# Reversibility of quantum resources through probabilistic protocols

Bartosz Regula<EMAIL_ADDRESS>Mathematical Quantum Information
RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR) and
RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan
Ludovico Lami<EMAIL_ADDRESS>QuSoft, Science Park 123, 1098 XG
Amsterdam, the Netherlands  Korteweg–de Vries Institute for Mathematics,
University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, the
Netherlands  Institute for Theoretical Physics, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, the Netherlands

###### Abstract

Among the most fundamental questions in the manipulation of quantum resources
such as entanglement is the possibility of reversibly transforming all
resource states. The most important consequence of this would be the
identification of a unique entropic resource measure that exactly quantifies
the limits of achievable transformation rates. Remarkably, previous results
claimed that such asymptotic reversibility holds true in very general
settings; however, recently those findings have been found to be incomplete,
casting doubt on the conjecture. Here we show that it is indeed possible to
reversibly interconvert all states in general quantum resource theories, as
long as one allows protocols that may only succeed probabilistically. Although
such transformations have some chance of failure, we show that their success
probability can be ensured to be bounded away from zero, even in the
asymptotic limit of infinitely many manipulated copies. As in previously
conjectured approaches, the achievability here is realised through operations
that are asymptotically resource non-generating. Our methods are based on
connecting the transformation rates under probabilistic protocols with strong
converse rates for deterministic transformations. We strengthen this
connection into an exact equivalence in the case of entanglement distillation.

How to measure and compare quantum resources? As evidenced by the plethora of
commonly used quantifiers of resources such as entanglement [1, 2], this
seemingly basic question has many possible answers, and it may appear as
though there is no unambiguous way to resolve it. However, it is important to
keep in mind that, instead of simply assigning some numerical value to a given
quantum state, one often wishes to compare quantum resources operationally:
how difficult is it to convert one resource state into another? Following this
pathway is reminiscent of the operational approach used to study
thermodynamics, where indeed a unique resource measure — the entropy — emerges
naturally from basic axioms [3, 4]. A phenomenon which underlies this
possibility is reversibility: two comparable states of equal entropy can
always be connected by a reversible adiabatic transformation [4].

Reversibility was observed also in the asymptotic manipulation of quantum
entanglement of pure states [5], prompting several conjectures about the
connections between entanglement and thermodynamics, and in particular about
the existence of a unique operational measure of entanglement [6, 7, 8, 9].
Here, the restrictions on resource manipulation are typically understood in
terms of asymptotic transformation rates $r(\rho\to\omega)$: given many copies
of a state $\rho$, how many copies of another state $\omega$ can we obtain per
each copy of $\rho$? The question of resource reversibility then asks whether
$r(\rho\to\omega)=r(\omega\to\rho)^{-1}$, meaning that exactly as many copies
of $\omega$ can be obtained in the transformation as is needed to transform
them back into $\rho$. Although entanglement of noisy states may exhibit
irreversibility in many contexts [10, 11, 12, 13], hopes persisted that an
operational approach allowing for universal reversibility can be constructed.

A remarkable axiomatic framework emerged, first for entanglement [8, 9] and
later for more general quantum resources [14], which claimed that
reversibility can indeed always be achieved under suitable assumptions. Such a
striking property would not only establish a unique entropic measure of
quantum resources, but also connect the broad variety of different resources
in a common operational formalism. However, issues have transpired in parts of
the proof of these results [15], putting this general reversibility into
question. As of now, there is no known framework that can establish the
reversibility of general quantum resource theories — and in particular quantum
entanglement — even under weaker assumptions. What is more, recent results
demonstrated an exceptionally strong type of irreversibility of entanglement
[13], casting doubt on the very possibility of recovering reversible
manipulation whatsoever.

Here we show that the reversibility of general quantum resource
transformations can be established in a setting that very closely resembles
the original assumptions of the reversibility conjectures [8, 9, 14], with
only one change: we allow probabilistic conversion protocols. That is, we
study transformations which allow for some probability of failure, and we
demonstrate that in this setting the conversion rates are exactly given by the
entropic resource measure known as the regularised relative entropy,
identifying it as the unique operational resource quantifier. To ensure that
the protocols are not unphysically difficult to realise, we constrain the
probability of failure so that it does not become prohibitively large: there
always remains a constant non-zero chance of successful conversion, even when
manipulating an unbounded number of quantum states.

We thus construct the first complete reversible framework for general quantum
resources, including entanglement. We stress that, although conceptually
similar, our approach does not exactly recover the reversibility conjectured
in [8, 9, 14], since only strictly deterministic transformations were employed
there. However, we consider it to be strong supporting evidence in favour of
reversibility being an achievable phenomenon in general quantum resource
theories.

On the technical side, the way we avoid issues associated with the generalised
quantum Stein’s lemma [16] that undermined the original reversibility claims,
is to use only the _strong converse_ part of the lemma, which is still valid
[15]. Strong converse rates are typically understood as general no-go
limitations on resource transformations, but here we turn them into achievable
rates precisely by employing probabilistic protocols. For the special case of
entanglement distillation, we show that these two concepts — strong converse
rates in deterministic transformations on one side, and probabilistic
conversion rates on the other — are exactly equivalent, which holds true also
in the most practically relevant settings of entanglement manipulation such as
under local operations and classical communication (LOCC).

Resource transformation rates. — Quantum resource theories represent various
settings of restricted quantum information processing [2]. Let us denote by
$\mathbb{O}$ the set of operations which are freely allowed within the
physical setting of the given resource theory. Our discussion of reversibility
will require a specific choice of $\mathbb{O}$, but for now it may be
understood as a general set of permitted operations.

The _deterministic transformation rate_ $r_{p=1}(\rho\to\omega)$ is defined as
the supremum of real numbers $r$ such that $n$ copies of the state $\rho$ can
be converted into $\left\lfloor rn\right\rfloor$ copies of the target state
$\omega$ using the free operations $\mathbb{O}$. The conversion here is
assumed to be deterministic, i.e. all transformations are realised by
completely positive and trace-preserving maps (quantum channels). However, the
process is only required to be approximate, in the sense that some error
$\varepsilon_{n}$ is allowed in the transformation, as long as it vanishes in
the limit as $n\to\infty$.

In many practical contexts, one may be willing to relax the assumption that
the error must tend to zero — it may, for instance, be appealing to tolerate
some manageable error in the transformation if it could lead to increased
transformation rates. The ultimate upper bound that constrains the
improvements that can be gained through such trade-offs is represented by the
_strong converse rate_ $r^{\dagger}_{p=1}(\rho\to\omega)$. It is defined as
the least rate $r$ such that, if we attempt the conversion $\rho^{\otimes
n}\to\omega^{\otimes\left\lfloor r^{\prime}n\right\rfloor}$ at any larger rate
$r^{\prime}>r$, then even approximate transformations with large error become
impossible.

Another common way to increase the capabilities in resource manipulation is to
allow for probabilistic transformations [17, 18, 19, 20, 21, 22, 23].
Probabilistic protocols in quantum information theory are represented by a
collection of completely positive, but not necessarily trace-preserving maps
$\\{\mathcal{E}^{(i)}\\}_{i}$, such that the total transformation
$\sum_{i}\mathcal{E}^{(i)}$ preserves trace. We say that $\rho$ can be
converted to $\omega$ if there exists a free probabilistic operation
$\mathcal{E}^{(i)}\in\mathbb{O}$ such that
$\frac{\mathcal{E}^{(i)}(\rho)}{\operatorname{Tr}\mathcal{E}^{(i)}(\rho)}=\omega$;
the probability of this transformation is
$p=\operatorname{Tr}\mathcal{E}^{(i)}(\rho)$. One can then define asymptotic
transformation rates analogously as before, by considering sequences of
probabilistic operations $(\mathcal{E}_{n})_{n}$ such that each
$\mathcal{E}_{n}\in\mathbb{O}$ converts $\rho^{\otimes n}$ to a state which is
$\varepsilon_{n}$-close to the target state $\omega^{\otimes\left\lfloor
rn\right\rfloor}$ and so that the error vanishes asymptotically.

In our definition of a probabilistic rate, the probability $p$ itself is not
counted as part of the rate, and we instead focus solely on the number of
copies of states that are being manipulated. There is a potential issue with
such an approach, as leaving the probability of the transformation
unconstrained effectively allows for postselecting on exponentially unlikely
events, which then makes possible transformations which are conventionally
known to be unachievable [23] — but only at the expense of the overall
probability of success $p_{n}=\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})$ becoming vanishingly small. To ensure that the protocol remains
practically realisable, here we forbid such a possibility. We thus consider
_probabilistic transformation rates with non-vanishing probability_ ,
$r_{p>0}(\rho\to\omega)$, where the probability is constrained to be bounded
away from zero even in the limit $n\to\infty$. Explicitly,

$\displaystyle r_{p>0}(\rho\to\omega)\coloneqq$
$\displaystyle\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}\;\mathcal{E}_{n}\in\mathbb{O},$
(1)
$\displaystyle\lim_{n\to\infty}F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)=1,$
$\displaystyle\liminf_{n\to\infty}\,\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})>0\Bigg{\\}},$

where $F$ denotes fidelity.

The fact that the deterministic transformation rate $r_{p=1}$ is the smallest
of the three types introduced above is clear from the definition. However,
there is no obvious relation between the strong converse rate and the
probabilistic one. We can nevertheless show that the rates actually form a
hierarchy:

$\displaystyle
r_{p=1}(\rho\to\omega)\,\leq\,r_{p>0}(\rho\to\omega)\,\leq\,r^{\dagger}_{p=1}(\rho\to\omega).$
(2)

This demonstrates in particular that the probabilistic rate $r_{p>0}$ is well
behaved, as it does not exceed conventional limitations imposed by strong
converse rates. It may also provide a tighter restriction on deterministic
transformation rates than those coming from strong converse bounds.

