# General Quantum Resource Theories: Distillation, Formation and Consistent
Resource Measures

Kohdai Kuroiwa<EMAIL_ADDRESS>Institute for Quantum Computing and
Department of Physics and Astronomy, University of Waterloo, 200 University
Avenue West, Waterloo, Ontario, N2L 3G1, Canada  Department of Applied
Physics, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo,
Bunkyo-ku, Tokyo 113–8656, Japan  Hayata Yamasaki<EMAIL_ADDRESS>Photon Science Center, Graduate School of Engineering, The University of
Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan  Institute for Quantum
Optics and Quantum Information — IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria

###### Abstract

Quantum resource theories (QRTs) provide a unified theoretical framework for
understanding inherent quantum-mechanical properties that serve as resources
in quantum information processing, but resources motivated by physics may
possess structure whose analysis is mathematically intractable, such as non-
uniqueness of maximally resourceful states, lack of convexity, and infinite
dimension. We investigate state conversion and resource measures in general
QRTs under minimal assumptions to figure out universal properties of
physically motivated quantum resources that may have such mathematical
structure whose analysis is intractable. In the general setting, we prove the
existence of maximally resourceful states in one-shot state conversion. Also
analyzing asymptotic state conversion, we discover catalytic replication of
quantum resources, where a resource state is infinitely replicable by free
operations. In QRTs without assuming the uniqueness of maximally resourceful
states, we formulate the tasks of distillation and formation of quantum
resources, and introduce distillable resource and resource cost based on the
distillation and the formation, respectively. Furthermore, we introduce
consistent resource measures that quantify the amount of quantum resources
without contradicting the rate of state conversion even in QRTs with non-
unique maximally resourceful states. Progressing beyond the previous work
showing a uniqueness theorem for additive resource measures, we prove the
corresponding uniqueness inequality for the consistent resource measures; that
is, consistent resource measures of a quantum state take values between the
distillable resource and the resource cost of the state. These formulations
and results establish a foundation of QRTs applicable in a unified way to
physically motivated quantum resources whose analysis can be mathematically
intractable.

##  1 Introduction

Advantages in quantum information processing compared to conventional
classical information processing arise from various inherent properties of
quantum states. A framework for systematically investigating quantum-
mechanical properties is essential for better understandings of quantum
mechanics and quantum information processing. Quantum resource theories (QRTs)
[1] give such a framework, in which the quantum properties are regarded as
resources for overcoming restrictions on operations on quantum systems;
especially, manipulation and quantification of resources are integral parts of
QRTs. QRTs have covered numerous aspects of quantum properties such as
entanglement [2, 3, 4], coherence [5, 6, 7, 8, 9, 10, 11], athermality [12,
13, 14, 15, 16], magic states [17, 18], asymmetry [19, 20, 21, 22, 23], purity
[24], non-Gaussianity [25, 26, 27, 28], and non-Markovianity [29, 30].
Recently, QRTs for a general resource have been studied to figure out common
structures shared among known QRTs and to understand the quantum properties
systematically [31, 32, 33, 34, 35, 36, 37, 38, 39, 40].

However, general QRTs are not necessarily mathematically tractable to analyze,
and simply extending the formulation of a known QRT such as bipartite
entanglement is insufficient. For example, maximal resources in the QRT of
magic states [17] and the QRT of coherence with physically incoherent
operations (PIO) [10] are not unique. Gaussian states [41, 42], quantum
discord [43], and quantum Markov chain [44] are quantum-mechanical properties
emerging in a non-convex quantum state space. Gaussian operations [41, 42] are
conventionally defined on a non-convex state space while existing QRTs of non-
Gaussianity [25, 26, 27, 28] are formulated on a convex state space.
Furthermore, the state spaces of QRTs of non-Gaussianity are infinite-
dimensional, and analysis of QRTs on finite-dimensional quantum systems is not
necessarily applicable to infinite-dimensional systems. The analyses of these
physically motivated quantum properties are mathematically intractable.

To analyze general quantum properties including those mentioned in the
previous paragraph, we investigate manipulation and quantification of quantum
resources in general QRTs that are physically motivated but hard to analyze.
We do not make mathematical assumptions such as the existence of a unique
maximal resource, a convex state space, and a finite-dimensional state space.

In this paper, we take a position that free operations determine free states.
A free operation is an element in a subset of quantum operations. The set of
free operations describes what is possible for free when we operate on a
quantum system. A quantum state that may not be obtained by free operations is
regarded as a resource state, while a quantum state freely obtained by free
operations is called a free state. Convertibility of quantum states under free
operations introduces a mathematical concept of order, preorder, of the states
in terms of resourcefulness. A maximally resourceful state is a special
resource state at the top of this ordering, regarded as a unit of the
resource. The existence of maximally resourceful states is essential for
quantifying quantum resources. Due to the generality of our formulation, the
existence of a maximally resourceful state is not obvious, but we prove that a
maximally resourceful state always exists by introducing compactness in our
framework. Furthermore, we analyze the set of free states, and clarify a
condition where a maximally resourceful state is not free.

To investigate manipulation of a quantum resource in general QRTs, we analyze
one-shot and asymptotic state conversion in the general framework of QRTs,
rather than specific resources. We discover a type of quantum resource with a
counter-intuitive property, which is a resource state that is not free to
generate but can be replicated infinitely by free operations using a given
copy catalytically. While catalytic conversion of quantum resources is
originally found in entanglement theory [45], our discovery provides another
form of catalytic property of quantum resources. We call this resource state a
catalytically replicable state. In addition, we formulate resource conversion
tasks in general QRTs, namely, distillation and formation of a resource [46],
and introduce general definitions of the distillable resource and the resource
cost through these tasks, which generalize those defined for bipartite
entanglement [46, 47], coherence [48], and athermality [13]. Formulation of
the distillation and the formation of a resource is not straightforward when
the QRT has non-unique maximally resourceful states. To overcome this issue,
we formulate the distillable resource as how many resources can be extracted
from the state in the worst-case scenario, and the resource cost as how many
resources are needed to generate the state in the best-case scenario. Under
this formulation, we identify a condition of the distillable resource being
smaller than the resource cost.

A resource measure is a tool for quantifying resources. In the QRT of
bipartite entanglement, it is known that a resource measure satisfying certain
properties given in Ref. [49] is lower-bounded by the distillable resource and
upper-bounded by the resource cost, which we call the uniqueness inequality.
In this paper, we show that the uniqueness inequality holds for a general QRT
under the same properties even in infinite-dimensional cases, but at the same
time show that these properties applicable to the QRT of bipartite
entanglement are too strong to be satisfied in known QRTs such as magic states
[17]. Motivated by this issue, we introduce a concept of consistent resource
measures, which provide quantification of quantum resources without
contradicting the rate of asymptotic state conversion. We prove that the
uniqueness inequality also holds for the consistent resource measure and
observe that this uniqueness inequality is more widely applicable than the
uniqueness inequality previously proved through the axiomatic approach.
Moreover, we show that the regularized relative entropy of resource serves as
a consistent resource measure, generalizing the existing results in reversible
QRTs [32].

These formulations and results establish a framework of general QRTs that are
applicable even to physically motivated restrictions on quantum operations
whose analysis is mathematically intractable. We here point out that whether a
given QRT is physically motivated or not is indeed a subjective issue,
depending on what operation we assume to be free and what quantum property we
regard as a resource. Remarkably, one can use our general results on
manipulation and quantification of quantum resources regardless of whether the
resources are physical or not from the subjective viewpoint; in particular,
our general results on QRTs are applicable to whatever QRTs that one may
consider to be “physical”. The significance of our contribution is the full
generality of the results, which opens a way to stop the controversy over what
QRTs are physically meaningful, so that anyone can suitably enjoy the benefit
of the general framework of QRTs for quantitatively analyzing quantum
properties of interest. These results lead to a theoretical foundation for
further understandings of quantum-mechanical phenomena through a systematic
approach based on QRTs.

The rest of this paper is organized as follows. In Sec. 2, we recall
descriptions of infinite-dimensional quantum mechanics and provide a framework
of general QRTs. In Sec. 3, we investigate maximally resourceful states and
free states in general QRTs. In Sec. 4, we analyze manipulation of quantum
states in general QRTs, especially, asymptotic state conversion. In Sec. 5, we
focus on the distillation of a resource from a quantum state and the formation
of a quantum state from a resource, and prove the uniqueness inequality. In
Sec. 6, we investigate the quantification of a resource, introducing and
analyzing a consistent resource measure. Our conclusion is given in Sec. 7.

##  2 Preliminaries

We provide preliminaries to quantum resource theories (QRTs) that we analyze
in this paper. In Sec. 2.1, we present notations on describing quantum
mechanics on infinite-dimensional quantum systems that QRTs in this paper
cover. In Sec. 2.2, we recall a formulation of QRTs. In Sec. 2.3, we recall
the definition of a preorder, and introduce a preorder introduced by free
operations in QRTs. The readers who are interested in QRTs on finite-
dimensional quantum systems and are familiar with finite-dimensional quantum
mechanics can skip Sec. 2.1, while we may use notions summarized in Sec. 2.1
to show our result in Sec. 3.1.

###  2.1 Quantum Mechanics on Infinite-Dimensional Quantum Systems

We provide mathematical notations of quantum mechanics that cover infinite-
dimensional quantum systems. Notice that some inherent properties of quantum
mechanics, such as non-Gaussianity [25, 26, 27, 28], are easier to formulate
on an infinite-dimensional quantum system than its approximation by a finite-
dimensional quantum system. As for proofs of mathematical facts that we use in
the following, see, e.g., Refs. [50, 51].

To represent a (finite- and infinite-dimensional) quantum system, we use a
complex Hilbert space $\mathcal{H}$, i.e., a complex inner product space that
is also a complete metric space with respect to the distance function induced
by the inner product. We represent a multipartite system as a tensor product
of the Hilbert spaces representing its subsystems. We may write an orthonormal
basis (i.e., a complete orthonormal system) of $\mathcal{H}$ as

$B_{\mathcal{H}}\coloneqq{\left\\{\Ket{k}\right\\}}_{k}.$ (1)

In cases where $\mathcal{H}$ represents a $D$-dimensional system,
$B_{\mathcal{H}}$ is a finite set with cardinality $D$, while
$B_{\mathcal{H}}$ can be an uncountable set in this paper.

We use a subset of operators on $\mathcal{H}$, in particular, the von Neumann
algebra, to describe quantum mechanics on the system represented by
$\mathcal{H}$, since $\mathcal{H}$ can be infinite-dimensional. Let
$\mathcal{L}\left(\mathcal{H}\right)$ denote the set of linear operators on
$\mathcal{H}$. Let
$\mathcal{B}\left(\mathcal{H}\right)\subset\mathcal{L}\left(\mathcal{H}\right)$
denote the set of bounded operators, that is, for any
$A\in\mathcal{B}\left(\mathcal{H}\right)$, the operator norm
$\left\|A\right\|_{\infty}\coloneqq\sup\left\\{{\left\|A\Ket{v}\right\|}_{\mathcal{H}}:\Ket{v}\in\mathcal{H},\,{\left\|\Ket{v}\right\|}_{\mathcal{H}}\leqq
1\right\\}$ is bounded, where ${\left\|\cdot\right\|}_{\mathcal{H}}$ denotes
the norm induced by the inner product of $\mathcal{H}$.

To define trace-class operators, we use the trace defined as
$\operatorname{Tr}T\coloneqq\sum_{\Ket{k}\in
B_{\mathcal{H}}}\Braket{k}{T}{k}$, where $B_{\mathcal{H}}$ is an orthonormal
basis of $\mathcal{H}$ defined in (1), each term on the right-hand side
denotes the inner product of $T\Ket{k}$ and $\Ket{k}$ on $\mathcal{H}$, and if
$B_{\mathcal{H}}$ is an infinite set, the summation on the right-hand side
means the limit of a net, i.e., a generalization of sequence. While a sequence
$(a_{n}:n\in\mathbb{N})$ is indexed by a natural number, that is, a countably
infinite and totally ordered set, a net $a_{n}$ is indexed by $n$ in a
directed set, a generalization of totally ordered sets, while this directed
set can be uncountable. In the definition of the trace, the net is indexed by
any finite subset $B_{\mathcal{H}}^{\prime}\subset B_{\mathcal{H}}$ and
defined as $\sum_{\Ket{k}\in B_{\mathcal{H}}^{\prime}}\Braket{k}{T}{k}$, which
approaches to the limit $\operatorname{Tr}T$ as $B_{\mathcal{H}}^{\prime}$
gets larger. Note that $\operatorname{Tr}T$ is independent of the choice of
$B_{\mathcal{H}}$. Let
$\mathcal{T}\left(\mathcal{H}\right)\subset\mathcal{B}\left(\mathcal{H}\right)$
denote the set of trace-class operators; that is, for any
$T\in\mathcal{T}\left(\mathcal{H}\right)$, we have finite and hence well-
defined $\operatorname{Tr}\left|T\right|$, where
$\left|T\right|\coloneqq\sqrt{T^{\dagger}T}$, and $T^{\dagger}$ denotes the
conjugation of $T$. For any $T\in\mathcal{T}\left(\mathcal{H}\right)$, the
trace norm of $T$ is defined as

$\left\|T\right\|_{1}\coloneqq\operatorname{Tr}\left|T\right|.$ (2)

To define the von Neumann algebra, we need to discuss convergence of bounded
operators mathematically. To discuss convergence of bounded operators, we need
a topology defined for $\mathcal{B}\left(\mathcal{H}\right)$, and we use the
ultraweak operator topology. The ultraweak operator topology of
$\mathcal{B}\left(\mathcal{H}\right)$ is a topology where any sequence
$A_{1},A_{2},\ldots\in\mathcal{B}\left(\mathcal{H}\right)$, or more generally,
any net $A_{i}$, converges to $A$ if and only if
$\operatorname{Tr}\left[TA_{i}\right]$ converges to
$\operatorname{Tr}\left[TA\right]$ for any
$T\in\mathcal{T}\left(\mathcal{H}\right)$. A von Neumann algebra $\mathcal{M}$
on $\mathcal{B}\left(\mathcal{H}\right)$ is a subset of
$\mathcal{B}\left(\mathcal{H}\right)$ (or
$\mathcal{B}\left(\mathcal{H}\right)$ itself) that contains the identity
operator $\mathbbm{1}$ on $\mathcal{H}$, is closed under linear combination,
product, and conjugation, and is also closed in terms of the ultraweak
operator topology.

A noncommutative von Neumann algebra can be used for describing a quantum
system, while a commutative von Neumann algebra for a classical system. To
describe quantum mechanics on $\mathcal{H}$, we use a set of operators
represented as a von Neumann algebra $\mathcal{M}$ on
$\mathcal{B}\left(\mathcal{H}\right)$. For example, for any finite-dimensional
Hilbert space $\mathcal{H}$, the algebra of all the linear operators
$\mathcal{L}\left(\mathcal{H}\right)=\mathcal{B}\left(\mathcal{H}\right)$ is a
von Neumann algebra, which suffices to describe the finite-dimensional quantum
mechanics. More generally, for any $\mathcal{H}$ that can be infinite-
dimensional, the algebra of all the bounded operators
$\mathcal{B}\left(\mathcal{H}\right)$ is a von Neumann algebra. In this paper,
a system $\mathcal{H}$ is always accompanied with a set of operators
$\mathcal{M}$ therein, where we may implicitly consider
$\mathcal{M}=\mathcal{B}\left(\mathcal{H}\right)$ unless stated otherwise.

Given that a quantum state associates a measurement of an observable with a
probability of a measurement outcome, we introduce a quantum state using a
linear functional from an operator to a scalar. In particular, for a system
$\mathcal{H}$ with $\mathcal{M}$, a state is defined as a linear functional
$f_{\psi}:\mathcal{M}\to\mathbb{C}$ that is positive semidefinite
$f_{\psi}\left(M^{\dagger}M\right)\geqq 0,\,\forall M\in\mathcal{M}$,
satisfies the normalization condition $f_{\psi}\left(\mathbbm{1}\right)=1$,
and is also normal, i.e., continuous in terms of the ultraweak operator
topology. We require this continuity in order for $f_{\psi}$ to be in the
predual $\mathcal{M}_{\ast}$ of $\mathcal{M}$, i.e., the space whose dual
${\left(\mathcal{M}_{\ast}\right)}^{\ast}$ equals (can be identified with)
$\mathcal{M}$. Given this duality and under the condition of
$\mathcal{M}=\mathcal{B}\left(\mathcal{H}\right)$, we identify $f_{\psi}$ with
the operator $\psi\in\mathcal{T}\left(\mathcal{H}\right)$ that satisfies for
any $M\in\mathcal{M}$

$\operatorname{Tr}\left[\psi M\right]=f_{\psi}\left(M\right).$ (3)

Note that we have one-to-one correspondence between $f_{\psi}$ and $\psi$ if
$\mathcal{M}=\mathcal{B}\left(\mathcal{H}\right)$. This operator $\psi$ is the
density operator representing the state $f_{\psi}$, and let
$\mathcal{D}\left(\mathcal{H}\right)\coloneqq\left\\{\psi\in\mathcal{T}\left(\mathcal{H}\right):\psi\geqq
0,\operatorname{Tr}\psi=1\right\\}$ denote the set of density operators on
$\mathcal{H}$ with $\mathcal{M}=\mathcal{B}\left(\mathcal{H}\right)$. For
simplicity, we may call $\psi$ a quantum state, rather than $f_{\psi}$.

