# Catalysis of entanglement and other quantum resources

Chandan Datta  Tulja Varun Kondra  Marek Miller  Alexander Streltsov
<EMAIL_ADDRESS>Centre for Quantum Optical Technologies, Centre of
New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland

###### Abstract

In chemistry, a catalyst is a substance which enables a chemical reaction or
increases its rate, while remaining unchanged in the process. Instead of
chemical reactions, _quantum catalysis_ enhances our ability to convert
quantum states into each other under physical constraints. The nature of the
constraints depends on the problem under study, and can arise, e.g., from
energy preservation. In this article we review the most recent developments of
quantum catalysis, and give also a historical overview of this research
direction. We focus on catalysis of quantum entanglement and coherence, and
also discuss this phenomenon in quantum thermodynamics and general quantum
resource theories. We review applications of quantum catalysis, and discuss
also the recent efforts on universal catalysis, where the quantum state of the
catalyst does not depend on the states to be transformed. Catalytic embezzling
is also considered, a phenomenon which occurs if the state of the catalyst can
change in the transition.

###### Contents

  1. I Introduction
  2. II Quantum resource theories
  3. III Exact catalysis
    1. III.1 Entanglement
    2. III.2 Quantum coherence
    3. III.3 Quantum thermodynamics
    4. III.4 Purity
    5. III.5 Other quantum resource theories
  4. IV Approximate catalysis
    1. IV.1 Quantum thermodynamics
    2. IV.2 Entanglement
    3. IV.3 Quantum coherence
    4. IV.4 Other quantum resource theories
  5. V Applications of quantum catalysis
    1. V.1 Quantum state merging
    2. V.2 Quantum teleportation
    3. V.3 Assisted distillation of entanglement and coherence
    4. V.4 Catalysis of noisy quantum channels
    5. V.5 Authentication protocols based on catalysis
    6. V.6 Catalytic decoupling
    7. V.7 Catalytic cooling
  6. VI Universal catalysis
    1. VI.1 Quantum thermodynamics
    2. VI.2 Pure state quantum entanglement
    3. VI.3 Comparison of universal catalysis of entanglement and quantum thermodynamics
  7. VII Catalytic embezzling phenomena
  8. VIII Perspectives and open problems

##  I Introduction

In the year 1835, Jacob Berzelius observed a common feature among several
chemical reactions: they all required an additional substance, which remained
unchanged in the process. He described this feature with the word “catalysis”,
which goes back to Greek, and can be translated as “down” or “loosen” [24,
156]. A systematic study of catalytic processes in chemistry has been
performed later by Wilhelm Ostwald, who gave a precise definition which is
also used today: “Catalysis is the acceleration of a slow chemical process by
the presence of a foreign substance” [149]. First known examples of chemical
catalysis used by humans are fermentation processes, which have been used for
thousands of years [156]. Today, chemical catalysis is broadly used in the
chemical industry and is essential for a large number of industrial processes.
In addition, the importance of catalytic reactions in biochemistry cannot be
overstated: to the point that the ability of living organisms to “catalyze
chemical reactions efficiently and selectively” via metabolic pathways could
be called one of the “fundamental conditions for life” [143].

Quantum catalysis is conceptually similar to chemical catalysis, but differs
from it in several important details. A simple analogy between quantum and
chemical catalysis can be established by replacing “chemical reaction” with
“quantum state transition”. With this, a quantum catalyst is a quantum system
which enables an otherwise impossible transition between quantum states. First
investigations of quantum catalysis go back to the work of Jonathan and Plenio
[105], who studied transitions between entangled states shared by two remote
parties (Alice and Bob). It is commonly assumed that the remote parties have
the complete freedom to manipulate their quantum systems locally, including
all kinds of transformations and measurements which are compatible with the
laws of quantum mechanics. Additionally, Alice and Bob can exchange the
outcomes of their measurements via a classical communication channel. This
allows each party to adjust their local setups to the outcomes of the previous
measurements obtained by the other party. This setup describes everything
Alice and Bob can do to their shared quantum state if they cannot exchange
other quantum particles [20, 103].

In the first example of entanglement catalysis given by Jonathan and Plenio
[105], Alice and Bob aim to convert their quantum state $\ket{\psi}^{AB}$ into
another state $\ket{\phi}^{AB}$ under the limitations described above. Few
months earlier, Nielsen [147] presented conditions for such state
transformations, which allow to check whether $\ket{\psi}^{AB}$ can be
converted into $\ket{\phi}^{AB}$ for any pair of states. Jonathan and Plenio
[105] found that for some states an otherwise impossible transition can be
enabled by using an additional pair of particles $A^{\prime}$ and $B^{\prime}$
held by Alice and Bob, respectively. More precisely, by choosing an
appropriate entangled state $\ket{\eta}^{A^{\prime}B^{\prime}}$, they showed
that $\ket{\psi}^{AB}\otimes\ket{\eta}^{A^{\prime}B^{\prime}}$ can be
converted into $\ket{\phi}^{AB}\otimes\ket{\eta}^{A^{\prime}B^{\prime}}$, even
though a direct conversion from $\ket{\psi}^{AB}$ into $\ket{\phi}^{AB}$ is
not possible. As a result the state $\ket{\eta}^{A^{\prime}B^{\prime}}$
remains unchanged and the two-particle system $A^{\prime}B^{\prime}$ can be
seen as a quantum catalyst.

In recent years, it became clear that entanglement is not the only quantum
feature to exhibit catalytic behavior, mostly thanks to the development of
quantum resource theories [47], which goes back to the systematic study of
quantum entanglement [103] as a resource for the emerging quantum technology.
Every quantum resource theory is based on a set of free states and free
operations, corresponding to states and transformations which are easy to
prepare and implement within well-defined physical constraints. Apart from
quantum entanglement, the framework of quantum resource theories gave
significant insight into the thermodynamical properties of quantum systems
[30, 77, 81], and the role of purity [99, 81] and quantum coherence [14, 167]
for technological applications. Catalytic effects have been found in most
resource theories studied in recent literature, allowing to overcome certain
constraints of the theory by adding a particle which remains unchanged in the
procedure.

The original formulation of quantum catalysis proposed by Jonathan and Plenio
[105] did not allow for any correlations between the catalyst and the primary
system as a result of a catalytic process. This constraint has been lifted in
recent literature, allowing the procedure to build up and preserve
correlations. This fruitful approach has significantly improved our
understanding of quantum catalysis, with applications in the resource theories
of quantum coherence [1], quantum thermodynamics [189, 139, 160, 161], purity
[26, 187], and entanglement [112]. In many cases, it has significantly
simplified the analysis of catalytic state transformations within the
corresponding resource theory [139, 160, 161, 26, 187, 112].

In this article, we review recent developments in the catalysis of quantum
entanglement and other quantum resource theories. The article is structured as
follows: In Section II we recall the main features of quantum resource
theories. Exact and approximate catalysis are discussed in detail in Sections
III and IV, respectively. In Section V we review applications of catalysis for
quantum information processing. Section VI presents recent results on
universal catalysis. Catalytic embezzling phenomena are discussed in Section
VII, and perspectives and open problems are presented in Section VIII.

##  II Quantum resource theories

The study of quantum resource theories [47] goes back to the early research in
quantum entanglement [103]. After the first observation of quantum
entanglement in nature [7, 8], it was suggested that entanglement might
eventually find interesting technological applications. Among others, Bennett
_et al._ [17] established that maximally entangled singlet states could be
regarded as a vehicle for quantum teleportation. And even before that, a
cryptographic scheme with singlets was described in [70]. Thus, from the point
of view of various technological applications, an important problem emerged,
how to find an optimal method of transforming arbitrary quantum states into
maximally entangled singlets:

$\ket{\phi_{2}^{+}}=\frac{1}{\sqrt{2}}\left(\ket{00}+\ket{11}\right).$ (1)

This question is highly relevant also in the context of mixed, or noisy,
quantum states, which are the right model of a quantum system in any real
experiment. One of the goals of quantum resource theories is to develop
optimal strategies of transforming arbitrary quantum states, states perhaps
not so useful for some applications into ones that are deemed more
“resourceful”.

Any quantum resource theory specifies a context in which certain states and
transformations are considered _free_. Free states are always available at no
cost – an additional subsystem as a free state can be added or discarded at
will. For example, the resource theory of entanglement regards all separable
states [186] as free:

$\rho_{\mathrm{sep}}=\sum_{i}p_{i}\rho_{i}^{A}\otimes\sigma_{i}^{B},$ (2)

where $\rho_{i}^{A}$ and $\sigma_{i}^{B}$ are local quantum states. A
composite quantum system can be brought to a separable state without the
requirement of having shared quantum entanglement in advance, using only local
operations and classical communication (LOCC). For a general resource theory,
free operations are those transformations of quantum systems that can be
implemented without the need of having access to resourceful, non-free states.
In the resource theory of entanglement, free operations are precisely the
class of LOCC.

It was the resource theory of quantum entanglement that was historically the
first to undergo a systematic study [103]. Intuitively, entanglement is a
valuable resource in systems that allow for clear spatial separation into
several discrete subsystems. Not all prominent applications of quantum
technologies, however, emphasise the role of such spatial separation. For
example, an instance of quantum computation is typically assumed to be a local
process. The role of entanglement in quantum computation has been the subject
of numerous publications, and it has been shown that under certain
assumptions, a quantum computer requires entanglement in order to exhibit an
exponential speed-up compared to classical computation [106]. As of today, it
remains unclear, however, whether it is quantum entanglement that is a crucial
component of quantum computation, especially when noisy quantum states are
considered. An interesting example to the contrary is the one clean qubit
model [110], which performs exponentially faster than the best-known classical
algorithm, with vanishing entanglement [53, 142]. Moreover, given certain
quantum computational tasks, large amounts of entanglement can have a
detrimental effect on the performance of a quantum processor [85, 142]. One
should also note that universal quantum computation is proven to be possible
with arbitrary little entanglement [144].

The framework of the general quantum resource theory allows for investigating
quantum systems beyond the constraints of particular resource manipulation
scenarios. Since not all quantum information processing tasks require
entanglement, a considerable amount of effort has been put in recent years
into the research of other resource theories [47], among others the resource
theories of purity [99, 81, 168], quantum thermodynamics [30, 77, 81],
coherence [14, 167, 2, 194], asymmetry [80, 82, 177, 134, 135], and
imaginarity [192, 193, 92]. As it has been shown that for every quantum
resource theory there exists a channel discrimination task showing a
quantitative advantage of utilising that particular resource [172, 171], the
newly developed theories may prove to be of considerable importance to future
applications of processing quantum information.

Figure 1: Quantum catalysis: Consider two system states $\rho^{S}$ and
$\sigma^{S}$ such that a conversion $\rho^{S}\rightarrow\sigma^{S}$ cannot be
achieved with free operations (left part of the figure). In many resource
theories, a conversion might still be enabled for some states by a catalyst in
the state $\tau^{C}$, allowing for the transformation
$\rho^{S}\otimes\tau^{C}\rightarrow\sigma^{S}\otimes\tau^{C}$ (right part of
the figure).

##  III Exact catalysis

Due to the rapid growth of quantum information science in recent years, we can
identify and quantify the quantum resources required for different quantum
information protocols. One of the important studies in this direction is to
learn how quantum states can be transformed into each other under some
physical constraints. More precisely, we are interested in knowing whether a
state $\sigma$ can be achieved from a given resourceful state $\rho$ as a
result of some physical operation allowed by the theory. Sometimes the
transformation is possible deterministically, i.e., there exists a free
operation $\Lambda_{f}$ such that $\sigma=\Lambda_{f}[\rho]$. When the
transformation from $\rho$ to $\sigma$ is not possible deterministically, we
consider the possibility of probabilistic conversion. Nonetheless, there exist
states which cannot be converted into each other even probabilistically. In
that case, there may exist an additional state that works as a catalyst
enabling the transformation while it remains unchanged in the process, see
also Fig. 1.

In this section, we will discuss exact catalysis, where no error is allowed in
the transformation. Historically, exact catalysis was first introduced for
quantum entanglement [105], and extended to other quantum resource theories
more recently. In the following, we will review recent results on quantum
catalysis for different resource theories. Before presenting the results, we
introduce the concept of exact catalysis. An exact catalytic transformation
between two states $\rho^{S}$ and $\sigma^{S}$ of a system $S$ is possible
when there exists a catalyst $C$ in a state $\tau^{C}$ and a free operation
$\Lambda_{f}$ acting on the system $S$ and the catalyst $C$ such that [105]

$\Lambda_{f}(\rho^{S}\otimes\tau^{C})=\sigma^{S}\otimes\tau^{C}.$ (3)

Exact catalytic transformation is very stringent, and over time relaxations of
it have been considered in the literature, allowing the catalyst to build up
correlations with the primary system [1, 129, 25, 153, 197]. In particular, an
exact correlated catalytic transformation between two states $\rho^{S}$ and
$\sigma^{S}$ is possible if and only if there exists a catalyst state
$\tau^{C}$, a correlated state $\sigma^{SC}$ and a free operation
$\Lambda_{f}$ such that

$\Lambda_{f}(\rho^{S}\otimes\tau^{C})=\sigma^{SC}\,\,\mbox{with}\,\,\operatorname{Tr}_{C}\sigma^{SC}=\sigma^{S}\,\,\mbox{and}\,\,\operatorname{Tr}_{S}\sigma^{SC}=\tau^{C}.$
(4)

Having defined exact catalysis for general quantum resource theories, we will
focus on catalysis of entanglement in the following.

###  III.1 Entanglement

Historically, Jonathan and Plenio [105] introduced the notion of exact
catalysis of quantum states first in the context of the resource theory of
entanglement. The idea is based on the following fundamental result, which
characterises fully the necessary and sufficient conditions to be met by all
admissible transformations of pure bipartite entangled states. Consider two
states $\ket{\psi}^{AB}=\sum_{i}\sqrt{\alpha_{i}}\ket{ii}$ and
$\ket{\phi}^{AB}=\sum_{i}\sqrt{\beta_{i}}\ket{ii}$, with their respective
Schmidt coefficients $\\{\alpha_{i}\\}$ and $\\{\beta_{i}\\}$ sorted in non-
increasing order.

######  Theorem 1 (Nielsen’s theorem, see [147]).

The transformation $\ket{\psi}^{AB}\rightarrow\ket{\phi}^{AB}$ is possible via
LOCC if and only if the reduced states $\psi^{A}$ and $\phi^{A}$ fulfill the
majorization relation $\psi^{A}\prec\phi^{A}$, i.e., their Schmidt
coefficients satisfy

$\sum_{i=0}^{n}\alpha_{i}\leq\sum_{i=0}^{n}\beta_{i}\,\,\,\mathrm{for\,all}\,\,\,0\leq
n\leq d-1,$ (5)

with $d=\min\\{d_{A},d_{B}\\}$, where $d_{A}$ and $d_{B}$ are the dimensions
of the Hilbert space of Alice and Bob, respectively.

The same result as in Theorem 1 has been derived independently by Hardy [88].

It is not difficult to find such states which do not satisfy the above
relations, and hence an LOCC transition between them is not possible. As an
example, consider the following two states [105]:

$\displaystyle\ket{\psi}$
$\displaystyle=\sqrt{0.4}\ket{00}+\sqrt{0.4}\ket{11}+\sqrt{0.1}\ket{22}+\sqrt{0.1}\ket{33},$
(6) $\displaystyle\ket{\phi}$
$\displaystyle=\sqrt{0.5}\ket{00}+\sqrt{0.25}\ket{11}+\sqrt{0.25}\ket{22}.$
(7)

It can be easily verified that the above states do not satisfy Nielsen’s
theorem, and as a result, a deterministic conversion is not possible. When a
deterministic transformation is not possible, one may consider probabilistic
transformation with maximum success probability given by [179]:

$P_{\max}(\ket{\psi}\rightarrow\ket{\phi})=\min_{n\in[0,d-1]}\frac{\sum_{i=n}^{d-1}\alpha_{i}}{\sum_{i=n}^{d-1}\beta_{i}}.$
(8)

For the states given in Eqs. (6) and (7), the optimal probability of
transforming $\ket{\psi}$ into $\ket{\phi}$ is $0.8$. Surprisingly, if the two
parties have access to an additional entangled state
$\ket{\eta}=\sqrt{0.6}\ket{00}+\sqrt{0.4}\ket{11}$, then they can perform the
transformation
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$ with
certainty [147, 105]. After the transformation the state $\ket{\eta}$ remains
unchanged and can be used again for the same purpose. We can say that the
state $\ket{\eta}$ acts as a catalyst and enables a transformation that
otherwise would not be possible. Note that in the context of quantum
entanglement, what is nowadays predominantly called “catalysis”, used to be
described as “entanglement-assisted LOCC operation” (ELOCC) [105, 69].