Free operations and reversibility. — The asymptotic transformation rates
depend heavily on the choice of the free operations $\mathbb{O}$. Typically,
practically relevant choices of free operations are subsets of _resource–non-
generating (RNG) operations_ , defined as those maps $\mathcal{E}$ (possibly
probabilistic ones) that satisfy
$\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\sigma)}\in\mathbb{F}$
for all $\sigma\in\mathbb{F}$. Here, $\mathbb{F}$ stands for the set of free
(resourceless) states of the given theory. The definition of RNG operations
then means that these maps are not allowed to generate any resources for free,
which is a very basic and undemanding assumption to make.

The framework of [8, 9, 14] studied the manipulation of quantum resources
under transformations which slightly relax the above constraint, imposing
instead that small amounts of resources may be generated, as long as they
vanish asymptotically. Specifically, let us consider the resource measure
known as _generalised (global) robustness_ $R_{\mathbb{F}}^{g}$, defined as
[24]

$\displaystyle R^{g}_{\mathbb{F}}(\rho)\coloneqq\inf\left\\{\,\lambda\geq
1\;\rule{0.0pt}{9.5pt}\left|\;\frac{\rho+\lambda\omega}{1+\lambda}\in\mathbb{F},\;\omega\in\mathbb{D}\right.\right\\},$
(3)

where $\mathbb{D}$ denotes the set of all states. The $\delta$-approximately
resource–non-generating operations $\mathbb{O}_{{{\rm RNG}_{\delta}}}$ are
then all maps $\mathcal{E}$ such that

$\displaystyle
R^{g}_{\mathbb{F}}\left(\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\sigma)}\right)\leq\delta\quad\forall\sigma\in\mathbb{F}.$
(4)

Finally, the transformation rates under _asymptotically resource–non-
generating maps_ $\mathbb{O}_{\rm ARNG}$, whether deterministic or
probabilistic, are defined as those where each transformation $\rho^{\otimes
n}\to\omega^{\otimes\left\lfloor rn\right\rfloor}$ is realised by a
$\delta_{n}$-approximately RNG operation, with $\delta_{n}\to 0$ in the limit
as $n\to\infty$. We will denote deterministic rates under such operations as
$r_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\omega)$, and analogously for the probabilistic rates $r_{p>0}$.

The main reason to study asymptotically RNG operations is their conjectured
reversibility [8, 9, 14], combined with the fact that reversibility of
entanglement has been ruled out under essentially all sets of operations
smaller than $\mathbb{O}_{\rm ARNG}$ [13]. Specifically, the claim is that the
deterministic rates always equal

$\displaystyle
r_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\omega)\stackrel{{\scriptstyle?}}{{=}}\frac{D^{\infty}_{\mathbb{F}}(\rho)}{D^{\infty}_{\mathbb{F}}(\omega)},$
(5)

where $D^{\infty}_{\mathbb{F}}$ denotes the _regularised relative entropy of a
resource_ ,

$\displaystyle
D^{\infty}_{\mathbb{F}}(\rho)\coloneqq\lim_{n\to\infty}\frac{1}{n}\,\inf_{\sigma\in\mathbb{F}}\,D(\rho^{\otimes
n}\|\sigma_{n})$ (6)

with $D(\rho\|\sigma)=\operatorname{Tr}\rho(\log\rho-\log\sigma)$ being the
quantum relative entropy. This would precisely identify
$D^{\infty}_{\mathbb{F}}$ as the unique resource measure in the asymptotic
setting. However, this conjecture relied crucially on the generalised quantum
Stein’s lemma [16], in whose proof a gap was recently discovered [15]. Hence,
the statement in Eq. (5) is not known to be true [15].

Probabilistic reversibility. — The conjectured resource reversibility in Eq.
(5) is remarkably general. The original claim applied not only to
entanglement, but also to more general quantum resources, as long as the set
$\mathbb{F}$ satisfies a number of mild assumptions — notably, it must be
convex, and it must be such that the tensor product of any two free states
remains free, as does their partial trace [16, 14]. These are weak assumptions
obeyed by the vast majority of theories of practical interest.

Our main result is a general probabilistic reversibility of quantum resources
under the exact same assumptions.

######  Theorem 1.

For all quantum states $\rho$ and $\omega$, the transformation rate with non-
vanishing probability of success under asymptotically resource–non-generating
operations satisfies

$\displaystyle
r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\omega)=\frac{D^{\infty}_{\mathbb{F}}(\rho)}{D^{\infty}_{\mathbb{F}}(\omega)}.$
(7)

This implies in particular a general reversibility of state transformations:
$r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\omega)=r_{p>0}(\omega\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\rho)^{-1}$ for any pair of states.

Both the converse and the achievability parts of this result make use of the
asymptotic equipartition property for the generalised robustness, which was
shown by Brandão and Plenio [16, Proposition II.1] and independently by Datta
[25, Theorem 1]. This property says that, under a suitable ‘smoothing’, the
generalised robustness $R^{g}_{\mathbb{F}}$ converges asymptotically to the
regularised relative entropy of the resource:

$\displaystyle\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\\!\min_{\frac{1}{2}\left\lVert\omega_{n}-\omega^{\otimes\left\lfloor
rn\right\rfloor}\right\rVert_{\,1}\leq\varepsilon}\\!\log\\!\left(1+R^{g}_{\mathbb{F}}(\omega_{n})\right)=r\,D^{\infty}_{\mathbb{F}}(\omega).$
(8)

Importantly, this finding directly leads to the strong converse of the
generalised quantum Stein’s lemma [16, Corollary III.3], but it does not
appear to be enough to deduce the main achievability part of the lemma [15].
Our main contribution here is to show that the strong converse part is
sufficient to show the reversibility of quantum resources, as long as
probabilistic protocols are allowed.

###### Proof sketch of Theorem 1.

The converse direction relies on the strong monotonicity properties of the
generalised robustness $R^{g}_{\mathbb{F}}$ as well as the aforementioned
asymptotic equipartition property (8). This follows a related approach that
was recently used to study postselected probabilistic transformation rates
[23], and here we extend it to asymptotically resource–non-generating
transformations ARNG. A point of note is that standard techniques based on the
asymptotic continuity of the relative entropy [1, 26, 27] do not seem to be
sufficient to establish a converse bound on probabilistic rates [23, Appendix
H], requiring the use of a different toolset that explicitly makes use of the
features of $R^{g}_{\mathbb{F}}$.

For the achievability part of the theorem, we use the exact calculation of the
strong converse exponent in the generalised quantum Stein’s lemma [16]. The
lemma is concerned with the distinguishability of many copies of a quantum
state $\rho^{\otimes n}$ against all states in the set of the free states
$\mathbb{F}$. The result of [16] then says that, for every resource theory,
there exists a sequence of measurement operators $(A_{n})_{n}$ such that
$\operatorname{Tr}\left(A_{n}\rho^{\otimes n}\right)\geq 1-\delta_{n}$ and

$\displaystyle-\frac{1}{n}\log\sup_{\sigma\in\mathbb{F}}\operatorname{Tr}(A_{n}\sigma)\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}D^{\infty}_{\mathbb{F}}(\rho).$
(9)

Here, $\delta_{n}$ denotes the probability of incorrectly guessing that
$\rho^{\otimes n}$ is a free state, while the quantity in Eq. (9)
characterises the opposite error of incorrectly guessing that a free state is
$\rho^{\otimes n}$. The issue here is that (9) is only valid in the strong
converse regime, which means that the error $\delta_{n}$ is not guaranteed to
vanish: it may actually tend to a constant arbitrarily close to 1. This
prevents a direct application of previous deterministic results [14].

What we do instead is define probabilistic operations of the form

$\displaystyle\mathcal{E}_{n}(\tau)\coloneqq\operatorname{Tr}(A_{n}\tau)\,\omega_{n}+\mu_{n}\operatorname{Tr}[(\mathbbm{1}-A_{n})\tau]\,\pi_{n},$
(10)

where: $\omega_{n}$ are states appearing in (8) which are
${\varepsilon_{n}}$-close to the target states $\omega^{\otimes\left\lfloor
n\,D^{\infty}_{\mathbb{F}}(\rho)/D^{\infty}_{\mathbb{F}}(\omega)\right\rfloor}$,
$\pi_{n}$ are some suitably chosen states, and $\mu_{n}\in[0,1]$ are
parameters to be fixed.

The basic idea is then that by decreasing $\mu_{n}$, we can make the output of
this operation closer to $\omega_{n}$, even when
$\operatorname{Tr}(A_{n}\rho^{\otimes n})\not\to 1$. However, one cannot just
decrease $\mu_{n}$ arbitrarily, as the maps $\mathcal{E}_{n}$ must be ensured
to be free operations. Our crucial finding is that $\mu_{n}$ can always be
chosen so that
$\mu_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}0$
while the operations $\mathcal{E}_{n}$ generate asymptotically vanishing
amounts of resources and the overall probability of success does not vanish.
This means precisely that the sequence $(\mathcal{E}_{n})_{n}$ is an ARNG
protocol that realises the desired conversion.∎

The complete proof of Theorem 1 can be found in Section II of the Appendix.

Entanglement distillation. — Two of the most important problems in the
understanding of asymptotic entanglement manipulation concern the tasks of
extracting ‘entanglement bits’ (ebits), i.e. copies of the maximally entangled
two-qubit singlet state $\Phi_{+}$, and the reverse task of converting ebits
into general noisy states. The rates of these two tasks are known as,
respectively, the _distillable entanglement_
$E_{d,\mathbb{O}}^{x}(\rho)\coloneqq r_{x}(\rho\to\Phi_{+})$ and the
_entanglement cost_ $E_{c,\mathbb{O}}^{x}(\rho)\coloneqq
r_{x}(\Phi_{+}\to\rho)^{-1}$, where $x$ stands for either $p\\!\\!=\\!\\!1$,
$p\\!>\\!0$, or the strong converse rate $p\\!\\!=\\!\\!1,\\!\dagger$.
Although exact expressions can be obtained for the entanglement cost in
various settings [28, 9, 29], the understanding of distillable entanglement
appears to be an extremely difficult problem that has so far resisted most
attempts at a conclusive solution, except in some special cases [30, 31]. Of
note is the conjectured result that [9, 15]

$\displaystyle E_{d,\rm
NE}^{p=1}(\rho)\stackrel{{\scriptstyle?}}{{=}}D^{\infty}_{\rm SEP}(\rho),$
(11)

where NE stands for the class of non-entangling operations (equivalent to RNG
maps in this theory) and $D^{\infty}_{\rm SEP}$ is the regularised relative
entropy of entanglement. Establishing this result would recover the
deterministic reversibility of entanglement theory (that is, Eq. (5)) [9, 15].
We note that distillation rates under NE operations are equal to rates under
asymptotically non-entangling operations (ANE) [13], which correspond to
$\mathbb{O}_{\rm ARNG}$ in the notation of this work [9].