We introduce a quantum channel on a system $\mathcal{H}$ with
$\mathcal{M}=\mathcal{B}\left(\mathcal{H}\right)$ in the Heisenberg picture as
a completely positive and unital linear map on $\mathcal{M}$, which
correspondingly yields the definition of the channel in the Schrödinger
picture as a completely positive and trace-preserving linear map of density
operators on $\mathcal{H}$. Given two systems
$\mathcal{H}^{\left(\mathrm{in}\right)}$ with
$\mathcal{M}^{\left(\mathrm{in}\right)}$ and
$\mathcal{H}^{\left(\mathrm{out}\right)}$ with
$\mathcal{M}^{\left(\mathrm{out}\right)}$ representing the spaces of the input
and the output respectively, a channel
$\tilde{\mathcal{E}}:\mathcal{M}^{\left(\mathrm{out}\right)}\to\mathcal{M}^{\left(\mathrm{in}\right)}$
in the Heisenberg picture is defined as a linear map that is completely
positive

$\displaystyle\sum_{j,k=0}^{n-1}M_{j}^{\left(\mathrm{in}\right)}{}^{\dagger}\tilde{\mathcal{E}}\left(M_{j}^{\left(\mathrm{out}\right)}{}^{\dagger}M_{k}^{\left(\mathrm{out}\right)}\right)M_{k}^{\left(\mathrm{in}\right)}\geqq
0,$ $\displaystyle\forall
M_{0}^{\left(\mathrm{in}\right)},\ldots,M_{n-1}^{\left(\mathrm{in}\right)}\in\mathcal{M}^{\left(\mathrm{in}\right)},$
$\displaystyle\forall
M_{0}^{\left(\mathrm{out}\right)},\ldots,M_{n-1}^{\left(\mathrm{out}\right)}\in\mathcal{M}^{\left(\mathrm{out}\right)},$
$\displaystyle\forall n\in\mathbb{N},$ (4)

unital
$\tilde{\mathcal{E}}\left(\mathbbm{1}^{\left(\mathrm{out}\right)}\right)=\mathbbm{1}^{\left(\mathrm{in}\right)}$,
and normal, i.e., continuous in terms of the ultraweak operator topologies of
$\mathcal{M}^{\left(\mathrm{out}\right)}$ and
$\mathcal{M}^{\left(\mathrm{in}\right)}$, where
$\mathbbm{1}^{\left(\mathrm{out}\right)}$ and
$\mathbbm{1}^{\left(\mathrm{in}\right)}$ are the identity operators on
$\mathcal{H}^{\left(\mathrm{out}\right)}$ and
$\mathcal{H}^{\left(\mathrm{in}\right)}$, respectively. In the same way as the
identification (3) of a functional $f_{\psi}$ of a state with the density
operator $\psi$ of the state, under the conditions of
$\mathcal{M}^{\left(\mathrm{in}\right)}=\mathcal{B}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\right)$
and
$\mathcal{M}^{\left(\mathrm{out}\right)}=\mathcal{B}\left(\mathcal{H}^{\left(\mathrm{out}\right)}\right)$,
we identify a channel
$\tilde{\mathcal{E}}:\mathcal{M}^{\left(\mathrm{out}\right)}\to\mathcal{M}^{\left(\mathrm{in}\right)}$
in the Heisenberg picture with the channel
$\mathcal{E}:\mathcal{T}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\right)\to\mathcal{T}\left(\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
in the Schrödinger picture that satisfies for any
$M^{\left(\mathrm{out}\right)}\in\mathcal{M}^{\left(\mathrm{out}\right)}$

$\operatorname{Tr}\left[\mathcal{E}\left(\psi\right)M^{\left(\mathrm{out}\right)}\right]=\left(f_{\psi}\circ\tilde{\mathcal{E}}\right)\left(M^{\left(\mathrm{out}\right)}\right),$
(5)

where $\psi$ and $f_{\psi}$ are related as (3). Note that $\mathcal{E}$ is a
completely positive and trace-preserving (CPTP) linear map by definition, and
if
$\mathcal{M}^{\left(\mathrm{in}\right)}=\mathcal{B}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\right)$
and
$\mathcal{M}^{\left(\mathrm{out}\right)}=\mathcal{B}\left(\mathcal{H}^{\left(\mathrm{out}\right)}\right)$,
we have one-to-one correspondence between $\tilde{\mathcal{E}}$ and
$\mathcal{E}$.

The set of channels from an input system
$\mathcal{H}^{\left(\mathrm{in}\right)}$ with
$\mathcal{M}^{\left(\mathrm{in}\right)}$ to an output system
$\mathcal{H}^{\left(\mathrm{out}\right)}$ with
$\mathcal{M}^{\left(\mathrm{out}\right)}$ is denoted by
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$,
which we may write $\mathcal{C}\left(\mathcal{H}\right)$ if
$\mathcal{H}=\mathcal{H}^{\left(\mathrm{in}\right)}=\mathcal{H}^{\left(\mathrm{out}\right)}$.
In this paper, we use the Schrödinger picture with
$\mathcal{M}^{\left(\mathrm{in}\right)}=\mathcal{B}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\right)$
and
$\mathcal{M}^{\left(\mathrm{out}\right)}=\mathcal{B}\left(\mathcal{H}^{\left(\mathrm{out}\right)}\right)$;
that is,
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
is the set of the CPTP linear maps, while it would be possible to use the
Heisenberg picture otherwise. We represent quantum operations as channels,
while it is possible to include measurements in our formulation as channels
from a quantum input system to a classical output system.

To discuss compactness of a set of states, we need further definitions of
topologies for the set of states. A compact set in terms of some topology is a
set where for any net in the set, there exists a subnet that converges in
terms of the topology, where a subnet generalizes a subsequence in the same
way as the net generalizing the sequence. A closed set in terms of some
topology is a set where for any net that converges in terms of this topology,
its limit point is in the set. Note that a compact set in a Hausdorff space is
a closed set, which holds in our case. Several different topologies can be
defined for the set of states. The weak operator topology of
$\mathcal{T}\left(\mathcal{H}\right)$ is a topology where any net $T_{i}$ in
$\mathcal{T}(\mathcal{H})$ converges to $T$ if and only if
$\operatorname{Tr}\left[T_{i}A\right]$ converges to
$\operatorname{Tr}\left[TA\right]$ for any
$A\in\mathcal{B}\left(\mathcal{H}\right)$. The trace norm topology of
$\mathcal{T}\left(\mathcal{H}\right)$ is a topology where any net $T_{i}$ in
$\mathcal{T}(\mathcal{H})$ converges to $T$ if and only if
${\left\|T_{i}-T\right\|}_{1}$ converges to $0$. The trace norm topology is
stronger than the ultraweak operator topology, and the ultraweak operator
topology is stronger than the weak operator topology, while these topologies
are the same in the finite-dimensional case. In general, whether a set is
compact or not may depend on the choice of the topology, but we show in
Appendix A that the compactness of a set of states in terms of these
topologies are equivalent. Thus, it suffices to consider the trace norm
topology when we discuss compactness of a set of states. Then, in the trace
norm topology, compactness is equivalent to sequential compactness, and hence
we may use a sequence rather than a net to discuss compactness of a set of
states. Note that if $\mathcal{H}$ is finite-dimensional,
$\mathcal{D}\left(\mathcal{H}\right)$ is compact, while
$\mathcal{D}\left(\mathcal{H}\right)$ for an infinite-dimensional system is
not compact in terms of the trace norm topology.

To discuss convergence and compactness of channels, we need a topology defined
for
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$,
and we use the bounded weak (BW) topology. The bounded weak topology of
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
is the weakest topology such that for any
$f_{\psi}\in\mathcal{M}^{\left(\mathrm{in}\right)}_{\ast}$ and
$M^{\left(\mathrm{out}\right)}\in\mathcal{M}^{\left(\mathrm{out}\right)}$, a
map
$\mathcal{S}_{\psi,M^{\left(\mathrm{out}\right)}}:\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)\to\mathbb{C}$
given by
$\mathcal{S}_{\psi,M^{\left(\mathrm{out}\right)}}\left(\mathcal{E}\right)=\left(f_{\psi}\circ\tilde{\mathcal{E}}\right)\left(M^{\left(\mathrm{out}\right)}\right)$
is continuous, where $\mathcal{E}$ and $\tilde{\mathcal{E}}$ are related as
(5). Note that if $\mathcal{H}^{\left(\mathrm{in}\right)}$ and
$\mathcal{H}^{\left(\mathrm{out}\right)}$ are finite-dimensional,
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
is compact, while
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
for infinite-dimensional systems is not compact in terms of the BW topology.

###  2.2 Framework of Quantum Resource Theories

In this section, we provide a formulation of quantum resource theories (QRTs)
starting from free operations with minimal assumptions. In the definition, we
consider a compact set of the free operations. We also present justification
of the compactness by examples from the perspective of indistinguishability.
To represent the state set of interest in a QRT, e.g., the set of pure states
on finite-dimensional $\mathcal{H}$, we consider a compact set of quantum
states chosen as desired

$\mathcal{S}\left(\mathcal{H}\right)\subseteq\mathcal{D}\left(\mathcal{H}\right).$
(6)

Note that the quantum system $\mathcal{H}$ can be infinite-dimensional as we
have introduced in Sec. 2.1.

Free operations in our formulation are introduced as follows [1]. Let
$\mathcal{O}$ be a mapping that takes two quantum systems
$\mathcal{H}^{\left(\mathrm{in}\right)}$ and
$\mathcal{H}^{\left(\mathrm{out}\right)}$ and outputs a compact set of
completely positive and trace-preserving (CPTP) maps from
$\mathcal{S}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\right)$ to
$\mathcal{S}\left(\mathcal{H}^{\left(\mathrm{out}\right)}\right)$. This set is
denoted by

$\mathcal{O}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)\subseteq\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right).$
(7)

A map contained in
$\mathcal{O}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
is called a free operation from $\mathcal{H}^{\left(\mathrm{in}\right)}$ to
$\mathcal{H}^{\left(\mathrm{out}\right)}$. If the input space and output space
are the same quantum system $\mathcal{H}$, we write the set of free operations
from $\mathcal{H}$ to $\mathcal{H}$ as
$\mathcal{O}\left(\mathcal{H}\right)\subseteq\mathcal{C}\left(\mathcal{H}\right)$.
We consider a compact set because two arbitrarily close CPTP maps are
indistinguishable by any protocol in a task of channel discrimination [52], as
we will discuss below by examples. We assume that $\mathcal{O}$ satisfies the
following axioms of QRTs:

  1. 1.

Let $\mathcal{H}_{1}$, $\mathcal{H}_{2}$ and $\mathcal{H}_{3}$ be arbitrary
quantum systems. For any
$\mathcal{M}\in\mathcal{O}\left(\mathcal{H}_{1}\to\mathcal{H}_{2}\right)$ and
$\mathcal{N}\in\mathcal{O}\left(\mathcal{H}_{2}\to\mathcal{H}_{3}\right)$, it
holds that
$\mathcal{N}\circ\mathcal{M}\in\mathcal{O}\left(\mathcal{H}_{1}\to\mathcal{H}_{3}\right)$,
where $\circ$ represents the composition.

  2. 2.

Let $\mathcal{H}^{\left(\mathrm{in}\right)}_{1}$,
$\mathcal{H}^{\left(\mathrm{out}\right)}_{1}$,
$\mathcal{H}^{\left(\mathrm{in}\right)}_{2}$ and
$\mathcal{H}^{\left(\mathrm{out}\right)}_{2}$ be arbitrary quantum systems.
For any
$\mathcal{M}\in\mathcal{O}(\mathcal{H}^{\left(\mathrm{in}\right)}_{1}\to\mathcal{H}^{\left(\mathrm{out}\right)}_{1})$
and
$\mathcal{N}\in\mathcal{O}(\mathcal{H}^{\left(\mathrm{in}\right)}_{2}\to\mathcal{H}^{\left(\mathrm{out}\right)}_{2})$,
it holds that
$\mathcal{M}\otimes\mathcal{N}\in\mathcal{O}(\mathcal{H}^{\left(\mathrm{in}\right)}_{1}\otimes\mathcal{H}^{(\textrm{in})}_{2}\to\mathcal{H}^{\left(\mathrm{out}\right)}_{1}\otimes\mathcal{H}^{(\textrm{out})}_{2})$.

  3. 3.

Let $\mathcal{H}$ be an arbitrary quantum system. Then, it holds that
$\operatorname{id}\in\mathcal{O}\left(\mathcal{H}\right)$, where
$\operatorname{id}$ is an identity map.

  4. 4.

Let $\mathcal{H}$ be an arbitrary quantum system. Then, it holds that
$\operatorname{Tr}\in\mathcal{O}\left(\mathcal{H}\to\mathbb{C}\right)$, where
$\operatorname{Tr}$ is the trace. Note that due to the above conditions, it is
necessary that the partial trace is also free.

The meanings of these axioms are as follows:

  1. 1.

We always have access to free operations and can use free operations as many
times as necessary.

  2. 2.

We can arbitrarily apply free operations to a quantum system regardless of
what free operations are applied to another quantum system.

  3. 3.

Doing nothing is free.

  4. 4.

Ignorance is free.

######  Remark 1 (Operations Not Satisfying the Axioms).

There can be classes of operations that do not satisfy the axioms stated
above. For example, Refs. [53] and [32] consider $\epsilon$-resource non-
generating operations. However, the composition of two $\epsilon$-resource
non-generating operations is not necessarily an $\epsilon$-resource non-
generating operation, which implies the set of $\epsilon$-resource non-
generating operations does not satisfy the first axiom. Hence, we do not
employ this class of operations as free operations since they are not free to
use multiple times. In addition, Ref. [54] considers separability preserving
(SEPP) operations. However, the set of SEPP operations is not closed under
tensor product, and hence does not satisfy the second axiom. We do not use
these operations as free operations since they are not free to apply to
multiple quantum systems simultaneously.

In the definition above, we use a compact set as the set of free operations.
Some classes of operations that are conventionally used as free operations do
not satisfy this compactness, such as local quantum operations and classical
communication (LOCC) in the QRT of bipartite entanglement [55]. However, in
this case, we take a position that the closure of LOCC, i.e., a compact
superset of LOCC, can be considered to be free in the sense that any channel
in the closure of LOCC is indistinguishable from a channel implementable in
the setting of LOCC, as discussed in Example 1. In the same way, Example 2
shows that we conventionally consider any unitary transformation to be
implementable by the Clifford+$T$ gate set in the sense that any unitary can
be approximated with arbitrary precision by this gate set. Note that the
compactness of the set of free operations is essential for guaranteeing the
existence of maximally resourceful states as we will see in Sec. 3.1.

######  Example 1 (LOCC and Closure of LOCC).

In the case of the QRT of entanglement, LOCC is conventionally considered to
be physically implementable operations, but our formulation of QRTs may use
the closure of LOCC in this case as a compact set of free operations instead
of LOCC. In particular, let
$\mathcal{O}_{\mathrm{LOCC}}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\textrm{out}}\right)$
be the set of LOCC from $\mathcal{H}^{\left(\mathrm{in}\right)}$ to
$\mathcal{H}^{\left(\mathrm{out}\right)}$. It is known that
$\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)$ is
not closed; that is,
$\overline{\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)}\neq\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)$
[55]. In this case, we use
$\mathcal{O}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)=\overline{\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)}$
as the set of free operation because for any CPTP map
$\mathcal{N}\in\overline{\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)}\setminus\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)$
and any $\epsilon>0$, we can construct a CPTP map
$\tilde{\mathcal{N}}\in\mathcal{O}_{\mathrm{LOCC}}\left(\mathbb{C}^{4}\to\mathbb{C}^{4}\right)$
that is indistinguishable from $\mathcal{N}$ up to an $\epsilon$ probability
by any protocol in a task of channel discrimination [52].

In the next example, we consider a situation of universal quantum computation
where any finite-depth quantum circuit composed of a universal gate set is
implementable as the free operations.

