# Catalytic and asymptotic equivalence for quantum entanglement

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

###### Abstract

Entanglement is a fundamental resource in quantum information processing, yet
understanding its manipulation and transformation remains a challenge. Many
tasks rely on highly entangled pure states, but obtaining such states is often
challenging due to the presence of noise. Typically, entanglement manipulation
procedures involving asymptotically many copies of a state are considered to
overcome this problem. These procedures allow for distilling highly entangled
pure states from noisy states, which enables a wide range of applications,
such as quantum teleportation and quantum cryptography. When it comes to
manipulating entangled quantum systems on a single copy level, using entangled
states as catalysts can significantly broaden the range of achievable
transformations. Similar to the concept of catalysis in chemistry, the
entangled catalyst is returned unchanged at the end of the state manipulation
procedure. Our results demonstrate that despite the apparent conceptual
differences between the asymptotic and catalytic settings, they are actually
strongly connected and fully equivalent for all distillable states. Our
methods rely on the analysis of many-copy entanglement manipulation procedures
which may establish correlations between different copies. As an important
consequence, we demonstrate that using an entangled catalyst cannot enhance
the asymptotic singlet distillation rate of a distillable quantum state. Our
findings provide a comprehensive understanding of the capabilities and
limitations of both catalytic and asymptotic state transformations of
entangled states, and highlight the importance of correlations in these
processes.

### Introduction

Entanglement is a key feature of quantum mechanics, and it plays a vital role
in many areas of quantum information science. Being a strong form of
correlations between quantum systems, entanglement enables a wide range of
applications and protocols that have the potential to revolutionize
information processing and communication [1, 2]. The study of entanglement and
its properties has led to significant advancements in our understanding of
quantum mechanics, and it has provided insights into how to manipulate and
harness its power for practical applications [3].

To understand the pivotal role of entanglement as a resource in quantum
information processing, we can consider the distant lab paradigm [4, 5]. This
scenario involves two parties, Alice and Bob, who are located in different
quantum laboratories and can exchange classical messages to communicate with
each other. In this setting, entangled states shared between Alice and Bob
become a valuable resource, allowing them to perform tasks that would
otherwise be impossible [3].

One of the most significant applications of entanglement is in the field of
quantum communication, including quantum teleportation [1] and quantum
cryptography [2]. These tasks typically rely on singlets, which are pure
highly entangled states of two qubits. However, in practice, Alice and Bob may
only have access to noisy states. In order to use noisy states for singlet-
based protocols, they can employ entanglement distillation [4, 6], which is a
special case of asymptotic state transformations. In this process, $n$ copies
of an initial state are transformed approximately into $rn$ copies of the
final state, where $r$ is the transformation rate. Quantum states which can be
distilled into singlets at a nonzero rate are called distillable. There exist
noisy entangled states which cannot be distilled into singlets, a phenomenon
known as bound entanglement [7].

Another way how Alice and Bob can gain access to singlets from noisy states is
to use entanglement catalysis. In this process, an auxiliary entangled state,
known as a catalyst, is employed to aid in the transformation of one entangled
state to another without altering the catalyst itself [8]. Recent work [9, 10,
11, 12] extended this idea to approximate catalysis, where the transformation
can be achieved with a certain degree of inaccuracy. This concept has proven
to be instrumental in advancing our understanding of catalytic entanglement
manipulation and its potential applications [13].

At first glance, catalytic and asymptotic transformations may seem like
distinct concepts, but recent research has uncovered a strong connection
between them. Initial evidence for a connection between these concepts was
presented in [14, 15], and subsequent work has made significant progress in
this direction, particularly through the use of approximate catalysis [9, 11].
Furthermore, it has been shown that in quantum thermodynamics, catalysis and
many-copy transformations with a unit rate are fully equivalent [16, 17].
Given the shared features between quantum entanglement and thermodynamics [18,
19, 20, 21], it is plausible that a similar equivalence may exist in
entanglement theory.

In this article, we resolve this question by considering catalytic and
asymptotic protocols which can establish a non-vanishing amount of
correlations. This provides a more flexible and practical approach for
studying catalysis and asymptotic transformations and their applications in
quantum information processing. In this setting, we prove that for distillable
states, catalysis and asymptotic transformations with unit rate are fully
equivalent notions of entangled state manipulation. We discuss several
applications of our results, including the crucial finding that the addition
of a catalyst cannot increase the distillable entanglement of a noisy
distillable state.

### Asymptotic entanglement manipulations and catalysis

As previously discussed, asymptotic transformations are a powerful tool for
understanding the structure and manipulation of quantum entanglement. For
instance, consider two bipartite pure states $\ket{\psi}$ and $\ket{\phi}$.
The objective is to use local operations and classical communication (LOCC) to
transform $n$ copies of $\ket{\psi}$ into $m$ copies of $\ket{\phi}$, allowing
for an error margin that vanishes in the limit of large $n$. The maximal ratio
$m/n$ defines the transformation rate. This framework is particularly useful
if the target is the singlet state
$\ket{\psi^{-}}=(\ket{01}-\ket{10})/\sqrt{2}$, in which case the optimal rate
is known as distillable entanglement [4, 6, 22]. It coincides with the
entanglement entropy $E(\ket{\psi})=S(\psi^{A})$ of the initial state, and
$S(\rho)=-\mathrm{Tr[\rho\log_{2}\rho]}$ is the von Neumann entropy [6]. In a
similar way, it is possible to define transformation rates for noisy states,
we refer to the Supplemental Material for more details. A state is called
asymptotically reducible onto another state if the transformation can be
achieved with a rate at least one [23]. This reflects the intuition that if
$\ket{\psi}$ is reducible onto $\ket{\phi}$, then $\ket{\psi}$ is at least as
valuable as $\ket{\phi}$ for any application that allows for asymptotic
transformations.

Entanglement catalysis is a phenomenon where an entangled catalyst is used to
facilitate the single-copy transformation of an entangled state into another
without changing the state of the catalyst [8, 24]. Given an entangled state
$\ket{\psi}$ and a target state $\ket{\phi}$, the aim is to find a catalytic
state $|\eta\rangle$ such that the transformation
$\ket{\psi}\otimes\ket{\eta}\rightarrow\ket{\phi}\otimes\ket{\eta}$ is
possible by LOCC. The catalyst is particularly useful if it enables the
transformation of $\ket{\psi}$ into $\ket{\phi}$ which is not possible without
the catalyst. Recently, the notion of catalysis has been extended to
approximate catalysis in [9, 10, 11, 12], which allows for some degree of
inaccuracy in the catalytic transformation. The notion of approximate
catalysis provides a more realistic model for practical implementations of
catalytic entanglement manipulation and enables a broader range of
applications [13]. It has been demonstrated in [9] that transformations
between bipartite pure states in this scenario are fully determined by the
entanglement entropy of the corresponding states. Therefore, for bipartite
pure states approximate catalysis is fully equivalent to reducibility.
Catalytic phenomena have been extensively studied not only in the context of
entanglement, but also in other areas of quantum physics, such as quantum
thermodynamics [25, 26, 16, 27, 17], where they are essential for
understanding and manipulating quantum systems subject to constraints imposed
by energy conservation.

As has been shown in [9], there is a close connection between asymptotic state
transformations and catalysis. More precisely, if a state $\rho$ is
asymptotically reducible to another state $\sigma$, then a transformation from
$\rho$ into $\sigma$ can also be achieved on the single-copy level with
approximate catalysis [9]. However, it has remained a crucial open question
whether the converse is also true, i.e., whether catalysis and asymptotic
reducibility are fully equivalent notions for entangled state transformations.
In this article, we introduce the frameworks of marginal asymptotic
transformations and correlated catalysis, which allows us to resolve this
question and establish the equivalence between catalysis and reducibility for
all distillable quantum states.

### Correlated catalysis and marginal reducibility

Figure 1:  Equivalence between catalysis and reducibility. The left part of
the figure shows a state $\rho$ being converted into another state $\sigma$
through a catalytic transformation by using a catalyst in the state $\tau$.
The right part of the figure shows that $\rho$ is asymptotically reducible
into $\sigma$. Our results demonstrate that the two processes are equivalent
for any pair of distillable states, assuming that both procedures can
establish correlations.