We now introduce a close relation that connects entanglement distillation
transformations in the probabilistic and strong converse regimes. Namely, we
show that one can always improve on the transformation error of a distillation
protocol by sacrificing some success probability, and vice versa. What this
means in particular is that every rate that can be achieved in the
deterministic strong converse regime (i.e. with a possibly large error
$\varepsilon<1$) can also be achieved probabilistically with error going to
zero. Crucially, to construct the new, modified protocol from the original one
we only need to employ local operations and classical communication (LOCC),
which are the standard class of free operations in entanglement theory,
meaning that the result applies to essentially all different types of
operations that extend LOCC.

######  Theorem 2.

Let $\mathbb{O}$ be any class of operations which is closed under composition
with LOCC, i.e. such that $\mathcal{E}\in\mathbb{O},\,\mathcal{F}\in\rm
LOCC\,\Rightarrow\,\mathcal{F}\circ\mathcal{E}\in\mathbb{O}$. This includes in
particular the set $\rm LOCC$ itself. Then, for all states $\rho$,

$\displaystyle
E_{d,\mathbb{O}}^{p=1,\dagger}(\rho)=E_{d,\mathbb{O}}^{p>0}(\rho).$ (12)

For the case of (asymptotically) non-entangling operations, we have that

$\displaystyle
E_{d,{\rm(A)NE}}^{p=1,\dagger}(\rho)=E_{d,{\rm(A)NE}}^{p>0}(\rho)=D^{\infty}_{\rm
SEP}(\rho).$ (13)

###### Proof sketch.

Assume that two spatially separated parties, conventionally called Alice and
Bob, share $n$ copies of an entangled state $\rho_{AB}$. Consider any sequence
of protocols which allows them to distill entanglement from such states at a
rate $r$ with with error
$\varepsilon_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}\varepsilon$
and probability
$p_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}p$.
(This includes the deterministic strong converse case where $p_{n}=1$.) After
performing the considered distillation protocol, they share a many-copy state
$\tau_{A^{\prime}B^{\prime}}$ which approximates
$\Phi_{+}^{\otimes\left\lfloor rn\right\rfloor}$. What they can do now is to
sacrifice a fixed number $k$ of their qubit pairs in order to perform a state
discrimination protocol: by testing whether the $k$-copy subsystem is in the
state $\Phi_{+}^{\otimes k}$ or not, they can probabilistically bring the
state of the rest of their shared system closer to
$\Phi_{+}^{\otimes\left\lfloor rn\right\rfloor-k}$. We show that this can be
done by a simple LOCC protocol wherein Alice and Bob perform measurements in
the computational basis and compare their outcomes. Since $k$ is arbitrary
here, Alice and Bob can perform the modified protocol without sacrificing the
transformation rate. Conversely, in a similar manner they can also increase
their probability of success by sacrificing some transformation fidelity. By
deriving the exact conditions for when distillation protocols can be
refashioned in such a away, we observe that another sequence of entanglement
distillation protocols with error
$\varepsilon^{\prime}_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}\varepsilon^{\prime}$
and probability
$p^{\prime}_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}p^{\prime}$
can exist _if and only if_

$\displaystyle p\,(1-\varepsilon)=p^{\prime}\,(1-\varepsilon^{\prime}).$ (14)

This directly implies Eq. (12).

To see Eq. (LABEL:eq:ent_equivalence_ne), it suffices to combine Theorem 1
with the known result that $D^{\infty}_{\rm SEP}(\rho)$ is a strong converse
rate for distillation under ANE [32, 9]. ∎

We have already shown the probabilistic reversibility of entanglement theory
in Theorem 1, so let us now discuss the deterministic case. Here,
reversibility is fully equivalent to the question of whether
$E^{p=1}_{d,\mathbb{O}}(\rho)=E^{p=1}_{c,\mathbb{O}}(\rho)$ holds for all
quantum states, which has been conjectured to be true for the class of
asymptotically non-entangling operations[9]. Combining our results with the
known findings of Brandão, Plenio, and Datta [9, 25], we have that

$\displaystyle E_{d,\rm ANE}^{p>0}(\rho)$
$\displaystyle\overset{\text{Thm.~{}\ref{thm:entanglement}}}{=\joinrel=}E_{d,\rm
ANE}^{p=1,\dagger}(\rho)\overset{\text{\cite[cite]{[\@@bibref{Number}{brandao_2010,berta_2022}{}{}]}}}{=\joinrel=}E_{c,\rm
ANE}^{p=1}(\rho)$ (15)
$\displaystyle\overset{\text{\cite[cite]{[\@@bibref{Number}{brandao_2010,datta_2009-2}{}{}]}}}{=\joinrel=}D^{\infty}_{\rm
SEP}(\rho)\overset{\text{Thm.~{}\ref{thm:reversibility}}}{=\joinrel=}E_{c,\rm
ANE}^{p>0}(\rho).$

The missing link is thus the question if $E_{d,\rm
ANE}^{p=1}(\rho)\stackrel{{\scriptstyle?}}{{=}}E_{d,\rm ANE}^{p>0}(\rho)$, or
the equivalent [15] question of whether $E_{c,\rm
ANE}^{p=1,\dagger}(\rho)\stackrel{{\scriptstyle?}}{{=}}E_{c,\rm
ANE}^{p>0}(\rho)$. Showing either of these statements would complete the proof
of the deterministic reversibility of quantum entanglement under
asymptotically non-entangling operations. An interesting consequence of the
above is that establishing the equivalent of Theorem 2 for entanglement
_dilution_ would be sufficient to recover a fully reversible entanglement
theory.

We remark that other quantum resource theories may not be amenable to a
characterisation in terms of distillation and dilution because they may not
possess a suitably well-behaved unit of a resource resembling the maximally
entangled state [14, 33, 34]. Nonetheless, reversibility in all resource
theories can be understood as in our Theorem 1.

Discussion. — We have shown that the conjectured reversibility as well as the
unique entropic measure of general quantum resources can be recovered,
provided one considers asymptotic transformation rates under asymptotically
resource non-generating operations realised as probabilistic protocols with
non-vanishing probability of success.

Although the setting departs slightly from the original conjectures of [9,
14], in view of the close relations between probabilistic and deterministic
rates (Eq. (2)) we regard our results as evidence that reversibility could
indeed be recovered also in the deterministic setting. This is further
motivated by the fact that, in many quantum information processing tasks,
strong converse rates actually coincide with the optimal achievable rates [35,
36, 37, 38, 39, 40, 41], meaning that $r_{p=1}=r^{\dagger}_{p=1}$ and the
hierarchy in Eq. (2) collapses. However, a complete proof of this fact in the
setting of resource manipulation remains elusive, and it is still possible
that one of the inequalities in Eq. (2) may be strict for some states, thus
ruling out deterministic reversibility. We hope that our results stimulate
further research in this direction, leading to an eventual resolution of the
open questions that cloud the understanding of asymptotic resource
manipulation and quantum hypothesis testing.

Acknowledgments. — We acknowledge discussions with Tulja Varun Kondra and
Alexander Streltsov. We are grateful to the Freie Universität Berlin for
hospitality.

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one´

— Appendix —

##  I Notation and basic properties

We begin by clarifying the assumptions and definitions used in this work.

###  I.1 Free operations and free states

Throughout this paper we assume that the underlying Hilbert space has finite
dimension. When discussing many-copy state transformations, we assume that
each $n$-copy space of quantum states has its own associated set of _free
states_ $\mathbb{F}_{n}$. We often use $\mathbb{F}$ to refer to the whole
family $(\mathbb{F}_{n})_{n}$ for simplicity.

Following [16, 14], we will assume the following basic axioms about the
resource theory in consideration:

  1. Axiom I.

Each $\mathbb{F}_{n}$ is convex and closed.

  2. Axiom II.

There exists a full-rank state $\sigma$ such that $\sigma^{\otimes
n}\in\mathbb{F}_{n}$ for all $n$.

  3. Axiom III.

The sets $\mathbb{F}_{n}$ are closed under partial trace: if
$\sigma\in\mathbb{F}_{n+1}$, then
$\operatorname{Tr}_{k}\sigma\in\mathbb{F}_{n}$ for every
$k\in\\{1,\ldots,n+1\\}$.

  4. Axiom IV.

The sets $\mathbb{F}_{n}$ are closed under tensor product: if
$\sigma\in\mathbb{F}_{n}$ and $\sigma^{\prime}\in\mathbb{F}_{m}$, then
$\sigma\otimes\sigma^{\prime}\in\mathbb{F}_{n+m}$.

The motivation for these assumptions is so that the theory is well behaved
asymptotically [16, 25].111In [16], an additional axiom of permutation
invariance was imposed. We do not need this here.