A maximally entangled state of any dimension cannot catalyse a transformation
between two arbitrary incomparable bipartite pure states [105]. A surprising
result that already suggests a rich structure of catalytic transformations of
quantum resources. Moreover, if for a pair of states $\ket{\psi}$ and
$\ket{\phi}$, there exist catalytic transformations
$\ket{\phi}\rightarrow\ket{\psi}$ and $\ket{\psi}\rightarrow\ket{\phi}$, i.e.,
if two states are interconvertible under catalytic LOCC, then there is a
_local_ unitary $U$ such that $\ket{\phi}=U\ket{\psi}$ [105]. As a
consequence, if a direct, non-catalytic transformation from $\ket{\psi}$ to
$\ket{\phi}$ is not possible, whereas a catalytic transformation exists in
that case, then the reverse transition $\ket{\phi}\rightarrow\ket{\psi}$
cannot be realised even as a catalytic one. Hence, catalysis can have an
advantage over direct transformation of quantum states only in the case when
the two states are incomparable: $\ket{\phi}\nleftrightarrow\ket{\psi}$, i.e.,
there is neither a direct transformation $\ket{\psi}\rightarrow\ket{\phi}$ nor
$\ket{\phi}\rightarrow\ket{\psi}$. However, note that the above result is
based on assuming pure state catalysts, and how the result behaves with mixed
catalyst states is not known. The result suggests that, as far as
transformations of two-qubit pure entangled states are concerned, there is no
advantage with a catalyst in a pure state, since it is always true that either
$\ket{\psi}\rightarrow\ket{\phi}$ or $\ket{\phi}\rightarrow\ket{\psi}$ in this
case.

A catalytic LOCC transformation with catalysts in pure states has no advantage
in the case of two-qutrit pure states as well. In order to see this, Jonathan
and Plenio [105] prove that a bipartite state $\ket{\psi}$ of local dimension
$d$ can be converted catalytically into another bipartite state $\ket{\phi}$,
only if the following inequalities hold

$\alpha_{0}\leq\beta_{0}\,\,\mbox{and}\,\,\alpha_{d-1}\geq\beta_{d-1},$ (9)

where the Schmidt coefficients $\\{\alpha_{i}\\}$ and $\\{\beta_{i}\\}$,
$i=0,1,\ldots,d-1$, of respectively $\ket{\psi}$ and $\ket{\phi}$ are sorted
in non-increasing order. In the case of $d=3$, the above necessary condition
for catalysis can be stated as

$\alpha_{0}\leq\beta_{0}\,\,\mbox{and}\,\,\alpha_{0}+\alpha_{1}\leq\beta_{0}+\beta_{1},$
(10)

since the numbers $\\{\alpha_{i}\\}$ and $\\{\beta_{i}\\}$ sum up to identity.
But this is exactly the condition (5) of Nielsen’s majorization theorem!
Hence, as in the two-qubit case, there is no advantage in catalysis for
bipartite systems with $d=3$: the existence of a catalytic conversion of
entangled states implies that these states can be converted directly via LOCC.
A genuine catalytic conversion of incomparable states occurs first in systems
of dimension $d=4$, as we have already seen in Eqs. (6) and (7).

When a catalytic transition from $\ket{\psi}$ to $\ket{\phi}$ is not possible
with certainty, it can happen that by adding a catalyst state we will increase
the probability of the conversion. As an example, consider two incomparable
states [105]:

$\displaystyle\ket{\psi}=\sqrt{0.6}\ket{00}+\sqrt{0.2}\ket{11}+\sqrt{0.2}\ket{22},$
(11)
$\displaystyle\ket{\phi}=\sqrt{0.5}\ket{00}+\sqrt{0.4}\ket{11}+\sqrt{0.1}\ket{22}.$
(12)

The optimal probability to convert $\ket{\psi}$ into $\ket{\phi}$ is
$P_{\max}(\ket{\psi}\rightarrow\ket{\phi})=0.8$ [179], whereas with the help
of another state $\ket{\eta}=\sqrt{0.65}\ket{00}+\sqrt{0.35}\ket{11}$, we have
$P_{\max}(\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta})=0.904$.
Of course, there are situations when even adding a catalyst in an arbitrary
pure state cannot increase the optimal probability of transformation. This is
the case, e.g. when the optimal probability of converting a bipartite state
$\ket{\psi}$ into $\ket{\phi}$ is
$P_{\max}(\ket{\psi}\rightarrow\ket{\phi})=\alpha_{d-1}/\beta_{d-1}$. Indeed,
let $\\{\gamma_{j}\\}_{j=0}^{d^{\prime}-1}$ be the ordered, non-zero Schmidt
coefficients of $\ket{\eta}$. Since the Schmidt coefficients, not necessarily
ordered, of the state $\ket{\psi}\otimes\ket{\eta}$ are given by
$\\{\alpha_{i}\gamma_{j}\\}_{i=0,j=0}^{d-1,d^{\prime}-1}$, and similarly:
$\\{\beta_{i}\gamma_{j}\\}_{i=0,j=0}^{d-1,d^{\prime}-1}$ for the state
$\ket{\phi}\otimes\ket{\eta}$, from Eq. (8) we obtain [105]

$\displaystyle
P_{\max}(\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta})$
$\displaystyle=\min\limits_{\begin{subarray}{c}l=0,1,\ldots,d-1\\\
k=0,1,\ldots,d^{\prime}-1\end{subarray}}\frac{\sum\limits_{i\geq l,j\geq
k}\alpha_{i}\gamma_{j}}{\sum\limits_{i\geq l,j\geq k}\beta_{i}\gamma_{j}}$
(13) $\displaystyle\leq\frac{\alpha_{d-1}}{\beta_{d-1}},$

where the inequality is obtained by setting $l=d-1,k=d^{\prime}-1$, instead of
minimising over all possible pairs $l,k$.

Of course, a pure catalyst state can also increase the transformation
probability. Feng  _et al._ [73, 74] have studied the advantage of pure
catalysts in probabilistic entangled state transformations. Here, the goal is
to find a pure catalyst state $\ket{\eta}$ increasing the optimal conversion
probability, i.e.,

$P_{\max}(\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta})>P_{\max}(\ket{\psi}\rightarrow\ket{\phi}).$
(14)

The following necessary and sufficient condition has been derived for the
existence of a catalyst $\ket{\eta}$ for such an enhancement [73]:

$P_{\max}(\ket{\psi}\rightarrow\ket{\phi})<\min\left\\{\frac{\alpha_{d-1}}{\beta_{d-1}},1\right\\},$
(15)

where $\alpha_{d-1}$ and $\beta_{d-1}$ are the smallest Schmidt coefficient of
$\ket{\psi}$ and $\ket{\phi}$ respectively. Furthermore, they also discussed
the scenario where the probability can be increased with the help of a two-
qubit pure catalyst. In addition, an efficient algorithm has been proposed to
find the most economical catalyst in such cases [74].

As mentioned earlier, if the local dimension of the given states are smaller
or equal than $3$, catalyst does not help in accomplishing a forbidden
transformation [105]. Now one may wonder that for a given final state
$\ket{\phi}$ of dimension at least 4 there exists an initial state
$\ket{\psi}$ with $\psi^{A}\nprec\phi^{A}$, such that $\ket{\psi}$ can be
converted catalytically into $\ket{\phi}$. It turns out that almost for all
final states $\ket{\phi}$ there exists a catalytic conversion from
$\ket{\psi}$ [50]. To be precise, the following has been shown: For a given
final state $\ket{\phi}$, there exist a pure state catalyst and an initial
state $\ket{\psi}$ with $\psi^{A}\nprec\phi^{A}$, such that the transformation
$\ket{\psi}\rightarrow\ket{\phi}$ is possible catalytically if and only if
$\beta_{0}\neq\beta_{l}$ and $\beta_{m}\neq\beta_{d-1}$ for some $l,m$ with
$0<l<m<d-1$ [50]. Here, $\beta_{i}$ are Schmidt coefficients of the state
$\ket{\phi}$. This result emphasises that for almost all final states of
dimension 4 or more, the sets of initial states to achieve a final state
$\ket{\phi}$ using a pure state catalyst and without a catalyst are not the
same, i.e., $T(\phi)\neq S(\phi)$. Here $T(\phi)$ and $S(\phi)$ represent the
sets of possible initial states $\ket{\psi}$ that can be converted into
$\ket{\phi}$ with a catalyst in an arbitrary pure state and without catalyst
respectively. Note that no restriction has been imposed on the dimension of
the catalysts. For a restriction on the dimension of the catalysts, the set of
initial states is represented by $T_{k}(\phi)$, where $k$ is the maximum
dimension of the catalysts. Then it has been shown that there exists no $k$
such that $T_{k}(\phi)=T(\phi)$ when $T(\phi)\neq S(\phi)$ [50]. Although the
set $T_{k}(\phi)$ is closed, the set $T(\phi)$ is not closed in general [50].
Therefore, the dimension of the catalyst is unbounded in general to determine
which states can be catalytically converted into a given final state.
Furthermore, as we have previously discussed maximally entangled state cannot
be a catalyst. Naturally, one may ask which states are useful to be catalyst.
As shown by Daftuar and Klimesh [50], it turns out that almost all bipartite
pure states are potential catalysts. Precisely the following has been shown:
For a bipartite state $\ket{\eta}=\sum_{i=0}^{d-1}\sqrt{\xi_{i}}\ket{ii}$ of
local dimension $d$ with a nonuniform distribution of Schmidt coefficients
there exist states $\ket{\psi}$ and $\ket{\phi}$ in dimension $4$ with
$\psi^{A}\nprec\phi^{A}$, such that
$\psi^{A}\otimes\eta^{A}\prec\phi^{A}\otimes\eta^{A}$ [50].

Since the discovery of entanglement catalysis by Jonathan and Plenio [105], an
important problem was to characterize all pairs of states $\ket{\psi}^{AB}$
and $\ket{\phi}^{AB}$ allowing for a catalytic LOCC transformation. Results in
this direction have been presented in [9, 10, 170, 175, 109], focusing on pure
catalyst states. When a catalyst enables a forbidden transformation between
two bipartite pure states, the respective Schmidt coefficients satisfy a
trumping condition. More precisely, a state $\ket{\psi}$ is trumped by another
state $\ket{\phi}$, i.e., $\psi^{A}\prec_{T}\phi^{A}$ if there exists a
catalyst in the state $\ket{\eta}$ such that
$\psi^{A}\otimes\eta^{A}\prec\phi^{A}\otimes\eta^{A}$ holds [50]. The trumping
condition is an extension of Nielsen’s majorization relation presented earlier
in Theorem 1.

A detailed characterization of the trumping condition is mathematically very
complex [50]. Sun  _et al._ [170] provided a necessary and sufficient
condition for the existence of a pure two-qubit catalyst state for $4\otimes
4$ incomparable pure states. Furthermore they have given an efficient
algorithm to decide whether a $k\otimes k$ pure catalyst state exists for a
pair of $n\otimes n$ incomparable pure states. Aubrun and Nechita [10, 9]
provided some more partial results in this direction by determining the
closure of the set $T(\phi)$ consisting of initial states $\ket{\psi}$ for a
given final state $\ket{\phi}$. Precisely they showed that the closure of
$T(\phi)$ in $l_{1}$ norm can be completely characterized by the following
inequalities: $||\psi^{A}||_{p}\leq||\phi^{A}||_{p}$ for all $p\geq 1$, where
$||\cdot||_{p}$ represents $l_{p}$ norm. Despite these efforts, the
characterization was not complete and a full characterization has been
provided by Turgut [175] and Klimesh [109]. To be precise, it has been shown
that the trumping condition is equivalent to [175, 109]

$f_{\alpha}(s(\phi))>f_{\alpha}(s(\psi))\,\,\forall\,\,\alpha\in(-\infty,\infty),$
(16)

where $s(\psi)$ and $s(\phi)$ represent the Schmidt coefficients of the states
$\ket{\psi}$ and $\ket{\phi}$ respectively. The function $f_{\alpha}(x)$ for a
$d$-dimensional probability vector $x$ is expressed as

$\displaystyle
f_{\alpha}(x)=\begin{cases}\ln\sum_{i=1}^{d}x_{i}^{\alpha}&(\alpha>1);\\\
\sum_{i=1}^{d}x_{i}\ln x_{i}&(\alpha=1);\\\
-\ln\sum_{i=1}^{d}x_{i}^{\alpha}&(0<\alpha<1);\\\ -\sum_{i=1}^{d}\ln
x_{i}&(\alpha=0);\\\ \ln\sum_{i=1}^{d}x_{i}^{\alpha}&(\alpha<0).\end{cases}$
(17)

If one allow infinitesimally small error in the initial state, i.e.,
$||\psi^{A}-\psi_{\varepsilon}^{A}||_{1}\leq\varepsilon$ with the trace norm
$||M||_{1}=\operatorname{Tr}\sqrt{M^{\dagger}M}$, then the trumping condition
$\psi_{\varepsilon}^{A}\prec_{T}\phi^{A}$ is equivalent to the monotonicity of
the Rényi entropies [29].

######  Theorem 2 (See Proposition 4 of the Supplementary Information in
[29]).

Consider two states $\ket{\psi}$ and $\ket{\phi}$ with Schmidt coefficients
$s(\psi)$ and $s(\phi)$ respectively. Then for arbitrary $\varepsilon>0$,
there exists a state $\psi_{\varepsilon}^{A}$ with
$||\psi^{A}-\psi_{\varepsilon}^{A}||_{1}\leq\varepsilon$ and
$\psi_{\varepsilon}^{A}\prec_{T}\phi^{A}$ if and only if

$\displaystyle H_{\alpha}(s(\psi))\geq H_{\alpha}(s(\phi))$ (18)

is true for all $\alpha\in(-\infty,\infty)$.

Here, $H_{\alpha}$ represent the Rényi entropies which can be expressed as

$H_{\alpha}(x)=\frac{\mathrm{sgn}(\alpha)}{1-\alpha}\ln\left(\sum_{i=1}^{d}x_{i}^{\alpha}\right)\,\,\mbox{for}\,\,\alpha\in\mathbb{R}\setminus\\{0,1\\}.$
(19)

$H_{\alpha}(x)$ for $\alpha\in\\{-\infty,0,1,\infty\\}$ are defined by taking
limits and expressed as

$\displaystyle H_{0}(x)$
$\displaystyle=\ln\mbox{rank}(x),\,\,H_{1}(x)=-\sum_{i=1}^{d}x_{i}\ln x_{i}$
(20) $\displaystyle H_{-\infty}(x)$ $\displaystyle=\ln
x_{\min},\,\,H_{\infty}(x)=-\ln x_{\max},$ (21)

where ${x_{\max}}$ and $x_{\min}$ are the maximum and minimum element of the
vector $x$.

In real life, noise is inevitable, and as a result, we are forced to deal with
mixed states rather than pure states. Mixed state transformations are much
more complicated [174] as majorization relations described in Theorem 1 are
not applicable in general. Consequently, the conversion of mixed states with
the help of a catalyst gets convoluted. The reason is that we need to first
check whether a mixed state transformation $\rho\rightarrow\sigma$ is
possible. In general it is a formidable task. Nevertheless, the problem has
been solved for some classes of mixed states. Eisert and Wilkens [69] explored
transformations between a special class of rank two mixed states of the form

$\displaystyle\rho$
$\displaystyle=\nu\ket{\psi}\\!\bra{\psi}+(1-\nu)\ket{\omega}\\!\bra{\omega}\,\,\mbox{and}$
(22) $\displaystyle\sigma$
$\displaystyle=\gamma\ket{\phi}\\!\bra{\phi}+(1-\gamma)\ket{\omega}\\!\bra{\omega},$
(23)

where $\gamma=\nu\mbox{Tr}[\Omega]$, $\Omega=\Pi\ket{\psi}\\!\bra{\psi}\Pi$,
$\Pi=\openone-\ket{\omega}\\!\bra{\omega}$ and $\ket{\omega}$ is a product
state. Moreover, we take $|\bra{\phi}\omega\rangle|^{2}=0$. With this the LOCC
transformation from $\rho$ to $\sigma$ implies that [69]

$\frac{\mbox{Tr}_{A}\Omega}{\mbox{Tr}\Omega}\prec\phi^{A}.$ (24)

Now let us choose

$\displaystyle\ket{\psi}$ $\displaystyle=$
$\displaystyle\sqrt{0.38}\ket{00}+\sqrt{0.38}\ket{11}+\sqrt{0.095}\ket{22}$
(25) $\displaystyle+$ $\displaystyle\sqrt{0.095}\ket{33}+\sqrt{0.05}\ket{44},$
$\displaystyle\ket{\phi}$ $\displaystyle=$
$\displaystyle\sqrt{0.5}\ket{00}+\sqrt{0.25}\ket{11}+\sqrt{0.25}\ket{22},$
(26) $\displaystyle\ket{\omega}$ $\displaystyle=$ $\displaystyle\ket{44},$
(27)

and $\gamma=0.95\nu$, where $1>\nu>0$ [69]. With this one can easily verify
that Eq. (24) is not satisfied, and hence the transformation
$\rho\rightarrow\sigma$ is not possible.