In the context of entanglement catalysis, an important generalization is to
consider _correlated catalysis_ , where the catalyst is allowed to have non-
vanishing correlations with the system throughout the transformation process.
This means that the system and the catalyst can remain correlated in the final
state. More precisely, we say that $\rho$ can be converted into $\sigma$ via
correlated catalysis if for any error margin $\varepsilon>0$ there is an LOCC
protocol $\Lambda$ and a catalyst state $\tau$ such that

$\displaystyle\mu^{SC}$ $\displaystyle=\Lambda(\rho^{S}\otimes\tau^{C}),$ (1)
$\displaystyle||\mu^{S}-\sigma^{S}||_{1}$
$\displaystyle<\varepsilon,\,\,\,\mu^{C}=\tau^{C}.$

Here, $S$ denotes a possibly multipartite system, $C$ denotes the catalyst,
and $||M||_{1}=\mathrm{Tr}\sqrt{M^{\dagger}M}$ is the trace norm. In other
words, the state $\mu^{SC}$ is obtained by applying an LOCC protocol $\Lambda$
to the state $\rho^{S}\otimes\tau^{C}$, such that the marginal on $C$ is
preserved and the resulting state on $S$ can be made arbitrarily close to the
target state $\sigma$. Previous studies in quantum thermodynamics have
explored the significance of correlations for catalytic state transformations,
revealing that the presence of correlations between the system and catalyst
can increase the transformation power of the procedure [28, 27, 17, 16].

We now introduce the notion of _marginal reducibility_. We say that $\rho$ can
be reduced into $\sigma$ in the marginals if for any arbitrarily small error
margin, there exists an LOCC protocol which can transform $n$ copies of $\rho$
into a state with approximately $m$ marginals, and each marginal being close
to the desired state $\sigma$. Specifically, we require for any
$\varepsilon,\delta>0$ there exists an LOCC protocol $\Lambda$ and integers
$m\leq n$ such that the following conditions hold for all $i\leq m$:

$\displaystyle\Lambda\left(\rho^{\otimes n}\right)$ $\displaystyle=\mu_{m},$
$\displaystyle\left\|\mu_{m}^{(i)}-\sigma\right\|_{1}$
$\displaystyle<\varepsilon,$ (2) $\displaystyle\frac{m}{n}+\delta$
$\displaystyle>1.$

Here, $\mu_{m}$ is a state on $m$ subsystems, each shared by Alice and Bob,
and $\mu_{m}^{(i)}$ is the reduced state of $\mu_{m}$ on $i$-th subsystem.
Marginal asymptotic transformations have been previously studied in continuous
variable systems in [29].

It is worth to discuss the difference between marginal reducibility and the
notion of reducibility introduced in [23]. The latter is more stringent as it
requires that the final state $\mu_{m}$ is close to $m$ copies of $\sigma$ as
a whole. However, for many quantum information processing tasks that rely on
pure states $\ket{\phi}$, such as singlets in the bipartite case or GHZ states
in the multipartite case, small perturbations of the state do not
significantly affect its usefulness. In other words, the state
$\mu_{\varepsilon}=(1-\varepsilon)\ket{\phi}\\!\bra{\phi}+\varepsilon\openone/d$
is also useful for small enough $\varepsilon$. For marginal reducibility, it
suffices that $\rho^{\otimes n}$ can be approximately converted into
$\mu_{\varepsilon}^{\otimes n}$ for any $\varepsilon>0$, which as we have
argued above is enough for many tasks based on pure states. Therefore, we
suggest that the framework of marginal reducibility is particularly suitable
when one aims to produce pure entangled states of high quality that are
intended to be used independently.

In the remainder of this article we will focus on the relationship between
correlated catalysis and marginal reducibility. Unless otherwise specified, we
will refer to these concepts simply as catalysis and reducibility,
respectively.

### Catalysis-reducibility equivalence

As previously noted, there have been indications that catalysis and
reducibility are interchangeable concepts for entangled state transformations
[14, 9, 11, 15]. The key contribution of this article is to establish this
equivalence for any pair of distillable states, using the notions of catalysis
and reducibility which include correlations, as outlined above in this
article. We recall that a distillable state is a quantum state which can be
converted into singlets at nonzero rate in the asymptotic limit.

######  Theorem 1.

For any pair of distillable states $\rho$ and $\sigma$ reducibility and
catalysis are fully equivalent.

We present a brief overview of the techniques employed for proving the
theorem, more details can be found in the Supplemental Material. Firstly, we
demonstrate that if the state $\rho$ can be reduced to the state $\sigma$,
then it is possible to achieve a catalytic transformation from $\rho$ to
$\sigma$, using techniques similar to those presented in prior works [16, 9,
10, 11]. Subsequently, we establish the converse by explicitly constructing a
reduction protocol that utilizes a catalytic conversion protocol from a
distillable state $\rho$ into $\sigma$. This involves several technical steps
that are described in detail in the Supplemental Material. By combining these
two results, we conclusively demonstrate the full equivalence of reducibility
and catalysis for any pair of distillable states $\rho$ and $\sigma$, see also
Fig. 1.

Since all entangled two-qubit states are distillable [30], Theorem 1 implies
that catalysis and reducibility are fully equivalent for all two-qubit states.
For states beyond two qubits, Theorem 1 also applies if the target state
$\sigma$ is not distillable. Moreover, catalysis is generally at least as
powerful as reducibility. With this in mind, Theorem 1 leaves open the
possibility that there exist bound entangled states $\rho$ that cannot be
reduced to some state $\sigma$, yet a catalytic conversion from $\rho$ to
$\sigma$ is possible. Thus, if catalysis and reducibility are not equally
powerful on all quantum states, catalysis must show an advantage on some bound
entangled initial states. This underscores the importance of investigating the
relationship between these concepts in the general case, as it can provide
insights into the nature of bound entanglement and the power of entanglement
catalysis. Additionally, our findings can have practical implications for
quantum information processing tasks where bound entangled states are known to
play a significant role [31, 32, 33].

Going one step further, we investigate the role of catalysis for asymptotic
transformation rates. Our findings reveal that the addition of a catalyst does
not alter the asymptotic rate of transformation from a distillable state
$\rho$ into another state $\sigma$, again under the assumption that
correlations can be established in the procedure. An important application of
this result pertains to the scenario where the target state is a singlet
$\ket{\psi^{-}}$. In this context, our analysis reveals that the correlations,
which are typically established in the catalytic and asymptotic procedures
considered earlier, vanish. This property allows us to explore the features of
distillable entanglement when a catalyst is incorporated into the
transformation, bringing us to the second main result of this article.

######  Theorem 2.

Catalysis cannot increase the distillable entanglement of a distillable state.

The proof of the theorem combines the previously mentioned results on
asymptotic transformation rates with the additional finding that correlations
usually established in the involved procedures disappear if the target state
is pure. We refer to the Supplemental Material for the proof and more details.
Recalling that all entangled two-qubit states can be distilled into singlets
[30], it follows that Theorem 2 applies to all two-qubit states. In general,
our results leave open the possibility that bound entangled states could be
activated into singlets through catalysis.

Figure 2: Distillation of multipartite entanglement. We call a multipartite
state distillable if it can be asymptotically converted into states comprising
a singlet between each of the parties. The figure shows the desired final
state for $3$ parties. The multipartite distillable entanglement is the
maximal transformation rate into states of this form. With these definitions,
Theorems 1 and 2 extend to the multipartite setting.

Our results have implications also beyond the scope of bipartite systems. It
is worth noting that Theorem 1 can be generalized to the multipartite
scenario. To this end, we consider multipartite distillable states, which are
those multipartite states that can be distilled into singlets between each
pair of parties with some nonzero rate in the asymptotic limit, see also Fig.
2. This includes all pure states which are entangled across any bipartition
[34, 35]. With this in mind, we can extend Theorem 1 to state that for any
pair of multipartite distillable states, reducibility and catalysis are fully
equivalent. Furthermore, Theorem 2 is also applicable to this scenario,
indicating that the addition of a catalyst cannot enhance the multipartite
distillable entanglement of any multipartite distillable state, we refer to
the Supplemental Material for more details. The results obtained in the
multipartite setting are in line with those in the bipartite setting and imply
that if catalysis offers any benefit over reducibility, it can only be
observed when the initial state is not distillable.

The limitation to distillable states in Theorem 1 can be overcome by allowing
the borrowing of a pure state, that is, considering transformations from
$\rho\otimes\psi$ to $\sigma\otimes\psi$ with some entangled pure state
$\ket{\psi}$. In this case, the state $\rho\otimes\psi$ is distillable,
leading to the equivalence of catalysis and reducibility for any $\rho$ and
$\sigma$. Interestingly, this applies even if the borrowed state $\ket{\psi}$
has arbitrarily little entanglement. Similarly, we can extend Theorem 2 to
state that catalysis cannot increase the distillable entanglement of the state
$\rho\otimes\psi$, where $\rho$ does not need to be distillable.

These findings offer a better understanding of the relationship between
entanglement catalysis and many-copy transformations, and can have practical
implications for the exploitation of entanglement in quantum information
processing tasks.

### Conclusions

In conclusion, our work establishes the complete equivalence between
reducibility and catalysis for any pair of distillable states, which extends
and confirms previous indications that these concepts are interchangeable for
entangled state transformations. Furthermore, we have demonstrated that the
addition of a catalyst does not alter the rate of asymptotic transformations
between distillable states.

