††thanks: Both authors contributed equally to this work.††thanks: Both authors
contributed equally to this work.

# One-Shot Manipulation of Dynamical Quantum Resources

Bartosz Regula  School of Physical and Mathematical Sciences, Nanyang
Technological University, 637371, Singapore<EMAIL_ADDRESS>Ryuji
Takagi  School of Physical and Mathematical Sciences, Nanyang Technological
University, 637371, Singapore  Center for Theoretical Physics and Department
of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA<EMAIL_ADDRESS>
###### Abstract

We develop a unified framework to characterize one-shot transformations of
dynamical quantum resources in terms of resource quantifiers, establishing
universal conditions for exact and approximate transformations in general
resource theories. Our framework encompasses all dynamical resources
represented as quantum channels, including those with a specific structure —
such as boxes, assemblages, and measurements — thus immediately applying in a
vast range of physical settings. For the particularly important manipulation
tasks of distillation and dilution, we show that our conditions become
necessary and sufficient for broad classes of important theories, enabling an
exact characterization of these tasks and establishing a precise connection
between operational problems and resource monotones based on entropic
divergences. We exemplify our results by considering explicit applications to:
quantum communication, where we obtain exact expressions for one-shot quantum
capacity and simulation cost assisted by no-signalling, separability-
preserving, and positive partial transpose–preserving codes; as well as to
nonlocality, contextuality, and measurement incompatibility, where we present
operational applications of a number of relevant resource measures.

Introduction. — Manipulating different resources under physical restrictions
underpins many quantum technologies, and the precise understanding of
physically realizable transformations is a fundamental question both from
theoretical and practical points of view. Quantum resource theories Horodecki
and Oppenheim (2012); Coecke  _et al._ (2016); Chitambar and Gour (2019)
provide a platform through which the quantification and manipulation of
resources can be explicitly considered, enabling the study of resource
manipulation in a range of settings of interest Horodecki  _et al._ (2009);
Streltsov  _et al._ (2017); Gour and Spekkens (2008); Brandão and Gour (2015).

Due to the inherent generality of the framework of resource theories, the
formalism could be expected to provide unifying results on resource
manipulation that hold for diverse classes of resources. Developing such
versatile general resource theories allows one to extract common features
shared by different physical phenomena and clarify the peculiarities that
differentiate one setting from others Brandão and Gour (2015); Liu  _et al._
(2017); Gour (2017); Anshu  _et al._ (2018); Regula (2018); Lami  _et al._
(2018); Takagi  _et al._ (2019); Takagi and Regula (2019); Uola  _et al._
(2019); Liu  _et al._ (2019); Takagi and Tajima (2020); Regula  _et al._
(2020); Fang and Liu (2020a); Zhou and Buscemi (2020); Vijayan  _et al._
(2020); Regula and Takagi (2021); Fang and Liu (2020b); Kuroiwa and Yamasaki
(2020); Regula  _et al._ (2021). However, previous approaches to these
problems suffer from a number of limitations. Many works have focused on the
static, rather than dynamical resources, in the sense that only the
manipulation of quantum states was considered, and a more general approach
that incorporates the ability to manipulate the dynamics of the systems was
not established. Recent works have begun to describe channel-based theories
Takagi and Regula (2019); Liu and Yuan (2020); Liu and Winter (2019); Gour and
Winter (2019); Regula and Takagi (2021); Fang and Liu (2020b); Takagi  _et
al._ (2020); Takagi (2020), but often focused on the investigation of specific
theories such as entanglement, coherence, or quantum memories Ben Dana  _et
al._ (2017); Díaz  _et al._ (2018); Rosset  _et al._ (2018); Bäuml  _et al._
(2019); Gour and Scandolo (2020); Saxena  _et al._ (2020); Pirandola  _et al._
(2017); Faist and Renner (2018); Theurer  _et al._ (2019, 2020); Yuan  _et
al._ (2021); Wang  _et al._ (2019a); Wang and Wilde (2019), or obtained
results that only apply in the idealized asymptotic limit Liu and Yuan (2020).
The characterization of dynamical resource theories is significantly more
complex than the manipulation of underlying states, and many questions remain
unanswered. Notably, the precise characterization of convertibility between
two quantum channels in the practical one-shot scenario has been an
outstanding problem to be addressed.

An important aspect of dynamical resources is that they can describe a much
broader range of settings than the commonly considered manipulation of quantum
states with general quantum channels. For instance, Bell nonlocality Bell
(1964); Brunner  _et al._ (2014) as well as quantum contextuality Kochen and
Specker (1967); Cabello  _et al._ (2014) have been investigated within formal
resource-theoretic settings Geller and Piani (2014); de Vicente (2014); Wolfe
_et al._ (2020); Horodecki  _et al._ (2015); Duarte and Amaral (2018), but the
specific restrictions on the structure of channels allowed in these settings
prevented them from being integrated into most of the previous general quantum
resource frameworks. The recent works of Refs. Schmid  _et al._ (2020); Rosset
_et al._ (2020) considered an approach to quantum nonlocality that encompasses
any type of input and output systems, suggesting that the establishment of a
broader framework of channel resource theories might be possible, and opening
up the potential to address the manipulation of dynamical resources in a
unified manner.

Here, we achieve such a unified description by providing a universal
characterization of one-shot resource transformations with finite error in
terms of fundamental resource measures, valid in general resource theories of
quantum channels. By allowing arbitrary restrictions on the set of channels
under consideration, our results apply not only to settings previously studied
in resource theories of channels — e.g., entanglement, coherence, magic,
quantum thermodynamics, and quantum communication — but also to other
dynamical resources such as Bell nonlocality, contextuality, steering Gallego
and Aolita (2015); Uola  _et al._ (2020), measurement incompatibility
Heinosaari  _et al._ (2016), and many others Schmid  _et al._ (2020); Rosset
_et al._ (2020), offering a very general quantitative description of the
fundamental task of resource manipulation. As an important subclass of
manipulation tasks, we study resource distillation and dilution, where we
present additional bounds and results that provide necessary and sufficient
conditions for resource conversion in many relevant cases. This provides an
exact characterization of optimal one-shot rates achievable in these tasks, in
which case important resource measures — such as resource robustness and
hypothesis testing relative entropy — are endowed with explicit operational
meaning. We further establish tight benchmarks on the achievable fidelity of
distillation. Finally, we discuss insights provided by our general results
into several physical scenarios. We first apply our results to quantum
communication and find exact expressions for quantum capacity and simulation
cost for communication assisted by no-signalling codes and codes preserving
separability or the positivity of the partial transpose (PPT). We then discuss
the application to nonlocality, contextuality, and measurement
incompatibility, where we link resource measures previously introduced in
other settings to approximate resource transformations.

We focus on discussing our main results below, and the technical proofs are
deferred to the Supplemental Material 111See the Supplemental Material for
detailed proofs of our main results, which includes Refs. Sion (1958); Fuchs
and van de Graaf (1999); Kretschmann and Werner (2004); Johnston  _et al._
(2018); Brandão and Plenio (2010); Brandão and Datta (2011).

Manipulation of dynamical resources. — Let $\mathbb{O}_{\rm all}$ be the set
of valid channels allowed in the given physical setting; this can be the set
of all quantum channels or a subset thereof, allowing us to take into
consideration possible restrictions on the types of channels being
manipulated. Each resource theory also designates a subset of channels that
are considered to be available for free, and we denote the given free channels
as $\mathbb{O}\subseteq\mathbb{O}_{\rm all}$. We impose mild assumptions that
the underlying Hilbert spaces are always finite dimensional and, for a fixed
dimension, the set of free channels $\mathbb{O}$ is convex and closed Brandão
and Gour (2015).

General transformations of quantum channels are described by quantum
superchannels Chiribella  _et al._ (2008). Since superchannels need not
preserve specific channel structures in general Schmid  _et al._ (2020), we
consider the subset of superchannels that map the set of allowed channels
$\mathbb{O}_{\rm all}$ to allowed output channels $\mathbb{O}_{\rm
all}^{\prime}$, defined as $\SS_{\rm all}\coloneqq\\{\Theta:\mathbb{O}_{\rm
all}\rightarrow\mathbb{O}_{\rm all}^{\prime}\\}$. We can now take a subset of
$\mathbb{O}_{\rm all}^{\prime}$ and consider it as the free channels in the
output space, which we denote by $\mathbb{O}^{\prime}$. A subset of $\SS_{\rm
all}$ serves as the set of free transformation that can be used for the
manipulation of resources. The standard requirement for any free operation is
that it should not generate any resource, i.e., it should not create any
costly channel out of a free one. We consider the maximal set satisfying this
condition, $\SS\coloneqq\left\\{\left.\Theta\in\SS_{\rm
all}\;\rule{0.0pt}{9.5pt}\right|\;\Theta(\mathcal{M})\in\mathbb{O}^{\prime}\;\forall\mathcal{M}\in\mathbb{O}\right\\}$.

Our goal is to find the conditions for the transformation from $\mathcal{E}$
to $\mathcal{N}$ using free superchannels in $\SS$, given any two channels
$\mathcal{E}\in\mathbb{O}_{\rm all}$ and $\mathcal{N}\in\mathbb{O}_{\rm
all}^{\prime}$. In practice, the transformation can often only be achieved
approximately, especially in nonasymptotic resource manipulation. To evaluate
the inevitable error, we consider the worst-case fidelity Belavkin  _et al._
(2005); Gilchrist  _et al._ (2005) defined for two channels
$\mathcal{E}_{1},\mathcal{E}_{2}\in\mathbb{O}_{\rm all}$ as
$F(\mathcal{E}_{1},\mathcal{E}_{2})\coloneqq\min_{\rho}F(\mathrm{id}\otimes\mathcal{E}_{1}(\rho),\mathrm{id}\otimes\mathcal{E}_{2}(\rho))$
where
$F(\rho,\sigma)\coloneqq\|\sqrt{\rho}\sqrt{\vphantom{\rho}\sigma}\|_{1}^{2}$
is the fidelity.

Our aim will be to characterize the conditions on resource transformation
through the resource contents of the given channels. To this end, we introduce
two types of resource measures. They both belong to, or are closely related
to, the class of one-shot entropic quantities Renner (2005) and in particular
channel divergences Yuan (2019); Gour and Wilde (2021). The first class is
known as the robustness measures Vidal and Tarrach (1999) defined for any
$\mathcal{E}\in\mathbb{O}_{\rm all}$ as

$\displaystyle
R_{\mathbb{O};\tilde{\mathbb{O}}}(\mathcal{E})\coloneqq\inf\left\\{\left.1+r\;\rule{0.0pt}{9.5pt}\right|\;\frac{\mathcal{E}+r\mathcal{M}}{1+r}\in\mathbb{O},\mathcal{M}\in\tilde{\mathbb{O}}\right\\},$
(1)

where $\tilde{\mathbb{O}}\subseteq\mathbb{O}_{\rm all}$ is some set of
channels containing $\mathbb{O}$. The extreme case
$\tilde{\mathbb{O}}=\mathbb{O}_{\rm all}$ is known as generalized robustness
Steiner (2003); Harrow and Nielsen (2003); Takagi and Regula (2019) and
corresponds to the max-relative entropy Datta (2009), hence we denote it by
$R_{\max,\mathbb{O}}\coloneqq R_{\mathbb{O};\mathbb{O}_{\rm all}}$. The other
case of interest is the standard robustness $R_{s,\mathbb{O}}\coloneqq
R_{\mathbb{O};\mathbb{O}}$ Yuan  _et al._ (2021); Takagi (2020). We also
define the smooth robustness
$R_{\mathbb{O};\tilde{\mathbb{O}}}^{\epsilon}(\mathcal{E})\coloneqq\min\\{R_{\mathbb{O};\tilde{\mathbb{O}}}(\mathcal{E}^{\prime})|F(\mathcal{E}^{\prime},\mathcal{E})\geq
1-\epsilon,\mathcal{E}^{\prime}\in\mathbb{O}_{\rm all}\\}$ for
$0\leq\epsilon\leq 1$. The other type of measure, based on the hypothesis
testing relative entropy Buscemi and Datta (2010); Wang and Renner (2012);
Yuan (2019); Gour and Wilde (2021), is defined for
$\mathcal{E}\in\mathbb{O}_{\rm all}$ as

$\displaystyle
R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})\coloneqq\min_{\mathcal{M}\in\mathbb{O}}\max_{\psi}R_{H}^{\epsilon}(\mathrm{id}\otimes\mathcal{E}(\psi)||\mathrm{id}\otimes\mathcal{M}(\psi))$
(2)

where $R_{H}^{\epsilon}(\rho||\sigma)\coloneqq\max_{\begin{subarray}{c}0\leq
P\leq\mathbbm{1}\\\ \operatorname{Tr}(P\rho)\geq
1-\epsilon\end{subarray}}\operatorname{Tr}(P\sigma)^{-1}$ and the optimization
is restricted to pure input states $\psi$ without loss of generality. This
entropic quantity characterizes the distinguishability between $\mathcal{E}$
and the channels in the set $\mathbb{O}$. The case of $\epsilon=0$ is known as
the min-relative entropy Datta (2009), denoted as
$R_{\min,\mathbb{O}}(\mathcal{E})$.