######  Example 2.

For any finite-dimensional quantum systems
$\mathcal{H}^{\left(\mathrm{in}\right)}$ and
$\mathcal{H}^{\left(\mathrm{out}\right)}$, we define
$\mathcal{O}^{\prime}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
as the set of CPTP maps that can be realized by a finite-depth circuit
composed of the identity gate, the partial trace, the Hadamard gate $H$, the
controlled-NOT gate, the $\pi/8$ phase gate $T$, appending $\ket{0}$, and the
measurement in the computational basis. Here, a finite-depth circuit refers to
a $d$-depth circuit for some integer $d\geqq 0$, and operations conditioned on
measurement outcomes are allowed. Conventionally, combination of these
operations can be considered to be universal because
$\mathcal{O}^{\prime}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
is dense in the set of all the channels
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$,
but
$\mathcal{O}^{\prime}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
is different from
$\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$
since a finite number of these gates cannot exactly implement qubit rotation
at an arbitrary angle [52, 56]. In this case, our framework may use the
closure of the set of operations implemented by all the $d$-depth circuits for
any $d$ as the set of the free operations; that is, we take the set of free
operations in this case as the set of all the channels
$\mathcal{O}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)=\mathcal{C}\left(\mathcal{H}^{\left(\mathrm{in}\right)}\to\mathcal{H}^{\left(\mathrm{out}\right)}\right)$.

We do not assume the convexity of the set of free operations in our framework.
Convex QRTs are a class of QRTs where the set of free operations is convex.
For instance, the QRT of bipartite entanglement [55], the QRT of coherence
[48] and the QRT of magic states [17] are known as convex QRTs. We can achieve
a convex combination of operations using classical randomness. In general,
randomness is regarded as a resource [57, 58], and randomness generation [59]
is indeed a promising application of noisy intermediate-scale quantum (NISQ)
devices [60]; therefore, we also consider non-convex QRTs in our framework,
such as the following example.

######  Example 3 (Non-Convex QRT).

The QRT of non-Markovianity [29, 30] is known as a non-convex QRT, where the
set of free operations is not convex.

###  2.3 Preorder in Quantum Resource Theories

In this section, we recall the definition of a preorder in order theory, and
provide the definition of the preorder of resourcefulness of quantum states
introduced by free operations.

Suppose that $S$ is a set and $\succeq$ is a binary relation on $S$. Then, the
relation $\succeq$ is called a preorder if it holds that

$\displaystyle a$ $\displaystyle\succeq a,$ (8) $\displaystyle a$
$\displaystyle\succeq b,b\succeq c\Rightarrow a\succeq c$ (9)

for all $a,b,c\in S$. This preorder introduces maximal elements and minimal
elements in the set $S$. If $a\in S$ satisfies

$b\succeq a\Rightarrow a\succeq b$ (10)

for all $b\in S$, $a$ is called a maximal element. Similarly, if $a\in S$
satisfies

$a\succeq b\Rightarrow b\succeq a$ (11)

for all $b\in S$, $a$ is called a minimal element. Intuitively, an element
$a\in S$ is maximal/minimal if $a$ is the largest/smallest among all the
elements that can be compared with $a$.

For any quantum system $\mathcal{H}$, free operations introduce a preorder
$\succeq$ on $\mathcal{S}\left(\mathcal{H}\right)$. This preorder is
introduced in terms of the exact one-shot state conversion under free
operations. Given two states
$\phi,\psi\in\mathcal{S}\left(\mathcal{H}\right)$, the exact one-shot state
conversion from $\phi$ to $\psi$ is a task of transforming a single $\phi$
exactly into a single $\psi$ by a free operation
$\mathcal{N}\in\mathcal{O}\left(\mathcal{H}\right)$. Formally, we write

$\phi\succeq\psi$ (12)

if there exists a free operation
$\mathcal{N}\in\mathcal{O}\left(\mathcal{H}\right)$ such that

$\mathcal{N}\left(\phi\right)=\psi.$ (13)

This relation $\succeq$ is indeed a preorder because it holds that

$\displaystyle\phi$ $\displaystyle\succeq\phi,$ (14) $\displaystyle\phi$
$\displaystyle\succeq\psi,\psi\succeq\sigma\Rightarrow\phi\succeq\sigma$ (15)

for any states $\phi,\psi,\sigma\in\mathcal{S}\left(\mathcal{H}\right)$.

With respect to this preorder, two states
$\phi,\psi\in\mathcal{S}\left(\mathcal{H}\right)$ are said to be equivalent if
both $\phi\succeq\psi$ and $\phi\preceq\psi$ hold. If $\phi$ and $\psi$ are
equivalent, we write

$\phi\sim\psi.$ (16)

##  3 Maximally Resourceful States and Free States

In this section, we analyze properties of maximally resourceful states and
free states in general quantum resource theories (QRTs). In Sec. 3.1, we
provide the definition of maximally resourceful states based on the preorder
mentioned in Sec. 2.3, and prove the existence of maximally resourceful states
in QRTs under our formulation in Sec. 2.2. In Sec. 3.2, we provide the
definition of free states, and give a condition under which a maximally
resourceful state is not free.

###  3.1 Maximally Resourceful States

In this section, we analyze maximally resourceful states defined by the
preorder of resourcefulness of states mentioned in Sec. 2.3. The existence of
maximally resourceful states is not trivial in general, although it is desired
for quantification of the resource. We prove the existence of maximally
resourceful states in any QRT that satisfies the four axioms and the
compactness given in Sec. 2.2.

The preorder of states introduces maximal elements in the set of states in
terms of order theory. Given a quantum system $\mathcal{H}$, we let
$\mathcal{G}\left(\mathcal{H}\right)$ denote the set of the maximal states of
$\mathcal{S}\left(\mathcal{H}\right)$ in terms of the preorder defined as
(12), that is,

$\mathcal{G}\left(\mathcal{H}\right)\coloneqq\left\\{\phi\in\mathcal{S}\left(\mathcal{H}\right):\forall\psi\in\mathcal{S}\left(\mathcal{H}\right),\psi\succeq\phi\Rightarrow\phi\succeq\psi\right\\}.$
(17)

The elements of $\mathcal{G}\left(\mathcal{H}\right)$ are called maximally
resourceful states. Note that there may be several non-equivalent maximally
resourceful states, which are not comparable with each other. Here, we recall
two QRTs that have two or more non-equivalent maximally resourceful states.

######  Example 4 (QRTs with Non-Equivalent Maximally Resourceful States).

The first example is the QRT of magic for qutrits [17]. In this QRT, there
exist two non-equivalent maximally resourceful states, which are called the
Norrell state and the Strange state. The second example is the QRT of
coherence with physically incoherent operations (PIO) [10]. In the QRT of
coherence with PIO, a free operation cannot change diagonal elements of a
quantum state represented in the standard basis. Therefore, there exist
infinitely many non-equivalent maximally resourceful states, which have
different diagonal elements from each other.

We here prove that a maximally resourceful state always exists in any QRT
satisfying axioms and the compactness of the set of states discussed in Sec.
2.2. Maximally resourceful states are regarded as a unit of resource [46, 8,
61]. For example, in the QRT of bipartite entanglement, the amount of
entanglement of the Bell state is defined as one ebit. Therefore, it is
crucial for QRTs to have a maximally resourceful state. In general, whether a
maximally resourceful state exists is not obvious. For example, a maximally
entangled state does not necessarily exist in a QRT of bipartite entanglement
for an infinite-dimensional system with a non-compact set of free operations
such as LOCC while a unique maximally entangled state exists for a finite-
dimensional system. Theorem 1 shows that for any given state, there exists a
maximally resourceful state that is more resourceful than the state, which
ensures the existence of maximally resourceful states in our framework.

######  Theorem 1 (Existence of a Maximally Resourceful State).

Let $\mathcal{H}$ is a quantum system. For any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$, there exists a state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ that upper-bounds $\psi$; that
is, $\psi\preceq\phi$.

###### Proof.

It is known that a compact space $\mathcal{X}$ with a preorder $\succeq$ has a
maximal element if the upper closure
$\mathcal{U}_{x}\coloneqq\left\\{y\in\mathcal{X}|x\preceq y\right\\}$ is
closed for any $x\in\mathcal{X}$ [62] (Proposition VI-1.6.(i)). Thus, it
suffices to show
$\mathcal{U}_{\psi}\coloneqq\left\\{\phi\in\mathcal{S}\left(\mathcal{H}\right)|\psi\preceq\phi\right\\}$
is closed in terms of the weak operator topology, or equivalently closed in
terms of the trace norm topology due to Lemma 28 in Appendix A. We take a
sequence ${(\phi_{n})}_{n\in\mathbb{N}}$ in $\mathcal{U}_{\psi}$ convergent to
$\phi\in\mathcal{S}\left(\mathcal{H}\right)$ in terms of the trace norm
topology and prove $\phi\in\mathcal{U}_{\psi}$. By the definition of the
preorder $\preceq$, for each $n\in\mathbb{N}$, there exists a free operation
$\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}\right)$ such that
$\psi=\mathcal{N}_{n}(\phi_{n})$. Since $\mathcal{O}\left(\mathcal{H}\right)$
is compact in terms of the bounded weak (BW) topology, there exists a subnet
${\left(\mathcal{N}_{n(i)}\right)}_{i\in I}$ convergent to some
$\mathcal{N}\in\mathcal{O}\left(\mathcal{H}\right)$ with respect to the BW
topology. In the following, we show $\psi=\mathcal{N}(\phi)$ to prove the
theorem.

Take an arbitrary $\epsilon>0$ and an arbitrary
$A\in\mathcal{B}\left(\mathcal{H}\right)\setminus\\{0\\}$, which satisfies
${\left\|A\right\|}_{\infty}>0$. Since $\mathcal{S}\left(\mathcal{H}\right)$
is compact in terms of the trace norm topology, there exists a finite subset
$\left\\{\chi_{k}:k\in\left\\{1,\ldots,N_{\epsilon}\right\\}\right\\}$ of
$\mathcal{S}\left(\mathcal{H}\right)$ such that for any
$\chi\in\mathcal{S}(\mathcal{H})$

$\min_{k\in\left\\{1,\ldots,N_{\epsilon}\right\\}}{\left\|\chi-\chi_{k}\right\|}_{1}<\frac{\epsilon}{{\|A\|}_{\infty}}.$
(18)

By definition of the convergence
$\mathcal{N}_{n(i)}\xrightarrow{\mathrm{BW}}\mathcal{N}$ in terms of the BW
topology with respect to $i$, there exists $i_{\epsilon,A}\in I$ such that for
any $i\geqq i_{\epsilon,A}$

$\max_{k\in\left\\{1,\ldots,N_{\epsilon}\right\\}}\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi_{k}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\chi_{k}\right)A\right)\right|<\epsilon.$
(19)

Thus, for any $i\geqq i_{\epsilon,A}$ and any
$\chi\in\mathcal{S}(\mathcal{H})$, we have

$\displaystyle\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\chi\right)A\right)\right|$
$\displaystyle=\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi-\chi_{k}\right)A\right)+\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi_{k}\right)A\right)\right.$
$\displaystyle\quad\quad\left.-\operatorname{Tr}\left(\mathcal{N}\left(\chi-\chi_{k}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\chi_{k}\right)A\right)\right|$
$\displaystyle\leqq\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi-\chi_{k}\right)A\right)\right|+\left|\operatorname{Tr}\left(\mathcal{N}\left(\chi-\chi_{k}\right)A\right)\right|$
$\displaystyle\quad\quad+\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi_{k}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\chi_{k}\right)A\right)\right|$
$\displaystyle<\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi-\chi_{k}\right)A\right)\right|+\left|\operatorname{Tr}\left(\mathcal{N}\left(\chi-\chi_{k}\right)A\right)\right|+\epsilon,$
(20)

where $\chi_{k}$ is an element in the finite subset
$\left\\{\chi_{k}\right\\}$ of $\mathcal{S}\left(\mathcal{H}\right)$ in (18)
satisfying

${\left\|\chi-\chi_{k}\right\|}_{1}<\frac{\epsilon}{{\|A\|}_{\infty}},$ (21)

and we use (19) in the last line. With

${\left\|\mathcal{N}_{n(i)}\right\|}_{\infty}\coloneqq\sup\left\\{{\left\|\mathcal{N}_{n(i)}(T)\right\|}_{1}:T\in\mathcal{T}(\mathcal{H}),{\|T\|_{1}}\leqq
1\right\\}$ (22)

denoting the operator norm of the linear map
$\mathcal{N}_{n(i)}:\mathcal{T}(\mathcal{H})\to\mathcal{T}(\mathcal{H})$, we
have

$\displaystyle\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi-\chi_{k}\right)A\right)\right|$
$\displaystyle\leqq{\left\|\mathcal{N}_{n(i)}\left(\chi-\chi_{k}\right)\right\|}_{1}\cdot{\left\|A\right\|}_{\infty}$
$\displaystyle\leqq{\left\|\mathcal{N}_{n(i)}\right\|}_{\infty}\cdot{\left\|\chi-\chi_{k}\right\|}_{1}\cdot{\left\|A\right\|}_{\infty}$
$\displaystyle<1\cdot\frac{\epsilon}{{\left\|A\right\|}_{\infty}}\cdot{\left\|A\right\|}_{\infty}=\epsilon,$
(23)

where the last inequality follows from the fact that any CPTP map
$\mathcal{N}_{n(i)}$ satisfies

${\left\|\mathcal{N}_{n(i)}\right\|}_{\infty}=\sup{\left\|\mathcal{N}_{n(i)}(T)\right\|}_{1}\leqq\sup{\|T\|}_{1}\leqq
1.$ (24)

In the same way as (3.1) by substituting $\mathcal{N}_{n(i)}$ with
$\mathcal{N}$, it holds that

$\left|\operatorname{Tr}\left(\mathcal{N}\left(\chi-\chi_{k}\right)A\right)\right|<\epsilon.$
(25)

Therefore, applying (3.1) and (25) to (3.1), for any $i\geqq i_{\epsilon,A}$
and any $\chi\in\mathcal{S}(\mathcal{H})$, we have

$\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\chi\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\chi\right)A\right)\right|<\epsilon+\epsilon+\epsilon=3\epsilon.$
(26)

Consequently, for any $i\geqq i_{\epsilon,A}$, we obtain

$\displaystyle\left|\operatorname{Tr}\left(\left(\psi-\mathcal{N}\left(\phi\right)\right)A\right)\right|$
$\displaystyle=\left|\operatorname{Tr}\left(\psi
A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\phi\right)A\right)\right|$
$\displaystyle=\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\phi_{n(i)}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\phi\right)A\right)\right|$
$\displaystyle\leqq\left|\operatorname{Tr}\left(\mathcal{N}_{n(i)}\left(\phi_{n(i)}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\phi_{n(i)}\right)A\right)\right|$
$\displaystyle\quad\quad+\left|\operatorname{Tr}\left(\mathcal{N}\left(\phi_{n(i)}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\phi\right)A\right)\right|$
$\displaystyle<3\epsilon+{\left\|\mathcal{N}\right\|}_{\infty}\cdot{\left\|\phi_{n(i)}-\phi\right\|}_{1}\cdot{\left\|A\right\|}_{\infty}\to
3\epsilon,$ (27)

where the last inequality follows from (26) by substituting $\chi$ with
$\phi_{n(i)}$ and from the inequality shown in the same way as (3.1)

$\displaystyle\left|\operatorname{Tr}\left(\mathcal{N}\left(\phi_{n(i)}\right)A\right)-\operatorname{Tr}\left(\mathcal{N}\left(\phi\right)A\right)\right|$
$\displaystyle\leqq{\left\|\mathcal{N}\right\|}_{\infty}\cdot{\left\|\phi_{n(i)}-\phi\right\|}_{1}\cdot{\left\|A\right\|}_{\infty},$
(28)

and the limit in the last line in terms of $i$ yields
$\left\|\phi_{n(i)}-\phi\right\|_{1}\to 0$. Since $\epsilon>0$ and
$A\in\mathcal{B}\left(\mathcal{H}\right)\setminus\\{0\\}$ are arbitrary, this
shows $\psi=\mathcal{N}(\phi)$. ∎

######  Remark 2.

In a similar manner, we can prove that the set of minimal elements

$\left\\{\phi\in\mathcal{S}\left(\mathcal{H}\right):\forall\psi\in\mathcal{S}\left(\mathcal{H}\right),\phi\succeq\psi\Rightarrow\psi\succeq\phi\right\\}$
(29)

is not empty as well. The set of minimal elements is considered as the set of
the least resourceful states. If the set of free states, which is defined in
the following section, is not empty, the set of minimal set is identical to
the set of free states. However, the set of minimal elements and that of free
states may be different because the set of free states can be empty for some
QRTs as we will show in Example 5.