Our results shed new light on the nature of entanglement catalysis and
entanglement-based protocols. The full equivalence between catalysis and
reducibility for distillable states provides a clearer understanding of the
limitations and capabilities of these tasks. We emphasize that our results
assume that correlations can be established in the transformation procedures
involved. This suggests that taking correlations into account can provide a
more complete and accurate understanding of quantum information processing
tasks which make use of catalysis. The methods developed in this article can
guide the design of new protocols, where catalysis and correlations play a
significant role. On the other hand, correlations disappear naturally for
transformations into pure target states. This allows us to conclude that the
addition of a catalyst cannot increase the asymptotic singlet distillation
rate, provided that the initial state has non-zero distillable entanglement to
begin with.

The manipulation of entanglement in the multipartite setting is a complex and
challenging problem [36], and further research in this direction is required
to fully understand and effectively utilize the power of multipartite
entanglement in quantum information processing. Our findings are particularly
relevant in this context, as they demonstrate the full equivalence of
catalysis and reducibility for transformations between multipartite
distillable states. These findings have significant implications for
understanding the role of catalysis in communication protocols that rely on
multipartite entangled states, such as quantum secret sharing [37, 38].

Furthermore, our work opens up new avenues for research into the relationship
between reducibility and catalysis in the general case, where the assumption
of distillability cannot be made. Investigating this relationship can help us
better understand the nature of bound entanglement and unlock the full
potential of entanglement and general quantum resources in quantum information
processing.

_Note added._ In an independent work [39] it has been shown with different
methods that by using entanglement catalysis it is not possible to distill
singlets from bound entangled states having positive partial transpose.

We thank Ludovico Lami for insightful comments on our manuscript. This work
was supported by the “Quantum Optical Technologies” project, carried out
within the International Research Agendas programme of the Foundation for
Polish Science co-financed by the European Union under the European Regional
Development Fund, the “Quantum Coherence and Entanglement for Quantum
Technology” project, carried out within the First Team programme of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund, and the National Science Centre, Poland,
within the QuantERA II Programme (No 2021/03/Y/ST2/00178, acronym ExTRaQT)
that has received funding from the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement No 101017733.

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## Supplemental Material

### Proof of Theorem 1

Theorem 1 of the main text states that marginal reducibility and correlated
catalysis are equivalent for bipartite distillable states. The proof of the
theorem follows from Propositions 3 and 4 which are given below. In the
following, $S$ denotes a possibly multipartite quantum system.

######  Proposition 3.

Marginal reducibility from $\rho^{S}$ onto $\sigma^{S}$ implies that
$\rho^{S}$ can be converted into $\sigma^{S}$ via correlated catalysis.

###### Proof.

Let $\Lambda$ be an LOCC protocol converting $n$ copies of an initial state
$\rho$ into a state

$\Gamma=\Lambda\left[\rho^{\otimes n}\right],$ (3)

which is a quantum state of the system $S_{1}\otimes S_{2}\otimes\cdots\otimes
S_{n}$, and each $S_{i}$ is a copy of the system $S$. In the following,
$\Gamma_{i}$ denotes the reduced state of $\Gamma$ on $S_{1}\otimes
S_{2}\otimes\cdots\otimes S_{i}$ with $\Gamma_{0}=1$. Moreover,
$\Gamma_{i}^{(j)}$ is the reduced state of $\Gamma_{i}$ on $S_{j}$ for $j\leq
i$.

Marginal reducibility of the state $\rho$ into $\sigma$ implies that for any
$\varepsilon>0$ and any $\delta>0$ there are integers $m\leq n$, and an LOCC
protocol $\Lambda$ such that

$\displaystyle\left\|\Gamma_{j}^{(j)}-\sigma\right\|_{1}$
$\displaystyle<\varepsilon,$ (4a) $\displaystyle\frac{m}{n}+\delta$
$\displaystyle>1$ (4b)

for all $j\in[1,m]$.

We are now ready to present a state of the catalyst $\tau$ achieving the
transformation $\rho\rightarrow\sigma$:

$\tau=\frac{1}{n}\sum_{k=1}^{n}\rho^{\otimes(k-1)}\otimes\Gamma_{n-k}\otimes\ket{k}\\!\bra{k},$
(5)

in analogy to the construction presented in [9] (see also [10, 15]). Here, the
states $\ket{k}$ are orthonormal states of an auxiliary system $K$ maintained
by Alice. Using the LOCC protocol described above Eq. (7) in [9], the state
$\rho^{S}\otimes\tau^{C}$ is transformed into a state $\mu^{SC}$ with the
property that $\mathrm{Tr}_{S}[\mu^{SC}]=\tau^{C}$. Here, $C$ denotes the
system of the catalyst.

What remains is to show that $\norm{\mu^{S}-\sigma^{S}}_{1}$ can be made
arbitrarily small. Indeed, we note that the state $\mu^{S}$ can be written as

$\mu^{S}=\frac{1}{n}\sum_{k=1}^{n}\Gamma_{k}^{(k)}=\frac{1}{n}\sum_{k=1}^{m}\Gamma_{k}^{(k)}+\frac{1}{n}\sum_{k=m+1}^{n}\Gamma_{k}^{(k)}.$
(6)

Using Eqs. (4) we can further write

$\displaystyle\left\|\mu^{S}-\sigma^{S}\right\|_{1}$
$\displaystyle=\frac{1}{n}\left\|\sum_{k=1}^{n}\left(\Gamma_{k}^{(k)}-\sigma^{S}\right)\right\|_{1}$
(7)
$\displaystyle\leq\frac{1}{n}\left\|\sum_{k=1}^{m}\left(\Gamma_{k}^{(k)}-\sigma^{S}\right)\right\|_{1}$
$\displaystyle+\frac{1}{n}\left\|\sum_{k=m+1}^{n}\left(\Gamma_{k}^{(k)}-\sigma^{S}\right)\right\|_{1}$
$\displaystyle<\frac{m}{n}\varepsilon+2\frac{n-m}{n}<\varepsilon+2\delta.$

The proof is complete by noting that $\varepsilon>0$ and $\delta>0$ can be
chosen arbitrarily. ∎

To complete the proof of Theorem 1 we also need to prove the converse, which
is established in the following. We will focus on bipartite systems in the
following, i.e., $S=AB$. Extension to multipartite settings will be discussed
below.

######  Proposition 4.

If a distillable state $\rho^{S}$ can be converted into $\sigma^{S}$ via
correlated catalysis, then $\rho^{S}$ is reducible onto $\sigma^{S}$ in the
marginals.

###### Proof.

Let $\tau$ be a state of the catalyst such that

$\displaystyle\mu^{SC}$
$\displaystyle=\Lambda\left(\rho^{S}\otimes\tau^{C}\right),$ (8a)
$\displaystyle\left\|\mu^{S}-\sigma^{S}\right\|_{1}$
$\displaystyle<\delta,\,\,\,\mu^{C}=\tau^{C}$ (8b)

for some $\delta>0$. We will now show that the conditions for marginal
reducibility in Eqs. (Correlated catalysis and marginal reducibility) are
fulfilled in this case.

Since the state $\rho$ is distillable, it is possible to distill some singlets
and therefore approximate any state $\tau$ via LOCC from a finite number of
copies of $\rho$. In more detail, for any $\varepsilon>0$ there is an integer
$k$ and an LOCC protocol $\Lambda^{\prime}$ such that

$\left\|\Lambda^{\prime}\left(\rho^{\otimes
k}\right)-\tau\right\|_{1}<\varepsilon.$ (9)

In the following, the state $\tau_{\varepsilon}=\Lambda^{\prime}(\rho^{\otimes
k})$ will be called _$\varepsilon$ -approximation of the catalyst_.

Consider now the following protocol, acting on $\rho^{\otimes n}$:

  1. 1.

The last $k$ copies of the state $\rho^{\otimes n}$ are converted into
$\tau_{\varepsilon}$ via LOCC, i.e.,

$\rho^{\otimes
n}\overset{\mathrm{LOCC}}{\longrightarrow}\rho^{\otimes(n-k)}\otimes\tau_{\varepsilon}.$
(10)

  2. 2.

Each of the remaining $n-k$ copies of $\rho$ is converted approximately into
the desired state, making repeated use of the state $\tau_{\varepsilon}$.

We will now analyze in more detail the procedure described above. After the
first state $\rho^{S_{1}}$ is converted using the $\varepsilon$-approximation
of the catalyst $\tau_{\varepsilon}^{C}$, Alice and Bob share the state
$\mu_{1}\otimes\rho^{\otimes(n-k-1)}$, where the state $\mu_{1}$ is given as

$\mu_{1}^{S_{1}C}=\Lambda\left(\rho^{S_{1}}\otimes\tau^{C}_{\varepsilon}\right),$
(11)

and $\Lambda$ is the same LOCC protocol as in Eqs. (8). Note that

$\left\|\mu_{1}^{S_{1}C}-\mu^{S_{1}C}\right\|_{1}\leq\left\|\rho^{S_{1}}\otimes\tau_{\varepsilon}^{C}-\rho^{S_{1}}\otimes\tau^{C}\right\|_{1}<\varepsilon.$
(12)

Recalling that $\mu^{C}=\tau^{C}$, we obtain the inequalities

$\displaystyle\left\|\mu_{1}^{S_{1}}-\mu^{S_{1}}\right\|_{1}$
$\displaystyle<\varepsilon,$ (13)
$\displaystyle\left\|\mu_{1}^{C}-\tau^{C}\right\|_{1}$
$\displaystyle<\varepsilon.$ (14)

Thus, the state $\mu_{1}^{C}$ is also an $\varepsilon$-approximation of the
catalyst state, having the same precision as $\tau_{\varepsilon}$.