It is also useful to introduce two classifications for the given theory
depending on the properties of $\mathbb{O}$. We say that $\mathbb{O}$ is full
dimensional if $\mathrm{span}(\mathbb{O})$ contains all channels in
$\mathbb{O}_{\rm all}$, and reduced dimensional otherwise Regula  _et al._
(2020). Intuitively, in full-dimensional theories, the set $\mathbb{O}$ is of
full measure, meaning that $R_{s,\mathbb{O}}(\mathcal{E})<\infty$ for all
$\mathcal{E}$; examples include the theory of entanglement or local operations
and shared randomness. On the other hand, reduced-dimensional theories are
equipped with a set of free channels $\mathbb{O}$ of zero measure, and the
standard robustness $R_{s,\mathbb{O}}$ can diverge, examples of which include
the theory of coherence, asymmetry, and quantum thermodynamics. In order to
characterize such resources, we will often need to consider an optimization
with respect to $\operatorname{aff}(\mathbb{O})$, the affine hull of
$\mathbb{O}$ Regula  _et al._ (2018, 2020); not , and we define in particular
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}$ (respectively,
$R_{\min,\operatorname{aff}(\mathbb{O})}$) as the hypothesis testing entropy
(min-relative entropy) minimized over $\operatorname{aff}(\mathbb{O})$ instead
of $\mathbb{O}$.

We can now state our main results which connect the resource monotones with
general one-shot resource transformations.

######  Theorem 1.

Let $\mathcal{E}\in\mathbb{O}_{\rm all}$ and $\mathcal{N}\in\mathbb{O}_{\rm
all}^{\prime}$. If there exists a free superchannel $\Theta\in\SS$ such that
$F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon$, then for any monotone
$\mathfrak{R}_{\mathbb{O}}$ it holds that
$\mathfrak{R}_{\mathbb{O}}(\mathcal{E})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$,
as well as that
$\mathfrak{R}_{\mathbb{O}}^{\delta}(\mathcal{E})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{2(\sqrt{\delta}+\sqrt{\epsilon})}(\mathcal{N})$
for any $0\leq\delta\leq 1$ where
$\mathfrak{R}_{\mathbb{O}}^{\epsilon}(\mathcal{E})\coloneqq\min\\{\mathfrak{R}_{\mathbb{O}}(\mathcal{E}^{\prime})\,|\,F(\mathcal{E}^{\prime},\mathcal{E})\geq
1-\epsilon,\,\mathcal{E}^{\prime}\in\mathbb{O}_{\rm all}\\}$.

Conversely, for any choices of $\epsilon,\delta\geq 0$ such that
$\epsilon+2\delta<1$, there exists a free superchannel $\Theta\in\SS$ such
that $F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon-2\delta$ if
$R_{H,\mathbb{O}}^{\delta}(\mathcal{E})\geq
R_{s,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$ or if
$R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})\geq
R_{\max,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$.

Here, the parameters $\epsilon,\delta$ can be used to study the trade-offs
between the error allowed in the transformation and the values of the smoothed
monotones $\mathfrak{R}_{\mathbb{O}}^{\epsilon}$ of both the input and the
output channel.

Notice that we provided two alternative achievability conditions: one using
the hypothesis testing measure $R_{H,\mathbb{O}}$ and the standard robustness
$R_{s,\mathbb{O}^{\prime}}$, and one using the affine hypothesis testing
measure $R_{H,\operatorname{aff}(\mathbb{O})}$ and the robustness
$R_{\max,\mathbb{O}^{\prime}}$. The reason for this is that the former
condition will typically trivialize in reduced-dimensional theories, while the
latter condition trivializes for full-dimensional theories.

Thm. 1 establishes the conditions for general resource transformation
universally applicable to any resource theory, including ones with specific
structures of allowed channels, reflected by choosing appropriate sets
$\mathbb{O}_{\rm all}$ and $\mathbb{O}_{\rm all}^{\prime}$. In the special
case of manipulating quantum states (channels with trivial input), we recover
the results of Ref. Zhou and Buscemi (2020) and extend them to reduced-
dimensional theories. We stress that the monotones are all convex optimization
problems and they reduce to computable semidefinite programs when $\mathbb{O}$
is characterized by semidefinite constraints Note (1).

Distillation and dilution — Two of the most important classes of resource
transformation tasks are resource distillation, where general resources are
transformed into some reference target resources, and dilution, where the
reference target resources are used to synthesize a given channel through free
transformations. In particular, one is often interested in two quantities: the
distillable resource $d^{\epsilon}_{\mathbb{O}}(\mathcal{E})$ and the resource
cost $c^{\epsilon}_{\mathbb{O}}(\mathcal{E})$; choosing a suitable class of
target reference channels $\mathbb{T}\subseteq\mathbb{O}^{\prime}_{\rm all}$,
we can define

$\displaystyle d^{\epsilon}_{\mathbb{O}}(\mathcal{E})$
$\displaystyle\coloneqq\sup\left\\{\left.\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\\!\;\rule{0.0pt}{9.5pt}\right|\;\\!F(\Theta(\mathcal{E}),\mathcal{T})\geq
1\\!-\\!\epsilon,\;\mathcal{T}\\!\in\mathbb{T},\;\Theta\in\SS\right\\}$ (3)
$\displaystyle c^{\epsilon}_{\mathbb{O}}(\mathcal{E})$
$\displaystyle\coloneqq\inf\left\\{\left.\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\\!\;\rule{0.0pt}{9.5pt}\right|\;\\!F(\mathcal{E},\Theta(\mathcal{T}))\geq
1\\!-\\!\epsilon,\;\mathcal{T}\\!\in\mathbb{T},\;\Theta\in\SS\right\\}$

where $\mathfrak{R}_{\mathbb{O}^{\prime}}$ refers to any chosen monotone — for
example, $R_{\min,\mathbb{O}^{\prime}}$, $R_{\max,\mathbb{O}^{\prime}}$, or
$R_{s,\mathbb{O}^{\prime}}$. In the discussion below, we will fix
$\mathfrak{R}_{\mathbb{O}^{\prime}}=R_{\min,\mathbb{O}^{\prime}}$ for
simplicity. The target channels $\mathbb{T}$ are often chosen as multiple
copies of some fixed reference channel, but we allow for broader types of
targets.

Notably, under suitable conditions on the reference channels $\mathbb{T}$, the
necessary and sufficient conditions of Thm. 1 coincide, yielding a precise
characterization of the resource cost.

######  Corollary 2.

If the chosen reference set satisfies
$R_{\min,\mathbb{O}^{\prime}}(\mathcal{T})=R_{s,\mathbb{O}^{\prime}}(\mathcal{T})\;\forall\mathcal{T}\in\mathbb{T}$,
then it holds that $c^{\epsilon}_{\mathbb{O}}(\mathcal{E})=\lceil
R_{s,\mathbb{O}}^{\epsilon}(\mathcal{E})\rceil_{\mathbb{T}}$.

Similarly, if the chosen reference set obeys
$R_{\min,\operatorname{aff}(\mathbb{O}^{\prime})}(\mathcal{T})=R_{\max,\mathbb{O}^{\prime}}(\mathcal{T})\;\forall\mathcal{T}\in\mathbb{T}$,
then it holds that $c^{\epsilon}_{\mathbb{O}}(\mathcal{E})=\lceil
R_{\max,\mathbb{O}}^{\epsilon}(\mathcal{E})\rceil_{\mathbb{T}}$.

Here, we used the notation $\lceil\cdot\rceil_{\mathbb{T}}$ to indicate the
smallest value greater than or equal to the argument for which there exists a
corresponding channel $\mathcal{T}\in\mathbb{T}$ — this is required, for
instance, when $\mathbb{T}$ forms a discrete set (see Ref. Note (1)).

The result establishes an operational meaning of the measures
$R_{s,\mathbb{O}}^{\epsilon}$ or $R_{\max,\mathbb{O}}^{\epsilon}$ as long as
the conditions are satisfied. This raises the question: when does a choice of
reference channels $\mathbb{T}$ satisfying
$R_{\min,\mathbb{O}}(\mathcal{T})=R_{s,\mathbb{O}}(\mathcal{T})$ or
$R_{\min,\operatorname{aff}(\mathbb{O})}(\mathcal{T})=R_{\max,\mathbb{O}}(\mathcal{T})$
exist? It emerges that this is a commonly occurring feature of resource
theories. For instance, it is satisfied by relevant choices of target channels
in resource theories of quantum memories and communication, immediately
providing a characterization of one-shot simulation cost of channels. When
discussing the transformations of quantum states, it was shown that in any
convex resource theory there exists maximal “golden states” $\phi_{\rm gold}$
such that $R_{\min,\mathbb{O}}(\phi_{\rm gold})=R_{\max,\mathbb{O}}(\phi_{\rm
gold})$ Regula  _et al._ (2020), and indeed such states satisfy all
requirements of the Corollary in theories such as quantum entanglement or
quantum coherence. In the case of quantum state manipulation, our result
recovers the considerations of Ref. Liu  _et al._ (2019), where a general
framework for quantum state resources was established.

We now turn to the case of distillation. Here, we can improve the bound in
Thm. 1 and obtain an alternative necessary condition. Importantly,
distillation is often understood as the purification of noisy resources, in
which case it is natural to consider pure reference channels $\mathbb{T}$ —
for instance, if the input space and output spaces coincide, unitary channels
can serve as targets, while if the input space is trivial, then pure-state
preparation channels can be regarded as targets. The property needed for
$\mathcal{N}$ to serve as the reference resource is that the output states for
pure input states remain pure. In such cases, we obtain the following general
conditions.

######  Theorem 3.

Let $\mathcal{E}\in\mathbb{O}_{\rm all}$, $\mathcal{N}\in\mathbb{O}_{\rm
all}^{\prime}$ and suppose $\mathrm{id}\otimes\mathcal{N}(\psi)$ is pure for
any pure state $\psi$. If there exists a free superchannel $\Theta\in\SS$ such
that $F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon$, then it holds that
$R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})\geq
R_{\min,\mathbb{O}^{\prime}}(\mathcal{N})$ and
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})\geq
R_{\min,\operatorname{aff}(\mathbb{O}^{\prime})}(\mathcal{N})$.

Conversely, there exists a free superchannel $\Theta\in\SS$ such that
$F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon-\delta$ if
$R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})\geq
R_{s,\mathbb{O}^{\prime}}^{\delta}(\mathcal{N})$ or if
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})\geq
R_{\max,\mathbb{O}^{\prime}}^{\delta}(\mathcal{N})$.

Thm. 3 adds a useful alternative characterization for distillation to the
general condition provided by Thm. 1. In particular, similarly to the case of
dilution, we can obtain the following.

######  Corollary 4.

Consider any reference set $\mathbb{T}$ such that
$\mathrm{id}\otimes\mathcal{T}(\psi)$ is pure for any pure $\psi$ and any
$\mathcal{T}\in\mathbb{T}$.