One may ask whether the forbidden transformation can be accomplished with the
help of a catalyst. The answer is yes and the protocol is as follows [69].
Alice performs a von Neumann measurement on her subsystem with Kraus operators
$K_{1}=\sum_{i=0}^{3}\ket{i}\\!\bra{i}$ and $K_{2}=\ket{4}\\!\bra{4}$. When
she gets an outcome associated with $K_{2}$, no further operations are needed
as we have state $\ket{\omega}$ with probability $(1-\gamma)$. In the case of
other outcome the final state is

$\ket{\varphi}=\sqrt{0.4}\ket{00}+\sqrt{0.4}\ket{11}+\sqrt{0.1}\ket{22}+\sqrt{0.1}\ket{33}.$
(28)

Note that, the reduced state of $\ket{\varphi}$ is not majorized by the
reduced state of $\ket{\phi}$. Therefore, a direct transformation is not
possible and a catalyst could help in the transformation
$\ket{\varphi}\rightarrow\ket{\phi}$. In fact by taking a catalyst in the
state $\ket{\zeta}=\sqrt{0.4}\ket{00}+\sqrt{0.6}\ket{11}$, we obtain

$\varphi^{A}\otimes\zeta^{A}\prec\phi^{A}\otimes\zeta^{A}.$ (29)

Finally, discarding the classical information, we achieve the desired state
$\sigma$.

In [117], the authors analyse catalytic transformation of mixed state of two
qubits. Building on the results from [79, 118], their work sheds some light on
the question whether mixed catalyst state could provide an advantage for
entanglement catalysis. For a given mixed state $\rho$, the authors define a
family of entanglement monotones:

$\hat{E}_{k,l}(\rho)=\min\limits_{\\{p_{j},\ket{\psi_{j}}\\}}\,\sum\limits_{j}p_{j}\,\hat{E}_{k,l}(\ket{\psi_{j}}),$
(30)

with $l=2,3,\ldots,d$ and $k\in(0,1]$, where the minimum is taken over all
ensembles $\\{p_{j},\ket{\psi_{j}}\\}$ of probability distributions $p_{j}$
and bipartite pure states $\ket{\psi_{j}}$ such that
$\rho=\sum_{j}p_{j}\ket{\psi_{j}}$, and

$\hat{E}_{k,l}(\ket{\psi})=\min\left\\{1,\sum\limits_{i=l}^{d}\frac{\alpha_{i}}{k}\right\\},$
(31)

for any bipartite pure state
$\ket{\psi}=\sum_{i}^{d}\sqrt{\alpha_{i}}\ket{ii}$ with ordered Schmidt
coefficients. The ensemble that minimises Eq. (30) is called _optimal_.

In another paper, [118], the same authors give a non-standard definition of a
LOCC transformation between mixed states. According to Definition 2.6 in
[118], a mixed state $\rho$ can be converted into another mixed state $\sigma$
via LOCC if there is an LOCC operation between optimal ensembles of $\rho$ and
$\sigma$. For a precise definition of a transformation between ensembles of
pure states, see [118], p. 97. For our discussion here, it is enough to notice
that if there is an LOCC map $\Lambda$, such that $\Lambda(\rho)=\sigma$ in
the ordinary sense, then $\rho$ can be converted to $\sigma$ via LOCC
according to Definition 2.6 in [118] – but not necessarily the other way
round.

The main result from [117], namely Proposition 2.2 therein, can be now stated
as follows: Suppose $\rho$, $\sigma$ and $\omega$ are rank 2 mixed bipartite
states of a two-qubit system. Then the state $\rho\otimes\omega$ can be
converted into $\sigma\otimes\omega$ via LOCC according to Definition 2.6 in
[118] if and only if

$\hat{E}_{k,l}(\rho\otimes\omega)\geq\hat{E}_{k,l}(\sigma\otimes\omega),$ (32)

for all $k\in(0,1]$ and $l=2,3,4$. In other words, as long as we consider rank
2 mixed states, a catalytic transformation between two-qubit systems is
governed by the family of entanglement monotones defined in Eq. (30). The
result is a generalisation of the trumping condition for pure states, see the
discussion above, p. III.1.

Another important result in [117], Proposition 2.5, follows quickly from the
proof of the Proposition 2.2, and gives us a partial answer to the question of
whether mixed catalyst states could provide an advantage over pure states: If
$\rho$, $\sigma$ are rank 2 mixed bipartite states of a two-qubit system that
_cannot_ be converted into each other via LOCC, but for which a catalysts
state $\omega$ exists as in Eq. (32), then $\omega$ _cannot_ be pure, even if
we allowed for $\omega$ to have arbitrary Schmidt rank. Note that this
statement assumes that all LOCC transformations between mixed states are
according to Definition 2.6 in [118], but as we have already discussed above,
that implies that there is no catalytic LOCC transformation in the ordinary
sense as well.

Lastly, the authors give an example of a mixed catalyst state that facilitates
a transformation between rank 2 mixed two-qubit states as above, see Sec. 3 in
[117]. However, again, since the authors’ definition of a LOCC transformation
between mixed states describes a larger family of maps than standard LOCC on
the convex set of mixed states, the question of whether an ordinary map and a
mixed catalyst state exist remains open. Given that we know that no pure two-
qubit state can be a catalyst in this case, a positive answer to that question
would be a strong argument for the advantages of mixed-state catalysis.

Next, we will discuss the advantage of a catalyst for small transformations
where we want to change the state slightly. In particular, the goal is to
convert a state $\rho$ into another state $\sigma$ with
$F(\rho,\sigma)>1-\varepsilon$, where
$F(\rho,\sigma)=\left(\mbox{Tr}[\sqrt{\sigma}\rho\sqrt{\sigma}]^{1/2}\right)^{2}$
is the fidelity [176]. In such a scenario, there exist states $\rho$ and
$\sigma$ such that the transformation $\rho\rightarrow\sigma$ is impossible
without catalyst and can be accomplished with a suitable catalyst. Indeed,
such states can be designed using Eqs. (22) and (23) [69]. On the contrary, a
surprising result exists when we want to transform a pure state slightly. For
all pure states $\ket{\psi}$ there exists an $\varepsilon>0$, such that the
following statement holds for all states $\ket{\phi}$ with
$|\braket{\psi}{\phi}|^{2}>1-\varepsilon$: the catalytic transformation
$\ket{\psi}\rightarrow\ket{\phi}$ is impossible with a pure catalyst if it is
impossible without the catalyst [69].

The structure of a catalyst for a successful transformation between two
incomparable pure states was studied by Zhou and Guo [200]. To be precise, the
authors showed that successful transformation of
$\ket{\psi}=\sum_{i=0}^{d-1}\sqrt{\alpha_{i}}\ket{ii}$ into
$\ket{\phi}=\sum_{i=0}^{d-1}\sqrt{\beta_{i}}\ket{ii}$ can be accomplished with
the help of a pure catalyst state having $m$ non-zero Schmidt coefficients,
i.e., $\ket{\eta}=\sum_{i=0}^{m-1}\sqrt{\gamma_{i}}\ket{ii}$, only if [200]

$m\gamma_{m-1}\leq P_{\max}(\ket{\psi}\rightarrow\ket{\phi}),$ (33)

where $P_{\max}(\ket{\psi}\rightarrow\ket{\phi})$ is described in Eq. (8),
represents the optimal probability of transforming $\ket{\psi}$ into
$\ket{\phi}$ [179]. Furthermore, they extended the above result for stochastic
scenario. As a matter of fact they showed that the probability of the
transformation $P_{\max}(\ket{\psi}\rightarrow\ket{\phi})$ can be increased
with the help of a pure state catalyst $\ket{\eta}$, only if [200]

$m\gamma_{m-1}\leq P_{\max}(\ket{\psi}\rightarrow\ket{\phi})/P^{\prime},$ (34)

where $P^{\prime}\geq P_{\max}(\ket{\psi}\rightarrow\ket{\phi})$ represents
the enhancement in the probability for catalytic transformation. Their results
provide some insight about the structure of the catalyst.

In [72], the concept of mutual catalysis has been introduced. For this,
suppose there are two pairs of states $(\ket{\psi},\ket{\phi})$ and
$(\ket{\eta},\ket{\zeta})$. The transformations
$\ket{\psi}\rightarrow\ket{\phi}$ and $\ket{\eta}\rightarrow\ket{\zeta}$ are
not possible deterministically. However, sometimes it is possible that joint
LOCC can transform $\ket{\psi}\otimes\ket{\eta}$ to
$\ket{\phi}\otimes\ket{\zeta}$. Therefore, $\ket{\psi}$ and $\ket{\eta}$ help
each other to achieve the desired target states. As an example, consider the
following states [72]

$\displaystyle\ket{\psi}$
$\displaystyle=\sqrt{0.4}\ket{00}+\sqrt{0.36}\ket{11}+\sqrt{0.14}\ket{22}$
(35) $\displaystyle\quad+\sqrt{0.1}\ket{33},$ $\displaystyle\ket{\phi}$
$\displaystyle=\sqrt{0.5}\ket{00}+\sqrt{0.25}\ket{11}+\sqrt{0.25}\ket{22},$
(36) $\displaystyle\ket{\eta}$
$\displaystyle=\sqrt{0.6}\ket{00}+\sqrt{0.4}\ket{11},$ (37)
$\displaystyle\ket{\zeta}$
$\displaystyle=\sqrt{0.55}\ket{00}+\sqrt{0.45}\ket{11}.$ (38)

It is easy to check that $\ket{\psi}\nleftrightarrow\ket{\phi}$ and
$\ket{\eta}\nrightarrow\ket{\zeta}$ [147]. However, they can be transformed
jointly to the desired target states. Necessary conditions for such
transformations have been derived in [72]. Furthermore, the authors have
discussed the scenario where a conversion from $\ket{\psi}$ to $\ket{\phi}$ is
not possible even with the help of a catalyst, but can be realized with mutual
catalysis [72].

According to Theorem 1, there exist states $\ket{\psi}$ and $\ket{\phi}$, such
that $\ket{\psi}\rightarrow\ket{\phi}$ cannot be realized with certainty.
However, in some cases, if we possess multiple copies of $\ket{\psi}$, they
can be transformed to the same number of $\ket{\phi}$ [12]. Along the same
line, a concept of multicopy catalysis has been introduced by Duan  _et al._
[64]. A state $\ket{\eta}$ is called a multiple-copy catalyst when it cannot
catalyze the transformation $\ket{\psi}\rightarrow\ket{\phi}$, however the
same can be accomplished with the help of $\ket{\eta}^{\otimes k}$ for some
$k>1$. As an example Duan  _et al._ [64] consider the following case. Consider
the initial state $\ket{\psi}$ as given in Eq. (6) and the target state is

$\ket{\phi}=\sqrt{0.5}\ket{00}+\sqrt{0.25}\ket{11}+\sqrt{0.22}\ket{22}+\sqrt{0.03}\ket{33}.$
(39)

The transformation $\ket{\psi}\rightarrow\ket{\phi}$ cannot be realized with a
pure two-qubit catalyst state [170]. However, note that the above
transformation can be realized with a pure two-qutrit catalyst state. Now
consider a situation where Alice and Bob only have access to some copies of a
two-qubit pure state. One may wonder whether the above transformation can be
realized with multiple copies of a two-qubit catalyst. Indeed there exists a
two-qubit state $\ket{\eta}=\sqrt{0.6}\ket{00}+\sqrt{0.4}\ket{11}$, such that
$\psi^{A}\otimes(\eta^{A})^{\otimes 5}\prec\phi\otimes(\eta^{A})^{\otimes 5}$
[64]. Furthermore, they have derived some necessary conditions for an
arbitrary pure state to be a catalyst for a given transformation
$\ket{\psi}\rightarrow\ket{\phi}$. In addition, an advantage in probabilistic
transformation with multiple copy catalysis has also been shown in [64].
Instead of a single copy of the initial state, if we have multiple copies of
it and want to transform it into the same number of copies of the final state
with the help of a multicopy catalyst, then a study has been done on the
tradeoff between the number of copies of the source state and that of the
catalyst state. To be precise, they have shown that the number of copies
required for a catalyst state decreases with the number of copies of the
source state [64]. In [63, 65] it has been shown that any multiple copy
entangled state transformation [12] can be realized in single copy level with
the help of a suitable pure catalyst. It is natural to wonder whether the
opposite is also true. But the answer is negative. As shown in [75], the
catalytic assisted transformation is strictly more powerful than multiple copy
transformations. Now, it is natural to ask whether they are equivalent in the
asymptotic scenario. Indeed it has been shown that catalytically assisted
transformation is asymptotically equivalent to the multiple-copy
transformation, when the dimension of the catalyst and number of copies tend
to infinity [66]. More precisely, these two kinds of transformations are
equivalent in the sense that they can simulate each other’s ability to
transform a given initial state to a given target state with the same optimal
success probability. Optimal probability for multiple-copy state
transformation from $\ket{\psi}$ to $\ket{\phi}$ can be defined as [66]

$P_{M}(\ket{\psi}\rightarrow\ket{\phi})=\sup_{m}[P_{\max}(\ket{\psi}^{\otimes
m}\rightarrow\ket{\phi}^{\otimes m})]^{1/m}.$ (40)

On the other hand, the optimal conversion probability for catalytically
assisted transformation is given by [66]

$P_{C}(\ket{\psi}\rightarrow\ket{\phi})=\sup_{\ket{\eta}}P_{\max}(\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}).$
(41)

In [66], it has been shown that
$P_{M}(\ket{\psi}\rightarrow\ket{\phi})=P_{C}(\ket{\psi}\rightarrow\ket{\phi})$.
Furthermore, Duan  _et al._ [63] have shown that multiple copy transformation,
with the aid of a pure catalyst, is equivalent to the single copy catalytic
transformation with arbitrary pure catalyst states. Let $V(\phi)$ be the set
of all initial states $\\{\ket{\psi}\\}$ such that
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$ for some
$\ket{\eta}$, and $V_{M}(\phi)$ is the set of all initial states
$\\{\ket{\psi}\\}$ such that $\ket{\psi}^{\otimes
m}\otimes\ket{\eta}\rightarrow\ket{\phi}^{\otimes m}\otimes\ket{\eta}$ for
some $m\geq 1$ and $\ket{\eta}$. Obviously $V(\phi)\subseteq V_{M}(\phi)$. In
fact, it has been shown that they are equal $V(\phi)=V_{M}(\phi)$ [63].
Therefore, the power of catalytic assisted transformation (assuming pure
catalyst states) cannot be elevated by increasing the number of copies of the
original state.

While the previous discussion showed when a catalytic transformation is
possible, it does not provide any information about the catalyst. To be
precise, it does not give information about the entanglement content or the
dimension of the catalyst required for a transformation. Sanders and Gour
[155] provided a lower bound on the dimension of a possible catalyst state
$\ket{\eta}$ for a transformation
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$. Suppose
$\ket{\psi}$, $\ket{\phi}$ are two incomparable pure states with $d$ non-zero
Schmidt coefficients and the transformation
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$ is
possible with the help of a pure catalyst state $\ket{\eta}$ that has $r$ non-
zero Schmidt coefficients. Then it has been shown that the minimum number of
non-zero Schmidt coefficients for the catalyst should be [155]

$r\geq
1+\frac{d-1}{d}\frac{\log_{2}C_{d-1}(\phi)-\log_{2}C_{d-1}(\psi)}{\log_{2}C_{d}(\psi)-\log_{2}C_{d}(\phi)},$
(42)

where $C_{k}$ represents generalized $k$-th concurrence and can be expressed
as

$C_{k}(\psi)=\left(\frac{f_{k}[s(\psi)]}{f_{k}[s(\psi^{+})]}\right)^{1/k},$
(43)

where $\psi^{+}$ is a maximally entangled state of dimension $d$ and
$f_{k}(x)$ represents $k$th elementary symmetric function of $d$ variables:
$f_{k}(x)=\sum_{i_{1}<\cdots<i_{k}}x_{i_{1}}\cdots x_{i_{k}}$ [78].

So far we did not discuss which amount of entanglement is required for a
catalyst to enable a transformation between a pair of incomparable states.
However, the answer is not fully known and only some partial results are
available in the literature. Duan  _et al._ [64] derived some necessary
conditions for a bipartite pure entangled state to be a catalyst. In fact, the
following has been proved: Suppose $\ket{\psi}$ and $\ket{\phi}$ are two
incomparable states of Schmidt rank $d$ with Schmidt coefficients $\\{p\\}$
and $\\{q\\}$ respectively, and a catalyst state $\ket{\eta}$ that has $k$
number of non-zero Schmidt coefficients represented by $\\{t\\}$ enables the
transformation from $\ket{\psi}$ to $\ket{\phi}$. Then for any $l\in
L_{\psi,\phi}$ and $0\leq i\leq k-2$, the Schmidt coefficients of the catalyst
state satisfy the following three conditions:

$\displaystyle\frac{t_{0}}{t_{k-1}}>\frac{q_{l}}{q_{l+1}},$ (44a)
$\displaystyle\frac{t_{0}}{t_{i}}>\frac{q_{l}}{q_{l+1}}\,\,\,\,\mbox{or}\,\,\,\,\frac{t_{i}}{t_{i+1}}<\frac{q_{0}}{q_{l}},$
(44b)
$\displaystyle\frac{t_{i+1}}{t_{d-1}}>\frac{q_{l}}{q_{l+1}}\,\,\,\,\mbox{or}\,\,\,\,\frac{t_{i}}{t_{i+1}}<\frac{q_{l+1}}{q_{d-1}},$
(44c)

where $L_{\psi,\phi}$ represents the set of all $l$ such that

$L_{\psi,\phi}=\left\\{l:0\leq l\leq
d-1\,\,\mbox{and}\,\,\sum_{j=0}^{l}p_{j}>\sum_{j=0}^{l}q_{j}\right\\}.$ (45)

These restrictions on the Schmidt coefficients represent the deviation of the
state $\ket{\eta}$ from the product and the maximally entangled state.