Alice and Bob now convert the second copy of the state $\rho^{S_{2}}$, using
the catalyst approximation $\mu_{1}^{C}$. If we define
$\mu_{2}^{S_{2}C}=\Lambda(\rho^{S_{2}}\otimes\mu_{1}^{C})$, then by the same
arguments as above we will find that

$\displaystyle\left\|\mu_{2}^{S_{2}}-\mu^{S_{2}}\right\|_{1}$
$\displaystyle<\varepsilon,$ (15)
$\displaystyle\left\|\mu_{2}^{C}-\tau^{C}\right\|_{1}$
$\displaystyle<\varepsilon.$ (16)

Iterating this procedure for each copy of $\rho$, we arrive at a state
$\nu^{S_{1}\ldots S_{n-k}}$ on $S_{1}\ldots S_{n-k}$, such that the reduced
states on $S_{i}$ fulfill

$\left\|\nu^{S_{i}}-\mu^{S_{i}}\right\|_{1}<\varepsilon.$ (17)

Due to Eqs. (8) and the triangle inequality we further find that

$\left\|\nu^{S_{i}}-\sigma\right\|_{1}<\varepsilon+\delta.$ (18)

The above arguments show that for every $\varepsilon>0$ and $\delta>0$ we can
convert $\rho^{\otimes n}$ into a state $\nu^{S_{1}\ldots S_{n-k}}$ fulfilling
Eq. (18). Interestingly, while the integer $k$ depends on $\varepsilon$ and
$\delta$, the integer $n$ does not depend on these parameters. Thus, we can
choose $n$ large enough, making $(n-k)/n$ arbitrarily close to 1. This proves
marginal reducibility from $\rho$ to $\sigma$, and the proof of the
proposition is complete. ∎

### Marginal and catalytic asymptotic transformation rates

A key quantity in asymptotic entanglement theory is the transformation rate,
describing the optimal performance of an asymptotic transformation of a state
$\rho^{S}$ into another state $\sigma^{S}$, where $S$ denotes a possibly
multipartite systems. We say that an asymptotic transformation from $\rho$
into $\sigma$ is possible at rate $r$ if for any $\varepsilon,\delta>0$ there
are integers $m$, $n$, and an LOCC protocol $\Lambda$ such that

$\displaystyle\left\|\Lambda\left(\rho^{\otimes n}\right)-\sigma^{\otimes
m}\right\|_{1}$ $\displaystyle<\varepsilon,$ (19a)
$\displaystyle\frac{m}{n}+\delta$ $\displaystyle>r.$ (19b)

The supremum over such achievable rates $r$ is the asymptotic transformation
rate $R(\rho\rightarrow\sigma)$. Rates of this form have been initially
studied in the context of singlet distillation [4, 6, 22], which will also be
discussed in more detail below. A state $\rho$ is reducible to $\sigma$ in the
notion of [23] if $R(\rho\rightarrow\sigma)\geq 1$.

Moreover, we say that $\rho$ can be converted into $\sigma$ with correlated
catalysis at rate $r$, if for any $\varepsilon>0$ and any $\delta>0$ there
exist integers $m$, $n$, a catalyst state $\tau$, and an LOCC protocol
$\Lambda$ such that

$\displaystyle\Lambda\left(\rho^{\otimes n}\otimes\tau^{C}\right)$
$\displaystyle=\mu^{S_{1}\ldots S_{m}C},$ (20a)
$\displaystyle\left\|\mu^{S_{1}\ldots S_{m}}-\sigma^{\otimes m}\right\|_{1}$
$\displaystyle<\varepsilon,$ (20b) $\displaystyle\mu^{C}$
$\displaystyle=\tau^{C},$ (20c) $\displaystyle\frac{m}{n}+\delta$
$\displaystyle>r.$ (20d)

The supremum over all such rates will be called _catalytic transformation
rate_ $R_{\mathrm{c}}(\rho\rightarrow\sigma)$.

Analogously, we say that a _marginal asymptotic transformation_ from $\rho$ to
$\sigma$ is possible with rate $r$, if for any $\varepsilon>0$ and any
$\delta>0$ there exist integers $m$, $n$, and an LOCC protocol $\Lambda$ such
that the following equations hold:

$\displaystyle\Lambda\left(\rho^{\otimes n}\right)=$
$\displaystyle\mu^{S_{1}\ldots S_{m}},$ (21a)
$\displaystyle\left\|\mu^{S_{i}}-\sigma\right\|_{1}$
$\displaystyle<\varepsilon\,\,\,\forall i\leq m,$ (21b)
$\displaystyle\frac{m}{n}+\delta$ $\displaystyle>r.$ (21c)

Here, $\mu^{S_{1}\ldots S_{m}}$ is a state of the system $S_{1}\otimes
S_{2}\otimes\cdots\otimes S_{m}$, where each $S_{i}$ is a copy of the system
$S$. The largest value of $r$ fulfilling these properties will be called
_marginal transformation rate_ $R_{\mathrm{m}}(\rho\rightarrow\sigma)$. We
note that rate of this form has been defined previously in [29]. A state
$\rho$ is said to be reducible to $\sigma$ in the marginals if
$R_{\mathrm{m}}(\rho\rightarrow\sigma)\geq 1$.

Finally, we say that a state $\rho$ can be converted into $\sigma$ via
_marginal asymptotic transformations with correlated catalysis_ with rate $r$
if for any $\varepsilon,\delta>0$ there exist integers $m$, $n$, a state of
the catalyst $\tau^{C}$, and an LOCC protocol $\Lambda$ such that

$\displaystyle\Lambda\left(\rho^{\otimes n}\otimes\tau^{C}\right)$
$\displaystyle=\mu^{S_{1}\ldots S_{m}C},$ (22a)
$\displaystyle\left\|\mu^{S_{i}}-\sigma\right\|_{1}$
$\displaystyle<\varepsilon\,\,\,\forall i\leq m,$ (22b) $\displaystyle\mu^{C}$
$\displaystyle=\tau^{C},$ (22c) $\displaystyle\frac{m}{n}+\delta$
$\displaystyle>r.$ (22d)

The maximal such rate will be called _marginal catalytic transformation rate_
$R_{\mathrm{mc}}(\rho\rightarrow\sigma)$. It is straightforward to see that

$\displaystyle R_{\mathrm{mc}}(\rho\rightarrow\sigma)$ $\displaystyle\geq
R_{\mathrm{m}}(\rho\rightarrow\sigma)\geq R(\rho\rightarrow\sigma),$ (23)
$\displaystyle R_{\mathrm{mc}}(\rho\rightarrow\sigma)$ $\displaystyle\geq
R_{\mathrm{c}}(\rho\rightarrow\sigma)\geq R(\rho\rightarrow\sigma).$ (24)

We will now prove that for bipartite distillable states, catalysis cannot
enhance marginal asymptotic transformation rates.

######  Proposition 5.

For any two bipartite distillable states $\rho$ and $\sigma$ it holds that

$R_{\mathrm{mc}}\left(\rho\rightarrow\sigma\right)=R_{\mathrm{m}}\left(\rho\rightarrow\sigma\right).$
(25)

###### Proof.

We will show that a marginal catalytic protocol achieving the rate
$R_{\mathrm{mc}}$ can always be used to construct a marginal protocol without
the catalyst, achieving the same rate. For this let $\tau$ be the state of the
catalyst such that Eqs. (22) are fulfilled. In analogy to the proof of
Proposition 4, recall that the state $\tau$ can be approximated by a state
$\tau_{\varepsilon^{\prime}}$, which can be obtained via LOCC from a finite
number of copies of the initial state $\rho$, i.e.,
$\tau_{\varepsilon^{\prime}}=\Lambda^{\prime}(\rho^{\otimes k})$ and
$||\tau_{\varepsilon^{\prime}}-\tau||_{1}<\varepsilon^{\prime}$.

