Fermat's Library | q-2020-11-01-355 annotated/explained version.

General Quantum Resource Theories: Distillation, Forma-

tion and Consistent Resource Measures

Kohdai Kuroiwa

1,2

and Hayata Yamasaki

3,4

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

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

Quantum resource theories (QRTs) provide a

uniﬁed theoretical framework for understand-

ing inherent quantum-mechanical properties

that serve as resources in quantum information

processing, but resources motivated by physics

may possess structure whose analysis is math-

ematically intractable, such as non-uniqueness

of maximally resourceful states, lack of con-

vexity, and inﬁnite dimension. We investigate

state conversion and resource measures in gen-

eral QRTs under minimal assumptions to ﬁg-

ure out universal properties of physically mo-

tivated quantum resources that may have such

mathematical structure whose analysis is in-

tractable. In the general setting, we prove

the existence of maximally resourceful states

in one-shot state conversion. Also analyzing

asymptotic state conversion, we discover cat-

alytic 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 forma-

tion of quantum resources, and introduce dis-

tillable resource and resource cost based on

the distillation and the formation, respectively.

Furthermore, we introduce consistent resource

measures that quantify the amount of quan-

tum resources without contradicting the rate

of state conversion even in QRTs with non-

unique maximally resourceful states. Pro-

gressing beyond the previous work showing

a uniqueness theorem for additive resource

measures, we prove the corresponding unique-

ness inequality for the consistent resource mea-

sures; that is, consistent resource measures of

a quantum state take values between the dis-

tillable resource and the resource cost of the

state. These formulations and results estab-

Kohdai Kuroiwa: kkuroiwa@uwaterloo.ca

Hayata Yamasaki: hayata.yamasaki@gmail.com

lish a foundation of QRTs applicable in a uni-

ﬁed way to physically motivated quantum re-

sources whose analysis can be mathematically

intractable.

1 Introduction

Advantages in quantum information processing com-

pared to conventional classical information process-

ing 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 quan-

tum properties are regarded as resources for over-

coming restrictions on operations on quantum sys-

tems; especially, manipulation and quantiﬁcation of

resources are integral parts of QRTs. QRTs have cov-

ered numerous aspects of quantum properties such

as entanglement [2–4], coherence [5–11], athermal-

ity [12–16], magic states [17, 18], asymmetry [19–

23], purity [24], non-Gaussianity [25–28], and non-

Markovianity [29, 30]. Recently, QRTs for a gen-

eral resource have been studied to ﬁgure out common

structures shared among known QRTs and to under-

stand the quantum properties systematically [31–40].

However, general QRTs are not necessarily math-

ematically tractable to analyze, and simply extend-

ing the formulation of a known QRT such as bi-

partite entanglement is insuﬃcient. For example,

maximal resources in the QRT of magic states [17]

and the QRT of coherence with physically incoher-

ent 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. Gaus-

sian operations [41, 42] are conventionally deﬁned on

a non-convex state space while existing QRTs of non-

Gaussianity [25–28] are formulated on a convex state

space. Furthermore, the state spaces of QRTs of non-

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arXiv:2002.02458v3 [quant-ph] 29 Oct 2020

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 in-

vestigate manipulation and quantiﬁcation of quantum

resources in general QRTs that are physically moti-

vated but hard to analyze. We do not make mathe-

matical 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 opera-

tions 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 quan-

tum state that may not be obtained by free opera-

tions 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 op-

erations 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 re-

source. 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. Fur-

thermore, 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 gener-

ate but can be replicated inﬁnitely by free operations

using a given copy catalytically. While catalytic con-

version of quantum resources is originally found in

entanglement theory [45], our discovery provides an-

other 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 forma-

tion of a resource [46], and introduce general deﬁni-

tions 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 resource-

ful states. To overcome this issue, we formulate the

distillable resource as how many resources can be ex-

tracted 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 distil-

lable resource being smaller than the resource cost.

A resource measure is a tool for quantifying re-

sources. 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 re-

source cost, which we call the uniqueness inequality.

In this paper, we show that the uniqueness inequal-

ity holds for a general QRT under the same prop-

erties 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 quantiﬁ-

cation 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 in-

equality 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 re-

source measure, generalizing the existing results in

reversible QRTs [32].

These formulations and results establish a frame-

work of general QRTs that are applicable even to

physically motivated restrictions on quantum oper-

ations whose analysis is mathematically intractable.

We here point out that whether a given QRT is phys-

ically 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 ma-

nipulation 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 contro-

versy over what QRTs are physically meaningful, so

that anyone can suitably enjoy the beneﬁt of the gen-

eral 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 system-

atic 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 gen-

eral QRTs. In Sec. 3, we investigate maximally re-

sourceful states and free states in general QRTs. In

Sec. 4, we analyze manipulation of quantum states in

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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 quan-

tum 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 the-

ories (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 def-

inition of a preorder, and introduce a preorder intro-

duced by free operations in QRTs. The readers who

are interested in QRTs on ﬁnite-dimensional quantum

systems and are familiar with ﬁnite-dimensional quan-

tum 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 me-

chanics that cover inﬁnite-dimensional quantum sys-

tems. Notice that some inherent properties of quan-

tum mechanics, such as non-Gaussianity [25–28],

are easier to formulate on an inﬁnite-dimensional

quantum system than its approximation by a ﬁnite-

dimensional quantum system. As for proofs of math-

ematical 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 H,

i.e., a complex inner product space that is also a com-

plete metric space with respect to the distance func-

tion 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 sys-

tem) of H as

= {|ki}

. (1)

In cases where H represents a D-dimensional system,

is a ﬁnite set with cardinality D, while B

can

be an uncountable set in this paper.

We use a subset of operators on H, in partic-

ular, the von Neumann algebra, to describe quan-

tum mechanics on the system represented by H,

since H can be inﬁnite-dimensional. Let L(H)

denote the set of linear operators on H. Let

B(H) ⊂ L(H) denote the set of bounded oper-

ators, that is, for any A ∈ B (H), the operator

norm kAk

∞

= sup {kA |vik

: |vi ∈ H, k|vik

5 1}

is bounded, where k·k

denotes the norm induced by

the inner product of H.

To deﬁne trace-class operators, we use the trace de-

ﬁned as Tr T

|ki∈B

hk |T |ki, where B

is an

orthonormal basis of H deﬁned in (1), each term on

the right-hand side denotes the inner product of T |ki

and |ki on H, and if B

is an inﬁnite set, the summa-

tion on the right-hand side means the limit of a net,

i.e., a generalization of sequence. While a sequence

: n ∈ N) is indexed by a natural number, that

is, a countably inﬁnite and totally ordered set, a net

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

⊂ B

and deﬁned

|ki∈B

hk |T |ki, which approaches to the limit

Tr T as B

gets larger. Note that Tr T is independent

of the choice of B

. Let T (H) ⊂ B(H) denote the set

of trace-class operators; that is, for any T ∈ T (H),

we have ﬁnite and hence well-deﬁned Tr |T |, where

|T |

√

†

T , and T

†

denotes the conjugation of T .

For any T ∈ T (H), the trace norm of T is deﬁned as

kT k

= Tr |T |. (2)

To deﬁne the von Neumann algebra, we need to

discuss convergence of bounded operators mathemat-

ically. To discuss convergence of bounded operators,

we need a topology deﬁned for B(H), and we use the

ultraweak operator topology. The ultraweak operator

topology of B(H) is a topology where any sequence

, A

, . . . ∈ B(H), or more generally, any net A

converges to A if and only if Tr [ T A

] converges to

Tr [T A] for any T ∈ T (H) . A von Neumann alge-

bra M on B (H) is a subset of B(H) (or B(H) itself)

that contains the identity operator on 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 com-

mutative von Neumann algebra for a classical system.

To describe quantum mechanics on H, we use a set

of operators represented as a von Neumann algebra

M on B (H). For example, for any ﬁnite-dimensional

Hilbert space H, the algebra of all the linear opera-

tors L(H) = B(H) is a von Neumann algebra, which

suﬃces to describe the ﬁnite-dimensional quantum

mechanics. More generally, for any H that can be

inﬁnite-dimensional, the algebra of all the bounded

operators B(H) is a von Neumann algebra. In this

paper, a system H is always accompanied with a set

of operators M therein, where we may implicitly con-

sider M = B(H) unless stated otherwise.

Given that a quantum state associates a measure-

ment of an observable with a probability of a measure-

ment outcome, we introduce a quantum state using a

linear functional from an operator to a scalar. In par-

ticular, for a system H with M, a state is deﬁned as a

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linear functional f

: M → C that is positive semidef-

inite f



†



= 0, ∀M ∈ M, satisﬁes the normal-

ization condition f

( ) = 1, and is also normal, i.e.,

continuous in terms of the ultraweak operator topol-

ogy. We require this continuity in order for f

to be

in the predual M

∗

of M, i.e., the space whose dual

∗

)

∗

equals (can be identiﬁed with) M. Given this

duality and under the condition of M = B (H), we

identify f

with the operator ψ ∈ T (H) that satisﬁes

for any M ∈ M

Tr [ψM ] = f

(M) . (3)

Note that we have one-to-one correspondence between

and ψ if M = B(H). This operator ψ is the

density operator representing the state f

, and let

D(H)

= {ψ ∈ T (H) : ψ = 0, Tr ψ = 1} denote the

set of density operators on H with M = B(H). For

simplicity, we may call ψ a quantum state, rather than

We introduce a quantum channel on a system H

with M = B(H) in the Heisenberg picture as a com-

pletely positive and unital linear map on 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 H.

Given two systems H

(in)

with M

(in)

and H

(out)

with

(out)

representing the spaces of the input and the

output respectively, a channel

E : M

(out)

→ M

(in)

the Heisenberg picture is deﬁned as a linear map that

is completely positive

n−1

j,k=0

(in)

†



(out)

†

(out)



(in)

= 0,

∀M

(in)

, . . . , M

(in)

n−1

∈ M

(in)

∀M

(out)

, . . . , M

(out)

n−1

∈ M

(out)

∀n ∈ N, (4)

unital



(out)



(in)

, and normal, i.e., continu-

ous in terms of the ultraweak operator topologies of

(out)

and M

(in)

, where

(out)

and

(in)

are the iden-

tity operators on H

(out)

and H

(in)

, respectively. In the

same way as the identiﬁcation (3) of a functional f

of a state with the density operator ψ of the state, un-

der the conditions of M

(in)

= B



(in)



and M

(out)



(out)



, we identify a channel

E : M

(out)

→

(in)

in the Heisenberg picture with the channel

E : T



(in)



→ T



(out)



in the Schrödinger pic-

ture that satisﬁes for any M

(out)

∈ M

(out)

E (ψ) M

(out)



◦





(out)



, (5)

where ψ and f

are related as (3). Note that E is a

completely positive and trace-preserving (CPTP) lin-

ear map by deﬁnition, and if M

(in)

= B



(in)



and

(out)

= B



(out)



, we have one-to-one correspon-

dence between

E and E.