###  3.2 Free States

In this section, we analyze properties of free states. A free state is defined
as a state that can be generated from any other state by a free operation. Let
$\mathcal{F}\left(\mathcal{H}\right)$ denote the set of free states; that is,

$\displaystyle\mathcal{F}\left(\mathcal{H}\right)\coloneqq$
$\displaystyle\Big{\\{}\psi\in\mathcal{S}\left(\mathcal{H}\right):\forall\mathcal{H}^{\prime},\forall\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right),$
(30)
$\displaystyle\exists\mathcal{N}\in\mathcal{O}\left(\mathcal{H}^{\prime}\to\mathcal{H}\right)\,\mathrm{s.t.}\,\psi=\mathcal{N}\left(\phi\right)\Big{\\}}.$

A state $\psi\in\mathcal{S}(\mathcal{H})\setminus\mathcal{F}(\mathcal{H})$
that is not free is called a resourceful state or a resource state. Since
$\operatorname{Tr}$ is a free operation, the set of free states is equal to
the set of states that can be generated from the scalar
$1\in\mathcal{S}\left(\mathbb{C}\right)$ as shown in the following
proposition.

######  Proposition 2.

Let $\mathcal{H}$ be a quantum system. Then, it holds that

$\displaystyle\mathcal{F}\left(\mathcal{H}\right)=$ (31)
$\displaystyle\Big{\\{}\psi\in\mathcal{S}\left(\mathcal{H}\right):\exists\mathcal{N}\in\mathcal{O}\left(\mathbb{C}\to\mathcal{H}\right)\,\mathrm{s.t.}\,\psi=\mathcal{N}\left(1\right)\Big{\\}}.$

###### Proof.

By the definition (30) of $\mathcal{F}\left(\mathcal{H}\right)$, it trivially
holds that

$\mathcal{F}\left(\mathcal{H}\right)\subseteq\Big{\\{}\psi\in\mathcal{S}\left(\mathcal{H}\right):\exists\mathcal{N}\in\mathcal{O}\left(\mathbb{C}\to\mathcal{H}\right)\,\mathrm{s.t.}\,\psi=\mathcal{N}\left(1\right)\Big{\\}}.$
(32)

To show the converse inclusion, assume that

$\psi\in\Big{\\{}\psi\in\mathcal{S}\left(\mathcal{H}\right):\exists\mathcal{N}\in\mathcal{O}\left(\mathbb{C}\to\mathcal{H}\right)\,\mathrm{s.t.}\,\psi=\mathcal{N}\left(1\right)\Big{\\}}.$
(33)

Let $\mathcal{N}\in\mathcal{O}\left(\mathbb{C}\to\mathcal{H}\right)$ be a free
operation such that $\psi=\mathcal{N}\left(1\right)$. Consider an arbitrary
quantum system $\mathcal{H}^{\prime}$ and an arbitrary state
$\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$. Since
$\operatorname{Tr}\in\mathcal{O}\left(\mathcal{H}^{\prime}\to\mathbb{C}\right)$,
it holds that

$\psi=\mathcal{N}\circ\operatorname{Tr}\left(\phi\right).$ (34)

Therefore, $\psi\in\mathcal{F}\left(\mathcal{H}\right)$, which yields the
conclusion. ∎

The set of free states $\mathcal{F}\left(\mathcal{H}\right)$ may be empty for
some $\mathcal{H}$ while the set of minimal elements defined in (29) is not
empty as seen in Remark 2. For example, if the set of free operations
$\mathcal{O}\left(\mathbb{C}\to\mathcal{H}\right)$ does not contain any
operation for a quantum system $\mathcal{H}$, then
$\mathcal{F}\left(\mathcal{H}\right)=\emptyset$. The following example gives a
more concrete scenario, where we take the logical 2-dimensional space of the
Gottesman-Kitaev-Preskill (GKP) code [63] as
$\mathcal{S}\left(\mathbb{C}^{2}\right)$. In this paper, to investigate
constraints and properties of QRTs in as general a setup as possible, we do
not make any assumption on whether $\mathcal{F}\left(\mathcal{H}\right)$ is
empty or not.

######  Example 5 (QRT of Non-Gaussianity on GKP Code).

The QRT of non-Gaussianity has applications to analyzing continuous-variable
quantum computation using the Gottesman-Kitaev-Preskill (GKP) code as shown in
Ref. [28]. The GKP code encodes a qubit into an infinite-dimensional
oscillator of an optical mode, and the logical 2-dimensional space can be
defined by dividing the Hilbert space of the bosonic mode into a logical qubit
and a gauge mode [64]. Gaussian operations [65] at a physical level suffice to
implement logical Clifford gates for the GKP code [63]. Suppose that
$\mathcal{S}\left(\mathbb{C}^{2}\right)$ is the set of logical states in the
logical 2-dimensional space of the GKP code. Take the quantum operations on
$\mathcal{S}\left(\mathbb{C}^{2}\right)$ implementable by the Gaussian
operations as the free operations. Any physical state of the GKP code is non-
Gaussian, and hence in this case, $\mathcal{F}$ only has the trivial element
$1$; that is, $\mathcal{F}(\mathcal{H})=\left\\{1\right\\}$ if
$\dim\mathcal{H}=1$, and $\mathcal{F}(\mathcal{H})=\emptyset$ otherwise.

The following proposition guarantees that a maximally resourceful state cannot
be a free state if a resource state exists.

######  Proposition 3.

Let $\mathcal{H}$ be a quantum system. Suppose that the set of resource state
is not empty; that is
$\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)\neq\emptyset$.
Then, it holds that

$\mathcal{G}\left(\mathcal{H}\right)\cap\mathcal{F}\left(\mathcal{H}\right)=\emptyset.$
(35)

###### Proof.

The proof is by contradiction. To prove (35), assume that
$\phi\in\mathcal{G}\left(\mathcal{H}\right)\cap\mathcal{F}\left(\mathcal{H}\right)$.
Take a resource state

$\psi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right).$
(36)

Since $\phi\in\mathcal{F}\left(\mathcal{H}\right)$, it holds that
$\psi\succeq\phi$. Then, since $\phi\in\mathcal{G}\left(\mathcal{H}\right)$,
it holds that $\phi\succeq\psi$. Therefore, $\psi$ is also a free state; that
is, $\psi\in\mathcal{F}\left(\mathcal{H}\right)$, which contradicts (36). ∎

We can observe that some properties of the set of free states
$\mathcal{F}\left(\mathcal{H}\right)$ are inherent in the set of the free
operations $\mathcal{O}\left(\mathcal{H}\right)$. The compactness of
$\mathcal{O}\left(\mathcal{H}\right)$ leads to the compact set of free states
$\mathcal{F}\left(\mathcal{H}\right)$. If
$\mathcal{O}\left(\mathcal{H}\right)$ is convex,
$\mathcal{F}\left(\mathcal{H}\right)$ is also convex.

##  4 Asymptotic State Conversion

In this section, we characterize the asymptotic state conversion in general
quantum resource theories (QRTs). Asymptotic state conversion gives a
fundamental limit of large-scale quantum information processing exploiting
quantum resources, and it has been widely discussed for known QRTs [1]. We
provide a general definition of a state conversion rate in Sec. 4.1. In terms
of the conversion rate, we find a class of resources that cannot be generated
from any free state with any free operation but can be replicated infinitely
by free operations. We call this state a catalytically replicable state. We
give the definition and an example of catalytically replicable states in Sec.
4.2. In Sec. 4.3, we formulate relations between asymptotic state conversion
and one-shot state conversion that hold in general QRTs, which may have
catalytically replicable states. In the following, the ceiling function is
denoted by $\lceil{}\cdots{}\rceil$, and the floor function is denoted by
$\lfloor{}\cdots{}\rfloor$.

###  4.1 Formulation of State Conversion Rate

We recall the concept of asymptotic state conversion and provide possible two
definitions of asymptotic state conversion rates. We show the equivalence of
these two definitions.

For two quantum systems $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, and two
quantum states $\phi\in\mathcal{S}\left(\mathcal{H}_{1}\right)$ and
$\psi\in\mathcal{S}\left(\mathcal{H}_{2}\right)$, asymptotic state conversion
from $\phi$ to $\psi$ is a task of transforming infinitely many copies of
$\phi$ into as many copies of $\psi$ as possible by a sequence of free
operations $\mathcal{N}_{1},\mathcal{N}_{2},\ldots$ within a vanishing error.
There are two possible ways to define state conversion rates from $\phi$ to
$\psi$: how many $\psi$’s can be generated from a single $\phi$, and how many
$\phi$’s are necessary to generate a single $\psi$. We write the first
conversion rate as $r_{\textup{conv}}\left(\phi\to\psi\right)$, and the second
conversion rate as $r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$. As
will be shown in Theorem 4, these two conversion rates are related to each
other in such a way that $r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$
is the inverse of $r_{\textup{conv}}\left(\phi\to\psi\right)$. Therefore, we
consider $r_{\textup{conv}}\left(\phi\to\psi\right)$ as the asymptotic state
conversion rate in this paper.

More formally, $r_{\textup{conv}}\left(\phi\to\psi\right)$ is defined as
follows. A set of asymptotic achievable rates is defined as

$\displaystyle\mathcal{R}\left(\phi\to\psi\right)\coloneqq$ (37)
$\displaystyle\Big{\\{}r\geqq
0:\exists\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}_{1}^{\otimes
n}\to\mathcal{H}_{2}^{\otimes\left\lceil
rn\right\rceil}\right):n\in\mathbb{N}\right),$
$\displaystyle\quad\liminf_{n\to\infty}\left\|\mathcal{N}_{n}\left(\phi^{\otimes
n}\right)-\psi^{\otimes\left\lceil rn\right\rceil}\right\|_{1}=0\Big{\\}},$

where $\phi^{\otimes 0}\coloneqq 1$. Intuitively, achievable rate $r\geqq 0$
is a positive number for which we can generate $rn$ copies of $\psi$ from $n$
copies of $\phi$. However, $rn$ is not necessarily an integer in general;
therefore, in (37), we regard $r$ as an achievable rate when we can generate
$\lceil rn\rceil$ copies of $\psi$, which guarantees that we can obtain $rn$
or more $\psi$’s. An asymptotic state conversion rate
$r_{\textup{conv}}\left(\phi\to\psi\right)$ is defined as

$r_{\textup{conv}}\left(\phi\to\psi\right)\coloneqq\sup\mathcal{R}\left(\phi\to\psi\right).$
(38)

Similarly, we can consider the other definition of a state conversion rate
$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$. Here, we define another
set of asymptotic achievable rates

$\displaystyle\mathcal{R}^{\prime}\left(\phi\to\psi\right)\coloneqq$ (39)
$\displaystyle\Big{\\{}r\geqq
0:\exists\left(\mathcal{N}^{\prime}_{n}\in\mathcal{O}\left(\mathcal{H}_{1}^{\otimes\left\lfloor
rn\right\rfloor}\to\mathcal{H}_{2}^{\otimes n}\right):n\in\mathbb{N}\right),$
$\displaystyle\quad\liminf_{n\to\infty}\left\|\mathcal{N}^{\prime}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\psi^{\otimes n}\right\|_{1}=0\Big{\\}}.$

Here, achievable rate $r^{\prime}\geqq 0$ is a positive number for which we
can generate $n$ copies of $\psi$ from $r^{\prime}n$ copies of $\phi$.
However, $r^{\prime}n$ is not necessarily an integer in general; therefore, in
(39), we regard $r^{\prime}$ as an achievable rate when we can generate $n$
copies of $\psi$ from $\lfloor r^{\prime}n\rfloor$ copies of $\phi$, which
guarantees that $r^{\prime}n$ or more copies of $\phi$ suffice to generate $n$
copies of $\psi$. With respect to this definition of achievable rates, an
asymptotic conversion rate
$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$ is defined as

$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)\coloneqq\inf\mathcal{R}^{\prime}\left(\phi\to\psi\right),$
(40)

where $r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$ is infinity if the
set on the right-hand side is empty.

These two conversion rates $r_{\textup{conv}}\left(\phi\to\psi\right)$ and
$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$ are related to each other
as shown in the following theorem. Hereafter, we will use
$r_{\textup{conv}}\left(\phi\to\psi\right)$ as the asymptotic states
conversion rate rather than
$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$.

######  Theorem 4 (Relation Between Two Conversion Rates).

Let $\mathcal{H}$ and $\mathcal{H}^{\prime}$ be quantum systems. For any
states $\phi\in\mathcal{S}\left(\mathcal{H}\right)$ and
$\psi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$, it holds that,

$r_{\textup{conv}}\left(\phi\to\psi\right)=\frac{1}{r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)},$
(41)

where we regard $1/0=\infty$.

###### Proof.

It suffices to show that

$r\in\mathcal{R}\left(\phi\to\psi\right)\Rightarrow\frac{1}{r}\in\mathcal{R}^{\prime}\left(\phi\to\psi\right),$
(42)

and that

$r\in\mathcal{R}^{\prime}\left(\phi\to\psi\right)\Rightarrow\frac{1}{r}\in\mathcal{R}\left(\phi\to\psi\right).$
(43)

First, assume that $r\in\mathcal{R}\left(\phi\to\psi\right)$ to show (42).
Choose a fixed positive real number $\epsilon>0$. Let $n$ be an arbitrary
positive integer such that

$\left\|\mathcal{N}_{n}\left(\phi^{\otimes n}\right)-\psi^{\otimes\left\lceil
rn\right\rceil}\right\|_{1}<\epsilon.$ (44)

Let $n^{\prime}=\left\lceil rn\right\rceil$. Because $n\leqq\lfloor
n^{\prime}/r\rfloor$, we can define a free operation
$\mathcal{M}_{n^{\prime}}$ as the partial trace over $\lfloor
n^{\prime}/r\rfloor-n$ systems so that

$\mathcal{M}_{n^{\prime}}\left(\phi^{\otimes\left\lfloor
n^{\prime}/r\right\rfloor}\right)=\phi^{\otimes n}.$ (45)

From (44) and (45), it holds that

$\left\|\mathcal{N}_{n}\circ\mathcal{M}_{n^{\prime}}\left(\phi^{\otimes\left\lfloor
n^{\prime}/r\right\rfloor}\right)-\psi^{\otimes
n^{\prime}}\right\|_{1}<\epsilon,$ (46)

and $1/r\in\mathcal{R}^{\prime}\left(\phi\to\psi\right)$ follows.

On the other hand, assume that
$r\in\mathcal{R^{\prime}}\left(\phi\to\psi\right)$ to show (43). Choose a
fixed positive real number $\epsilon>0$. Let $n$ be an arbitrary positive
integer such that

$\left\|\mathcal{N}^{\prime}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\psi^{\otimes n}\right\|_{1}<\epsilon$ (47)

Let $n^{\prime}=\left\lfloor rn\right\rfloor$. Because $n\geqq\lceil
n^{\prime}/r\rceil$, we can define a free operation
$\mathcal{M}^{\prime}_{n^{\prime}}$ as the partial trace over $n-\lceil
n^{\prime}/r\rceil$ systems so that

$\mathcal{M}^{\prime}_{n^{\prime}}\left(\psi^{\otimes
n}\right)=\psi^{\otimes\left\lceil n^{\prime}/r\right\rceil}.$ (48)

From (47) and (48), it holds that

$\displaystyle\left\|\mathcal{M}^{\prime}_{n^{\prime}}\circ\mathcal{N}^{\prime}_{n}\left(\phi^{\otimes
n^{\prime}}\right)-\psi^{\otimes\left\lceil
n^{\prime}/r\right\rceil}\right\|_{1}$ (49)
$\displaystyle=\left\|\mathcal{M}^{\prime}_{n^{\prime}}\circ\mathcal{N}^{\prime}_{n}\left(\phi^{\otimes
n^{\prime}}\right)-\mathcal{M}^{\prime}_{n^{\prime}}\left(\psi^{\otimes
n}\right)\right\|_{1}$
$\displaystyle\leqq\left\|\mathcal{N}^{\prime}_{n}\left(\phi^{\otimes
n^{\prime}}\right)-\psi^{\otimes n}\right\|_{1}$ $\displaystyle<\epsilon,$

and $1/r\in\mathcal{R}\left(\phi\to\psi\right)$ follows. ∎

######  Remark 3.

Conventionally, a conversion rate may also be defined as [1, 5]

$\tilde{r}_{\textup{conv}}\left(\phi\to\psi\right)\coloneqq\sup\tilde{\mathcal{R}}\left(\phi\to\psi\right),$
(50)

where

$\displaystyle\tilde{\mathcal{R}}\left(\phi\to\psi\right)\coloneqq$ (51)
$\displaystyle\Big{\\{}r\geqq
0:\exists\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}_{1}^{\otimes
n}\to\mathcal{H}_{2}^{\otimes\left\lfloor
rn\right\rfloor}\right):n\in\mathbb{N}\right),$
$\displaystyle\quad\liminf_{n\to\infty}\left\|\mathcal{N}_{n}\left(\phi^{\otimes
n}\right)-\psi^{\otimes\left\lfloor rn\right\rfloor}\right\|_{1}=0\Big{\\}},$

instead of a conversion rate $r_{\textup{conv}}\left(\phi\to\psi\right)$
defined in (37) and (38). In (51), the floor function is used instead of the
ceiling function used in (37).