Consider now the following LOCC protocol, acting on $n+k$ copies of $\rho$. In
the first step, $k$ copies of the state $\rho$ are converted into
$\tau_{\varepsilon^{\prime}}$ via LOCC. After this step, the total state is
given by $\rho^{\otimes n}\otimes\tau_{\varepsilon^{\prime}}$. In the next
step, Alice and Bob apply the LOCC protocol from Eqs. (22). The resulting
state will be denoted by $\mu_{1}$, and can be explicitly written as

$\mu_{1}^{S_{1}\ldots S_{m}C}=\Lambda\left(\rho^{\otimes
n}\otimes\tau_{\varepsilon^{\prime}}^{C}\right).$ (26)

Note that

$\left\|\mu_{1}^{S_{1}\ldots S_{m}C}-\mu^{S_{1}\ldots
S_{m}C}\right\|_{1}\leq\left\|\rho^{\otimes
n}\otimes\tau_{\varepsilon^{\prime}}-\rho^{\otimes
n}\otimes\tau_{\varepsilon}\right\|_{1}<\varepsilon^{\prime}$ (27)

which implies the inequalities

$\displaystyle\left\|\mu_{1}^{S_{i}}-\mu^{S_{i}}\right\|_{1}$
$\displaystyle<\varepsilon^{\prime},$ (28)
$\displaystyle\left\|\mu_{1}^{C}-\tau^{C}\right\|_{1}$
$\displaystyle<\varepsilon^{\prime}.$ (29)

The latter inequality implies that $\mu_{1}^{C}$ approximates the state $\tau$
with same precision as $\tau_{\varepsilon^{\prime}}$. Using Eqs. (22) and the
triangle inequality we further obtain

$\left\|\mu_{1}^{S_{i}}-\sigma\right\|_{1}<\varepsilon+\varepsilon^{\prime}$
(30)

for all $i\leq m$.

We will now extend our analysis to $2n+k$ copies of the initial state $\rho$.
Again, $k$ copies of $\rho$ will be used to establish the state
$\tau_{\varepsilon^{\prime}}$, resulting in the total state $\rho^{\otimes
n}\otimes\rho^{\otimes n}\otimes\tau_{\varepsilon^{\prime}}$. The first $n$
copies of $\rho$ together with $\tau_{\varepsilon^{\prime}}$ are converted
into the state $\mu_{1}$, as described above in this proof, leading to the
total state $\mu_{1}^{S_{1}\ldots S_{m}C}\otimes\rho^{\otimes n}$. The
remaining $n$ copies of $\rho$ are now converted with the LOCC protocol given
in Eqs. (22), using $\mu_{1}^{C}$ as the catalyst state. Recall that
$\mu_{1}^{C}$ approximates the state $\tau^{C}$ with the error
$\varepsilon^{\prime}$, which is the same as for
$\tau_{\varepsilon^{\prime}}$. The total state of the systems $S_{m+1}\ldots
S_{2m}$ after this transformation will be denoted by $\mu_{2}^{S_{m+1}\ldots
S_{2m}C}=\Lambda(\rho^{\otimes n}\otimes\mu_{1}^{C})$. By the same arguments
as above, we find that

$\displaystyle\left\|\mu_{2}^{C}-\tau^{C}\right\|_{1}$
$\displaystyle<\varepsilon^{\prime},$ (31)
$\displaystyle\left\|\mu_{2}^{S_{i}}-\sigma\right\|_{1}$
$\displaystyle<\varepsilon+\varepsilon^{\prime}$ (32)

for all $i\in[m+1,2m]$.

Iterating the above procedure $l$ times, we see that it is possible to convert
the state $\rho^{\otimes ln+k}$ into the state $\nu^{S_{1}\ldots S_{lm}}$
having the property that
$||\nu^{S_{i}}-\sigma||_{1}<\varepsilon+\varepsilon^{\prime}$ for all
$i\in[1,lm]$. Since this procedure works for any $l$, choosing $l$ large
enough we can make $\frac{lm}{ln+k}$ arbitrarily close to $\frac{m}{n}$, and
thus also arbitrarily close to $R_{\mathrm{mc}}$. This proves that it is
possible to convert $\rho$ into $\sigma$ with rate $R_{\mathrm{mc}}$ via
marginal asymptotic transformations, and the proof of the proposition is
complete. ∎

We note that the role of catalysis for many-copy transformations between
bipartite pure states has been studied earlier in [15]. In particular, it was
shown that multiple copy transformations, with the aid of a pure catalyst, are
equivalent to the single copy catalytic transformation with arbitrary pure
catalyst states [15].

### Marginal and catalytic transformations for pure target states

Here we will focus on a bipartite setting with a pure target state
$\ket{\phi}$. A case of particular interest is when the target state is a
singlet, in which case $R(\rho\rightarrow\psi^{-})$ is known as the
distillable entanglement $E_{\mathrm{d}}(\rho)$ [4, 6, 22]. Using the fact
that pure state transformations are asymptotically reversible [6], it is
straightforward to see that for pure target states the following holds:

$R(\rho\rightarrow\phi)=\frac{E_{\mathrm{d}}(\rho)}{S(\phi^{A})}.$ (33)

We will now show that $R_{\mathrm{m}}$ and $R$ coincide for pure target
states.

######  Proposition 6.

The marginal transformation rate coincides with the standard transformation
rate for pure target states:

$R_{\mathrm{m}}(\rho\rightarrow\phi)=R(\rho\rightarrow\phi).$ (34)

###### Proof.

If $S(\phi^{A})=0$, the target state is not entangled, and both
$R(\rho\to\phi)$ and $R_{\mathrm{m}}(\rho\to\phi)$ diverge in this case.
Without loss of generality we assume $S(\phi^{A})>0$.

We introduce a slightly different task of transforming a state $\rho$
asymptotically into a state with marginals having distillable entanglement
close to $E_{\mathrm{d}}(\phi)=S(\phi^{A})$. Here, we say that a
transformation with rate $r$ is possible if for any $\varepsilon,\delta>0$
there exist integers $m$, $n$ and an LOCC protocol $\Lambda$ such that

$\displaystyle\Lambda\left(\rho^{\otimes n}\right)$
$\displaystyle=\mu^{S_{1}\ldots S_{m}},$ (35a)
$\displaystyle\left|E_{\mathrm{d}}(\mu^{S_{i}})-S(\phi^{A})\right|$
$\displaystyle<\varepsilon\,\,\,\forall i\leq m,$ (35b)
$\displaystyle\frac{m}{n}+\delta$ $\displaystyle>r.$ (35c)

The maximal such rate will be denoted $\tilde{R}_{\mathrm{m}}(\rho,\phi)$.
Recall that the distillable entanglement is bounded as [40, 41]

$S(\rho^{A})-S(\rho^{AB})\leq E_{\mathrm{d}}(\rho^{AB})\leq S(\rho^{A}).$ (36)

This implies that $E_{\mathrm{d}}$ is continuous in the vicinity of any pure
state, and therefore $\tilde{R}_{\mathrm{m}}(\rho,\phi)\geq
R_{\mathrm{m}}(\rho\rightarrow\phi)$.

Consider now an LOCC protocol achieving Eqs. (35). Using Eq. (33) and the
properties of distillable entanglement (see also Proposition 11) we find

$\displaystyle nR\left(\rho\rightarrow\phi\right)$
$\displaystyle=n\frac{E_{\mathrm{d}}(\rho)}{S(\phi^{A})}=\frac{E_{\mathrm{d}}\left(\rho^{\otimes
n}\right)}{S(\phi^{A})}\geq\frac{E_{\mathrm{d}}\left(\mu^{S_{1}\ldots
S_{m}}\right)}{S(\phi^{A})}$ (37)
$\displaystyle\geq\frac{1}{S(\phi^{A})}\sum_{i=1}^{m}E_{\mathrm{d}}\left(\mu^{S_{i}}\right).$

Using this inequality and Eqs. (35) we further obtain

$nR\left(\rho\rightarrow\phi\right)>m\left(1-\frac{\varepsilon}{S(\phi^{A})}\right),$
(38)

which implies that

$\frac{m}{n}<\frac{R\left(\rho\rightarrow\phi\right)}{1-\frac{\varepsilon}{S(\phi^{A})}}.$
(39)

Using Eqs. (35) once again we arrive at

$r<\frac{R\left(\rho\rightarrow\phi\right)}{1-\frac{\varepsilon}{S(\phi^{A})}}+\delta.$
(40)

Recalling that $\varepsilon,\delta>0$ can be chosen arbitrarily, we conclude
that $\tilde{R}_{\mathrm{m}}(\rho,\phi)\leq R(\rho\rightarrow\phi)$.
Collecting the above arguments we have

$\tilde{R}_{\mathrm{m}}(\rho,\phi)\geq R_{\mathrm{m}}(\rho\rightarrow\phi)\geq
R(\rho\rightarrow\phi)\geq\tilde{R}_{\mathrm{m}}(\rho,\phi),$ (41)

which shows that these inequalities are actually equalities. ∎

We note that this proposition can be extended also to quantum resource
theories different from entanglement, we refer to Proposition 13.

From the above proposition, it follows that in this setting marginal
reducibility is equivalent to reducibility as defined in [23]. This means that
for pure target states in the bipartite setting allowing for correlations
between the marginals does not improve the transformation rate. Combining the
above results, we can now prove the following proposition.