If $\mathbb{T}$ also satisfies
$R_{\min,\mathbb{O}}(\mathcal{T})=R_{s,\mathbb{O}}(\mathcal{T})\;\forall\mathcal{T}\in\mathbb{T}$,
then it holds that $d^{\epsilon}_{\mathbb{O}}(\mathcal{E})=\lfloor
R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})\rfloor_{\mathbb{T}}$.

Similarly, if the chosen reference set obeys
$R_{\min,\operatorname{aff}(\mathbb{O})}(\mathcal{T})=R_{\max,\mathbb{O}}(\mathcal{T})\;\forall\mathcal{T}\in\mathbb{T}$,
then it holds that $d^{\epsilon}_{\mathbb{O}}(\mathcal{E})=\lfloor
R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})\rfloor_{\mathbb{T}}$.

This establishes a precise characterization of distillable resource in any
resource theory for suitable target channels $\mathbb{T}$, and furthermore
gives an exact operational meaning to the resource measures
$R_{H,\mathbb{O}}^{\epsilon}$ and
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}$ in the task of distillation.
When the manipulated objects are quantum states, we recover the results of
Refs. Liu  _et al._ (2019); Regula  _et al._ (2020).

In some cases, distillation with the desired precision might not be possible.
It is then of interest to instead ask how close one can approximate the chosen
target channel, that is, characterize the maximal achievable fidelity of
distillation. We can adapt our methods to obtain close upper and lower bounds
for this quantity. Importantly, the bounds become tight for relevant reference
states obeying conditions as in Cor. 4, allowing us to provide an exact
expression for the fidelity of distillation. We discuss the full details in
Ref. Note (1).

Another bound on distillation fidelity was recently presented with respect to
the so-called resource weight Regula and Takagi (2021); Fang and Liu (2020b).
Our approach allows one to extend the insight from these works to theories
with arbitrary channel structures $\mathbb{O}_{\rm all}$, enabling an
operational application of the corresponding weight measures, some of which
were previously introduced in other contexts Lewenstein and Sanpera (1998); de
Vicente (2014); Skrzypczyk  _et al._ (2014); Grudka  _et al._ (2014); Abramsky
_et al._ (2017); Pusey (2015).

One can obtain additional results in the characterization of distillation and
dilution in special cases of resource theories, for example, when the given
theory is concerned with an underlying state-based resource. We discuss such
cases in Ref. Note (1).

Quantum communication — A central problem in quantum communication is to
manipulate a given channel to enhance its communication capabilities using
resources available to both parties. Quantum capacity Bennett  _et al._
(1996); Lloyd (1997); Bennett  _et al._ (2002) and simulation cost Bennett
_et al._ (2002, 2014); Berta  _et al._ (2011) are important figures of merit
to evaluate the operational capability of quantum channels, and their one-shot
characterization received considerable attention recently Leung and Matthews
(2015); Tomamichel  _et al._ (2016); Wang  _et al._ (2019b); Fang  _et al._
(2020); Takagi  _et al._ (2020); Hsieh (2021). These tasks are precisely
channel distillation and dilution where the reference resource is the identity
channel, and the sets of free channels and free superchannels specify the
accessible resources for the sender and receiver. Formally, we define the one-
shot quantum capacity and simulation cost with the set of free superchannels
$\SS$ as

$\displaystyle Q^{\epsilon}_{\SS}(\mathcal{E})$
$\displaystyle\coloneqq\max\left\\{\left.\log
d\;\rule{0.0pt}{9.5pt}\right|\;\exists\Theta\in\SS,F(\Theta(\mathcal{E}),\mathrm{id}_{d})\geq
1-\epsilon\right\\}$ (4) $\displaystyle C^{\epsilon}_{\SS}(\mathcal{E})$
$\displaystyle\coloneqq\min\left\\{\left.\log
d\;\rule{0.0pt}{9.5pt}\right|\;\exists\Theta\in\SS,F(\Theta(\mathrm{id}_{d}),\mathcal{E})\geq
1-\epsilon\right\\}.$

We can then use our results to immediately obtain an exact characterization of
these quantities in relevant settings. For example, when $\mathbb{O}$ is the
set of separable channels $\mathbb{O}_{\rm SEP}$ whose Choi states are
separable, this setting corresponds to communication with separability-
preserving codes $\SS=\SS_{\rm SEP}$. This is a full-dimensional theory and it
holds that $R_{\min,\mathbb{O}_{\rm
SEP}}(\mathrm{id}_{d})=R_{s,\mathbb{O}_{\rm SEP}}(\mathrm{id}_{d})=d$ Yuan
_et al._ (2021). Then, Cors. 2 and 4 provide a complete characterization of
one-shot quantum capacity and simulation cost as $Q_{\SS_{\rm
SEP}}^{\epsilon}(\mathcal{E})=\log\lfloor R_{H,\mathbb{O}_{\rm
SEP}}^{\epsilon}(\mathcal{E})\rfloor$ and $C_{\SS_{\rm
SEP}}^{\epsilon}(\mathcal{E})=\log\lceil R_{s,\mathbb{O}_{\rm
SEP}}^{\epsilon}(\mathcal{E})\rceil$, the latter of which recovers a result of
Ref. Yuan  _et al._ (2021). Similar results apply to the setting of
communication assisted by codes preserving the positivity of the partial
transpose (PPT), where $\mathbb{O}_{\rm PPT}$ is the set of PPT channels Rains
(2001); Leung and Matthews (2015). We analogously obtain $Q_{\SS_{\rm
PPT}}^{\epsilon}(\mathcal{E})=\log\lfloor R_{H,\mathbb{O}_{\rm
PPT}}^{\epsilon}(\mathcal{E})\rfloor$ and $C_{\SS_{\rm
PPT}}^{\epsilon}(\mathcal{E})=\log\lceil R_{s,\mathbb{O}_{\rm
PPT}}^{\epsilon}(\mathcal{E})\rceil$. Interestingly, $R_{H,\mathbb{O}_{\rm
PPT}}^{\epsilon}$ appeared as a bound in Ref. Tomamichel  _et al._ (2016).

Another example is the case when $\mathbb{O}$ is the set of replacement
channels
$\mathbb{O}_{R}\coloneqq\\{\mathcal{R}_{\sigma}|\mathcal{R}_{\sigma}(\cdot)=\operatorname{Tr}(\cdot)\sigma\\}$,
where $\SS$ becomes the set of channel transformations assisted by no-
signalling (NS) correlations, $\SS_{\rm NS}$ Duan and Winter (2016); Takagi
_et al._ (2020). Since $\mathbb{O}_{R}$ is closed under linear combinations,
it is reduced dimensional. We also have
$R_{\min,\operatorname{aff}({\mathbb{O}_{R}})}(\mathrm{id}_{d})=R_{\max,\mathbb{O}_{R}}(\mathrm{id}_{d})=d^{2}$,
and our results immediately give $Q_{\SS_{\rm
NS}}^{\epsilon}(\mathcal{E})=\frac{1}{2}\log\lfloor
R_{H,\operatorname{aff}(\mathbb{O}_{R})}^{\epsilon}(\mathcal{E})\rfloor$ and
$C_{\SS_{\rm NS}}^{\epsilon}(\mathcal{E})=\frac{1}{2}\log\lceil
R_{\max,\mathbb{O}_{R}}^{\epsilon}(\mathcal{E})\rceil$. The one-shot NS-
assisted quantum capacity was obtained in Ref. Wang  _et al._ (2019b) in the
form of a semidefinite program; our result identifies it with the affine
hypothesis testing relative entropy, providing an operational meaning to this
resource measure. The one-shot NS simulation cost was obtained in Refs. Takagi
_et al._ (2020); Fang  _et al._ (2020), which is recovered by our general
approach as a special case. Furthermore, we can quantify exactly the fidelity
of NS-assisted coding Note (1), recovering a result of Ref. Leung and Matthews
(2015).

Our methods apply also to the study of the entanglement of bipartite channels
Gour and Scandolo (2021); Bäuml  _et al._ (2019), where the target resources
are maximally entangled states in the underlying state-based resource theory.
We then establish exact one-shot rates of channel manipulation under PPT- and
separability-preserving superchannels (see Ref. Note (1)).

Nonlocality and contextuality — Quantum nonlocality has been a major subject
of study not only as a key feature of quantum theory, but also as a useful
resource in a number of operational tasks Buhrman  _et al._ (2010); Cubitt
_et al._ (2010); Barrett  _et al._ (2005); Acín  _et al._ (2006); Pironio  _et
al._ (2010); Vazirani and Vidick (2014). The latter view motivates a precise
understanding of the manipulation of nonlocal resources de Vicente (2014);
Forster  _et al._ (2009); Nery  _et al._ (2020); Gallego and Aolita (2015),
but the characterization of such one-shot transformations has remained
elusive. Our framework encompasses this scenario by choosing $\mathbb{O}_{\rm
all}$ to be the set of no-signalling channels, where classical input or output
systems are represented through dephasing in a given basis, and taking
$\mathbb{O}$ to be the channels that can be constructed by local operations
and shared randomness. This includes not only the standard setting of Bell
nonlocality (where such channels are the classical “boxes”), but also more
general resources such as steering (where channels represent “assemblages”)
Schmid  _et al._ (2020); Rosset  _et al._ (2020). Our results then provide an
operational application of nonlocality measures individually introduced in
different settings of nonlocality de Vicente (2014); Piani and Watrous (2015)
and unified in Refs. Schmid  _et al._ (2020); Rosset  _et al._ (2020), which
we relate to one-shot resource transformations (see also Ref. Wolfe  _et al._
(2020)). We also introduce new monotones to this setting, which can add
further insights into feasible manipulation of nonlocal resources. For
instance: since a noisy box of the form
$\mathcal{B}=(1-\epsilon)\mathcal{B}_{\rm PR}+\epsilon\mathcal{B}_{\rm free}$,
where $\mathcal{B}_{\rm PR}$ is the Popescu-Rohrlich (PR) box Popescu and
Rohrlich (1994) and $\mathcal{B}_{\rm free}\in\mathbb{O}$ is some local box,
has $R_{\min,\mathbb{O}}(\mathcal{B}^{\otimes n})=1\;\forall n$, we recover
the fact that it cannot be distilled to other fundamental resources such as
the PR box even when multiple copies of the box $\mathcal{B}$ are available
Dukaric and Wolf (2008); Forster  _et al._ (2009). In Ref. Note (1), we
numerically evaluate the resource measures for a special class known as
isotropic boxes Masanes  _et al._ (2006) where we explicitly observe this
property.

Notably, our framework can also be applied to another related phenomenon known
as quantum contextuality, which also serves as an operational resource Hameedi
_et al._ (2017); Raussendorf (2013); Howard  _et al._ (2014); Bermejo-Vega
_et al._ (2017); Kleinmann  _et al._ (2011); Cabello  _et al._ (2018); Schmid
and Spekkens (2018). Namely, we consider the set of all classical-classical
channels for consistent boxes Grudka  _et al._ (2014) as $\mathbb{O}_{\rm
all}$ and the set of channels corresponding to noncontextual boxes as
$\mathbb{O}$. Our results characterize the exact and approximate box
transformations with operations that do not create contextuality, offering a
new perspective to the recent resource-theoretic framework Horodecki  _et al._
(2015); Duarte and Amaral (2018); Amaral  _et al._ (2018), as well as
providing operational application of the robustness of contextuality Meng  _et
al._ (2016a, b); Li  _et al._ (2020), contextual fraction Abramsky and
Brandenburger (2011); Grudka  _et al._ (2014); Abramsky  _et al._ (2017), and
hypothesis testing measures introduced in this work.

Measurement incompatibility — Measurement incompatibility refers to the
impossibility of simultaneous measurement and is closely related to the
aforementioned phenomena such as Bell nonlocality and steering Quintino  _et
al._ (2014); Uola  _et al._ (2014); Pusey (2015). The set of POVMs
$\\{M_{a|x}\\}$, where $M_{a|x}$ is the POVM element with outcome $a$ for the
measurement labeled by setting $x$, is called compatible (or jointly
measurable) if there exists a parent measurement $\\{P_{i}\\}$ and a
conditional probability distribution $\\{q(a|x,i)\\}$ such that
$M_{a|x}=\sum_{i}q(a|x,i)P_{i}$.