We use CPTP to denote completely positive and trace-preserving maps (quantum
channels), and CPTNI to denote completely positive and trace–non-increasing
maps (probabilistic quantum operations). A _probabilistic protocol_ is a
collection of CPTNI maps $\\{\mathcal{E}^{(i)}\\}_{i}$ that forms a valid
quantum instrument, i.e. the overall transformation
$\sum_{i}\mathcal{E}^{(i)}$ is trace preserving. The latter condition simply
means that the outcome probabilities necessarily add up to one. Given some
class of _free operations_ $\mathbb{O}$, we say that a protocol is free if
$\mathcal{E}^{(i)}\in\mathbb{O}\;\forall i$. When discussing transformations
between two fixed states $\rho\to\omega$, we may coarse-grain the outcomes of
the instrument and consider only two-outcome protocols
$\\{\mathcal{E}^{(i)}\\}_{i\in\\{0,1\\}}$ — either the protocol is successful
in performing the given transformation, or it fails. We can then say that
$\rho$ can be converted to $\omega$ probabilistically if there exists a free
protocol $\\{\mathcal{E}^{(i)}\\}_{i\in\\{0,1\\}}$ such that
$\mathcal{E}^{(0)}(\rho)\propto\omega$.

_Resource non-generating (RNG) operations_ are the maps which can never
transform a resourceless state into a resourceful one:

$\displaystyle\mathbb{O}_{\rm
RNG}\coloneqq\left\\{\,\mathcal{E}\in{\mathrm{CPTNI}}\;\rule{0.0pt}{9.5pt}\left|\;\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\sigma)}\in\mathbb{F}\;\forall\sigma\in\mathbb{F}\right.\right\\}.$
(A1)

Their approximate variant is defined as

$\displaystyle\mathbb{O}_{{\rm
RNG},\delta}\coloneqq\left\\{\,\mathcal{E}\in{\mathrm{CPTNI}}\;\rule{0.0pt}{9.5pt}\left|\;R^{g}_{\mathbb{F}}\left(\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\sigma)}\right)\leq\delta\;\forall\sigma\in\mathbb{F}\right.\right\\}.$
(A2)

We note here that the latter definition depends on the choice of the measure
with which one quantifies the generated resources — in this case, this is the
_generalised robustness_ [24]

$\displaystyle R^{g}_{\mathbb{F}}(\rho)$
$\displaystyle\coloneqq\inf\left\\{\,\lambda\geq
1\;\rule{0.0pt}{9.5pt}\left|\;\frac{\rho+\lambda\omega}{1+\lambda}\in\mathbb{F},\;\omega\in\mathbb{D}\right.\right\\}$
(A3) $\displaystyle\hphantom{:}=\inf\left\\{\,\lambda\geq
1\;\rule{0.0pt}{9.5pt}\left|\;\rho\leq(\lambda+1)\sigma,\;\sigma\in\mathbb{F}\right.\right\\}.$

This quantity can also be identified with the max-relative entropy of a
resource [25],
$D_{\max,\mathbb{F}}(\rho)\coloneqq\log\left(1+R^{g}_{\mathbb{F}}(\rho)\right)$.
Other choices of measures may lead to completely different asymptotic
behaviour [13], and our choice here is motivated by the framework of [8, 9]
that conjectured reversibility of quantum resources under operations defined
in this way.

Under resource non-generating transformations $\mathbb{O}_{\rm RNG}$ (or
$\mathbb{O}_{{\rm RNG},\delta}$), the existence of a probabilistic protocol
that performs a transformation $\rho\to\omega$ is fully equivalent to the
existence of a map $\mathcal{E}\in\mathbb{O}_{\rm RNG}$ such that
$\mathcal{E}(\rho)/\operatorname{Tr}\mathcal{E}(\rho)=\omega$. This is because
any such map can always be completed to a free instrument: we simply define
$\mathcal{E}^{\prime}(X)\coloneqq\left[\operatorname{Tr}X-\operatorname{Tr}\mathcal{E}(X)\right]\sigma$
for some $\sigma\in\mathbb{F}$, yielding
$(\mathcal{E},\mathcal{E}^{\prime})\in\mathbb{O}$. For other sets of
operations, it may be the case that a map $\mathcal{E}\in\mathbb{O}$ cannot
always be completed to a free instrument $(\mathcal{E},\mathcal{E}^{\prime})$
— e.g. for separable operations in entanglement theory [42] or stabiliser
operations in the theory of magic states [43] — so more care may need to be
taken when discussing probabilistic transformations.

###  I.2 Transformation rates

Given a class of completely positive and trace non-increasing maps
$\mathbb{O}$, the various transformation rates that we study are defined as
follows.

The _deterministic conversion rate_ is

$\displaystyle
r_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)\coloneqq\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}$
$\displaystyle\lim_{n\to\infty}F\\!\left(\mathcal{E}_{n}(\rho^{\otimes
n}),\,\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)=1,\quad\mathcal{E}_{n}\in\mathbb{O}\cap{\rm
CPTP}\Bigg{\\}},$ (A4)

where the optimisation is over all sequences $(\mathcal{E}_{n})_{n}$ of maps
satisfying the given constraints. Here,
$F(\rho,\sigma)\coloneqq\left\lVert\sqrt{\rho}\sqrt{\vphantom{\rho}\sigma}\right\rVert_{\,1}^{2}$
is the fidelity.

The _probabilistic rate with non-vanishing probability of success_ is

$\displaystyle
r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)\coloneqq\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}$
$\displaystyle\lim_{n\to\infty}F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)=1,$ (A5)
$\displaystyle\mathcal{E}_{n}\in\mathbb{O},\quad\liminf_{n\to\infty}\,\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})>0\Bigg{\\}}.$

We refer to $1-F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)$ as the
_transformation error_ and to $\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})$ as the _transformation probability_.

The _deterministic strong converse rate_ is

$\displaystyle
r_{p=1}^{\dagger}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)$
$\displaystyle\coloneqq\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}\;\liminf_{n\to\infty}\,F\\!\left(\mathcal{E}_{n}(\rho^{\otimes
n}),\,\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)>0,\quad\mathcal{E}_{n}\in\mathbb{O}\cap{\rm
CPTP}\Bigg{\\}}.$ (A6)
$\displaystyle\hphantom{:}=\inf\Bigg{\\{}\,r^{\prime}\;\Bigg{|}\;\liminf_{n\to\infty}\sup_{\mathcal{E}_{n}\in\mathbb{O}\,\cap\,{\rm
CPTP}}F\\!\left(\mathcal{E}_{n}(\rho^{\otimes
n}),\,\omega^{\otimes\left\lfloor
r^{\prime}n\right\rfloor}\right)=0\Bigg{\\}}.$

We note here that there are differences in the precise definition of a strong
converse in the literature, and stronger definitions can sometimes be
encountered. One such variant is

$\displaystyle
r_{p=1}^{\ddagger}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)$
$\displaystyle\coloneqq\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}\;\limsup_{n\to\infty}\,F\\!\left(\mathcal{E}_{n}(\rho^{\otimes
n}),\,\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)>0,\quad\mathcal{E}_{n}\in\mathbb{O}\cap{\rm
CPTP}\Bigg{\\}}$ (A7)
$\displaystyle\hphantom{:}=\inf\Bigg{\\{}\,r^{\prime}\;\Bigg{|}\;\lim_{n\to\infty}\sup_{\mathcal{E}_{n}\in\mathbb{O}\,\cap\,{\rm
CPTP}}F\\!\left(\mathcal{E}_{n}(\rho^{\otimes
n}),\,\omega^{\otimes\left\lfloor
r^{\prime}n\right\rfloor}\right)=0\Bigg{\\}}.$

The difference between $r^{\dagger}$ and $r^{\ddagger}$ is that the former
maximises over rates which are truly achievable, albeit with a large error; in
other words, for all sufficiently large $n$, the fidelity with the target
state must be larger than some non-zero constant. On the other hand, the
maximisation in (A7) only requires that the fidelity be non-zero infinitely
often. It can be noticed that $r_{p=1}^{\dagger}\leq r_{p=1}^{\ddagger}$ in
general, although we do not know if this inequality can ever be strict when
considering i.i.d. state transformations as we do here. We choose to employ
the definition of $r_{p=1}^{\dagger}$ in Eq. (A6) as it will lead to tighter
statements of our results.

For completeness, we will also briefly consider the _strong converse
probabilistic rate_

$\displaystyle
r^{\dagger}_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)\coloneqq\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}$
$\displaystyle\liminf_{n\to\infty}\,F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)>0,$ (A8)
$\displaystyle\mathcal{E}_{n}\in\mathbb{O},\quad\liminf_{n\to\infty}\,\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})>0\Bigg{\\}}.$

Finally, our main object of study are probabilistic transformation rates under
_asymptotically resource–non-generating transformations_ $\mathbb{O}_{\rm
ARNG}$, which are sequences of operations in $\mathbb{O}_{{\rm
RNG},\delta_{n}}$ such that
$\delta_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}0$.
Specifically,

$\displaystyle
r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\omega)\coloneqq\sup_{(\mathcal{E}_{n})_{n}}\Bigg{\\{}\,r\;\Bigg{|}$
$\displaystyle\lim_{n\to\infty}F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)=1,$ (A9)
$\displaystyle\lim_{n\to\infty}\,\sup_{\sigma\in\mathbb{F}_{n}}\,R^{g}_{\mathbb{F}_{\left\lfloor
rn\right\rfloor}}\\!\left(\frac{\mathcal{E}_{n}(\sigma)}{\operatorname{Tr}\mathcal{E}_{n}(\sigma)}\right)=0,\quad\liminf_{n\to\infty}\,\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})>0\Bigg{\\}}.$

When employing the above definitions, we will make use of the fact that, due
to the Fuchs–van de Graaf inequalities [44], the transformation error can be
equivalently defined with the trace distance
$\frac{1}{2}\left\lVert\rho-\sigma\right\rVert_{\,1}$:

$\displaystyle\lim_{n\to\infty}\,F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)=1$
$\displaystyle\iff\;\lim_{n\to\infty}\,\frac{1}{2}\left\lVert\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}-\omega^{\otimes\left\lfloor rn\right\rfloor}\right\rVert_{\,1}=0,$ (A10)
$\displaystyle\liminf_{n\to\infty}\,F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)>0$
$\displaystyle\iff\;\limsup_{n\to\infty}\frac{1}{2}\left\lVert\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}-\omega^{\otimes\left\lfloor rn\right\rfloor}\right\rVert_{\,1}<1.$

The relations between the different transformation rates are encapsulated in
the following lemma.