Very recently, Grabowecky and Gour [83] presented some stronger conditions
than Eqs. (44). They showed that for any two incomparable states $\ket{\psi}$,
$\ket{\phi}$ with $d$ non-zero Schmidt coefficients and a catalyst state
$\ket{\eta}$ having $k$ non-zero Schmidt coefficients, if
$\psi^{A}\otimes\eta^{A}\prec\phi^{A}\otimes\eta^{A}$, then [83]

$\displaystyle\max_{i\in\\{0,\ldots,k-2\\}}\frac{t_{i}}{t_{i+1}}$
$\displaystyle<\min\left\\{\frac{q_{0}}{q_{m}},\frac{q_{n+1}}{q_{d-1}}\right\\}\equiv
a,$ (46a) $\displaystyle\frac{t_{0}}{t_{k-1}}$ $\displaystyle>\max_{l\in
L_{\psi,\phi}}\frac{q_{l}}{q_{l+1}}\equiv b,$ (46b)

where $m=\min L_{\psi,\phi}$ and $n=\max L_{\psi,\phi}$. These two conditions
show how much a catalyst state can differ from both a product state and a
maximally entangled state. Previously in Eq. (42), we have seen a lower bound
on the dimension of the catalyst in terms of the generalised $k$-th
concurrence [155]. However, calculating $k$-th concurrence for a higher-
dimensional state is not easy in general. In that context, Grabowecky and Gour
[83] provided the following lower bound on the dimension of the catalyst,
which can be easily calculated:

$k>\frac{\ln b}{\ln a}+1.$ (47)

The constraints on the Schmidt coefficients of the catalyst state presented in
Eqs. (44) and (46) depend only on the Schmidt coefficients of the target
state. Recently, Guo  _et al._ [86] studied a similar scenario and showed that
for $\psi^{A}\otimes\eta^{A}\prec\phi^{A}\otimes\eta^{A}$ the following result
holds:

$\frac{t_{0}}{t_{k-1}}>\max_{l\in
L_{\psi,\phi}}\left(\min\left\\{\frac{p_{l}}{p_{l+1}},\frac{q_{l}}{q_{l+1}}\right\\}\right)\equiv
c,$ (48)

where $k$ is the number of non-zero Schmidt coefficients of the state
$\ket{\eta}$ and also represents the dimension of the catalyst. Furthermore,
they put a bound on the minimum dimension of the catalyst state that can be
expressed as [86]

$k>\frac{\ln c}{\ln(a\sqrt{b})}+1.$ (49)

Unlike Eq. (47), the above bound depends on both $\\{p\\}$ and $\\{q\\}$. One
can compare and check which bound provides a good lower bound.

Duarte  _et al._ [67] explored an interesting question about self-catalysis.
Given two incomparable states $\ket{\psi}$ and $\ket{\phi}$, they studied
whether $\ket{\psi}$ can catalyze the transformation
$\ket{\psi}\otimes\ket{\psi}\rightarrow\ket{\phi}\otimes\ket{\psi}$. As an
example, consider the following incomparable states [67]:

$\displaystyle\ket{\psi}$
$\displaystyle=\sqrt{0.9}\ket{00}+\sqrt{0.081}\ket{11}+\sqrt{0.01}\ket{22}$
$\displaystyle+\sqrt{0.009}\ket{33},$ (50) $\displaystyle\ket{\phi}$
$\displaystyle=\sqrt{0.95}\ket{00}+\sqrt{0.03}\ket{11}+\sqrt{0.02}\ket{22}.$
(51)

The transformation $\ket{\psi}$ into $\ket{\phi}$ can be accomplished with the
help of $\ket{\psi}$ as catalyst, i.e.,
$\ket{\psi}\otimes\ket{\psi}\rightarrow\ket{\phi}\otimes\ket{\psi}$. One can
modify it further by considering multi-copy self catalysis, where
$\ket{\psi}\otimes\ket{\psi}\nrightarrow\ket{\phi}\otimes\ket{\psi}$ is not
possible but $\ket{\psi}\otimes\ket{\psi}^{\otimes
N}\rightarrow\ket{\phi}\otimes\ket{\psi}^{\otimes N}$ is possible for some
$N>1$ [67]. For instance, consider the initial state to be

$\ket{\psi_{1}}=\sqrt{0.918}\ket{00}+\sqrt{0.07}\ket{11}+\sqrt{0.006}\ket{22}+\sqrt{0.006}\ket{33}$
(52)

and the target state to be $\ket{\phi}$ in Eq. (51). Then the transformation
$\ket{\psi_{1}}$ into $\ket{\phi}$ can be achieved using
$\ket{\psi_{1}}^{\otimes 4}$ as a catalyst. Duarte  _et al._ [67] tried to
answer the question of how typical the self catalysis phenomenon is using
numerical methods. Although they found that self-catalysis occurs in systems
of any dimension, the phenomenon is relatively rare [67]. They also explored
self-catalysis of stochastic transformations and concluded that self-catalytic
in this case occurs more often than in the deterministic one [67]. In a recent
paper, Acácio and Duarte [3] discuss in detail whether neural network models
can detect catalysis or self-catalysis for bipartite pure entangled state
conversions.

The practical implementation of majorization relations was first studied by
Gagatsos  _et al._ [76]. In fact they showed that beam splitter can be used to
achieve several classes of majorization relations. Furthermore, they construct
incomparable states and showed that these states can be catalytically
converted. Such catalyst states can be easily implemented by using single
photon states or two mode squeezed vacuum states [76].

Recently, Santra and Malinovsky [157] studied the catalytic entanglement
concentration of two-qubit pure states. They derived the formula for the
maximum probability of converting a finite number of copies of a general two-
qubit pure state into an singlet with the help of a two-qubit pure catalyst
state. Furthermore, they have shown that any pure two-qubit entangled state
can be a potential catalyst.

In a recent numerical study, entanglement properties of the pure catalyst
states have been explored [87]. The average amount of entanglement required to
catalyze a forbidden transformation decreases with the increment of
entanglement difference between the pure initial and target state.
Furthermore, for multiple copy state transformations, the amount of
entanglement required to catalyze the transformation is lower compared to the
single copy catalytic transformation.

Kay and Ericsson [108] explored the cloning of non-maximally entangled states
with LOCC and a copy of maximally entangled state. The authors completely
characterised the set of clonable states. Furthermore, it has been shown that
catalysts do not provide any advantage in the cloning process [108].

In general, multipartite entangled state transformation is more complex in
nature [182] and that makes catalytic transformation even more complex. Unlike
catalytic transformation in bipartite scenarios, very few results are known
for the catalytic state transformations in multipartite scenarios. One of the
first result in this direction was provided by Chen  _et al._ [42] where the
advantage of catalysis in stochastic LOCC (SLOCC) transformation of
multipartite pure states has been studied. In fact, it has been shown that
there exist state transformations that cannot be accomplished by SLOCC, but
can be realised by both catalytic SLOCC and multicopy SLOCC. Catalytic SLOCC
represents the probabilistic transformation of $\ket{\psi}$ to $\ket{\phi}$
with the help of a catalyst state $\ket{\eta}$, such that the transformation
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$ has non-
vanishing probability. On the other hand, a transformation is possible via
multicopy SLOCC if there exists a $k$, such that $\ket{\psi}^{\otimes k}$ can
be converted into $\ket{\phi}^{\otimes k}$ with non-vanishing probability. As
specific examples, consider the following transformations [42]:
i)
$\ket{\psi}_{ABC}\rightarrow\ket{\phi^{+}_{2}}_{AB}\otimes\ket{\phi^{+}_{2}}_{BC}\otimes\ket{\phi^{+}_{2}}_{CA}$,
where $\ket{\psi}_{ABC}$ is any generalised GHZ-type state of tensor rank 6.
ii) $\ket{GHZ_{n}^{n-1}}\rightarrow\ket{W_{n}}$, where
$\ket{GHZ_{n}^{d}}=\sum_{i=0}^{d-1}\ket{i}^{\otimes n}/\sqrt{d}$ and
$\ket{W_{n}}$ is $n$-partite generalisation of
$\ket{W_{3}}=(\ket{001}+\ket{010}+\ket{100})/\sqrt{3}$.
Both the above state transformations are not possible even stochastically, but
can be accomplished via both catalytic SLOCC and multicopy SLOCC. Furthermore,
it has been shown that if $\ket{\psi}$ can be converted to $\ket{\phi}$ via
multicopy SLOCC then the same can be realised with catalytic SLOCC [42].

Recently Neven  _et al._ [145] found a majorization condition for a class of
GHZ-like states, which completely characterises LOCC transformations. For
this, suppose $g$ and $h$ are two invertible, complex and diagonal matrices
satisfy $\operatorname{Tr}(g^{\dagger}g)=\operatorname{Tr}(h^{\dagger}h)$.
Then the transformation

$\openone\otimes\cdots\otimes\openone\otimes
g\ket{\mathrm{GHZ}_{n}^{d}}\rightarrow\openone\otimes\cdots\otimes\openone\otimes
h\ket{\mathrm{GHZ}_{n}^{d}}$ (53)

is possible by LOCC if and only if

$\left(|g_{1}|^{2},\ldots,|g_{d}|^{2}\right)^{T}\prec\left(|h_{1}|^{2},\ldots,|h_{d}|^{2}\right)^{T}$
(54)

holds. Here, $\left(g_{1},\ldots,g_{d}\right)$ and
$\left(h_{1},\ldots,h_{d}\right)$ are diagonal elements of the matrices $g$
and $h$ respectively. Note that $g$ and $h$ correspond to the GHZ-like states
of the form $\sum_{i=0}^{d-1}\sqrt{a_{i}}\ket{i}^{\otimes n}$. As a direct
consequence of this we see that $\ket{\psi}$ and $\ket{\phi}$ characterised by

$\displaystyle g$
$\displaystyle=\mathrm{diag}(\sqrt{0.45},\sqrt{0.35},\sqrt{0.12},\sqrt{0.08}),$
(55) $\displaystyle h$
$\displaystyle=\mathrm{diag}(\sqrt{0.56},\sqrt{0.21},\sqrt{0.17},\sqrt{0.06}),$
(56)

respectively, are incomparable as they fail to satisfy Eq. (54). However, the
transformation can be accomplished with another GHZ-like state $\ket{\varphi}$
characterised by

$f=\mathrm{diag}(\sqrt{0.63},\sqrt{0.27},\sqrt{0.07},\sqrt{0.03}),$ (57)

that acts as a catalyst in the process. In fact, the transformation
$\ket{\psi}\otimes\ket{\varphi}\rightarrow\ket{\phi}\otimes\ket{\varphi}$ is
equivalent to the following transformation:

$\openone\otimes\cdots\otimes\openone\otimes(g\otimes
f)\ket{\mathrm{GHZ}_{n}^{16}}\rightarrow\openone\otimes\cdots\otimes\openone\otimes(h\otimes
f)\ket{\mathrm{GHZ}_{n}^{16}}.$ (58)

This completes our discussion of exact entanglement catalysis, and we will
consider exact catalysis of coherence in the following.

###  III.2 Quantum coherence

Recent studies suggest that quantum coherence might be another useful resource
for many information processing tasks that are otherwise impossible without it
[167]. There are two different notions of coherence in the literature:
_unspeakable_ and _speakable_ [136]. In the case of unspeakable coherence, the
levels of eigenspaces are important for the quantification of coherence, for
example, energy. Whereas for the latter one, it is irrelevant which
eigenspaces are responsible for the coherence. While the former type of
coherence is characterized using the resource theory of asymmetry [133], the
latter is characterized using the usual notion of resource theory of coherence
introduced by Baumgratz  _et al._ [14]. In the following, we will discuss
catalysis in both cases.

Quantum catalysis in the resource theory of coherence was introduced in [1].
In this paper, the author studied the question of how to use a catalyst to
create superposition between different energy levels under energy preserving
unitary operations. As energy must remain conserved, a system in a definite
energy level cannot be changed to a superposition of different energy levels.
However, an additional system or reservoir with high degree of coherence can
help to achieve this. Due to this, the amount of coherence in the reservoir
can decrease. In contrast, Åberg [1] showed that the coherence in the
reservoir can be used to turn the state of the system into a superposition of
energy levels without consuming it in the process. Hence, coherence in this
reservoir acts like a catalyst in the process. The advantage has been shown by
constructing two examples: the doubly-infinite ladder and the half-infinite
ladder [1]. Furthermore, as a direct consequence it has been shown that this
catalytic procedure has application to work extraction [1]. Note that in
Åberg’s protocol the state of the reservoir does not remain the same.
Nonetheless, the argument for catalysis is that the coherence resource in the
reservoir does not degrade in the process and can be reused on another system
to apply the same transformation.

The change in the reservoir’s state is unusual to the definition of catalysis.
Hence, an alternative description to [1] has been provided by Korzekwa  _et
al._ [114], who preferred the term “repeatability” over catalysis as the state
of the reservoir changes. They studied work extraction from the coherence of
the given system while the reservoir can be used repeatedly. The amount of
work that can be extracted from a system in the state $\rho$ is equal to the
difference of free energy $\Delta F(\rho)$ when the allowed operations
conserve the energy only on average [164]. Here $\Delta
F(\rho)=F(\rho)-F(\gamma)$ with the Helmholtz free energy

$F(\rho)=\operatorname{Tr}[\rho H]-kTS(\rho),$ (59)

where $H$ and $\gamma$ correspond to the system Hamiltonian and thermal state
respectively, and $S(\rho)=-\operatorname{Tr}[\rho\log\rho]$ is the von
Neumann entropy.111If not stated otherwise, we define the von Neumann entropy
with the natural logarithm. However, in [114] the authors are interested in
the work extraction only from the coherence of the system, and in that case,

$F(\rho)-F(D[\rho])=kTS(\rho||D[\rho])$ (60)

is the amount of free energy available for the extraction from the coherence
of the state $\rho$, where $D$ is a dephasing operation in the energy
eigenbasis and $S(\rho||D[\rho])$ represents a measure of coherence [80] with
the quantum relative entropy
$S(\rho||\sigma)=\operatorname{Tr}[\rho\log\rho]-\operatorname{Tr}[\rho\log\sigma]$.
Korzekwa  _et al._ [114] showed that the amount of work extracted from the
coherence of the state $\rho$ is arbitrarily close to the free energy
extraction limit, i.e., $kTS(\rho||D[\rho])$. However, all the coherence
cannot be converted to work as some of the work will be needed to repump the
reservoir such that it can be reused again. Nevertheless, they showed that the
amount of extracted work can be arbitrarily close to the coherence provided
when the coherence of the reference system is sufficiently large.

Recently Vaccaro  _et al._ [178] analyzed the proposal of Åberg [1] to check
whether coherence can be really used repeatedly without any degradation in its
performance. In particular, Vaccaro  _et al._ [178] showed that if the
reservoir is used repeatedly then the final systems have correlations which
has not been considered by Åberg [1]. Correlations between the final systems
actually affect the efficiency of the repeated operations. To be precise, the
coherence drawn from the reservoir by uncorrelated systems is completely
different compared to the coherence drawn by correlated systems. Furthermore,
they showed that neglecting these correlations among final systems could lead
to unphysical phenomena, for example, two non-orthogonal reservoir states can
be discriminated almost perfectly [178].

In the context of whether coherence is catalytic an important study was done
by Lostaglio and Müller [129]. In fact it has been shown that coherence can
neither be cloned nor broadcasted in finite dimensional systems. The coherence
of a finite-dimensional system $\rho^{E}$ can be broadcasted if there exist a
time-translation covariant channel $\Lambda$ and an incoherent state
$\rho^{S}$, such that $[\rho^{S},H^{S}]=0$ with

$\Lambda(\rho^{E}\otimes\rho^{S})=\sigma^{ES},$ (61)

where $\rho^{E}=\operatorname{Tr}_{S}(\sigma^{ES})$,
$\sigma^{S}=\operatorname{Tr}_{E}(\sigma^{ES})$ and $[\sigma^{S},H^{S}]\neq
0$. The action of a time-translation covariant channel on a state $\rho$ with
Hamiltonian $H$ is expressed as
$\Lambda(\rho)=\operatorname{Tr}_{A}[U(\rho\otimes\sigma^{A})U^{\dagger}]$,
where $\sigma^{A}$ is some ancillary state with Hamiltonian $H_{A}$ and $U$
represents some global unitary with $[U,H+H_{A}]=0$. On the other hand, the
coherence of the system $\rho^{S}$ can be cloned when there is no correlation
between the sub-systems, i.e.,

$\Lambda(\rho^{E}\otimes\rho^{S})=\sigma^{ES}=\rho^{E}\otimes\sigma^{S}.$ (62)

Here, the system $E$ acts like a catalyst. In such a scenario, correlated
catalyst under covariant operations cannot broadcast or clone coherence in an
initially incoherent state $\rho^{S}$ [129]. However, the question of
coherence cloning and broadcasting remain open for infinite dimensional
systems. Note that the results of Åberg [1] suggest that a weaker version of
coherence broadcasting, where the state $\rho^{E}$ is allowed to change in the
procedure, is possible for infinite dimensional systems. Hence it seems like
broadcasting as defined earlier in Eq. (61), where the state of the $\rho^{E}$
remains unchanged may not be possible.