Our formalism can handle scenarios where resources take the form of ensembles
by incorporating the classical labels of the ensembles into the description of
$\mathbb{O}_{\rm all}$. In the case of measurement incompatibility,
$\mathbb{O}_{\rm all}$ represents the set of channels corresponding to POVMs,
while $\mathbb{O}\subseteq\mathbb{O}_{\rm all}$ represents the set of
compatible POVMs (see Ref. Note (1)). This form allows one to apply our
results to this setting and characterize the approximate one-shot
transformation of incompatible sets of measurements, complementing the
previous works which focused on exact transformation with smaller sets of free
operations in different approaches Skrzypczyk  _et al._ (2019); Buscemi  _et
al._ (2020) and providing operational applications of the related measures
Haapasalo (2015); Uola  _et al._ (2015); Pusey (2015); Designolle  _et al._
(2019). Although the robustness and weight measures are usually defined at the
level of POVMs, we show in Ref. Note (1) that they coincide with the channel-
based measures defined in our framework, allowing one to carry over the
previous analyses to characterize resource transformations. We also note that
the discussion here can be straightforwardly extended to channel
incompatibility Heinosaari  _et al._ (2016); Heinosaari and Miyadera (2017),
which includes measurement incompatibility as a special case.

Conclusions — We established fundamental bounds on the transformations between
general dynamical quantum resources in the one-shot regime. We tightly
characterized the ability to manipulate resources by providing conditions for
convertibility in terms of the robustness and hypothesis testing measures. In
particular, under suitable assumptions, we established an exact quantification
of the one-shot distillable resource and one-shot resource cost of general
channels, giving a precise operational interpretation to the considered
monotones in these important tasks. This not only extends and unifies previous
specialized results Leung and Matthews (2015); Wang  _et al._ (2019b); Yuan
_et al._ (2021); Gour and Scandolo (2021); Bäuml  _et al._ (2019); Saxena  _et
al._ (2020), but also sheds light on the general structure of dynamical
resource theories by providing a common description of their operational
aspects. Besides contributing to the theory of quantum resources, our methods
find direct practical use, as we exemplified with several explicit
applications.

###### Acknowledgements.

Note —  During the completion of this manuscript, we became aware of two
related works, Ref. Kim  _et al._ (2021) by Kim et al. and Ref. Yuan  _et al._
(2020) by Yuan et al., where the authors independently obtained results
overlapping with some of our findings. The former work considers one-shot
distillation and dilution of quantum channel entanglement, which coincides
with our characterization of these tasks in Corollaries 2 and 4 (see Ref. Note
(1)), while the latter work introduces a general framework for one-shot
channel distillation and dilution which again corresponds to our Cors. 2 and 4
for the cases of quantum channel resources. We thank David Schmid for sharing
useful references with us, which helped us expand the scope of our results. We
are also grateful to the authors of Refs. Kim  _et al._ (2021); Yuan  _et al._
(2020) for coordinating the submission of our works. B.R. was supported by the
Presidential Postdoctoral Fellowship from Nanyang Technological University,
Singapore. R.T. acknowledges the support of NSF, ARO, IARPA, AFOSR, the
Takenaka Scholarship Foundation, the National Research Foundation (NRF)
Singapore, under its NRFF Fellow programme (Award No. NRF-NRFF2016-02), and
the Singapore Ministry of Education Tier 1 Grant 2019-T1-002-015. Any
opinions, findings and conclusions or recommendations expressed in this
material are those of the author(s) and do not reflect the views of National
Research Foundation, Singapore.

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##  Appendix A Properties of the resource measures

We will use the notation $\left\langle\cdot,\cdot\right\rangle$ for the
Hilbert-Schmidt inner product $\left\langle
A,B\right\rangle=\operatorname{Tr}(A^{\dagger}B)$.

Recall that

$\displaystyle
R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})=\min_{\mathcal{M}\in\mathbb{O}}\max_{\psi}R_{H}^{\epsilon}(\mathrm{id}\otimes\mathcal{E}(\psi)||\mathrm{id}\otimes\mathcal{M}(\psi))$
(5)

where $R_{H}^{\epsilon}(\rho||\sigma)\coloneqq\max_{\begin{subarray}{c}0\leq
P\leq\mathbbm{1}\\\ \left\langle P,\rho\right\rangle\geq
1-\epsilon\end{subarray}}\left\langle P,\sigma\right\rangle^{-1}$. Here, for a
channel which takes operators acting on an input Hilbert space
$\mathcal{H}_{\rm in}$ to operators acting on $\mathcal{H}_{\rm out}$, the
optimization $\max_{\psi}$ means that we are optimizing over all pure states
acting on the Hilbert space $\mathcal{H}_{\rm
in^{\prime}}\otimes\mathcal{H}_{\rm in}$ with $\mathcal{H}_{\rm
in^{\prime}}\cong\mathcal{H}_{\rm in}$.

While the quantities $R_{H,\mathbb{O}}^{\epsilon}$, $G_{\mathbb{O}}$, and
their affine equivalents might seem like daunting optimization tasks, they
actually admit a number of useful properties and can be expressed as convex
optimization problems. To see this, we first establish a useful minimax
result.

######  Lemma 5.

For any closed and convex set $\mathbb{O}$, it holds that

$\displaystyle R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\psi}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle\leq\lambda\;\forall\mathcal{M}\in\mathbb{O},\;0\leq
P\leq\mathbbm{1},\;\left\langle
P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq 1-\epsilon\right\\}$
(6) $\displaystyle
R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\psi}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle=\lambda\;\forall\mathcal{M}\in\mathbb{O},\;0\leq
P\leq\mathbbm{1},\;\left\langle
P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq 1-\epsilon\right\\}.$

###### Proof.

By definition, we have that

$\displaystyle R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\max_{\mathcal{M}\in\mathbb{O}}\min_{\psi}\min_{\begin{subarray}{c}0\leq
P\leq\mathbbm{1}\\\ \left\langle
P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq
1-\epsilon\end{subarray}}\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle$ (7)
$\displaystyle=\min_{\psi}\min_{\begin{subarray}{c}0\leq P\leq\mathbbm{1}\\\
\left\langle P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq
1-\epsilon\end{subarray}}\max_{\mathcal{M}\in\mathbb{O}}\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle$ (8)
$\displaystyle=\min_{\psi}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle\leq\lambda\;\forall\mathcal{M}\in\mathbb{O},\;0\leq
P\leq\mathbbm{1},\;\left\langle
P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq 1-\epsilon\right\\}$
(9)

where in the second line we used the linearity of $\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle$ to apply Sion’s minimax
theorem Sion (1958). In the case of
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}$, the theorem applies
analogously ($\operatorname{aff}(\mathbb{O})$ is no longer compact, in
general, but it still is a convex set) and so we get

$\displaystyle
R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\psi}\min_{\begin{subarray}{c}0\leq P\leq\mathbbm{1}\\\
\left\langle P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq
1-\epsilon\end{subarray}}\sup_{\mathcal{M}\in\operatorname{aff}(\mathbb{O})}\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle$ (10)
$\displaystyle=\min_{\psi}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle=\lambda\;\forall\mathcal{M}\in\mathbb{O},\;0\leq
P\leq\mathbbm{1},\;\left\langle
P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq 1-\epsilon\right\\}$
(11)

The last line follows from a characterization of affine hulls found e.g. in
Regula  _et al._ (2020), and can be seen explicitly as follows. If there exist
$\mathcal{M}_{1},\mathcal{M}_{2}\in\mathbb{O}$ such that $\left\langle
P,\mathrm{id}\otimes\mathcal{M}_{1}(\psi)\right\rangle\neq\left\langle
P,\mathrm{id}\otimes\mathcal{M}_{2}(\psi)\right\rangle$, then the supremum
over $\operatorname{aff}(\mathbb{O})$ becomes unbounded. On the other hand,
the whole quantity is upper bounded by 1 as can be checked by taking
$P=\mathbbm{1}$. Thus, optimal $P$ and $\psi$ necessarily satisfy
$\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle=\lambda,\forall\mathcal{M}\in\mathbb{O}$
for some constant $\lambda$, also implying
$\sup_{\mathcal{M}\in\operatorname{aff}(\mathbb{O})}\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle=\lambda$. ∎

Using the above, we can express the hypothesis testing measures as

$\displaystyle R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\rho}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
Q,J_{\mathcal{M}}\right\rangle\leq\lambda\;\forall\mathcal{M}\in\mathbb{O},\;0\leq
Q\leq\rho\otimes\mathbbm{1},\;\left\langle Q,J_{\mathcal{E}}\right\rangle\geq
1-\epsilon\right\\},$ (12) $\displaystyle
R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\rho}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
Q,J_{\mathcal{M}}\right\rangle=\lambda\;\forall\mathcal{M}\in\mathbb{O},\;0\leq
Q\leq\rho\otimes\mathbbm{1},\;\left\langle Q,J_{\mathcal{E}}\right\rangle\geq
1-\epsilon\right\\},$

where the optimization is over all density matrices $\rho$ acting on the input
space $\mathcal{H}_{\rm in}$ of dimension $d_{\rm in}$, and
$J_{\mathcal{E}}=d_{\rm in}(\mathrm{id}\otimes\mathcal{E})(\psi^{+})$, with
$\psi^{+}$ being the maximally entangled state, is the Choi operator of
$\mathcal{E}$. Here, we used the fact that any input $\psi$ acting on
$\mathcal{H}_{\rm in^{\prime}}\otimes\mathcal{H}_{\rm in}$ can be written as
$\psi=(\rho^{1/2}\otimes\mathbbm{1})\psi^{+}(\rho^{1/2}\otimes\mathbbm{1})$
for some density operator $\rho$ ($\psi$ being the canonical purification of
$\rho$).

In addition to the hypothesis testing measures, we will see later that the
following measure characterizes the fidelity of distillation:

$\displaystyle G_{\mathbb{O}}(\mathcal{E};m)$
$\displaystyle=\max_{\psi}\left\\{\,\left\langle\mathrm{id}\otimes\mathcal{E}(\psi),\,W\right\rangle\,\rule{0.0pt}{9.5pt}\left|\;0\leq
W\leq\mathbbm{1},\left\langle\mathrm{id}\otimes\mathcal{M}(\psi),W\right\rangle\leq\frac{1}{m}\;\forall\mathcal{M}\in\mathbb{O}\right.\right\\},$
(13)

which is parameterized by $m\in\mathbb{R}$, and we also define
$G_{\operatorname{aff}(\mathbb{O})}(\mathcal{E};m)$ analogously. In a similar
manner, we can also express the measures $G_{\mathbb{O}}$ and
$G_{\operatorname{aff}(\mathbb{O})}$ as

$\displaystyle G_{\mathbb{O}}(\mathcal{E};m)$
$\displaystyle=\max_{\rho}\left\\{\,\left\langle
W,J_{\mathcal{E}}\right\rangle\,\rule{0.0pt}{9.5pt}\left|\;0\leq
W\leq\rho\otimes\mathbbm{1},\;\left\langle
W,J_{\mathcal{M}}\right\rangle\leq\frac{1}{m}\;\forall\mathcal{M}\in\mathbb{O}\right.\right\\},$
(14) $\displaystyle G_{\operatorname{aff}{\mathbb{O}}}(\mathcal{E};m)$
$\displaystyle=\sup_{\rho}\left\\{\,\left\langle
W,J_{\mathcal{E}}\right\rangle\,\rule{0.0pt}{9.5pt}\left|\;0\leq
W\leq\rho\otimes\mathbbm{1},\;\left\langle
W,J_{\mathcal{M}}\right\rangle=\frac{1}{m}\;\forall\mathcal{M}\in\mathbb{O}\right.\right\\}.$

To more explicitly see the computability of these problems, let us define the
dual cones

$\displaystyle\mathbb{O}\textup{*}$
$\displaystyle\coloneqq\left\\{\left.X\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
X,J_{\mathcal{M}}\right\rangle\geq
0\;\forall\mathcal{M}\in\mathbb{O}\right\\}$ (15)
$\displaystyle\operatorname{aff}(\mathbb{O})\textup{*}$
$\displaystyle\coloneqq\left\\{\left.X\;\rule{0.0pt}{9.5pt}\right|\;\left\langle
X,J_{\mathcal{M}}\right\rangle=0\;\forall\mathcal{M}\in\mathbb{O}\right\\},$

using which we can write

$\displaystyle R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\rho}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\lambda\frac{\mathbbm{1}}{d_{A}}-Q\in\mathbb{O}\textup{*},\;0\leq
Q\leq\rho\otimes\mathbbm{1},\;\left\langle Q,J_{\mathcal{E}}\right\rangle\geq
1-\epsilon\right\\},$ (16) $\displaystyle G_{\mathbb{O}}(\mathcal{E};m)$
$\displaystyle=\max_{\rho}\left\\{\,\left\langle
W,J_{\mathcal{E}}\right\rangle\,\rule{0.0pt}{9.5pt}\left|\;0\leq
W\leq\rho\otimes\mathbbm{1},\;\frac{\mathbbm{1}}{d_{A}}-mW\in\mathbb{O}\textup{*}\right.\right\\},$

and analogously for $\operatorname{aff}(\mathbb{O})$. Whenever the dual cone
of operators $\mathbb{O}\textup{*}$ (respectively,
$\operatorname{aff}(\mathbb{O})\textup{*}$) is characterized by semidefinite
constraints, the programs $R_{H,\mathbb{O}}^{\epsilon}$ and $G_{\mathbb{O}}$
(respectively, $R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}$ and
$G_{\operatorname{aff}(\mathbb{O})}$) are SDPs, allowing for an efficient
evaluation in relevant cases. A similar characterization can be extended to
related settings, such as when instead of optimizing over a set of channels
$\mathbb{O}$ one optimizes over a larger set, as is the case when dealing with
PPT maps Leung and Matthews (2015); Wang  _et al._ (2019b).