The set of channels from an input system H

(in)

with M

(in)

to an output system H

(out)

with M

(out)

is denoted by C



(in)

→ H

(out)



, which we may write

C (H) if H = H

(in)

= H

(out)

. In this paper, we use

the Schrödinger picture with M

(in)

= B



(in)



and

(out)

= B



(out)



; that is, C



(in)

→ H

(out)



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 classi-

cal 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 con-

verges in terms of the topology, where a subnet gen-

eralizes 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 con-

verges 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 dif-

ferent topologies can be deﬁned for the set of states.

The weak operator topology of T (H) is a topology

where any net T

in T (H) converges to T if and only if

Tr [T

A] converges to Tr [T A] for any A ∈ B(H). The

trace norm topology of T (H) is a topology where any

net T

in T (H) converges to T if and only if kT

− T k

converges to 0. The trace norm topology is stronger

than the ultraweak operator topology, and the ultra-

weak operator topology is stronger than the weak op-

erator 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 com-

pactness 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, com-

pactness 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 H

is ﬁnite-dimensional, D(H) is compact, while D (H)

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



(in)

→ H

(out)



, and we use the bounded weak

(BW) topology. The bounded weak topology of



(in)

→ H

(out)



is the weakest topology such

that for any f

∈ M

(in)

∗

and M

(out)

∈ M

(out)

a map S

ψ,M

(out)

: C



(in)

→ H

(out)



→ C given

by S

ψ,M

(out)

(E) =



◦



(out)



is continuous,

where E and

E are related as (5). Note that if H

(in)

and H

(out)

are ﬁnite-dimensional, C



(in)

→ H

(out)



is compact, while C



(in)

→ H

(out)



for inﬁnite-

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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 con-

sider a compact set of the free operations. We also

present justiﬁcation of the compactness by examples

from the perspective of indistinguishability. To rep-

resent the state set of interest in a QRT, e.g., the set

of pure states on ﬁnite-dimensional H, we consider a

compact set of quantum states chosen as desired

S (H) ⊆ D(H) . (6)

Note that the quantum system H can be inﬁnite-

dimensional as we have introduced in Sec. 2.1.

Free operations in our formulation are introduced

as follows [1]. Let O be a mapping that takes two

quantum systems H

(in)

and H

(out)

and outputs a com-

pact set of completely positive and trace-preserving

(CPTP) maps from S



(in)



to S



(out)



. This set

is denoted by



(in)

→ H

(out)



⊆ C



(in)

→ H

(out)



. (7)

A map contained in O



(in)

→ H

(out)



is called a free

operation from H

(in)

to H

(out)

. If the input space

and output space are the same quantum system H,

we write the set of free operations from H to H as

O(H) ⊆ C (H). We consider a compact set because

two arbitrarily close CPTP maps are indistinguish-

able by any protocol in a task of channel discrimina-

tion [52], as we will discuss below by examples. We

assume that O satisﬁes the following axioms of QRTs:

1. Let H

, H

and H

be arbitrary quantum

systems. For any M ∈ O(H

→ H

) and

N ∈ O(H

→ H

), it holds that N ◦ M ∈

O(H

→ H

), where ◦ represents the composi-

tion.

2. Let H

(in)

, H

(out)

, H

(in)

and H

(out)

be arbitrary

quantum systems. For any M ∈ O(H

(in)

→

(out)

) and N ∈ O(H

(in)

→ H

(out)

), it holds that

M ⊗ N ∈ O(H

(in)

⊗ H

(in)

→ H

(out)

⊗ H

(out)

3. Let H be an arbitrary quantum system. Then,

it holds that id ∈ O (H), where id is an identity

map.

4. Let H be an arbitrary quantum system. Then,

it holds that Tr ∈ O(H → C), where 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. We always have access to free operations and can

use free operations as many times as necessary.

2. We can arbitrarily apply free operations to a

quantum system regardless of what free opera-

tions are applied to another quantum system.

3. Doing nothing is free.

4. Ignorance is free.

Remark 1 (Operations Not Satisfying the Axioms).

There can be classes of operations that do not sat-

isfy the axioms stated above. For example, Refs. [53]

and [32] consider -resource non-generating opera-

tions. However, the composition of two -resource

non-generating operations is not necessarily an -

resource non-generating operation, which implies the

set of -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 op-

erations 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 imple-

mentable in the setting of LOCC, as discussed in Ex-

ample 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 ar-

bitrary precision by this gate set. Note that the com-

pactness 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 imple-

mentable 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 O

LOCC



(in)

→ H

out



be the set

of LOCC from H

(in)

to H

(out)

. It is known

that O

LOCC



→ C



is not closed; that is,

LOCC

→ C

) 6= O

LOCC



→ C



[55]. In this

case, we use O



→ C



LOCC

→ C

) as

the set of free operation because for any CPTP

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map N ∈ O

LOCC

→ C

) \O

LOCC



→ C



and

any  > 0, we can construct a CPTP map

N ∈

LOCC



→ C



that is indistinguishable from N

up to an  probability by any protocol in a task of

channel discrimination [52].

In the next example, we consider a situation of uni-

versal 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 sys-

tems H

(in)

and H

(out)

, we deﬁne O



(in)

→ H

(out)



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 π/8 phase gate T , ap-

pending |0i, and the measurement in the computa-

tional basis. Here, a ﬁnite-depth circuit refers to

a d-depth circuit for some integer d = 0, and op-

erations conditioned on measurement outcomes are

allowed. Conventionally, combination of these op-

erations can be considered to be universal because



(in)

→ H

(out)



is dense in the set of all the chan-

nels C



(in)

→ H

(out)



, but O



(in)

→ H

(out)



diﬀerent from C



(in)

→ H

(out)



since a ﬁnite num-

ber of these gates cannot exactly implement qubit ro-

tation at an arbitrary angle [52, 56]. In this case, our

framework may use the closure of the set of opera-

tions 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 O



(in)

→ H

(out)



= C



(in)

→ H

(out)



We do not assume the convexity of the set of free op-

erations 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 pre-

order of resourcefulness of quantum states introduced

by free operations.

Suppose that S is a set and  is a binary relation

on S. Then, the relation  is called a preorder if it

holds that

a  a, (8)

a  b, b  c ⇒ a  c (9)

for all a, b, c ∈ S. This preorder introduces maximal

elements and minimal elements in the set S. If a ∈ S

satisﬁes

b  a ⇒ a  b (10)

for all b ∈ S, a is called a maximal element. Similarly,

if a ∈ S satisﬁes

a  b ⇒ b  a (11)

for all b ∈ S, a is called a minimal element. Intu-

itively, an element a ∈ S is maximal/minimal if a is

the largest/smallest among all the elements that can

be compared with a.

For any quantum system H, free operations intro-

duce a preorder  on S (H). This preorder is intro-

duced in terms of the exact one-shot state conversion

under free operations. Given two states φ, ψ ∈ S (H),

the exact one-shot state conversion from φ to ψ is a

task of transforming a single φ exactly into a single ψ

by a free operation N ∈ O (H). Formally, we write

φ  ψ (12)

if there exists a free operation N ∈ O (H) such that

N (φ) = ψ. (13)

This relation  is indeed a preorder because it holds

that

φ  φ, (14)

φ  ψ, ψ  σ ⇒ φ  σ (15)

for any states φ, ψ, σ ∈ S (H).

With respect to this preorder, two states φ, ψ ∈

S (H) are said to be equivalent if both φ  ψ and

φ  ψ hold. If φ and ψ are equivalent, we write

φ ∼ ψ. (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 ex-

istence 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.

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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 max-

imally resourceful states is not trivial in general,

although it is desired for quantiﬁcation of the re-

source. We prove the existence of maximally resource-

ful 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 H, we let G (H) denote the set of

the maximal states of S (H) in terms of the preorder

deﬁned as (12), that is,

G (H)

= {φ ∈ S (H) : ∀ψ ∈ S (H) , ψ  φ ⇒ φ  ψ}.

(17)

The elements of G (H) are called maximally resource-

ful 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 maxi-

mally 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 ex-

ist 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 can-

not change diagonal elements of a quantum state rep-

resented 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 [8, 46, 61]. For example, in the QRT of bipar-

tite 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 H is a quantum system. For any state

ψ ∈ S (H), there exists a state φ ∈ G (H) that upper-

bounds ψ; that is, ψ  φ.

Proof. It is known that a compact space X with a pre-

order  has a maximal element if the upper closure

= {y ∈ X|x  y} is closed for any x ∈ X [62]

(Proposition VI-1.6.(i)). Thus, it suﬃces to show

= {φ ∈ S (H) |ψ  φ} 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 (φ

)

n∈N

in U

convergent to φ ∈ S (H) in terms of the trace norm

topology and prove φ ∈ U

. By the deﬁnition of the

preorder , for each n ∈ N, there exists a free op-

eration N

∈ O(H) such that ψ = N

(φ

). Since

O(H) is compact in terms of the bounded weak (BW)

topology, there exists a subnet



n(i)



i∈I

convergent

to some N ∈ O(H) with respect to the BW topol-

ogy. In the following, we show ψ = N(φ) to prove the

theorem.

Take an arbitrary  > 0 and an arbitrary A ∈

B(H) \ {0}, which satisﬁes kAk

∞

> 0. Since S (H)

is compact in terms of the trace norm topology, there

exists a ﬁnite subset {χ

: k ∈ {1, . . . , N



}} of S (H)

such that for any χ ∈ S(H)

min

k∈{1,...,N



}

kχ − χ



kAk

∞

. (18)

By deﬁnition of the convergence N

n(i)

−−→ N in

terms of the BW topology with respect to i, there

exists i

,A

∈ I such that for any i = i

,A

max

k∈{1,...,N



}





n(i)

(χ

) A



− Tr (N (χ

) A)



< .