######  Proposition 7.

For any bipartite distillable state $\rho$ and any bipartite pure state
$\ket{\phi}$ it holds that

$\displaystyle R_{\mathrm{mc}}(\rho\rightarrow\phi)$
$\displaystyle=R_{\mathrm{m}}(\rho\rightarrow\phi)=R_{\mathrm{c}}(\rho\rightarrow\phi)$
(42)
$\displaystyle=R(\rho\rightarrow\phi)=\frac{E_{\mathrm{d}}(\rho)}{S(\phi^{A})}.$

###### Proof.

The proof follows by combining Propositions 5 and 6 with Eqs. (24) and (33). ∎

Interestingly, it remains unclear whether Proposition 6 extends to the
multipartite setting, if $\ket{\phi}$ is a general multipartite pure state.
However, as we will see below, Proposition 6 also applies if $\ket{\phi}$ is a
specific multipartite state, comprising a singlet shared between each of the
parties.

We will now investigate catalytic transformations with pure target states. We
say that a state $\rho$ can be converted into $\sigma$ via _correlated
catalysis with decoupling_ if for any $\varepsilon>0$ there is a catalyst
state $\tau$ and an LOCC protocol $\Lambda$ such that [9, 12]

$\displaystyle\left\|\Lambda\left(\rho^{S}\otimes\tau^{C}\right)-\sigma^{S}\otimes\tau^{C}\right\|_{1}$
$\displaystyle<\varepsilon,$ (43a)
$\displaystyle\mathrm{Tr}_{S}\left[\Lambda\left(\rho^{S}\otimes\tau^{C}\right)\right]$
$\displaystyle=\tau^{C}.$ (43b)

Note that in this framework, the correlations between the primary system $S$
and the catalyst $C$ can be made vanishingly small. As we show in the
following proposition, for pure target states $\sigma=\phi$, correlated
catalysis is equivalent to correlated catalysis with decoupling. Here, $S$
denotes a possibly multipartite system.

######  Proposition 8.

A state $\rho^{S}$ can be converted into a pure state $\ket{\psi}^{S}$ via
correlated catalysis if and only if the conversion is possible via correlated
catalysis with decoupling.

###### Proof.

Assume that the transformation $\rho\rightarrow\phi$ is possible via
correlated catalysis, i.e., for any $\varepsilon>0$ there exists a catalyst
state $\tau$ and an LOCC protocol $\Lambda$ such that Eqs. (1) are fulfilled.
Using Lemma 10 which is given below, we see that the catalyst decouples in
this procedure, and moreover

$\left\|\mu^{SC}-\phi^{S}\otimes\tau^{C}\right\|_{1}<\varepsilon+6\sqrt{\frac{\varepsilon}{2}}.$
(44)

This shows that the existence of a correlated catalytic transformation from
$\rho$ into $\ket{\phi}$ implies that the transformation is also possible via
correlated catalysis with decoupling. The converse is straightforward, noting
that correlated catalysis is at least as powerful as correlated catalysis with
decoupling in general. ∎

Finally, we will now show that for pure target states in the bipartite
setting, correlated catalysis with decoupling is equivalent to the notion of
reducibility defined in [23].

######  Proposition 9.

The following statements are equivalent for any bipartite distillable state
$\rho$ and any bipartite pure state $\ket{\phi}$:

  1. 1.

$\rho$ is reducible onto $\ket{\phi}$

  2. 2.

$\rho$ can be converted into $\ket{\phi}$ via correlated catalysis with
decoupling

  3. 3.

$E_{\mathrm{d}}(\rho)\geq S(\phi^{A})$

###### Proof.

From Theorem 1, we see that marginal reducibility from $\rho$ to $\ket{\phi}$
is equivalent to the existence of a correlated catalytic transformation from
$\rho$ to $\ket{\phi}$. Proposition 6 implies that in this setting marginal
reducibility is equivalent to the notion of reducibility defined in [23].
Proposition 8 further implies that correlated catalysis is equivalent to
correlated catalysis with decoupling. This proves that conditions 1 and 2 are
equivalent. The equivalence of condition 3 follows from Proposition 7. ∎

We complete this section with the following lemma.

######  Lemma 10.

For any quantum state $\mu^{SC}$ the inequality

$\left\|\mu^{S}-\ket{\phi}\\!\bra{\phi}^{S}\right\|_{1}<\varepsilon$ (45)

implies that

$\left\|\mu^{SC}-\ket{\phi}\\!\bra{\phi}^{S}\otimes\mu^{C}\right\|_{1}<\varepsilon+6\sqrt{\frac{\varepsilon}{2}}.$
(46)

###### Proof.

Note that Eq. (45) implies the inequality

$\displaystyle F\left(\mu^{S},\ket{\phi}\\!\bra{\phi}^{S}\right)$
$\displaystyle\geq\sqrt{1-\frac{1}{2}\left\|\mu^{S}-\ket{\phi}\\!\bra{\phi}^{S}\right\|_{1}}$
$\displaystyle>\sqrt{1-\frac{\varepsilon}{2}}.$ (47)

with fidelity $F(\rho,\sigma)=\mathrm{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$.
The state $\mu^{S}$ has a purification

$\ket{\mu}^{ST}=\sum_{i}\lambda_{i}\ket{i}^{S}\ket{i}^{T}$ (48)

with the Schmidt coefficient $\lambda_{i}$ sorted in decreasing order. Due to
Eq. (47) we have

$\lambda_{0}>\sqrt{1-\frac{\varepsilon}{2}}.$ (49)

Let now $\ket{\nu}^{SCD}$ be a purification of $\mu^{SC}$, and observe that it
can be written as

$\ket{\nu}^{SCD}=\sum_{i}\lambda_{i}\ket{i}^{S}\ket{\alpha_{i}}^{CD},$ (50)

where $\lambda_{i}$ are the same Schmidt coefficients as in Eq. (48) and
$\\{\ket{\alpha_{i}}\\}$ is an orthonormal basis on $CD$. Noting that

$F\left(\ket{\nu}\\!\bra{\nu}^{SCD},\ket{0}\\!\bra{0}^{S}\otimes\ket{\alpha_{0}}\\!\bra{\alpha_{0}}^{CD}\right)=\lambda_{0},$
(51)

and using the fact that the fidelity does not decrease under partial trace we
obtain

$F\left(\mu^{SC},\ket{0}\\!\bra{0}^{S}\otimes\mathrm{Tr}_{D}\left[\ket{\alpha_{0}}\\!\bra{\alpha_{0}}^{CD}\right]\right)>\sqrt{1-\frac{\varepsilon}{2}}.$
(52)

Using the inequality $||\rho-\sigma||_{1}/2\leq\sqrt{1-F(\rho,\sigma)^{2}}$ we
arrive at

$\frac{1}{2}\left\|\mu^{SC}-\ket{0}\\!\bra{0}^{S}\otimes\mathrm{Tr}_{D}\left[\ket{\alpha_{0}}\\!\bra{\alpha_{0}}^{CD}\right]\right\|_{1}<\sqrt{\frac{\varepsilon}{2}}.$
(53)

Noting that the trace norm does not increase under partial trace we obtain

$\displaystyle\left\|\mu^{C}-\mathrm{Tr}_{D}\left[\ket{\alpha_{0}}\\!\bra{\alpha_{0}}^{CD}\right]\right\|_{1}<2\sqrt{\frac{\varepsilon}{2}}.$
(54)

We now use the triangle inequality, arriving at

$\displaystyle\left\|\mu^{SC}-\ket{0}\\!\bra{0}^{S}\otimes\mu^{C}\right\|_{1}\leq\left\|\mu^{SC}-\ket{0}\\!\bra{0}^{S}\otimes\mathrm{Tr}_{D}\left[\ket{\alpha_{0}}\\!\bra{\alpha_{0}}^{CD}\right]\right\|_{1}$
$\displaystyle+\left\|\ket{0}\\!\bra{0}^{S}\otimes\mathrm{Tr}_{D}\left[\ket{\alpha_{0}}\\!\bra{\alpha_{0}}^{CD}\right]-\ket{0}\\!\bra{0}^{S}\otimes\mu^{C}\right\|_{1}<4\sqrt{\frac{\varepsilon}{2}}.$
(55)

Using again Eq. (53) we find

$\left\|\mu^{S}-\ket{0}\\!\bra{0}^{S}\right\|_{1}<2\sqrt{\frac{\varepsilon}{2}},$
(56)

which together with Eq. (45) and triangle inequality implies that

$\left\|\ket{\phi}\\!\bra{\phi}^{S}-\ket{0}\\!\bra{0}^{S}\right\|_{1}<\varepsilon+2\sqrt{\frac{\varepsilon}{2}}.$
(57)

Using once again the triangle inequality we obtain Eq. (46):

$\displaystyle\left\|\mu^{SC}-\ket{\phi}\\!\bra{\phi}^{S}\otimes\mu^{C}\right\|_{1}\leq\left\|\mu^{SC}-\ket{0}\\!\bra{0}^{S}\otimes\mu^{C}\right\|_{1}$
$\displaystyle+\left\|\ket{0}\\!\bra{0}^{S}\otimes\mu^{C}-\ket{\phi}\\!\bra{\phi}^{S}\otimes\mu^{C}\right\|_{1}<\varepsilon+6\sqrt{\frac{\varepsilon}{2}}.$
(58)

∎

### Proof of Theorem 2

Let us introduce the _catalytic distillable entanglement_

$E_{\mathrm{cd}}(\rho)=R_{\mathrm{c}}(\rho\rightarrow\psi^{-}),$ (59)

that is, the optimal rate of obtaining singlets with the help of a correlated
catalyst. Theorem 2 of the main text states that catalytic distillable
entanglement coincides with the standard distillable entanglement for any
distillable state, i.e.,

$E_{\mathrm{d}}(\rho)=E_{\mathrm{cd}}(\rho).$ (60)

This is a direct consequence of Proposition 7.