We note also that, for $\epsilon=0$, the hypothesis testing relative entropy
$R_{H,\mathbb{O}}^{\epsilon}$ reduces to an optimization based on the min-
relative entropy Datta (2009), given by

$\displaystyle R_{\min,\mathbb{O}}(\mathcal{E})^{-1}$
$\displaystyle=\min_{\psi}\max_{\mathcal{M}\in\mathbb{O}}\left\langle\Pi_{\mathrm{id}\otimes\mathcal{E}(\psi)},\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle$
(17)

where $\Pi_{\rho}$ denotes the projection onto the support of $\rho$. Of
interest to us will be the case when $\mathrm{id}\otimes\mathcal{N}(\psi)$ is
a pure state for any pure input state, in which case

$\displaystyle R_{\min,\mathbb{O}}(\mathcal{N})^{-1}$
$\displaystyle=\min_{\psi}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle\mathrm{id}\otimes\mathcal{N}(\psi),\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle\leq\lambda\;\forall\mathcal{M}\in\mathbb{O}\right\\}$
(18)

We analogously define the min-relative entropy with respect to
$\operatorname{aff}(\mathbb{O})$, given by

$\displaystyle R_{\min,\operatorname{aff}(\mathbb{O})}(\mathcal{N})^{-1}$
$\displaystyle=\inf_{\psi}\left\\{\left.\lambda\;\rule{0.0pt}{9.5pt}\right|\;\left\langle\mathrm{id}\otimes\mathcal{N}(\psi),\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle=\lambda\;\forall\mathcal{M}\in\mathbb{O}\right\\}$
(19)

when $\mathrm{id}\otimes\mathcal{N}(\psi)$ is pure, where we take
$\inf\emptyset=\infty$.

##  Appendix B Proofs of the main results

######  Theorem 1.

Let $\mathcal{E}\in\mathbb{O}_{\rm all}$ and $\mathcal{N}\in\mathbb{O}_{\rm
all}^{\prime}$. If there exists a free superchannel $\Theta\in\SS$ such that
$F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon$, then for any monotone
$\mathfrak{R}_{\mathbb{O}}$ it holds that
$\mathfrak{R}_{\mathbb{O}}(\mathcal{E})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$,
as well as that
$\mathfrak{R}_{\mathbb{O}}^{\delta}(\mathcal{E})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{2(\sqrt{\delta}+\sqrt{\epsilon})}(\mathcal{N})$
for any $0\leq\delta\leq 1$ where
$\mathfrak{R}_{\mathbb{O}}^{\epsilon}(\mathcal{E})\coloneqq\min\\{\mathfrak{R}_{\mathbb{O}}(\mathcal{E}^{\prime})\,|\,F(\mathcal{E}^{\prime},\mathcal{E})\geq
1-\epsilon,\,\mathcal{E}^{\prime}\in\mathbb{O}_{\rm all}\\}$.

Conversely, for any choices of $\epsilon,\delta\geq 0$ such that
$\epsilon+2\delta<1$, there exists a free superchannel $\Theta\in\SS$ such
that $F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon-2\delta$ if
$R_{H,\mathbb{O}}^{\delta}(\mathcal{E})\geq
R_{s,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$ or if
$R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})\geq
R_{\max,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$.

###### Proof.

To show the first part, we note that the smoothed version of any monotone
$\mathfrak{R}_{\mathbb{O}}$ also serves as a monotone because for any
$\Theta\in\SS$, it holds that

$\displaystyle\mathfrak{R}_{\mathbb{O}}^{\epsilon}(\mathcal{E})=\mathfrak{R}_{\mathbb{O}}(\mathcal{E}^{\prime})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}(\Theta(\mathcal{E}^{\prime}))\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{\epsilon}(\Theta(\mathcal{E}))$
(20)

where in the equality we defined $\mathcal{E}^{\prime}\in\mathbb{O}_{\rm all}$
as a channel such that $F(\mathcal{E}^{\prime},\mathcal{E})\geq 1-\epsilon$
and
$\mathfrak{R}_{\mathbb{O}}^{\epsilon}(\mathcal{E})=\mathfrak{R}_{\mathbb{O}}(\mathcal{E}^{\prime})$,
and in the first inequality we used the monotonicity of
$\mathfrak{R}_{\mathbb{O}}$. The last inequality is obtained by noting
$F(\Theta(\mathcal{E}^{\prime}),\Theta(\mathcal{E}))\geq
F(\mathcal{E}^{\prime},\mathcal{E})\geq 1-\epsilon$ due to the data processing
inequality of the fidelity under any superchannel Belavkin  _et al._ (2005);
Gilchrist  _et al._ (2005), and by using the definition of the smooth measure.

Now, suppose $\mathcal{N}^{\prime}$ is a channel such that
$F(\mathcal{N}^{\prime},\Theta(\mathcal{E}))\geq 1-\delta$ and
$\mathfrak{R}_{\mathbb{O}^{\prime}}^{\delta}(\Theta(\mathcal{E}))=\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{N}^{\prime})$.
Then, we use the relation between the fidelity and diamond norm distance Fuchs
and van de Graaf (1999) to get

$\displaystyle 1-F(\mathcal{N},\mathcal{N}^{\prime})\leq
1-\left(1-\frac{1}{2}\|\mathcal{N}-\mathcal{N}^{\prime}\|_{\diamond}\right)^{2}\leq\|\mathcal{N}-\mathcal{N}^{\prime}\|_{\diamond}$
$\displaystyle\leq\|\mathcal{N}-\Theta(\mathcal{E})\|_{\diamond}+\|\Theta(\mathcal{E})-\mathcal{N}^{\prime}\|_{\diamond}$
(21) $\displaystyle\leq
2\left(\sqrt{1-F(\mathcal{N},\Theta(\mathcal{E}))}+\sqrt{1-F(\Theta(\mathcal{E}),\mathcal{N}^{\prime})}\right)$
$\displaystyle\leq 2(\sqrt{\epsilon}+\sqrt{\delta}),$

implying that
$\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{N}^{\prime})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{2(\sqrt{\delta}+\sqrt{\epsilon})}(\mathcal{N})$.
The statement is shown by combining this with (20). When $\delta=0$,
$\mathcal{N}^{\prime}$ coincides with $\Theta(\mathcal{E})$. Using the
assumption that $1-F(\Theta(\mathcal{E}),\mathcal{N})\leq\epsilon$, we get
$\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{N}^{\prime})\geq\mathfrak{R}_{\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$,
resulting in the improved bound.

Let us now show the second part. We start with the condition using
$R_{H,\mathbb{O}}$. Let $\tilde{\mathcal{N}}\in\mathbb{O}_{\rm all}^{\prime}$
be a channel such that $F(\tilde{\mathcal{N}},\mathcal{N})\geq 1-\epsilon$ and
$R_{s,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})=R_{s,\mathbb{O}^{\prime}}(\tilde{\mathcal{N}})$.
Then, there exists channel $\mathcal{Q}\in\mathbb{O}^{\prime}$ such that
$\tilde{\mathcal{N}}+(R_{s,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})-1)\mathcal{Q}\in
R_{s,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})\mathbb{O}^{\prime}$. Let us
also take $P^{\star}$ and $\psi^{\star}$ as the optimal operators achieving
the optimization in the hypothesis testing entropy measure for $\mathcal{E}$
as
$R_{H,\mathbb{O}}^{\delta}(\mathcal{E})=\min_{\mathcal{M}\in\mathbb{O}}\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{M}(\psi^{\star})\right\rangle^{-1}$ with
$0\leq P^{\star}\leq\mathbbm{1}$ and $\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{E}(\psi^{\star})\right\rangle\geq
1-\delta$ (recall Lemma 5). Then, we define the following map:

$\displaystyle\Theta(\mathcal{L})\coloneqq\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{L}(\psi^{\star})\right\rangle\tilde{\mathcal{N}}+\left\langle\mathbbm{1}-P^{\star},\mathrm{id}\otimes\mathcal{L}(\psi^{\star})\right\rangle\mathcal{Q}.$
(22)

The fact that this is a valid superchannel follows since $\Theta$ consists of
the preprocessing channel $\omega\mapsto\omega\otimes\psi^{\star}$, followed
by an application of $\mathrm{id}\otimes\mathcal{L}$, and the postprocessing
channel

$\displaystyle\omega^{\prime}\mapsto$
$\displaystyle\left[\tilde{\mathcal{N}}\otimes\left\langle
P^{\star},\cdot\right\rangle\right](\omega^{\prime})$ (23) $\displaystyle+$
$\displaystyle\left[\mathcal{Q}\otimes\left\langle\mathbbm{1}-P^{\star},\cdot\right\rangle\right](\omega^{\prime}),$

where the latter can be seen to be a valid CPTP map since
$\\{P^{\star},\mathbbm{1}-P^{\star}\\}$ defines a valid POVM. Now, since
$\tilde{\mathcal{N}},\mathcal{Q}\in\mathbb{O}_{\rm all}$, it further holds
that $\Theta\in\SS_{\rm all}$. One can also check that $\Theta\in\SS$ by
noting that for any $\mathcal{M}\in\mathbb{O}$, it holds that $\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{M}(\psi^{\star})\right\rangle\leq
R_{H,\mathbb{O}}^{\delta}(\mathcal{E})^{-1}\leq
R_{s,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})^{-1}=R_{s,\mathbb{O}^{\prime}}(\tilde{\mathcal{N}})^{-1}$,
ensuring $\Theta(\mathcal{M})\in\mathbb{O}^{\prime}$. Using the concavity of
the (square root of) fidelity, we obtain

$\displaystyle\sqrt{F(\Theta(\mathcal{E}),\mathcal{N})}$
$\displaystyle\geq\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{E}(\psi^{\star})\right\rangle\sqrt{F(\tilde{\mathcal{N}},\mathcal{N})}+\left\langle\mathbbm{1}-P^{\star},\mathrm{id}\otimes\mathcal{E}(\psi^{\star})\right\rangle\sqrt{F(\mathcal{Q},\mathcal{N})}$
(24) $\displaystyle\geq(1-\delta)\sqrt{1-\epsilon},$

which leads to
$F(\Theta(\mathcal{E}),\mathcal{N})\geq(1-\delta)^{2}(1-\epsilon)\geq
1-\epsilon-2\delta$.