(19)

Thus, for any i = i

,A

and any χ ∈ S(H), we have





n(i)

(χ) A



− Tr (N (χ) A)





n(i)

(χ − χ

) A



+ Tr



n(i)

(χ

) A



−Tr (N (χ − χ

) A) − Tr (N (χ

) A)|





n(i)

(χ − χ

) A





+ |Tr (N (χ − χ

) A)|





n(i)

(χ

) A



− Tr (N (χ

) A)





n(i)

(χ − χ

) A





+ |Tr (N (χ − χ

) A)| + ,

(20)

where χ

is an element in the ﬁnite subset {χ

} of

S (H) in (18) satisfying

kχ − χ



kAk

∞

, (21)

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



n(i)



∞

= sup





n(i)

(T )



: T ∈ T (H), kT k

5 1



(22)

denoting the operator norm of the linear map N

n(i)

T (H) → T (H), we have





n(i)

(χ − χ

) A





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

n(i)

(χ − χ

)



· kAk

∞



n(i)



∞

· kχ − χ

· kAk

∞

< 1 ·



kAk

∞

· kAk

∞

= , (23)

where the last inequality follows from the fact that

any CPTP map N

n(i)

satisﬁes



n(i)



∞

= sup



n(i)

(T )



5 sup kT k

5 1. (24)

In the same way as (23) by substituting N

n(i)

with

N, it holds that

|Tr (N (χ −χ

) A)| < . (25)

Therefore, applying (23) and (25) to (20), for any

i = i

,A

and any χ ∈ S(H), we have





n(i)

(χ) A



− Tr (N (χ) A)



<  +  +  = 3.

(26)

Consequently, for any i = i

,A

, we obtain

|Tr ((ψ − N (φ)) A)|

= |Tr (ψA) −Tr (N (φ) A)|





n(i)



n(i)



− Tr (N (φ) A)





n(i)



n(i)



− Tr



n(i)







n(i)



− Tr (N (φ) A)



< 3 + kNk

∞



n(i)

− φ



· kAk

∞

→ 3, (27)

where the last inequality follows from (26) by substi-

tuting χ with φ

n(i)

and from the inequality shown in

the same way as (23)





n(i)



− Tr (N (φ) A)



5 kNk

∞



n(i)

− φ



· kAk

∞

, (28)

and the limit in the last line in terms of i yields



n(i)

− φ



→ 0. Since  > 0 and A ∈ B(H) \ {0}

are arbitrary, this shows ψ = N(φ). Q.E.D.

Remark 2. In a similar manner, we can prove that the

set of minimal elements

{φ ∈ S (H) : ∀ψ ∈ S (H) , φ  ψ ⇒ ψ  φ} (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 fol-

lowing 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 F (H)

denote the set of free states; that is,

F (H)

ψ ∈ S (H) : ∀H

, ∀φ ∈ S (H

) ,

∃N ∈ O(H

→ H) s.t. ψ = N (φ)

(30)

A state ψ ∈ S(H) \ F(H) that is not free is called a

resourceful state or a resource state. Since 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 ∈ S (C)

as shown in the following proposition.

Proposition 2. Let H be a quantum system. Then,

it holds that

F (H) =

ψ ∈ S (H) : ∃N ∈ O(C → H) s.t. ψ = N (1)

(31)

Proof. By the deﬁnition (30) of F (H), it trivially

holds that

F (H) ⊆

ψ ∈ S (H) : ∃N ∈ O(C → H) s.t. ψ = N (1)

(32)

To show the converse inclusion, assume that

ψ ∈

ψ ∈ S (H) : ∃N ∈ O(C → H) s.t. ψ = N (1)

(33)

Let N ∈ O(C → H) be a free operation such that

ψ = N (1). Consider an arbitrary quantum system

and an arbitrary state φ ∈ S (H

). Since Tr ∈

O(H

→ C), it holds that

ψ = N ◦ Tr (φ) . (34)

Therefore, ψ ∈ F (H), which yields the conclusion.

Q.E.D.

The set of free states F (H) may be empty for some

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 O(C → H) does not contain any

operation for a quantum system H, then F (H) = ∅.

The following example gives a more concrete sce-

nario, where we take the logical 2-dimensional space

of the Gottesman-Kitaev-Preskill (GKP) code [63] as





. 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 F (H) is

empty or not.

Example 5 (QRT of Non-Gaussianity on GKP

Code). The QRT of non-Gaussianity has applica-

tions to analyzing continuous-variable quantum com-

putation using the Gottesman-Kitaev-Preskill (GKP)

code as shown in Ref. [28]. The GKP code en-

codes 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

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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 S





is the set of logical

states in the logical 2-dimensional space of the GKP

code. Take the quantum operations on S





im-

plementable by the Gaussian operations as the free

operations. Any physical state of the GKP code is

non-Gaussian, and hence in this case, F only has the

trivial element 1; that is, F(H) = {1} if dim H = 1,

and F(H) = ∅ otherwise.

The following proposition guarantees that a max-

imally resourceful state cannot be a free state if a

resource state exists.

Proposition 3. Let H be a quantum system. Suppose

that the set of resource state is not empty; that is

S (H) \ F (H) 6= ∅. Then, it holds that

G (H) ∩F (H) = ∅. (35)

Proof. The proof is by contradiction. To prove (35),

assume that φ ∈ G (H)∩F (H). Take a resource state

ψ ∈ S (H) \ F (H) . (36)

Since φ ∈ F (H), it holds that ψ  φ. Then, since

φ ∈ G (H), it holds that φ  ψ. Therefore, ψ is also a

free state; that is, ψ ∈ F (H), which contradicts (36).

Q.E.D.

We can observe that some properties of the set of

free states F (H) are inherent in the set of the free

operations O (H). The compactness of O (H) leads

to the compact set of free states F (H). If O(H) is

convex, F (H) 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 funda-

mental limit of large-scale quantum information pro-

cessing 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 re-

sources 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 exam-

ple 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 func-

tion is denoted by d···e, and the ﬂoor function is

denoted by b···c.

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 H

and H

, and two

quantum states φ ∈ S (H

) and ψ ∈ S (H

), asymp-

totic state conversion from φ to ψ is a task of trans-

forming inﬁnitely many copies of φ into as many

copies of ψ as possible by a sequence of free oper-

ations N

, N

, . . . within a vanishing error. There

are two possible ways to deﬁne state conversion rates

from φ to ψ: how many ψ’s can be generated from

a single φ, and how many φ’s are necessary to gen-

erate a single ψ. We write the ﬁrst conversion rate

as r

conv

(φ → ψ), and the second conversion rate as

conv

(φ → ψ). As will be shown in Theorem 4, these

two conversion rates are related to each other in such a

way that r

conv

(φ → ψ) is the inverse of r

conv

(φ → ψ).

Therefore, we consider r

conv

(φ → ψ) as the asymp-

totic state conversion rate in this paper.

More formally, r

conv

(φ → ψ) is deﬁned as follows.

A set of asymptotic achievable rates is deﬁned as

R(φ → ψ)

r = 0 : ∃



∈ O



⊗n

→ H

⊗drne



: n ∈ N



lim inf

n→∞





⊗n



− ψ

⊗drne



= 0

(37)

where φ

⊗0

= 1. Intuitively, achievable rate r = 0

is a positive number for which we can generate rn

copies of ψ from n copies of φ. However, rn is not

necessarily an integer in general; therefore, in (37), we

regard r as an achievable rate when we can generate

drne copies of ψ, which guarantees that we can obtain

rn or more ψ’s. An asymptotic state conversion rate

conv

(φ → ψ) is deﬁned as

conv

(φ → ψ)

= sup R(φ → ψ) . (38)

Similarly, we can consider the other deﬁnition of a

state conversion rate r

conv

(φ → ψ). Here, we deﬁne

another set of asymptotic achievable rates

(φ → ψ)

r = 0 : ∃



∈ O



⊗brnc

→ H

⊗n



: n ∈ N



lim inf

n→∞





⊗brnc



− ψ

⊗n



= 0

(39)

Here, achievable rate r

= 0 is a positive number for

which we can generate n copies of ψ from r

n copies of

φ. However, r

n is not necessarily an integer in gen-

eral; therefore, in (39), we regard r

as an achievable

rate when we can generate n copies of ψ from br

copies of φ, which guarantees that r

n or more copies

of φ suﬃce to generate n copies of ψ. With respect

to this deﬁnition of achievable rates, an asymptotic

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conversion rate r

conv

(φ → ψ) is deﬁned as

conv

(φ → ψ)

= inf R

(φ → ψ) , (40)

where r

conv

(φ → ψ) is inﬁnity if the set on the right-

hand side is empty.

These two conversion rates r

conv

(φ → ψ) and

conv

(φ → ψ) are related to each other as shown

in the following theorem. Hereafter, we will use

conv

(φ → ψ) as the asymptotic states conversion rate

rather than r

conv

(φ → ψ).

Theorem 4 (Relation Between Two Conversion

Rates). Let H and H

be quantum systems. For any

states φ ∈ S (H) and ψ ∈ S (H

), it holds that,

conv

(φ → ψ) =

conv

(φ → ψ)

, (41)

where we regard 1/0 = ∞.

Proof. It suﬃces to show that

r ∈ R(φ → ψ) ⇒

∈ R

(φ → ψ) , (42)

and that

r ∈ R

(φ → ψ) ⇒

∈ R(φ → ψ) . (43)

First, assume that r ∈ R(φ → ψ) to show (42).

Choose a ﬁxed positive real number  > 0. Let n

be an arbitrary positive integer such that





⊗n



− ψ

⊗drne



< . (44)

Let n

= drne. Because n 5 bn

/rc, we can deﬁne a

free operation M

as the partial trace over bn

/rc−n

systems so that



⊗



= φ

⊗n

. (45)

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



◦ M



⊗



− ψ

⊗n



< , (46)

and 1/r ∈ R

(φ → ψ) follows.

On the other hand, assume that r ∈ R

(φ → ψ) to

show (43). Choose a ﬁxed positive real number  > 0.

Let n be an arbitrary positive integer such that





⊗brnc



− ψ

⊗n



<  (47)

Let n

= brnc. Because n = dn

/re, we can deﬁne a

free operation M

as the partial trace over n−dn

/re

systems so that



⊗n



= ψ

⊗

. (48)

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



◦ N



⊗n



− ψ

⊗



◦ N



⊗n



− M



⊗n







⊗n



− ψ

⊗n



< ,

(49)

and 1/r ∈ R(φ → ψ) follows. Q.E.D.