######  Lemma A1.

Let $\mathbb{O}$ be any class of free operations such that, for an instrument
$\\{\mathcal{E}^{(i)}\\}_{i}$ with $\mathcal{E}^{(i)}\in\mathbb{O}$, it holds
that $\sum_{i}\mathcal{E}^{(i)}\in\mathbb{O}$. This can be, for instance, the
classes $\mathbb{O}_{\rm RNG}$ or $\mathbb{O}_{\rm RNG,\delta}$ in any convex
resource theory, or any suitable class of free operations in entanglement
theory such as LOCC.

Then, for all quantum states $\rho$ and $\omega$, the transformation rates
satisfy

$\displaystyle
r_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)\,\leq\,r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)$
$\displaystyle\,\leq\,r^{\dagger}_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)\,=\,r^{\dagger}_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega).$
(A11)

######  Remark.

Let us comment briefly on the assumption of this lemma. If there exists a
classical system $C$ with an orthonormal basis $\\{\ket{i}_{C}\\}_{i}$ such
that each element of the basis is a free state, one can equivalently
understand the realisation of any probabilistic protocol
$\\{\mathcal{E}^{(i)}\\}_{i}$ as a CPTP map

$\displaystyle\rho\mapsto\sum_{i}\mathcal{E}^{(i)}(\rho)\otimes\ket{i}\\!\bra{i}_{C}$
(A12)

where the classical system is used to keep track of the probabilistic outcome
of the protocol. If we are free to ‘forget’ this outcome — that is, if tracing
out the classical system is an allowed operation — then the given set
$\mathbb{O}$ satisfies the conditions of the lemma. This is a very weak
assumption that is satisfied by essentially all types of free operations in
the majority of resource theories, but there are special cases in which it may
not be obeyed, e.g. for thermal operations in the theory of thermodynamics
[45].

###### Proof.

As deterministic protocols are a special case of probabilistic ones, the first
inequality is immediate. For the second inequality, assume that there exists a
sequence of free probabilistic operations $(\mathcal{E}_{n})_{n}$ which
performs the transformation $\rho\to\omega$ at a rate $r$ with error
$\varepsilon_{n}$ such that $\lim_{n\to\infty}\varepsilon_{n}=0$ and
probability $p_{n}$ such that $\liminf_{n\to\infty}p_{n}=p>0$. By definition
of a probabilistic protocol, there exist operations
$\mathcal{E}^{\prime}_{n}\in\mathbb{O}$ which complete each $\mathcal{E}_{n}$
to a valid quantum instrument, that is, such that

$\displaystyle\mathcal{D}_{n}(X)\coloneqq\mathcal{E}_{n}(X)+\mathcal{E}^{\prime}_{n}(X)$
(A13)

is trace preserving. (When $\mathbb{O}$ denotes resource non-generating
operations, we can simply take
$\mathcal{E}^{\prime}_{n}(X)=\left[\operatorname{Tr}X-\operatorname{Tr}\mathcal{E}_{n}(X)\right]\sigma_{n}$
for some $\sigma_{n}\in\mathbb{F}$.) By hypothesis, we have that
$\mathcal{D}_{n}\in\mathbb{O}$.

Since $p_{n}=\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes n})$, the
concavity of the fidelity then gives

$\displaystyle\liminf_{n\to\infty}\,F\\!\left(\mathcal{D}_{n}(\rho^{\otimes
n}),\omega^{\otimes\left\lfloor rn\right\rfloor}\right)$
$\displaystyle\geq\liminf_{n\to\infty}\,p_{n}\,F\\!\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{p_{n}},\,\omega^{\otimes\left\lfloor rn\right\rfloor}\right)$ (A14)
$\displaystyle=\liminf_{n\to\infty}\,p_{n}(1-\varepsilon_{n})$
$\displaystyle=p>0,$

hence $r\leq r^{\dagger}_{p=1}(\rho\to\omega)$.

Let now $r$ be any feasible rate for
$r^{\dagger}_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)$,
that is, such that there exists a sequence of free probabilistic operations
$(\mathcal{E}_{n})_{n}$ with non-vanishing error
$\limsup_{n\to\infty}\varepsilon_{n}=\varepsilon<1$ and probability of success
$\liminf_{n\to\infty}p_{n}=p>0$. The same argument as above can be used to
construct a deterministic protocol with

$\displaystyle\liminf_{n\to\infty}\,F\\!\left(\mathcal{D}_{n}(\rho^{\otimes
n}),\omega^{\otimes\left\lfloor rn\right\rfloor}\right)$
$\displaystyle\geq\liminf_{n\to\infty}\,p_{n}(1-\varepsilon_{n})\geq
p(1-\varepsilon)>0,$ (A15)

yielding $r\leq
r^{\dagger}_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)$
and hence $r^{\dagger}_{p>0}\leq r^{\dagger}_{p=1}$. Since the opposite
inequality between the strong converse rates follows straightforwardly from
the fact that any deterministic protocol is a special case of a probabilistic
one, the two rates must be equal. ∎

######  Remark.

If one defines the strong converse rate as in $r^{\ddagger}_{p=1}$ in Eq. (A7)
and introduces an analogous definition of a probabilistic strong converse
$r^{\ddagger}_{p>0}$, then it also holds that
$r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)\leq
r^{\ddagger}_{p=1}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)=r^{\ddagger}_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[21.69447pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}$}}}}}}\omega)$.

##  II Probabilistic reversibility of general quantum resources

######  Theorem 1.

In any resource theory satisfying axioms I–IV of Sec. I.1, it holds that

$\displaystyle
r_{p>0}(\rho\mathrel{\stackunder[0pt]{\xrightarrow{\makebox[33.53615pt]{$\scriptstyle$}}}{\scriptstyle\vphantom{a}\smash{{\raisebox{0.0pt}{\ssmall{$\mathbb{O}_{\rm
ARNG}$}}}}}}\omega)=\frac{D^{\infty}_{\mathbb{F}}(\rho)}{D^{\infty}_{\mathbb{F}}(\omega)}$
(A16)

provided that $D^{\infty}_{\mathbb{F}}(\rho)\neq 0\neq
D^{\infty}_{\mathbb{F}}(\omega)$.

###### Proof.

Converse. — An analogous result when only resource–non-generating operations
$\mathbb{O}_{\rm RNG}$ are considered was already shown in [23, Proposition
6]. The extension of the result to $\mathbb{O}_{\rm ARNG}$ will follow a
similar idea, but will require some modifications that explicitly make use of
the properties of the generalised robustness $R^{g}_{\mathbb{F}}$ — or, in
fact, the closely related max-relative entropy
$D_{\max,\mathbb{F}}(\rho)=\log\left(1+R^{g}_{\mathbb{F}}(\rho)\right)$.

Assume that $r$ is an achievable probabilistic rate, that is, there exists a
sequence of protocols $(\mathcal{E}_{n})_{n}\in\mathbb{O}_{{{\rm
RNG}_{\delta_{n}}}}$ such that
$\frac{1}{2}\left\lVert\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}-\omega^{\otimes\left\lfloor
rn\right\rfloor}\right\rVert_{\,1}\eqqcolon\varepsilon_{n}$ with
$\lim_{n\to\infty}\varepsilon_{n}=0$,
$\liminf_{n\to\infty}\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})\eqqcolon p>0$, and furthermore $\lim_{n\to\infty}\delta_{n}=0$. For all
$\zeta<p$ and all sufficiently large $n$, it then holds that
$\zeta/\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes n})+\varepsilon_{n}<1$.
Letting

$\displaystyle
D^{\zeta}_{\max,\mathbb{F}}(\rho)\coloneqq\min_{\frac{1}{2}\left\lVert\rho-\rho^{\prime}\right\rVert_{\,1}\leq\zeta}D_{\max,\mathbb{F}}(\rho^{\prime}),$
(A17)

we use the probabilistic monotonicity of the max-relative entropy (Lemma A3
below) to get

$\displaystyle D^{\zeta}_{\max,\mathbb{F}}(\rho^{\otimes n})$
$\displaystyle\geq D^{\zeta/\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}_{\max,\mathbb{F}}\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}\right)+\log\left(\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})-\zeta\right)-\log(1+\delta_{n})$ (A18) $\displaystyle\geq
D^{\zeta/\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})+\varepsilon_{n}}_{\max,\mathbb{F}}\left(\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)+\log\left(\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})-\zeta\right)-\log(1+\delta_{n}).$

Here, in the last line we used the fact that
$\frac{1}{2}\left\lVert\tau-\omega^{\otimes\left\lfloor
rn\right\rfloor}\right\rVert_{\,1}\leq\frac{1}{2}\left\lVert\tau-\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}\right\rVert_{\,1}+\frac{1}{2}\left\lVert\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}-\omega^{\otimes\left\lfloor rn\right\rfloor}\right\rVert_{\,1}$ for any
state $\tau$.