For the condition using $R_{H,\operatorname{aff}(\mathbb{O})}$, recall from
Lemma 5 that

$\displaystyle
R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})^{-1}$
$\displaystyle=\min\Big{\\{}\lambda\;\Big{|}\;\left\langle
P,\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle=\lambda\;\forall\mathcal{M}\in\mathbb{O},$
(25) $\displaystyle\hphantom{=\min\Big{\\{}\lambda\;\Big{|}\;}0\leq
P\leq\mathbbm{1},\ \left\langle
P,\mathrm{id}\otimes\mathcal{E}(\psi)\right\rangle\geq 1-\delta\Big{\\}}.$
(26)

Then, let $P^{\star}$ and $\psi^{\star}$ be the optimal $P$ and $\psi$ in the
above, and $\tilde{\mathcal{N}}\in\mathbb{O}_{\rm all}^{\prime}$ be a channel
such that $F(\tilde{\mathcal{N}},\mathcal{N})\geq 1-\epsilon$ and
$R_{\max,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})=R_{\max,\mathbb{O}^{\prime}}(\tilde{\mathcal{N}})$.
Since $R_{\max,\mathbb{O}}^{\epsilon}(\mathcal{N})\leq
R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})$, there exists a
channel $\mathcal{Q}\in\mathbb{O}_{\rm all}^{\prime}$ such that
$\tilde{\mathcal{N}}+(R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})-1)\,\mathcal{Q}\in
R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})\tilde{\mathbb{O}}^{\prime}$,
which allows us to construct the superchannel

$\displaystyle\Theta(\mathcal{L})\coloneqq\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{L}(\psi^{\star})\right\rangle\tilde{\mathcal{N}}+\left\langle\mathbbm{1}-P^{\star},\mathrm{id}\otimes\mathcal{L}(\psi^{\star})\right\rangle\mathcal{Q}.$
(27)

Now, since $\left\langle
P^{\star},\mathrm{id}\otimes\mathcal{M}(\psi^{\star})\right\rangle=R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})^{-1}$
for any $\mathcal{M}\in\mathbb{O}$ due to the conditions in (26), we have that

$\displaystyle\Theta(\mathcal{M})$
$\displaystyle=R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})^{-1}\tilde{\mathcal{N}}+(1-R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})^{-1})\mathcal{Q}$
(28)
$\displaystyle\propto\tilde{\mathcal{N}}+(R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})-1)\,\mathcal{Q}\in\mathbb{O}^{\prime}$

and so $\Theta\in\SS$. Noticing that $F(\Theta(\mathcal{E}),\mathcal{N})\geq
1-\epsilon-2\delta$ analogously as in the previous case completes the proof.

∎

######  Remark.

The quantity $R_{H,\mathbb{O}}^{\delta}$ reduces to $R_{\min,\mathbb{O}}$ when
$\delta=0$, as we mentioned before. However, for
$R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}$ and
$R_{\min,\operatorname{aff}(\mathbb{O})}$, there is a potential gap between
the two: we always have that
$R_{H,\operatorname{aff}(\mathbb{O})}^{\delta}(\mathcal{E})\geq 1$ for any
$\delta$, but the quantity $R_{\min,\operatorname{aff}(\mathbb{O})}$ can be
equal to 0 when there is no input state $\psi$ such that that
$\left\langle\Pi_{\mathrm{id}\otimes\mathcal{E}(\psi)},\mathrm{id}\otimes\mathcal{M}(\psi)\right\rangle$
is constant for all $\mathcal{M}\in\mathbb{O}$. Interestingly, either
$R_{H,\operatorname{aff}(\mathbb{O})}^{0}$ or
$R_{\min,\operatorname{aff}(\mathbb{O})}$ can be used in the proof in the case
$\delta=0$ (in the sense that we can require either
$R_{H,\operatorname{aff}(\mathbb{O})}^{0}(\mathcal{E})\geq
R_{\max,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$ or
$R_{\min,\operatorname{aff}(\mathbb{O})}(\mathcal{E})\geq
R_{\max,\mathbb{O}^{\prime}}^{\epsilon}(\mathcal{N})$), but using the
hypothesis testing relative entropy provides a potentially more robust
condition.

######  Theorem 3.

Let $\mathcal{E}\in\mathbb{O}_{\rm all}$, $\mathcal{N}\in\mathbb{O}_{\rm
all}^{\prime}$ and suppose $\mathrm{id}\otimes\mathcal{N}(\psi)$ is pure for
any pure state $\psi$. If there exists a free superchannel $\Theta\in\SS$ such
that $F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon$, then it holds that
$R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})\geq
R_{\min,\mathbb{O}^{\prime}}(\mathcal{N})$ and
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})\geq
R_{\min,\operatorname{aff}(\mathbb{O}^{\prime})}(\mathcal{N})$.

Conversely, there exists a free superchannel $\Theta\in\SS$ such that
$F(\Theta(\mathcal{E}),\mathcal{N})\geq 1-\epsilon-\delta$ if
$R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})\geq
R_{s,\mathbb{O}^{\prime}}^{\delta}(\mathcal{N})$ or if
$R_{H,\operatorname{aff}(\mathbb{O})}^{\epsilon}(\mathcal{E})\geq
R_{\max,\mathbb{O}^{\prime}}^{\delta}(\mathcal{N})$.

###### Proof.

Let $\tilde{\mathcal{N}}$ be a channel such that
$\Theta(\mathcal{E})=\tilde{\mathcal{N}}$. Using the data processing
inequality for hypothesis testing relative entropy Wang and Renner (2012);
Yuan (2019); Gour and Scandolo (2021), we get

$\displaystyle R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})$
$\displaystyle=\min_{\mathcal{M}\in\mathbb{O}}R^{\epsilon}_{H}(\mathcal{E}\|\mathcal{M})$
(29)
$\displaystyle\geq\min_{\mathcal{M}\in\mathbb{O}}R^{\epsilon}_{H}(\Theta(\mathcal{E})\|\Theta(\mathcal{M}))$
$\displaystyle\geq\min_{\mathcal{M}^{\prime}\in\mathbb{O}^{\prime}}R^{\epsilon}_{H}(\tilde{\mathcal{N}}\|\mathcal{M}^{\prime})$
$\displaystyle=\min_{\mathcal{M}^{\prime}\in\mathbb{O}^{\prime}}\max_{\psi^{\prime}}R^{\epsilon}_{H}(\mathrm{id}\otimes\tilde{\mathcal{N}}(\psi^{\prime})\|\mathrm{id}\otimes\mathcal{M}^{\prime}(\psi^{\prime}))$
$\displaystyle=\min_{\mathcal{M}^{\prime}\in\mathbb{O}^{\prime}}\max_{\psi^{\prime}}\max_{\begin{subarray}{c}0\leq
P\leq\mathbbm{1}\\\ \left\langle
P,\mathrm{id}\otimes\tilde{\mathcal{N}}(\psi^{\prime})\right\rangle\geq
1-\epsilon\end{subarray}}\left\langle
P,\mathrm{id}\otimes\mathcal{M}^{\prime}(\psi^{\prime})\right\rangle^{-1}.$

Using the assumption on $\mathcal{N}$, for all pure states $\psi^{\prime}$ it
holds that

$\displaystyle\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\prime}),\mathrm{id}\otimes\tilde{\mathcal{N}}(\psi^{\prime})\right\rangle\geq\min_{\psi^{\prime}}\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\prime}),\mathrm{id}\otimes\tilde{\mathcal{N}}(\psi^{\prime})\right\rangle=F(\mathcal{N},\tilde{\mathcal{N}})\geq
1-\epsilon$ (30)

and so we can always choose $P=\mathrm{id}\otimes\mathcal{N}(\psi^{\prime})$
as a feasible solution in the innermost maximization above. We thus get

$\displaystyle R_{H,\mathbb{O}}^{\epsilon}(\mathcal{E})$
$\displaystyle\geq\min_{\mathcal{M}^{\prime}\in\mathbb{O}^{\prime}}\max_{\psi^{\prime}}\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\prime}),\mathrm{id}\otimes\mathcal{M}^{\prime}(\psi^{\prime})\right\rangle^{-1}$
(31) $\displaystyle=R_{\min,\mathbb{O}^{\prime}}(\mathcal{N}).$

The proof for the condition using $R_{H,\operatorname{aff}(\mathbb{O})}$
follows in the same way since $R_{H}^{\epsilon}$ satisfies the data processing
inequality even when the second argument is not a CP map (this is due to the
fact that any feasible solution for
$R^{\epsilon}_{H}(\mathcal{E}\|\mathcal{M})^{-1}$ provides a feasible solution
for $R^{\epsilon}_{H}(\Theta(\mathcal{E})\|\Theta(\mathcal{M}))^{-1}$
regardless of what $\mathcal{M}$ is, and the objective function of
$R^{\epsilon}_{H}(\mathcal{E}\|\mathcal{M})^{-1}$ is linear in $\mathcal{M}$).

To show the second part, let $\tilde{\mathcal{N}}\in\mathbb{O}_{\rm
all}^{\prime}$ be the channel such that
$F(\tilde{\mathcal{N}},\mathcal{N})\geq 1-\delta$ and
$R_{s,\mathbb{O}^{\prime}}^{\delta}(\mathcal{N})=R_{s,\mathbb{O}^{\prime}}(\tilde{\mathcal{N}})$.
Then, the construction in (22) as well as the purity of
$\mathrm{id}\otimes\mathcal{N}(\psi)$ gives that

$\displaystyle F(\Theta(\mathcal{E}),\mathcal{N})$
$\displaystyle=\min_{\psi}\left\langle\mathrm{id}\otimes\Theta(\mathcal{E})(\psi),\mathrm{id}\otimes\mathcal{N}(\psi)\right\rangle$
(32)
$\displaystyle\geq\left<P^{\star},\mathrm{id}\otimes\mathcal{E}(\psi^{\star})\right>\min_{\psi}\left\langle\mathrm{id}\otimes\tilde{\mathcal{N}}(\psi),\mathrm{id}\otimes\mathcal{N}(\psi)\right\rangle$
$\displaystyle\geq(1-\epsilon)(1-\delta)$ $\displaystyle\geq
1-\epsilon-\delta.$

The case of $R_{H,\operatorname{aff}(\mathbb{O})}$ can be shown analogously.

∎

##  Appendix C Remarks on distillable resource and resource cost

In the main text, we defined the notions of one-shot distillable resource and
resource cost as

$\displaystyle d^{\epsilon}_{\mathbb{O}}(\mathcal{E})$
$\displaystyle\coloneqq\sup\left\\{\left.\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\\!\;\rule{0.0pt}{9.5pt}\right|\;\\!F(\Theta(\mathcal{E}),\mathcal{T})\geq
1\\!-\\!\epsilon,\;\mathcal{T}\\!\in\mathbb{T},\;\Theta\in\SS\right\\}$ (33)
$\displaystyle c^{\epsilon}_{\mathbb{O}}(\mathcal{E})$
$\displaystyle\coloneqq\inf\left\\{\left.\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\\!\;\rule{0.0pt}{9.5pt}\right|\;\\!F(\mathcal{E},\Theta(\mathcal{T}))\geq
1\\!-\\!\epsilon,\;\mathcal{T}\\!\in\mathbb{T},\;\Theta\in\SS\right\\}$

where $\mathbb{T}$ is some given set of “target channels” and
$\mathfrak{R}_{\mathbb{O}^{\prime}}$ is a choice of a monotone (that we fixed
as $R_{\min,\mathbb{O}^{\prime}}$ for simplicity). In many practical settings,
the set of targets is typically chosen simply as

$\displaystyle\mathbb{T}=\left\\{\left.\mathcal{T}^{\otimes
m}\;\rule{0.0pt}{9.5pt}\right|\;m\in\mathbb{N}\right\\}$ (34)

for some fixed channel $\mathcal{T}$, such as a qubit resourceful channel. The
quantities $d^{\epsilon}_{\mathbb{O}}$ and $c^{\epsilon}_{\mathbb{O}}$ then
reduce to asking: what is the largest number of copies of $\mathcal{T}$ that
can be distilled from $\mathcal{E}$, and what is the least number of copies of
$\mathcal{T}$ required to synthesize $\mathcal{E}$ with free transformations?
This corresponds exactly, for instance, to the familiar notion of distillable
entanglement and entanglement cost known from entanglement theory.