Remark 3. Conventionally, a conversion rate may also

be deﬁned as [1, 5]

˜r

conv

(φ → ψ)

= sup

R(φ → ψ) , (50)

where

R(φ → ψ)

r = 0 : ∃



∈ O



⊗n

→ H

⊗brnc



: n ∈ N



lim inf

n→∞





⊗n



− ψ

⊗brnc



= 0

(51)

instead of a conversion rate r

conv

(φ → ψ) deﬁned in

(37) and (38). In (51), the ﬂoor function is used in-

stead of the ceiling function used in (37).

As we show in the following, these two conversion

rates r

conv

(φ → ψ) and ˜r

conv

(φ → ψ) are identical;

however, considering the meaning of r

conv

(φ → ψ)

discussed below (38), we adopt r

conv

(φ → ψ) in this

paper. To see that the two conversion rates are identi-

cal, ﬁrst suppose that r ∈ R(φ → ψ) for φ, ψ ∈ S(H).

Since drne = brnc and since the partial trace is free, it

holds that r ∈

R(φ → ψ). Conversely, suppose that

r ∈

R(φ → ψ). Then, the fact that

brnc = drne − 1 =



r −





(52)

implies that r − 1/n can be regarded as a achievable

rate with respect to R(φ → ψ). Since r − 1/n ap-

proaches to r as n becomes large, considering the

deﬁnition (37) of R(φ → ψ), it can be concluded

that r ∈ R(φ → ψ). Therefore, r

conv

(φ → ψ) and

˜r

conv

(φ → ψ) are identical.

In the same way, we may replace the ﬂoor function

in (39) with the ceiling function to obtain the identi-

cal 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 conver-

sion rates given in Ref. [66]. Let H

, H

and H

quantum systems. Let ρ ∈ S (H

), σ ∈ S (H

) and

ω ∈ S (H

) be quantum states. Suppose that we ﬁrst

asymptotically generate σ from ρ, then we generate

ω from σ to achieve conversion from ρ to ω. While

this protocol generate ω from ρ, the protocol is not

necessarily optimal. In fact, it is known that

conv

(ρ → ω) = r

conv

(ρ → σ) r

conv

(σ → ω) . (53)

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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 re-

source. One of the fundamental principles of quan-

tum 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 ﬁg-

ure 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 re-

source state is known but the operation is restricted

to the set of free operations. In terms of the asymp-

totic state conversion, the replication of a resource is

regarded as a catalytic state conversion between the

same state similarly to catalytic transformation of en-

tanglement [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 de-

pends on a resource measure, which will be discussed

in detail in Sec. 6.

Theorem 5 (Replication of State). Let H be a quan-

tum system. For any state ψ ∈ S (H), r

conv

(ψ → ψ)

is equal to either 1 or +∞.

Proof. It trivially holds that

conv

(ψ → ψ) = 1 (54)

because id ∈ O(H).

Assume that r

conv

(ψ → ψ) > 1, that is, there ex-

ists r > 1 such that r ∈ R(ψ → ψ). To prove

conv

(ψ → ψ) = ∞, it suﬃces to show that

2r −1 ∈ R(ψ → ψ) (55)

because if (55) holds, an arbitrarily large rate can be

achieved by exploiting (55) repeatedly.

Choose a ﬁxed positive real number .

There exists a sequence of free operations



∈ O



⊗n

→ H

⊗drne



: n ∈ N



such that





⊗n



− ψ

⊗drne





(56)

holds for an inﬁnitely large subset of N.

Deﬁne

= inf

, (57)

where

= max {r

= 0 : dr

ne = 2 drne − n}, (58)

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

For n satisfying (56), it holds that

= 2r − 1 +

2α

, (59)

where α

is a real number satisfying 0 5 α

< 1 and

drne = rn + α

. The term 2α

/n approaches to zero

as n approaches to inﬁnity. Then, it holds that

inf

= inf



2r −1 +

2α



= 2r − 1.

(60)

Therefore, it holds that

= 2r − 1. (61)

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

∈ R(ψ → ψ) . (62)

Now, observe that for any n,

drne > n (63)

always holds. This implies that for any n,

2 drne − n > drne (64)

holds.

For n satisfying (56) and (63), we deﬁne a free oper-

ation M

as the partial trace over (2 drne− n)−dr

systems so that we can obtain



⊗2drne−n



= ψ

⊗dr

. (65)

Therefore, from the triangle inequality, it follows that



◦ (N

⊗ id) ◦ N



⊗n



− ψ

⊗dr



◦ (N

⊗ id) ◦ N



⊗n



−M

◦ (N

⊗ id)



⊗drne





◦ (N

⊗ id)



⊗drne



− ψ

⊗dr



(66)

where id is the identity map on drne − n systems.

Since the trace distance is non-increasing for quantum

operations, it holds that



◦ (N

⊗ id) ◦ N



⊗n



−M

◦ (N

⊗ id)



⊗drne







⊗n



− ψ

⊗drne



(67)

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and that



◦ (N

⊗ id)



⊗drne



− ψ

⊗dr



⊗ id)



⊗(n+(drne−n))



− ψ

⊗(2drne−n)







⊗n



− ψ

⊗drne



⊗ ψ

⊗(drne−n)





⊗n



− ψ

⊗drne



(68)

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



◦ (N

⊗ id) ◦ N



⊗n



− ψ

⊗dr



5 2





⊗n



− ψ

⊗drne



< ,

(69)

which implies that r

∈ R(ψ → ψ). Q.E.D.

Remarkably, we here give an example where

conv

(ψ → ψ) = ∞, but ψ is not a free state, that

is, ψ 6∈ F (H). In this paper, we call a state ψ that

satisﬁes r

conv

(ψ → ψ) = ∞ and ψ /∈ F (H) a catalyt-

ically replicable state. A catalytically replicable state

is regarded as a form of catalytic property of quantum

resources, which are similar to catalytic state conver-

sion in the entanglement theory [45]. Any free state

ψ is a trivial example of r

conv

(ψ → ψ) = ∞, but the

following example shows that this is not the whole

story; that is, r

conv

(ψ → ψ) = ∞ implies that ψ is

free or catalytically replicable.

Example 6 (Catalytically Replicable Resource).

Suppose that S





= {|0ih0|, |1ih1|}. Further sup-

pose that the set of free operations O consists of op-

erations that are realized by circuits composed of the

identity gate, the partial trace, the controlled-NOT

gate, the preparation of an auxiliary qubit in |0i state.

For any integer n = 0, the set of free states is







⊗n



|0ih0|

⊗n

. (70)

In this case, whereas |1ih1| 6∈ F





conv

(|1ih1| → |1ih1|) = +∞ because we can

convert |1ih1| into |1ih1|

⊗n

for any n by appending

an auxiliary system prepared in |0i and applying

controlled-NOT repeatedly.

4.3 Relations Between One-Shot State Con-

version 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 φ to ψ is a task transforming inﬁnitely many

copies of φ into many copies of ψ with a vanishing

error, while the exact one-shot conversion φ to ψ is a

task transforming a single φ into a single ψ exactly.

We prove two propositions both of which give re-

lations between the asymptotic state conversion and

the exact one-shot state conversion. The ﬁrst proposi-

tion 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 proposi-

tion 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 H

be quantum systems. Let

φ, ψ ∈ S (H

) and ρ ∈ S (H

) be quantum states. If

φ  ψ, then it holds that

conv

(ρ → φ) 5 r

conv

(ρ → ψ) (71)

conv

(ψ → ρ) 5 r

conv

(φ → ρ) . (72)

Proof. To prove (71), it suﬃces to show that

R(ρ → φ) ⊆ R(ρ → ψ). Suppose that r ∈

R(ρ → φ). Then, there exists a sequence of free oper-

ations



∈ O



⊗n

→ H

⊗drne



: n ∈ N



such that

for arbitrary  > 0,





⊗n



− φ

⊗drne



<  (73)

holds for an inﬁnitely large subset of N. As φ  ψ,

there exists a free operation N such that

N (φ) = ψ. (74)

Deﬁne a sequence of free operations



∈ O



⊗n

→ H

⊗drne



: n ∈ N



= N

⊗drne

◦ N

. (75)

Then, for any n satisfying (73),





⊗n



− ψ

⊗drne



⊗drne

◦ N



⊗n



− N

⊗drne



⊗drne







⊗n



− φ

⊗drne



< 

(76)

holds, and this implies that r ∈ R(ρ → ψ).

Then, we prove (72). Note that (72) is equiv-

alent to r

conv

(φ → ρ) 5 r

conv

(ψ → ρ) because of

Theorem 1. It suﬃces to show that R

(ψ → ρ) ⊆

(φ → ρ). Suppose that r ∈ R

(ψ → ρ).

Then, there exists a sequence of free operations



∈ O



⊗brnc→H

⊗n



: n ∈ N



such that for arbi-

trary  > 0,





⊗brnc



− ρ

⊗n



<  (77)

holds for an inﬁnitely large subset of N. As φ  ψ,

there exists a free operation N

such that

(φ) = ψ. (78)

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Deﬁne a sequence of free operations



∈ O



⊗brnc

→ H

⊗n



: n ∈ N



= N

◦ N

0⊗brnc

. (79)

Then, for any n satisfying (77),





⊗brnc



− ρ

⊗n



◦ N

0⊗brnc



⊗brnc



− ρ

⊗n





⊗brnc



− ρ

⊗n



< 

(80)

holds and this implies that r ∈ R

(φ → ρ). Q.E.D.

Next, we investigate the other relation between

the asymptotic conversion and the exact one-shot

conversion, i.e., the asymptotic state conversion be-

tween two equivalent states. Asymptotically, we may

achieve conversion between states that are not con-

vertible to each other in the one-shot state conver-

sion. One may wonder whether we can achieve a bet-

ter asymptotic conversion rate between states that are

equivalent under the one-shot conversion. The follow-

ing 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 H be a quantum system. Let

ψ, φ ∈ S (H) be quantum states such that ψ ∼ φ.