Applying the above, we obtain that

$\displaystyle\inf_{(\zeta_{n})_{n}}\left\\{\left.\limsup_{n\to\infty}\frac{1}{n}D^{\zeta_{n}}_{\max,\mathbb{F}}(\rho^{\otimes
n})\;\rule{0.0pt}{9.5pt}\right|\;\lim_{n\to\infty}\zeta_{n}=0\right\\}$
$\displaystyle\geq\inf_{(\zeta_{n})_{n}}\left\\{\left.\limsup_{n\to\infty}\frac{1}{n}D^{\zeta_{n}/\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})+\varepsilon_{n}}_{\max,\mathbb{F}}\left(\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)\;\rule{0.0pt}{9.5pt}\right|\;\lim_{n\to\infty}\zeta_{n}=0\right\\}$
(A19)
$\displaystyle\geq\inf_{(\eta_{n})_{n}}\left\\{\left.\limsup_{n\to\infty}\frac{1}{n}D^{\eta_{n}}_{\max,\mathbb{F}}\left(\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)\;\rule{0.0pt}{9.5pt}\right|\;\lim_{n\to\infty}\eta_{n}=0\right\\},$

where in the third line we observed that
$\zeta_{n}/\mathcal{E}_{n}(\rho^{\otimes n})+\varepsilon_{n}$ tends to $0$ for
any sequence $(\zeta_{n})_{n}$ tending to $0$. Using the asymptotic
equipartition property [16, 25]

$\displaystyle\inf_{(\zeta_{n})_{n}}\left\\{\left.\limsup_{n\to\infty}\frac{1}{n}D^{\zeta_{n}}_{\max,\mathbb{F}}(\omega^{\otimes\left\lfloor
rn\right\rfloor})\;\rule{0.0pt}{9.5pt}\right|\;\lim_{n\to\infty}\zeta_{n}=0\right\\}=r\,D^{\infty}_{\mathbb{F}}(\omega)$
(A20)

on both sides of (A19) concludes the converse part of the Theorem.

Achievability. — From the results of Brandão–Plenio [16], and in particular
from the aforementioned equipartition property, we know two things:

  1. 1)

([16, Proposition II.1 and Corollary III.2]; also [25, Theorem 1] ) For any
$\omega$, any $r>0$ and any $c>D^{\infty}_{\mathbb{F}}(\omega)$, there exists
a sequence of states $(\omega_{n})_{n}$ such that
$F\\!\left(\omega_{n},\,\omega^{\otimes\left\lfloor
rn\right\rfloor}\right)\geq 1-\varepsilon_{n}$ with
$\lim_{n\to\infty}\varepsilon_{n}=0$ and such that

$\displaystyle\limsup_{n\to\infty}\frac{1}{n}\log\left(1+R^{g}_{\mathbb{F}}(\omega_{n})\right)=\limsup_{n\to\infty}\frac{1}{n}D_{\max,\mathbb{F}}(\omega_{n})=cr.$
(A21)

  2. 2)

([16, Corollary III.3; cf. also proof of Theorem I therein]) For any $\rho$
and any $d<D^{\infty}_{\mathbb{F}}(\rho)$, there exists a sequence of POVM
elements $(A_{n})_{n}$ satisfying $0\leq A_{n}\leq\mathbbm{1}$ for all $n$,
and moreover $\operatorname{Tr}\left(A_{n}\rho^{\otimes n}\right)\geq
1-\delta_{n}$, where the sequence $(\delta_{n})_{n}$ is such that

$\displaystyle\limsup_{n\to\infty}\delta_{n}=\delta<1$ (A22)

and222The proof of [16] actually makes the stronger statement that we can take
$-\frac{1}{n}\log\sup_{\sigma\in\mathbb{F}_{n}}\operatorname{Tr}(A_{n}\sigma)=d$
for all $n$. We do not need this here, hence we leave the statement in the
above more general form.

$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\log\sup_{\sigma\in\mathbb{F}_{n}}\operatorname{Tr}(A_{n}\sigma)=d.$
(A23)

We define the shorthands

$\displaystyle a_{n}$
$\displaystyle\coloneqq\sup_{\sigma\in\mathbb{F}_{n}}\operatorname{Tr}(A_{n}\sigma),$
(A24) $\displaystyle\lambda_{n}$ $\displaystyle\coloneqq
1+R^{g}_{\mathbb{F}}(\omega_{n}).$

Let us use the above to construct a probabilistic protocol which transforms
$n$ copies of $\rho$ to $\left\lfloor rn\right\rfloor$ copies of $\omega$ for
some $r$. We start by defining the sequence of maps

$\displaystyle\mathcal{E}_{n}(X)\coloneqq\operatorname{Tr}(A_{n}X)\,\omega_{n}+\mu_{n}\operatorname{Tr}[(\mathbbm{1}-A_{n})X]\,\pi_{n},$
(A25)

where $\mu_{n}\in[0,1]$ are some parameters to be chosen later, $\omega_{n}$
are the states appearing in (A21), and $\pi_{n}$ are states such that

$\displaystyle\frac{\omega_{n}+(\lambda_{n}-1)\pi_{n}}{\lambda_{n}}=\sigma_{n}\in\mathbb{F}_{\left\lfloor
rn\right\rfloor},$ (A26)

which exist by definition of $R^{g}_{\mathbb{F}}$. The maps are clearly
completely positive and trace non-increasing by construction.

We will now show that, for any choice of a sequence $(\mu_{n})_{n}$ such that

$\displaystyle\mu_{n}\geq\frac{\lambda_{n}-1}{a_{n}^{-1}-1}$ (A27)

for all $n$, the sequence of maps $(\mathcal{E}_{n})_{n}$ is asymptotically
resource non-generating. This can be seen by observing that, for any
$\sigma\in\mathbb{F}_{n}$, it holds that

$\displaystyle\mathcal{E}_{n}(\sigma)$
$\displaystyle=\operatorname{Tr}(A_{n}\sigma)\,\omega_{n}+\mu_{n}(1-\operatorname{Tr}(A_{n}\sigma)\,)\pi_{n}$
(A28)
$\displaystyle=\operatorname{Tr}(A_{n}\sigma)\,\lambda_{n}\frac{\omega_{n}+(\lambda_{n}-1)\pi_{n}}{\lambda_{n}}+\big{[}\mu_{n}\big{(}1-\operatorname{Tr}(A_{n}\sigma)\big{)}-\operatorname{Tr}(A_{n}\sigma)\,(\lambda_{n}-1)\big{]}\pi_{n}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(i)}}}}{{\leq}}\operatorname{Tr}(A_{n}\sigma)\,\lambda_{n}\,\sigma_{n}+\big{[}\mu_{n}\big{(}1-\operatorname{Tr}(A_{n}\sigma)\big{)}-\operatorname{Tr}(A_{n}\sigma)\,(\lambda_{n}-1)\big{]}\frac{\lambda_{n}}{\lambda_{n}-1}\sigma_{n}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(ii)}}}}{{\leq}}\operatorname{Tr}(A_{n}\sigma)\,\lambda_{n}\frac{\lambda_{n}}{\lambda_{n}-1}\,\sigma_{n}+\big{[}\mu_{n}\big{(}1-\operatorname{Tr}(A_{n}\sigma)\big{)}-\operatorname{Tr}(A_{n}\sigma)\,(\lambda_{n}-1)\big{]}\frac{\lambda_{n}}{\lambda_{n}-1}\sigma_{n}$
$\displaystyle=\big{[}\mu_{n}\big{(}1-\operatorname{Tr}(A_{n}\sigma)\big{)}+\operatorname{Tr}(A_{n}\sigma)\big{]}\frac{\lambda_{n}}{\lambda_{n}-1}\sigma_{n}$
$\displaystyle=\big{[}\\!\operatorname{Tr}\mathcal{E}_{n}(\sigma)\big{]}\,\frac{\lambda_{n}}{\lambda_{n}-1}\sigma_{n}.$

Here, in (i) we used the fact that
$\pi_{n}\leq\frac{\lambda_{n}}{\lambda_{n}-1}\sigma_{n}$, which follows from
Eq. (A26), as well as that
$\mu_{n}\big{(}1-\operatorname{Tr}(A_{n}\sigma)\big{)}-\operatorname{Tr}(A_{n}\sigma)\,(\lambda_{n}-1)\geq
0$, which is due to the assumption in Eq. (A27). In (ii), we simply used that
$\frac{\lambda_{n}}{\lambda_{n}-1}\geq 1$. The above guarantees that

$\displaystyle\lim_{n\to\infty}\,R^{g}_{\mathbb{F}}\left(\frac{\mathcal{E}_{n}(\sigma)}{\operatorname{Tr}\mathcal{E}_{n}(\sigma)}\right)\leq\lim_{n\to\infty}\left(\frac{\lambda_{n}}{\lambda_{n}-1}-1\right)=0,$
(A29)

where the last identity follows from the observation that $\lambda_{n}$
diverges exponentially, as is apparent from (A21) and (A24).

Crucially, the probability of success of these operations satisfies

$\displaystyle p_{n}=\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes n})$
$\displaystyle=\operatorname{Tr}\left(A_{n}\rho^{\otimes
n}\right)+\mu_{n}\left(1-\operatorname{Tr}\left(A_{n}\rho^{\otimes
n}\right)\right)$ (A30) $\displaystyle\geq 1-\delta_{n}(1-\mu_{n}),$

while the transformation fidelity is at least

$\displaystyle F\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\omega^{\otimes\left\lfloor rn\right\rfloor}\right)$
$\displaystyle\geq\frac{\operatorname{Tr}\left(A_{n}\rho^{\otimes
n}\right)}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})}\,F(\omega_{n},\omega^{\otimes\left\lfloor rn\right\rfloor})$ (A31)
$\displaystyle\geq\frac{1-\delta_{n}}{1-\delta_{n}+\mu_{n}\delta_{n}}(1-\varepsilon_{n}),$

where the first line follows by the concavity of the fidelity. It follows
that, by decreasing $\mu_{n}$, we can decrease the error in this
transformation; on the other hand, the probability of success will never go
below $1-\delta>0$ no matter how much $\mu_{n}$ is decreased, since
$\delta_{n}\xrightarrow{\vphantom{.}\smash{{\raisebox{-1.8pt}{\scriptsize{$n\\!\to\\!\infty$}}}}}\delta<1$.

