations:

  1. 1.

The intersection $\Delta_{d}\cap t\nabla_{d}$ has a single vertex per
$k$-dimensional face of $\Delta_{d}$.

  2. 2.

The intersection $\Delta_{d}\cap t\nabla_{d}$ has exactly $(d-k)$ vertices per
$k$-dimensional face of $\Delta_{d}$.

This results in either $\binom{d}{k}$ vertices or $k\binom{d}{k}$ vertices in
total, with the latter being the larger one and attaining maximum at
$k=\left\lceil\frac{d}{2}\right\rceil$. This completes the reasoning for
simplices with a common center $\boldsymbol{c}$.

The heuristic approach for the translations defined as

$\Delta_{d}+\boldsymbol{\delta}=\operatorname{conv}\quantity(\quantity{\boldsymbol{v}_{i}+\boldsymbol{\delta}}_{i=1}^{d}).$
(74)

is based on the following reasoning. Let us focus on the intersection with a
single face $F$ from $\Delta_{d}$ and assume that $t_{d-1}<\frac{2}{d-2}$. Any
shift in a direction parallel to the plane the face is contained in,
$\boldsymbol{\delta}\parallel F$, will result in translating the intersection
points between $t\nabla_{d}$ and $F+\boldsymbol{\delta}$. Similarly, the shift
in the perpendicular direction will result in scaling the “intersection
simplex”. This observation indicates that the general structure of the
intersections remains unchanged, confirming that the upper bound on the number
of vertices from concentric $\Delta_{d}$ and $t_{d}\nabla_{d}$ holds.

##  Appendix D Proof of Proposition 14

In this section, we assume that the Gibbs state is given by
$\boldsymbol{\gamma}=\quantity(\gamma_{1},\,\gamma_{2},\,\gamma_{2},\,\gamma_{4})$.
As a result, arbitrary unitary operation in the subspace
$\operatorname{span}(\ket{01},\,\ket{10})$ is energy-preserving, and thus an
admissible thermal operation (TO). Under these assumptions there exists a
subset of states that can become entangled under TOs. In particular, for
2-qubit states with populations $\boldsymbol{p}$ a state can be entangled
using energy-preserving unitaries if and only if
$4p_{1}p_{4}-(p_{2}-p_{3})^{2}<0$. Furthermore, as shown in [48], a state
$\boldsymbol{p}$ can be entangled by TOs if and only if the extreme point of
its future thermal cone
$\boldsymbol{p}^{\pi}\in\mathcal{T}_{+}(\boldsymbol{p})$ with $\beta$-order
given by $\pi=(2134)$. We denote by $\mathbb{TN}$ the set of states
$\boldsymbol{p}$ for which no point of the future cone
$\mathcal{T}_{+}(\boldsymbol{p})$ can become entangled under energy-preserving
unitaries; this set is called thermally non-entanglable set. Similarly, we
define a set of catalytically nonentaglable set $\mathbb{CN}$ as set of states
for which no state from either their future cone or the catalysable set
$\mathcal{C}_{+}(\boldsymbol{p})$ can become entangled under energy-preserving
unitaries.

In order to prove Proposition 14, we need to show that the boundary of set
$\mathbb{CN}$ always contains a subspace-thermalised state of the form

$\partial\mathbb{CN}\ni\boldsymbol{p}^{*}=(1-t^{\prime})\quantity(0,0,0,1)+\frac{t^{\prime}}{Z_{123}}\quantity(\gamma_{1},\gamma_{2},\gamma_{2},0),$
(75)

with $Z_{123}=\gamma_{1}+2\gamma_{2}$. This state by itself is clearly non-
entanglable by using only energy-preserving unitaries, as it is proportional
to identity on the subspace $\operatorname{span}(\ket{01},\,\ket{10})$ and as
such, does not admit generation of any coherences within the aforementioned
subspace. However, its future thermal cone
$\mathcal{T}_{+}(\boldsymbol{p}^{*})$ may contain states that are entanglable.

In particular, one of the extreme points of
$\mathcal{T}_{+}(\boldsymbol{p}^{*})$ is a subspace-thermalised state of the
form

$\boldsymbol{p}^{**}=(1-t)\quantity(0,1,0,0)+\frac{t}{Z_{124}}\quantity(\gamma_{1},0,\gamma_{2},\gamma_{4}),$
(76)

with $Z_{123}=\gamma_{1}+\gamma_{2}+\gamma_{4}$. This admits the distinguished
$\beta$-order of $\boldsymbol{\pi}=(2,1,3,4)$, and thus, by Theorem 2 from
[48], if this is not an element of the entanglable set,
$\boldsymbol{p}^{**}\in\mathbb{TN}$ It is easy to find $t$ for which this
state becomes non-entanglable by solving,

$4\frac{t^{2}}{Z_{124}^{2}}\gamma_{1}\gamma_{4}-\quantity(1-t-\frac{t}{Z_{124}}\gamma_{2})^{2}=0,$
(77)

which is solved by setting

$t=\frac{\gamma_{1}+\gamma_{2}+\gamma_{4}}{2\sqrt{\gamma_{1}}\sqrt{\gamma_{4}}+1}.$
(78)

Now we would like to find a state $\boldsymbol{p}^{*}$ which tightly majorises
the state $\boldsymbol{p}^{**}$,
$\boldsymbol{p}^{*}\succcurlyeq\boldsymbol{p}^{**}$. It is achieved by setting
$t^{\prime}=\frac{Z_{123}}{Z_{124}}$. By reinstating
$\gamma_{1}=1,\,\gamma_{2}=\gamma_{3}=e^{-\beta},\,\gamma_{4}=e^{-2\beta}$, we
retrieve Proposition 14.

According to the above derivation, we can say that
$\mathcal{T}_{+}(\boldsymbol{p}^{*})\subset\mathbb{TN}$, but also
$\mathcal{T}_{+}(\boldsymbol{p}^{*})\subset\mathbb{CN}$, as
$\mathcal{C}_{+}(\boldsymbol{p}^{*})=\emptyset$ and for all
$\boldsymbol{q}\prec\boldsymbol{p}^{*}$ we find that
$\mathcal{C}_{+}(\boldsymbol{q})\subset\mathcal{T}_{+}(\boldsymbol{p}^{*})$.
However, by explicit check one can verify that for
$\boldsymbol{q}^{\prime}\in\mathcal{B}(\boldsymbol{p}^{*},\epsilon)\cap\mathcal{T}_{+}(\boldsymbol{p}^{*})$
and $\epsilon\leq\epsilon_{*}$, where
$\mathcal{B}(\boldsymbol{p}^{*},\epsilon)$ is a ball of radius $\epsilon$
centered at $\boldsymbol{p}^{*}$ and $\epsilon_{*}\ll 1$ a certain critical
value, we find $\boldsymbol{q}^{\prime}\in\mathbb{CE}$.

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