Suppose that r

conv

(ψ → ψ) = 1. Then, it holds that

conv

(ψ → φ) = r

conv

(φ → ψ) = 1. (81)

Proof. Since r

conv

(ψ → ψ) = 1, it follows that

conv

(ψ → φ) r

conv

(φ → ψ) 5 r

conv

(ψ → ψ) = 1

(82)

from (53). On the other hand, since ψ ∼ φ, there exist

free operations M ∈ O(H) and N ∈ O(H) such that

M(ψ) = φ (83)

N (φ) = ψ, (84)

which implies that

conv

(ψ → φ) = 1 (85)

conv

(φ → ψ) = 1. (86)

Therefore, both r

conv

(ψ → φ) and r

conv

(φ → ψ) must

be equal to 1. Q.E.D.

5 Distillable Resource and Resource

Cost

In this section, we analyze properties of the distillable

resource R

and the resource cost R

, which rep-

resent 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 de-

ﬁne the distillable resource R

as how many resource-

ful states can be generated from a state in the worst-

case scenario, and we deﬁne the resource cost R

how many resourceful states are needed to generate

a state in the best-case scenario. Our deﬁnition for-

mulates distillation and formation of a resource even

in cases where maximally resourceful states are not

unique. In Sec. 5.2, we analyze distillation and for-

mation 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 investi-

gate the resource cost, and prove that an upper bound

of the resource cost is achievable by a maximally re-

sourceful state if the number of non-equivalent max-

imally resourceful states is ﬁnite. In Sec. 5.5, gen-

eralizing 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 Distill-

able Resource

In this section, we provide a formulation of distilla-

tion and formation of a resource, and give the def-

initions of the distillable resource and the resource

cost, generalizing those in known QRTs such as bipar-

tite entanglement [46, 47], coherence [48], and ather-

mality [13]. In contrast with the deﬁnition in these

previous works, our deﬁnitions are applicable to gen-

eral QRTs, where maximally resourceful states are not

necessarily unique. Our deﬁnition of the distillable

resource represents how many resources can be gen-

erated in the worst case, and the deﬁnition of the re-

source 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 H and a

state ψ ∈ S (H), distillation from the state ψ is a task

of extracting many copies of a state φ ∈ S (H) from

many copies of ψ, where φ is a state that is the most

diﬃcult to generate from ψ. More formally, distilla-

tion is regarded as state conversion from ψ to a state φ

for which r

conv

(ψ → φ) takes a minimum value. Sim-

ilarly, formation of ψ is a task of generating many

copies of ψ from many copies of a state φ ∈ S (H)

where φ is a state that can the most easily generate

ψ. Formation is regarded as state conversion from φ

to a state ψ, where r

conv

(φ → ψ) takes a maximum

value for φ.

The distillable resource R

represents the amount

of resource obtained by distillation; the resource cost

represents the amount of resource needed for for-

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mation of a state. Formally, the distillable resource

of any state ψ ∈ S (H) is deﬁned as

(ψ)

= inf

φ∈S(H)

conv

(ψ → φ) R

(H)

max

, (87)

where R

(H)

max

= 0 is a normalization constant. If the

dimension of H is ﬁnite, we typically take R

(H)

max

as the

required number of qubits for representing the system

H; that is,

(H)

max

= log

(dim H) , (88)

where dim H denotes the dimension of H, which im-

plies that R

(H)

max

represents the maximum amount of a

resource in S (H). Similarly, the resource cost of any

state ψ ∈ S (H) is deﬁned as

(ψ)

= inf

φ∈S(H)

(

(H)

max

conv

(φ → ψ)

)

. (89)

Note that if we set the normalization constant R

(H)

max

log

(dim H) as mentioned in (88) in the QRT of bi-

partite entanglement, R

and R

reduce to the distil-

lable entanglement and the entanglement cost [46, 47],

respectively.

We obtain the following proposition for QRTs with-

out 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 H be a quantum system, and let

ψ ∈ S (H) be a state. Suppose that S (H)\F (H) 6= ∅.

If ψ is not a catalytically replicable state, it holds that

0 5 R

(ψ) 5 R

(H)

max

, (90)

0 5 R

(ψ) 5 R

(H)

max

. (91)

Especially, if ψ is a free state, it holds that

(ψ) = 0, (92)

(ψ) = 0. (93)

Proof. Note that by the deﬁnitions (87) of R

and

(89) of R

, 0 5 R

(ψ) and 0 5 R

(ψ) trivially hold.

First, we prove the statement for a free state. Let

ψ ∈ F (H) be a free state. Since the set of free states

is closed, for any resource state φ ∈ S (H) \ F (H), it

holds that

conv

(ψ → φ) = 0. (94)

Therefore, it holds that

0 5 R

(ψ) 5 R

(H)

max

conv

(ψ → φ) = 0, (95)

which shows R

(ψ) = 0. On the other hand, by

Proposition 2, there exists a free operation N ∈

O(C → H) such that N (1) = ψ. Therefore, it holds

that

⊗n

(1) = ψ

⊗n

(96)

for any positive integer n, which implies that

conv

(1 → ψ) = ∞. (97)

Therefore, it holds that

0 5 R

(ψ) 5

(H)

max

conv

(1 → ψ)

= 0, (98)

which shows R

(ψ) = 0.

Next, we prove (90) for a resource state ψ ∈ S (H)\

F (H). By Theorem 5, we have r

conv

(ψ → ψ) = 1

because ψ is not a catalytically replicable state. Then,

from (87), we obtain

(ψ) 5 R

(H)

max

conv

(ψ → ψ)

= R

(H)

max

(99)

We can show (91) by replacing R

with R

and

conv

(ψ → ψ) with 1/r

conv

(ψ → ψ) respectively in

the proof of (90). Q.E.D.

In fact, from the relation between the preorder in-

troduced 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

and R

, rather than

the inﬁmum over the whole set of states.

Theorem 9 (Maximally Resourceful States are Suf-

ﬁcient for Distillable Resource and Resource Cost).

Let ψ ∈ S (H) be an arbitrary state. It is suﬃcient

to consider G (H) instead of S (H) when we take the

inﬁmum in the deﬁnitions (87) and (89) of R

and

; that is, it holds that

(ψ) = inf

φ∈G(H)

conv

(ψ → φ) R

(H)

max

, (100)

(ψ) = inf

φ∈G(H)

(

(H)

max

conv

(φ → ψ)

)

. (101)

Proof. To show (101), it suﬃces to show that

(ψ) = inf

φ∈G(H)

conv

(φ → ψ) R

(H)

max

. (102)

By Proposition 6 and Theorem 1, for any state ρ ∈

S (H), there always exists a maximally resourceful

state φ ∈ G (H) such that

conv

(φ → ψ) R

(H)

max

5 r

conv

(ρ → ψ) R

(H)

max

. (103)

Therefore, (102) holds. Equation (100) can be shown

by replacing R

with R

and r

conv

(ρ → ψ) with

conv

(ρ → ψ) in (102). Q.E.D.

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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

G (H) / ∼

= {C

: φ ∈ G (H)} (104)

be the set of equivalence classes of the maximally re-

sourceful states, where

= {ψ ∈ G (H) : ψ ∼ φ} (105)

is the equivalence class of φ. Suppose that the num-

ber of non-equivalent maximally resourceful states is

ﬁnite; that is, |G (H) / ∼| < ∞. For example, in the

QRT of bipartite entanglement, |G (H) / ∼| = 1; in

the QRT of magic states for qutrits, |G (H) / ∼| =

2 [17]. In these cases, the inﬁmum is achievable

by a maximally resourceful state because of Propo-

sition 7. Thus, for these existing QRTs, we can ac-

tually 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 inﬁ-

mum with the minimum for QRTs with inﬁnitely

many non-equivalent maximally resourceful states,

i.e., |G (H) / ∼| = ∞.

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 H, let ψ, φ ∈

S (H) be quantum states such that φ  ψ. Then, it

holds that

(ψ) 5 R

(φ) , (106)

(ψ) 5 R

(φ) . (107)

Proof. Let  be an arbitrary positive number. Due to

Theorem 9, we can take a maximally resourceful state

ρ ∈ G (H) such that

(φ) +  = R

(H)

max

conv

(φ → ρ) . (108)

Then, by Proposition 6,

(ψ) = inf

σ∈G(H)

(H)

max

conv

(ψ → σ)

(109)

5 R

(H)

max

conv

(ψ → ρ) (110)

5 R

(H)

max

conv

(φ → ρ) (111)

5 R

(φ) +  (112)

holds. As we can take an arbitrarily small , R

(ψ) 5

(φ) holds.

Similarly, due to Theorem 9, we can take a maxi-

mally resourceful state ρ ∈ G (H) such that

(φ) +  =

(H)

max

conv

(ρ → φ)

. (113)

Then, by Proposition 6,

(ψ) = inf

σ∈G(H)

(

(H)

max

conv

(σ → ψ)

)

(114)

(H)

max

conv

(ρ → ψ)

(115)

(H)

max

conv

(ρ → φ)

(116)

5 R

(φ) +  (117)

holds. As we can take an arbitrarily small , R

(ψ) 5

(φ) holds. Q.E.D.

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 re-

source 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 ψ ∈ S (H) be a state satisfying

conv

(ψ → ψ) = ∞. Then,

(ψ) = 0. (118)

holds. Moreover, if R

(ψ) > 0,

(ψ) = ∞ (119)

holds.

Proof. Note that 0 5 R

(ψ) and 0 5 R

(ψ) hold by

the deﬁnitions. Since r

conv

(ψ → ψ) = ∞,

(ψ) 5

(H)

max

conv

(ψ → ψ)

= 0

(120)

holds. Therefore, it holds that R

(ψ) = 0.

Recall that for quantum states ρ, σ and ω, it holds

that r

conv

(ρ → ω) = r

conv

(ρ → σ) r

conv

(σ → ω) as

shown in (53). Take an arbitrary positive number

. Let φ ∈ S (H) be a state such that

(ψ) +  = r

conv

(ψ → φ) R

(H)

max

= R

(ψ). (121)

Then, it holds that

(ψ) +  = r

conv

(ψ → φ) R

(H)

max

(122)

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= r

conv

(ψ → ψ) r

conv

(ψ → φ) R

(H)

max

(123)

conv

(ψ → ψ) R

(ψ) (124)

= ∞. (125)

As we can take an arbitrarily small , it holds that

(ψ) = ∞. Q.E.D.

