# Resource Preservability

Chung-Yun Hsieh<EMAIL_ADDRESS>ICFO - Institut de Ciències
Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Spain

###### Abstract

Resource theory is a general, model-independent approach aiming to understand
the qualitative notion of resource quantitatively. In a given resource theory,
free operations are physical processes that do not create the resource and are
considered zero-cost. This brings the following natural question: For a given
free operation, what is its ability to preserve a resource? We axiomatically
formulate this ability as the resource preservability, which is constructed as
a channel resource theory induced by a state resource theory. We provide two
general classes of resource preservability monotones: One is based on state
resource monotones, and another is based on channel distance measures.
Specifically, the latter gives the robustness monotone, which has been
recently found to have an operational interpretation. As examples, we show
that athermality preservability of a Gibbs-preserving channel can be related
to the smallest bath size needed to thermalize all its outputs, and it also
bounds the capacity of a classical communication scenario under certain
thermodynamic constraints. We further apply our theory to the study of
entanglement preserving local thermalization (EPLT) and provide a new family
of EPLT which admits arbitrarily small nonzero entanglement preservability and
free entanglement preservation at the same time. Our results give the first
systematic and general formulation of the resource preservation character of
free operations.

##  1 Introduction

An important goal in the study of physics is to understand and identify
different resources: It may be an effect, an object, or a phenomenon, which
enables us to achieve something that can never be achieved in its absence.
Before consuming the resource and triggering the advantages, one needs to make
sure the given systems have the resource. Hence, the first important question
is: How to probe it? Tremendous efforts have been made in this line of
research for various resources. For instance, the positive partial transpose
criterion for entanglement is a representative result for entanglement
detection [1, 2, 3]. Also, various Bell inequalities and steering inequalities
provide alternative ways of probing different quantum resources [4, 5, 6, 7,
8, 9].

Knowing merely the existence of the resource is, however, insufficient for all
applications. This is because one may not only need the resource, but also
need it to be strong enough: To demonstrate quantum advantages in
teleportation [10, 11], to witness a stronger than classical heat back-flow
[12], or to violate a Bell/steering inequality, strong enough quantum
correlations are necessary. A quantitative understanding of qualitative
resources is therefore crucial. This question can be answered by a generic
approach called resource theory, aiming to provide a general strategy to
quantitatively formulate a given resource.

A resource theory can be interpreted as a triplet, consisting of the resource
itself (e.g. entanglement), quantities without the resource (e.g. separable
states), and physical processes that will not create the resource (e.g. local
operation and classical communication channels [13]). A resource theory
provides a method to quantify the resource: With reasonable postulates, a
resource monotone can be introduced, which can be interpreted as a quantifier
attributing numbers to the resource content. This important feature allows us
to know more than just whether the resource exists or not: It also enables us
to know the amount of resource. Many results have been established in (but not
limited to) the resource theories of entanglement [1, 14], coherence [15, 16],
nonlocality [5, 17], steering [8, 18, 19, 20, 9], asymmetry [21, 22], and
athermality [23, 25, 24, 26, 28, 27]. Also, several general features of
resource theories have been reported [29, 30, 31, 32, 33, 35, 38, 43, 34, 37,
36, 41, 42, 39, 40]. Notably, resource theories of quantum channels (or simply
channels, which are also known as completely-positive trace-preserving maps
[13]) and related topics have drawn much attention recently [44, 45, 47, 46,
43, 50, 48, 51, 56, 60, 53, 49, 54, 55, 57, 58, 52, 61, 62, 59].

Figure 1:  In this work, we say a channel preserves a resource if it does not
completely destroy the resource for every inputs. To illustrate this, consider
the above figure. Each battery icon represents a state, and the height of the
colored column is a resource measure. The purple color indicates the existence
of the resource, and the yellow means the state is free. Then we have three
examples: (a) The channel maintains the resourcefulness for every input. (b)
The channel partially degrades the resource content but can still output the
resource for certain inputs. (c) The channel totally destroys resource for
every input. In this work, channels in (a) and (b) are considered to have the
ability to preserve the resource, but not the one in (c).

One important ingredient in a resource theory is the allowed physical
processes that will not create the resource, which are called free operations.
An ultimate goal for a resource theory is to identify under which conditions a
quantity can be transformed into another via free operations. A proper answer
can tell us how resourceful the output quantities can be after free
operations, giving useful information for both theoretical and practical
purposes. This is conceptually related to channel’s ability to preserve a
resource, which is a phenomenon lacking a quantitative understanding. To be
precise, we say a channel preserves a resource if it does not completely
destroy the resource for every input. In other words, it can partially degrade
the resource, while there must be certain output states that are resourceful
(see Fig. 1). This motivates us to ask the following question:

Given a free operation, how to quantify its ability to preserve the given
resource?

In other words, we are asking for a quantitative study of the qualitative
behavior (i.e. the ability to preserve the resource) of free operations, which
can be interpreted as a resource theory inherited from the given resource
theory. With a rigorous answer, one will be able to identify the efficiency of
the given free operation to protect the resource, which will clarify the
fundamental structure of free operations in a general resource theory. This
question is also motivated by other purposes: For example, a suitable measure
of the ability of a given dynamics to preserve entanglement can provide new
insights to the study of the interplay between entanglement and thermalization
[63]. Also, some previous results have addressed similar issues for
entanglement [64], while a general treatment for free operations with
arbitrary state resources is still unknown.

In this work, we axiomatically formulate the ability of free operations to
preserve a resource of quantum states. This ability, termed resource
preservability, is formulated as a channel resource theory induced by the
given state resource theory. We provide general assumptions of the
formulation, discussing the corresponding free operation, and introducing
axioms on the resource preservability monotones.

Two classes of resource preservability monotones are provided: One is induced
by the resource monotones of the given state resource theory, with the
intuition behind as the maintained resource during the process; another is
based on the channel distance from the set of free operations that will
destroy the resource. Moreover, the one based on channel distance will induce
a robustness-like monotone, with an operational interpretation as the erasure
cost of resource preservability due to Ref. [54].

As an example, we consider the resource theory of athermality and show that an
resource preservability monotone of a given Gibbs-preserving channel is
related to the smallest bath size needed to thermalize all its output states.
We further show that the robustness-like monotone serves as an upper bound of
the classical capacity of a classical communication scenario subject to
certain thermodynamic constraints. These connect thermodynamics and classical
communication to our current study. As another application, we apply our
theory to the study of entanglement preserving local thermalizations (EPLTs)
[63], which are local operation plus shared randomness channels that can
locally thermalize subsystems for arbitrary inputs, while keep the global
entanglement for certain inputs. We show that EPLTs can admit arbitrarily
small entanglement preservability at finite temperatures and preservation of
free entanglement [65] simultaneously. This reveals the fact that EPLT is a
concept compatible with arbitrarily small ability of entanglement
preservation, and can still preserve distillable entanglement at the same
time.

This work is structured as follows. We start with basic notions of a general
state resource theory and general setup of resource preservability in Sec. 2.
After the formal setup, we formulate free super-channel in Sec. 3, and in Sec.
4 we axiomatically introduce resource preservability monotones. In Sec. 5, we
consider examples with the resource theory of athermality and apply the theory
of resource preservability to EPLT. Finally, we conclude in Sec. 6.

##  2 Setup and Assumptions

A resource theory of quantum states, or simply a state resource theory, can be
understood as a combination of the following three ingredients: The resource
itself (denoted by $R$), states without the resource (the free states; denote
the set of all free states by $\mathcal{F}_{R}$), and channels that can be
applied freely and cannot create the resource (the free operations; denote the
set of all free operations by $\mathcal{O}_{R}$). Hence, a state resource
theory can be written as the triplet $(R,\mathcal{F}_{R},\mathcal{O}_{R}).$ A
channel resource theory can be defined in a similar way with a state resource
theory by replacing states by channels, and the corresponding free operations
($\mathcal{O}_{R}$) will be super-channels [66, 67].

In this work, the only class of channel resource theories will be the one of
resource preservability induced by different state resource theories. Hence,
for convenience, from now on $R$-theory means the resource theory of the given
resource $R$ of quantum states. The corresponding channel resource theory of
resource preservability (abbreviated as $R$-preservability) will be called an
$R$-preservability theory.

To formulate $R$-preservability as a channel resource theory inherited from a
given $R$-theory, the first thing is to identify the free channels. To this
end, we consider free operations of the given $R$-theory that cannot preserve
resource for every input:

$\displaystyle\mathcal{O}_{R}^{N}\coloneqq\\{\mathcal{E}\in\mathcal{O}_{R}\;|\;\mathcal{E}(\rho)\in\mathcal{F}_{R}\;\forall\,\rho\\}.$
(1)

Channels of this kind will be called resource-annihilating channels
(abbreviated as $R$-annihilating channels) which is inspired by the name of
entanglement-annihilating channel [64]. This set gives the free channels of
the $R$-preservability theory. In view of this notion, every element in
$\mathcal{O}_{R}\setminus\mathcal{O}_{R}^{N}$ will be understood to have
certain ability to preserve the given resource111We remark that the setting
here is consistent while not the same with the channel resource theory
introduced in Ref. [55]: Since in the study of $R$-preservability the identity
channel will be the most resourceful one, some results of Refs. [55, 54]
cannot apply. Also, our approach is genuinely different from the resource
destroying maps [34], which leave free states invariant and map resourceful
states to some free states. (see also Fig. 1).

It remains to specify the corresponding free operations and quantifiers of
$R$-preservability, which are the main tasks of this work. Before that, we
need to impose some basic assumptions and constraints on the given $R$-theory
in order to have a reasonable study.

At the beginning of the formulation, one may wonder whether we should assume
the following property in a bipartite system ${\rm SS^{\prime}}$:

$\Lambda_{\rm S}\otimes\Lambda_{\rm S^{\prime}}\in\mathcal{O}_{R}^{N}$ if
$\Lambda_{\rm S},\Lambda_{\rm S^{\prime}}\in\mathcal{O}_{R}^{N}$?

This property forbids any possibility to activate the $R$-preservability. This
is, however, not true due to the existence of activation properties of certain
resources [68, 69, 70, 71, 72]. More precisely, in Appendix A we show that in
some $R$-theories, one can construct a free operation
$\widetilde{\mathcal{T}}\in\mathcal{O}_{R}^{N}$ such that
$\widetilde{\mathcal{T}}^{\otimes k}\notin\mathcal{O}_{R}^{N}$ for some
integer $k>0$. This means if we want to formulate $R$-preservability theory in
a general way applicable to different $R$-theories, we need to respect certain
properties such as the activation of the $R$-preservability. To impose basic
assumptions on $R$-theory, we need the following concept first:

######  Definition 1.

(Absolutely Free State) A free state $\widetilde{\eta}$ is said to be an
absolutely free state for the given $R$-theory if

$\displaystyle\widetilde{\eta}\otimes\eta\in\mathcal{F}_{R}\quad\forall\;\eta\in\mathcal{F}_{R}.$
(2)

We denote the set of all absolutely free states by
$\widetilde{\mathcal{F}}_{R}$.

In other words, absolutely free states are those without hidden resource [71,
72]. For example, in the $R$-theory of entanglement, all the separable states
are absolutely free states. However, as we have mentioned, there also exist
$R$-theories with states that are not absolutely free: This can be seen by the
superactivation of nonlocality [68] and steering [69, 70]. We remark that
$\widetilde{\mathcal{F}}_{R}$ is closed under tensor product; that is,
$\widetilde{\eta}_{1}\otimes\widetilde{\eta}_{2}\in\widetilde{\mathcal{F}}_{R}$
if $\widetilde{\eta}_{1},\widetilde{\eta}_{2}\in\widetilde{\mathcal{F}}_{R}$.

With the above notion, we consider $R$-theories with the following properties
in this work:

  1. (R1)

$\widetilde{\mathcal{F}}_{R}\neq\emptyset$ and $\mathcal{F}_{R}$ is convex.

  2. (R2)

Identity and partial trace are free operations.

  3. (R3)

Tensoring with absolutely free states [i.e.
$(\cdot)\mapsto(\cdot)\otimes\widetilde{\eta}$ for a given
$\widetilde{\eta}\in\widetilde{\mathcal{F}}_{R}$] are free operations.

  4. (R4)

Free operations are closed under tensor products, convex sums, and
compositions: If $\mathcal{E}_{1},\mathcal{E}_{2}\in\mathcal{O}_{R}$, then
$\mathcal{E}_{1}\otimes\mathcal{E}_{2}\in\mathcal{O}_{R}$,
$p\mathcal{E}_{1}+(1-p)\mathcal{E}_{2}\in\mathcal{O}_{R}$
$\forall\;p\in[0,1]$, and
$\mathcal{E}_{1}\circ\mathcal{E}_{2}\in\mathcal{O}_{R}$.

Let us briefly comment on the above properties. We assume property (R1)
because we aim to study $R$-preservability, which is a comparison of
resourceless states and resourceful states. Also, we expect genuinely
resourceless states exist and convex sums of resourceless states will not be
resourceful, which are common features shared by many $R$-theories. Property
(R2) is assumed because in an $R$-theory, identity map and partial trace can
never increase the amount of resource and will usually fulfill other
conditions of a free operation: Conceptually, it means “doing noting” and
“ignoring part of the system” are both free and costless. Property (R3) makes
sure the resource content will not increase after an extension with an
absolutely free state $\widetilde{\eta}$. Property (R4) is imposed because,
for two channels which cannot create the resource, we expect their
simultaneous applications (tensor product), classical mixture (convex sum),
and sequential applications (composition) still will not have the ability to
create the resource. As expected, it is a common property possessed by many
choices of free operations in $R$-theories such as the ones of entanglement
[1], nonlocality [49, 17], and athermality [23, 26] (note that there do exist
examples which cannot satisfy this property222To see a counterexample,
consider the $R$-theory of nonlocality with the nonlocality non-generating
channels as free operations. Suppose $\rho_{0}$ is a local state such that
$\rho_{0}^{\otimes 2}$ is nonlocal [68]. Then the state preparation channel
$\Phi_{\rho_{0}}:(\cdot)\mapsto\rho_{0}$ is a nonlocality non-generating
channel, while $\Phi_{\rho_{0}}\otimes\Phi_{\rho_{0}}$ will always have
nonlocal output, thereby being able to generate nonlocality. This again shows
the activation property may lead to unexpected results.). This also implies
that in this work the set $\mathcal{O}_{R}^{N}$ is always convex.

Before the formulation of $R$-preservability, it is important to introduce the
following analog concept of absolutely free states for channels.

######  Definition 2.

(Absolutely $R$-Annihilating Channel) We say
$\widetilde{\Lambda}\in\mathcal{O}_{R}^{N}$ is an absolutely $R$-annihilating
channel if

$\displaystyle\widetilde{\Lambda}\otimes\Lambda\in\mathcal{O}_{R}^{N}\quad\forall\,\Lambda\in\mathcal{O}_{R}^{N}.$
(3)

We denote the set of all such channels by $\widetilde{\mathcal{O}}_{R}^{N}$.

This definition means the $R$-preservability of absolutely $R$-annihilating
channels cannot be activated. As an example of an absolutely $R$-annihilating
channel, consider again the $R$-theory of entanglement. Then every local
operation and classical communication (LOCC) channel that is entanglement-
annihilating [64] and entanglement-breaking [73] will be absolutely
$R$-annihilating channels (see Appendix B for the detailed explanation).

We also remark the following facts for a given $R$-theory:

$\displaystyle\widetilde{\Lambda}\circ\mathcal{E}\in\widetilde{\mathcal{O}}_{R}^{N}\quad\&\quad\mathcal{E}\circ\widetilde{\Lambda}\in\widetilde{\mathcal{O}}_{R}^{N}\quad\forall\;\mathcal{E}\in\mathcal{O}_{R};$
$\displaystyle\widetilde{\Lambda}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}\quad\forall\;\widetilde{\Lambda}_{\rm
S},\widetilde{\Lambda}_{\rm S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}.$
(4)

According to the first line in Eq. (2), a sequential application of free
operations cannot preserve any resource (even with the assistance of ancillary
$R$-annihilating channels) if one has already added one absolutely
$R$-annihilating channel in the sequence. Also, since absolutely
$R$-annihilating channels forbid activation, the second line in Eq. (2) means
simultaneous applications of two such channels still do not allow activation.

Before introducing the main results, we specify notations. In this work we
ignore the dependency of system size of the notations $\mathcal{O}_{R}^{N}$
and $\mathcal{O}_{R}$. To emphasize the contrast between the main systems and
ancillary systems, we use subscripts ${\rm S,S^{\prime}}$ for the main systems
and A,B for the ancillary systems. When only bipartition needs to be
addressed, we use the common notations A,B for subsystems. The meaning of
subscripts will be clear from the context.