Let us then fix

$\displaystyle\mu_{n}\coloneqq\frac{\lambda_{n}-1}{a_{n}^{-1}-1}$ (A32)

and pick a rate

$\displaystyle
r=\frac{D^{\infty}_{\mathbb{F}}(\rho)}{D^{\infty}_{\mathbb{F}}(\omega)}-\chi$
(A33)

for some $\chi>0$. For all $\xi,\xi^{\prime}>0$ and for all large enough $n$,
it then holds that

$\displaystyle\log\lambda_{n}$ $\displaystyle\leq
rn\,D^{\infty}_{\mathbb{F}}(\omega)+n\xi$ (A34) $\displaystyle\leq
n\,D^{\infty}_{\mathbb{F}}(\rho)-n\chi D^{\infty}_{\mathbb{F}}(\rho)+n\xi$

and

$\displaystyle\log a_{n}^{-1}$ $\displaystyle\geq
n\,D^{\infty}_{\mathbb{F}}(\rho)-n\xi^{\prime}.$ (A35)

Picking $\xi^{\prime}=\xi=\chi D^{\infty}_{\mathbb{F}}(\rho)/4$ , we get

$\displaystyle\lim_{n\to\infty}\,\mu_{n}$
$\displaystyle=\lim_{n\to\infty}\,\frac{\lambda_{n}-1}{a_{n}^{-1}-1}$ (A36)
$\displaystyle\leq\lim_{n\to\infty}\,\frac{2^{nD^{\infty}_{\mathbb{F}}(\rho)\left(1-\chi+\chi/4\right)}-1}{2^{nD^{\infty}_{\mathbb{F}}(\rho)\left(1-\chi/4\right)}-1}$
$\displaystyle=\lim_{n\to\infty}\,\frac{2^{nD^{\infty}_{\mathbb{F}}(\rho)\left(1-\chi/4\right)-nD^{\infty}_{\mathbb{F}}(\rho)\chi/2}-1}{2^{nD^{\infty}_{\mathbb{F}}(\rho)\left(1-\chi/4\right)}-1}$
$\displaystyle=\lim_{n\to\infty}\,2^{-nD^{\infty}_{\mathbb{F}}(\rho)\chi/2}$
$\displaystyle=0.$

Altogether, the probabilistic protocol $(\mathcal{E}_{n})_{n}$ with the choice
of $\mu_{n}$ as in Eq. (A32) performs the transformation $\rho\to\omega$ at
the rate
$\frac{D^{\infty}_{\mathbb{F}}(\rho)}{D^{\infty}_{\mathbb{F}}(\omega)}-\chi$
with error satisfying

$\displaystyle\limsup_{n\to\infty}\,1-F\left(\frac{\mathcal{E}_{n}(\rho^{\otimes
n})}{\operatorname{Tr}\mathcal{E}_{n}(\rho^{\otimes
n})},\omega^{\otimes\left\lfloor rn\right\rfloor}\right)$ $\displaystyle\leq
1-\liminf_{n\to\infty}\,\frac{1-\delta_{n}}{1-\delta_{n}+\mu_{n}\delta_{n}}(1-\varepsilon_{n})$
(A37)
$\displaystyle=1-\liminf_{n\to\infty}\left(1+\frac{\mu_{n}\delta_{n}}{1-\delta_{n}}\right)^{-1}(1-\varepsilon_{n})$
$\displaystyle=0$

and probability of success

$\displaystyle\liminf_{n\to\infty}\,p_{n}$ $\displaystyle\geq
1-\limsup_{n\to\infty}\,\delta_{n}(1-\mu_{n})$ (A38) $\displaystyle=1-\delta$
$\displaystyle>0.$

Since $\chi$ was arbitrary, the result follows. ∎

######  Lemma A3 (Probabilistic monotonicity of max-relative entropy).

For any probabilistic operation $\mathcal{E}\in\mathbb{O}_{{\rm RNG},\delta}$
and any $\eta<\operatorname{Tr}\mathcal{E}(\rho)$, it holds that

$\displaystyle D^{\eta}_{\max,\mathbb{F}}(\rho)\geq
D^{\eta/\operatorname{Tr}\mathcal{E}(\rho)}_{\max,\mathbb{F}}\left(\frac{\mathcal{E}(\rho)}{\operatorname{Tr}\mathcal{E}(\rho)}\right)+\log\left(\operatorname{Tr}\mathcal{E}(\rho)-\eta\right)-\log(1+\delta).$
(A39)

###### Proof.

Consider any state $\rho^{\prime}$ with
$\frac{1}{2}\left\lVert\rho-\rho^{\prime}\right\rVert_{\,1}\leq\eta$. The
first step is to leverage the strong monotonicity of the robustness [46].
Explicitly, write
$2^{D_{\max,\mathbb{F}}(\rho^{\prime})}=\inf\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\rho^{\prime}\leq\lambda\sigma,\;\sigma\in\mathbb{F}\right\\}$
and let $\sigma\in\mathbb{F}$ be any feasible solution to this problem. Then

$\displaystyle\frac{\mathcal{E}(\rho^{\prime})}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}$
$\displaystyle\leq\lambda\,\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}$
(A40)
$\displaystyle=\lambda\,\frac{\operatorname{Tr}\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}\,\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\sigma)}$
$\displaystyle\leq\lambda\,(1+\delta)\,\frac{\operatorname{Tr}\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}\,\sigma^{\prime}$
$\displaystyle\leq\lambda\,(1+\delta)\,\frac{1}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}\,\sigma^{\prime},$

where we used the fact that $\mathcal{E}$ is a positive and trace–non-
increasing map, as well as that
$\frac{\mathcal{E}(\sigma)}{\operatorname{Tr}\mathcal{E}(\sigma)}\leq(1+\delta)\sigma^{\prime}$
for some $\sigma^{\prime}\in\mathbb{F}$ by definition of $\mathbb{O}_{{\rm
RNG},\delta}$. Optimising over all feasible $\lambda$ gives

$\displaystyle
D_{\max}\left(\frac{\mathcal{E}(\rho^{\prime})}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}\right)\leq
D_{\max,\mathbb{F}}(\rho^{\prime})-\log\operatorname{Tr}\mathcal{E}(\rho^{\prime})+\log(1+\delta).$
(A41)

Now, since
$\frac{1}{2}\left\lVert\rho-\rho^{\prime}\right\rVert_{\,1}\leq\eta$, we have
$\lvert\operatorname{Tr}\mathcal{E}(\rho)-\operatorname{Tr}\mathcal{E}(\rho^{\prime})\rvert\leq\eta$,
so we can bound
$\operatorname{Tr}\mathcal{E}(\rho^{\prime})\geq\operatorname{Tr}\mathcal{E}(\rho)-\eta$.
Furthermore, for any two states and any trace non-increasing positive map
$\mathcal{E}$ an application of the data processing inequality for the
generalised trace distance [47, Proposition 3.8] shows that [23, proof of
Proposition 6]

$\displaystyle\frac{1}{2}\left\lVert\frac{\mathcal{E}(\rho)}{\operatorname{Tr}\mathcal{E}(\rho)}-\frac{\mathcal{E}(\rho^{\prime})}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}\right\rVert_{\,1}\leq\frac{\frac{1}{2}\left\lVert\rho-\rho^{\prime}\right\rVert_{\,1}}{\operatorname{Tr}\mathcal{E}(\rho)},$
(A42)

which altogether gives

$\displaystyle
D^{\eta/\operatorname{Tr}\mathcal{E}(\rho)}_{\max,\mathbb{F}}\left(\frac{\mathcal{E}(\rho)}{\operatorname{Tr}\mathcal{E}(\rho)}\right)\leq
D_{\max,\mathbb{F}}\left(\frac{\mathcal{E}(\rho^{\prime})}{\operatorname{Tr}\mathcal{E}(\rho^{\prime})}\right)\leq
D_{\max,\mathbb{F}}(\rho^{\prime})-\log(\operatorname{Tr}\mathcal{E}(\rho)-\eta)+\log(1+\delta).$
(A43)

Optimising over all feasible $\rho^{\prime}$ yields the statement of the
Lemma. ∎

##  III Equivalence of strong converse and probabilistic distillation in
entanglement theory

Recall that $E_{d,\mathbb{O}}^{p=1}(\rho)\coloneqq r_{p=1}(\rho\to\Phi_{+})$,
$E_{d,\mathbb{O}}^{p=1,\dagger}(\rho)\coloneqq
r_{p=1}^{\dagger}(\rho\to\Phi_{+})$, and
$E_{d,\mathbb{O}}^{p>0}(\rho)\coloneqq r_{p>0}(\rho\to\Phi_{+})$.

In order to study transformations of entangled states under different free
operations in entanglement theory, in particular LOCC, it is important to note
that now it does not suffice to study sub-normalised quantum operations
$\mathcal{E}$ like we did previously for RNG operations, as we must ensure
that the overall instrument $(\mathcal{E},\mathcal{E}^{\prime})$ can be
realised as a free protocol. To simplify the considerations, we make two
observations:

  1. (i)

Let $\mathbb{O}$ be any class of operations that includes local operations and
shared randomness (LOSR). Due to isotropic twirling [48], the output of any
distillation protocol with $\mathbb{O}$ can be assumed without loss of
generality to be of the form

$\displaystyle\varsigma(m,\varepsilon)\coloneqq\left(1-\varepsilon\right)\Phi_{2}^{\otimes
m}+\varepsilon\tau_{2^{m}},$ (A44)

where

$\displaystyle\Phi_{d}\coloneqq\ket{\Phi_{d}}\\!\bra{\Phi_{d}}\,,\qquad\ket{\Phi_{d}}\coloneqq\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii}\,,\qquad\tau_{d}\coloneqq\frac{\mathbbm{1}-\Phi_{d}}{d^{2}-1}\,.$
(A45)

Now, given a positive integer $m\in\mathbb{N}$ representing the number of
local qubits and two error probabilities $\xi,\delta\in[0,1]$ such that
$\xi+\delta\leq 1$, let us define the $(2^{m}+1)\times(2^{m}+1)$ bipartite
state

$\displaystyle\omega(m,\xi,\delta)\coloneqq$ $\displaystyle\
\left(1-\xi-\delta\right)\Phi_{2}^{\otimes
m}+\xi\,\tau_{2^{m}}+\delta\ket{ee}\\!\bra{ee}$ (A46) $\displaystyle=$
$\displaystyle\ \left(1-\frac{\xi}{1-4^{-m}}-\delta\right)\Phi_{2}^{\otimes
m}+\frac{\xi}{1-4^{-m}}\left(\frac{\mathbbm{1}}{4}\right)^{\otimes
m}+\delta\ket{ee}\\!\bra{ee},$

where $\ket{e}$ represents a local error flag that is orthogonal to the
Hilbert space corresponding to the $m$ qubits.