As we show in the following, these two conversion rates
$r_{\textup{conv}}\left(\phi\to\psi\right)$ and
$\tilde{r}_{\textup{conv}}\left(\phi\to\psi\right)$ are identical; however,
considering the meaning of $r_{\textup{conv}}\left(\phi\to\psi\right)$
discussed below (38), we adopt $r_{\textup{conv}}\left(\phi\to\psi\right)$ in
this paper. To see that the two conversion rates are identical, first suppose
that $r\in\mathcal{R}\left(\phi\to\psi\right)$ for
$\phi,\psi\in\mathcal{S}(\mathcal{H})$. Since $\lceil rn\rceil\geqq\lfloor
rn\rfloor$ and since the partial trace is free, it holds that
$r\in\tilde{\mathcal{R}}\left(\phi\to\psi\right)$. Conversely, suppose that
$r\in\tilde{\mathcal{R}}\left(\phi\to\psi\right)$. Then, the fact that

$\lfloor rn\rfloor\geqq\lceil
rn\rceil-1=\left\lceil\left(r-\frac{1}{n}\right)n\right\rceil$ (52)

implies that $r-1/n$ can be regarded as a achievable rate with respect to
$\mathcal{R}\left(\phi\to\psi\right)$. Since $r-1/n$ approaches to $r$ as $n$
becomes large, considering the definition (37) of
$\mathcal{R}\left(\phi\to\psi\right)$, it can be concluded that
$r\in\mathcal{R}\left(\phi\to\psi\right)$. Therefore,
$r_{\textup{conv}}\left(\phi\to\psi\right)$ and
$\tilde{r}_{\textup{conv}}\left(\phi\to\psi\right)$ are identical.

In the same way, we may replace the floor function in (39) with the ceiling
function to obtain the identical conversion rate, but we adopt the definition
(39) and (40) due to the meaning of this conversion rate discussed below (39).

Finally, we recall a useful relation of state conversion rates given in Ref.
[66]. Let $\mathcal{H}_{1}$, $\mathcal{H}_{2}$ and $\mathcal{H}_{3}$ be
quantum systems. Let $\rho\in\mathcal{S}\left(\mathcal{H}_{1}\right)$,
$\sigma\in\mathcal{S}\left(\mathcal{H}_{2}\right)$ and
$\omega\in\mathcal{S}\left(\mathcal{H}_{3}\right)$ be quantum states. Suppose
that we first asymptotically generate $\sigma$ from $\rho$, then we generate
$\omega$ from $\sigma$ to achieve conversion from $\rho$ to $\omega$. While
this protocol generate $\omega$ from $\rho$, the protocol is not necessarily
optimal. In fact, it is known that

$r_{\textup{conv}}\left(\rho\to\omega\right)\geqq
r_{\textup{conv}}\left(\rho\to\sigma\right)r_{\textup{conv}}\left(\sigma\to\omega\right).$
(53)

Note that the equality of (53) does not necessarily hold. For example, a bound
entangled state cannot generate maximally entangled states, but needs them to
be formed [67]. On the other hand, it is known that the equality of (53) holds
in special cases such as the conversion between pure states in the QRT of
bipartite entanglement [68, 69].

###  4.2 Catalytic Replication of Resource

In this section, we analyze the replication of a resource. One of the
fundamental principles of quantum mechanics is the no-cloning theorem [70],
which shows that we cannot clone a quantum state if we do not know the
description of the state. The no-cloning theorem gives a fundamental
limitation of quantum mechanics, and contributes to understanding what is
achievable in quantum mechanics. Similarly, to figure out what is capable in
our framework of QRTs, we consider replication of a quantum resource. In the
task of the replication, we generate tensor products of a resource state,
where the description of the resource state is known but the operation is
restricted to the set of free operations. In terms of the asymptotic state
conversion, the replication of a resource is regarded as a catalytic state
conversion between the same state similarly to catalytic transformation of
entanglement [45].

We prove that the replication of a resource has only two scenarios: we cannot
replicate the resource, or we can replicate the resource infinitely.
Furthermore, we find a counter-intuitive example where a resource state is
replicable infinitely. Note that the infinite replication of a resource does
not necessarily mean that the amount of the resource increases under free
operations because the quantification of a resource depends on a resource
measure, which will be discussed in detail in Sec. 6.

######  Theorem 5 (Replication of State).

Let $\mathcal{H}$ be a quantum system. For any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$,
$r_{\textup{conv}}\left(\psi\to\psi\right)$ is equal to either 1 or $+\infty$.

###### Proof.

It trivially holds that

$r_{\textup{conv}}\left(\psi\to\psi\right)\geqq 1$ (54)

because $\operatorname{id}\in\mathcal{O}\left(\mathcal{H}\right)$.

Assume that $r_{\textup{conv}}\left(\psi\to\psi\right)>1$, that is, there
exists $r>1$ such that $r\in\mathcal{R}\left(\psi\to\psi\right)$. To prove
$r_{\textup{conv}}\left(\psi\to\psi\right)=\infty$, it suffices to show that

$2r-1\in\mathcal{R}\left(\psi\to\psi\right)$ (55)

because if (55) holds, an arbitrarily large rate can be achieved by exploiting
(55) repeatedly.

Choose a fixed positive real number $\epsilon$. There exists a sequence of
free operations $\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}^{\otimes
n}\to\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}\right):n\in\mathbb{N}\right)$ such that

$\left\|\mathcal{N}_{n}\left(\psi^{\otimes n}\right)-\psi^{\otimes\left\lceil
rn\right\rceil}\right\|_{1}<\frac{\epsilon}{2}$ (56)

holds for an infinitely large subset of $\mathbb{N}$.

Define

$r_{0}\coloneqq\inf_{n}r_{n},$ (57)

where

$r_{n}\coloneqq\max\left\\{r^{\prime}\geqq 0:\left\lceil
r^{\prime}n\right\rceil=2\left\lceil rn\right\rceil-n\right\\},$ (58)

and the infimum is taken over $n$ satisfying (56).

For $n$ satisfying (56), it holds that

$r_{n}=2r-1+\frac{2\alpha_{n}}{n},$ (59)

where $\alpha_{n}$ is a real number satisfying $0\leqq\alpha_{n}<1$ and
$\left\lceil rn\right\rceil=rn+\alpha_{n}$. The term $2\alpha_{n}/n$
approaches to zero as $n$ approaches to infinity. Then, it holds that

$\displaystyle\inf_{n}r_{n}$
$\displaystyle=\inf_{n}\left\\{2r-1+\frac{2\alpha_{n}}{n}\right\\}$ (60)
$\displaystyle=2r-1.$

Therefore, it holds that

$r_{0}=2r-1.$ (61)

Then, due to (55) and (61), it suffices to show that

$r_{0}\in\mathcal{R}\left(\psi\to\psi\right).$ (62)

Now, observe that for any $n$,

$\left\lceil rn\right\rceil>n$ (63)

always holds. This implies that for any $n$,

$2\left\lceil rn\right\rceil-n>\left\lceil rn\right\rceil$ (64)

holds.

For $n$ satisfying (56) and (63), we define a free operation $\mathcal{M}_{n}$
as the partial trace over $\left(2\left\lceil
rn\right\rceil-n\right)-\left\lceil r_{0}n\right\rceil$ systems so that we can
obtain

$\mathcal{M}_{n}\left(\psi^{\otimes 2\left\lceil
rn\right\rceil-n}\right)=\psi^{\otimes\left\lceil r_{0}n\right\rceil}.$ (65)

Therefore, from the triangle inequality, it follows that

$\displaystyle\left\|\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\circ\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)-\psi^{\otimes\left\lceil r_{0}n\right\rceil}\right\|_{1}$ (66)
$\displaystyle\leqq\left\|\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\circ\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)\right.$
$\displaystyle\quad\quad\left.-\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\left(\psi^{\otimes\left\lceil
rn\right\rceil}\right)\right\|_{1}$
$\displaystyle\quad+\left\|\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\left(\psi^{\otimes\left\lceil
rn\right\rceil}\right)-\psi^{\otimes\left\lceil
r_{0}n\right\rceil}\right\|_{1},$

where $\operatorname{id}$ is the identity map on $\left\lceil
rn\right\rceil-n$ systems. Since the trace distance is non-increasing for
quantum operations, it holds that

$\displaystyle\left\|\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\circ\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)\right.$ (67)
$\displaystyle\quad\left.-\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\left(\psi^{\otimes\left\lceil
rn\right\rceil}\right)\right\|_{1}$
$\displaystyle\leqq\left\|\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)-\psi^{\otimes\left\lceil rn\right\rceil}\right\|_{1},$

and that

$\displaystyle\left\|\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\left(\psi^{\otimes\left\lceil
rn\right\rceil}\right)-\psi^{\otimes\left\lceil
r_{0}n\right\rceil}\right\|_{1}$ (68)
$\displaystyle\leqq\left\|\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\left(\psi^{\otimes\left(n+\left(\left\lceil
rn\right\rceil-n\right)\right)}\right)-\psi^{\otimes\left(2\left\lceil
rn\right\rceil-n\right)}\right\|_{1}$
$\displaystyle=\left\|\left(\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)-\psi^{\otimes\left\lceil
rn\right\rceil}\right)\otimes\psi^{\otimes\left(\left\lceil
rn\right\rceil-n\right)}\right\|_{1}$
$\displaystyle=\left\|\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)-\psi^{\otimes\left\lceil rn\right\rceil}\right\|_{1}.$

Therefore, by (66), (67), and (68), we obtain

$\displaystyle\left\|\mathcal{M}_{n}\circ\left(\mathcal{N}_{n}\otimes\operatorname{id}\right)\circ\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)-\psi^{\otimes\left\lceil r_{0}n\right\rceil}\right\|_{1}$ (69)
$\displaystyle\leqq 2\left\|\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)-\psi^{\otimes\left\lceil rn\right\rceil}\right\|_{1}$
$\displaystyle<\epsilon,$

which implies that $r_{0}\in\mathcal{R}\left(\psi\to\psi\right)$. ∎

Remarkably, we here give an example where
$r_{\textup{conv}}\left(\psi\to\psi\right)=\infty$, but $\psi$ is not a free
state, that is, $\psi\not\in\mathcal{F}\left(\mathcal{H}\right)$. In this
paper, we call a state $\psi$ that satisfies
$r_{\textup{conv}}(\psi\to\psi)=\infty$ and
$\psi\notin\mathcal{F}\left(\mathcal{H}\right)$ a catalytically replicable
state. A catalytically replicable state is regarded as a form of catalytic
property of quantum resources, which are similar to catalytic state conversion
in the entanglement theory [45]. Any free state $\psi$ is a trivial example of
$r_{\textup{conv}}\left(\psi\to\psi\right)=\infty$, but the following example
shows that this is not the whole story; that is,
$r_{\textup{conv}}(\psi\to\psi)=\infty$ implies that $\psi$ is free or
catalytically replicable.

######  Example 6 (Catalytically Replicable Resource).

Suppose that
$\mathcal{S}\left(\mathbb{C}^{2}\right)=\left\\{\ket{0}\bra{0},\ket{1}\bra{1}\right\\}$.
Further suppose that the set of free operations $\mathcal{O}$ consists of
operations that are realized by circuits composed of the identity gate, the
partial trace, the controlled-NOT gate, the preparation of an auxiliary qubit
in $\ket{0}$ state. For any integer $n\geqq 0$, the set of free states is

$\mathcal{F}\left({\left(\mathbb{C}^{2}\right)}^{\otimes
n}\right)=\left\\{\ket{0}\bra{0}^{\otimes n}\right\\}.$ (70)

In this case, whereas
$\ket{1}\bra{1}\not\in\mathcal{F}\left(\mathbb{C}^{2}\right)$,
$r_{\textup{conv}}\left(\ket{1}\bra{1}\to\ket{1}\bra{1}\right)=+\infty$
because we can convert $\ket{1}\bra{1}$ into $\ket{1}\bra{1}^{\otimes n}$ for
any $n$ by appending an auxiliary system prepared in $\ket{0}$ and applying
controlled-NOT repeatedly.

###  4.3 Relations Between One-Shot State Conversion and Asymptotic State
Conversion

In this section, we analyze relations between the asymptotic state conversion
and the exact one-shot state conversion. The asymptotic state conversion from
$\phi$ to $\psi$ is a task transforming infinitely many copies of $\phi$ into
many copies of $\psi$ with a vanishing error, while the exact one-shot
conversion $\phi$ to $\psi$ is a task transforming a single $\phi$ into a
single $\psi$ exactly.

We prove two propositions both of which give relations between the asymptotic
state conversion and the exact one-shot state conversion. The first
proposition provides the relation that holds for inequivalent states. On the
other hand, the second proposition characterizes the asymptotic conversion
rate between two equivalent states. Firstly, the following proposition shows
that the more resourceful a state is, the harder it is to distill the state
and the easier it is to form another state from the state.

######  Proposition 6.

Let $\mathcal{H}_{1}$,$\mathcal{H}_{2}$ be quantum systems. Let
$\phi,\psi\in\mathcal{S}\left(\mathcal{H}_{1}\right)$ and
$\rho\in\mathcal{S}\left(\mathcal{H}_{2}\right)$ be quantum states. If
$\phi\succeq\psi$, then it holds that

$\displaystyle r_{\textup{conv}}\left(\rho\to\phi\right)$ $\displaystyle\leqq
r_{\textup{conv}}\left(\rho\to\psi\right)$ (71) $\displaystyle
r_{\textup{conv}}\left(\psi\to\rho\right)$ $\displaystyle\leqq
r_{\textup{conv}}\left(\phi\to\rho\right).$ (72)

###### Proof.

To prove (71), it suffices to show that
$\mathcal{R}\left(\rho\to\phi\right)\subseteq\mathcal{R}\left(\rho\to\psi\right)$.
Suppose that $r\in\mathcal{R}\left(\rho\to\phi\right)$. Then, there exists a
sequence of free operations
$\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}_{2}^{\otimes
n}\to\mathcal{H}_{1}^{\otimes\left\lceil
rn\right\rceil}\right):n\in\mathbb{N}\right)$ such that for arbitrary
$\epsilon>0$,

$\left\|\mathcal{N}_{n}\left(\rho^{\otimes n}\right)-\phi^{\otimes\left\lceil
rn\right\rceil}\right\|_{1}<\epsilon$ (73)

holds for an infinitely large subset of $\mathbb{N}$. As $\phi\succeq\psi$,
there exists a free operation $\mathcal{N}$ such that

$\mathcal{N}\left(\phi\right)=\psi.$ (74)

Define a sequence of free operations
$\left(\mathcal{M}_{n}\in\mathcal{O}\left(\mathcal{H}_{2}^{\otimes
n}\to\mathcal{H}_{1}^{\otimes\left\lceil
rn\right\rceil}\right):n\in\mathbb{N}\right)$ as

$\mathcal{M}_{n}\coloneqq\mathcal{N}^{\otimes\left\lceil
rn\right\rceil}\circ\mathcal{N}_{n}.$ (75)

Then, for any $n$ satisfying (73),

$\displaystyle\left\|\mathcal{M}_{n}\left(\rho^{\otimes
n}\right)-\psi^{\otimes\left\lceil rn\right\rceil}\right\|_{1}$ (76)
$\displaystyle=\left\|\mathcal{N}^{\otimes\left\lceil
rn\right\rceil}\circ\mathcal{N}_{n}\left(\rho^{\otimes
n}\right)-\mathcal{N}^{\otimes\left\lceil
rn\right\rceil}\left(\phi^{\otimes\left\lceil
rn\right\rceil}\right)\right\|_{1}$
$\displaystyle\leqq\left\|\mathcal{N}_{n}\left(\rho^{\otimes
n}\right)-\phi^{\otimes\left\lceil rn\right\rceil}\right\|_{1}$
$\displaystyle<\epsilon$

holds, and this implies that $r\in\mathcal{R}\left(\rho\to\psi\right)$.