##  3 Free Operation of Resource Preservability

To specify the free operation of $R$-preservability, we need to know first how
to map a channel in $\mathcal{O}_{R}$ into another channel in
$\mathcal{O}_{R}$. The general structure of such mapping (which maps channels
to channels) is shown to take the following form [66]:

$\displaystyle\mathcal{E}\mapsto\mathcal{M}\circ(\mathcal{E}\otimes\mathcal{I}_{\rm
A})\circ\mathcal{N},$ (5)

where ${\rm A}$ stands for the ancillary system, and $\mathcal{M}$,
$\mathcal{N}$ are some quantum channels. Such mappings are called super-
channels [66, 67, 36, 54]. One potential way to introduce free operations of
$R$-preservability, or simply free super-channels, is to consider all super-
channels that will not increase $R$-preservability. This gives the largest
possible set of free super-channels, while it may not always have intuitive
and clear physical interpretation (see Ref. [59] for an exception). Also,
whether all such mappings can always map elements of $\mathcal{O}_{R}$ into
$\mathcal{O}_{R}$ is still unclear333We remark that the structure of free
supper channels depends on the structure of $\mathcal{O}_{R}$. For example, in
the $R$-theory of entanglement, if we set $\mathcal{O}_{R}$ to be all LOSR
channels (see Appendix A.1 for the definition), then LOCC channels are outside
our consideration. This means a super-channel that maps some LOSR channels
into LOCC channels will not be a suitable free super-channel in this case.
Hence, the set of all super-channels that will not generate $R$-preservability
may not always be a well-defined set of free super-channels of
$R$-preservability.. Hence, in this work we prefer a different approach: We
try to impose conditions on Eq. (5) and focus on free super-channels with
clear physical meanings.

To this end, we interpret Eq. (5) as a three-step process consisting of a pre-
processing ($\mathcal{N}$), an ancillary process ($\mathcal{I}_{\rm A}$), and
a post-processing ($\mathcal{M}$). The first condition to be imposed is that
free super-channels should be realized freely in the given $R$-theory, since
we expect them to be implementable without the assistance of the resource $R$.
This suggests that all steps in Eq. (5) should be free operations of the given
$R$-theory; that is, $\mathcal{N},\mathcal{M}\in\mathcal{O}_{R}$. The second
condition to be imposed is that free super-channels cannot create
$R$-preservability. However, since identity map has the best
$R$-preservability, this may fail if one uses identity map for the ancillary
process in Eq. (5). This suggests that the ancillary system should perform
certain processes to ensure it is impossible to create $R$-preservability.
Concerning the existence of activation properties discussed in Appendix A, we
ask the ancillary system to perform only absolutely $R$-annihilating channels.
The above discussions motivate us to consider the following notion as the free
operation of an $R$-preservability theory in this work:

######  Definition 3.

(Free Super-Channel of $R$-Preservability) In this work, the free operation of
$R$-preservability, or say the free super-channel
$F:\mathcal{O}_{R}\to\mathcal{O}_{R}$, is of the form

$\displaystyle
F_{\mathcal{E}}\coloneqq\Lambda_{+}\circ(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
A})\circ\Lambda_{-},$ (6)

where $\Lambda_{+},\Lambda_{-}\in\mathcal{O}_{R}$ are free operations of the
$R$-theory and $\widetilde{\Lambda}_{\rm A}\in\widetilde{\mathcal{O}}_{R}^{N}$
is an absolutely $R$-annihilating channel.

For the generality of the $R$-preservability theory, we allow different
input/output dimensions of the free super-channels444One can also formulate
the theory with the fixed system dimension and forbid the ancillary systems,
while in our approach we prefer a more general version. This is similar to the
case of channel discrimination: One can use either trace norm or diamond norm.
The trace norm gives an intuitive description of channel discrimination with
the focus only on the given system, while the performance can be improved when
one switches to the diamond norm. In this work, we try to capture the spirit
of the latter., which means the $R$-preservability of the given channel on the
main system S may be assisted by channels acting on ancillary systems, while
the ancillary channels need to obey the rules: They cannot provide additional
$R$-preservability, and they cannot be assisted by the given state resource
$R$.

Note that if one simply assumes $\Lambda_{+},\Lambda_{-}$ to possess zero
$R$-preservability, then the output will only be $R$-annihilating channels.
Hence, we allow $\Lambda_{+},\Lambda_{-}$ to be arbitrary free operations.
Also, we have $F_{\Lambda}\in\mathcal{O}_{R}^{N}$ if
$\Lambda\in\mathcal{O}_{R}^{N}$, which is because $\widetilde{\Lambda}_{\rm
A}\in\widetilde{\mathcal{O}}_{R}^{N}$. This ensures that Eq. (6) is a suitable
free operation even with the activation property of $R$-preservability
(Appendix A).

##  4 Resource Preservability Monotone

An important feature of a resource theory is that it provides a way to
quantify the resource [36]. Let $\mathbb{Q}$ be the set of all states or all
channels. Then a resource monotone of the given resource $R$ is a function
$Q_{R}:\mathbb{Q}\to[0,\infty]$ satisfying properties (M1) and (M2):

  1. (M1)

$Q_{R}(q)\geq 0\quad\forall\,q\in\mathbb{Q}$ and $Q_{R}(q)=0$ if
$q\in\mathcal{F}_{R}$.

  2. (M2)

$Q_{R}[\Lambda(q)]\leq
Q_{R}(q)\quad\forall\,q\in\mathbb{Q}\quad\&\quad\forall\Lambda\in\mathcal{O}_{R}$.

  3. (M3)

$Q_{R}[pq_{1}+(1-p)q_{2}]\leq pQ_{R}(q_{1})+(1-p)Q_{R}(q_{2})$
$\forall\,q_{1},q_{2}\in\mathbb{Q}\quad\&\quad\forall p\in[0,1]$.

  4. (M4)

$Q_{R}(q)=0$ if and only if $q\in\mathcal{F}_{R}$.

It is called convex if it also satisfies property (M3), and it is called
faithful if it also satisfies property (M4). To avoid trivial case, we always
assume $Q_{R}(q)>0$ for some $q$ in this work. With the above notions, we are
now in position to introduce the $R$-preservability monotones.

######  Definition 4.

(Resource Preservability Monotone) In an $R$-preservability theory, an
$R$-preservabil-ity monotone $P_{R}$ is a channel resource monotone satisfying
the following additional property:

$\displaystyle P_{R}(\mathcal{E}\otimes\mathcal{E}^{\prime})\geq
P_{R}(\mathcal{E})\quad\forall\;\mathcal{E},\mathcal{E}^{\prime}\in\mathcal{O}_{R},$
(7)

and the equality holds if
$\mathcal{E}^{\prime}\in\widetilde{\mathcal{O}}_{R}^{N}$.

This additional property illustrates the basic expectation of a good
quantifier of $R$-preservability: $R$-preservability will not decrease under
tensor product, and it will not increase under tensor product with absolutely
$R$-annihilating channels. Note again that we do not impose the property
$P_{R}(\mathcal{E}_{\rm S}\otimes\Lambda_{\rm S^{\prime}})\leq
P_{R}(\mathcal{E}_{\rm S})\quad\forall\,\Lambda_{\rm
S^{\prime}}\in\mathcal{O}_{R}^{N}\;\&\;\mathcal{E}_{\rm S}\in\mathcal{O}_{R}$
due to the existence of the activation property discussed in Appendix A. It is
still possible for an $R$-preservability monotone to satisfy this property,
which simply means that monotone cannot witness activated $R$-preservability.

We introduce two classes of $R$-preservability monotones, whose underlying
intuitions are stated as follows:

  * •

Interpret $R$-preservability as the ability to maintain resource during the
operation.

  * •

Interpret $R$-preservability as the channel distance from the set of
$R$-annihilating channels.

While they originate from different concepts, in the following sections we
will show that both of them admit $R$-preservability monotones.

###  4.1 Resource Preservability Monotone: The Maintained Resource

For a given resourceful state $\rho$ and a given state resource monotone
$Q_{R}$, an intuitive way to quantify the ability of a free operation
$\mathcal{E}_{\rm S}$ to preserve the resource $R$ of $\rho$ is to compare the
difference between $Q_{R}(\rho)$ and $Q_{R}[\mathcal{E}_{\rm S}(\rho)]$; that
is, $\frac{Q_{R}[\mathcal{E}_{\rm S}(\rho)]}{Q_{R}(\rho)}$. This proposes the
following general candidate induced by $Q_{R}$: (we use subscript to denote
the corresponding subsystems)

$\displaystyle P_{Q_{R}}^{(f,g)}(\mathcal{E}_{\rm
S})\coloneqq\overline{\sup_{{\rm A}}}\;\frac{(f\circ Q_{R})[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})]}{(g\circ
Q_{R})(\rho_{\rm SA})},$ (8)

where $f$ is a finite-valued strictly increasing function with $f(0)=0$, $g$
is a non-decreasing function satisfying $g^{-1}(\\{0\\})\subseteq\\{0\\}$
[this means the only $x$ that may achieve $g(x)=0$ is $x=0$]. Here we use the
following abbreviation:

$\displaystyle\overline{\sup_{\rm A}}\coloneqq\sup_{{\rm
A};\widetilde{\Lambda}_{\rm A}\in\widetilde{\mathcal{O}}_{R}^{N};\rho_{\rm
SA}},$ (9)

where the maximization is taken over all possible finite dimensional ancillary
systems ${\rm A}$, all absolutely $R$-annihilating channels
$\widetilde{\Lambda}_{\rm A}\in\widetilde{\mathcal{O}}_{R}^{N}$ on the
ancillary system A, and all states $\rho_{\rm SA}$ on the composite system
${\rm SA}$. In the maximization we allow the ancillary system to have zero
dimension, corresponding to the original system ${\rm S}$. We stress that the
maximization in Eq. (8) is restricted to $\rho_{\rm SA}$ achieving non-zero
$Q_{R}$ values. This makes sure the value is always finite.

The idea behind Eq. (8) is to consider a general ratio between the input and
the output of the given free operation. By considering particular combinations
of $f$ and $g$, we have the following candidates:

$\displaystyle\overline{\sup_{{\rm
A}}}\,\frac{Q_{R}[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})]}{Q_{R}(\rho_{\rm SA})}\;;\;\overline{\sup_{{\rm
A}}}\,Q_{R}[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})].$
(10)

The first one can be interpreted as the optimal maintained resource during the
process $\mathcal{E}$, and the second one can be understood as the optimal
remaining amount of resource in the end of the process $\mathcal{E}$.

Note that we do not use identity map $\mathcal{I}_{\rm A}$ for the ancillary
systems in the above definition. This is because identity channel is the most
resourceful channel, and considering ancillary system with it may create
“artificial $R$-preservability”. For example, if one uses identity for the
ancillary systems in the $R$-theory of entanglement, then one will have non-
zero $R$-preservability for entanglement-annihilating channels that are not
entanglement-breaking [64]. Merely using $R$-annihilating channels
$\mathcal{O}_{R}^{N}$ for the extension is still not enough due to the
existence of the activation property (Appendix A). This explains the need of
introducing absolutely $R$-annihilating channels.

We now present the first main result, whose proof is given in Appendix C.
Recall that $R$-theory represents a state resource theory with resource $R$.

######  Theorem 1.

Given an $R$-theory and a state resource monotone $Q_{R}$. Then
$P_{Q_{R}}^{(f,g)}$ defined by Eq. (8) is an $R$-preservability monotone.
Moreover, It is faithful if $Q_{R}$ is faithful, and it is convex if $f\circ
Q_{R}$ is convex.

As a remark, the assumption $\widetilde{\mathcal{F}}_{R}\neq\emptyset$ is only
used in the proof of Eq. (7), and this assumption can be dropped when $g$ is a
positive constant. We state this special case in Corollary C.1. Also, it will
be an interesting future research topic to study specific operational
interpretations of different combinations of $f,g$ with different
$R$-theories.

###  4.2 Resource Preservability Monotone: The Channel Distance

One intuitive way to quantify a resource is to consider the distance away from
the set consisting of quantities without the resource. Here we use the similar
way to interpret $R$-preservability. To this end, we consider a general
distance measure on states defined as a function
$D:\mathcal{S}\times\mathcal{S}\to[0,\infty]$ satisfying $D(\rho,\sigma)\geq
0$ and equality holds if and only if $\rho=\sigma$ ($\mathcal{S}$ is the set
of quantum states). Now, we introduce the following candidates induced by $D$
to quantify $R$-preservability:

$\displaystyle P_{D}(\mathcal{E})\coloneqq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA}),(\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})\right];$ (11)
$\displaystyle\bar{P}_{D}(\mathcal{E})\coloneqq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\sup_{{\rm A};\rho_{\rm
SA}}D\left[(\mathcal{E}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA}),(\Lambda_{\rm S}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})\right].$ (12)

Again, we use the abbreviation introduced in Eq. (9), and $\sup_{{\rm
A};\rho_{\rm SA}}$ means the maximization taken over all the ancillary systems
${\rm A}$ and the states $\rho_{\rm SA}$ on ${\rm SA}$. Note that unlike the
previous section, since now we only compare the distance between two channels,
using identity to extend the system is allowed, and this is the reason why we
list two candidates here. Before introducing the main result, we say a set
$\mathcal{A}$ is closed under the distance measure $D$ if for every sequence
$\\{\Lambda_{k}\\}_{k=1}^{\infty}\subseteq\mathcal{A}$ satisfying
$\lim_{k\to\infty}\sup_{\rho}D[\mathcal{E}(\rho),\Lambda_{k}(\rho)]=0$, we
will have $\mathcal{E}\in\mathcal{A}$. We now provide the following result,
whose proof is given in Appendix D.

######  Theorem 2.

Given an $R$-theory and a distance measure $D$ satisfying the property

$\displaystyle D[\Lambda(\rho),\Lambda(\sigma)]\leq
D(\rho,\sigma)\quad\forall\;\rho,\sigma\;\&\;\forall\;\Lambda\in\mathcal{O}_{R}.$
(13)

Then $P_{D}$ and $\bar{P}_{D}$ are $R$-preservability monotones. Moreover,
they are faithful if $\mathcal{O}_{R}^{N}$ is closed under $D$.

Note that Eq. (13) is a relaxed version of the data-processing inequality. As
a remark, Eq. (13) and condition (R4) imply the ordering
$P_{D}\leq\bar{P}_{D}$.

###  4.3 Resource Preservability Monotone: The Robustness

We will provide a detailed example in this section to illustrate Theorem 2. In
short, with a specific distance measure, a robustness-like monotone can be
obtained. To start with, consider the max-relative entropy defined by [74]:

$\displaystyle D_{\rm
max}(\rho\|\sigma)\coloneqq\log_{2}\inf\\{\lambda\,|\,\rho\leq\lambda\sigma\\},$
(14)

where the minimization is taken over all non-negative integer $\lambda$, and
in this work we always consider logarithm to the base 2. $D_{\rm max}$
fulfills [74] (1) $D_{\rm max}(\rho\|\sigma)\geq 0$ and the equality holds if
and only if $\rho=\sigma$, (2) (data-processing inequality) $D_{\rm
max}[\mathcal{E}(\rho)\|\mathcal{E(\sigma)}]\leq D_{\rm max}(\rho\|\sigma)$
for all channels $\mathcal{E}$ and states $\rho,\sigma$. Hence, it satisfies
Eq. (13). Theorem 2 means $P_{D_{\rm max}}$ and $\bar{P}_{D_{\rm max}}$ are
both $R$-preservability monotone, and they are faithful if
$\mathcal{O}_{R}^{N}$ is closed under $D_{\rm max}$.

It turns out that this fact implies a direct robustness form and the
corresponding operational interpretation based on Ref. [54]. To see this,
define the $R$-preservability log-robustness according to Ref. [54]:

$\displaystyle
L_{R}(\mathcal{E})=-\log_{2}\sup\\{p\in[0,1]\,|\,p\mathcal{E}+(1-p)\mathcal{C}\in\mathcal{O}_{R}^{N}\\},$
(15)

where the optimization is taken over all channels $\mathcal{C}$. This quantity
depicts how robust the $R$-preservability of $\mathcal{E}$ is when it is
interrupted by another channel. From Ref. [54] we learn that $\bar{P}_{D_{\rm
max}}=L_{R}$. This means $\bar{P}_{D_{\rm max}}$ may have the same operational
interpretation with $L_{R}$. To formally illustrate this, we now translate the
Definition 9 in Ref. [54] into the following version for $R$-preservability
(in what follows, the diamond norm is defined by $\left\|\mathcal{E}_{\rm
S}\right\|_{\diamond}\coloneqq\sup_{{\rm A};\rho_{\rm
SA}}\left\|(\mathcal{E}_{\rm S}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA})\right\|_{1}$, where the maximization is taken over all ancillary systems
${\rm A}$ and states $\rho_{\rm SA}$ on the system ${\rm SA}$, and
$\left\|\rho\right\|_{1}\coloneqq{\rm tr}|\rho|={\rm
tr}\sqrt{\rho^{\dagger}\rho}$ is the trace norm):

######  Definition 5.

($R$-Preservability Destruction Cost) For a given channel $\mathcal{E}_{\rm
S}\in\mathcal{O}_{R}$ and $0<\epsilon\leq 1$, we say a channel
$\bar{\Lambda}_{\rm S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}$ together
with an ensemble of reversible unitary free operations
$\\{\mathcal{U}_{i},\mathcal{V}_{i},p_{i}\\}_{i=1}^{k}$ (i.e.
$\mathcal{U}_{i},\mathcal{V}_{i}\in\mathcal{O}_{R}$ and also their inverses
are in $\mathcal{O}_{R}$) form an $\epsilon$-destruction process of
$R$-preservability for $\mathcal{E}_{\rm S}$ if for some $\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}$ we have

$\displaystyle\frac{1}{2}\left\|\sum_{i=1}^{k}p_{i}\mathcal{U}_{i}\circ(\mathcal{E}_{\rm
S}\otimes\bar{\Lambda}_{\rm S^{\prime}})\circ\mathcal{V}_{i}-\Lambda_{\rm
SS^{\prime}}\right\|_{\diamond}\leq\epsilon.$ (16)

The $\epsilon$-destruction cost for $R$-preservability is defined by
$C_{R}^{\epsilon}(\mathcal{E}_{\rm S})\coloneqq\log_{2}\min k$, where the
minimization is taken over all $\epsilon$-destruction process of
$R$-preservability for $\mathcal{E}_{\rm S}$.