Although we allow more general notions of target channels, it is clear that
the set $\mathbb{T}$ need not be a continuous set, which is precisely why we
defined the notation

$\displaystyle\left\lfloor x\right\rfloor_{\mathbb{T}}$
$\displaystyle\coloneqq\sup\left\\{\left.\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\;\rule{0.0pt}{9.5pt}\right|\;\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\leq
x\right\\}$ (35) $\displaystyle\left\lceil x\right\rceil_{\mathbb{T}}$
$\displaystyle\coloneqq\inf\left\\{\left.\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\;\rule{0.0pt}{9.5pt}\right|\;\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})\geq
x\right\\}.$

This allows us to express the notions of distillable resource and resource
cost in a concise form, and it reduces to the usual notions of floor
$\lfloor\cdot\rfloor$ and ceiling $\lceil\cdot\rceil$ functions when
$\mathfrak{R}_{\mathbb{O}^{\prime}}(\mathcal{T})$ only takes values in
$\mathbb{N}$.

##  Appendix D Details of the fidelity of distillation

Here, we provide results and detailed discussions on the characterization of
the maximal achievable fidelity, providing practical benchmarks on the
distillation error. Recalling the definitions of $G_{\mathbb{O}}$ and
$G_{\operatorname{aff}(\mathbb{O})}$ introduced in (13), we have the following
result.

######  Theorem 6.

For a channel $\mathcal{N}\in\mathbb{O}_{\rm all}^{\prime}$, suppose that
$\mathrm{id}\otimes\mathcal{N}(\psi)$ is pure for any pure state $\psi$. Then,
for any channel $\mathcal{E}\in\mathbb{O}_{\rm all}$, it holds that

$\displaystyle
G_{\mathbb{O}}(\mathcal{E};R_{\min,\mathbb{O}^{\prime}}(\mathcal{N}))\geq\max_{\Theta\in\SS}F(\Theta(\mathcal{E}),\mathcal{N})\geq
G_{\mathbb{O}}(\mathcal{E};R_{s,\mathbb{O}^{\prime}}(\mathcal{N}))$ (36)

and

$\displaystyle
G_{\operatorname{aff}(\mathbb{O})}(\mathcal{E};R_{\min,\operatorname{aff}(\mathbb{O}^{\prime})}(\mathcal{N}))$
$\displaystyle\geq\max_{\Theta\in\SS}F(\Theta(\mathcal{E}),\mathcal{N})\geq
G_{\operatorname{aff}(\mathbb{O})}(\mathcal{E};R_{\max,\mathbb{O}^{\prime}}(\mathcal{N})).$
(37)

###### Proof.

Let us consider $G_{\mathbb{O}}$ first. Let $\Theta\in\SS$ be any free
transformation, and $\psi^{\star}$ be a pure state such that
$R_{\min,\mathbb{O}^{\prime}}(\mathcal{N})^{-1}=\max_{\mathcal{M}\in\mathbb{O}^{\prime}}\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\star}),\mathrm{id}\otimes\mathcal{M}(\psi^{\star})\right\rangle$.
Since $\Theta$ is a quantum superchannel, we can write it as
$\Theta(\cdot)=\mathcal{E}_{\rm
post}\circ\mathrm{id}\otimes\cdot\circ\mathcal{E}_{\rm pre}$ Chiribella  _et
al._ (2008) for some CPTP maps $\mathcal{E}_{\rm pre},\mathcal{E}_{\rm post}$.
Define

$\displaystyle W\coloneqq[\mathrm{id}\otimes\mathcal{E}^{\dagger}_{\rm
post}]\left(\mathrm{id}\otimes\mathcal{N}(\psi^{\star})\right).$ (38)

The complete positivity of $\mathcal{E}_{\rm post}$ immediately gives that
$W\geq 0$, and coupled with the fact that $\mathcal{E}_{\rm post}$ preserves
trace we get for any state $\sigma$ that

$\displaystyle\left\langle W,\sigma\right\rangle$
$\displaystyle=\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\star}),\mathrm{id}\otimes\mathcal{E}_{\rm
post}(\sigma)\right\rangle\leq 1$ (39)

which means that $W\leq\mathbbm{1}$. Finally, for any
$\mathcal{M}\in\mathbb{O}$, it holds that

$\displaystyle\left\langle
W,\mathrm{id}\otimes\mathcal{M}\left(\mathrm{id}\otimes\mathcal{E}_{\rm
pre}(\psi^{\star})\right)\right\rangle$
$\displaystyle=\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\star}),\mathrm{id}\otimes\Theta[\mathcal{M}](\psi^{\star})\right\rangle$
(40)
$\displaystyle\leq\max_{\mathcal{M}^{\prime}\in\mathbb{O}^{\prime}}\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\star}),\mathrm{id}\otimes\mathcal{M}^{\prime}(\psi^{\star})\right\rangle$
$\displaystyle=R_{\min,\mathbb{O}^{\prime}}(\mathcal{N})^{-1}.$

Altogether, we have shown that $W$ and the state
$\mathrm{id}\otimes\mathcal{E}_{\rm pre}(\psi^{\star})$ form a feasible
solution to $G_{\mathbb{O}}(\mathcal{E};R_{\min,\mathbb{O}}(\mathcal{N}))$.
Thus

$\displaystyle G_{\mathbb{O}}(\mathcal{E};R_{\min,\mathbb{O}}(\mathcal{N}))$
$\displaystyle\geq\left\langle
W,\mathrm{id}\otimes\mathcal{E}\left(\mathrm{id}\otimes\mathcal{E}_{\rm
pre}(\psi^{\star})\right)\right\rangle$ (41)
$\displaystyle=\left\langle\mathrm{id}\otimes\mathcal{N}(\psi^{\star}),\mathrm{id}\otimes\Theta[\mathcal{E}](\psi^{\star})\right\rangle$
$\displaystyle\geq F(\Theta(\mathcal{E}),\mathcal{N}).$

Since this holds for any $\Theta\in\SS$, the upper bound follows.

For the other inequality, take any feasible $W^{\prime}$ and $\psi^{\prime}$
in the optimization problem
$G_{\mathbb{O}}(\mathcal{E};R_{s,\mathbb{O}^{\prime}}(\mathcal{N}))$, and
define the superchannel

$\displaystyle\Theta(\mathcal{L})\coloneqq\left\langle
W^{\prime},\mathrm{id}\otimes\mathcal{L}(\psi^{\prime})\right\rangle\mathcal{N}+\left\langle\mathbbm{1}-W^{\prime},\mathrm{id}\otimes\mathcal{L}(\psi^{\prime})\right\rangle\mathcal{Q}$
(42)

where $\mathcal{Q}$ is a channel such that
$\mathcal{N}+(R_{s,\mathbb{O}^{\prime}}(\mathcal{N})-1)\mathcal{Q}\in
R_{s,\mathbb{O}^{\prime}}(\mathcal{N})\mathbb{O}^{\prime}$. By an argument
analogous to the one in the proof of Thm. 1, we have that $\Theta\in\SS$ and
get

$\displaystyle F(\Theta(\mathcal{E}),\mathcal{N})$
$\displaystyle\geq\left\langle
W^{\prime},\mathrm{id}\otimes\mathcal{E}(\psi^{\prime})\right\rangle\min_{\psi}\left\langle\mathrm{id}\otimes\mathcal{N}(\psi),\mathrm{id}\otimes\mathcal{N}(\psi)\right\rangle$
(43) $\displaystyle=\left\langle
W^{\prime},\mathrm{id}\otimes\mathcal{E}(\psi^{\prime})\right\rangle.$

Since this holds for any feasible $W^{\prime}$ and $\psi^{\prime}$, we get the
desired inequality.

The proof for $G_{\operatorname{aff}(\mathbb{O})}$ proceeds analogously (cf.
Thm. 1). ∎

In the case of manipulating quantum states, Thm. 6 recovers the results of
Ref. Regula  _et al._ (2020).

######  Remark.

Our proofs can be straightforwardly adapted to a related notion of fidelity
between channels often used in quantum communication theory, where the input
state is fixed as the maximally entangled state $\psi^{+}$:

$\displaystyle\widetilde{F}(\mathcal{E},\mathcal{F})\coloneqq
F(\mathrm{id}\otimes\mathcal{E}(\psi^{+}),\mathrm{id}\otimes\mathcal{F}(\psi^{+})).$
(44)

If one is interested in approximate transformations with respect to this
notion of channel distance, our main results can be recovered by replacing
$R_{\min,\mathbb{O}}$ with the quantity

$\displaystyle\widetilde{R}_{\min,\mathbb{O}}(\mathcal{E})$
$\displaystyle\coloneqq\min_{\mathcal{M}\in\mathbb{O}}R_{\min}(\mathrm{id}\otimes\mathcal{E}(\psi^{+})||\mathrm{id}\otimes\mathcal{M}(\psi^{+})),$
(45)

and analogously for $R_{\min,\operatorname{aff}(\mathbb{O})}$. It is known
that in settings such as quantum communication, the different notions of
fidelity are asymptotically equivalent Kretschmann and Werner (2004).
Interestingly, they can lead to the same quantitative statements already at
the one-shot level: in the setting of no-signalling–assisted communication, we
get from Thm. 6 that the maximal achievable fidelity is given by
$G_{\mathbb{O}_{\rm R}}(\mathcal{E})$, and this exactly matches the fidelity
computed in Ref. Leung and Matthews (2015) using the alternative fidelity
measure $\widetilde{F}$ (in fact, it is also equivalent to an analogous figure
of merit in classical communication with non-signalling assistance Leung and
Matthews (2015)).

##  Appendix E Remarks on state-based resource theories

Some resource theories are concerned with the manipulation of channels in
relation to some underlying state-based resource, characterized by a convex
set of free states $\mathbb{F}$, and the channels $\mathbb{O}$ corresponds to
some class of free operations for the state theory. In such cases, one can
define monotones $R_{\max,\mathbb{F}},R_{\min,\mathbb{F}},R_{s,\mathbb{F}}$
etc. in full analogy with the channel cases, but without the need to optimize
over input states. There is then a natural choice of reference (target)
channels: the preparation channel $\mathcal{P}_{\phi}$ with trivial input that
simply prepares a chosen resourceful pure state $\phi$. Indeed, this is the
way that entanglement Gour and Scandolo (2021); Bäuml  _et al._ (2019),
coherence Saxena  _et al._ (2020), and more general channel resources based on
states Liu and Yuan (2020) are often approached.

Our results immediately apply to this setting. Because of the simpler
structure of preparation channels, a number of simplifications will occur —
for instance, the quantification of the resourcefulness of
$\mathcal{P}_{\phi}$ will reduce to monotones defined at the level of states.

######  Lemma 7.

Let $\mathbb{O}$ be any class of free operations such that
$\mathcal{P}_{\sigma}\in\mathbb{O}\;\forall\sigma\in\mathbb{F}$. Then, for any
preparation channel $\mathcal{P}_{\phi}$, it holds that

$\displaystyle R_{\max,\mathbb{O}}(\mathcal{P}_{\phi})$
$\displaystyle=R_{\max,\mathbb{F}}(\phi)$ (46) $\displaystyle
R_{s,\mathbb{O}}(\mathcal{P}_{\phi})$ $\displaystyle=R_{s,\mathbb{F}}(\phi)$
$\displaystyle R_{\min,\mathbb{O}}(\mathcal{P}_{\phi})$
$\displaystyle=R_{\min,\mathbb{F}}(\phi).$

The Lemma follows immediately since the trivial input space of
$\mathcal{P}_{\phi}$ forces all channels in the optimization to also be
preparation channels, which means that we are optimizing over
$\mathcal{P}_{\sigma}\in\mathbb{O}$ s.t. $\sigma\in\mathbb{F}$. The result can
also be extended to more general replacement channels Regula and Takagi
(2021).