In the above result we see that broadcasting of coherence to an initially
incoherent state is not possible from a finite dimensional system [129]. On
the contrary, Åberg’s result suggests that it could be possible with infinite-
dimensional systems [1]. Recently Ding  _et al._ [58] tried to create a
connection between the two results by studying the role of a finite-
dimensional catalyst in enlarging the set of accessible states under time-
translational covariant operations. In their protocol they allow correlation
between the system and the catalyst after the transformation, however the
reduced state of the catalyst remains unchanged. As correlation is allowed,
they first showed that pure state catalysts do not help in any state
transformations under covariant operations. Precisely, if $\rho$ cannot be
transformed into $\rho^{\prime}$ under covariant operations, then there exists
no pure catalyst state $\ket{\phi}$ that can catalyse the transformation
$\rho\otimes\phi\rightarrow\rho^{\prime}\otimes\phi$ under covariant catalytic
transformations. This result is a generalisation to the previously studied
result by Marvian and Spekkens [133], where $\rho$ and $\rho^{\prime}$ are
considered to be pure. Now if one can allow correlation between the system and
the catalyst then there exist coherent states such that the achievable set of
states from it is strictly larger under covariant operation when a correlated
catalyst has been used in the process. This has been shown explicitly for
qubit states [58]. Furthermore, with the dimension of the catalyst the
accessible set of states increases [58].

In the same spirit as Nielsen’s theorem in entanglement theory (see Theorem
1), a counterpart has been derived in coherence theory to address the
conversions among pure states [61, 190, 201]. Before presenting the result, we
introduce incoherent states and incoherent operations. To introduce these, we
first fix a basis $\\{\ket{i}\\}$ (called as incoherent basis) in a finite-
dimensional Hilbert space. The incoherent states are diagonal in the
incoherent basis and can be represented as $\sum_{i}p_{i}\ket{i}\\!\bra{i}$,
where $p_{i}\geq 0$ and $\sum_{i}{p_{i}}=1$. The set of all such incoherent
states are represented by $\mathcal{I}$. A quantum operation characterised by
a set of Kraus operators $\\{K_{n}\\}$ satisfying
$\sum_{n}K_{n}^{\dagger}K_{n}=\openone$, is called incoherent operation if it
satisfy $K_{n}\rho K_{n}^{\dagger}/\operatorname{Tr}\left(K_{n}\rho
K_{n}^{\dagger}\right)\in\mathcal{I}$ for all $n$ and $\rho\in\mathcal{I}$
[14]. Let us consider a pair of states:
$\ket{\psi}=\sum_{i=0}^{d-1}\alpha_{i}\ket{i}$ and
$\ket{\phi}=\sum_{i=0}^{d-1}\beta_{i}\ket{i}$, where $\alpha_{i}$ and
$\beta_{i}$ are arranged in the following way
$|\alpha_{0}|^{2}\geq\cdots\geq|\alpha_{d-1}|^{2}$ and
$|\beta_{0}|^{2}\geq\cdots\geq|\beta_{d-1}|^{2}$. Then $\ket{\psi}$ can be
converted to $\ket{\phi}$ via incoherent operations iff
$\Delta[\psi]\prec\Delta[\phi]$, or equivalently the following holds [61, 190,
201]

$\sum_{i=0}^{k}|\alpha_{i}|^{2}\leq\sum_{i=0}^{k}|\beta_{i}|^{2}\,\,\,\,\forall
k=\\{0,\ldots,d-1\\},$ (63)

where $\psi=\ket{\psi}\\!\bra{\psi}$ and
$\Delta[\rho]=\sum_{i}\braket{i}{\rho}{i}\ket{i}\\!\bra{i}$ represents
complete dehpasing in the incoherent basis. Note that the above result also
holds true for other classes of operations considered in coherence theory
[190].

Following similar ideas as for entanglement, it is not difficult to find
examples of incomparable states where the above conditions are not satisfied.
However, in some cases an additional coherent state can help in accomplishing
the forbidden transformation without being changed in the process [61]. The
above result in Eq. (63) sheds some light on the problem of coherence
catalysis, similar to the previously discussed cases for entanglement [105].
In fact, maximally coherent states cannot catalyze a forbidden transformation
between two arbitrary incomparable pure states [61]. Furthermore, if a
forbidden transformation $\ket{\psi}\nrightarrow\ket{\phi}$ can be
accomplished with the help of a pure catalyst state, then the reverse
transformation $\ket{\phi}\nrightarrow\ket{\psi}$ cannot be catalyzed with a
pure catalyst state [61]. Hence, pure catalysts can only catalyse
transformations between incomparable pure states. Another important
consequence of Eq. (63) is that conversion of $\ket{\psi}$ into $\ket{\phi}$
is possible with the help of a pure catalyst if $|\alpha_{0}|\leq|\beta_{0}|$
and $|\alpha_{d-1}|\geq|\beta_{d-1}|$ [61].

Bu  _et al._ [35] studied the necessary and sufficient conditions for
catalytic coherence transformations considering pure catalyst states. They
derived necessary and sufficient conditions for the existence of a catalyst
for a pair of incomparable pure states. Suppose $\ket{\psi}$ and $\ket{\phi}$
are two incomparable states of dimension $d$. Then there exists a catalyst
state $\ket{\eta}$ accomplishing the transformation
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$, if and
only if [35]

$\displaystyle H_{\alpha}(\Delta[\psi])$
$\displaystyle>H_{\alpha}(\Delta[\phi])\,\,\,\forall\alpha\in(-\infty,\infty)/0\,\,\,\mbox{and}$
(64) $\displaystyle\operatorname{Tr}(\log\Delta[\psi])$
$\displaystyle>\operatorname{Tr}(\log\Delta[\phi]).$ (65)

Here

$H_{\alpha}(\rho)=\frac{\mbox{sgn}(\alpha)}{1-\alpha}\log\operatorname{Tr}[\rho^{\alpha}]$
(66)

represents Rényi entropy. The strict inequalities in the above equations can
be made less strict by allowing an infinitesimally small error in the initial
or in the final state. In fact, for arbitrary $\varepsilon>0$, there exists a
state $\ket{\psi_{\varepsilon}}$ with
$||\psi-\psi_{\varepsilon}||_{1}\leq\varepsilon$ such that
$\ket{\psi_{\varepsilon}}$ can be converted into $\ket{\phi}$ via catalytic
incoherent operations (assuming pure catalyst states), if and only if [35]

$\displaystyle H_{\alpha}(\Delta[\psi])$ $\displaystyle\geq
H_{\alpha}(\Delta[\phi])\,\,\,\forall\alpha\in(-\infty,\infty)/0\,\,\,\mbox{and}$
(67) $\displaystyle\operatorname{Tr}(\log\Delta[\psi])$
$\displaystyle\geq\operatorname{Tr}(\log\Delta[\phi]).$ (68)

Note the analogy between this result and the case of entanglement as discussed
in Eq. (18). Furthermore, Bu  _et al._ [35] explored the structure of qubit
catalyst states for four dimensional state transformations. The necessary and
sufficient conditions for the existence of a qubit catalyst
$\ket{\eta}=\sqrt{a}\ket{0}+\sqrt{1-a}\ket{1}$ for two four dimensional
incomparable states $\ket{\psi}=\sum_{i=0}^{3}\sqrt{\alpha_{i}}\ket{i}$ and
$\ket{\phi}=\sum_{i=0}^{3}\sqrt{\beta_{i}}\ket{i}$ are

$\displaystyle\alpha_{0}$ $\displaystyle\leq\beta_{0},$ (69)
$\displaystyle\alpha_{0}+\alpha_{1}$ $\displaystyle>\beta_{0}+\beta_{1},$ (70)
$\displaystyle\alpha_{0}+\alpha_{1}+\alpha_{2}$
$\displaystyle\leq\beta_{0}+\beta_{1}+\beta_{2},$ (71)

and additionally

$\displaystyle\max\left\\{\frac{\alpha_{0}+\alpha_{1}-\beta_{0}}{\beta_{1}+\beta_{2}},1-\frac{\alpha_{3}-\beta_{3}}{\beta_{2}-\alpha_{2}}\right\\}$
$\displaystyle\leq
a\leq\min\left\\{\frac{\beta_{0}}{\alpha_{0}+\alpha_{1}},\frac{\beta_{0}-\alpha_{0}}{\alpha_{1}-\beta_{1}},1-\frac{\beta_{3}}{\alpha_{2}+\alpha_{3}}\right\\}.$
(72)

Here, without loss of generality we assume that $\\{\alpha_{i}\\}$ and
$\\{\beta_{i}\\}$ are real and they are arranged in descending order. Hence,
Eq. (III.2) provides a range of the catalyst for the above transformation.
Similar to the result in entanglement theory [105], catalytic conversions
(assuming pure catalysts) of incomparable pure states are not possible for two
and three dimensional systems, and the first nontrivial result can be found in
dimension four as shown in Eq. (III.2), see [35].

So far we have discussed deterministic state transformations. If a pair of
states cannot be transformed deterministically, i.e, they violate Eq. (63),
they still can be transformed stochastically with some optimal probability
$P_{\max}$ [62]. Bu  _et al._ [35] studied the advantage of a catalyst in
stochastic state transformations. In fact, they showed that for two
incomparable pure states $\ket{\psi}=\sum_{i=0}^{d-1}\alpha_{i}\ket{i}$ and
$\ket{\phi}=\sum_{i=0}^{d-1}\beta_{i}\ket{i}$, there exists a catalyst state
$\ket{\eta}$ such that
$P_{\max}(\ket{\psi}\rightarrow\ket{\phi})<P_{\max}(\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta})$,
if and only if

$P_{\max}(\ket{\psi}\rightarrow\ket{\phi})<\min\left\\{\frac{|\alpha_{d-1}|^{2}}{|\beta_{d-1}|^{2}},1\right\\}.$
(73)

Note that a similar result is also present in entanglement theory [105]. By
analogy to self-catalysis in entanglement [67], in [35] the authors also
explored self-catalysis in the context of coherence theory. As an example,
consider the states

$\displaystyle\ket{\psi}$
$\displaystyle=\sqrt{0.9}\ket{0}+\sqrt{0.081}\ket{1}+\sqrt{0.01}\ket{2}+\sqrt{0.009}\ket{3},$
(74) $\displaystyle\ket{\phi}$
$\displaystyle=\sqrt{0.95}\ket{0}+\sqrt{0.03}\ket{1}+\sqrt{0.02}\ket{2}.$ (75)

It is easy to check that $\ket{\psi}\nrightarrow\ket{\phi}$, while at the same
time $\ket{\psi}\otimes\ket{\psi}\rightarrow\ket{\phi}\otimes\ket{\psi}$ [35].

Earlier, we discussed the advantage of catalysis-assisted probabilistic
transformation of pure states. Naturally, one may wonder whether a catalyst
exists such that it can enhance the probability of transforming a mixed state
to a pure state. Recently, Liu and Zhou [126] explored this scenario for
strictly incoherent operations and derived necessary and sufficient conditions
for the existence of a pure catalyst such that the optimal transformation
probability is enhanced. A quantum operation characterised by Kraus operators
$\\{K_{n}\\}$ is called strictly incoherent operation if all $K_{n}$ and
$K_{n}^{\dagger}$ are incoherent in the given incoherent basis [190, 198]. For
such operations Liu and Zhou [126] showed that
$P_{\max}(\rho\otimes\eta\rightarrow\psi\otimes\eta)>P_{\max}(\rho\rightarrow\psi)$
is possible, if and only if the following holds at least for one $\mu$:

$P_{\max}(\phi^{\mu}\rightarrow\psi)<\min\left\\{\frac{\phi^{\mu}_{n}}{\psi_{n}},1\right\\},$
(76)

where $\\{\ket{\phi^{\mu}}\\}$ correspond to the maximally dimensional pure
coherent state subspaces of $\rho$,
$\ket{\phi^{\mu}}=\sum_{i=0}^{d_{1}-1}\phi^{\mu}_{i}\ket{i}$,
$\ket{\psi}=\sum_{i=0}^{d_{2}-1}\psi_{i}\ket{i}$,
$n=\max\\{d_{1}-1,d_{2}-1\\}$, and $d_{1}$ and $d_{2}$ denote the dimensions
of $\ket{\phi}^{\mu}$ and $\ket{\psi}$ respectively. The coefficients of the
pure states are arranged in descending order. A state $\rho$ has a pure
coherent state subspace of dimension $n$ if there exists an incoherent
projector $\mathbb{P}$ such that
$\frac{\mathbb{P}_{\mu}\rho\mathbb{P}_{\mu}}{\mbox{Tr}\mathbb{P}_{\mu}\rho\mathbb{P}_{\mu}}=\phi^{\mu}$
and $\phi^{\mu}$ has coherence rank $n$ [126]. Coherence rank corresponds to
the number of nonzero terms $\phi_{i}^{\mu}$ in $\ket{\phi}^{\mu}$. The
subspace is maximally coherent when the rank of $\phi^{\mu}$ cannot be
increased.

Catalytic coherence transformation of some classes of mixed states has been
studied by Xing and Yang [195]. Consider the following rank-2 mixed states of
the form

$\displaystyle\rho=\alpha\ket{\psi}\\!\bra{\psi}+(1-\alpha)\ket{4}\\!\bra{4}\,\,\mbox{and}$
(77)
$\displaystyle\sigma=\beta\ket{\phi}\\!\bra{\phi}+(1-\beta)\ket{4}\\!\bra{4},$
(78)

where

$\displaystyle\ket{\psi}$
$\displaystyle=\sqrt{0.38}\ket{0}+\sqrt{0.38}\ket{1}+\sqrt{0.095}\ket{2}+\sqrt{0.095}\ket{3}$
$\displaystyle+\sqrt{0.05}\ket{4},$ (79) $\displaystyle\ket{\phi}$
$\displaystyle=\sqrt{0.5}\ket{0}+\sqrt{0.25}\ket{1}+\sqrt{0.25}\ket{2}.$ (80)

The transformation of $\rho$ into $\sigma$ is not possible via incoherent
transformation [195]. However, as shown in [195], the transformation can be
achieved with the help of a catalyst in the state
$\ket{\varphi}=\sqrt{0.6}\ket{0}+\sqrt{0.4}\ket{1}$. Note that the above
states and the catalytic conversion protocol is similar to the entanglement
case discussed in [69]. The above transformation of rank-2 mixed states can be
translated to some special class of higher rank states as well. In addition,
for general mixed states catalytic coherence transformation is also discussed
in [195].

Xing and Yang [196] studied _supercatalytic_ coherence transformation. Suppose
we have two incomparable states $\ket{\psi}$ and $\ket{\phi}$, and we want to
achieve the transformation $\ket{\psi}\rightarrow\ket{\phi}$. We already see
that such transformations can be realized with the help of some auxiliary
state if Eqs. (64) and (65) are satisfied. In contrast to this, the
transformation can be realised in another way where the auxiliary state
changes in the process, but the coherence of the final auxiliary system is
increased. More precisely,
$\ket{\psi}\otimes\ket{\varphi_{1}}\rightarrow\ket{\phi}\otimes\ket{\varphi_{2}}$,
where $\ket{\varphi_{1}}$ and $\ket{\varphi_{2}}$ represent auxiliary states
with $C_{r}(\varphi_{2})\geq C_{r}(\varphi_{1})$. Here $C_{r}$ represents the
relative entropy of coherence [14]. Examples and some interesting results of
such supercatalytic coherence transformation are discussed in [196].

###  III.3 Quantum thermodynamics

Catalysis has also been broadly explored in quantum thermodynamics,
investigating transformations of quantum systems under thermal operations in
the presence of catalysts. Thermal operations model transformations of a
quantum system $S$ with Hamiltonian $H_{S}$ interacting with a heat reservoir
$B$ with Hamiltonian $H_{B}$ and in a thermal state $\gamma^{B}=e^{-\beta
H_{B}}/Z_{B}$, where $\beta=1/kT$ is the inverse temperature and
$Z_{B}=\operatorname{Tr}e^{-\beta H_{B}}$ is the partition function. The
interaction is described by a unitary $U_{SB}$ that conserves the total
energy: $[U_{SB},H_{S}+H_{B}]=0$. A thermal operation $\Lambda$ on a state
$\rho^{S}$ is defined as [104]

$\Lambda(\rho^{S})=\operatorname{Tr}_{B}\left(U_{SB}\left[\rho^{S}\otimes\gamma^{B}\right]U_{SB}^{\dagger}\right).$
(81)

See also [100] for a detailed discussion on thermal operations.

To describe catalytic thermal operations, Brandão  _et al._ [29] allow for a
additional system $\sigma^{C}$ as a catalyst. Precisely, a transformation from
$\rho^{S}$ to $\mu^{S}$ is possible via catalytic thermal operations, if and
only if there exist a thermal operation $\Lambda$ and a state $\sigma^{C}$
taking $\rho\otimes\sigma$ to $\mu\otimes\sigma$ [29], i.e.,

$\Lambda(\rho^{S}\otimes\sigma^{C})=\operatorname{Tr}_{B}\left(U_{SCB}\left[\rho^{S}\otimes\sigma^{C}\otimes\gamma^{B}\right]U_{SCB}^{\dagger}\right)=\mu^{S}\otimes\sigma^{C},$
(82)

where the unitary $U_{SCB}$ satisfies $[U_{SCB},H_{S}+H_{C}+H_{B}]=0$ and
$H_{C}$ represents the Hamiltonian of the catalyst system $C$. Moreover, the
authors find necessary and sufficient conditions for such transformation to be
possible when the initial and the final states are block-diagonal in energy
eigenbasis [29].