  2. (ii)

Instead of probabilistic transformations into isotropic states as in (A44), we
can then consider deterministic transformations into $\omega(m,\xi,q)$. This
is because, on the one hand, for any probabilistic distillation protocol which
results in the state $\varsigma(m,\varepsilon)$ with some probability $p$, if
the protocol fails, Alice and Bob can simply prepare the state
$\ket{ee}\\!\bra{ee}$ with probability $1-p$, thus obtaining the state
$\omega(m,\varepsilon p,1-p)$ deterministically. On the other hand, any
deterministic protocol which results in the state $\omega(m,\xi,\delta)$ can
be easily modified to yield the state $\varsigma(m,\xi/(1-\delta))$ with
probability $1-\delta$ by simply measuring whether an error occurred or not.
In particular, for any non-vanishing $p$, the error $\varepsilon$ vanishes in
the limit $m\to\infty$ if and only if so does $\xi$.

Let $\mathbb{O}$ be a class of free operations in entanglement theory that
includes LOCC. Given as above $\xi,\delta\in[0,1)$ such that $\xi+\delta<1$,
let us define the corresponding one-shot probabilistically distillable
entanglement by

$\displaystyle
E_{d,\mathbb{O}}^{(1),\xi,\delta}(\rho)\coloneqq\max\left\\{\,m\in\mathbb{N}\;\rule{0.0pt}{9.5pt}\left|\;\exists\
\mathcal{E}\in\mathbb{O}\cap{\rm CPTP}:\
\mathcal{E}(\rho)=\omega(m,\xi,\delta)\right.\right\\},$ (A47)

where $\rho$ is a generic state, and $\omega(m,\xi,\delta)$ is defined by
(A46). Asymptotically, we can set

$\displaystyle
E_{d,\mathbb{O}}^{\xi,\delta}(\rho)\coloneqq\liminf_{n\to\infty}\frac{1}{n}\,E_{d,\mathbb{O}}^{(1),\xi,\delta}(\rho^{\otimes
n}).$ (A48)

######  Lemma A4.

Let $m\in\mathbb{N}$ be a positive integer, and consider $\xi,\delta\in[0,1]$
with $\xi+\delta\leq 1$. Then, for all $\lambda\in[0,1]$ and all
$k\in\mathbb{N}$ with $k<m$ the following transitions are possible via LOCC:

$\displaystyle\omega(m,\xi,\delta)\to\omega(m^{\prime}\\!,\xi^{\prime}\\!,\delta^{\prime})\,,\qquad$
$\displaystyle m^{\prime}=m\,,\quad$
$\displaystyle\xi^{\prime}=\xi+\lambda\delta(1-2^{-m})\,,\quad$
$\displaystyle\delta^{\prime}=\delta(1-\lambda)\,;$ (A49)
$\displaystyle\omega(m,\xi,\delta)\to\omega(m^{\prime\prime}\\!\\!,\xi^{\prime\prime}\\!\\!,\delta^{\prime\prime})\,,\qquad$
$\displaystyle m^{\prime\prime}=m-k\,,\quad$
$\displaystyle\xi^{\prime\prime}=\frac{1-4^{-m+k}}{1-4^{-m}}\,\frac{\xi}{2^{k}}\,,\quad$
$\displaystyle\delta^{\prime\prime}=\delta+\frac{1-2^{-k}}{1-4^{-m}}\,\xi\,.$
(A50)

###### Proof.

The transformation of type (A49) is obtained by the following procedure:

  1. (i)

Alice and Bob check if the system is in the error state $\ket{e}$; if it is
not, they do nothing;

  2. (ii)

if it is, with some probability $\lambda$ they prepare the separable [48]
state $2^{-m}\Phi_{2}^{\otimes m}+(1-2^{-m})\tau_{2^{m}}$; with probability
$1-\lambda$, they do nothing.

The transformation of type (A50), instead, is obtained by a different
procedure. Intuitively, we sacrifice $k$ pairs of qubits to try and determine
whether the system is in the maximally entangled state or nor. More precisely,
we run the following protocol:

  1. (i)

Alice and Bob check if the system is in the error state $\ket{e}$; if it is,
they do nothing;

  2. (ii)

if it is not, then they measure the first $r$ qubits in the computational
basis;

  3. (iii)

if upon communicating the outcomes they find that some of them differ, they
declare an error, re-preparing the rest of the system in the state $\ket{ee}$;

  4. (iv)

if instead all outcomes are found to coincide pairwise, they do nothing on the
remaining $m-k$ pairs of qubits.

The elementary computations needed to verify that these protocols yield the
claimed transformations are left to the reader. ∎

######  Proposition A5.

Let $\mathbb{O}$ be a class of free operations in entanglement theory that is
closed under composition with LOCCs. Then, for a fixed $\rho$, the distillable
entanglement $E_{d,\mathbb{O}}^{\xi,\delta}(\rho)$ depends on
$\xi,\delta\in[0,1)$ only through the sum $\xi+\delta$, or equivalently
through $p(1-\varepsilon)$.

###### Proof.

For asymptotically large $m$ and fixed $k$, transformations (A49)–(A50)
simplify considerably, becoming

$\displaystyle\omega(m,\xi,\delta)\to\omega(m^{\prime}\\!,\xi^{\prime}\\!,\delta^{\prime})\,,\qquad$
$\displaystyle m^{\prime}=m\,,\quad$
$\displaystyle\xi^{\prime}\approx\xi+\lambda\delta\,,\quad$
$\displaystyle\delta^{\prime}=\delta(1-\lambda)\,;$ (A51)
$\displaystyle\omega(m,\xi,\delta)\to\omega(m^{\prime\prime}\\!\\!,\xi^{\prime\prime}\\!\\!,\delta^{\prime\prime})\,,\qquad$
$\displaystyle m^{\prime\prime}=m-k\,,\quad$
$\displaystyle\xi^{\prime\prime}\approx\frac{\xi}{2^{k}}\,,\quad$
$\displaystyle\delta^{\prime\prime}\approx\delta+(1-2^{-k})\,\xi\,.$ (A52)

With these transformations, it is clear that we can turn a pair $(\xi,\delta)$
with $\xi+\delta<1$ into another pair $(\xi^{\prime},\delta^{\prime})$ with
$\xi^{\prime}+\delta^{\prime}<1$ if and only if
$\xi+\delta=\xi^{\prime}+\delta^{\prime}$. ∎

A corollary of the above is Theorem 2 in the main text, whose extended
statement we present here.

######  Theorem 2.

Let $\mathbb{O}$ be any class of operations which is closed under composition
with LOCC, i.e. such that $\mathcal{E}\in\mathbb{O},\,\mathcal{F}\in\rm
LOCC\,\Rightarrow\,\mathcal{F}\circ\mathcal{E}\in\mathbb{O}$. This includes in
particular the set $\rm LOCC$ itself.

Consider then any sequence of entanglement transformation protocols
$(\mathcal{E}_{n})_{n}$ with associated error
$\limsup_{n\to\infty}\varepsilon_{n}=\varepsilon<1$ and probability of success
$\liminf_{n\to\infty}p_{n}=p>0$. Then, then there exists another sequence of
entanglement distillation protocols $(\mathcal{E}^{\prime}_{n})_{n}$ with
error
$\limsup_{n\to\infty}\varepsilon^{\prime}_{n}\eqqcolon\varepsilon^{\prime}$
and probability $\liminf_{n\to\infty}p^{\prime}_{n}\eqqcolon p^{\prime}$ if
and only if

$\displaystyle p\,(1-\varepsilon)=p^{\prime}\,(1-\varepsilon^{\prime}).$ (A53)

Hence, for all states $\rho$ and all sets of operations $\mathbb{O}$ as above,

$\displaystyle
E_{d,\mathbb{O}}^{p>0}(\rho)=E_{d,\mathbb{O}}^{p=1,\dagger}(\rho)=E_{d,\mathbb{O}}^{p>0,\dagger}(\rho).$
(A54)

In particular, for all non-entangling operations it holds that

$\displaystyle D^{\infty}_{\rm SEP}(\rho)$ $\displaystyle=E_{d,{\rm
NE}}^{p>0}(\rho)=E_{d,{\rm NE}}^{p=1,\dagger}(\rho)=E_{d,{\rm
NE}}^{p=1,\ddagger}(\rho)$ (A55) $\displaystyle=E_{d,{\rm
ANE}}^{p>0}(\rho)=E_{d,{\rm ANE}}^{p=1,\dagger}(\rho)=E_{d,{\rm
ANE}}^{p=1,\ddagger}(\rho).$

The last statement in the above can be seen from the fact that
$D^{\infty}_{\rm
SEP}(\rho)=E^{p=1,\dagger}_{d,\rm(A)NE}(\rho)=E^{p=1,\ddagger}_{d,\rm(A)NE}(\rho)$
[9] (cf. [16, Corollary III.3]), where the latter term denotes the strong
converse distillable entanglement defined using the slightly stronger notion
of a strong converse rate found in Eq. (A7).333Whether this equivalence
between the two definitions of the strong converse holds also for other sets
of free operations $\mathbb{O}$ is an interesting open problem. A more direct
proof of the fact that the rate $D^{\infty}_{\rm SEP}(\rho)$ can be achieved
by probabilistic non-entangling operations (as opposed to _asymptotically_
non-entangling ones as in Theorem 1) can be obtained by following the proof of
Theorem 1 but choosing $\omega_{n}=\Phi_{2}^{\otimes\left\lfloor
rn\right\rfloor}$ and realising that we can then pick $\pi_{n}$ to be
separable states [24, 49], so no resources need to be generated whatsoever.

We sincerely appreciate all readers who actually read all the way until this
point and didn’t just scroll down to see how long the paper is.