Definition 5 is slightly different from the Definition 9 in Ref. [54]. In Ref.
[54], $\mathcal{U}_{i}$ and $\mathcal{V}_{i}$ are asked to be free channels,
which will correspond to $\mathcal{O}_{R}^{N}$ in our current study. While
this will always lead to zero $R$-preservability for the output channel, we
relax this condition in this work. Also, we require the ancillary channel
$\bar{\Lambda}_{\rm S^{\prime}}$ to be an absolutely $R$-annihilating channel.

To state the result, we also consider the smooth version of $P_{D_{\rm max}}$
and $\bar{P}_{D_{\rm max}}$ [54]:

$\displaystyle P_{D_{\rm max}}^{\epsilon}(\mathcal{E})$
$\displaystyle\coloneqq\inf_{\frac{1}{2}\left\|\mathcal{E}^{\prime}-\mathcal{E}\right\|_{\diamond}\leq\epsilon}P_{D_{\rm
max}}(\mathcal{E}^{\prime});$ (17) $\displaystyle\bar{P}_{D_{\rm
max}}^{\epsilon}(\mathcal{E})$
$\displaystyle\coloneqq\inf_{\frac{1}{2}\left\|\mathcal{E}^{\prime}-\mathcal{E}\right\|_{\diamond}\leq\epsilon}\bar{P}_{D_{\rm
max}}(\mathcal{E}^{\prime}).$ (18)

Now we state the following result when the given $R$-theory admits no
activation of $R$-preservability. We note that although we write it as a
theorem, conceptually this result is a corollary of Theorem 10 in Ref. [54].
We give the proof in Appendix E for the completeness of this work.

######  Theorem 3.

Given an $R$-theory satisfying the following three conditions:

  1. (i)

$\mathcal{O}_{R}^{N}=\widetilde{\mathcal{O}}_{R}^{N}\neq\emptyset$.

  2. (ii)

$\mathcal{O}_{R}$ is closed in the diamond norm topology for all possible
input/output dimensions.

  3. (iii)

In a multipartite case, the pair-wise permutation unitaries between two local
systems are in $\mathcal{O}_{R}$.

Then for a given $\mathcal{E}\in\mathcal{O}_{R}$ and for any
$0<\eta\leq\epsilon<1$, we have

$\displaystyle\bar{P}_{D_{\rm
max}}^{\sqrt{\epsilon(2-\epsilon)}}(\mathcal{E})\leq
C^{\epsilon}_{R}(\mathcal{E})\leq\bar{P}_{D_{\rm
max}}^{\epsilon-\eta}(\mathcal{E})+2\log_{2}\frac{1}{\eta}-1.$ (19)

Theorem 3 provides a clear operational meaning of $\bar{P}_{D_{\rm
max}}(\mathcal{E})$: It shows how robust the $R$-preservability of the given
free operation $\mathcal{E}$ is when it is randomized over reversible free
unitary operations together with an ancillary absolutely R-annihilating
channel. This can also be interpreted as the erasure cost of
$R$-preservability. Note that we assume no activation property of
$R$-preservability. When the given $R$-preservability can be activated, the
lower bounds in Theorem 3 can still be proved, while it is so far unclear
whether the upper bound can also be obtained.

##  5 Applications to Thermodynamics

After introducing the general framework, one may ask for specific examples to
illustrate $R$-preservability. Specially, a natural question is whether there
is any application. These issues will be addressed in the following two
sections with the focus on thermodynamics. We remark that detailed studies of
coherence preservability have been reported in Ref. [75] recently.

###  5.1 Thermodynamic Implications of Athermality Preservability

We will give two examples by considering the $R$-theory of athermality with
Gibbs-preserving maps as the free operations. It turns out that the
$R$-preservability monotones of athermality (or simply athermality
preservability) given by Eqs. (11) and (12) with max-relative entropy $D_{\rm
max}$ can be directly related to two recently reported results: For a given
Gibbs-preserving channel $\mathcal{N}$, $P_{D_{\rm max}}(\mathcal{N})$ [Eq.
(11)] is operationally related to the smallest bath size needed to thermalize
all outputs of $\mathcal{N}$ [76], and $\bar{P}_{D_{\rm max}}(\mathcal{N})$
[Eq. (12)] is an upper bound of the single-shot classical capacity of
$\mathcal{N}$ in a classical communication scenario subject to thermodynamic
constraints [59]. These illustrate how $R$-preservability can be related to
existing results and provide new physical messages.

To start with, we define the $R$-theory of athermality. The term “athermality”
means the status that a system is out of thermal equilibrium. With a fixed
system size, the only state without this resource is the unique thermal state.
Formally, consider a given system ${\rm S}$ with dimension $d$. Suppose the
system Hamiltonian is $H_{\rm S}$ and a temperature $T$ is also given. Then
the corresponding thermal state reads

$\displaystyle\gamma=\frac{e^{-\beta H_{\rm S}}}{{\rm tr}(e^{-\beta H_{\rm
S}})},$ (20)

where $\beta=\frac{1}{k_{B}T}$ is the inverse temperature and $k_{B}$ is the
Boltzmann constant. In this $R$-theory, all free states take the form
$\\{\gamma^{\otimes k}\,|\,k\in\mathbb{N}\\}$, where we only consider
dimensions of the form $d^{k}$ with some positive integer $k$. The free
operations will be the Gibbs-preserving maps, which are channels $\mathcal{E}$
satisfying $\mathcal{E}(\gamma^{\otimes k})=\gamma^{\otimes l}$ for some
$k,l\in\mathbb{N}$ (note that $k,l$ are uniquely determined by the
input/output dimensions). Intuitively, these channels are those which cannot
drive thermal equilibrium states out of equilibrium, thereby being unable to
create athermality. Equipped with Gibbs-preserving maps, the corresponding
$R$-theory will satisfy properties (R1), (R2), (R3), and (R4).

####  5.1.1 Athermality Preservability and Bath Size

As the first example, we will demonstrate that athermality preservability of a
Gibbs-preserving channel can be naturally linked to the bath size needed to
thermalize a system. To this end, we use the framework and a thermalization
model introduced by Ref. [76]. We will briefly explain the ingredients
relevant to this work, and we refer the readers to Ref. [76] for further
details.

We begin by specifying the system and the bath in the thermalization scenario.
Consider a system ${\rm S}$ with Hilbert space $\mathcal{H}_{\rm S}$ and a
bath ${\rm B}$ with Hilbert space $\mathcal{H}_{\rm S}^{\otimes(n-1)}$
($n\in\mathbb{N}$). The bath is assumed to possess the temperature $T$ and the
Hamiltonian $H_{\rm
B}=\sum_{i=1}^{n-1}\mathbb{I}_{1}\otimes...\otimes\mathbb{I}_{i-1}\otimes
H_{\rm S}\otimes\mathbb{I}_{i+1}\otimes...\otimes\mathbb{I}_{n-1}$, where
$H_{\rm S}$ is the Hamiltonian of the given system ${\rm S}$. Let $\gamma$ be
the thermal state associated with $T$ and $H_{\rm S}$. Then we assume the bath
is initially in the state $\gamma^{\otimes(n-1)}$. The central question is to
study how large the bath needs to be in order to successfully thermalize the
system ${\rm S}$ in a given state $\rho_{\rm S}$.

To this end, a global channel $\mathcal{E}_{\rm SB}:{\rm SB}\to{\rm SB}$ is
said to $\epsilon$-thermalize the system state $\rho_{\rm S}$ if [76]

$\displaystyle\left\|\mathcal{E}_{\rm SB}\left[\rho_{\rm
S}\otimes\gamma^{\otimes(n-1)}\right]-\gamma^{\otimes
n}\right\|_{1}\leq\epsilon.$ (21)

Here the aim is to study thermalization of a fixed and given input state in
the sense that the channel will globally thermalize the system ${\rm SB}$. To
model the system-bath interaction for thermalization, we consider the
following master equation introduced by Ref. [76]:

$\displaystyle\frac{\partial\rho_{\rm SB}(t)}{\partial
t}=\sum_{k}\lambda_{k}\left[U^{(k)}_{\rm SB}\rho_{\rm
SB}(t)U^{(k),\dagger}_{\rm SB}-\rho_{\rm SB}(t)\right],$ (22)

where $\rho_{\rm SB}(t)$ is the state on the global system ${\rm SB}$ at time
$t$, $U^{(k)}_{\rm SB}$ is an energy-preserving unitary on the global system
satisfying $[U^{(k)}_{\rm SB},H_{\rm S}+H_{\rm B}]=0$, and $\lambda_{k}$ is
the rate for the corresponding unitary channel to happen [one can see this by
checking Eqs. (A2) and (A3) in the Appendix A of Ref. [76], which imply that
each unitary operator $U^{(k)}_{\rm SB}$ occurs according to a Poisson
distribution with mean value $\lambda_{k}t$]. Each unitary $U^{(k)}_{\rm SB}$
models an elastic collision between certain subsystems in ${\rm SB}$. We refer
the reader to Ref. [76] for the detailed framework.

Now, let $\mathcal{C}_{n}$ be the set of all channels acting as ${\rm
SB}\to{\rm SB}$ that can be generated by the model Eq. (22) with a bath of
size $n-1$ and a realization time $t$. Then consider the following quantity
[76]:

$\displaystyle n_{\epsilon}(\rho_{\rm
S})\coloneqq\inf\\{n\in\mathbb{N}\,|\,\exists\,\mathcal{E}_{\rm
SB}\in\mathcal{C}_{n}\;{\rm s.t.\;Eq.~{}\eqref{Eq:Thermalize-Def}\;holds}\\}.$
(23)

This quantity can be interpreted as the smallest bath size needed to
$\epsilon$-thermalize the given state $\rho_{\rm S}$ with the thermalization
model Eq. (22). It turns out that this concept can be generalized to channels.
Define

$\displaystyle\mathcal{B}^{\epsilon}(\mathcal{N})\coloneqq\sup_{\rho}n_{\epsilon}[\mathcal{N}(\rho)]-1,$
(24)

which is the maximization over all the smallest bath sizes among all outputs
of $\mathcal{N}$ (note that here we only consider channel $\mathcal{N}$ with
the output space ${\rm S}$, which is the main system admitting the given
thermal state $\gamma$). Then this can be interpreted as the smallest bath
size needed to $\epsilon$-thermalize all outputs of $\mathcal{N}$ under the
given thermalization model. One can therefore understand this quantity as the
minimal bath size associated with the channel $\mathcal{N}$.

Now we are in the position to provide the following bounds, whose proof is
given in Appendix F. This can be regarded as a generalization of the main
results of Ref. [76] to channels:

######  Theorem 4.

Given a Gibbs-preserving map $\mathcal{N}$ and $0\leq\epsilon<1$, we have

$\displaystyle\mathcal{B}^{\epsilon}(\mathcal{N})\leq\frac{1}{\epsilon^{2}}2^{P_{D_{\rm
max}}(\mathcal{N})}.$ (25)

Moreover, if we further assume that $\gamma$ is full-rank, $\mathcal{N}$ is
coherence-annihilating, and the system Hamiltonian $H_{\rm S}$ satisfies the
energy subspace condition (Definition F.1), then we have

$\displaystyle 2^{P_{D_{\rm
max}}(\mathcal{N})}\leq\mathcal{B}^{\epsilon}(\mathcal{N})+\frac{2\sqrt{\epsilon}}{p_{\rm
min}(\gamma)}+1.$ (26)

where $p_{\rm min}(\gamma)$ is the smallest eigenvalue of $\gamma$.

We remark that by saying “coherence-annihilating” we mean the channel can only
output states diagonal in the given energy eigenbasis (i.e. no coherence can
survive). This requirement plus the energy subspace condition (Definition F.1)
are necessary for the proof of the lower bound derived in Ref. [76]
(specifically, it is required by Lemma 17 in the Appendix C of Ref. [76]). An
open question in this research line is whether one can derive a similar lower
bound without these constraints.

As expected, for a Gibbs-preserving channel $\mathcal{N}$, the quantity
$\mathcal{B}^{\epsilon}(\mathcal{N})$ can be understood as a measure of the
robustness of the channel $\mathcal{N}$ against thermalization. From the upper
bound, we learn that the weaker the channel’s ability to preserve athermality
is, the smaller a heat bath needs to be to thermalize every output of
$\mathcal{N}$. Theorem 4 builds a link between the ability to preserve
athermality and the resource needed to thermalize all the outputs of a given
channel, and it also gives athermality preservability a different
thermodynamic interpretation.

####  5.1.2 Athermality Preservability and Classical Communication

It turns out that the robustness-like monotone $\bar{P}_{D_{\rm max}}$ can be
related to a classical communication scenario subject to certain thermodynamic
constraints, which is the second example in this section. To start with,
consider the communication scenario in which we want to send classical
information (in terms of classical bits, which can be written as an
orthonormal basis $\\{|m\rangle\\}_{m=0}^{M-1}$) via a channel $\mathcal{N}$.
Again, we always assume the channel $\mathcal{N}$ has the output space ${\rm
S}$, which is associated with a thermal state $\gamma$ and hence an $R$-theory
of athermality. Here are two constraints for this communication setup:

  * •

The whole process (including encoding and decoding) cannot create athermality.

In other words, this means the corresponding physical system used to implement
the channel and transmit the classical information can only get closer and
closer to thermal equilibrium.

  * •

The dynamics of $\mathcal{N}$ has a time scale much longer than the
thermalization time scale in the environment.

Theoretically, this motivates us to approximate the dynamics of the
surrounding system ${\rm A}$ by the full thermalization $\Phi_{\gamma_{\rm
A}}:(\cdot)\mapsto\gamma_{\rm A}$, where $\gamma_{\rm A}$ is the thermal state
of the environment (that is, $\gamma_{\rm A}=\gamma^{\otimes k}$ for a
positive integer $k$). Together with the non-signalling constraints discussed
in Ref. [59] (whose framework is briefly introduced in Appendix G), we have
the following theoretical model for this communication scenario:

$\displaystyle\mathcal{E}_{d}\circ(\mathcal{N}\otimes\Phi_{\gamma_{\rm
A}})\circ\mathcal{E}_{e},$ (27)

where $\mathcal{E}_{e},\mathcal{E}_{d},\mathcal{N}$ are all Gibbs-preserving
maps ($\mathcal{E}_{e}$ and $\mathcal{E}_{d}$ can be interpreted as the
encoding and decoding maps, respectively), and in the ancillary system ${\rm
A}$ there is a full thermalization channel $\Phi_{\gamma_{\rm A}}$. We call
this a thermalized classical communication scenario, which is identical to the
communication scenario given in Ref. [59] (see also Appendix G) subject to the
thermodynamic constraints given above. The central goal is to understand how
much classical information can be sent within a given error.

Formally, the classical information is indicated by an orthonormal basis
$\\{|m\rangle\\}_{m=0}^{M-1}$, and we are interested in how many basis
elements can be recovered in the end of the whole process. To this end, we use
the averaged error
$\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm A})$ of a
given combination $(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm
A})$ to evaluate the faithfulness of the output:

$\displaystyle\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm
A})\coloneqq 1-\frac{1}{M}\sum_{m=0}^{M-1}\langle
m|\mathcal{E}_{d}\circ(\mathcal{N}\otimes\Phi_{\gamma_{\rm
A}})\circ\mathcal{E}_{e}(|m\rangle\langle m|)|m\rangle.$ (28)

We can now define the following single-shot classical capacity with an error
$0<\epsilon<1$:

$\displaystyle
C_{\gamma,(1)}^{\epsilon}(\mathcal{N})\coloneqq\sup_{\mathcal{E}_{e},\mathcal{E}_{d},{\rm
A}}\\{\log_{2}M\,|\,\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm
A})\leq\epsilon\\},$ (29)

where the maximization is taken over all the possible Gibbs-preserving
channels $\mathcal{E}_{e},\mathcal{E}_{d}$ and ancillary systems ${\rm A}$.
This quantity tells us the optimal performance of the channel $\mathcal{N}$ in
a thermalized classical communication scenario: It is the highest amount of
classical information that can be transmitted within the given error
$\epsilon$. As a remark, in the realistic situation the size of the
environment cannot be as large as we want. This means in the practical setup
one cannot optimize over all the possible ancillary systems, and the above
quantity is an upper bound of the realistic capacity in general.

It turns out that this quantity is upper bounded by the athermality
preservability monotone $\bar{P}_{D_{\rm max}}$. Hence, when the given channel
$\mathcal{N}$ has a weak ability to maintain athermality, then it can neither
have a good performance in a thermalized classical communication scenario.
Formally, we have the following result, which is conceptually a corollary of
Theorem 3 in Ref. [59]. The proof can be found in Appendix H.

######  Theorem 5.

For a Gibbs-preserving map $\mathcal{N}$ and $0<\epsilon,\delta<1$, we have

$\displaystyle
C^{\epsilon}_{\gamma,(1)}(\mathcal{N})\leq\bar{P}^{\delta}_{D_{\rm
max}}(\mathcal{N})+\log_{2}\frac{1}{1-\epsilon-\delta}.$ (30)

Theorem 5 gives an alternative operational interpretation of Eq. (12) in the
case of athermality: It is an upper bound of the optimal performance in a
thermalized classical communication scenario. Being consistent with the
intuition, this result implies that if the given Gibbs-preserving channel has
highly thermalized output states, then it can hardly keep the encoded
classical messages through a scenario that cannot drive the system away from
thermal equilibrium.