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 Dis-

tillable Resource). As shown in Proposition 8 and

Proposition 11, the distillable resource of a catalyt-

ically replicable state may be inﬁnity while that of a

free state is zero. Consider the same setup as Exam-

ple 6. In this case,

(|1ih1|) = ∞, (126)

(|1ih1|) = 0 (127)

follows from G





= {|1ih1|} and

conv

(|1ih1| → |1ih1|) = ∞. In contrast, for

any free state ψ, Proposition 8 shows that

(ψ) = 0, (128)

(ψ) = 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 H is ﬁnite-dimensional and the normal-

ization constant is set as R

(H)

max

= log

(dim H). The

deﬁnitions of additivity and subadditivity are as fol-

lows.

Deﬁnition 12 (Additivity and Subadditivity). Let

be a family of functions from S (H) to R, where

H is a quantum system. We may omit the subscript

of f

to write f for brevity. Then, f is said to be

additive if it holds that

f (ψ ⊗ φ) = f (ψ) + f (φ) (130)

for any states ψ ∈ S (H) and φ ∈ S (H

). On the

other hand, f is said to be weakly additive if it holds

that



⊗n



= nf (ψ) (131)

for any state ψ ∈ S (H) and for any positive integer

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

f (ψ ⊗ φ) 5 f (ψ) + f (φ) (132)

for any states ψ ∈ S (H) and φ ∈ S (H

). On the

other hand, f is said to be weakly subadditive if it

holds that



⊗n



5 nf (ψ) (133)

for any state ψ ∈ S (H) and for any positive integer

Note that in some cases, e.g., in Ref. [49], the prop-

erty mentioned in (131), which we call weak additivity

here, is reﬀered to as additivity. However, in this pa-

per, we follow the convention of Ref. [1].

The proof of weak subadditivity exploits the follow-

ing proposition.

Proposition 13. Let H and H

be quantum systems.

Let ψ ∈ S (H) and φ ∈ S (H

) be quantum states.

Then for any n ∈ N, it holds that

conv

(ψ → φ) = r

conv



⊗n

→ φ

⊗n



. (134)

Proof. It suﬃces to show that R(ψ → φ) =

R(ψ

⊗n

→ φ

⊗n

). First, assume r ∈ R(ψ

⊗n

→ φ

⊗n

)

to show R(ψ

⊗n

→ φ

⊗n

) ⊆ R(ψ → φ).

Choose a positive number  > 0. Then,

there exists a sequence of free operations



(n)

∈ O



⊗nm

→ H

0⊗ndrme



: m ∈ N



such

that



(n)





⊗n



⊗m



−



⊗n



⊗drme



<  (135)

holds for an inﬁnitely large subset of N. Since

n drme = drnme holds, we can deﬁne N

(n)

∈



0⊗ndrme

→ H

0⊗drnme



as the partial trace over

n drme − drnme systems. Then, we have

(n)



⊗ndrme



= φ

⊗drnme

. Therefore, it holds that



(n)

◦ M

(n)



⊗nm



− φ

⊗drnme



(n)

◦ M

(n)





⊗n



⊗m



− N

(n)



⊗ndrme





(n)





⊗n



⊗m



− φ

⊗ndrme



< .

(136)

Therefore, for an integer k = nm with a suﬃciently

large m, there exists a free operation L

= N

(n)

◦

(n)

such that





⊗k



− φ

⊗drke



< .

(137)

Therefore, r ∈ R(ψ → φ), which implies

R(ψ

⊗n

→ φ

⊗n

) ⊆ R(ψ → φ).

On the other hand, to show R(ψ → φ) ⊆

R(ψ

⊗n

→ φ

⊗n

), assume r ∈ R(ψ → φ).

Choose a positive number . Then,

there exists a sequence of free operations



∈ O



⊗m

→ H

0⊗drme



: m ∈ N



such that

for a ﬁxed positive integer n,





⊗m



− φ

⊗drme





(138)

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holds for an inﬁnitely large subset of N. Therefore,

for any n satisfying (138), it holds that



⊗n





⊗m



⊗n



−



⊗drme



⊗n



< , (139)

which implies that R(ψ → φ) ⊆ R(ψ

⊗n

→ φ

⊗n

Q.E.D.

Using Proposition 13, we show Theorem 14. The

meaning of this theorem will be discussed after the

proof.

Theorem 14 (Weak Subadditivity of Distillable Re-

source and Resource Cost). Let H be an arbitrary

ﬁnite-dimensional system. Set the normalization con-

stant as R

(H)

max

= log

(dim H) as stated in (88). For

any n ∈ N and for any state ψ ∈ S (H),



⊗n



5 nR

(ψ) , (140)



⊗n



5 nR

(ψ) . (141)

Proof. First, we prove (140). Let n be a ﬁxed posi-

tive integer, and let  be an arbitrary positive number.

Due to Theorem 9, we can take a maximally resource-

ful state φ ∈ G (H) such that

(ψ) +



= R

(H)

max

conv

(ψ → φ) . (142)

Since φ

⊗n

∈ S (H

⊗n

(ψ) +  = nR

(H)

max

conv

(ψ → φ) (143)

= R

⊗n

)

max

conv



⊗n

→ φ

⊗n



(144)

= R



⊗n



(145)

holds where (144) follows from Proposition 13. As we

can take an arbitrarily small , (140) holds. We can

show (140) in a similar way by replacing R

with R

and r

conv

(ψ → φ) with 1/r

conv

(φ → ψ). Q.E.D.

In the statement of Theorem 14, (140) means that

for a maximally resourceful state φ ∈ G (H), there

may be φ

∈ G (H

⊗n

) that is harder to distill than

⊗n

. On the other hand, (141) means that for a max-

imally resourceful state φ ∈ G (H), there may be a

more resourceful state φ

∈ G (H

⊗n

) in resource for-

mation than φ

⊗n

for n = 2.

5.4 Maximally Resourceful State Maximizing

Resource Cost

In this section, we prove that the upper bound R

(H)

max

of the resource cost R

shown in Proposition 8 is

indeed achievable by a maximally resourceful state

if the number of equivalence classes of the maxi-

mally resourceful states is ﬁnite and if there is no

catalytically replicable state. Note that this prop-

erty holds even in inﬁnite-dimensional cases; that is,

(H)

max

= log

(dim H) for ﬁnite-dimensional H is not

assumed in this section.

Using Proposition 7, we prove Theorem 15. Re-

call the set of equivalence classes of the maximally re-

sourceful states G (H) / ∼ 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,

|G (H) / ∼| < ∞. (146)

In this case, the following theorem shows that the up-

per bound R

(H)

max

of the resource cost given in Proposi-

tion 8 is actually achievable by a maximally resource-

ful 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, S (H) \

F (H) 6= ∅. If |G (H) / ∼| < ∞ where G (H) / ∼ is

deﬁned in (104), then there exists a maximally re-

sourceful state φ ∈ G (H) such that

(φ) = R

(H)

max

(147)

holds.

Proof. Assume that for all states φ ∈ G (H), R

(φ) <

(H)

max

. Because of Theorem 9, this assumption im-

plies that for all states φ ∈ G (H), there exists a

maximally resourceful state ρ ∈ G (H) such that

conv

(ρ → φ) > 1. According to Proposition 7, ρ must

be in an equivalence class diﬀerent from C

. Write

this relation as φ 7→ ρ; that is, for φ ∈ G (H) / ∼ and

ρ ∈ G (H) / ∼, we write φ 7→ ρ if r

conv

(ρ → φ) > 1.

Since |G (H) / ∼| < ∞, there must exist a loop of el-

ements in G (H) / ∼

7→ ρ

7→ ··· 7→ ρ

7→ ρ

. (148)

Therefore, Theorem 5 shows that

conv

(ρ

→ ρ

) × r

conv

(ρ

→ ρ

n−1

) ×

··· × r

conv

(ρ

→ ρ

)

> 1.

On the other hand, note that for any maxi-

mally resourceful state ρ ∈ G (H), it holds that

conv

(ρ → ρ) = 1 because there is no catalytically

replicable state and because ρ 6∈ F (H) due to Propo-

sition 3. From (53), it follows that

conv

(ρ

→ ρ

) × ··· ×r

conv

(ρ

→ ρ

)

5 r

conv

(ρ

→ ρ

)

= 1,

(149)

which contradicts (149). Therefore, there exists a

maximally resourceful state φ ∈ G (H) such that

(φ) = R

(H)

max

. Q.E.D.

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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 assump-

tion on the existence of additive measures, and hence

is simpler and has more applicability than the existing

technique.

Theorem 16 (Condition for Distillable Resource Up-

per-Bounded by Resource Cost). Let H be a quan-

tum system. Suppose that S (H) \ F (H) 6= ∅. Fur-

ther suppose that there is no catalytically replicable

state; that is, r

conv

(φ → φ) = 1 for any resource state

φ ∈ S (H) \ F (H). Then, for any state ψ ∈ S (H), it

holds that

(ψ) 5 R

(ψ) . (150)

Proof. As shown in Proposition 8, for a free state

ψ ∈ F (H), it holds that R

(ψ) = R

(ψ) = 0.

Therefore, (150) trivially holds for a free state. Then,

let ψ ∈ S (H) \ F (H) be an arbitrary resource state.

Let  > 0 be an arbitrary positive number. Due to

Theorem 9, we take a maximally resourceful state

φ ∈ G (H) such that

(ψ) +  =

(H)

max

conv

(φ → ψ)

. (151)

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

holds that

(ψ) 5 r

conv

(ψ → φ) R

(H)

max

. (152)

As r

conv

(ψ → φ) r

conv

(φ → ψ) 5 r

conv

(ψ → ψ) = 1

shown in (53), it holds that

(ψ) 5 r

conv

(ψ → φ) R

(H)

max

(153)

(H)

max

conv

(φ → ψ)

(154)

5 R

(ψ) + . (155)

As we can take an arbitrarily small , it holds that

(ψ) 5 R

(ψ). Q.E.D.

6 Resource Measures

In this section, we investigate a formulation of re-

source 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 over-

come 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 maxi-

mally resourceful states exist. Furthermore, we prove

a similar uniqueness inequality to the previous ap-

proach; 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 def-

inition 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. [20, 32, 72–75], and recall conventionally stud-

ied 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 quan-

tify the resource without contradicting the fact that

free operations cannot generate resources by them-

selves, 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 sys-

tems H

(in)

and H

(out)

, any state ψ ∈ S



(in)



, and

any free operation N ∈ O



(in)

→ H

(out)



, it holds

that

(in)

(ψ) = R

(out)

(N(ψ)). (156)

Here, we recall the deﬁnition of a resource measure.