As an immediate consequence, from Cors. 2 and 4 we obtain that the one-shot
distillable resource $d_{\mathbb{O}}^{\epsilon}$ and one-shot resource cost of
any channel can be quantified exactly, when the reference states $\phi$
satisfy $R_{s,\mathbb{F}}(\phi)=R_{\min,\mathbb{F}}(\phi)$ or
$R_{\max,\mathbb{F}}(\phi)=R_{\min,\operatorname{aff}(\mathbb{F})}(\phi)$.
Notably, in Refs. Liu  _et al._ (2019); Regula  _et al._ (2020), it was shown
that in every convex resource theory of quantum states there exists a set of
so-called golden states ${\phi_{\rm gold}}$, such that each ${\phi_{\rm
gold}}$ in the set maximizes both $R_{\max,\mathbb{F}}$ and
$R_{\min,\mathbb{F}}$ among all states in the given space, and crucially that
$R_{\max,\mathbb{F}}({\phi_{\rm gold}})=R_{\min,\mathbb{F}}({\phi_{\rm
gold}})$. In many relevant resource theories, such golden states satisfy
$R_{s,\mathbb{F}}({\phi_{\rm gold}})=R_{\min,\mathbb{F}}({\phi_{\rm gold}})$ —
e.g. in bi- and multipartite entanglement Vidal and Tarrach (1999); Steiner
(2003); Regula  _et al._ (2020), entanglement of Schmidt number $k$ Johnston
_et al._ (2018), multi-level coherence Johnston  _et al._ (2018), and non-
positive partial transpose — or $R_{\max,\mathbb{F}}({\phi_{\rm
gold}})=R_{\min,\operatorname{aff}(\mathbb{F})}({\phi_{\rm gold}})$ — e.g. in
coherence and thermodynamics, thus fulfilling the required conditions.

Two direct applications of our results are then in the theory of bipartite
channel entanglement Gour and Scandolo (2021); Bäuml  _et al._ (2019), where
the aim is to distill copies of the maximally entangled state, or to simulate
the action of a given channel by using maximally entangled states.
Specifically, the distillable entanglement and entanglement cost can be
defined as

$\displaystyle d^{\epsilon}_{E}(\mathcal{E})$
$\displaystyle\coloneqq\max\left\\{\left.\log
d\;\rule{0.0pt}{9.5pt}\right|\;\exists\Theta\in\SS,F(\Theta(\mathcal{E}),\phi^{+}_{d})\geq
1-\epsilon\right\\}$ (47) $\displaystyle c^{\epsilon}_{E}(\mathcal{E})$
$\displaystyle\coloneqq\min\left\\{\left.\log
d\;\rule{0.0pt}{9.5pt}\right|\;\exists\Theta\in\SS,F(\Theta(\phi^{+}_{d}),\mathcal{E})\geq
1-\epsilon\right\\},$

where $\phi^{+}_{d}$ is the $d$-dimensional maximally entangled state, and the
allowed free transformations $\SS$ will depend on which class of channels we
wish to preserve. When the free channels are all separable channels
$\mathbb{O}_{\rm SEP}$, Cors. 2 and 4 establish exact expressions for the
distillable entanglement and entanglement cost of a channel under
separability-preserving superchannels:

$\displaystyle d^{\epsilon}_{E}(\mathcal{E})=\log\lfloor R_{H,\mathbb{O}_{\rm
SEP}}^{\epsilon}(\mathcal{E})\rfloor,\qquad
c^{\epsilon}_{E}(\mathcal{E})=\log\lceil R_{s,\mathbb{O}_{\rm
SEP}}^{\epsilon}(\mathcal{E})\rceil.$ (48)

Analogously, when the free channels are all PPT channels $\mathbb{O}_{\rm
PPT}$, we get $d^{\epsilon}_{E}(\mathcal{E})=\log\lfloor R_{H,\mathbb{O}_{\rm
PPT}}^{\epsilon}(\mathcal{E})\rfloor$ and
$c^{\epsilon}_{E}(\mathcal{E})=\log\lceil R_{s,\mathbb{O}_{\rm
PPT}}(\mathcal{E})\rceil$. These results generalize our considerations in
quantum communication theory to the bipartite channel setting, and can be
compared with the results of Refs. Gour and Scandolo (2021); Bäuml  _et al._
(2019) where the manipulation of channel entanglement under so-called PPT
superchannels was considered. We note that the corresponding results in the
special case of manipulating the entanglement of states were first reported in
Brandão and Plenio (2010); Brandão and Datta (2011).

##  Appendix F Remarks on the theory of nonlocality

For a number of settings for quantum nonlocality, the monotones discussed in
this work can be straightforwardly computed by exploiting the fact that
$\mathbb{O}$ can be represented by linear or semidefinite constraints Rosset
_et al._ (2020). We use this to evaluate the measures for a class of boxes
which includes the important cases of the Popescu-Rohrlich (PR) box and the
Tsirelson box.

Figure 1: Measures of nonlocality of the isotropic box
$\mathcal{B}_{p}=(1-p)\mathcal{B}_{\rm PR}+p\mathcal{B}_{\rm random}$, where
$\mathcal{B}_{\rm PR}$ is the PR box and $\mathcal{B}_{\rm random}$ denotes
white noise. The Tsirelson box is obtained for $p=1-\frac{\sqrt{2}}{2}\approx
0.29$.

We see in Fig. 1 that the generalized robustness $R_{\max,\mathbb{O}}$
linearly decreases from the value of $R_{\max,\mathbb{O}}(\mathcal{B}_{\rm
PR})=\frac{4}{3}$, while $R_{s,\mathbb{O}}$ exhibits similar behavior from the
value of $R_{s,\mathbb{O}}(\mathcal{B}_{\rm PR})=\frac{3}{2}$. Notice in
particular that $R_{\max,\mathbb{O}}(\mathcal{B}_{\rm
PR})=R_{\min,\mathbb{O}}(\mathcal{B}_{\rm PR})$ (analogously to the property
for golden states Liu  _et al._ (2019); Regula  _et al._ (2020) in state-based
theories), but the full-rank noise in the form of $\mathcal{B}_{\rm random}$
makes the min-entropy measure $R_{\min,\mathbb{O}}(\mathcal{B}_{p})$ equal to
1 for all other boxes in this family. This means, in fact, that the
transformation $\mathcal{B}_{p}^{\otimes n}\to\mathcal{B}_{\rm PR}$ with $p>0$
is impossible for any $n$, since the Choi matrix $J_{\mathcal{B}_{p}^{\otimes
n}}$ is always full rank and hence
$R_{\min,\mathbb{O}}(\mathcal{B}_{p}^{\otimes n})=1$.

##  Appendix G Remarks on robustness measures

Due to their general formulation, robustness and other related measures have
been considered in various settings such as nonlocality, contextuality, and
measurement incompatibility, which impose specific structures on the resources
of interest. We will show that the robustness measures defined based on the
mixture of the original resources studied in previous works coincide with the
channel-based robustness defined in our formalism. Here, we focus on the case
of the generalized robustness of measurement incompatibility, but similar
arguments can be applied to other types of resources as well as other variants
of the measures such as the standard robustness and weight measure.

Let $\mathfrak{M}$ be the set of all sets of POVMs, and $\mathfrak{C}$ be the
set of compatible POVMs. For any set of POVMs $\\{M_{a|x}\\}$, the robustness
of measurement incompatibility was originally defined as Haapasalo (2015);
Uola  _et al._ (2015)

$\displaystyle
R_{\mathfrak{C}}(\\{M_{a|x}\\})\coloneqq\min\left\\{\left.1+r\;\rule{0.0pt}{9.5pt}\right|\;\frac{M_{a|x}+rN_{a|x}}{1+r}=F_{a|x}\,\forall
a,x,\ \\{F_{a|x}\\}\in\mathfrak{C},\ \\{N_{a|x}\\}\in\mathfrak{M}\right\\}.$
(49)

On the other hand, our formalism represents the sets of POVMs in the form of
bipartite channels. Namely, the theory of measurement incompatibility can be
formulated by taking $\mathbb{O}_{\rm all}$ to be the set of channels
representing the sets of POVMs as

$\displaystyle\mathbb{O}_{\rm
all}\coloneqq\left\\{\left.\mathcal{E}_{\\{M_{a|x}\\}}\;\rule{0.0pt}{9.5pt}\right|\;\\{M_{a|x}\\}\in\mathfrak{M}\right\\}$
(50)

where we defined

$\displaystyle\mathcal{E}_{\\{M_{a|x}\\}}(\sigma\otimes\rho)\coloneqq\sum_{x,a}\bra{x}\sigma\ket{x}\left\langle
M_{a|x},\rho\right\rangle\ket{a}\\!\bra{a}$ (51)

where $\\{\ket{x}\\}$ and $\\{\ket{a}\\}$ are orthonormal bases representing
classical variables for measurement settings and measurement outcomes
respectively. Then, the set of free channels
$\mathbb{O}\subseteq\mathbb{O}_{\rm all}$ can naturally be defined using the
set of compatible POVMs $\mathfrak{C}$ in the place of $\mathfrak{M}$ in (50).

With this formalism, the robustness for channel $\mathcal{E}_{\\{M_{a|x}\\}}$
is given by (1). Below, we show the equivalence between this and the original
robustness measure (49).

######  Lemma 8.

Let $\mathbb{O}_{\rm all}$ be the set of bipartite channels defined in (50)
and $\mathbb{O}$ be the set of free channels that represent the compatible
POVMs. Then, it holds that

$\displaystyle
R_{\mathfrak{C}}(\\{M_{a|x}\\})=R_{\mathbb{O}}(\mathcal{E}_{\\{M_{a|x}\\}}).$
(52)

###### Proof.

Following the definition of $R_{\mathbb{O}}$,

$\displaystyle R_{\mathbb{O}}(\mathcal{E}_{\\{M_{a|x}\\}})$
$\displaystyle=\min\left\\{\left.1+r\;\rule{0.0pt}{9.5pt}\right|\;\frac{\mathcal{E}_{\\{M_{a|x}\\}}+r\mathcal{N}}{1+r}=\mathcal{F},\
\mathcal{N}\in\mathbb{O}_{\rm all},\mathcal{F}\in\mathbb{O}\right\\}$ (53)
$\displaystyle=\min\left\\{\left.1+r\;\rule{0.0pt}{9.5pt}\right|\;\frac{\mathcal{E}_{\\{M_{a|x}\\}}+r\mathcal{E}_{\\{N_{a|x}\\}}}{1+r}=\mathcal{E}_{\\{F_{a|x}\\}},\
\\{N_{a|x}\\}\in\mathfrak{M},\\{F_{a|x}\\}\in\mathfrak{C}\right\\}$
$\displaystyle=\min\left\\{\left.1+r\;\rule{0.0pt}{9.5pt}\right|\;\forall\sigma,\rho,\sum_{x,a}\bra{x}\sigma\ket{x}\left\langle\frac{M_{a|x}+rN_{a|x}}{1+r}-F_{a|x},\rho\right\rangle\ket{a}\\!\bra{a}=0,\
\\{N_{a|x}\\}\in\mathfrak{M},\\{F_{a|x}\\}\in\mathfrak{C}\right\\}$
$\displaystyle=\min\left\\{\left.1+r\;\rule{0.0pt}{9.5pt}\right|\;\frac{M_{a|x}+rN_{a|x}}{1+r}=F_{a|x}\,\forall
a,x,\ \\{N_{a|x}\\}\in\mathfrak{M},\\{F_{a|x}\\}\in\mathfrak{C}\right\\}$

where the last equality is obtained by observing that in the expression of the
the third line, the fact that the equality holds for any $\sigma$, as well as
that $\ket{a}\\!\bra{a}$ is a classical state, implies that
$\left\langle\frac{M_{a|x}+rN_{a|x}}{1+r}-F_{a|x},\rho\right\rangle=0,\forall
a,x$, and further imposing that this holds for any $\rho$ gives
$\frac{M_{a|x}+rN_{a|x}}{1+r}-F_{a|x}=0,\forall a,x$. On the other hand,
$\frac{M_{a|x}+rN_{a|x}}{1+r}-F_{a|x}=0,\forall a,x$ clearly implies the
expression in the third line. The proof is completed by noting that the last
expression is nothing but the one for $R_{\mathfrak{C}}(\\{M_{a|x}\\})$. ∎