###  5.2 Application to Entanglement Preserving Local Thermalization

As another application, we apply the theory of $R$-preservability to the study
of entanglement preserving local thermalization (EPLT) [63], which is a topic
aiming to understand the interplay between globally distributed quantum
correlation and locally performed thermalizations. The central question of
EPLT is: Can entanglement survive subsystem thermalizations? To formulate the
question, suppose an unknown input state is distributed to two local agents A
and B. We assume that the agents can neither use quantum resources (e.g.
sharing a maximally entangled state), nor can they communicate with each
other. Both of them possess an individual local heat bath (and hence a given
local thermal state $\gamma_{\rm X}$; ${\rm X=A,B}$), and we allow classical
correlation between the local heat baths. When both local agents let their
local systems interact with the local heat baths and thermalize, the question
is whether we can have global entanglement after thermalizations are achieved
locally, at least for certain input states.

The above question can be formulated information-theoretically as follows.
Formally, a local operations plus shared randomness (LOSR; see Appendix A.1
for the definition) channel $\mathcal{E}$ is called a local thermalization to
a pair of single party thermal states $(\gamma_{\rm A},\gamma_{\rm B})$ if

$\displaystyle{\rm tr}_{\rm A}\circ\mathcal{E}(\cdot)=\gamma_{\rm B};\;{\rm
tr}_{\rm B}\circ\mathcal{E}(\cdot)=\gamma_{\rm A}.$ (31)

In other words, it is a full thermalization channel to $\gamma_{\rm X}$ in the
local system ${\rm X}$ [which is different from the state-dependent definition
given by Eq. (21)]. An EPLT is defined to be a local thermalization that can
preserve entanglement for certain inputs; that is, it is a local
thermalization with non-zero entanglement preservability. A physical message
from the existence of EPLT is when local agents couple to a global heat bath
that only admits classical correlations within, it is still possible for
global entanglement to survive after subsystem thermalizations: Classical
correlations in the bath are enough to protect entanglement from being
destroyed by locally performed thermalizations.

Recently, the existence of EPLT has been proved for all nonzero local
temperatures and finite-energy local Hamiltonians in a bipartite setup with
the help of shared randomness555The shared randomness is necessary for the
existence of EPLT. This is because the full thermalization channel
$\Phi_{\gamma}:(\cdot)\mapsto\gamma$ is an entanglement-breaking channel, and
every product local thermalization to $(\gamma_{\rm A},\gamma_{\rm B})$ will
take be form $(\cdot)\mapsto\gamma_{\rm A}\otimes\gamma_{\rm B}$ [63]. We also
remark that the EPLT constructed in Ref. [63] can be interpreted as a
combination of twirling operation plus a generalized depolarizing channel in
finite temperatures. See the beginning of Appendix K for the detail. [63].
This means there is no temperature and energy thresholds for the existence of
EPLT; that is, EPLT has no “thermodynamic threshold” to exist. From this a
natural question is to ask whether there is any “correlation threshold”; in
other words, is it true that EPLT can exist only when its ability to preserve
entanglement is strong enough? With the formulation of $R$-preservability in
hand, we can now answer this question. As proved in Appendix I, the result we
found suggests that EPLT is a phenomenon generic for different values of the
entanglement preservability [note that the $R$-theory of entanglement with
LOSR channels as free operations will satisfy properties (R1), (R2), (R3), and
(R4)].

######  Theorem 6.

For every full-rank $\gamma_{\rm A},\gamma_{\rm B}$ and every $\delta>0$,
there exists an entanglement preserving local thermalization $\mathcal{E}$ to
$(\gamma_{\rm A},\gamma_{\rm B})$ such that

$\displaystyle\bar{P}_{\left\|\cdot\right\|_{1}}(\mathcal{E})<\delta.$ (32)

Hence, the existence of EPLT is generic both in thermodynamic measures
(temperature and energy) and correlation measure (entanglement
preservability). Moreover, this physical message can actually be generalized
to the preservability of free entanglement [65]. Before stating the main
result, we specify terminologies. In what follows, the normalized temperature
of the given local system ${\rm X}$ is defined by $\tau_{\rm
X}\coloneqq\frac{k_{B}T_{\rm X}}{\left\|H_{\rm X}\right\|_{\infty}}$, where
$T_{\rm X}$ is the local temperature, $k_{B}$ is the Boltzmann constant, and
$\left\|H_{\rm X}\right\|_{\infty}$ is the highest local energy (here the sup
norm is defined by
$\left\|\rho\right\|_{\infty}\coloneqq\sup_{|\psi\rangle}|\langle\psi|\rho|\psi\rangle|$).
$\mathcal{O}_{\rm FE}^{N}$ is the set of all LOSR channels that cannot
preserve free entanglement [65]. Using a new family of EPLT constructed in
Appendix J [Eq. (105)], we prove the following result in Appendix K ($d$ is
the common local dimensions of both subsystems):

######  Theorem 7.

For every pair $(\gamma_{\rm A},\gamma_{\rm B})$ there exists a local
thermalization $\mathcal{E}_{+}$ such that

$\displaystyle\inf_{\Lambda\in\mathcal{O}_{\rm
FE}^{N}}\left\|\mathcal{E}_{+}-\Lambda\right\|_{\diamond}\geq(3d-1)p_{\rm
min}-2,$ (33)

where $p_{\rm min}$ is the smallest eigenvalue among $\gamma_{\rm A}$ and
$\gamma_{\rm B}$.

For every $\delta>0$, there exists a finite value $\tau_{\delta}>0$ such that
for every pair $(\gamma_{\rm A},\gamma_{\rm B})$ with $\min_{\rm X}\tau_{\rm
X}>\tau_{\delta}$, there exists an entanglement preserving local
thermalization $\mathcal{E}_{-}$ to $(\gamma_{\rm A},\gamma_{\rm B})$ such
that

$\displaystyle\bar{P}_{\left\|\cdot\right\|_{1}}(\mathcal{E}_{-})<\delta\quad\&\quad\mathcal{E}_{-}\notin\mathcal{O}_{\rm
FE}^{N}.$ (34)

That is, $\mathcal{E}_{-}$ can preserve free entanglement.

Hence, for arbitrarily small entanglement preservability, there always exists
a finite temperature EPLT that can also preserve free entanglement. In other
words, while they preserve arbitrarily little entanglement, many copies of
some output can be distilled back to a maximally entangled state by LOCC
channels. Hence, the conclusion that EPLT exists without a correlation
threshold is the same even when we use the preservability of free entanglement
as the measure.

We also remark that since

$\displaystyle\bar{P}_{\left\|\cdot\right\|_{1}}(\mathcal{E}_{+})\coloneqq\inf_{\Lambda\in\mathcal{O}_{\rm
E}^{N}}\left\|\mathcal{E}_{+}-\Lambda\right\|_{\diamond}\geq\inf_{\Lambda\in\mathcal{O}_{\rm
FE}^{N}}\left\|\mathcal{E}_{+}-\Lambda\right\|_{\diamond},$ (35)

Eq. (33) automatically implies a lower bound of the entanglement
preservability. For high normalized temperatures, we have $p_{\rm
min}\rightarrow\frac{1}{d}$ and the bound in Eq. (33) becomes arbitrarily
close to $1-\frac{1}{d}$, as expected since $\inf_{\Lambda\in\mathcal{O}_{\rm
E}^{N}}\left\|\mathcal{T}-\Lambda\right\|_{\diamond}\geq 1-\frac{1}{d}$ [see
Eq. (K.1)] and $\mathcal{T}$ is an EPLT at infinite normalized temperature
[63].

##  6 Conclusions

In a given resource theory of quantum states, we quantify the ability of free
operations to preserve the resource. To this end, we formulate this ability,
termed resource preservability, as a channel resource induced by the given
state resource. Two classes of resource preservability monotones are proved:
One is induced by state resource quantifiers, and another is based on channel
distance measures. The latter also induces a robustness-like measure with
operational interpretation as the erasure cost of resource preservability
[54].

To illustrate the connection between resource preservability and other
research directions, we consider the resource theory of athermality. We
provide physical interpretations of two ahtermality preservability monotones
induced by max-relative entropy. One has a thermodynamic interpretation
directly related to the bath size needed for thermalization: The ability of a
Gibbs-preserving channel to preserve athermality is physically connected to
the minimal bath size needed to thermalize all its outputs. Another monotone
is shown to bound the capacity of a classical communication scenario under
certain thermodynamic constraints. By adding thermodynamic conditions to a
general classical communication setup, the ability for the given Gibbs-
preserving channel to preserve athermality tells us the highest possible
amount of transmissible classical messages.

As another application, we study the entanglement preservability of
entanglement preserving local thermalizations (EPLTs) [63], which is a family
of local operation plus shared randomness channels that locally behave as
thermalization for arbitrary inputs, while globally have the ability to
preserve certain amounts of entanglement. In this work, we show that EPLT can
exist with arbitrarily small entanglement preservability for every positive
temperatures. Hence, EPLT’s existence is independent of both temperature
constraints and the ability to preserve entanglement. We further provide a new
family of EPLTs that has the ability to preserve free entanglement, even
though its entanglement preservability can be arbitrarily small at finite
temperatures. This suggests the existence of EPLT is generic in various values
of free entanglement preservability.

Several open questions remain. From the operational perspective, it will be
interesting to know whether there is any operational interpretation of
$R$-preservability monotones induced by state resource monotones introduced in
Sec. 4.1. Also, the robustness-like measure introduced in Sec. 4.3 is shown to
have an operational interpretation [54] when the given $R$-preservability
theory has no activation property, while it is unknown whether this
operational interpretation can still hold when the given $R$-preservability
allows activation. Regarding the structure of channel resource theory, it is
so far unknown how to characterize the largest set of free super-channels of
$R$-preservability (since it may not always be the set of all super-channels
that cannot generate $R$-preservability as discussed in footnote 3). Finally,
it is also an open question whether one can drop the temperature dependency of
entanglement preserving local thermalizations in Theorem 7. We hope this work
can initiate the interest in the study of resource preservation properties in
different state resource theories.

Note added. Recently, we became aware of the related work Ref. [75] which
consider the preservation of coherence as a channel resource.

## Acknowledgements

We thank (in alphabetical order) Antonio Acín, Stefan B$\ddot{\rm a}$uml,
Daniel Cavalcanti, Marcus Huber, Yeong-Cherng Liang, Matteo Lostaglio, Ryuji
Takagi, Marco Túlio Quintino, Gabriel Senno, Andreas Winter for fruitful
discussions and comments. This project is part of the ICFOstepstone - PhD
Programme for Early-Stage Researchers in Photonics, funded by the Marie
Skłodowska-Curie Co-funding of regional, national and international programmes
(GA665884) of the European Commission, as well as by the ‘Severo Ochoa
2016-2019’ program at ICFO (SEV-2015-0522), funded by the Spanish Ministry of
Economy, Industry, and Competitiveness (MINECO). We also acknowledge support
from the Spanish MINECO (Severo Ochoa SEV-2015-0522), Fundació Cellex and Mir-
Puig, Generalitat de Catalunya (SGR1381 and CERCA Programme).

##  Appendix A Remark on the Activation Property of Resource Preservability

In this section, we provide an example of activation property of
$R$-preservability. Consider the $R$-theory of nonlocality [4, 5] (and we
write $R={\rm NL}$) on a bipartite system ${\rm SS^{\prime}}$ with equal
finite local dimension $D$, and local operations plus shared randomness (LOSR)
channels as the free operations [49, 17] (in Appendix A.1 we briefly explain
the reason). First, we recall a phenomenon called superactivation, which is
proved for nonlocality [68] and generalized to quantum steering [69, 70] (and
we also mention other activation properties of nonlocality in Refs. [71, 72]).
Formally, a local state $\rho$ (with local dimension $D=d$) is said to admit
superactivation of nonlocality if there exists a finite $k\in\mathbb{N}$ such
that $\rho^{\otimes k}$ is nonlocal (in the bipartition ${\rm SS^{\prime}}$
and local dimension $D=d^{k}$). We refer the readers to Appendix A.1 for the
definition of local/nonlocal states. In ${\rm SS^{\prime}}$ with $D=d$, it is
shown that a state can demonstrate superactivation of nonlocality if its fully
entangled fraction (FEF) is higher than $\frac{1}{d}$ [77], where for the
given bipartite system the FEF is defined by [11, 78]:

$\displaystyle\mathcal{F}(\rho_{\rm
SS^{\prime}})\coloneqq\sup_{|\Phi_{d}\rangle}\langle\Phi_{d}|\rho_{\rm
SS^{\prime}}|\Phi_{d}\rangle.$ (36)

The maximization is taken over all maximally entangled states
$|\Phi_{d}\rangle$ on the given bipartite system ${\rm SS^{\prime}}$. FEF is
well-known for its capacity to characterize various quantum properties [11,
78, 79, 77, 5, 70, 69, 80, 1, 81].

To construct the example, we make use of the $(U\otimes U^{*})$-twirling
operation on ${\rm SS^{\prime}}$ defined by [82, 83]

$\displaystyle\mathcal{T}(\cdot)\coloneqq\int_{U(d)}(U\otimes
U^{*})(\cdot)(U\otimes U^{*})^{\dagger}dU,$ (37)

where the integration is taken over the group of $d\times d$ unitary operators
$U(d)$ with the Haar measure $dU$. The twirling operation $\mathcal{T}$ is by
definition an LOSR channel, thereby being a free operation. It has the
property to preserve entanglement:

$\displaystyle\langle\Psi_{d}^{+}|\mathcal{T}(\rho_{\rm
SS^{\prime}})|\Psi_{d}^{+}\rangle=\langle\Psi_{d}^{+}|\rho_{\rm
SS^{\prime}}|\Psi_{d}^{+}\rangle.$ (38)

Also, the output of $\mathcal{T}$ will always be an isotropic state [82]:

$\displaystyle\rho_{\rm iso}(p)\coloneqq
p|\Psi_{d}^{+}\rangle\langle\Psi_{d}^{+}|+(1-p)\frac{\mathbb{I}_{\rm
SS^{\prime}}}{d^{2}},$ (39)

where
$|\Psi_{d}^{+}\rangle\coloneqq\frac{1}{\sqrt{d}}\sum_{n=0}^{d-1}|n\rangle\otimes|n\rangle$
is a maximally entangled state, and $p\in\left[-\frac{1}{d^{2}-1},1\right]$
due to the positivity of quantum states. Now we consider the following
channel:

$\displaystyle\widetilde{\mathcal{T}}(\cdot)\coloneqq\widetilde{p}\mathcal{T}(\cdot)+(1-\widetilde{p})\frac{\mathbb{I}_{\rm
SS^{\prime}}}{d^{2}},$ (40)

and we choose $\widetilde{p}$ such that the output state cannot have FEF
larger than the threshold for nonlocality of isotropic states [5], while can
still have FEF larger than $\frac{1}{d}$ for certain entangled inputs. More
precisely, we choose [5, 84]

$\displaystyle\frac{1}{d+1}<\widetilde{p}<\frac{(d-1)^{(d-1)}(3d-1)}{(d+1)d^{d}},$
(41)

which will guarantee the above claim. Being an LOSR channel, this means
$\widetilde{\mathcal{T}}\in\mathcal{O}_{\rm NL}^{N}$. Also, when the input
state is $|\Psi_{d}^{+}\rangle$,
$\widetilde{\mathcal{T}}(|\Psi_{d}^{+}\rangle\langle\Psi_{d}^{+}|)$ will be an
entangled isotropic state, thereby having FEF $>\frac{1}{d}$ and hence
admitting superactivation of nonlocality. Hence, when one consider
$\widetilde{\mathcal{T}}^{\otimes k}$ with a large enough $k$, it is possible
to output nonlocal states (on the given bipartition ${\rm SS^{\prime}}$ with
local dimension $D=d^{k}$), which means $\widetilde{\mathcal{T}}^{\otimes
k}\notin\mathcal{O}_{\rm NL}^{N}$. This illustrates the existence of
superactivation property of nonlocality preservability, which also teaches us
that for a general formulation, the assumption $\Lambda_{\rm
S}\otimes\Lambda_{\rm S^{\prime}}\in\mathcal{O}_{R}^{N}$ if $\Lambda_{\rm
S},\Lambda_{\rm S^{\prime}}\in\mathcal{O}_{R}^{N}$ cannot be imposed.

As a remark, we note that there do exist examples without activation property.
For instance, if we use Gibbs-preserving map as the free operation in the
$R$-theory of athermality, then the only $R$-annihilating channel is the state
preparation channel of the given thermal state [Eq. (20)]. Because product
local thermalization cannot preserve any correlation [63], we learn that it is
impossible to activate resource preservability in this case.

###  A.1 Local Operations Plus Shared Randomness Channels

In this section, we briefly explain why LOSR channels can be free operations
of nonlocality. It suffices to consider a bipartite system ${\rm AB}$.
Formally, an LOSR channel is defined to take the following form:

$\displaystyle\mathcal{E}=\int(\mathcal{E}_{\lambda}^{\rm
A}\otimes\mathcal{E}_{\lambda}^{\rm B})p_{\lambda}d\lambda,$ (42)

where the integration is taken over the variable $\lambda$ and
$\mathcal{E}_{\lambda}^{\rm A},\mathcal{E}_{\lambda}^{\rm B}$ are local
channels. In what follows we will write $\\{E_{a|x}\\}$ as a set of local
positive operator-valued measures (POVMs) [13]; that is, for each input value
$x$, $E_{a|x}$’s form an POVM: $E_{a|x}\geq 0\;\forall\,a$ and
$\sum_{a}E_{a|x}=\mathbb{I}_{\rm A}\;\forall\,x$. We use the notation
$\\{E_{b|y}\\}$ for the POVMs in the subsystem ${\rm B}$.