######  Theorem 3 (See family of second laws in [29]).

There exists a catalytic thermal operation mapping a diagonal state $\rho$ to
another diagonal state $\rho^{\prime}$, if and only if

$F_{\alpha}(\rho,\gamma_{\beta})\geq
F_{\alpha}(\rho^{\prime},\gamma_{\beta})\,\,\textrm{for}\,\,\alpha\geq 0.$
(83)

Here, the generalized free energies are defined as

$F_{\alpha}(\rho,\gamma_{\beta})=kTS_{\alpha}(\rho||\gamma_{\beta})-kT\log Z,$
(84)

where

$S_{\alpha}(\rho||\sigma)=\begin{cases}\frac{1}{\alpha-1}\log\operatorname{Tr}\left(\rho^{\alpha}\sigma^{1-\alpha}\right),\,\,\alpha\in[0,1)\\\
\frac{1}{\alpha-1}\log\operatorname{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right],\,\,\alpha>1\end{cases}$
(85)

is the Rényi relative entropy and
$S_{1}(\rho||\sigma)=\operatorname{Tr}[\rho(\log\rho-\log\sigma)]$ is found by
taking the limit $\alpha\rightarrow 1$. Also note that, for
$\lim_{\alpha\rightarrow 1}F_{\alpha}(\rho,\gamma_{\beta})$ we obtain the
ordinary Helmholtz free energy, see also Eq. (59). The family of inequalities
in Eq. (83) is known as the _second laws of quantum thermodynamics_ [29]. In
contrast, note that at the macroscopic scale or in the standard thermodynamic
limit, transitions are governed by free energy only, and its monotonicity
provides the standard second law. Furthermore, Brandão  _et al._ [29]
discussed a generalisation of the above result by considering transformations
of arbitrary states and provided some necessary conditions. A state $\rho$ can
be transformed into $\rho^{\prime}$ with arbitrary accuracy only if [29]

$F^{\prime}_{\alpha}(\rho,\gamma_{\beta})\geq
F^{\prime}_{\alpha}(\rho,\gamma_{\beta})\,\,\forall\alpha\geq\frac{1}{2},$
(86a) $F^{\prime}_{\alpha}(\gamma_{\beta},\rho)\geq
F^{\prime}_{\alpha}(\gamma_{\beta},\rho^{\prime})\,\,\mathrm{for}\,\,\frac{1}{2}\leq\alpha\leq
1,\,\,\mbox{and},$ (86b) $F^{\prime\prime}_{\alpha}(\rho,\gamma_{\beta})\geq
F^{\prime\prime}_{\alpha}(\rho^{\prime},\gamma_{\beta})\,\,\mathrm{for}\,\,0\leq\alpha\leq
2,$ (86c)

where $F^{\prime}$ and $F^{\prime\prime}$ are defined as

$\displaystyle F^{\prime}(\rho,\gamma_{\beta})$
$\displaystyle=\frac{kT}{\alpha-1}\log\operatorname{Tr}\left[\left(\gamma_{\beta}^{\frac{1-\alpha}{2\alpha}}\rho\gamma_{\beta}^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]-kT\log
Z,$ (87) $\displaystyle F^{\prime\prime}(\rho,\gamma_{\beta})$
$\displaystyle=\frac{kT\mathrm{sgn}(\alpha)}{\alpha-1}\log\operatorname{Tr}\left(\rho^{\alpha}\gamma_{\beta}^{1-\alpha}\right)-kT\log
Z.$ (88)

van der Meer  _et al._ [138] studied how the above mentioned free energies
behave in the asymptotic limit for block-diagonal states. As a result of their
study, they find out that if $F(\rho)>F(\rho^{\prime})$, then for any
$\varepsilon>0$ there exits a sufficiently large $n$ and a catalytic thermal
operation $\Lambda$, taking $\rho^{\otimes n}$ arbitrarily close to
$\rho^{\prime\otimes n}$, i.e., $||\Lambda(\rho^{\otimes
n})-\rho^{\prime\otimes n}||_{1}\leq\varepsilon$.

In contrast to the previous result by Brandão  _et al._ [29], Lostaglio  _et
al._ [130] showed that only the monotonicity of free energy is required for
state transformations if one allow correlations among the particles of the
catalyst and an arbitrary small error in the final state $\rho^{\prime}$. In
their definition of catalytic state transformation, a block-diagonal state
$\rho$ is transformed to another block-diagonal state $\rho^{\prime}$ when for
all $\varepsilon>0$, there exist a thermal operation $\Lambda$, a state
$\rho_{\varepsilon}^{\prime}$ with
$||\rho^{\prime}-\rho_{\varepsilon}^{\prime}||<\varepsilon$ and a correlated
state $\sigma_{C_{1}\cdots C_{N}}$ with marginals
$\sigma_{C_{1}},\cdots,\sigma_{C_{N}}$ such that [130]

$\displaystyle\Lambda(\rho\otimes\sigma_{C_{1}}\otimes\cdots\otimes\sigma_{C_{N}})=\rho_{\varepsilon}^{\prime}\otimes\sigma_{C_{1}\cdots
C_{N}}.$ (89)

Such a transformation is possible, if and only if $F(\rho)\geq
F(\rho^{\prime})$ [130]. However, note that although the local states of the
particles remain the same, the global state of the catalyst changes in this
procedure. This may lead to embezzling, a phenomenon discussed in more detail
in Section VII.

The above result emphasises that if a transformation is possible in the
thermodynamic limit then it is also possible in the single copy settings.
Hence, it provides an operational meaning to the free energy. Motivated by
this result, Müller and Pastena [140] derived a necessary and sufficient
condition for the transformations of probability distributions. For this, they
studied transformations of probability distributions under bistochastic maps,
which are represented by bistochastic matrices that are composed of non-
negative real numbers with every individual row and column summing up to one.
In particular, Müller and Pastena [140] considered transformations of
probability distributions of the form

$p\otimes r_{1}\otimes\cdots\otimes r_{N}\rightarrow q\otimes r_{1,\cdots,N},$
(90)

where the probability distribution $p=(p_{1},\cdots,p_{m})$ is mapped to
another probability distribution $q=(q_{1},\cdots,q_{m})$ with the help of
auxiliary systems that get correlated in the procedure, but at the same time,
the marginals remain unchanged. In such a scenario, if $p$ and $q$ are not
identical up to permutation and both contain non-zero elements, then the
transformation from $p$ to $q$ is possible if and only if $H(p)<H(q)$ [140],
where $H(p)=\sum_{i}p_{i}\log p_{i}$ represents the Shannon entropy.

In most of the previous studies [29, 138, 130], only block-diagonal states are
considered, which contain no coherence in the eigenbasis of the Hamiltonian.
Although the conditions in Eqs. (86) hold for arbitrary states, the role of
coherence in state transformations is not apparent from them. On the contrary,
in [128] the authors considered arbitrary states and derived necessary
conditions for state transformation under catalytic thermal operation. They
considered exact catalytic thermal operation presented in Eq. (82). If a state
$\rho$ can be converted to another state $\rho^{\prime}$ via a catalytic
thermal operation where the catalyst state is in the block-diagonal form, then
we have [128]

$A_{\alpha}(\rho)\geq A_{\alpha}(\rho^{\prime})\,\,\forall\,\alpha\geq 0,$
(91)

where $A_{\alpha}(\rho)=S_{\alpha}(\rho||D[\rho])$ represents free coherence
of the state $\rho$ [128] and $D$ corresponds to the dephasing operation in
the given energy eigenbasis.

In [189], the authors considered marginal-catalytic transformation and
correlated-catalytic transformation under Gibbs preserving operations, and
showed that the free energy is the only monotonic function under such free
operations. In their definition, the marginal-catalytic transformation has a
similar structure as presented in Eq. (89), with the only difference that the
final state is returned exactly and the maps are Gibbs preserving [189]. On
the other hand, in the case of correlated-catalytic transformation of
$\rho^{S}$ into $\mu^{S}$, we have
$\rho^{S}\otimes\sigma^{C}\rightarrow\mu^{SC}$, where the final state
$\mu^{SC}$ is correlated with marginals $\mu^{S}$ and $\sigma^{C}$ [189]. Then
it has been shown that any monotone under these two kinds of catalytic
transformations, which is additive on tensor product and continuous, is
proportional to the free energy difference [189]. Hence, free energy is the
only monotone for such kind of catalytic transformations.

###  III.4 Purity

In the resource theory of purity, a key question is to find the minimum amount
of noise required to perform a given physical process on a quantum system [81,
25]. In this theory, the free operations are noisy operations. In general, a
noisy operation can be implemented in the following way: i) first a given
system is attached with an ancillary system prepared in a completely mixed
state, ii) then a unitary operation is performed to couple them, and iii)
finally we discard the ancillary system by taking partial trace over the
ancillary system. The overall transformation on a state $\rho^{S}$ in the
Hilbert space $\mathcal{H}_{S}$ can be written as

$\Lambda[\rho^{S}]=\operatorname{Tr}_{A}\left[U\left(\rho^{S}\otimes\frac{\openone_{A}}{d_{A}}\right)U^{\dagger}\right],$
(92)

where $U$ is in $\mathcal{H}_{S}\otimes\mathcal{H}_{A}$, $\mathcal{H}_{A}$
represents the Hilbert space of the ancillary system, and $d_{A}$ corresponds
to the dimension of the ancillary system. Note that this kind of operation was
first reported in [94, 99]. All the states except the maximally mixed state
are considered resource in this theory. Furthermore, note that without loss of
generality, we can assume the states to be diagonal matrices as unitary
operations are free in this resource theory. The necessary and sufficient
conditions for exact state transformations are given by the majorization
relation. More precisely, a transformation from $\rho$ to $\sigma$ is possible
under noisy operation if and only if $\sigma\prec\rho$ [81]. In contrast to
the entanglement theory, the majorization relation is in the opposite
direction.

An immediate consequence is that there exist transformations which are not
possible via noisy operation but could be accomplished with the help of a
catalyst [81]. To analyse state transformations under catalytic noisy
operation one needs to consider a trumping relation $\sigma\prec_{T}\rho$,
which states that there exists a catalyst $\eta$ that helps in converting
$\rho$ into $\sigma$ via noisy operations, i.e.,
$\sigma\otimes\eta\prec\rho\otimes\eta$ [81]. The noisy trumping relation is
equivalent to [109, 81]

$f_{\alpha}(\rho)>f_{\alpha}(\sigma)\,\,\forall\,\,\alpha\in(-\infty,\infty),$
(93)

where the functions $f_{\alpha}$ are the extensions of the functions in Eq.
(17) to density matrices, i.e.,

$\displaystyle
f_{\alpha}(\rho)=\begin{cases}\ln\operatorname{Tr}[\rho^{\alpha}]&(\alpha>1);\\\
\operatorname{Tr}[\rho\ln\rho]&(\alpha=1);\\\
-\ln\operatorname{Tr}[\rho^{\alpha}]&(0<\alpha<1);\\\
-\operatorname{Tr}[\ln\rho]&(\alpha=0);\\\
\ln[\operatorname{Tr}\rho^{\alpha}]&(\alpha<0).\end{cases}$ (94)

Similar to the case of entanglement and coherence, an alternative definition
of noisy trumping is introduced [81], where an infinitesimally small error is
allowed in the initial or final state. More formally a state $\rho$ trumps
noisily $\sigma$, if there exists a state $\eta$ such that
$\rho\otimes\nu_{\delta}\otimes\eta\rightarrow\sigma\otimes\eta$ for all
$\delta>0$. Here $\nu_{\delta}=\left(1/m,\cdots,1/m,0,\cdots,0\right)$ is a
sharp state of dimension $n$ with $m\leq n$ and satisfies
$||\nu_{\delta}-\openone_{n}/n||_{1}\leq\delta$. With this, Gour  _et al._
[81] provided necessary and sufficient conditions for noisy trumping as
following: A state $\rho$ noisy trumps $\sigma$, if and only if

$H_{\alpha}(\rho)\leq H_{\alpha}(\sigma)\,\,\forall\,\,\alpha\geq 0,$ (95)

where $H_{\alpha}$ represent the Rényi entropies defined earlier in Eq. (66).
Furthermore, they also explored approximate catalysis where a state $\rho$ is
transformed into $\sigma$ approximately via noisy operations and a catalyst
$\eta$. However, no further restriction is imposed on the catalyst except that
the final state is $\varepsilon$ close to $\sigma\otimes\eta$. Therefore, the
catalyst might change infinitesimally in the procedure. The slight change in
the catalyst leads to embezzling, which is discussed in more detail in Section
VII. In such a scenario, for a given $\varepsilon>0$ the noisy operations are
trivial, i.e., every state can be converted to every other state [81, 29]. We
also refer to Section IV for a more detailed analysis of approximate
catalysis.

Previously, we discussed the necessary and sufficient conditions for state
transformations under catalytic noisy operations. Now, one may wonder what is
the minimum amount of randomness required to implement state transitions via
noisy operations and is there any difference between classical and quantum
randomness. Before answering these questions, understanding of classical and
quantum randomness is important. The nature of the source of randomness
defines whether it is quantum or classical. A transformation from $\rho$ to
$\sigma$ using $m$ units of _classical randomness_ is possible if there exists
a set of unitaries $\\{U_{i}\\}$ such that [25]

$\sigma=\frac{1}{m}\sum_{i=1}^{m}U_{i}\rho U^{\dagger}_{i}$ (96)

and the transformation is denoted by $\rho\xrightarrow[m]{C}\sigma$. On the
other hand, a transformation from $\rho$ to $\sigma$ using $m$ units of
_quantum randomness_ is possible if there exists a unitary $U$ and an
ancillary system $A$ of dimension $m$ in the maximally mixed state
$\openone/m$ such that [25]

$\sigma=\operatorname{Tr}_{A}U\left(\rho\otimes\frac{\openone}{m}\right)U^{\dagger},$
(97)

and the transformation is denoted by $\rho\xrightarrow[m]{Q}\sigma$. When
there is no bound on the amount of available randomness (quantified by $m$),
the set of transitions that can be achieved with quantum randomness sources
can also be achieved with classical randomness sources [132, 25]. Equivalently
we have,

$\rho\xrightarrow[\infty]{C}\sigma\iff\rho\xrightarrow[\infty]{Q}\sigma\iff\rho\succ\sigma.$
(98)

The transformations of the form (97) in the limit $m\rightarrow\infty$ have
been previously studied in [81], where noise is treated as free or in other
words we can use a maximally mixed state of arbitrary dimension to implement a
noisy operation. On the contrary, Boes  _et al._ [25] treated noise as a
resource and investigated the minimum amount of noise required to accomplish a
given state transformation. For a given Hilbert space $\mathcal{H}_{S}$ of
dimension $d$, the minimum amount of noise or randomness can be quantified as

$m_{X}^{*}(d)=\mbox{arg\,min}_{m}\left[\rho\xrightarrow[m]{X}\sigma\,\,\forall\,\,\rho,\sigma\in\mathcal{H_{S}}\,\,\mbox{with}\,\,\rho\succ\sigma\right],$
(99)

where $X$ corresponds to either $C$ or $Q$. It is important to note that for a
specific transition the required randomness could be smaller than the above
specified value in Eq. (99). Boes  _et al._ [25] provide an analytical
expression for these quantities as following

$\displaystyle m_{C}^{*}(d)$ $\displaystyle=d,$ (100) $\displaystyle
m_{Q}^{*}(d)$ $\displaystyle=\lceil\sqrt{d}\rceil.$ (101)

Here, we will focus mainly on the quantum randomness. In proving the
optimality of $m_{Q}^{*}$, the authors of [25], use the Schur-Horn lemma [93]
with the majorisation relation $\rho\succ\sigma$, which states that for every
state transition under noisy operation there exists a map of the form

$\sigma=\Lambda(\rho)=\mathcal{U}^{\prime}\circ\Delta_{B}\circ\mathcal{U}(\rho)$
(102)

where, $\mathcal{U}$, $\mathcal{U^{\prime}}$ are unitary channels that are
specific to the states $\rho$ and $\sigma$. Here $\Delta_{B}$ represents a
dephasing map in a fixed basis $B=\\{\ket{i}\\}_{i=1}^{d}$ of dimension $d$
and defined as $\bra{i}\Delta_{B}(\rho)\ket{j}=\bra{i}\rho\ket{j}\delta_{ij}$.
As the implementations of unitary channels do not require any noise, all the
noise needed for a state transition is required to implement the dephasing map
$\Delta_{B}$ in Eq. (102). Next Boes  _et al._ [25] provide an explicit
protocol implementing dephasing map $\Delta_{B}$ in any basis $B$, using a
noise model and requiring $m_{Q}^{*}(d)=\lceil\sqrt{d}\rceil$. Specifically,
for any dimension $d$ and basis $B$, Boes  _et al._ [25] showed that there
exists a unitary $U$, such that

$\displaystyle\operatorname{Tr}_{A}\left[U\left(\rho\otimes\frac{\openone}{\lceil\sqrt{d}\rceil}\right)U^{\dagger}\right]=\Delta_{B}(\rho),$
(103)
$\displaystyle\operatorname{Tr}_{S}\left[U\left(\rho\otimes\frac{\openone}{\lceil\sqrt{d}\rceil}\right)U^{\dagger}\right]=\frac{\openone}{\lceil\sqrt{d}\rceil}\,\,\forall\,\,\rho.$
(104)

Although the final state may have correlations between the system and the
ancilla, the second condition implies that the ancillary system $A$ is
returned exactly in the same state as before and thus acts as a catalyst in
the process. As mentioned previously, Boes  _et al._ [25] considered noise as
a resource, and interestingly, here the noise is not consumed in the process
and can be reused again. Furthermore, note that Eq. (102) emphasises that
every possible state transition utilising randomness can be accomplished if a
dephasing map can be constructed and Boes  _et al._ [25] provided an explicit
protocol for that.