Deﬁnition 17 (Resource Measure). A resource mea-

sure R

is a family of real functions from S (H) for a

quantum system H to R satisfying monotonicity. We

may omit the subscript of R

to write R for brevity.

By monotonicity, a resource measure R

for a

quantum system H quantiﬁes the resource consis-

tently with the preorder introduced by free opera-

tions. For two states satisfying

φ  ψ, (157)

it holds that

(φ) = R

(ψ) . (158)

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Note that if two states satisfy

φ ∼ ψ (159)

then we have

(φ) = R

(ψ) . (160)

Furthermore, using a resource measure, we can eval-

uate the resource amounts of two diﬀerent states that

cannot be compared in terms of the preorder intro-

duced 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 re-

source measures as shown in the following proposition.

Proposition 18. Suppose that for any quantum sys-

tem H, there exists a maximally resourceful state

φ ∈ G(H) such that for any quantum system H

and

for any maximally resourceful state φ

∈ G(H

), it

holds that

conv

(φ → φ

) =

(H)

max

)

max

. (161)

Then, the distillable resource R

and the resource cost

satisfy monotonicity.

Proof. Choose any δ > 0. Let H

and H

be quantum

systems. Let ψ ∈ S(H

) be a quantum state and

N ∈ O(H

→ H

) be a free operation. Suppose that

∈ G(H

) is a maximally resourceful state satisfying

the assumption given in (161).

Now, suppose that φ

∈ G(H

) is a maximally re-

sourceful state such that

(ψ) = R

)

max

conv

(ψ → φ

) − δ. (162)

Then, it holds that

(N(ψ)) 5 R

)

max

conv

(N(ψ) → φ

)

5 R

)

max

conv

(ψ → φ

)

5 R

)

max

conv

(ψ → φ

conv

(φ

→ φ

)

5 R

)

max

conv

(ψ → φ

)

5 R

(ψ) + δ.

(163)

The ﬁrst inequality holds by deﬁnition. The second

inequality follows from Proposition 6. The third in-

equality follows from (161). Using (53), we have the

fourth inequality. Since we can take an arbitrarily

small δ, we have R

(N(ψ)) 5 R

(ψ).

Next, suppose that φ

∈ G(H

) is a maximally re-

sourceful state such that

(ψ) =

)

max

conv

(φ

→ ψ)

− δ. (164)

Then, it holds that

(N(ψ)) 5

)

max

conv

(φ

→ N(ψ))

)

max

conv

(φ

→ ψ)

)

max

conv

(φ

→ φ

)

conv

(φ

→ ψ)

)

max

conv

(φ

→ ψ)

5 R

(ψ) + δ.

(165)

Since we can take an arbitrarily small δ, we have

(N(ψ)) 5 R

(ψ). Q.E.D.

Now, we recall several properties of a resource mea-

sure that are conventionally studied.

Additivity: Strong superadditivity refers to





= R





+ R





, (166)

and superadditivity refers to

R (ψ ⊗ φ) = R (ψ) + R (φ) . (167)

Subadditivity refers to

R (ψ ⊗ φ) 5 R (ψ) + R (φ) . (168)

Additivity refers to

R (ψ ⊗ φ) = R (ψ) + R (φ) , (169)

while weak additivity refers to



⊗n



= nR (ψ) . (170)

Regularization of R provides a measure that is addi-

tive for tensor product of the same states

∞

(ψ)

= lim

n→∞

R (ψ

⊗n

)

, (171)

as long as the right-hand side exists. The following

proposition shows that additivity of a resource mea-

sure implies that free states have zero resource, which

is a generalization of the statement shown for entan-

glement in Ref. [49] to general QRTs.

Proposition 19. If a resource measure R is weakly

additive, R (φ) = 0 for any free state φ.

Proof. Suppose that ψ is a free state. Then, there ex-

ists a free operation M such that M(1) = ψ. There-

fore, for any n ∈ N, M

⊗n

(1) = ψ

⊗n

holds, which

implies ψ

⊗n

is also a free state. Then, there exist free

operations N

and N

such that



⊗n



= ψ,

(ψ) = ψ

⊗n

(172)

hold. Therefore, it holds that R (ψ) = R (ψ

⊗n

). Since

R is weakly additive, R (ψ) = nR (ψ) for any n, which

implies R (ψ) = 0. Q.E.D.

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One conventional way of normalizing resource mea-

sures such as that in the entanglement theory is as

follows, which we call conventional normalization:

• For any free state σ,

R (σ) = 0. (173)

• For any maximally resourceful state φ ∈ G (H),

R (φ) = R

(H)

max

. (174)

Because of monotonicity of a resource measure, a re-

source 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 re-

source measure to R

(H)

max

in the same way that we nor-

malize the distillable resource (87) and the resource

cost (89) with the normalization constant R

(H)

max

. In

a ﬁnite-dimensional case with R

(H)

max

= log

(dim H)

shown in (88), (174) provides the normalization gen-

eralizing that of entanglement measures in the en-

tanglement theory, but our deﬁnition is applicable to

inﬁnite-dimensional cases. We here remark that gen-

eral 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 H,

is asymptotically continuous if for any sequence

of positive integers (n

)

i∈N

, and any sequences of

states (φ

∈ S (H

⊗n

))

and (ψ

∈ S (H

⊗n

))

sat-

isfying lim

i→∞

kφ

− ψ

= 0, it holds that

lim

i→∞

⊗n

(φ

) − R

⊗n

(ψ

= 0. (175)

Our deﬁnition of asymptotic continuity is applicable

to an inﬁnite-dimensional system. If we take R

(H)

max

log

(dim H) as stated in (88) for a ﬁnite-dimensional

system H, our deﬁnition (175) corresponds to Con-

dition (E3) in Ref. [49]. Note that our deﬁnition in-

cludes asymptotic continuity discussed in [76] (Def-

inition 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 sys-

tems, we employ asymptotic continuity of a family

of functions as a desired property of resource mea-

sures rather than continuity of a single function R

for a ﬁxed quantum system H deﬁned as follows. A

function R

: S (H) → R is continuous if for any

sequences of states (φ

∈ S (H))

and (ψ

∈ S (H))

satisfying lim

n→∞

kφ

− ψ

= 0, it holds that

lim

n→∞

(φ

) − R

(ψ

)| = 0. (176)

Our deﬁnition of asymptotic continuity implies conti-

nuity as a special case.

6.2 Generalization of Uniqueness Inequality

In this section, we show that we have the inequal-

ity R

5 R 5 R

for a resource measure R if R

satisﬁes conventional normalization, asymptotic con-

tinuity, and weak additivity. We call this inequality

the uniqueness inequality. The uniqueness inequality

is originally proved in the QRT of bipartite entangle-

ment in ﬁnite-dimensional cases [49]. We show that

the proof of this uniqueness inequality can be gener-

alized 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.

We prove the following uniqueness inequality for

general QRTs in our framework.

Proposition 20 (Uniqueness Inequality). Let H be

a quantum system. Suppose that there is no catalyt-

ically replicable state; that is, r

conv

(φ → φ) = 1 for

any resource state φ ∈ S (H) \ F (H). If a resource

measure R

satisﬁes the conventional normalization,

asymptotic continuity, and weak additivity, then for

any state ψ ∈ S (H), R

satisﬁes

(ψ) 5 R

(ψ) . (177)

Proof. First, we prove R

(ψ) 5 R

(ψ) for any state

ψ ∈ S (H). Let δ be an arbitrary positive number.

Due to Theorem 9, we take a maximally resourceful

state φ ∈ G (H) such that

(ψ) 5 r

conv

(ψ → φ) R

(H)

max

. (178)

Let r

= r

conv

(ψ → φ). For any positive integer n, it

holds that

(H)

max

drne

(H)

max

. (179)

Then, by conventional normalization and weak addi-

tivity, it holds that

drne

(H)

max

drne

(φ)

⊗drne



⊗drne



(180)

By the deﬁnition of r

conv

(ψ → φ) shown in (38), there

exist free operations



∈ O



⊗n

→ H

⊗drne



such

that for any , it holds that





⊗n



− φ

⊗drne



< , (181)

for an inﬁnitely large subset of N. Because of asymp-

totic continuity, there exists suﬃciently large n such

that

⊗drne



⊗drne



⊗drne

(ψ

⊗n

))

+ δ. (182)

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By monotonicity and weak additivity, it holds that

⊗drne

(ψ

⊗n

))

⊗n

(ψ

⊗n

)

(ψ)

= R

(ψ) .

(183)

Therefore, by (178), (179), (180), (182), and (183), it

holds that

(ψ) 5 R

(ψ) + δ. (184)

Since we can take arbitrarily small δ, it holds that

(ψ) 5 R

(ψ) holds.

Next, we prove R

(ψ) = R

(ψ) for any state ψ ∈

S (H). Let δ be an arbitrary positive number. Due

to Theorem 9, we take a maximally resourceful state

φ ∈ G (H) such that

(ψ) + δ =

(H)

max

conv

(φ → ψ)

= r

conv

(φ → ψ) R

(H)

max

(185)

Let r

= r

conv

(φ → ψ). For any positive integer n, it

holds that

(H)

max

brnc

(H)

max

. (186)

By the conventional normalization, weak additivity,

and monotonicity, it holds that

brnc

(H)

max

brnc

(φ)

⊗brnc



⊗brnc



⊗n



⊗brnc



(187)

for any free operation M

. By the deﬁnition of

conv

(φ → ψ) shown in (40), there exist free opera-

tions



∈ O



⊗brnc

→ H

⊗n



such that for any

, it holds that





⊗brnc



− ψ

⊗n



< , (188)

for an inﬁnitely large subset of N. Because of asymp-

totic continuity, there exists suﬃciently large n such

that

⊗n



⊗brnc



⊗n

(ψ

⊗n

)

− δ. (189)

Therefore, by (185), (186), (187), (189), and weak ad-

ditivity, it holds that

(ψ) + δ =

⊗n

(ψ

⊗n

)

− δ

(ψ)

− δ

= R

(ψ) − δ.

(190)

Since we can take arbitrarily small δ, it holds that

(ψ) = R

(ψ) . (191)

Q.E.D.