With the above setting, a quantum state $\rho_{\rm AB}$ is said to be local if
for every local sets of POVMs $\\{E_{a|x}\\},\\{E_{b|y}\\}$ one can write [4,
5]

$\displaystyle{\rm tr}\left[(E_{a|x}\otimes E_{b|y})\rho_{\rm
AB}\right]=\int_{\lambda\in\Lambda_{\rm
LHV}}P(a|x,\lambda)P(b|y,\lambda)p_{\lambda}d\lambda$ (43)

for some variable $\lambda$ in a set $\Lambda_{\rm LHV}$ and some probability
distributions $P(a|x,\lambda),P(b|y,\lambda),p_{\lambda}$. In other words, a
state is local if all the possible combinations of local POVMs cannot
distinguish it with a local hidden-variable model, as depicted by
$\Lambda_{\rm LHV}$. Any state that is not local is said to be nonlocal.

Now we explain that LOSR channel will map local states to local states. To see
this, we note that for a given LOSR channel $\mathcal{E}$, we have

$\displaystyle{\rm tr}\left[(E_{a|x}\otimes E_{b|y})\mathcal{E}(\rho_{\rm
AB})\right]$ $\displaystyle=\int{\rm tr}\left[(E_{a|x}\otimes
E_{b|y})(\mathcal{E}_{\lambda}^{\rm A}\otimes\mathcal{E}_{\lambda}^{\rm
A})(\rho_{\rm AB})\right]p_{\lambda}d\lambda$ $\displaystyle=\int{\rm
tr}\left\\{\left[\mathcal{E}_{\lambda}^{{\rm
A},\dagger}(E_{a|x})\otimes\mathcal{E}_{\lambda}^{{\rm
B},\dagger}(E_{b|y})\right](\rho_{\rm AB})\right\\}p_{\lambda}d\lambda,$ (44)

where for ${\rm X=A,B}$, $\mathcal{E}_{\lambda}^{{\rm X},\dagger}$’s are
completely-positive unital map since $\mathcal{E}_{\lambda}^{\rm X}$’s are
completely-positive trace-preserving map. This means
$\mathcal{E}_{\lambda}^{{\rm A},\dagger}(E_{a|x})$ and
$\mathcal{E}_{\lambda}^{{\rm B},\dagger}(E_{b|y})$ again form local sets of
POVMs. Since $\rho_{\rm AB}$ is local, the quantity ${\rm
tr}\left\\{\left[\mathcal{E}_{\lambda}^{{\rm
A},\dagger}(E_{a|x})\otimes\mathcal{E}_{\lambda}^{{\rm
B},\dagger}(E_{b|y})\right](\rho_{\rm AB})\right\\}$ must take the form of Eq.
(43). This shows that LOSR channels map local states to local states, and
hence form a suitable candidate of free operations for nonlocality.

##  Appendix B Example of Absolutely Resource Annihilating Channels

Using the $R$-theory of entanglement, we will show that every channel that is
entanglement-annihilating and entanglement-breaking will be an absolutely
$R$-annihilating channel (and we also say it is absolutely entanglement-
annihilating).

######  Fact B.1.

If a bipartite channel $\mathcal{E}$ is entanglement-annihilating and
entanglement-breaking, then it is absolutely entanglement-annihilating.

###### Proof.

We rewrite this channel as $\mathcal{E}_{\rm A_{1}B_{1}}$, where the subscript
means that it is in the bipartite system ${\rm A_{1}B_{1}}$. Then it suffices
to show that there will be no entanglement in the ${\rm AB}$ bipartition after
the product channel $\mathcal{E}_{\rm A_{1}B_{1}}\otimes\Lambda_{\rm
A_{2}B_{2}}$, where $\Lambda_{\rm A_{2}B_{2}}$ is an entanglement-annihilating
channel in the bipartite system ${\rm A_{2}B_{2}}$ and it annihilates
entanglement in the ${\rm AB}$ bipartition.

Because $\mathcal{E}_{\rm A_{1}B_{1}}$ is entanglement-breaking, this means
$\mathcal{E}_{\rm A_{1}B_{1}}\otimes\mathcal{I}_{\rm A_{2}B_{2}}$ is
entanglement-annihilating in the $12$ bipartition. In other words, there will
be no entanglement in the $12$ bipartition after $\mathcal{E}_{\rm
A_{1}B_{1}}\otimes\Lambda_{\rm A_{2}B_{2}}$. Hence, the remaining possibility
for the preserved entanglement are in the bipartite systems ${\rm A_{1}B_{1}}$
and ${\rm A_{2}B_{2}}$. But since both $\mathcal{E}_{\rm A_{1}B_{1}}$ and
$\Lambda_{\rm A_{2}B_{2}}$ are entanglement-annihilating in the ${\rm AB}$
bipartition, we conclude that no entanglement exists in the bipartite systems
${\rm A_{1}B_{1}}$ and ${\rm A_{2}B_{2}}$. This shows that the output states
cannot be entangled in the ${\rm AB}$ bipartition. ∎

##  Appendix C Proof of Theorem 1

###### Proof.

To show property (M1), note that for a given $\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}$ we have $\Lambda_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A}\in\mathcal{O}_{R}^{N}$ for all
$\widetilde{\Lambda}_{\rm A}\in\widetilde{\mathcal{O}}_{R}^{N}$. This means
$(f\circ Q_{R})[(\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})]=0$ for all $\widetilde{\Lambda}_{\rm
A}\in\widetilde{\mathcal{O}}_{R}^{N}$ and for all $\rho_{\rm SA}$. Hence,
property (M1) is proved.

To show property (M2), we recall from Definition 3 that for a given free
super-channel $F_{\mathcal{E}}$ acting on free operations
$\mathcal{E}\in\mathcal{O}_{R}$, there exist an ancillary system ${\rm B}$,
two free operations $\Lambda_{+},\Lambda_{-}\in\mathcal{O}_{R}$, and an
absolutely $R$-annihilating channel $\widetilde{\Lambda}_{\rm
B}\in\widetilde{\mathcal{O}}_{R}^{N}$ such that
$F_{\mathcal{E}}=\Lambda_{+}\circ(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
B})\circ\Lambda_{-}$. In what follows, because the input/output dimensions of
$\Lambda_{-}$ do not need to be the same, we write ${\rm S^{\prime}}$ as the
input space and ${\rm SB}$ as the output space of $\Lambda_{-}$; namely, we
have $\Lambda_{-}:{\rm S^{\prime}}\to{\rm SB}$. Then we have [note that the
maximization is taken over $\rho_{\rm SA}$ satisfying $Q_{R}(\rho_{\rm SA})>0$
according to our definition; see explanations below Eq. (9)]

$\displaystyle P_{Q_{R}}^{(f,g)}(F_{\mathcal{E}})$
$\displaystyle=\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left\\{[(\Lambda_{+}\otimes\mathcal{I}_{\rm
A})\circ(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
B}\otimes\widetilde{\Lambda}_{\rm A})\circ(\Lambda_{-}\otimes\mathcal{I}_{\rm
A})](\rho_{\rm S^{\prime}A})\right\\}}{(g\circ Q_{R})(\rho_{\rm
S^{\prime}A})}$ $\displaystyle\leq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left\\{[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
B}\otimes\widetilde{\Lambda}_{\rm A})\circ(\Lambda_{-}\otimes\mathcal{I}_{\rm
A})](\rho_{\rm S^{\prime}A})\right\\}}{(g\circ Q_{R})(\rho_{\rm
S^{\prime}A})}$ $\displaystyle\leq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left\\{[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
B}\otimes\widetilde{\Lambda}_{\rm A})\circ(\Lambda_{-}\otimes\mathcal{I}_{\rm
A})](\rho_{\rm S^{\prime}A})\right\\}}{(g\circ
Q_{R})[(\Lambda_{-}\otimes\mathcal{I}_{\rm A})(\rho_{\rm S^{\prime}A})]}$
$\displaystyle\leq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
B}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SBA})\right]}{(g\circ
Q_{R})(\rho_{\rm SBA})}$ $\displaystyle\leq\overline{\sup_{\rm
A}}\;\frac{(f\circ Q_{R})\left[(\mathcal{E}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SA})\right]}{(g\circ Q_{R})(\rho_{\rm SA})}$
$\displaystyle=P_{Q_{R}}^{(f,g)}(\mathcal{E}).$ (45)

The second line is because $Q_{R}$ is non-increasing under free operation
$(\Lambda_{+}\otimes\mathcal{I}_{\rm A})$, which is due to the properties
(R2), (R4), (M2), and the fact that $f$ is strictly increasing. The same
reasons imply the third line (while with some subtleties explained below). The
fourth line is because maximizing over all states of the form
$(\Lambda_{-}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})$ is sub-optimal than
the range of all states on the system ${\rm SBA}$. The fifth line is because
$\widetilde{\Lambda}_{\rm B}\otimes\widetilde{\Lambda}_{\rm A}$ gives a range
that is sub-optimal than all the possible $\widetilde{\Lambda}_{\rm A}$ when
one maximizes over all the ancillary systems ${\rm A}$ [recall from Eq. (2)
that the set of absolutely $R$-annihilating channels for an $R$-theory
satisfying properties (R1), (R2), (R3), and (R4) is closed under tensor
product].

Here we note that the ranges of optimization in the second line and the third
line are different. In the second line, the optimization is taken over
$\rho_{\rm S^{\prime}A}$ with $Q_{R}(\rho_{\rm S^{\prime}A})>0$, which implies
two different cases. The first case is when the optimization over this range
is zero [$\overline{\sup}_{\rm A}(...)=0$ in the second line]. Then in this
case the desired inequality holds. This means we can assume the second case
without loss of generality; that is, we can assume the optimization in the
second line over $Q_{R}(\rho_{\rm S^{\prime}A})>0$ gives nonzero value. Hence,
the range for the second line can be rewritten as $\rho_{\rm S^{\prime}A}$
with $Q_{R}(\rho_{\rm S^{\prime}A})>0$ and
$Q_{R}[(\Lambda_{-}\otimes\mathcal{I}_{\rm A})(\rho_{\rm S^{\prime}A})]>0$,
since the latter inequality is necessary for a nonzero numerator (note that
actually the latter inequality implies the former one, while we still write
them both explicitly for understanding). Then one can proceed to the third
line with this condition. This proves property (M2).

To prove the property given by Eq. (7), we first note the following: (the
maximization is again taken over states with non-zero $Q_{R}$ values)

$\displaystyle P_{Q_{R}}^{(f,g)}(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}})$ $\displaystyle=\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SS^{\prime}A})\right]}{(g\circ Q_{R})(\rho_{\rm SS^{\prime}A})}$
$\displaystyle\geq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})\right]}{(g\circ Q_{R})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})}$
$\displaystyle\geq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})\right]}{(g\circ Q_{R})(\rho_{\rm SA}\otimes\widetilde{\eta}_{\rm
S^{\prime}})}$ $\displaystyle\geq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})\right]}{(g\circ Q_{R})(\rho_{\rm SA})}$
$\displaystyle=P_{Q_{R}}^{(f,g)}(\mathcal{E}_{\rm S}).$ (46)

Note that $\overline{\sup}_{\rm A}$ in the first line is maximizing over the
system ${\rm SS^{\prime}A}$. The second line is because fixing an absolutely
free state $\widetilde{\eta}_{\rm S^{\prime}}\in\widetilde{\mathcal{F}}_{R}$
[here we use the assumption $\widetilde{\mathcal{F}}_{R}\neq\emptyset$ in
property (R1)] will make the maximization sub-optimal than the original one,
and we note that since this line $\overline{\sup}_{\rm A}$ is maximizing over
${\rm SA}$ [with $Q_{R}(\rho_{\rm SA})>0$]. The third line is because $f$ is
strictly increasing and $Q_{R}$ is a resource monotone [property (R2)]. The
fourth line is because $g$ is non-decreasing and $Q_{R}$ is a resource
monotone [property (R3)]. This proves the inequality in Eq. (7) for general
$\mathcal{E}_{\rm S}$ and $\mathcal{E}_{\rm S^{\prime}}$.

In the case that $\mathcal{E}_{\rm S^{\prime}}=\widetilde{\Lambda}_{\rm
S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}$, we have

$\displaystyle P_{Q_{R}}^{(f,g)}(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm S^{\prime}})$
$\displaystyle=\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SS^{\prime}A})\right]}{(g\circ Q_{R})(\rho_{\rm SS^{\prime}A})}$
$\displaystyle\leq\overline{\sup_{\rm A}}\;\frac{(f\circ
Q_{R})\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})\right]}{(g\circ Q_{R})(\rho_{\rm SA})}$
$\displaystyle=P_{Q_{R}}^{(f,g)}(\mathcal{E}_{\rm S}),$ (47)

where the second line is because the range $\widetilde{\Lambda}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A}$ with the fixed
$\widetilde{\Lambda}_{\rm S^{\prime}}$ is sub-optimal than all the possible
absolutely $R$-annihilating channel $\widetilde{\Lambda}_{\rm A}$ when one
maximizes over all the ancillary systems ${\rm A}$ [recall again from Eq. (2)
that the set of absolutely $R$-annihilating channels will be closed under
tensor product in the current case]. This shows the equality in Eq. (7).

Finally, when $f\circ Q_{R}$ is convex, $P_{Q_{R}}^{(f,g)}$ is by definition
convex. This proves property (M3). To address property (M4), for a given
$\mathcal{E}_{\rm S}\in\mathcal{O}_{R}$ we note that
$P_{Q_{R}}^{(f,g)}(\mathcal{E}_{\rm S})=0$ implies
$Q_{R}\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})\right]=0$ for all $\rho_{\rm SA}$, all $\widetilde{\Lambda}_{\rm
A}\in\widetilde{\mathcal{O}}_{R}^{N}$, and all ancillary systems ${\rm A}$. By
considering the ancillary system as the trivial one (i.e. with zero
dimension), we have $Q_{R}\left[\mathcal{E}_{\rm S}(\rho_{\rm S})\right]=0$
for all $\rho_{\rm S}$. when $Q_{R}$ is faithful, this means $\mathcal{E}_{\rm
S}(\rho_{\rm S})\in\mathcal{F}_{R}$ for all $\rho_{\rm S}$, thereby implying
$\mathcal{E}_{\rm S}\in\mathcal{O}_{R}^{N}$. This shows property (M4) and also
completes the whole proof. ∎

We remark that the assumption $\widetilde{\mathcal{F}}_{R}\neq\emptyset$ is
only used in the proof of Eq. (C). In other words, this assumption can be
dropped if $g$ maps every input to a positive constant. Write
$g_{c}(\cdot)=c$, this means the following corollary for an $R$-theory
satisfying the rest of properties (R1), (R2), (R3), and (R4):

######  Corollary C.1.

Given an $R$-theory and a state resource monotone $Q_{R}$. $f$ is a finite-
valued strictly increasing function with $f(0)=0$ and $c>0$ is a positive
constant. Then $P_{Q_{R}}^{(f,g_{c})}$is an $R$-preservability monotone.
Moreover, It is faithful if $Q_{R}$ is faithful, and it is convex if $f\circ
Q_{R}$ is convex.

##  Appendix D Proof of Theorem 2

###### Proof.

Property (M1) holds automatically according to the definition. To prove
property (M2), for a given free super-channel $F_{\mathcal{E}_{\rm
S}}=\Lambda_{+}\circ(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
B})\circ\Lambda_{-}$ with $\Lambda_{+},\Lambda_{-}\in\mathcal{O}_{R}$ and
$\widetilde{\Lambda}_{\rm B}\in\widetilde{\mathcal{O}}_{R}^{N}$, the direct
computation shows (we again adapt the notation $\Lambda_{-}:{\rm S^{\prime}\to
SB}$)

$\displaystyle P_{D}(F_{\mathcal{E}_{\rm S}})$
$\displaystyle=\inf_{\Lambda_{\rm
S^{\prime}}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left[(F_{\mathcal{E}_{\rm S}}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm S^{\prime}A}),(\Lambda_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm S^{\prime}A})\right]$
$\displaystyle\leq\inf_{\Lambda_{\rm
SB}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left\\{(F_{\mathcal{E}_{\rm S}}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm S^{\prime}A}),\left[(\Lambda_{+}\circ\Lambda_{\rm
SB}\circ\Lambda_{-})\otimes\widetilde{\Lambda}_{\rm A}\right](\rho_{\rm
S^{\prime}A})\right\\}$ $\displaystyle\leq\inf_{\Lambda_{\rm
SB}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left\\{\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
B}\otimes\widetilde{\Lambda}_{\rm A})\circ(\Lambda_{-}\otimes\mathcal{I}_{\rm
A})\right](\rho_{\rm S^{\prime}A}),\left[(\Lambda_{\rm
SB}\otimes\widetilde{\Lambda}_{\rm A})\circ(\Lambda_{-}\otimes\mathcal{I}_{\rm
A})\right](\rho_{\rm S^{\prime}A})\right\\}$
$\displaystyle\leq\inf_{\Lambda_{\rm
SB}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}\;D\left[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm B}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SBA}),(\Lambda_{\rm SB}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SBA})\right]$ $\displaystyle\leq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}\;D\left[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm B}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SBA}),(\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm
B}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SBA})\right]$
$\displaystyle\leq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}\;D\left[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA}),(\Lambda_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})\right]$
$\displaystyle=P_{D}(\mathcal{E}_{\rm S}).$ (48)

The second line is because $\Lambda_{+}\circ\Lambda_{\rm
SB}\circ\Lambda_{-}\in\mathcal{O}_{R}^{N}$ [which is true because of the
assumptions that we made for $R$-theories in this work] forms a sub-optimal
range compared with $\Lambda_{\rm S^{\prime}}\in\mathcal{O}_{R}^{N}$. The
third line is because of the properties (R2) and (R4), plus the fact that $D$
satisfies Eq. (13). The fourth line is because
$(\Lambda_{-}\otimes\mathcal{I}_{\rm A})(\rho_{\rm S^{\prime}A})$ forms a sub-
optimal range for the maximization $\overline{\sup}_{\rm A}$. The fifth line
is because $\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm
B}\in\mathcal{O}_{R}^{N}$ (this is true due to the definition of the
absolutely $R$-annihilating channels) with the fixed map
$\widetilde{\Lambda}_{\rm B}\in\widetilde{\mathcal{O}}_{R}^{N}$ and the
variable $\Lambda_{\rm S}$ forms a sub-optimal range for the minimization
$\inf_{\Lambda_{\rm SB}\in\mathcal{O}_{R}^{N}}$. The sixth line is because
$\widetilde{\Lambda}_{\rm B}\otimes\widetilde{\Lambda}_{\rm A}$ forms a sub-
optimal range for the maximization $\overline{\sup}_{\rm A}$ [recall Eq. (2)].
This proves property (M2).