In [25], only maximally mixed states has been used as a source of randomness.
In contrast Lie  _et al._ [119] considered any randomness source $\eta$ and
derived a necessary and sufficient condition such that the randomness source
$\eta$ can be used to dephase an arbitrary input state $\rho$. In fact, it has
bee shown that any randomness source $\eta$ can dephase a $d^{2}$ dimensional
input states catalytically iff the following holds [119]

$S_{\min}(\eta)\geq\log_{2}d,$ (105)

where $S_{\min}(\eta)=-\log_{2}||\eta||$ represents min-entropy and the
operator norm $||\eta||$ corresponds to the largest singular value of $\eta$.
However, note that in their formulation, the state of the catalyst $\eta$ can
change, but the same catalyst can be used repeatedly to dephase arbitrary
input states of the same dimension. In validation of this, the authors of
[119] showed that randomness described by min-entropy is not consumed in the
process and can be reused again.

Based on the result in [25], Boes  _et al._ [26] proved that von Neumann
entropy completely characterises single-shot state transitions under unitary
operations. To achieve this, the authors in [26] used an ancillary system as a
catalyst and an environment which dephases a system in a preferred basis. In
particular, the following has been shown [26].

######  Theorem 4 (See Theorem 1 in [26]).

For any two states $\rho$ and $\sigma$ of the same finite dimension and
different spectra, the following statements are equivalent
(i) $S(\sigma)>S(\rho)$ and $rank(\sigma)\geq rank(\rho)$
(ii) There exists a unitary $U$ and a finite dimensional catalyst in the state
$\eta$, which can be chosen to be diagonal in any basis $B$ such that

$\displaystyle\operatorname{Tr}_{A}U[\rho\otimes\eta]U^{\dagger}=\sigma,$
(106)
$\displaystyle\Delta_{B}(\operatorname{Tr}_{S}U[\rho\otimes\eta]U^{\dagger})=\eta.$
(107)

Note that the result corresponding to Eqs. (103) and (104) is a special case
of the above theorem. Naturally one may wonder whether the dephasing map given
in Eq. (107), is at all necessary to implement all state transitions such that
the von Neumann entropy does not decrease. In support of this, Boes  _et al._
[26] provided the following _catalytic entropy conjecture_.

######  Conjecture 1 (See Conjecture 1 in [26]).

For any two states $\rho$ and $\sigma$ of the same finite dimension and
different spectra, the following statements are equivalent:
(i) $S(\sigma)>S(\rho)$ and $\mathrm{rank}(\sigma)\geq\mathrm{rank}(\rho)$,
(ii) There exists a finite dimensional state $\eta$ and a unitary $U$ such
that,

$\displaystyle\operatorname{Tr}_{A}U[\rho\otimes\eta]U^{\dagger}=\sigma,$
(108) $\displaystyle\operatorname{Tr}_{S}U[\rho\otimes\eta]U^{\dagger}=\eta.$
(109)

This conjecture has been proven very recently by Wilming [188]. In fact,
Wilming [188] showed the following: Let $\rho$ and $\rho^{\prime}$ be two
states of dimension $d$ in $\mathcal{H}_{d}$ that are not unitarily
equivalent. Then there exist a natural number $n$ and a state
$\rho_{n}^{\prime}$ on $\mathcal{H}_{d}^{\otimes n}$ such that

$\displaystyle\rho^{\otimes
n}\succ\rho_{n}^{\prime}\,\,\mbox{and}\,\,\operatorname{Tr}_{\\{1,\cdots,n\\}\setminus
i}\rho_{n}^{\prime}=\rho^{\prime}\,\,\forall i\in\\{1,\cdots,n\\}$ (110)

holds true if and only if $S(\rho)<S(\rho^{\prime})$ and
$\mbox{rank}(\rho)\leq\mbox{rank}(\rho^{\prime})$. The validity of the
catalytic entropy conjecture is a consequence of this result [188]

Recently, in [121, 120], randomness to implement quantum channels has been
explored. In particular, Lie and Jeong [121] are using an ancillary system as
a source of randomness such that when this ancillary system is used to
implement a quantum channel no information is leaked in it. They named this a
_randomness-utilising process_ , as no information is leaked into ancillary
system while implementing a quantum channel. As a consequence, the ancillary
system after the implementation of the process should not depend on the input
state of the channel. A randomness-utilising channel on
$\mathcal{B}(\mathcal{H}_{S})$ can be expressed as [121]

$\Lambda(\rho)=\operatorname{Tr}_{A}\left[U\left(\rho\otimes\sigma\right)U^{\dagger}\right],$
(111)

with a randomness source $\sigma$ on $\mathcal{H}_{A}$, some unitary $U$ on
$\mathcal{H}_{S}\otimes\mathcal{H}_{A}$, and the state
$\operatorname{Tr}_{S}[U(\rho\otimes\sigma)U^{\dagger}]$ is independent of the
input state. Lie and Jeong [121] called this state of the ancillary system
after the implementation of the randomness-utilising process _residue
randomness_. If the residue randomness has the same spectrum as $\sigma$, the
process is called catalytic. Hence, the randomness source can be reused to
implement the same process on another input state. Equipped with this Lie and
Jeong [121] showed that any dimension-preserving randomness-utilising process
is catalytic. By “dimension-preserving” the authors of [121] meant that the
input and output systems are of the same dimension. Naturally one may wonder
whether all dimension-preserving channels can be implemented by randomness-
utilising process. It turns out that only unital quantum channels can be
implemented using random-utilising process [121]. A unital quantum channel
$\Lambda_{U}$ keeps the identity operator invariant, i.e.,
$\Lambda_{U}(\openone)=\openone$. However, whether all unital quantum channels
are catalytic, or in other words, all unital quantum channels can be
implemented using randomness-utilizing processes, remains open in [121]. A
negative answer to this problem is given in [120] by showing that not all
unital quantum channels are catalytic. Furthermore, Lie and Jeong [120]
defined catalytic entropy to quantify the maximum amount of randomness that
can be extracted catalytically from a randomness source $\sigma$. Catalytic
entropy of a randomness source $\sigma$ with eigenspace decomposition
$\sigma=\sum_{i}\lambda_{i}P_{i}$ is given in [120] by

$S_{C}(\sigma):=-\sum_{i}\lambda_{i}r_{i}\log_{2}\frac{\lambda_{i}}{r_{i}}.$
(112)

Here, $P_{i}$ are the orthogonal projectors associated with the eigenvalue
$\lambda_{i}$ and satisfy $P_{i}P_{j}=0$ if $\lambda_{i}\neq\lambda_{j}$, and
$r_{i}=\operatorname{Tr}P_{i}$ represents degeneracy of $\lambda_{i}$.
$S_{C}(\sigma)$ is the maximum amount of entropy or randomness that can be
produced from $\sigma$ [120]. In another recent work, the authors extended the
role of catalytic quantum randomness for dynamical maps [122].

Henao and Uzdin [90] explored the role of catalysts in reducing the impact of
information erasure on the environment. In any information erasing process, an
initial state $\rho_{s}$ of a system $s$ with von Neumann entropy
$S(\rho_{s})$ is taken to another state $\rho_{s}^{\prime}$ that has less
entropy than the initial state $\rho_{s}$, i.e., $\Delta
S_{s}=S(\rho_{s})-S(\rho_{s}^{\prime})>0$, and the change in entropy $-\Delta
S_{s}$ is dumped as heat in the environment. For a finite-dimensional
environment, the authors in [90] showed that the correlation between the
system and the environment can be maneuvered catalytically to reduce the
entropy or heat waste in the environment. To set the scene, we introduce an
environment $e$ in the state $\rho_{e}$ that is attached with the state
$\rho_{s}$ to reduce the entropy of the system $s$. By applying a global
unitary transformation $U_{se}$ on the system and the environment, the
uncorrelated state $\rho_{s}\otimes\rho_{e}$ is transformed to a final state
$\sigma_{se}=U(\rho_{s}\otimes\rho_{e})U^{\dagger}$, such that the entropy of
the system decreases, i.e., $S(\sigma_{s})-S(\rho_{s})<0$, where
$\sigma_{s}=\operatorname{Tr}_{e}\sigma_{se}$. As a result the heat dumped in
the environment is $Q_{e}=\operatorname{Tr}[H_{e}(\sigma_{e}-\rho_{e})]$,
where $H_{e}$ represents the Hamiltonian associated with the environment and
$\sigma_{e}=\operatorname{Tr}_{s}\sigma_{se}$. Now with the help of a catalyst
$c$ in the state $\rho_{c}$, Henao and Uzdin [90] aimed to reduce the heat
dissipation in the environment to a value $Q_{e}^{\prime}$ such that
$Q_{e}^{\prime}<Q_{e}$. This leads to a two-step process where
$\rho_{s}\otimes\rho_{e}$ is transformed to $\sigma_{se}$ and then
$\sigma_{se}$ is transformed to $\rho_{sec}^{\prime}$ with the help of a
suitable catalyst state $\rho_{c}$ and a global unitary $U_{sec}$ such that
$Q_{e}^{\prime}=\operatorname{Tr}[H_{e}(\rho_{e}^{\prime}-\rho_{e})]<Q_{e}$.
Here,
$\rho_{sec}^{\prime}=U_{sec}(\sigma_{se}\otimes\rho_{c})U_{sec}^{\dagger}$ and
$\rho_{e}^{\prime}=\operatorname{Tr}_{sc}\rho_{sec}^{\prime}$. Apart from
ensuring
$\rho_{s}^{\prime}=\operatorname{Tr}_{ec}\rho_{sec}^{\prime}=\sigma_{s}$, we
need to make sure that the transformation is catalytic, i.e.,
$\operatorname{Tr}_{se}\rho_{sec}^{\prime}=\rho_{c}$. With this, Henao and
Uzdin [90] proved the following: There exist a unitary transformation
$U_{sev}$ and a catalyst state $\rho_{c}$ such that
$\rho_{e}^{\prime}\succ\sigma_{s}$, if and only if $\sigma_{se}$ is a
correlated state of the system $s$ and the environment $e$. Furthermore, they
showed that if $\rho_{e}^{\prime}$ majorises $\sigma_{s}$, then $Q_{e}\geq
Q_{e}^{\prime}$ [90]. Note that as unitary is free in this process, the
catalyst is the only party responsible for the reduction in heat dissipation
to the environment. Furthermore, they showed $S(\rho_{e}^{\prime})\leq
S(\sigma_{e})$ which reflects that the entropy change of the environment can
also be reduced. Therefore, in any information erasure process involving a
correlated state $\sigma_{se}$ it is possible to reduce the heat and entropy
dissipation to the environment by using a suitable catalyst state.

###  III.5 Other quantum resource theories

Symmetry arguments play a crucial part to understand many important results in
physics. For example, in many cases when the evolution of a system under some
dynamics gets complicated to analyse, one can look for symmetry arguments to
derive some useful results. In quantum theory, Marvian and Spekkens [133]
explored the role that symmetry plays in shaping the dynamics of a quantum
system. More precisely, they explored state transformations in the context of
time evolution with a given symmetry. To analyse this feature systematically,
they introduced the resource theory of asymmetry in [133]. Suppose a group $G$
with elements $g$ represents a set of symmetry transformations. The action of
each group element $g$ is described by a unitary transformation $U_{g}$. In
this resource theory, the free states are those states that remain invariant
under the action of all $g$, i.e., $\mathcal{U}_{g}(\rho)=\rho$ $\forall g\in
G$, where $\mathcal{U}_{g}(\rho)=U_{g}(\rho)U_{g}^{\dagger}$. In addition, a
quantum operation $\Lambda$ is free if it respects the symmetry, i.e.,

$\Lambda[U_{g}^{i}\rho(U_{g}^{i})^{\dagger}]=U_{g}^{o}\Lambda[\rho](U_{g}^{o})^{\dagger}\,\,\,\,\,\,\,\forall
g\in G,$ (113)

where $U_{g}^{i}$ and $U_{g}^{o}$ represent the input and output
representation of $G$ respectively. This type of operations are named as
$G$-covariant operations in the literature [133].

In such a resource theory, Marvian and Spekkens [133] explored the role of
catalysis for pure states. If a state $\ket{\psi}$ cannot be transformed into
$\ket{\phi}$ under any deterministic $G$-covariant operation but the same
transformation can be achieved deterministically with a $G$-covariant
operation and a catalyst in a state $\ket{\eta}$ such that
$\ket{\phi}\otimes\ket{\eta}\rightarrow\ket{\psi}\otimes\ket{\eta}$, then such
a transformation from $\ket{\psi}$ to $\ket{\phi}$ has been called a
nontrivial example of catalysis [133]. With this Marvian and Spekkens [133]
showed that no nontrivial example of catalysis is possible using a finite-
dimensional catalyst when the symmetries are associated with compact connected
Lie groups. However, this does not work for finite groups and a nontrivial
example of catalysis has been shown for finite groups [133].

Recently Rethinasamy and Wilde [153] studied the role of catalysis in relative
majorisation for classical probability distributions. A pair
$(p^{\prime},q^{\prime})$ of probability distributions $p^{\prime}$ and
$q^{\prime}$ is relatively majorised by another pair $(p,q)$ of probability
distributions $p$ and $q$, i.e., $(p,q)\succ(p^{\prime},q^{\prime})$ if and
only if there exists a classical channel or bistochastic matrix that transform
$p$ and $q$ to $p^{\prime}$ and $q^{\prime}$ respectively [51, 152, 38]. In
[153], the authors reformulated the above mentioned transformation protocol in
the presence of a catalyst pair $(r,s)$ of probability distributions $r$ and
$s$. Now the input pair $(p,q)$ is used jointly with the catalyst pair $(r,s)$
to achieve the desired output pair $(p^{\prime},q^{\prime})$ by applying a
classical channel $\Lambda$ on the joint pair $(p\otimes r,q\otimes r)$.
Although, they allowed correlation on $\Lambda(p\otimes r)=t^{\prime}$, but
restrict that the first and second marginals of $t^{\prime}$ should be $p$ and
$r$ respectively. Furthermore, for the second one they impose no correlation,
i.e., $\Lambda(q\otimes s)=q^{\prime}\otimes s$. As the catalyst is returned
exactly it can be reused again to transform another independent set of
$(p,q)$. With this the authors in [153] showed that relative entropy
completely determines exact catalytic pair transformations. Suppose
$p,q,p^{\prime},q^{\prime}$ are probability distributions on the same
alphabet, with $(p,q)$ and $(p^{\prime},q^{\prime})$ have different relative
spectra, and $q,q^{\prime}$ have only rational entries and full support. Then
the following two statements are equivalent: i)
$D(p||q)>D(p^{\prime}||q^{\prime})$ and $D_{0}(p||q)\geq
D_{0}(p^{\prime}||q^{\prime})$. ii) For all $\delta>0$, there exist a
probability distribution $r$, and a joint probability distribution
$t^{\prime}$ with marginals $p^{\prime}$ and $r$ such that $(p\otimes
r,q\otimes\nu)\succ(t^{\prime},q^{\prime}\otimes\nu)$, and
$D(t^{\prime}||p^{\prime}\otimes r)<\delta$ [153]. Here
$D(p||q)=\sum_{x}p(x)\ln(p(x)/q(x))$, $D_{0}(p||q)=-\ln\sum_{x:p(x)\neq
0}q(x)$, and $\nu$ is a uniform distribution that is used in place of $s$. The
relative spectrum of a pair $(p,q)$ is defined as $\\{p(x)/q(x)\\}_{x}$ [153].
Therefore, relative entropy completely characterises exact transformations
under correlated catalyst.