### Extending Theorems 1 and 2 to multipartite settings

We can generalize Theorem 1 and 2 into a multipartite setting as follows: let
us consider distillation into singlets shared between any pair of parties. Let
us denote the parties as $A_{1},\ldots,A_{n}$, and let us call all pairs
$\mathbb{P}=\Bqty{\pqty{A_{i},A_{j}}\left|\,i<j\right.}$. The state that we
would like to distill is $\Phi=\bigotimes_{p\in\mathbb{P}}\psi^{-}_{p}$ (see
also Fig. 2), and it lives in
$\mathcal{H}_{\textrm{total}}=\bigotimes_{p\in\mathbb{P}}\mathcal{H}_{p}=\bigotimes_{\pqty{A_{i},A_{j}}\in\mathbb{P}}A_{i}\otimes
A_{j}$. Note that to each party, we associate a different subsystem for every
pair, so the total number of subsystems is $n(n-1)/2$. Let us call the optimal
standard asymptotic rate $E_{\mathrm{d}}(\rho)=R(\rho\to\Phi)$, and states for
which $E_{\mathrm{d}}\pqty{\rho}>0$ distillable. Theorem 1 follows from
Proposition 3 and 4. Since Proposition 3 does not assume any bipartite
structure, it also holds in the multipartite setting. Furthermore, for any
state that is distillable, the construction provided in the proof of
Proposition 4 also works, so in combination we have Theorem 1.

To obtain Theorem 2, we relied on Proposition 5 and 6. It is clear that
Proposition 5 extends easily to this setting, so we only have to show that
Proposition 6 also holds, at least around $\Phi$. In a multipartite setting,
$E_{\mathrm{d}}$ is still monotonic, additive under tensor products and
superadditive. Therefore, it is enough to show that it is lower semi-
continuous near $\Phi$. Now, suppose we have an LOCC map $T$ that maps a state
$\rho$ into $\mathcal{H}_{\textrm{total}}$. Let
$r=\min_{p\in\mathbb{P}}E_{\mathrm{d}}^{p}(T(\rho))$, where
$E_{\mathrm{d}}^{p}$ is the bipartite distillable entanglement between the
factors in $p$. Then, for any pair of parties we can obtain at least $r$
singlets per copy of $\rho$. Furthermore, these distillation protocols can be
run independently because they are acting on different factors of
$\mathcal{H}_{\textrm{total}}$. Therefore, we have
$E_{\mathrm{d}}\pqty{\rho}\geq\min_{p\in\mathbb{P}}E_{\mathrm{d}}^{p}(T(\rho))$.
Since $E_{\mathrm{d}}^{p}$ is simply bipartite distillable entanglement, the
hashing bound [41] provides a continuous lower bound. Therefore we would be
finished if we can show that it is tight for $\Phi$. Now, $\Phi$ contains a
singlet for every pair of parties, so we can choose $T$ that separates these
singlets for each party. We can verify that in this case, the hashing bound
gives $E_{\mathrm{d}}\pqty{\Phi}\geq 1$, which is indeed tight. Therefore,
Theorem 2 also holds in a multipartite setting.

### Strong superadditivity of asymptotic transformation rates

We will show that the transformation rate to go to any pure state $\phi$ is
superadditive. In particular, this means that the bipartite distillable
entanglement is superadditive, and the same holds true for the multipartite
distillable entanglement as defined above. In the following, $S_{1}$ and
$S_{2}$ denote two (possibly multipartite) systems.

######  Proposition 11.

For any state $\mu^{S_{1}S_{2}}$ and any pure state $\phi$, we have

$\displaystyle R\pqty{\mu^{S_{1}S_{2}}\to\phi}$ $\displaystyle\geq
R\pqty{\mu^{S_{1}}\to\phi}+R\pqty{\mu^{S_{2}}\to\phi}.$ (61)

###### Proof.

Let us take a feasible rate $r_{i}<R\pqty{\mu^{S_{i}}\to\phi}$ and show that
$r_{1}+r_{2}$ is a feasible rate for the transformation
$\mu^{S_{1}S_{2}}\to\phi$. To do so, we have to show that for any
$\varepsilon,\delta>0$, there exist $m,n$, and an LOCC protocol $\Lambda$ such
that

$\displaystyle\norm{\Lambda\pqty{\pqty{\mu^{S_{1}S_{2}}}^{\otimes
n}}-\outerproduct{\phi}{\phi}^{\otimes m}}_{1}<\varepsilon,$ (62a)
$\displaystyle\frac{m}{n}+\delta>r_{1}+r_{2}.$ (62b)

Fix an arbitrary $\varepsilon,\delta>0$. Without loss of generality, let us
assume $\varepsilon<1$. In the following, we denote the space of
$S_{i}^{\otimes n}$ as $\mathbb{S}_{i}$, where $i\in\\{1,2\\}$. Note that
since $r_{i}$ are feasible, there exist $m_{i}$, $n$ and some LOCC protocol
$\Lambda_{i}$ such that

$\displaystyle\left\|\Lambda_{i}\left(\left(\mu^{S_{i}}\right)^{\otimes
n}\right)-\outerproduct{\phi}{\phi}^{\otimes
m_{i}}\right\|_{1}<\frac{\varepsilon^{2}}{100}<\frac{\varepsilon}{2},$ (63a)
$\displaystyle\frac{m_{i}}{n}+\frac{\delta}{2}>r_{i}.$ (63b)

Here, $\Lambda_{i}$ is a LOCC map acting on the space $\mathbb{S}_{i}\equiv
S_{i}^{\otimes n_{i}}$ where $i\in\\{1,2\\}$. Note that without loss of
generality, we can choose the same $n$ for both systems since otherwise we can
take the product $n=n_{1}n_{2}$ and update $m_{i},\Lambda_{i}$ accordingly.
From, Lemma 10 and Eq. (63a), it follows that

$\displaystyle\left\|\Lambda_{1}\otimes\openone_{2}\left(\left(\mu^{S_{1}S_{2}}\right)^{\otimes
n}\right)-\outerproduct{\phi}{\phi}^{\otimes
m_{1}}\otimes\left(\mu^{S_{2}}\right)^{\otimes n}\right\|_{1}$
$\displaystyle<\frac{\varepsilon^{2}}{100}+6\frac{\varepsilon}{\sqrt{200}}$
$\displaystyle<\frac{\varepsilon}{2}.$

Using the data processing inequality of trace norm, it follows that

$\displaystyle\left\|\Lambda_{1}\otimes\Lambda_{2}\left(\left(\mu^{S_{1}S_{2}}\right)^{\otimes
n}\right)-\outerproduct{\phi}{\phi}^{\otimes
m_{1}}\otimes\Lambda_{2}\left(\left(\mu^{S_{2}}\right)^{\otimes
n}\right)\right\|_{1}<\varepsilon/2.$

Using the triangle inequality along with Eq. (63a), we get

$\displaystyle\left\|\Lambda_{1}\otimes\Lambda_{2}\left(\left(\mu^{S_{1}S_{2}}\right)^{\otimes
n}\right)-\outerproduct{\phi}{\phi}^{\otimes(m_{1}+m_{2})}\right\|_{1}\leq$
$\displaystyle\left\|\Lambda_{1}\otimes\Lambda_{2}\left(\left(\mu^{S_{1}S_{2}}\right)^{\otimes
n}\right)-\outerproduct{\phi}{\phi}^{\otimes
m_{1}}\otimes\Lambda_{2}\left(\left(\mu^{S_{2}}\right)^{\otimes
n}\right)\right\|_{1}$
$\displaystyle+\left\|\outerproduct{\phi}{\phi}^{\otimes
m_{1}}\otimes\Lambda_{2}\left(\left(\mu^{S_{2}}\right)^{\otimes
n}\right)-\outerproduct{\phi}{\phi}^{\otimes(m_{1}+m_{2})}\right\|_{1}$
$\displaystyle<\varepsilon.$ (64)

Also note that, from Eq. (63b)

$\displaystyle\frac{m_{1}+m_{2}}{n}+\delta>r_{1}+r_{2}.$ (65)

We see that the Eqs. (62) are satisfied by choosing $m=m_{1}+m_{2}$,
$\Lambda=\Lambda_{1}\otimes\Lambda_{2}$. Therefore,
$R\pqty{\mu^{S_{1}}\to\phi}+R\pqty{\mu^{S_{2}}\to\phi}$ is a feasible rate for
the transformation $\mu^{S_{1}S_{2}}\to\phi$, and the claim is shown. ∎

### Properties of catalytic and marginal transformation rates

In the following we will provide a general upper bound on the catalytic and
marginal transformation rates in terms of the squashed entanglement of the
corresponding quantum states. Squashed entanglement is defined as [42]

$E_{\mathrm{sq}}(\rho^{AB})=\inf\left\\{\frac{1}{2}I(A;B|E):\rho^{ABE}\mathrm{\,\,extension\,\,of\,\,}\rho^{AB}\right\\},$
(66)

with the quantum conditional mutual information
$I(A;B|E)=S(\rho^{AE})+S(\rho^{BE})-S(\rho^{ABE})-S(\rho^{E})$.