To prove Eq. (7), we first compute the following

$\displaystyle P_{D}(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm S^{\prime}})$
$\displaystyle=\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left[(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SS^{\prime}A}),(\Lambda_{\rm SS^{\prime}}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SS^{\prime}A})\right]$ $\displaystyle\geq\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left[(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}}),(\Lambda_{\rm
SS^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})\right]$
$\displaystyle\geq\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left\\{(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SA}),{\rm tr}_{\rm S^{\prime}}\left[(\Lambda_{\rm
SS^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})\right]\right\\}$
$\displaystyle\geq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}\;D\left[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA}),(\Lambda_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})\right]$
$\displaystyle=P_{D}(\mathcal{E}_{\rm S}).$ (49)

In the second line we pick a fixed absolutely free state
$\widetilde{\eta}_{\rm S^{\prime}}\in\widetilde{\mathcal{F}}_{R}$, which is
possible due to the property (R1). Then the second line follows from the fact
that $\rho_{\rm SA}\otimes\widetilde{\eta}_{\rm S^{\prime}}$ forms a sub-
optimal range for the maximization $\overline{\sup}_{\rm A}$. The third line
is because of Eq. (13). The fourth line is because the mapping ${\rm tr}_{\rm
S^{\prime}}\left\\{\Lambda_{\rm
SS^{\prime}}[(\cdot)\otimes\widetilde{\eta}_{\rm S^{\prime}}]\right\\}$ will
be an $R$-annihilating channel [properties (R2), (R3), and (R4)]. This
consequently implies a sup-optimal range for the minimization compared with
$\inf_{\Lambda_{\rm S}\in\mathcal{O}_{R}^{N}}$. Then the inequality in Eq. (7)
is proved.

To show the equality, we compute the following for a given
$\widetilde{\Lambda}_{\rm S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}$:

$\displaystyle P_{D}(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
S^{\prime}})$ $\displaystyle=\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm
A}}\;D\left[(\mathcal{E}_{\rm S}\otimes\widetilde{\Lambda}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SS^{\prime}A}),(\Lambda_{\rm SS^{\prime}}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SS^{\prime}A})\right]$ $\displaystyle\leq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}\;D\left[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm S^{\prime}}\otimes\widetilde{\Lambda}_{\rm
A})(\rho_{\rm SS^{\prime}A}),(\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm
S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SS^{\prime}A})\right]$ $\displaystyle\leq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}\;D\left[(\mathcal{E}_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA}),(\Lambda_{\rm
S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})\right]$
$\displaystyle=P_{D}(\mathcal{E}_{\rm S}).$ (50)

The second line is because $\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm
S^{\prime}}$ with the fixed $\widetilde{\Lambda}_{\rm S^{\prime}}$ forms a
sub-optimal range for the minimization compared with $\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}$. The third line is because
$\widetilde{\Lambda}_{\rm S^{\prime}}\otimes\widetilde{\Lambda}_{\rm A}$ forms
a sub-optimal range for the maximization of $\overline{\sup}_{\rm A}$ [Eq.
(2)]. This proves the equality and Eq. (7).

Finally, suppose for a given channel $\mathcal{E}$ we have
$P_{D}(\mathcal{E})=0$. By definition this implies

$\displaystyle\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\sup_{\rho}D[\mathcal{E}(\rho),\Lambda_{\rm
S}(\rho)]=0$ (51)

since we are allowed to consider the zero-dimensional ancillary system. Hence,
there exists a sequence
$\\{\Lambda_{k}\\}_{k=1}^{\infty}\subseteq\mathcal{O}_{R}^{N}$ such that
$\lim_{k\to\infty}\sup_{\rho}D[\mathcal{E}(\rho),\Lambda_{k}(\rho)]=0$. If
$\mathcal{O}_{R}^{N}$ is closed under $D$, this implies
$\mathcal{E}\in\mathcal{O}_{R}^{N}$. This proves property (M4), and the proof
for $P_{D}$ is completed.

The case for $\bar{P}_{D}$ is almost the same: One simply needs to replace
$\widetilde{\Lambda}_{\rm A}$ and $\overline{\sup}_{\rm A}$ by
$\mathcal{I}_{\rm A}$ and $\sup_{{\rm A};\rho_{\rm SA}}$, respectively. Also
we remark that the proof of property (M2) for $\bar{P}_{D}$ is a direct
application of Theorem 1 in Ref. [55]. This also means $\bar{P}_{D}$ can be a
monotone if we consider the largest set of possible free super-channels,
whenever this set is well-defined (see Ref. [App:Well-defined] for the
explanation). ∎

##  Appendix E Proof of Theorem 3

To sketch the proof, we note that Theorem 10 in Ref. [54] is true even without
assumptions 3 in their paper, which is crucial for $R$-preservability theories
since the identity channel can never be a free channel. Using all the listed
assumptions in Theorem 3, one can prove the upper bound by the same strategy
in Ref. [54]. Also, the small difference between Definition 5 in this work and
Definition 9 in Ref. [54] will not change the proof of the lower bound.

For the completeness of this work, we still state the detailed proof in this
section. Before the proof, we recall the Generalized Convex-Split Lemma for
completely-positive maps [54]:

######  Lemma E.1.

(Generalized Convex-Split Lemma) [54] Let $\alpha,\beta$ be completely-
positive maps with
$\left\|\alpha\right\|_{\diamond}=\left\|\beta\right\|_{\diamond}=1$. Suppose
there exists a completely-positive map $\alpha^{\prime}$ with
$\left\|\alpha^{\prime}\right\|_{\diamond}\leq 1$ and $p\in(0,1]$ such that
$\beta=p\alpha+(1-p)\alpha^{\prime}$. Then the validity of the inequality
$\log_{2}{n}\geq\log_{2}{\frac{1}{p}}+2\log_{2}{\frac{1}{\delta}}$ implies the
following estimate

$\displaystyle\left\|\sum_{i=1}^{n}\frac{1}{n}\beta^{\otimes(i-1)}\otimes\alpha\otimes\beta^{\otimes(n-i)}-\beta^{\otimes
n}\right\|_{\diamond}\leq\delta.$ (52)

Before the main proof, we note the following two facts. The first one has
$\bar{P}_{D_{\rm max}}=L_{R}$ [Eq. (15)] as a direct consequence.

######  Fact E.2.

Given two channels $\mathcal{E}$ and $\Lambda$. Then we have

$\displaystyle\sup_{{\rm A};\rho_{\rm
SA}}\inf\left\\{\lambda\,|\,0\leq[(\lambda\Lambda-\mathcal{E})\otimes\mathcal{I}_{\rm
A}](\rho_{\rm
SA})\right\\}=\inf\left\\{\lambda\,|\,0\leq[(\lambda\Lambda-\mathcal{E})\otimes\mathcal{I}_{\rm
A}](\rho_{\rm SA})\;\forall{\rm A}\,\&\,\rho_{\rm SA}\right\\}.$ (53)

###### Proof.

Define the set $\mathcal{L}_{\bf
A}\coloneqq\left\\{\lambda\,|\,0\leq[(\lambda\Lambda-\mathcal{E})\otimes\mathcal{I}_{\rm
A}](\rho_{\rm SA})\right\\}$ with ${\bf A}$ denotes a particular combination
of an ancillary system ${\rm A}$ and a state $\rho_{\rm SA}$ on the system
${\rm SA}$. Then the left-hand-side can be written as $\sup_{\rm\bf
A}\inf\\{\lambda\,|\,\lambda\in\mathcal{L}_{\bf A}\\}$, and the right-hand-
side can be written as $\inf\left\\{\lambda\,|\,\lambda\in\bigcap_{\bf
A}\mathcal{L}_{\bf A}\right\\}$. With the above notations, the inequality
“$\leq$” follows by the fact that $\bigcap_{\bf A}\mathcal{L}_{\bf
A}\subseteq\mathcal{L}_{{\bf A}^{\prime}}$ for all ${{\bf A}^{\prime}}$. On
the other hand, consider a given $k\in\mathbb{N}$. Then there exists an ${\bf
A}_{k}$ such that $\inf\left\\{\lambda\,|\,\lambda\in\mathcal{L}_{{\bf
A}_{k}}\right\\}+\frac{1}{k}>\sup_{\rm\bf
A}\inf\\{\lambda\,|\,\lambda\in\mathcal{L}_{\bf
A}\\}\geq\inf\left\\{\lambda\,|\,\lambda\in\mathcal{L}_{{\bf
A}_{k}}\right\\}$. Also, there exists $\lambda_{k}\in\mathcal{L}_{{\bf
A}_{k}}$ such that
$\lambda_{k}-\frac{1}{k}<\inf\left\\{\lambda\,|\,\lambda\in\mathcal{L}_{{\bf
A}_{k}}\right\\}\leq\lambda_{k}$. This means
$\lambda_{k}+\frac{1}{k}>\inf\\{\lambda\,|\,\lambda\in\mathcal{L}_{\bf A}\\}$
for all ${\bf A}$. In other words, this means
$\lambda_{k}+\frac{1}{k}\in\bigcap_{\bf A}\mathcal{L}_{\bf A}$, which also
implies

$\displaystyle\inf\left\\{\lambda\,|\,\lambda\in\bigcap_{\bf
A}\mathcal{L}_{\bf A}\right\\}$ $\displaystyle\leq\lambda_{k}+\frac{1}{k}$
$\displaystyle\leq\inf\left\\{\lambda\,|\,\lambda\in\mathcal{L}_{{\bf
A}_{k}}\right\\}+\frac{2}{k}$ $\displaystyle\leq\sup_{\rm\bf
A}\inf\\{\lambda\,|\,\lambda\in\mathcal{L}_{\bf A}\\}+\frac{2}{k}.$ (54)

Since this is true for all $k\in\mathbb{N}$, the result follows. ∎

The second fact is a property similar to Eq. (7) for the smooth version of
$\bar{P}_{D}$ defined similarly to Eq. (18).

######  Fact E.3.

Given an $R$-theory and a distance measure $D$ satisfying Eq. (13). Then for
every $\delta\geq 0$ and channels $\mathcal{E}_{\rm S},\mathcal{E}_{\rm
S^{\prime}}\in\mathcal{O}_{R}$, we have

$\displaystyle\bar{P}_{D}^{\delta}(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}})\geq\bar{P}_{D}^{\delta}(\mathcal{E}_{\rm S}).$ (55)

###### Proof.

First, we have the following definition similar to Eq. (18):

$\displaystyle\bar{P}_{D}^{\delta}(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}})\coloneqq\inf_{\frac{1}{2}\left\|\mathcal{C}_{\rm
SS^{\prime}}-\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\right\|_{\diamond}\leq\delta}\bar{P}_{D}(\mathcal{C}_{\rm
SS^{\prime}}).$ (56)

A direct computation shows

$\displaystyle\bar{P}_{D}(\mathcal{C}_{\rm SS^{\prime}})$
$\displaystyle=\inf_{\Lambda_{\rm SS^{\prime}}\in\mathcal{O}_{R}^{N}}\sup_{\rm
A;\rho_{\rm SS^{\prime}A}}D\left[(\mathcal{C}_{\rm
SS^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SS^{\prime}A}),(\Lambda_{\rm
SS^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SS^{\prime}A})\right]$
$\displaystyle\geq\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}\sup_{\rm A;\rho_{\rm
SA}}D\left[(\mathcal{C}_{\rm SS^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}}),(\Lambda_{\rm
SS^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})\right]$
$\displaystyle\geq\inf_{\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}}\sup_{\rm A;\rho_{\rm
SA}}D\left[(\mathcal{C}_{\rm S}^{\prime}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA}),(\Lambda_{\rm S}^{\prime}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA})\right]$ $\displaystyle\geq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\sup_{\rm A;\rho_{\rm SA}}D\left[(\mathcal{C}_{\rm
S}^{\prime}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA}),(\Lambda_{\rm
S}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})\right]$
$\displaystyle=\bar{P}_{D}(\mathcal{C}_{\rm S}^{\prime}).$ (57)

The second line is because fixing an absolutely free state
$\widetilde{\eta}_{\rm S^{\prime}}$ [which is possible due to property (R1)]
forms a sub-optimal range for the maximization. The third line is a
consequence of the data-processing inequality under partial trace [namely, Eq.
(13) and property (R2)], where we define $\mathcal{C}_{\rm
S}^{\prime}(\cdot)\coloneqq{\rm tr}_{\rm S^{\prime}}\circ\mathcal{C}_{\rm
SS^{\prime}}[(\cdot)\otimes\widetilde{\eta}_{\rm S^{\prime}}]$ and
$\Lambda_{\rm S}^{\prime}(\cdot)\coloneqq{\rm tr}_{\rm
S^{\prime}}\circ\Lambda_{\rm SS^{\prime}}[(\cdot)\otimes\widetilde{\eta}_{\rm
S^{\prime}}]$. Using properties (R2), (R3), and (R4), we learn that
$\Lambda_{\rm S}^{\prime}\in\mathcal{O}_{R}^{N}$ and hence all possible such
channels form a sub-optimal range for the minimization compared with all
elements in $\mathcal{O}_{R}^{N}$. This explains the fourth line.

Now we note that

$\displaystyle\left\|\mathcal{C}_{\rm SS^{\prime}}-\mathcal{E}_{\rm
S}\otimes\mathcal{E}_{\rm S^{\prime}}\right\|_{\diamond}$
$\displaystyle\coloneqq\sup_{\rm A;\rho_{\rm
SS^{\prime}A}}\left\|(\mathcal{C}_{\rm SS^{\prime}}\otimes\mathcal{I}_{\rm
A})(\rho_{\rm SS^{\prime}A})-(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SS^{\prime}A})\right\|_{1}$
$\displaystyle\geq\sup_{\rm A;\rho_{\rm SA}}\left\|(\mathcal{C}_{\rm
SS^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})-(\mathcal{E}_{\rm
S}\otimes\mathcal{E}_{\rm S^{\prime}}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA}\otimes\widetilde{\eta}_{\rm S^{\prime}})\right\|_{1}$
$\displaystyle\geq\sup_{\rm A;\rho_{\rm SA}}\left\|(\mathcal{C}_{\rm
S}^{\prime}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})-(\mathcal{E}_{\rm
S}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})\right\|_{1}$
$\displaystyle=\left\|\mathcal{C}_{\rm S}^{\prime}-\mathcal{E}_{\rm
S}\right\|_{\diamond}.$ (58)

Again, the second line is due to the sub-optimal range for the maximization
when we fix an absolutely free state $\widetilde{\eta}_{\rm S^{\prime}}$.
Also, the third line follows from the data-processing inequality (or
equivalently, the contractivity) of trace norm under quantum channels.

Finally, we have

$\displaystyle\bar{P}_{D}^{\delta}(\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}})$ $\displaystyle\geq\inf_{\frac{1}{2}\left\|\mathcal{C}_{\rm
SS^{\prime}}-\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\right\|_{\diamond}\leq\delta}\bar{P}_{D}(\mathcal{C}_{\rm
S}^{\prime})$ $\displaystyle\geq\inf_{\frac{1}{2}\left\|\mathcal{C}_{\rm
S}^{\prime}-\mathcal{E}_{\rm
S}\right\|_{\diamond}\leq\delta}\bar{P}_{D}(\mathcal{C}_{\rm S}^{\prime})$
$\displaystyle\geq\inf_{\frac{1}{2}\left\|\mathcal{E}^{\prime}-\mathcal{E}_{\rm
S}\right\|_{\diamond}\leq\delta}\bar{P}_{D}(\mathcal{E}^{\prime})$
$\displaystyle=\bar{P}_{D}^{\delta}(\mathcal{E}_{\rm S}).$ (59)

The second line follows from Eq. (E), which implies that all $\mathcal{C}_{\rm
SS^{\prime}}$ satisfying $\frac{1}{2}\left\|\mathcal{C}_{\rm
SS^{\prime}}-\mathcal{E}_{\rm S}\otimes\mathcal{E}_{\rm
S^{\prime}}\right\|_{\diamond}\leq\delta$ form a subset of the set of all
$\mathcal{C}_{\rm SS^{\prime}}$ satisfying $\frac{1}{2}\left\|\mathcal{C}_{\rm
S}^{\prime}-\mathcal{E}_{\rm S}\right\|_{\diamond}\leq\delta$. Since all
possible $\mathcal{C}_{\rm S}^{\prime}$ form a subset of all possible channels
$\mathcal{E}^{\prime}$ satisfying
$\frac{1}{2}\left\|\mathcal{E}^{\prime}-\mathcal{E}_{\rm
S}\right\|_{\diamond}\leq\delta$, we have the third line. The proof is
completed. ∎

Now we start the proof of Theorem 3:

###### Proof.

We follow the same strategy in the proof of Theorem 10 in Ref. [54]. We will
show the upper bound at first.