Note that the correlation can be made arbitrary small such that $t^{\prime}$
is almost the same as $p^{\prime}\otimes r$. The constraints on different
relative spectra, $q,q^{\prime}$ are rational, and $D_{0}(p||q)\geq
D_{0}(p^{\prime}||q^{\prime})$ can be lifted by allowing an infinitesimally
small error on the target probability distribution $p^{\prime}$ [153]. Suppose
$p,q,p^{\prime},q^{\prime}$ are probability distributions on the same alphabet
with $q,q^{\prime}$ have full support. Then the following two statements are
equivalent: i) $D(p||q)\geq D(p^{\prime}||q^{\prime})$. ii) For all $\delta>0$
and $\varepsilon>0$, there exist a probability distribution $r$, and a joint
probability distribution $t_{\varepsilon}^{\prime}$ with marginals
$p_{\varepsilon}^{\prime}$ and $r$, such that $(p\otimes r,q\otimes
r)\succ(t_{\varepsilon}^{\prime},q^{\prime}\otimes\nu)$,
$||p_{\varepsilon}^{\prime}-p^{\prime}||_{1}\leq 2\varepsilon$, and
$D(t_{\varepsilon}^{\prime}||p_{\varepsilon}^{\prime}\otimes r)<\delta$. Here,
$||p-q||_{1}=\sum_{i}|p_{i}-q_{i}|$. Considering $q$ and $q^{\prime}$ to be
the uniform distribution, a special case arises. Suppose $p,p^{\prime}$ are
probability distributions on the same alphabet with
$p\neq\mathrm{Per}(p^{\prime})$ for all permutation matrices $\mathrm{Per}$.
Then the following two statements are equivalent: i) $H(p)\geq H(p^{\prime})$.
ii) For all $\delta>0$ and $\varepsilon>0$, there exist a probability
distribution $r$, and a joint probability distribution
$t_{\varepsilon}^{\prime}$ with marginals $p_{\varepsilon}^{\prime}$ and $r$,
such that $p\otimes r\succ t_{\varepsilon}^{\prime}$,
$||p_{\varepsilon}^{\prime}-p^{\prime}||\leq 2\varepsilon$, and
$D(t_{\varepsilon}^{\prime}||p_{\varepsilon}^{\prime}\otimes r)<\delta$ [153].
Here $H$ represents the Shannon entropy of the probability distribution.
Therefore, a single entropic condition completely dictates approximate
catalytic relative majorisation of classical probabilities. Note that this
result can be applied to other quantum resource theories, for examples purity,
quasiclassical thermodynamics, and asymmetric distinguishability [153].

The role of catalysis has also been explored in the framework of Gaussian
thermal resource theory [197]. Let us consider an $n$-mode Bosonic continuous
variable quantum systems on the Hilbert space
$\mathcal{H}=\otimes_{k=1}^{n}\mathcal{H}_{k}$, with corresponding Bosonic
creation and annihilation field operators $\hat{a}_{k}$ and
$\hat{a}_{k}^{\dagger}$ respectively. From the field operators, the quadrature
operators (or position and momentum like operators) are constructed as
$\hat{x}_{k}=(\hat{a}_{k}+\hat{a}_{k}^{\dagger})/\sqrt{2}$ and
$\hat{p}_{k}=i(\hat{a}_{k}^{\dagger}-\hat{a}_{k})/\sqrt{2}$. For convenience
all the quadrature operators are grouped together to make a column vector
$\hat{\mathbb{r}}=(\hat{x}_{1},\hat{p}_{1},\cdots,\hat{x}_{n},\hat{p}_{n})^{T}$.
For a given quantum state $\rho$, the corresponding first and second moments
are defined as [184]

$\displaystyle\mathbb{r}=(\langle\hat{x}_{1}\rangle_{\rho},\langle\hat{p}_{1}\rangle_{\rho},\cdots,\langle\hat{x}_{n}\rangle_{\rho},\langle\hat{p}_{n}\rangle_{\rho})^{T}\,\,\mbox{and},$
(114)
$\displaystyle\sigma_{ij}=\frac{\langle\\{\hat{\mathbb{r}}_{i},\hat{\mathbb{r}}_{j}\\}\rangle_{\rho}}{2}-\langle\hat{\mathbb{r}}_{i}\rangle_{\rho}\langle\hat{\mathbb{r}}_{j}\rangle_{\rho}$
(115)

respectively, where the second moments $\sigma_{ij}$ are the elements of the
covariance matrix $\sigma$. Gaussian states are completely characterized by
its first moments and covariance matrix [184].

After defining Gaussian states, we introduce Gaussian operations which
preserve Gaussianity or in other words take Gaussian states to Gaussian
states. Generally, a Gaussian operation can be realised by allowing an
interaction between an environment in a Gaussian state and a given system
through a global Gaussian unitary, and then performing a projective
measurement on the environment [184]. In case of Gaussian thermal operations,
the global unitary and the state of the environment should be energy
preserving and in a thermal state respectively. As Gaussian states are
completely characterised by $\mathbb{r}$ and $\sigma$, so it is enough to
study the transformations of these two under Gaussian thermal operations. In
[197], the authors considered Gaussian states with $\mathbb{r}=0$ and studied
the transformation of $\sigma$ under strong and weak catalytic Gaussian
thermal operations. Note that the covariance matrix can be characterised by
two more matrices $M$ and $A$ with elements $M_{ij}=\langle
a_{j}^{\dagger}a_{i}\rangle-\delta_{ij}\nu$ and $A_{ij}=\langle
a_{j}a_{i}\rangle$ respectively [163] and this has been used to study state
transformations under Gaussian thermal operations [197, 159]. Here
$\nu=(\mathrm{e}^{\beta\omega}+1)/(\mathrm{e}^{\beta\omega}-1)$, $\omega$ is
the eigenfrequency of the modes and $\beta=1/kT$ is the inverse temperature. A
strong catalytic Gaussian transformation of $\rho$ to $\rho^{\prime}$ is
possible when there exists a catalyst state $\sigma$ such that
$\rho\otimes\sigma\rightarrow\rho^{\prime}\otimes\sigma$ is achieved under
Gaussian thermal operation. On the other hand, a weak catalytic Gaussian
transformation of $\rho$ to $\rho^{\prime}$ is possible when there exist a
catalyst state $\sigma$ and a joint final state $\mu^{SC}$ such that
$\rho\otimes\sigma\rightarrow\nu^{SC}$ with $\operatorname{Tr}_{S}\nu=\sigma$
and $\operatorname{Tr}_{C}\nu=\rho^{\prime}$ is accomplished under Gaussian
thermal operation. Equipped with these tools, the authors in [197] studied
transformations of single mode Bosonic Gaussian states. The transition from
$\sigma$ to $\sigma^{\prime}$ under strong catalytic Gaussian thermal
operation is possible if and only if there exists a $p\in[0,1]$ such that the
conditions

$\displaystyle\mu^{\prime}=p\mu\,\,\,\mbox{and}\,\,\,\alpha^{\prime}=p\alpha$
(116)

hold [197]. Here $\sigma$ and $\sigma^{\prime}$ are the covariance matrices of
a single mode Bosonic Gaussian state. For an $n$-mode Bosonic state, $\mu$ and
$\alpha$ correspond to the eigenvalues and singular values of the matrix $M$
and $A$ respectively. Note that the conditions in Eq. (116) are exactly the
same as the non-catalytic transition presented in [159]. Hence, strong
catalysis does not provide any advantage in this scenario. Now if we allow
correlations between the state and the catalyst then in the weak catalytic
scenario the above transition is possible if and only if there exist
$p,q\in[0,1]$ with $p\geq q$ such that the conditions

$\displaystyle\mu^{\prime}=p\mu\ \,\,\mbox{and}\,\,\alpha^{\prime}=q\alpha$
(117)

are satisfied [197]. Furthermore, in [197] multi-mode state transformations
have been discussed as well.

In a recent study [107], the role of catalysis has been explored in the
resource theory of contextuality [111, 36] and non-locality [15, 34]. Karvonen
[107] considered the following kind of catalytic transformation: $p\otimes
c\rightsquigarrow q\otimes c$, where $p$, $q$, and $c$ represent some
correlations, and $q$ can be simulated ($\rightsquigarrow$) from $p$ with the
help of a catalyst $c$ that remains intact in the process. Based on this,
Karvonen [107] proved that catalysis does not help in the resource theory of
contextuality and non-locality, or in other words, there exists no correlation
that can be used to catalyse an otherwise impossible transformation. More
precisely, if $p\otimes c\rightsquigarrow q\otimes c$ is possible in the
resource theory of contextuality and non-locality, then $p\rightsquigarrow q$
is also possible [107].

##  IV Approximate catalysis

The definition of exact catalysis, initially introduced for the resource
theory of entanglement [105], does not allow for any errors in the
transformation. In recent years, this notion has been extended, allowing for
an error in the initial or the final state [29, 130], at the same time
requiring that the catalyst is completely decoupled from the primary system in
the end of the process. In parallel, there has been a growing interest in
correlated catalysis, allowing the catalyst to build up correlations with the
system, which may also be preserved at the end of the procedure [1, 189, 139,
160, 161, 26, 187, 112].

Recently, the framework of _approximate catalysis_ has been proposed in [112,
54]. While this framework allows for correlations between the main system and
the catalyst, it additionally requires that these correlations can be made
arbitrarily small. In particular, an approximate catalytic transformation
between two states $\rho^{S}$ and $\sigma^{S}$ of a system $S$ is possible if
and only if for any $\varepsilon>0$ there exists a catalyst state $\tau^{C}$
and a free operation $\Lambda_{f}$ acting on the system and the catalyst such
that [112, 54]

$\displaystyle\left\|\Lambda_{f}\left(\rho^{S}\otimes\tau^{C}\right)-\sigma^{S}\otimes\tau^{C}\right\|_{1}$
$\displaystyle\leq\varepsilon,$ (118)
$\displaystyle\operatorname{Tr}_{S}\left[\Lambda_{f}\left(\rho^{S}\otimes\tau^{C}\right)\right]$
$\displaystyle=\tau^{C}.$ (119)

In contrast to exact catalysis discussed in Section III, the above definition
allows for correlations between the system and the catalyst in the final state
$\mu^{SC}=\Lambda_{f}(\rho^{S}\otimes\tau^{C})$. However, Eq. (118) ensures
that these correlations can be made arbitrarily small, by choosing an
appropriate state of the catalyst $\tau^{C}$ and free operation $\Lambda_{f}$.
The condition in Eq. (119) guarantees that the state of the catalyst does not
change in the procedure. This ensures that the catalyst can be reused for the
same transformation in the future. Moreover, this requirement makes it
impossible to “embezzle” quantum resources by using small changes of the
catalyst state, see also Section VII.

An equivalent formulation of approximate catalysis has been given by Rubboli
and Tomamichel [154]. While Eq. (118) requires that the correlations between
the system and the catalyst can be made arbitrarily small as quantified by the
trace norm, it is possible to formulate a similar requirement in terms of the
mutual information $I^{A:B}(\rho^{AB})=S(\rho^{A})+S(\rho^{B})-S(\rho^{AB})$.
Then, an approximate catalytic transformation from $\rho^{S}$ into
$\sigma^{S}$ is possible if and only if for any $\varepsilon>0$ and any
$\delta>0$ there exists a free operation $\Lambda_{f}$ and a catalyst state
$\tau^{C}$ such that [154]

$\displaystyle\left\|\operatorname{Tr}_{C}\left[\Lambda_{f}\left(\rho^{S}\otimes\tau^{C}\right)\right]-\sigma^{S}\right\|_{1}$
$\displaystyle\leq\varepsilon,$ (120)
$\displaystyle\operatorname{Tr}_{S}\left[\Lambda_{f}\left(\rho^{S}\otimes\tau^{C}\right)\right]$
$\displaystyle=\tau^{C},$ (121) $\displaystyle
I^{S:C}\left[\Lambda_{f}\left(\rho^{S}\otimes\tau^{C}\right)\right]$
$\displaystyle\leq\delta.$ (122)

Note that this formulation of approximate catalysis is equivalent to the
definition in Eqs. (118) and (119).

Note that the term “approximate catalysis” has been previously used by Gour
_et al._ [81] in the resource theory of purity, referring to transformations
which fulfill Eq. (118) for any $\varepsilon>0$. Since the condition in Eq.
(119) is not imposed, this type of transformations leads to embezzling
phenomena [81], see also Section VII. Throughout this article, whenever we
refer to approximate catalysis we mean transformations which obey Eqs. (118)
and (119).

In the following, we will review recent results on approximate catalysis for
concrete resource theories, including quantum thermodynamics, entanglement,
and quantum coherence.

###  IV.1 Quantum thermodynamics

Approximate catalysis has been first investigated in quantum thermodynamics by
Müller [139]. For the case that the initial and the final state are both block
diagonal in the energy eigenbasis, it has been shown that state
transformations are completely characterized by the Helmholtz free energy. In
particular, a block diagonal state $\rho$ can be converted into another block
diagonal state $\sigma$ via thermal operations with approximate catalysis if
and only if [139]

$F(\rho)\geq F(\sigma).$ (123)

Shiraishi and Sagawa [160, 161] have investigated a closely related setting,
replacing thermal operations by the more general class of Gibbs-preserving
operations. In this case, Eq. (123) completely characterizes transformations
between all quantum states, leading to the following theorem.

######  Theorem 5 (See Theorem 1 in [160]).

A state $\rho$ can be converted into another state $\sigma$ via Gibbs-
preserving operations with approximate catalysis if and only if $F(\rho)\geq
F(\sigma)$.

It is possible to express the Helmholtz free energy in terms of the quantum
relative entropy as follows:

$F(\rho)=kT[S(\rho||\gamma)-\log Z],$ (124)

where $Z$ is the partition function and $\gamma$ is the Gibbs state. Using Eq.
(124), we see that Eq. (123) is equivalent to

$S(\rho||\gamma)\geq S(\sigma||\gamma).$ (125)

The techniques developed by Shiraishi and Sagawa [160, 161] to prove the above
theorem turned out to be applicable to other resource theories, and we will
provide a sketch of the proof in the following. The major insight behind the
proof is a connection between asymptotic transformations and approximate
catalysis, which has been investigated earlier in [63]. Let us assume that
$\rho$ can be converted into $\sigma$ via Gibbs-preserving operations in the
asymptotic limit, i.e., for any $\delta>0$ there exists an integer $n$ and a
Gibbs-preserving operation $\Lambda$ acting on $n$ copies of $\rho$ such that

$\left\|\Lambda\left(\rho^{\otimes n}\right)-\sigma^{\otimes
n}\right\|_{1}\leq\delta.$ (126)

In the following $S_{i}$ will denote copies of the system $S$, and we further
define $\Gamma=\Lambda(\rho^{\otimes n})$, which is a quantum state on
$S_{1}\otimes S_{2}\otimes\cdots\otimes S_{n}$.

Shiraishi and Sagawa [160, 161] showed that Eq. (126) implies an approximate
catalytic transformation from $\rho$ into $\sigma$. The catalyst achieving
this transformation is given as

$\tau^{C}=\frac{1}{n}\sum_{k=1}^{n}\rho^{\otimes
k-1}\otimes\Gamma_{n-k}\otimes\ket{k}\\!\bra{k}.$ (127)

Here, $\Gamma_{0}=1$, and for $i>0$ the state $\Gamma_{i}$ is the reduced
state of $\Gamma$ on $S_{1}\otimes S_{2}\otimes\cdots\otimes S_{i}$. The
Hilbert space of the catalyst is in $S^{\otimes{n-1}}\otimes K$, where $K$
represents an ancillary system of dimension $n$ with a fully degenerated
Hamiltonian and basis states $\\{\ket{k}\\}_{k=1}^{n}$.

The Gibbs-preserving operation achieving the desired conversion consists of
the following three steps applied onto the total state
$\rho^{S}\otimes\tau^{C}$ [160, 161]:

  1. 1.

A controlled Gibbs-preserving operation is performed, with the ancillary
system $K$ as the control, and all other particles as the target. Conditioned
on the ancilla being in the state $\ket{n}$, the Gibbs-preserving operation
$\Lambda$ is applied onto $S_{1}\otimes S_{2}\otimes\cdots\otimes S_{n}$ such
that Eq. (126) is fulfilled.

  2. 2.

A unitary is applied onto the ancillary system that converts
$\ket{n}\xrightarrow{}\ket{1}$ and $\ket{i}\xrightarrow{}\ket{i+1}$.

  3. 3.

The systems are shifted as $S_{i}\xrightarrow{}S_{i+1}$ and
$S_{n}\xrightarrow{}S_{1}$.

The initial state of the system and the catalyst has the form

$\rho^{S}\otimes\tau^{C}=\frac{1}{n}\sum_{k=1}^{n}\rho^{\otimes
k}\otimes\Gamma_{n-k}\otimes\ket{k}\\!\bra{k}.$ (128)

After applying 1. step of the procedure, we obtain the following state:

$\mu_{1}=\frac{1}{n}\sum_{k=1}^{n-1}\rho^{\otimes
k}\otimes\Gamma_{n-k}\otimes\ket{k}\\!\bra{k}+\frac{1}{n}\Gamma\otimes\ket{n}\\!\bra{n}.$
(129)

In 2. step, $\mu_{1}$ is transformed into the state

$\mu_{2}=\frac{1}{n}\sum_{k=1}^{n}\rho^{\otimes
k-1}\otimes\Gamma_{n+1-k}\otimes\ket{k}\\!\bra{k}.$ (130)

Tracing out $S_{n}$ from $\mu_{2}$ gives the initial state of the catalyst
$\tau$, see Eq. (127). Therefore, the shift in 3. step transforms $\mu_{2}$ to
the final state $\mu^{SC}$ having the property
$\operatorname{Tr}_{S}[\mu^{SC}]$ = $\tau^{C}$, which implies that the
condition (119) for approximate catalysis is fulfilled.

Using the protocol presented above, Shiraishi and Sagawa [160] further showed
that the final system state $\mu^{S}$ can be made arbitrarily close to the
desired state $\sigma^{S}$, and also the mutual information between the system
and the catalyst in the total final state $\mu^{SC}$ can be made arbitrarily