Then, we prove (72). Note that (72) is equivalent to
$r^{\prime}_{\textup{conv}}\left(\phi\to\rho\right)\leqq
r^{\prime}_{\textup{conv}}\left(\psi\to\rho\right)$ because of Theorem 1. It
suffices to show that
$\mathcal{R}^{\prime}\left(\psi\to\rho\right)\subseteq\mathcal{R}^{\prime}\left(\phi\to\rho\right)$.
Suppose that $r\in\mathcal{R}^{\prime}\left(\psi\to\rho\right)$. Then, there
exists a sequence of free operations
$\left(\mathcal{N}^{\prime}_{n}\in\mathcal{O}\left(\mathcal{H}_{1}^{\otimes\left\lfloor
rn\right\rfloor\to\mathcal{H}_{2}^{\otimes n}}\right):n\in\mathbb{N}\right)$
such that for arbitrary $\epsilon>0$,

$\left\|\mathcal{N}^{\prime}_{n}\left(\psi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\rho^{\otimes n}\right\|_{1}<\epsilon$ (77)

holds for an infinitely large subset of $\mathbb{N}$. As $\phi\succeq\psi$,
there exists a free operation $\mathcal{N}^{\prime}$ such that

$\mathcal{N}^{\prime}\left(\phi\right)=\psi.$ (78)

Define a sequence of free operations
$\left(\mathcal{M}^{\prime}_{n}\in\mathcal{O}\left(\mathcal{H}_{1}^{\otimes\left\lfloor
rn\right\rfloor}\to\mathcal{H}_{2}^{\otimes n}\right):n\in\mathbb{N}\right)$
as

$\mathcal{M}^{\prime}_{n}\coloneqq\mathcal{N}^{\prime}_{n}\circ\mathcal{N}^{\prime\otimes\left\lfloor
rn\right\rfloor}.$ (79)

Then, for any $n$ satisfying (77),

$\displaystyle\left\|\mathcal{M}^{\prime}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\rho^{\otimes n}\right\|_{1}$ (80)
$\displaystyle=\left\|\mathcal{N}^{\prime}_{n}\circ\mathcal{N}^{\prime\otimes\left\lfloor
rn\right\rfloor}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\rho^{\otimes n}\right\|_{1}$
$\displaystyle\leqq\left\|\mathcal{N}^{\prime}_{n}\left(\psi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\rho^{\otimes n}\right\|_{1}$ $\displaystyle<\epsilon$

holds and this implies that
$r\in\mathcal{R}^{\prime}\left(\phi\to\rho\right)$. ∎

Next, we investigate the other relation between the asymptotic conversion and
the exact one-shot conversion, i.e., the asymptotic state conversion between
two equivalent states. Asymptotically, we may achieve conversion between
states that are not convertible to each other in the one-shot state
conversion. One may wonder whether we can achieve a better asymptotic
conversion rate between states that are equivalent under the one-shot
conversion. The following proposition shows that the asymptotic conversion
rate for two equivalent states is equal to $1$ in a QRT without catalytically
replicable states.

######  Proposition 7.

Let $\mathcal{H}$ be a quantum system. Let
$\psi,\phi\in\mathcal{S}\left(\mathcal{H}\right)$ be quantum states such that
$\psi\sim\phi$. Suppose that $r_{\textup{conv}}\left(\psi\to\psi\right)=1$.
Then, it holds that

$r_{\textup{conv}}\left(\psi\to\phi\right)=r_{\textup{conv}}\left(\phi\to\psi\right)=1.$
(81)

###### Proof.

Since $r_{\textup{conv}}\left(\psi\to\psi\right)=1$, it follows that

$r_{\textup{conv}}\left(\psi\to\phi\right)r_{\textup{conv}}\left(\phi\to\psi\right)\leqq
r_{\textup{conv}}\left(\psi\to\psi\right)=1$ (82)

from (53). On the other hand, since $\psi\sim\phi$, there exist free
operations $\mathcal{M}\in\mathcal{O}\left(\mathcal{H}\right)$ and
$\mathcal{N}\in\mathcal{O}\left(\mathcal{H}\right)$ such that

$\displaystyle\mathcal{M}\left(\psi\right)$ $\displaystyle=\phi$ (83)
$\displaystyle\mathcal{N}\left(\phi\right)$ $\displaystyle=\psi,$ (84)

which implies that

$\displaystyle r_{\textup{conv}}\left(\psi\to\phi\right)$ $\displaystyle\geqq
1$ (85) $\displaystyle r_{\textup{conv}}\left(\phi\to\psi\right)$
$\displaystyle\geqq 1.$ (86)

Therefore, both $r_{\textup{conv}}\left(\psi\to\phi\right)$ and
$r_{\textup{conv}}\left(\phi\to\psi\right)$ must be equal to $1$. ∎

##  5 Distillable Resource and Resource Cost

In this section, we analyze properties of the distillable resource
$R_{\textup{D}}$ and the resource cost $R_{\textup{C}}$, which represent how
many resources can be extracted from a state and how many resources are needed
to generate a state respectively. As noted in Sec. 3.1, maximally resourceful
states are not necessarily unique in general quantum resource theories (QRTs).
In Sec. 5.1, we define the distillable resource $R_{\textup{D}}$ as how many
resourceful states can be generated from a state in the worst-case scenario,
and we define the resource cost $R_{\textup{C}}$ as how many resourceful
states are needed to generate a state in the best-case scenario. Our
definition formulates distillation and formation of a resource even in cases
where maximally resourceful states are not unique. In Sec. 5.2, we analyze
distillation and formation of catalytically replicable states. In Sec. 5.3, we
prove weak subadditivity of the distillable resource and the resource cost. In
Sec. 5.4, we further investigate the resource cost, and prove that an upper
bound of the resource cost is achievable by a maximally resourceful state if
the number of non-equivalent maximally resourceful states is finite. In Sec.
5.5, generalizing the fact that the distillable entanglement is always smaller
than the entanglement cost [49], we prove that the distillable resource is
smaller than the resource cost in general QRTs without catalytically
replicable states.

###  5.1 Definitions of Resource Cost and Distillable Resource

In this section, we provide a formulation of distillation and formation of a
resource, and give the definitions of the distillable resource and the
resource cost, generalizing those in known QRTs such as bipartite entanglement
[46, 47], coherence [48], and athermality [13]. In contrast with the
definition in these previous works, our definitions are applicable to general
QRTs, where maximally resourceful states are not necessarily unique. Our
definition of the distillable resource represents how many resources can be
generated in the worst case, and the definition of the resource cost
represents how many resources are needed to form a state in the best case.

Our formulation of distillation and formation of a resource is as follows. For
a quantum system $\mathcal{H}$ and a state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$, distillation from the state
$\psi$ is a task of extracting many copies of a state
$\phi\in\mathcal{S}\left(\mathcal{H}\right)$ from many copies of $\psi$, where
$\phi$ is a state that is the most difficult to generate from $\psi$. More
formally, distillation is regarded as state conversion from $\psi$ to a state
$\phi$ for which $r_{\textup{conv}}\left(\psi\to\phi\right)$ takes a minimum
value. Similarly, formation of $\psi$ is a task of generating many copies of
$\psi$ from many copies of a state
$\phi\in\mathcal{S}\left(\mathcal{H}\right)$ where $\phi$ is a state that can
the most easily generate $\psi$. Formation is regarded as state conversion
from $\phi$ to a state $\psi$, where
$r_{\textup{conv}}\left(\phi\to\psi\right)$ takes a maximum value for $\phi$.

The distillable resource $R_{\textup{D}}$ represents the amount of resource
obtained by distillation; the resource cost $R_{\textup{C}}$ represents the
amount of resource needed for formation of a state. Formally, the distillable
resource of any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ is defined
as

$R_{\textup{D}}\left(\psi\right)\coloneqq\inf_{\phi\in\mathcal{S}\left(\mathcal{H}\right)}\left\\{r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}\right\\},$
(87)

where $R_{\max}^{(\mathcal{H})}\geqq 0$ is a normalization constant. If the
dimension of $\mathcal{H}$ is finite, we typically take
$R_{\max}^{(\mathcal{H})}$ as the required number of qubits for representing
the system $\mathcal{H}$; that is,

$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right),$ (88)

where $\dim\mathcal{H}$ denotes the dimension of $\mathcal{H}$, which implies
that $R_{\max}^{(\mathcal{H})}$ represents the maximum amount of a resource in
$\mathcal{S}\left(\mathcal{H}\right)$. Similarly, the resource cost of any
state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ is defined as

$R_{\textup{C}}\left(\psi\right)\coloneqq\inf_{\phi\in\mathcal{S}\left(\mathcal{H}\right)}\left\\{\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}\right\\}.$
(89)

Note that if we set the normalization constant
$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right)$ as mentioned
in (88) in the QRT of bipartite entanglement, $R_{\textup{D}}$ and
$R_{\textup{C}}$ reduce to the distillable entanglement and the entanglement
cost [46, 47], respectively.

We obtain the following proposition for QRTs without catalytically replicable
states, while QRTs with catalytically replicable states will be discussed in
the next subsection. This proposition provides general bounds of the
distillable resource and the resource cost, and we will also analyze
achievability of the bound of the resource cost in Sec. 5.4.

######  Proposition 8.

Let $\mathcal{H}$ be a quantum system, and let
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$ be a state. Suppose that
$\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)\neq\emptyset$.
If $\psi$ is not a catalytically replicable state, it holds that

$\displaystyle 0\leqq R_{\textup{D}}\left(\psi\right)\leqq
R_{\max}^{(\mathcal{H})},$ (90) $\displaystyle 0\leqq
R_{\textup{C}}\left(\psi\right)\leqq R_{\max}^{(\mathcal{H})}.$ (91)

Especially, if $\psi$ is a free state, it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle=0,$ (92)
$\displaystyle R_{\textup{C}}\left(\psi\right)$ $\displaystyle=0.$ (93)

###### Proof.

Note that by the definitions (87) of $R_{\textup{D}}$ and (89) of
$R_{\textup{C}}$, $0\leqq R_{\textup{D}}\left(\psi\right)$ and $0\leqq
R_{\textup{C}}\left(\psi\right)$ trivially hold. First, we prove the statement
for a free state. Let $\psi\in\mathcal{F}\left(\mathcal{H}\right)$ be a free
state. Since the set of free states is closed, for any resource state
$\phi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$,
it holds that

$r_{\textup{conv}}\left(\psi\to\phi\right)=0.$ (94)

Therefore, it holds that

$0\leqq R_{\textup{D}}\left(\psi\right)\leqq
R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\psi\to\phi\right)=0,$ (95)

which shows $R_{\textup{D}}\left(\psi\right)=0$. On the other hand, by
Proposition 2, there exists a free operation
$\mathcal{N}\in\mathcal{O}\left(\mathbb{C}\to\mathcal{H}\right)$ such that
$\mathcal{N}\left(1\right)=\psi$. Therefore, it holds that

$\mathcal{N}^{\otimes n}\left(1\right)=\psi^{\otimes n}$ (96)

for any positive integer $n$, which implies that

$r_{\textup{conv}}\left(1\to\psi\right)=\infty.$ (97)

Therefore, it holds that

$0\leqq
R_{\textup{C}}\left(\psi\right)\leqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(1\to\psi\right)}=0,$
(98)

which shows $R_{\textup{C}}\left(\psi\right)=0$.

Next, we prove (90) for a resource state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$.
By Theorem 5, we have $r_{\textup{conv}}\left(\psi\to\psi\right)=1$ because
$\psi$ is not a catalytically replicable state. Then, from (87), we obtain

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle\leqq
R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\psi\to\psi\right)$ (99)
$\displaystyle=R_{\max}^{(\mathcal{H})}.$

We can show (91) by replacing $R_{\textup{D}}$ with $R_{\textup{C}}$ and
$r_{\textup{conv}}\left(\psi\to\psi\right)$ with
$1/r_{\textup{conv}}\left(\psi\to\psi\right)$ respectively in the proof of
(90). ∎

In fact, from the relation between the preorder introduced by the free
operations and the asymptotic conversion rate shown in Proposition 6, we
obtain the following theorem, which shows that it is sufficient to take the
infimum over the maximally resourceful states in the definitions of
$R_{\textup{D}}$ and $R_{\textup{C}}$, rather than the infimum over the whole
set of states.

######  Theorem 9 (Maximally Resourceful States are Sufficient for Distillable
Resource and Resource Cost).

Let $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ be an arbitrary state. It is
sufficient to consider $\mathcal{G}\left(\mathcal{H}\right)$ instead of
$\mathcal{S}\left(\mathcal{H}\right)$ when we take the infimum in the
definitions (87) and (89) of $R_{\textup{D}}$ and $R_{\textup{C}}$; that is,
it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)$
$\displaystyle=\inf_{\phi\in\mathcal{G}\left(\mathcal{H}\right)}\left\\{r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}\right\\},$
(100) $\displaystyle R_{\textup{C}}\left(\psi\right)$
$\displaystyle=\inf_{\phi\in\mathcal{G}\left(\mathcal{H}\right)}\left\\{\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}\right\\}.$
(101)

###### Proof.

To show (101), it suffices to show that

$R_{\textup{D}}\left(\psi\right)=\inf_{\phi\in\mathcal{G}\left(\mathcal{H}\right)}\left\\{r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)R_{\max}^{(\mathcal{H})}\right\\}.$
(102)

By Proposition 6 and Theorem 1, for any state
$\rho\in\mathcal{S}\left(\mathcal{H}\right)$, there always exists a maximally
resourceful state $\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)R_{\max}^{(\mathcal{H})}\leqq
r^{\prime}_{\textup{conv}}\left(\rho\to\psi\right)R_{\max}^{(\mathcal{H})}.$
(103)

Therefore, (102) holds. Equation (100) can be shown by replacing
$R_{\textup{C}}$ with $R_{\textup{D}}$ and
$r^{\prime}_{\textup{conv}}\left(\rho\to\psi\right)$ with
$r_{\textup{conv}}\left(\rho\to\psi\right)$ in (102). ∎

######  Remark 4.

Due to Theorem 9, the infimum in the definitions of the distillable resource
and the resource cost is achieved in the following cases. Let

$\mathcal{G}\left(\mathcal{H}\right)/\sim\coloneqq\left\\{C_{\phi}:\phi\in\mathcal{G}\left(\mathcal{H}\right)\right\\}$
(104)

be the set of equivalence classes of the maximally resourceful states, where

$C_{\phi}\coloneqq\left\\{\psi\in\mathcal{G}\left(\mathcal{H}\right):\psi\sim\phi\right\\}$
(105)

is the equivalence class of $\phi$. Suppose that the number of non-equivalent
maximally resourceful states is finite; that is,
$\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|<\infty$. For example,
in the QRT of bipartite entanglement,
$\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|=1$; in the QRT of magic
states for qutrits, $\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|=2$
[17]. In these cases, the infimum is achievable by a maximally resourceful
state because of Proposition 7. Thus, for these existing QRTs, we can actually
replace the infimum in the definitions of the distillable resource (87) and
the resource cost (89) with the minimum, while further research is needed to
clarify whether or not we can replace the infimum with the minimum for QRTs
with infinitely many non-equivalent maximally resourceful states, i.e.,
$\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|=\infty$.

By using Theorem 9, we here prove that the more resourceful a state is, the
larger the distillable resource and the resource cost of the state are.

######  Proposition 10.

For a quantum system $\mathcal{H}$, let
$\psi,\phi\in\mathcal{S}\left(\mathcal{H}\right)$ be quantum states such that
$\phi\succeq\psi$. Then, it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle\leqq
R_{\textup{D}}\left(\phi\right),$ (106) $\displaystyle
R_{\textup{C}}\left(\psi\right)$ $\displaystyle\leqq
R_{\textup{C}}\left(\phi\right).$ (107)

###### Proof.

Let $\epsilon$ be an arbitrary positive number. Due to Theorem 9, we can take
a maximally resourceful state $\rho\in\mathcal{G}\left(\mathcal{H}\right)$
such that

$R_{\textup{D}}\left(\phi\right)+\epsilon\geqq
R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\phi\to\rho\right).$ (108)

Then, by Proposition 6,

$\displaystyle R_{\textup{D}}\left(\psi\right)$
$\displaystyle=\inf_{\sigma\in\mathcal{G}\left(\mathcal{H}\right)}\left\\{R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\psi\to\sigma\right)\right\\}$
(109) $\displaystyle\leqq
R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\psi\to\rho\right)$ (110)
$\displaystyle\leqq
R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\phi\to\rho\right)$ (111)
$\displaystyle\leqq R_{\textup{D}}\left(\phi\right)+\epsilon$ (112)

holds. As we can take an arbitrarily small $\epsilon$,
$R_{\textup{D}}\left(\psi\right)\leqq R_{\textup{D}}\left(\phi\right)$ holds.

Similarly, due to Theorem 9, we can take a maximally resourceful state
$\rho\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$R_{\textup{C}}\left(\phi\right)+\epsilon\geqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\rho\to\phi\right)}.$
(113)

Then, by Proposition 6,

$\displaystyle R_{\textup{C}}\left(\psi\right)$
$\displaystyle=\inf_{\sigma\in\mathcal{G}\left(\mathcal{H}\right)}\left\\{\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\sigma\to\psi\right)}\right\\}$
(114)
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\rho\to\psi\right)}$
(115)
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\rho\to\phi\right)}$
(116) $\displaystyle\leqq R_{\textup{C}}\left(\phi\right)+\epsilon$ (117)

holds. As we can take an arbitrarily small $\epsilon$,
$R_{\textup{C}}\left(\psi\right)\leqq R_{\textup{C}}\left(\phi\right)$ holds.
∎

###  5.2 Distillable Resource and Resource Cost of Catalytically Replicable
States

In this section, we analyze the distillable resource and the resource cost of
a catalytically replicable state. As the conversion rate between a
catalytically replicable state is infinite, we obtain a counter-intuitive
result, which shows that an infinitely large number of a resource can be
distilled from a catalytically replicable state and that a catalytically
replicable state can be generated without any cost.