######  Proposition 12.

The rates $R_{\mathrm{m}}$ and $R_{\mathrm{mc}}$ are bounded as

$R_{\mathrm{m}}\left(\rho\rightarrow\sigma\right)\leq
R_{\mathrm{mc}}\left(\rho\rightarrow\sigma\right)\leq\frac{E_{\mathrm{sq}}\left(\rho\right)}{E_{\mathrm{sq}}\left(\sigma\right)}.$
(67)

###### Proof.

We introduce a slightly different version of the transformation rate, which we
will call _squashed transformation rate_. In this framework, we say that a
state $\rho$ can be converted into $\sigma$ with rate $r$ if the following
inequalities are fulfilled for all $\varepsilon,\delta>0$:

$\displaystyle\Lambda\left(\rho^{\otimes n}\otimes\tau^{C}\right)$
$\displaystyle=\mu^{S_{1}\ldots S_{m}C},$ (68a)
$\displaystyle\left|E_{\mathrm{sq}}(\mu^{S_{i}})-E_{\mathrm{sq}}(\sigma)\right|$
$\displaystyle<\varepsilon\,\,\,\forall i\leq m,$ (68b) $\displaystyle\mu^{C}$
$\displaystyle=\tau^{C},$ (68c) $\displaystyle\frac{m}{n}+\delta$
$\displaystyle>r.$ (68d)

The squashed transformation rate is the maximal such rate, and it will be
denoted by $R_{\mathrm{sq}}$. By continuity of squashed entanglement [43], it
is clear that $R_{\mathrm{sq}}(\rho\rightarrow\sigma)\geq
R_{\mathrm{mc}}(\rho\rightarrow\sigma)$.

Consider now an LOCC protocol $\Lambda$ and a catalyst state $\tau$ achieving
Eqs. (68). Using the properties of squashed entanglement [42] we find

$\displaystyle
nE_{\mathrm{sq}}\left(\rho\right)+E_{\mathrm{sq}}\left(\tau\right)$
$\displaystyle=E_{\mathrm{sq}}\left(\rho^{\otimes n}\otimes\tau^{C}\right)\geq
E_{\mathrm{sq}}\left(\mu^{S_{1}\ldots S_{m}C}\right)$ (69)
$\displaystyle\geq\sum_{i=1}^{m}E_{\mathrm{sq}}\left(\mu^{S_{i}}\right)+E_{\mathrm{sq}}\left(\tau\right),$

and thus

$\displaystyle nE_{\mathrm{sq}}\left(\rho\right)$
$\displaystyle\geq\sum_{i=1}^{m}E_{\mathrm{sq}}\left(\mu^{S_{i}}\right).$ (70)

Using this inequality and Eqs. (68) we further obtain

$nE_{\mathrm{sq}}\left(\rho\right)>m\left[E_{\mathrm{sq}}\left(\sigma\right)-\varepsilon\right],$
(71)

which implies that

$\frac{m}{n}<\frac{E_{\mathrm{sq}}\left(\rho\right)}{E_{\mathrm{sq}}\left(\sigma\right)-\varepsilon}.$
(72)

Using Eqs. (68) once again we arrive at

$r<\frac{E_{\mathrm{sq}}\left(\rho\right)}{E_{\mathrm{sq}}\left(\sigma\right)-\varepsilon}+\delta.$
(73)

Recalling that $\varepsilon,\delta>0$ can be chosen arbitrarily, we conclude
that

$R_{\mathrm{sq}}\left(\rho\rightarrow\sigma\right)\leq\frac{E_{\mathrm{sq}}\left(\rho\right)}{E_{\mathrm{sq}}\left(\sigma\right)},$
(74)

and the proof is complete. ∎

We note that the inequality $R_{\mathrm{m}}(\rho\rightarrow\sigma)\leq
E_{\mathrm{sq}}(\rho)/E_{\mathrm{sq}}(\sigma)$ has been proven previously in
[29].

### General quantum resource theories

We will now show a generalized version of Proposition 6 for general quantum
resource theories. This, under some assumptions, will show an equivalence
between asymptotic rates and marginal asymptotic rates in general resource
theories.

Any quantum resource theory is defined by a set of free states ($\mathcal{F}$)
and a set of free operations ($\mathcal{O}$) [44], such that the following
property holds

$\Lambda_{f}(\rho_{f})\in\mathcal{F}\,\,\forall\,\,\rho_{f}\in\mathcal{F}\,\,\text{and}\,\,\Lambda_{f}\in\mathcal{O}.$
(75)

With this introduction, we now state our result. As a notation, we denote
$R(\rho\rightarrow\sigma)$ as $R_{\sigma}(\rho)$.

######  Proposition 13.

For any initial state $\rho$ and a target state $\sigma$,
$R_{\mathrm{m}}(\rho\rightarrow\sigma)=R(\rho\rightarrow\sigma)$ if the
following conditions hold

  * •

$R_{\sigma}(\mu^{SS^{\prime}})\geq
R_{\sigma}(\mu^{S})+R_{\sigma}(\mu^{S^{\prime}})$ for all $\mu^{SS^{\prime}}$
(super-additivity).

  * •

$R_{\sigma}(\cdot)$ is lower semi-continuous at $\sigma$.

###### Proof.

Note that, by definition $R_{\mathrm{m}}(\rho\rightarrow\sigma)\geq
R_{\sigma}(\rho)$. Therefore it is enough to show that
$R_{\mathrm{m}}(\rho\rightarrow\sigma)\leq R(\rho\rightarrow\sigma)$.

Since, $R_{\sigma}(\cdot)$ is lower semi-continuous at $\sigma$, there exists
a small enough $\varepsilon^{\prime}$, such that for every $\tau$ satisfying
$\left\|\tau-\sigma\right\|_{1}\leq\varepsilon^{\prime}$, we have

$\displaystyle R_{\sigma}(\tau)\geq
R_{\sigma}(\sigma)-f(\varepsilon^{\prime})=1-f(\varepsilon^{\prime}).$ (76)

Here, $f(\varepsilon^{\prime})\rightarrow 0$ as
$\varepsilon^{\prime}\rightarrow 0^{+}$. Here, marginal asymptotic
transformations can be defined analogous to Eq. (21), by replacing the LOCC
operation $\Lambda$, by a free operation.

Consider now a free operation ($\Lambda$) achieving Eq. (21). Applying
$R_{\sigma}(\cdot)$, on both sides of Eq. (21a) we get

$\displaystyle R_{\sigma}\left(\rho^{\otimes
n}\right)=nR_{\sigma}\left(\rho\right)$ $\displaystyle\geq
R_{\sigma}\left(\mu^{S_{1}\ldots S_{m}}\right)$ (77)
$\displaystyle\geq\sum_{i=1}^{m}R_{\sigma}\left(\mu^{S_{i}}\right)$
$\displaystyle\geq m\left(R_{\sigma}\left(\sigma\right)-f(\varepsilon)\right)$
$\displaystyle=m\left(1-f(\varepsilon)\right).$

Here, we used additivity, monotonicity, super-additivity (assumed) and lower
semi-continuity (assumed) of $R_{\sigma}(\cdot)$. This implies

$\frac{m}{n}\leq\frac{R_{\sigma}\left(\rho\right)}{1-f(\varepsilon)}.$ (78)

Using Eqs. (65) we arrive at

$r<\frac{R_{\sigma}\left(\rho\right)}{1-f(\varepsilon)}+\delta.$ (79)

Recalling that $\varepsilon,\delta>0$ can be chosen arbitrarily small, we
conclude that $R_{\mathrm{m}}(\rho\rightarrow\sigma)\leq
R(\rho\rightarrow\sigma)$. This completes the proof. ∎