Proof of the upper bound.– At the very beginning, consider an arbitrarily
given positive integer $l\in\mathbb{N}$. By definition, there exists a channel
$\mathcal{E}_{l}$ such that
$\left\|\mathcal{E}_{l}-\mathcal{E}\right\|_{\diamond}\leq 2(\epsilon-\eta)$
and

$\displaystyle\bar{P}_{D_{\rm
max}}^{\epsilon-\eta}(\mathcal{E})\leq\bar{P}_{D_{\rm
max}}(\mathcal{E}_{l})\leq\bar{P}_{D_{\rm
max}}^{\epsilon-\eta}(\mathcal{E})+\frac{1}{l}.$ (60)

Also, because we have $\mathcal{O}_{R}^{N}=\widetilde{\mathcal{O}}_{R}^{N}$,
there exists a channel $\bar{\Lambda}_{l}\in\widetilde{\mathcal{O}}_{R}^{N}$
such that

$\displaystyle\bar{P}_{D_{\rm max}}(\mathcal{E}_{l})$
$\displaystyle\leq\sup_{{\rm A};\rho_{\rm SA}}D_{\rm
max}\left[(\mathcal{E}_{l}\otimes\mathcal{I}_{\rm A})(\rho_{\rm
SA})\|(\bar{\Lambda}_{l}\otimes\mathcal{I}_{\rm A})(\rho_{\rm SA})\right]$
$\displaystyle\coloneqq\sup_{{\rm A};\rho_{\rm
SA}}\log_{2}\inf\left\\{\lambda\,|\,0\leq[(\lambda\bar{\Lambda}_{l}-\mathcal{E}_{l})\otimes\mathcal{I}_{\rm
A}](\rho_{\rm SA})\right\\}$
$\displaystyle=\log_{2}\inf\left\\{\lambda\,|\,0\leq[(\lambda\bar{\Lambda}_{l}-\mathcal{E}_{l})\otimes\mathcal{I}_{\rm
A}](\rho_{\rm SA})\;\forall{\rm A}\,\&\,\rho_{\rm SA}\right\\}$
$\displaystyle=-\log_{2}\sup\left\\{q\in[0,1]\,|\,0\leq[(\bar{\Lambda}_{l}-q\mathcal{E}_{l})\otimes\mathcal{I}_{\rm
A}](\rho_{\rm SA})\;\forall{\rm A}\,\&\,\rho_{\rm SA}\right\\}$
$\displaystyle\leq\bar{P}_{D_{\rm max}}(\mathcal{E}_{l})+\frac{1}{l},$ (61)

where in the third line we use Fact E.2 and the fourth line is due to the fact
that $\lambda<1$ is forbidden in the minimization range (otherwise there exist
quantum states $\sigma$ and $\sigma^{\prime}$ such that
$\lambda\sigma-\sigma^{\prime}\geq 0$ for some $\lambda<1$, which is
impossible since this implies $0\leq{\rm
tr}(\lambda\sigma-\sigma^{\prime})=\lambda-1<0$). Let $\mathcal{U}_{i}$ be the
pair-wise permutation unitary channel between the first and the $i$th
subsystems (that is, the swap unitary between the two subsystems). Then we
consider the destruction process with $\bar{\Lambda}_{l}^{\otimes(n-1)}$ and
$\left\\{\mathcal{U}_{i},\mathcal{U}_{i},p_{i}=\frac{1}{n}\right\\}_{i=1}^{n}$
666Note that in general the two pair-wise permutation channels are different
because they may act on different spaces. Here we simply use the same notation
to stress the fact that both of them are permutation unitary channels between
the first and the $i$th subsystems. More precisely, if
$\bar{\Lambda}_{l},\mathcal{E}_{l}:{\rm S}\to{\rm S^{\prime}}$, then we have
${\rm S}^{\otimes n}\to{\rm S}^{\otimes n}$ for the pre-processing
permutations and ${\rm S^{\prime}}^{\otimes n}\to{\rm S^{\prime}}^{\otimes n}$
for the post-processing permutations., which gives the following:

$\displaystyle\sum_{i=1}^{n}\frac{1}{n}\mathcal{U}_{i}\circ\left(\mathcal{E}_{l}\otimes\bar{\Lambda}_{l}^{\otimes(n-1)}\right)\circ\mathcal{U}_{i}=\frac{1}{n}\sum_{i=1}^{n}\bar{\Lambda}_{l}^{\otimes(i-1)}\otimes\mathcal{E}_{l}\otimes\bar{\Lambda}_{l}^{\otimes(n-i)}.$
(62)

From Eq. (E) we note that when $\log_{2}{n}>\bar{P}_{D_{\rm
max}}(\mathcal{E}_{l})+\frac{1}{l}+2\log_{2}\frac{1}{2\eta}$ [which
automatically implies $\bar{P}_{D_{\rm max}}(\mathcal{E}_{l})<\infty$], there
always exists an $p_{l}\in(0,1)$ such that

  * •

$\log_{2}{n}>\log_{2}\frac{1}{p_{l}}+2\log_{2}\frac{1}{2\eta}$.

  * •

$\bar{\Lambda}_{l}-p_{l}\mathcal{E}_{l}$ is completely-positive.

By defining
$\alpha^{\prime}_{l}\coloneqq\frac{1}{1-p_{l}}(\bar{\Lambda}_{l}-p_{l}\mathcal{E}_{l})$,
one can see that $\alpha^{\prime}_{l}$ is completely-positive and trace-
preserving [since both $\bar{\Lambda}_{l}$ and $\mathcal{E}_{l}$ are trace-
preserving and we have
$\bar{\Lambda}_{l}=p_{l}\mathcal{E}_{l}+(1-p_{l})\alpha^{\prime}_{l}$; note
that $p_{l}<1$]. This means $\alpha^{\prime}_{l}$ is also a quantum channel
(i.e. a completely-positive trace-preserving map), thereby having
$\left\|\alpha^{\prime}_{l}\right\|_{\diamond}=1$. Then Lemma E.1 (with
$\alpha=\mathcal{E}_{l}$ and $\beta=\bar{\Lambda}_{l}$) implies that when
$\log_{2}{n}>\bar{P}_{D_{\rm
max}}(\mathcal{E}_{l})+\frac{1}{l}+2\log_{2}\frac{1}{2\eta}$ holds, then we
have

$\displaystyle\left\|\frac{1}{n}\sum_{i=1}^{n}\bar{\Lambda}_{l}^{\otimes(i-1)}\otimes\mathcal{E}_{l}\otimes\bar{\Lambda}_{l}^{\otimes(n-i)}-\bar{\Lambda}_{l}^{\otimes
n}\right\|_{\diamond}\leq 2\eta;$ (63)

in other words, Eq. (62) forms an $\eta$-destruction process for
$\mathcal{E}_{l}$. (Note that $\bar{\Lambda}_{l}^{\otimes
n}\in\widetilde{\mathcal{O}}_{R}^{N}$ since we assume no activation property).
This also implies the existence of an $\epsilon$-destruction process for
$\mathcal{E}$ since

$\displaystyle\left\|\frac{1}{n}\sum_{i=1}^{n}\bar{\Lambda}_{l}^{\otimes(i-1)}\otimes\mathcal{E}\otimes\bar{\Lambda}_{l}^{\otimes(n-i)}-\bar{\Lambda}_{l}^{\otimes
n}\right\|_{\diamond}$
$\displaystyle\leq\left\|\mathcal{E}-\mathcal{E}_{l}\right\|_{\diamond}+\left\|\frac{1}{n}\sum_{i=1}^{n}\bar{\Lambda}_{l}^{\otimes(i-1)}\otimes\mathcal{E}_{l}\otimes\bar{\Lambda}_{l}^{\otimes(n-i)}-\bar{\Lambda}_{l}^{\otimes
n}\right\|_{\diamond}$ $\displaystyle\leq 2(\epsilon-\eta)+2\eta=2\epsilon,$
(64)

where we use the relation
$\left\|\mathcal{E}-\mathcal{E}_{l}\right\|_{\diamond}\leq 2(\epsilon-\eta)$,
data-processing inequality, and triangle inequality.

Finally, let
$n^{\prime}=\min\left\\{n\in\mathbb{N}\,|\,\log_{2}{n}>\bar{P}_{D_{\rm
max}}(\mathcal{E}_{l})+\frac{1}{l}+2\log_{2}{\frac{1}{2\eta}}\right\\}$. Since
$C_{R}^{\epsilon}(\mathcal{E})\coloneqq\min\log_{2}{n}$ and the minimization
is taken over all $\epsilon$-destruction processes, we conclude the following

$\displaystyle C_{R}^{\epsilon}(\mathcal{E})$
$\displaystyle\leq\log_{2}{n^{\prime}}$ $\displaystyle\leq\bar{P}_{D_{\rm
max}}(\mathcal{E}_{l})+\frac{1}{l}+2\log_{2}{\frac{1}{2\eta}}+\max_{x\in\mathbb{N}}\left[\log_{2}(x+1)-\log_{2}{x}\right]$
$\displaystyle\leq\bar{P}_{D_{\rm
max}}(\mathcal{E}_{l})+\frac{1}{l}+2\log_{2}{\frac{1}{2\eta}}+1$
$\displaystyle\leq\bar{P}_{D_{\rm
max}}^{\epsilon-\eta}(\mathcal{E})+\frac{2}{l}+2\log_{2}{\frac{1}{\eta}}-1,$
(65)

and the proof of the upper bound is completed since the above estimate works
for all positive integer $l$.

Proof of the lower bound.– The proof is completely the same with the proof of
the lower bound of Theorem 10 in Ref. [54], and we briefly sketch it. Consider
a given $\mathcal{E}_{\rm S}\in\mathcal{O}_{R}$. Then for a given
$\epsilon$-destruction process of $R$-preservability consisting of
$\bar{\Lambda}_{\rm S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}$ and
$\\{\mathcal{U}_{i},\mathcal{V}_{i},p_{i}\\}_{i=1}^{K}$, we have

$\displaystyle\left\|\sum_{i=1}^{K}p_{i}\mathcal{N}_{i}-\Lambda_{\rm
SS^{\prime}}\right\|_{\diamond}\leq 2\epsilon,$ (66)

where $\Lambda_{\rm SS^{\prime}}\in\mathcal{O}_{R}^{N}$ and
$\mathcal{N}_{i}\coloneqq\mathcal{U}_{i}\circ(\mathcal{E}_{\rm
S}\otimes\bar{\Lambda}_{\rm S^{\prime}})\circ\mathcal{V}_{i}$. This
$\epsilon$-destruction process of $R$-preservability can also be interpreted
as an $\epsilon$-destruction process defined in Ref. [54] by identifying
$\mathcal{O}_{R}$ as free channels in their framework (that is, in the proof
of the lower bound of Theorem 10 in Ref. [54] we set
$\mathcal{N}=\mathcal{E}_{\rm S}\in\mathcal{O}_{R}$,
$\mathcal{F}=\bar{\Lambda}_{\rm
S^{\prime}}\in\widetilde{\mathcal{O}}_{R}^{N}$, $\mathcal{M}=\Lambda_{\rm
SS^{\prime}}\in\mathcal{O}_{R}^{N}$, and consider the channel resource theory
with free channels as $\mathcal{O}_{R}$.). The same proof applies until we
reach the following inequality, which is the last inequality in the bottom of
page 15 in Ref. [54] (the assumptions made in Theorem 3 plus the properties
(R1), (R2), (R3), (R4) ensure the applicability of the proof of Theorem 10 in
Ref. [54] when we identify $\mathcal{O}_{R}$ as free channels in their
setting):

$\displaystyle\left\|\mathcal{E}_{\rm S}\otimes\bar{\Lambda}_{\rm
S^{\prime}}-\sum_{i=1}^{K}p_{i}\mathcal{U}_{i}^{\dagger}\circ\mathcal{M}_{i}\circ\mathcal{V}_{i}^{\dagger}\right\|_{\diamond}\leq\sqrt{\epsilon(2-\epsilon)},$
(67)

where $\mathcal{M}_{i}$’s are completely-positive maps satisfying
$\sum_{i=1}^{K}p_{i}\mathcal{M}_{i}=\Lambda_{\rm SS^{\prime}}$, which means
$p_{i}\mathcal{M}_{i}\leq\Lambda_{\rm SS^{\prime}}$ for all $i$ (by writing
$\mathcal{E}\leq\mathcal{E}^{\prime}$ for two channel $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ we means $\mathcal{E}^{\prime}-\mathcal{E}$ is
completely-positive). Hence, we have

$\displaystyle\sum_{i=1}^{K}p_{i}\mathcal{U}_{i}^{\dagger}\circ\mathcal{M}_{i}\circ\mathcal{V}_{i}^{\dagger}\leq\sum_{i=1}^{K}\mathcal{U}_{i}^{\dagger}\circ\Lambda_{\rm
SS^{\prime}}\circ\mathcal{V}_{i}^{\dagger}.$ (68)

Note that the left-hand-side is a channel. Because for any $R$-theory
considered in this work the set $\mathcal{O}_{R}^{N}$ is by definition convex,
we have $\frac{1}{K}\sum_{i=1}^{K}\mathcal{U}_{i}^{\dagger}\circ\Lambda_{\rm
SS^{\prime}}\circ\mathcal{V}_{i}^{\dagger}\in\mathcal{O}_{R}^{N}$. Hence, we
conclude

$\displaystyle\bar{P}_{D_{\rm
max}}^{\sqrt{\epsilon(2-\epsilon)}}(\mathcal{E}_{\rm
S}\otimes\bar{\Lambda}_{\rm S^{\prime}})\leq\bar{P}_{D_{\rm
max}}\left(\sum_{i=1}^{K}p_{i}\mathcal{U}_{i}^{\dagger}\circ\mathcal{M}_{i}\circ\mathcal{V}_{i}^{\dagger}\right)\leq\log_{2}{K}.$
(69)

Using Fact E.3, we conclude that $\bar{P}_{D_{\rm
max}}^{\sqrt{\epsilon(2-\epsilon)}}(\mathcal{E}_{\rm S})\leq\log_{2}K$ for all
possible $K$. This completes the proof. ∎

##  Appendix F Proof of Theorem 4

Before the proof, we need to recap certain key ingredients in Ref. [76]. The
first one is a central assumption called energy subspace condition (we use the
notation ${\bf m}=(m_{1},m_{2},...,m_{d})$ to denote a vector in
$\mathbb{N}^{d}$):

######  Definition F.1.

(Energy Subspace Condition) [76] A given Hamiltonian $H$ with energy levels
$\\{E_{i}\\}_{i=1}^{d}$ is said to fulfill the energy subspace condition if
for any positive integer $M$ and two different vectors $\\{{\bf m}\neq{\bf
m}^{\prime}\\}\subset\mathbb{N}^{d}$ satisfying
$\sum_{i=1}^{d}m_{i}=\sum_{i=1}^{d}m^{\prime}_{i}=M$, we have

$\displaystyle\sum_{i=1}^{d}m_{i}E_{i}\neq\sum_{i=1}^{d}m^{\prime}_{i}E_{i}.$
(70)

Roughly speaking, Definition F.1 means energy levels cannot be integer
multiples of each other. This condition also forbids the possibility of
degeneracy (otherwise one can simply switch the coefficients of a vector ${\bf
m}$ in a subspace with degeneracy to construct a counterexample).

Before mentioning the main results in Ref. [76], we define the smooth max-
relative entropy as [also recall Eq. (14)]

$\displaystyle D_{\rm
max}^{\epsilon}(\rho||\sigma)\coloneqq\inf_{\frac{1}{2}\left\|\rho^{\prime}-\rho\right\|_{1}\leq\epsilon}D_{\rm
max}(\rho^{\prime}||\sigma).$ (71)

Note that there is a difference of $\frac{1}{2}$ factor compared with Eq. (11)
in Ref. [76]. Then we have [76] ($\gamma$ is the thermal state associated with
the given bath temperature $T$ and system Hamiltonian $H_{\rm S}$):

######  Theorem F.2.

[76] For a given state $\rho_{\rm S}$, we have

$\displaystyle n_{\epsilon}(\rho_{\rm S})\leq\frac{1}{\epsilon^{2}}2^{D_{\rm
max}(\rho_{\rm S}||\gamma)}+1.$ (72)

Moreover, if the system Hamiltonian $H_{\rm S}$ satisfies the energy subspace
condition and $\rho_{\rm S}$ is diagonal in the energy eigenbasis of $H_{\rm
S}$, then we also have

$\displaystyle D_{\rm max}^{\sqrt{\epsilon}}(\rho_{\rm
S}||\gamma)\leq\log_{2}n_{\epsilon}(\rho_{\rm S}).$ (73)

Now the idea is to use the above theorem to prove Theorem 4. But before the
proof, we still need to establish the following lemma regarding the continuity
of the max-relative entropy (in a finite dimensional case, we say a quantum
state is full-rank if it has only positive eigenvalues; in other words, its
support coincides with the whole Hilbert space):

######  Lemma F.3.

Given three states $\rho,\rho^{\prime},\sigma$ and $\sigma$ is full-rank. Then
we have

$\displaystyle\left|2^{D_{\rm max}(\rho^{\prime}||\sigma)}-2^{D_{\rm
max}(\rho||\sigma)}\right|\leq\frac{\left\|\rho-\rho^{\prime}\right\|_{1}}{p_{\rm
min}(\sigma)},$ (74)

where $p_{\rm min}(\sigma)$ is the smallest eigenvalue of $\sigma$.