The following proposition shows that the resource cost needed to form a
catalytically replicable state is equal to zero. Moreover, if the distillable
resource of a catalytically replicable state is nonzero, an infinite amount of
a resource can be distilled from the state.

######  Proposition 11.

Let $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ be a state satisfying
$r_{\textup{conv}}\left(\psi\to\psi\right)=\infty$. Then,

$R_{\textup{C}}\left(\psi\right)=0.$ (118)

holds. Moreover, if $R_{\textup{D}}\left(\psi\right)>0$,

$R_{\textup{D}}\left(\psi\right)=\infty$ (119)

holds.

###### Proof.

Note that $0\leqq R_{\textup{C}}\left(\psi\right)$ and $0\leqq
R_{\textup{D}}\left(\psi\right)$ hold by the definitions. Since
$r_{\textup{conv}}\left(\psi\to\psi\right)=\infty$,

$\displaystyle R_{\textup{C}}\left(\psi\right)$
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\psi\to\psi\right)}$
(120) $\displaystyle=0$

holds. Therefore, it holds that $R_{\textup{C}}\left(\psi\right)=0$.

Recall that for quantum states $\rho$, $\sigma$ and $\omega$, it holds that
$r_{\textup{conv}}\left(\rho\to\omega\right)\geqq
r_{\textup{conv}}\left(\rho\to\sigma\right)r_{\textup{conv}}\left(\sigma\to\omega\right)$
as shown in (53). Take an arbitrary positive number $\epsilon$. Let
$\phi\in\mathcal{S}\left(\mathcal{H}\right)$ be a state such that

$R_{\textup{D}}\left(\psi\right)+\epsilon\geqq
r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}{\geqq
R_{\textup{D}}\left(\psi\right)}.$ (121)

Then, it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)+\epsilon$ $\displaystyle\geqq
r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}$ (122)
$\displaystyle\geqq
r_{\textup{conv}}\left(\psi\to\psi\right)r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}$
(123)
$\displaystyle{\geqq}r_{\textup{conv}}\left(\psi\to\psi\right)R_{\textup{D}}\left(\psi\right)$
(124) $\displaystyle=\infty.$ (125)

As we can take an arbitrarily small $\epsilon$, it holds that
$R_{\textup{D}}\left(\psi\right)=\infty$. ∎

In a QRT whose maximally resourceful states are catalytically replicable,
there may be a state of which the distillable resource is infinite, and the
resource cost is zero, as shown in the Example 7.

######  Example 7 (Zero Resource Cost and Infinite Distillable Resource).

As shown in Proposition 8 and Proposition 11, the distillable resource of a
catalytically replicable state may be infinity while that of a free state is
zero. Consider the same setup as Example 6. In this case,

$\displaystyle R_{\textup{D}}\left(\ket{1}\bra{1}\right)$
$\displaystyle=\infty,$ (126) $\displaystyle
R_{\textup{C}}\left(\ket{1}\bra{1}\right)$ $\displaystyle=0$ (127)

follows from
$\mathcal{G}\left(\mathbb{C}^{2}\right)=\left\\{\ket{1}\bra{1}\right\\}$ and
$r_{\textup{conv}}\left(\ket{1}\bra{1}\to\ket{1}\bra{1}\right)=\infty$. In
contrast, for any free state $\psi$, Proposition 8 shows that

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle=0,$ (128)
$\displaystyle R_{\textup{C}}\left(\psi\right)$ $\displaystyle=0.$ (129)

###  5.3 Weak Subadditivity of Distillable Resource and Resource Cost

In this section, we prove that the distillable resource and the resource cost
are weakly subadditive if the Hilbert space $\mathcal{H}$ is finite-
dimensional and the normalization constant is set as
$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right)$. The
definitions of additivity and subadditivity are as follows.

######  Definition 12 (Additivity and Subadditivity).

Let $f_{\mathcal{H}}$ be a family of functions from
$\mathcal{S}\left(\mathcal{H}\right)$ to $\mathbb{R}$, where $\mathcal{H}$ is
a quantum system. We may omit the subscript of $f_{\mathcal{H}}$ to write $f$
for brevity. Then, $f$ is said to be additive if it holds that

$f\left(\psi\otimes\phi\right)=f\left(\psi\right)+f\left(\phi\right)$ (130)

for any states $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and
$\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$. On the other hand, $f$
is said to be weakly additive if it holds that

$f\left(\psi^{\otimes n}\right)=nf\left(\psi\right)$ (131)

for any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and for any
positive integer $n$.

Similarly, $f$ is said to be subadditive if it holds that

$f\left(\psi\otimes\phi\right)\leqq f\left(\psi\right)+f\left(\phi\right)$
(132)

for any states $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and
$\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$. On the other hand, $f$
is said to be weakly subadditive if it holds that

$f\left(\psi^{\otimes n}\right)\leqq nf\left(\psi\right)$ (133)

for any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and for any
positive integer $n$.

Note that in some cases, e.g., in Ref. [49], the property mentioned in (131),
which we call weak additivity here, is reffered to as additivity. However, in
this paper, we follow the convention of Ref. [1].

The proof of weak subadditivity exploits the following proposition.

######  Proposition 13.

Let $\mathcal{H}$ and $\mathcal{H}^{\prime}$ be quantum systems. Let
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and
$\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$ be quantum states. Then
for any $n\in\mathbb{N}$, it holds that

$r_{\textup{conv}}\left(\psi\to\phi\right)=r_{\textup{conv}}\left(\psi^{\otimes
n}\to\phi^{\otimes n}\right).$ (134)

###### Proof.

It suffices to show that
$\mathcal{R}\left(\psi\to\phi\right)=\mathcal{R}\left(\psi^{\otimes
n}\to\phi^{\otimes n}\right)$. First, assume
$r\in\mathcal{R}\left(\psi^{\otimes n}\to\phi^{\otimes n}\right)$ to show
$\mathcal{R}\left(\psi^{\otimes n}\to\phi^{\otimes
n}\right)\subseteq\mathcal{R}\left(\psi\to\phi\right)$. Choose a positive
number $\epsilon>0$. Then, there exists a sequence of free operations
$\left(\mathcal{M}^{(n)}_{m}\in\mathcal{O}\left(\mathcal{H}^{\otimes
nm}\to\mathcal{H}^{\prime\otimes n\left\lceil
rm\right\rceil}\right):m\in\mathbb{N}\right)$ such that

$\left\|\mathcal{M}^{(n)}_{m}\left({\left(\psi^{\otimes n}\right)}^{\otimes
m}\right)-{\left(\phi^{\otimes n}\right)}^{\otimes\left\lceil
rm\right\rceil}\right\|_{1}<\epsilon$ (135)

holds for an infinitely large subset of $\mathbb{N}$. Since $n\left\lceil
rm\right\rceil\geqq\left\lceil rnm\right\rceil$ holds, we can define
$\mathcal{N}^{(n)}_{m}\in\mathcal{O}\left(\mathcal{H}^{\prime\otimes
n\left\lceil rm\right\rceil}\to\mathcal{H}^{\prime\otimes\left\lceil
rnm\right\rceil}\right)$ as the partial trace over $n\left\lceil
rm\right\rceil-\left\lceil rnm\right\rceil$ systems. Then, we have
$\mathcal{N}^{(n)}_{m}\left(\phi^{\otimes n\left\lceil
rm\right\rceil}\right)=\phi^{\otimes\left\lceil rnm\right\rceil}$. Therefore,
it holds that

$\displaystyle\left\|\mathcal{N}^{(n)}_{m}\circ\mathcal{M}^{(n)}_{m}\left(\psi^{\otimes
nm}\right)-\phi^{\otimes\left\lceil rnm\right\rceil}\right\|_{1}$ (136)
$\displaystyle=\left\|\mathcal{N}^{(n)}_{m}\circ\mathcal{M}^{(n)}_{m}\left({\left(\psi^{\otimes
n}\right)}^{\otimes m}\right)-\mathcal{N}^{(n)}_{m}\left(\phi^{\otimes
n\left\lceil rm\right\rceil}\right)\right\|_{1}$
$\displaystyle\leqq\left\|\mathcal{M}^{(n)}_{m}\left({\left(\psi^{\otimes
n}\right)}^{\otimes m}\right)-\phi^{\otimes n\left\lceil
rm\right\rceil}\right\|_{1}$ $\displaystyle<\epsilon.$

Therefore, for an integer $k=nm$ with a sufficiently large $m$, there exists a
free operation
$\mathcal{L}_{k}\coloneqq\mathcal{N}^{(n)}_{m}\circ\mathcal{M}^{(n)}_{m}$ such
that

$\displaystyle\left\|\mathcal{L}_{k}\left(\psi^{\otimes
k}\right)-\phi^{\otimes\left\lceil rk\right\rceil}\right\|_{1}<\epsilon.$
(137)

Therefore, $r\in\mathcal{R}\left(\psi\to\phi\right)$, which implies
$\mathcal{R}\left(\psi^{\otimes n}\to\phi^{\otimes
n}\right)\subseteq\mathcal{R}\left(\psi\to\phi\right)$.

On the other hand, to show
$\mathcal{R}\left(\psi\to\phi\right)\subseteq\mathcal{R}\left(\psi^{\otimes
n}\to\phi^{\otimes n}\right)$, assume
$r\in\mathcal{R}\left(\psi\to\phi\right)$. Choose a positive number
$\epsilon$. Then, there exists a sequence of free operations
$\left(\mathcal{M}_{m}\in\mathcal{O}\left(\mathcal{H}^{\otimes
m}\to\mathcal{H}^{\prime\otimes\left\lceil
rm\right\rceil}\right):m\in\mathbb{N}\right)$ such that for a fixed positive
integer $n$,

$\left\|\mathcal{M}_{m}\left(\psi^{\otimes m}\right)-\phi^{\otimes\left\lceil
rm\right\rceil}\right\|_{1}<\frac{\epsilon}{n}$ (138)

holds for an infinitely large subset of $\mathcal{N}$. Therefore, for any $n$
satisfying (138), it holds that

$\left\|\mathcal{M}_{m}^{\otimes n}\left({\left(\psi^{\otimes
m}\right)}^{\otimes n}\right)-{\left(\phi^{\otimes\left\lceil
rm\right\rceil}\right)}^{\otimes n}\right\|_{1}<\epsilon,$ (139)

which implies that
$\mathcal{R}\left(\psi\to\phi\right)\subseteq\mathcal{R}\left(\psi^{\otimes
n}\to\phi^{\otimes n}\right)$. ∎

Using Proposition 13, we show Theorem 14. The meaning of this theorem will be
discussed after the proof.

######  Theorem 14 (Weak Subadditivity of Distillable Resource and Resource
Cost).

Let $\mathcal{H}$ be an arbitrary finite-dimensional system. Set the
normalization constant as
$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right)$ as stated in
(88). For any $n\in\mathbb{N}$ and for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$,

$\displaystyle R_{\textup{D}}\left(\psi^{\otimes n}\right)$
$\displaystyle\leqq nR_{\textup{D}}\left(\psi\right),$ (140) $\displaystyle
R_{\textup{C}}\left(\psi^{\otimes n}\right)$ $\displaystyle\leqq
nR_{\textup{C}}\left(\psi\right).$ (141)

###### Proof.

First, we prove (140). Let $n$ be a fixed positive integer, and let $\epsilon$
be an arbitrary positive number. Due to Theorem 9, we can take a maximally
resourceful state $\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$R_{\textup{D}}\left(\psi\right)+\frac{\epsilon}{n}\geqq
R_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\psi\to\phi\right).$ (142)

Since $\phi^{\otimes n}\in\mathcal{S}\left(\mathcal{H}^{\otimes n}\right)$,

$\displaystyle nR_{\textup{D}}\left(\psi\right)+\epsilon$
$\displaystyle=nR_{\max}^{(\mathcal{H})}r_{\textup{conv}}\left(\psi\to\phi\right)$
(143) $\displaystyle=R_{\max}^{(\mathcal{H}^{\otimes
n})}r_{\textup{conv}}\left(\psi^{\otimes n}\to\phi^{\otimes n}\right)$ (144)
$\displaystyle\geqq R_{\textup{D}}\left(\psi^{\otimes n}\right)$ (145)

holds where (144) follows from Proposition 13. As we can take an arbitrarily
small $\epsilon$, (140) holds. We can show (140) in a similar way by replacing
$R_{\textup{D}}$ with $R_{\textup{C}}$ and
$r_{\textup{conv}}\left(\psi\to\phi\right)$ with
$1/r_{\textup{conv}}\left(\phi\to\psi\right)$. ∎

In the statement of Theorem 14, (140) means that for a maximally resourceful
state $\phi\in\mathcal{G}\left(\mathcal{H}\right)$, there may be
$\phi_{n}\in\mathcal{G}\left(\mathcal{H}^{\otimes n}\right)$ that is harder to
distill than $\phi^{\otimes n}$. On the other hand, (141) means that for a
maximally resourceful state $\phi\in\mathcal{G}\left(\mathcal{H}\right)$,
there may be a more resourceful state
$\phi_{n}\in\mathcal{G}\left(\mathcal{H}^{\otimes n}\right)$ in resource
formation than $\phi^{\otimes n}$ for $n\geqq 2$.

###  5.4 Maximally Resourceful State Maximizing Resource Cost

In this section, we prove that the upper bound $R_{\max}^{(\mathcal{H})}$ of
the resource cost $R_{\textup{C}}$ shown in Proposition 8 is indeed achievable
by a maximally resourceful state if the number of equivalence classes of the
maximally resourceful states is finite and if there is no catalytically
replicable state. Note that this property holds even in infinite-dimensional
cases; that is,
$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right)$ for finite-
dimensional $\mathcal{H}$ is not assumed in this section.

Using Proposition 7, we prove Theorem 15. Recall the set of equivalence
classes of the maximally resourceful states
$\mathcal{G}\left(\mathcal{H}\right)/\sim$ defined in (104). Consider a QRT
where the number of maximally resourceful states is finite up to the
equivalence with regard to the preorder, that is,

$\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|<\infty.$ (146)

In this case, the following theorem shows that the upper bound
$R_{\max}^{(\mathcal{H})}$ of the resource cost given in Proposition 8 is
actually achievable by a maximally resourceful state.

######  Theorem 15 (Maximally Resourceful State that Maximizes Resource Cost).

Suppose that there is no catalytically replicable state. Suppose further that
the set of resource states is not empty; that is,
$\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)\neq\emptyset$.
If $\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|<\infty$ where
$\mathcal{G}\left(\mathcal{H}\right)/\sim$ is defined in (104), then there
exists a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$R_{\textup{C}}\left(\phi\right)=R_{\max}^{(\mathcal{H})}$ (147)

holds.

###### Proof.

Assume that for all states $\phi\in\mathcal{G}\left(\mathcal{H}\right)$,
$R_{\textup{C}}\left(\phi\right)<R_{\max}^{(\mathcal{H})}$. Because of Theorem
9, this assumption implies that for all states
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$, there exists a maximally
resourceful state $\rho\in\mathcal{G}\left(\mathcal{H}\right)$ such that
$r_{\textup{conv}}\left(\rho\to\phi\right)>1$. According to Proposition 7,
$\rho$ must be in an equivalence class different from $C_{\phi}$. Write this
relation as $\phi\mapsto\rho$; that is, for
$\phi\in\mathcal{G}\left(\mathcal{H}\right)/\sim$ and
$\rho\in\mathcal{G}\left(\mathcal{H}\right)/\sim$, we write $\phi\mapsto\rho$
if $r_{\textup{conv}}\left(\rho\to\phi\right)>1$. Since
$\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|<\infty$, there must
exist a loop of elements in $\mathcal{G}\left(\mathcal{H}\right)/\sim$

$\rho_{0}\mapsto\rho_{1}\mapsto\cdots\mapsto\rho_{n}\mapsto\rho_{0}.$ (148)

Therefore, Theorem 5 shows that

$\displaystyle r_{\textup{conv}}\left(\rho_{0}\to\rho_{n}\right)\times
r_{\textup{conv}}\left(\rho_{n}\to\rho_{n-1}\right)\times$
$\displaystyle\quad\cdots\times
r_{\textup{conv}}\left(\rho_{1}\to\rho_{0}\right)$ $\displaystyle>1.$

On the other hand, note that for any maximally resourceful state
$\rho\in\mathcal{G}\left(\mathcal{H}\right)$, it holds that
$r_{\textup{conv}}\left(\rho\to\rho\right)=1$ because there is no
catalytically replicable state and because
$\rho\not\in\mathcal{F}\left(\mathcal{H}\right)$ due to Proposition 3. From
(53), it follows that

$\displaystyle
r_{\textup{conv}}\left(\rho_{0}\to\rho_{n}\right)\times\cdots\times
r_{\textup{conv}}\left(\rho_{1}\to\rho_{0}\right)$ (149) $\displaystyle\leqq
r_{\textup{conv}}\left(\rho_{0}\to\rho_{0}\right)$ $\displaystyle=1,$

which contradicts (5.4). Therefore, there exists a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that
$R_{\textup{C}}\left(\phi\right)=R_{\max}^{(\mathcal{H})}$. ∎

###  5.5 Condition for Distillable Resource Upper-Bounded by Resource Cost

In this section, we prove that the distillable resource is smaller than or
equal to the resource cost if the QRT does not have any catalytically
replicable state. The claim that the distillable resource is smaller than or
equal to the resource cost was proved in the QRT of bipartite entanglement
[49]. Progressing beyond this previous work, our proof does not make any
assumption on the existence of additive measures, and hence is simpler and has
more applicability than the existing technique.

######  Theorem 16 (Condition for Distillable Resource Upper-Bounded by
Resource Cost).

Let $\mathcal{H}$ be a quantum system. Suppose that
$\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)\neq\emptyset$.
Further suppose that there is no catalytically replicable state; that is,
$r_{\textup{conv}}\left(\phi\to\phi\right)=1$ for any resource state
$\phi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$.
Then, for any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$, it holds
that

$R_{\textup{D}}\left(\psi\right)\leqq R_{\textup{C}}\left(\psi\right).$ (150)

###### Proof.

As shown in Proposition 8, for a free state
$\psi\in\mathcal{F}\left(\mathcal{H}\right)$, it holds that
$R_{\textup{D}}\left(\psi\right)=R_{\textup{C}}\left(\psi\right)=0$.
Therefore, (150) trivially holds for a free state. Then, let
$\psi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$
be an arbitrary resource state. Let $\epsilon>0$ be an arbitrary positive
number. Due to Theorem 9, we take a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$\displaystyle
R_{\textup{C}}\left(\psi\right)+\epsilon\geqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}.$
(151)

By the definition (87) of the distillable resource, it holds that

$R_{\textup{D}}\left(\psi\right)\leqq
r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}.$ (152)

As
$r_{\textup{conv}}\left(\psi\to\phi\right)r_{\textup{conv}}\left(\phi\to\psi\right)\leqq
r_{\textup{conv}}\left(\psi\to\psi\right)=1$ shown in (53), it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle\leqq
r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}$ (153)
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}$
(154) $\displaystyle\leqq R_{\textup{C}}\left(\psi\right)+\epsilon.$ (155)

As we can take an arbitrarily small $\epsilon$, it holds that
$R_{\textup{D}}\left(\psi\right)\leqq R_{\textup{C}}\left(\psi\right)$. ∎

##  6 Resource Measures

In this section, we investigate a formulation of resource measures in general
QRTs and clarify general properties of the resource measures. The resource
measures quantify the amount of quantum resources, which is a central interest
in QRTs [1]. In Sec. 6.1 we provide the definition of a resource measure. In
Sec. 6.2, progressing beyond the existing result on bipartite entanglement
[49], we show in the general setting that a resource measure is upper-bounded
by the resource cost and lower-bounded by the distillable resource if it
satisfies the same properties as those given in Ref. [49]. At the same time,
we show that the QRT of magic for qutrits [71], which has several non-
equivalent maximally resourceful states, has no resource measure satisfying
the properties. To overcome this problem in the axiomatic approach based on
Ref. [49], we here introduce a concept of consistency of a resource measure in
Sec. 6.3. In contrast with the previous approach, a consistent resource
measure exists in the case where multiple non-equivalent maximally resourceful
states exist. Furthermore, we prove a similar uniqueness inequality to the
previous approach; that is, the consistent resource measure is bounded by the
distillable resource and the resource cost if it is normalized. In Sec. 6.4,
we provide the definition of the relative entropy of resource, and show that
the regularized relative entropy of resource serves as a consistent resource
measure.

###  6.1 Properties of Resource Measures

In this section, we provide a definition of resource measure as discussed in
previous works on studying resource measures in a wide class of QRTs, such as
Refs. [72, 32, 73, 20, 74, 75], and recall conventionally studied properties
of a resource measure some of which are also discussed in Refs. [1, 49]. A
resource measure $R$ quantifies the amount of the resource of a state. It
takes a state as an input and outputs a real number that represents the amount
of the resource. To quantify the resource without contradicting the fact that
free operations cannot generate resources by themselves, a resource measure
must satisfy a property called monotonicity; i.e., the amount of the resource
quantified by a resource measure does not increase through application of free
operations. Formally, monotonicity is defined as follows. For quantum systems
$\mathcal{H}^{(\mathrm{in})}$ and $\mathcal{H}^{(\mathrm{out})}$, any state
$\psi\in\mathcal{S}\left(\mathcal{H}^{(\mathrm{in})}\right)$, and any free
operation
$\mathcal{N}\in\mathcal{O}\left(\mathcal{H}^{(\mathrm{in})}\to\mathcal{H}^{(\mathrm{out})}\right)$,
it holds that

$R_{\mathcal{H}^{(\mathrm{in})}}(\psi)\geqq
R_{\mathcal{H}^{(\mathrm{out})}}(\mathcal{N}(\psi)).$ (156)

Here, we recall the definition of a resource measure.

######  Definition 17 (Resource Measure).

A resource measure $R_{\mathcal{H}}$ is a family of real functions from
$\mathcal{S}\left(\mathcal{H}\right)$ for a quantum system $\mathcal{H}$ to
$\mathbb{R}$ satisfying monotonicity. We may omit the subscript of
$R_{\mathcal{H}}$ to write $R$ for brevity.

By monotonicity, a resource measure $R_{\mathcal{H}}$ for a quantum system
$\mathcal{H}$ quantifies the resource consistently with the preorder
introduced by free operations. For two states satisfying

$\phi\succeq\psi,$ (157)

it holds that

$R_{\mathcal{H}}\left(\phi\right)\geqq R_{\mathcal{H}}\left(\psi\right).$
(158)

Note that if two states satisfy

$\phi\sim\psi$ (159)

then we have

$R_{\mathcal{H}}\left(\phi\right)=R_{\mathcal{H}}\left(\psi\right).$ (160)

Furthermore, using a resource measure, we can evaluate the resource amounts of
two different states that cannot be compared in terms of the preorder
introduced by free operations.

For example, in the QRT of bipartite entanglement, the distillable
entanglement and the entanglement cost are known to be entanglement measures
[46, 47]. Generalizing this fact, we find a condition where the distillable
resource and the resource cost become resource measures as shown in the
following proposition.

######  Proposition 18.

Suppose that for any quantum system $\mathcal{H}$, there exists a maximally
resourceful state $\phi\in\mathcal{G}(\mathcal{H})$ such that for any quantum
system $\mathcal{H}^{\prime}$ and for any maximally resourceful state
$\phi^{\prime}\in\mathcal{G}(\mathcal{H}^{\prime})$, it holds that

$r_{\textup{conv}}(\phi\to\phi^{\prime})\geqq\frac{R_{\max}^{(\mathcal{H})}}{R_{\max}^{(\mathcal{H}^{\prime})}}.$
(161)

Then, the distillable resource $R_{\textup{D}}$ and the resource cost
$R_{\textup{C}}$ satisfy monotonicity.

###### Proof.

Choose any $\delta>0$. Let $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ be quantum
systems. Let $\psi\in\mathcal{S}(\mathcal{H}_{1})$ be a quantum state and
$\mathcal{N}\in\mathcal{O}(\mathcal{H}_{1}\to\mathcal{H}_{2})$ be a free
operation. Suppose that $\phi_{2}\in\mathcal{G}(\mathcal{H}_{2})$ is a
maximally resourceful state satisfying the assumption given in (161).

Now, suppose that $\phi_{1}\in\mathcal{G}(\mathcal{H}_{1})$ is a maximally
resourceful state such that

$R_{\textup{D}}(\psi)\geqq
R_{\max}^{(\mathcal{H}_{1})}r_{\textup{conv}}(\psi\to\phi_{1})-\delta.$ (162)

Then, it holds that

$\displaystyle R_{\textup{D}}(\mathcal{N}(\psi))$ $\displaystyle\leqq
R_{\max}^{(\mathcal{H}_{2})}r_{\textup{conv}}(\mathcal{N}(\psi)\to\phi_{2})$
(163) $\displaystyle\leqq
R_{\max}^{(\mathcal{H}_{2})}r_{\textup{conv}}(\psi\to\phi_{2})$
$\displaystyle\leqq
R_{\max}^{(\mathcal{H}_{1})}r_{\textup{conv}}(\psi\to\phi_{2})r_{\textup{conv}}(\phi_{2}\to\phi_{1})$
$\displaystyle\leqq
R_{\max}^{(\mathcal{H}_{1})}r_{\textup{conv}}(\psi\to\phi_{1})$
$\displaystyle\leqq R_{\textup{D}}(\psi)+\delta.$

The first inequality holds by definition. The second inequality follows from
Proposition 6. The third inequality follows from (161). Using (53), we have
the fourth inequality. Since we can take an arbitrarily small $\delta$, we
have $R_{\textup{D}}(\mathcal{N}(\psi))\leqq R_{\textup{D}}(\psi)$.

Next, suppose that $\phi_{1}\in\mathcal{G}(\mathcal{H}_{1})$ is a maximally
resourceful state such that

$R_{\textup{C}}(\psi)\geqq\frac{R_{\max}^{(\mathcal{H}_{1})}}{r_{\textup{conv}}(\phi_{1}\to\psi)}-\delta.$
(164)

Then, it holds that

$\displaystyle R_{\textup{C}}(\mathcal{N}(\psi))$
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H}_{2})}}{r_{\textup{conv}}(\phi_{2}\to\mathcal{N}(\psi))}$
(165)
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H}_{2})}}{r_{\textup{conv}}(\phi_{2}\to\psi)}$
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H}_{1})}r_{\textup{conv}}(\phi_{2}\to\phi_{1})}{r_{\textup{conv}}(\phi_{2}\to\psi)}$
$\displaystyle\leqq\frac{R_{\max}^{(\mathcal{H}_{1})}}{r_{\textup{conv}}(\phi_{1}\to\psi)}$
$\displaystyle\leqq R_{\textup{C}}(\psi)+\delta.$

Since we can take an arbitrarily small $\delta$, we have
$R_{\textup{C}}(\mathcal{N}(\psi))\leqq R_{\textup{C}}(\psi)$. ∎

Now, we recall several properties of a resource measure that are
conventionally studied.

Additivity: Strong superadditivity refers to

$R\left(\psi^{AB}\right)\geqq R\left(\psi^{A}\right)+R\left(\psi^{B}\right),$
(166)

and superadditivity refers to

$R\left(\psi\otimes\phi\right)\geqq R\left(\psi\right)+R\left(\phi\right).$
(167)

Subadditivity refers to

$R\left(\psi\otimes\phi\right)\leqq R\left(\psi\right)+R\left(\phi\right).$
(168)

Additivity refers to

$R\left(\psi\otimes\phi\right)=R\left(\psi\right)+R\left(\phi\right),$ (169)

while weak additivity refers to

$R\left(\psi^{\otimes n}\right)=nR\left(\psi\right).$ (170)

Regularization of $R$ provides a measure that is additive for tensor product
of the same states

$R^{\infty}\left(\psi\right)\coloneqq\lim_{n\to\infty}\frac{R\left(\psi^{\otimes
n}\right)}{n},$ (171)

as long as the right-hand side exists. The following proposition shows that
additivity of a resource measure implies that free states have zero resource,
which is a generalization of the statement shown for entanglement in Ref. [49]
to general QRTs.

######  Proposition 19.

If a resource measure $R$ is weakly additive, $R\left(\phi\right)=0$ for any
free state $\phi$.

###### Proof.

Suppose that $\psi$ is a free state. Then, there exists a free operation
$\mathcal{M}$ such that $\mathcal{M}\left(1\right)=\psi$. Therefore, for any
$n\in\mathbb{N}$, $\mathcal{M}^{\otimes n}\left(1\right)=\psi^{\otimes n}$
holds, which implies $\psi^{\otimes n}$ is also a free state. Then, there
exist free operations $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ such that

$\displaystyle\mathcal{N}_{1}\left(\psi^{\otimes n}\right)=\psi,$ (172)
$\displaystyle\mathcal{N}_{2}\left(\psi\right)=\psi^{\otimes n}$

hold. Therefore, it holds that $R\left(\psi\right)=R\left(\psi^{\otimes
n}\right)$. Since $R$ is weakly additive,
$R\left(\psi\right)=nR\left(\psi\right)$ for any $n$, which implies
$R\left(\psi\right)=0$. ∎

One conventional way of normalizing resource measures such as that in the
entanglement theory is as follows, which we call conventional normalization:

  * •

For any free state $\sigma$,

$R\left(\sigma\right)=0.$ (173)

  * •

For any maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$,

$R\left(\phi\right)=R_{\max}^{(\mathcal{H})}.$ (174)

Because of monotonicity of a resource measure, a resource measure takes the
least value for free states. From Proposition 19, the least value is
automatically set to zero for an additive measure. We can assume this
normalization also for non-additive measures. Furthermore, we can set the
greatest value of a resource measure to $R_{\max}^{(\mathcal{H})}$ in the same
way that we normalize the distillable resource (87) and the resource cost (89)
with the normalization constant $R_{\max}^{(\mathcal{H})}$. In a finite-
dimensional case with
$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right)$ shown in (88),
(174) provides the normalization generalizing that of entanglement measures in
the entanglement theory, but our definition is applicable to infinite-
dimensional cases. We here remark that general QRTs do not necessarily have
resource measures satisfying this conventional normalization, as we will prove
in the next subsection.

Asymptotic continuity: For any quantum system $\mathcal{H}$, $R_{\mathcal{H}}$
is asymptotically continuous if for any sequence of positive integers
${\left(n_{i}\right)}_{i\in\mathbb{N}}$, and any sequences of states
${\left(\phi_{n_{i}}\in\mathcal{S}\left(\mathcal{H}^{\otimes
n_{i}}\right)\right)}_{i}$ and
${\left(\psi_{n_{i}}\in\mathcal{S}\left(\mathcal{H}^{\otimes
n_{i}}\right)\right)}_{i}$ satisfying
$\lim_{i\to\infty}\left\|\phi_{n_{i}}-\psi_{n_{i}}\right\|_{1}=0$, it holds
that

$\lim_{i\to\infty}\frac{\left|R_{\mathcal{H}^{\otimes
n_{i}}}\left(\phi_{n_{i}}\right)-R_{\mathcal{H}^{\otimes
n_{i}}}\left(\psi_{n_{i}}\right)\right|}{n_{i}}=0.$ (175)

Our definition of asymptotic continuity is applicable to an infinite-
dimensional system. If we take
$R_{\max}^{(\mathcal{H})}=\log_{2}\left(\dim\mathcal{H}\right)$ as stated in
(88) for a finite-dimensional system $\mathcal{H}$, our definition (175)
corresponds to Condition (E3) in Ref. [49]. Note that our definition includes
asymptotic continuity discussed in [76] (Definition 1) as a tighter bound
applicable to a finite-dimensional system.

######  Remark 5 (Continuity and Asymptotic Continuity).

Since a resource measure is a family of functions each of which may be defined
for different quantum systems, we employ asymptotic continuity of a family of
functions as a desired property of resource measures rather than continuity of
a single function $R_{\mathcal{H}}$ for a fixed quantum system $\mathcal{H}$
defined as follows. A function
$R_{\mathcal{H}}:\mathcal{S}\left(\mathcal{H}\right)\to\mathbb{R}$ is
continuous if for any sequences of states
${\left(\phi_{n}\in\mathcal{S}\left(\mathcal{H}\right)\right)}_{n}$ and
${\left(\psi_{n}\in\mathcal{S}\left(\mathcal{H}\right)\right)}_{n}$ satisfying
$\lim_{n\to\infty}\left\|\phi_{n}-\psi_{n}\right\|_{1}=0$, it holds that

$\lim_{n\to\infty}\left|R_{\mathcal{H}}\left(\phi_{n}\right)-R_{\mathcal{H}}\left(\psi_{n}\right)\right|=0.$
(176)

Our definition of asymptotic continuity implies continuity as a special case.

###  6.2 Generalization of Uniqueness Inequality

In this section, we show that we have the inequality $R_{\textup{D}}\leqq
R\leqq R_{\textup{C}}$ for a resource measure $R$ if $R$ satisfies
conventional normalization, asymptotic continuity, and weak additivity. We
call this inequality the uniqueness inequality. The uniqueness inequality is
originally proved in the QRT of bipartite entanglement in finite-dimensional
cases [49]. We show that the proof of this uniqueness inequality can be
generalized to all the QRTs in our framework that covers infinite-dimensional
cases. At the same time, we also show a QRT in which no resource measure
satisfies these properties; that is, the set of states satisfying the
uniqueness inequality becomes empty.