###### Proof.

Define the set $\mathcal{L}(\rho||\sigma)\coloneqq\\{\lambda\geq
0\;|\;\rho\leq\lambda\sigma\\}$. Then one can rewrite the definition of the
max-relative entropy as $D_{\rm
max}(\rho||\sigma)=\inf_{\lambda\in\mathcal{L}(\rho||\sigma)}\log_{2}\lambda.$
Now we note that $\lambda\in\mathcal{L}(\rho||\sigma)$ if and only if
$\lambda\sigma-\rho\geq 0$, which is true if and only if

$\displaystyle\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho)|\phi\rangle\geq
0.$ (75)

This implies the following estimate for any
$\lambda\in\mathcal{L}(\rho||\sigma)$:

$\displaystyle\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho)|\phi\rangle$
$\displaystyle=\inf_{|\phi\rangle}\left[\langle\phi|(\lambda\sigma-\rho^{\prime})|\phi\rangle+\langle\phi|(\rho^{\prime}-\rho)|\phi\rangle\right]$
$\displaystyle\geq\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho^{\prime})|\phi\rangle+\inf_{|\phi\rangle}\langle\phi|(\rho^{\prime}-\rho)|\phi\rangle$
$\displaystyle\geq\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho^{\prime})|\phi\rangle-\left\|\rho-\rho^{\prime}\right\|_{1},$
(76)

where in the last line we use the relation
$\inf_{|\phi\rangle}\langle\phi|(\rho^{\prime}-\rho)|\phi\rangle=-\sup_{|\phi\rangle}\langle\phi|(\rho-\rho^{\prime})|\phi\rangle\geq-\left\|\rho-\rho^{\prime}\right\|_{\infty}\geq-\left\|\rho-\rho^{\prime}\right\|_{1}$
(recall that
$\left\|\cdot\right\|_{\infty}\coloneqq\sup_{|\psi\rangle}|\langle\psi|\cdot|\psi\rangle|$
and $\left\|\cdot\right\|_{\infty}\leq\left\|\cdot\right\|_{1}$). Since the
argument also works when we exchange the roles of $\rho$ and $\rho^{\prime}$,
we conclude the following bound:

$\displaystyle\left|\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho^{\prime})|\phi\rangle-\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho)|\phi\rangle\right|\leq\left\|\rho-\rho^{\prime}\right\|_{1}.$
(77)

With the help of the above bound, we have the following computation for a
given $\lambda\in\mathcal{L}(\rho||\sigma)$:

$\displaystyle
0\leq\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho)|\phi\rangle$
$\displaystyle\leq\inf_{|\phi\rangle}\langle\phi|(\lambda\sigma-\rho^{\prime})|\phi\rangle+\left\|\rho-\rho^{\prime}\right\|_{1}$
$\displaystyle=\inf_{|\phi\rangle}\langle\phi|\left[\left(\lambda+\frac{\left\|\rho-\rho^{\prime}\right\|_{1}}{\langle\phi|\sigma|\phi\rangle}\right)\sigma-\rho^{\prime}\right]|\phi\rangle$
$\displaystyle\leq\inf_{|\phi\rangle}\langle\phi|\left[\left(\lambda+\frac{\left\|\rho-\rho^{\prime}\right\|_{1}}{p_{\rm
min}(\sigma)}\right)\sigma-\rho^{\prime}\right]|\phi\rangle,$ (78)

where in the second line we have $\langle\phi|\sigma|\phi\rangle>0$ since
$\sigma$ is full-rank and hence has no zero eigenvalue. Also,
$\langle\phi|\sigma|\phi\rangle$ is lower bounded by $p_{\rm
min}(\sigma)=\inf_{|\psi\rangle}\langle\psi|\sigma|\psi\rangle$ for all
$|\phi\rangle$. This computation implies
$\lambda+\frac{\left\|\rho-\rho^{\prime}\right\|_{1}}{p_{\rm
min}(\sigma)}\in\mathcal{L}(\rho^{\prime}||\sigma)$ whenever
$\lambda\in\mathcal{L}(\rho||\sigma)$ [recall Eq. (75)]. As a consequence, we
have

$\displaystyle 2^{D_{\rm max}(\rho^{\prime}||\sigma)}$
$\displaystyle=\inf_{\lambda\in\mathcal{L}(\rho^{\prime}||\sigma)}\lambda$
$\displaystyle\leq\inf_{\lambda\in\mathcal{L}(\rho||\sigma)}\left(\lambda+\frac{\left\|\rho-\rho^{\prime}\right\|_{1}}{p_{\rm
min}(\sigma)}\right)$ $\displaystyle=2^{D_{\rm
max}(\rho||\sigma)}+\frac{\left\|\rho-\rho^{\prime}\right\|_{1}}{p_{\rm
min}(\sigma)}.$ (79)

Then the desired bound can be proved by exchanging the roles of $\rho$ and
$\rho^{\prime}$ and apply the same arguement again. ∎

As a remark, we note that the above lemma implies the Lipschitz continuity of
the function $2^{D_{\rm max}(\cdot||\sigma)}$ when $\sigma$ is full-rank. With
Theorem F.2 and Lemma F.3 in hand, we now start the proof of Theorem 4.

###### Proof.

When we consider the $R$-theory of athermality equipped with Gibbs-preserving
maps, the only absolutely $R$-annihilating channel (with output space ${\rm
A}$ and output dimension $d^{k}$) is the full thermalization channel
$\Phi_{\gamma_{\rm A}}:(\cdot)\mapsto\gamma_{\rm A}$, where $\gamma_{\rm
A}=\gamma^{\otimes k}$ and $\gamma$ is the given thermal state in the system
${\rm S}$. Then direct computation shows [recall Eqs. (9) and (11) for
notations]

$\displaystyle P_{D_{\rm max}}(\mathcal{N})$
$\displaystyle\coloneqq\inf_{\Lambda_{\rm
S}\in\mathcal{O}_{R}^{N}}\overline{\sup_{\rm A}}D_{\rm
max}[(\mathcal{N}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm
SA})||(\Lambda_{\rm S}\otimes\widetilde{\Lambda}_{\rm A})(\rho_{\rm SA})]$
$\displaystyle=\sup_{{\rm A};\rho_{\rm SA}}D_{\rm
max}[(\mathcal{N}\otimes\Phi_{\gamma_{\rm A}})(\rho_{\rm
SA})||\Phi_{\gamma\otimes\gamma_{\rm A}}(\rho_{\rm SA})]$
$\displaystyle=\sup_{{\rm A};\rho_{\rm S}}D_{\rm max}[\mathcal{N}(\rho_{\rm
S})\otimes\gamma_{\rm A}||\gamma\otimes\gamma_{\rm A}]$
$\displaystyle=\sup_{\rho}D_{\rm max}[\mathcal{N}(\rho)||\gamma],$ (80)

where the last equality follows from the fact that for an operator $A$ and a
positive operator $E$ we have $A\geq 0$ if and only if $A\otimes E\geq 0$,
which implies the relation $D_{\rm
max}(\rho\otimes\eta||\sigma\otimes\eta)=D_{\rm max}(\rho||\sigma)$ for all
quantum states $\rho,\sigma,\eta$. Together with Theorem F.2 and the quantity
defined in Eq. (24), we conclude that

$\displaystyle\mathcal{B}^{\epsilon}(\mathcal{N})$
$\displaystyle\coloneqq\sup_{\rho}n_{\epsilon}[\mathcal{N}(\rho)]-1$
$\displaystyle\leq\sup_{\rho}\frac{1}{\epsilon^{2}}2^{D_{\rm
max}[\mathcal{N}(\rho)||\gamma]}$
$\displaystyle=\frac{1}{\epsilon^{2}}2^{P_{D_{\rm max}}(\mathcal{N})},$ (81)

and the upper bound is proved.

To see the lower bound, first we note that being coherence-annihilating for
the given Gibbs-preserving channel $\mathcal{N}$ means $\mathcal{N}(\rho)$ is
diagonal in the energy eigenbasis for all inputs $\rho$. Applying Theorem F.2
and Lemma F.3, one can conclude that

$\displaystyle 1+\mathcal{B}^{\epsilon}(\mathcal{N})$
$\displaystyle\coloneqq\sup_{\rho}n_{\epsilon}[\mathcal{N}(\rho)]$
$\displaystyle\geq\sup_{\rho}2^{D_{\rm
max}^{\sqrt{\epsilon}}[\mathcal{N}(\rho)||\gamma]}$
$\displaystyle=\sup_{\rho}\inf_{\frac{1}{2}\left\|\rho^{\prime}-\mathcal{N}(\rho)\right\|_{1}\leq\sqrt{\epsilon}}2^{D_{\rm
max}(\rho^{\prime}||\gamma)}$ $\displaystyle\geq\sup_{\rho}\left(2^{D_{\rm
max}[\mathcal{N}(\rho)|\gamma]}-\frac{2\sqrt{\epsilon}}{p_{\rm
min}(\gamma)}\right)$ $\displaystyle=2^{P_{D_{\rm
max}}(\mathcal{N})}-\frac{2\sqrt{\epsilon}}{p_{\rm min(\gamma)}},$ (82)

and the proof is completed. ∎

##  Appendix G Non-Signalling Assisted Classical Communication Scenario

Recently, the application of a channel resource theory to classical
communication scenario has been addressed [59]. The central question is how
much classical information (in terms of classical bits) can be transmitted
noiselessly (or up to certain error) via a channel of the following form:

$\displaystyle\mathcal{E}_{d}\circ(\mathcal{N}\otimes\mathcal{I}_{\rm
A})\circ\mathcal{E}_{e},$ (83)

where $\mathcal{N}$ is the noisy channel which is sending the information, and
$\mathcal{E}_{e},\mathcal{E}_{d}$ are the encoding and the decoding channels,
respectively. Note that we have $\mathcal{E}_{e}:{\rm C\to SA}$ and
$\mathcal{E}_{d}:{\rm SA\to C}$, where ${\rm C}$ is the space for the
classical bits, which are represented by the orthonormal basis
$\\{|m\rangle\\}_{m=0}^{M-1}$. As explained in Ref. [59], this structure gives
the non-signalling assisted classical communication scenario: The transmission
of the classical information is assisted by any possible non-signalling
structure, which has the general form given by Eq. (83). In this case, both
$\mathcal{E}_{e},\mathcal{E}_{d}$ can be arbitrary channels.

To quantify how much classical information is transmitted successfully within
a given error $\epsilon\in(0,1)$, we consider the following averaged error
associated to a given combination of encoding channel $\mathcal{E}_{e}$,
decoding channel $\mathcal{E}_{d}$, and transmitting channel $\mathcal{N}$
[59]:

$\displaystyle\varepsilon(\mathcal{E}_{e},\mathcal{E}_{d},\mathcal{N})\coloneqq
1-\frac{1}{M}\sum_{m=0}^{M-1}\langle
m|\mathcal{E}_{d}\circ(\mathcal{N}\otimes\mathcal{I}_{\rm
A})\circ\mathcal{E}_{e}(|m\rangle\langle m|)|m\rangle.$ (84)

Then one can define the corresponding single-shot classical capacity with
error $\epsilon$ as [59]:

$\displaystyle C^{\epsilon}_{\rm
NS,(1)}(\mathcal{N})\coloneqq\sup_{\mathcal{E}_{e},\mathcal{E}_{d}}\\{\log_{2}M\,|\,\varepsilon(\mathcal{E}_{e},\mathcal{E}_{d},\mathcal{N})\leq\epsilon\\}.$
(85)

Note that in the non-signalling assisted scenario, the above optimization is
taken over all possible channels $\mathcal{E}_{e},\mathcal{E}_{d}$. More
precisely, the encoding map $\mathcal{E}_{e}$ can be understood as an
effectively “classical-quantum” channel because we only consider inputs of the
form $|m\rangle\langle m|$. Similarly, the decoding map $\mathcal{E}_{d}$ can
be interpreted as an effectively “quantum-classical” channel. The above
classical capacity indicates the optimal amount of classical bits that can be
transmitted and recovered within the given error $\epsilon$ when the only
constraint is the non-signalling condition. From here one can observe that the
setup in Sec. 5.1.2 is equivalent to the above setup plus the two imposed
thermodynamic constraints.

##  Appendix H Proof of Theorem 5

The strategy is to follow the spirit of the proof of Theorem 3 in Ref. [59].
Before the main proof, we need to establish two facts. First, recall from Eq.
(28) the following expression for a given combination
$(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm A})$:

$\displaystyle
1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm
A})=\frac{1}{M}\sum_{m=0}^{M-1}\langle
m|\mathcal{E}_{d}\circ(\mathcal{N}\otimes\Phi_{\gamma_{\rm
A}})\circ\mathcal{E}_{e}(|m\rangle\langle m|)|m\rangle$ (86)

Here we note that $\gamma_{\rm A}=\gamma^{\otimes k}$ for a positive integer
$k$, where $\gamma$ is the thermal state associated with the given $R$-theory
of athermality in the main system ${\rm S}$. Again, $\Phi_{\gamma_{\rm
A}}:(\cdot)\mapsto\gamma_{\rm A}$ is the full thermalization (or,
equivalently, the state preparation channel) with the target thermal state
$\gamma_{\rm A}$. We remark that throughout this section we assume the
channels $\mathcal{N},\mathcal{N}^{\prime}$ have the same output space ${\rm
S}$. Now, we note the following result, which is similar to Lemma 4 in Ref.
[59]:

######  Fact H.1.

Given Gibbs-preserving channels
$\mathcal{N},\mathcal{N}^{\prime},\mathcal{E}_{e},\mathcal{E}_{d}$, and a
thermal state $\gamma_{\rm A}$. Then we have

$\displaystyle\left|\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]-\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N}^{\prime},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]\right|\leq\frac{1}{2}\left\|\mathcal{N}-\mathcal{N}^{\prime}\right\|_{\diamond}.$
(87)

###### Proof.

We assume
$\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]\geq\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N}^{\prime},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]$ without loss of generality. For every positive integer $k\in\mathbb{N}$,
let $\mathcal{E}^{(k)}$ be the Gibbs-preserving map satisfying
$\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]\leq
1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}^{(k)},\gamma_{\rm
A})+\frac{1}{k}$. Then we have

$\displaystyle\left|\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]-\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N}^{\prime},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]\right|$
$\displaystyle\leq\frac{1}{k}+1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}^{(k)},\gamma_{\rm
A})-[1-\varepsilon(\mathcal{N}^{\prime},\mathcal{E}_{e},\mathcal{E}^{(k)},\gamma_{\rm
A})]$ $\displaystyle=\frac{1}{k}+\frac{1}{M}\sum_{m=0}^{M-1}{\rm
tr}[E^{(k)}_{m}((\mathcal{N}-\mathcal{N}^{\prime})\otimes\Phi_{\gamma_{\rm
A}})\circ\mathcal{E}_{e}(|m\rangle\langle m|)],$ (88)

where $\\{E^{(k)}_{m}\\}_{m=0}^{M-1}$ is a POVM defined by
$E^{(k)}_{m}\coloneqq\mathcal{E}^{(k),\dagger}(|m\rangle\langle m|)$ since
each $\mathcal{E}^{(k),\dagger}$ is a completely-positive unital map.
Following the proof of Lemma 4 in Ref. [59], we note the following estimate:

$\displaystyle\left\|\mathcal{N}-\mathcal{N}^{\prime}\right\|_{\diamond}\geq\left\|(\mathcal{N}-\mathcal{N}^{\prime})(\rho)\right\|_{1}=\sup_{0\leq
E\leq\mathbb{I}}2{\rm tr}[E(\mathcal{N}-\mathcal{N}^{\prime})(\rho)].$ (89)

Then we conclude that

$\displaystyle\left|\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]-\sup_{\mathcal{E}\in\mathcal{O}_{R}}[1-\varepsilon(\mathcal{N}^{\prime},\mathcal{E}_{e},\mathcal{E},\gamma_{\rm
A})]\right|$
$\displaystyle\leq\frac{1}{k}+\frac{1}{2M}\sum_{m=0}^{M-1}\left\|(\mathcal{N}-\mathcal{N}^{\prime})\otimes\Phi_{\gamma_{\rm
A}}\right\|_{\diamond}$
$\displaystyle\leq\frac{1}{k}+\frac{1}{2}\left\|\mathcal{N}-\mathcal{N}^{\prime}\right\|_{\diamond},$
(90)

where the last inequality follows from the data-processing inequality of the
trace norm (or, equivalently, the contractivity under quantum channels). Since
this argument works for all positive integer $k$, the desired upper bound is
proved. ∎

We still need to show another fact, which is similar to Theorem 5 in Ref.
[41]:

######  Fact H.2.

$\displaystyle
1-\varepsilon(\mathcal{N},\mathcal{E}_{e},\mathcal{E}_{d},\gamma_{\rm
A})\leq\frac{1}{M}\times 2^{\bar{P}_{D_{\rm max}}(\mathcal{N})}.$ (91)

###### Proof.

Follow the proof of Theorem 5 in Ref. [41], we first recall from Eq. (15) that
with a $R$-theory of athermality we have (note that the output space of the
channel $\mathcal{N}$ is ${\rm S}$, and $\gamma$ is the corresponding thermal
state)