Remark 6. If we can assume strong superadditivity

of the resource measure R, the uniqueness inequality

in Proposition 20 holds for R satisfying the conven-

tional normalization, weak additivity, strong super-

additivity, and lower semi-continuity for each quan-

tum system H, instead of assuming asymptotic con-

tinuity. A function R

: S (H) → R is lower semi-

continuous if for any ﬁxed state ψ ∈ S (H) and any

sequence of states (ψ

∈ S (H))

converging to ψ, i.e.,

lim

n→∞

kψ − ψ

= 0, it holds that

lim inf

n→∞

(ψ

) = R

(ψ) . (192)

Strong superadditivity and lower semi-continuity do

not necessarily imply asymptotic continuity, and

asymptotic continuity does not necessarily imply

strong superadditivity and lower semi-continuity ei-

ther; that is, these assumptions are independent.

We prove the uniqueness inequality for R satisfy-

ing the conventional normalization, weak additivity,

strong superadditivity, and lower semi-continuity in

Appendix B. The combination of strong superadditiv-

ity and lower semi-continuity is ﬁrst used in Ref. [77]

to bound a resource cost in the context of a QRT of

coherence, but our contribution is to generalize it to

the full uniqueness inequality by completing proofs of

both bounds, that is, the bounds for the distillable

resource as well as the resource cost.

Despite the general uniqueness inequality shown in

Proposition 20, we show a condition of QRTs where

no resource measure satisﬁes the conventional normal-

ization, asymptotic continuity, and weak additivity si-

multaneously. In these QRTs, Proposition 20 is not

applicable. Note that because of Proposition 7, the

condition (194) in the following theorem is not sat-

isﬁed for QRTs with a unique maximally resource-

ful state, but may hold for QRTs with two or more

diﬀerent equivalence classes of maximally resourceful

states; that is,

|G (H) / ∼| = 2. (193)

Theorem 21 (Inconsistency of Conventional Prop-

erties of Resource Measure). Suppose that there exist

maximally resourceful states φ

, φ

∈ G (H) such that

conv

(φ

→ φ

) > 1. (194)

If R

(H)

max

> 0, then there exists no resource measure

satisfying the conventional normalization, asymptotic

continuity, and weak additivity simultaneously.

Proof. The proof is by contradiction. Assume that

there exists a resource measure R that satisﬁes all

of the conventional normalization, asymptotic conti-

nuity, and weak additivity. Let φ

, φ

∈ G (H) be

maximally resourceful states such that

= r

conv

(φ

→ φ

) > 1. (195)

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By the deﬁnition of r

conv

(φ

→ φ

)

shown in (38), there exist free operations



∈ O



⊗n

→ H

⊗drne



such that for any ,

it holds that





⊗n



− φ

⊗drne



< , (196)

for an inﬁnitely large subset of N. Then, by asymp-

totic continuity, for a ﬁxed  > 0, there exists suﬃ-

ciently large n such that

⊗drne



⊗n



⊗drne



⊗drne



− .

(197)

By monotonicity, it holds that

⊗n



⊗n



⊗drne



⊗drne



− . (198)

Then, from weak additivity, it follows that

(φ

) =

drne

(φ

) − . (199)

Therefore, due to the conventional normalization, it is

necessary that for any  and n such that (197) holds,

it holds that

(H)

max

−

drne

(H)

max

= −. (200)

Since R

(H)

max

> 0, we have

1 −

drne

= −



(H)

max

. (201)

Since r > 1, this inequality does not hold for suﬃ-

ciently small  or suﬃciently large n, which implies

that there is no such R. Q.E.D.

The following example shows that the QRT of

magic has no measure with the conventional normal-

ization, asymptotic continuity, and weak additivity.

Example 8 (QRT without Measure with Conven-

tional Normalization, Asymptotic Continuity, and

Weak Additivity). The QRT of magic for qutrits [17]

shown in Example 4 does not have any conventionally

normalized, asymptotically continuous and weakly

additive measure. It has been proved that the asymp-

totic conversion rate from the Strange state to the

Norrell state is larger than 1 [71]. Therefore, from

Theorem 21, it follows that no measure can satisfy

the conventional normalization, asymptotic continu-

ity and weak additivity simultaneously in this QRT.

6.3 Consistency of Resource Measures

In this section, we introduce consistent resource mea-

sures in place of resource measures in the previous

axiomatic approach in Secs. 6.1 and 6.2. As The-

orem 21 and Example 8 suggest, the conventional

normalization, asymptotic continuity and weak ad-

ditivity do not necessarily hold simultaneously with

monotonicity in general QRTs. On the other hand,

a consistent resource measure is compatible with the

state conversion rate. We prove that the uniqueness

inequality (177) also holds for a consistent resource

measure that is appropriately normalized. Note that

this normalization respects the state conversion rate,

and hence can be diﬀerent from conventional normal-

ization given by (173) and (174).

First, we introduce a deﬁnition of a consistent re-

source measure. A consistent resource measure quan-

tiﬁes the amount of a resource without contradicting

the state conversion rate.

Deﬁnition 22 (Consistent Resource Measure). For

quantum systems H and H

, a resource measure R is

called a consistent resource measure if for any states

ψ ∈ S (H) and φ ∈ S (H

), it holds that

(ψ) r

conv

(φ → ψ) 5 R

(φ) . (202)

The following proposition shows that a consistent

resource measure must be weakly additive for any

non-catalytically replicable resource states.

Proposition 23. Let R be a consistent resource mea-

sure. Then, for any resource state ψ ∈ S (H) \ F (H)

that is not catalytically replicable and for any positive

integer n, it holds that



⊗n



= nR (ψ) . (203)

Proof. By (53), it holds that

conv



ψ → ψ

⊗n



conv



⊗n

→ ψ



5 r

conv

(ψ → ψ)

= 1.

(204)

Since the identity map is a free operation, we have

conv



ψ → ψ

⊗n



, (205)

conv



⊗n

→ ψ



= n. (206)

Thus, we have

conv



ψ → ψ

⊗n



, (207)

conv



⊗n

→ ψ



= n. (208)

By the deﬁnition of a consistent resource measure

combined with the equations above, it holds that

nR (ψ) 5 R



⊗n



, (209)



⊗n



5 R (ψ) . (210)

Therefore, it holds that



⊗n



= nR (ψ) . (211)

Q.E.D.

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We prove that the uniqueness inequality holds for a

consistent resource measure that satisﬁes normaliza-

tions in the following propositions.

Proposition 24. Let R

be a consistent resource

measure. Suppose that 0 5 R

(ψ) 5 R

(H)

max

for any

state ψ ∈ S (H). Then, R

satisﬁes

(ψ) 5 R

(ψ) . (212)

Proof. Let  be an arbitrary positive number. Due

to Theorem 9, we take a maximally resourceful state

φ ∈ G (H) such that

(ψ) +  =

(H)

max

conv

(φ → ψ)

. (213)

Then, it holds that

(ψ) +  =

(H)

max

conv

(φ → ψ)

(214)

(φ)

conv

(φ → ψ)

(215)

= R

(ψ) , (216)

where the second inequality follows from the deﬁnition

of consistent resource measures. As we can take an

arbitrarily small , R

(ψ) 5 R

(ψ) holds. Q.E.D.

Proposition 25. Let R

be a consistent resource

measure. Suppose that there exists a maximally re-

sourceful state φ ∈ G (H) such that R

(φ) = R

(H)

max

Then, R

satisﬁes

(ψ) = R

(ψ) (217)

for any state ψ ∈ S (H).

Proof. By the deﬁnition of the distillable resource, it

holds that

(ψ) 5 r

conv

(ψ → φ) R

(H)

max

. (218)

Then, it holds that

(ψ) 5 r

conv

(ψ → φ) R

(H)

max

(219)

= r

conv

(ψ → φ) R

(φ) (220)

5 R

(ψ) . (221)

Therefore, R

(ψ) 5 R

(ψ) holds. Q.E.D.

From Proposition 24 and Proposition 25, we obtain

the following corollary.

Corollary 26. Let R

be a consistent resource mea-

sure. Suppose that R

satisﬁes the following assump-

tions:

• For any state ψ ∈ S (H), 0 5 R

(ψ) 5 R

(H)

max

;

• There exists φ ∈ G (H) such that R

(φ) = R

(H)

max

Then, it holds that

(ψ) 5 R

(ψ) , (222)

for any state ψ ∈ S (H).

Remark 7. The second condition of Corollary 26 can

be replaced by the existence of a state ρ ∈ S (H)

(which is not necessarily maximally resourceful) such

that R

(ρ) = R

(H)

max

. Suppose a resource measure R

satisﬁes the following conditions:

• The measure R

is normalized in such a way that

there exists a positive real number C such that

0 5 R

(ψ) 5 C for any state ψ ∈ S (H);

• The upper-bound of R

is achieved by some

state; that is, there exists ρ ∈ S (H) such that

(ρ) = C.

Then, there exists a maximally resourceful state φ ∈

G (H) such that R

(φ) = C by monotonicity of the

resource measure.

To observe whether a consistent resource measure

satisﬁes asymptotic continuity, take any two states

φ, ψ ∈ S (H) such that r

conv

(φ → ψ) 5 1. Con-

sider a consistent resource measure R satisfying 0 5

(ψ) 5 R

(H)

max

for any ψ ∈ S(H). By the deﬁnition

of a consistent resource measure, it holds that

(ψ) − R

(φ)

(H)

max

5 1 −r

conv

(φ → ψ) . (223)

Assume that the quantum system H is ﬁnite-

dimensional, and take R

(H)

max

= log

(dim H) as stated

in (88). Inequality (223) suggests that a consis-

tent resource measure has asymptotic continuity if

1 − r

conv

(φ

→ ψ

) converges to zero for any se-

quence of positive integers (n

)

i∈N

and any sequences

of states (φ

∈ S(H

⊗n

)) and (ψ

∈ S(H

⊗n

)) satis-

fying lim

i→∞

kφ

− ψ

= 0. However, in general,

1 − r

conv

(φ

→ ψ

) is not necessarily small even if

kφ

− ψ

is small, because the convertibility of

two states under free operations is not related to the

distance between the two states. Therefore, a con-

sistent resource measure is not necessarily asymptot-

ically continuous. More generally, the diﬀerence in

the amount of resources present in two states is not

necessarily related to the distance between the states

since the preorder and the distance are independent.

Thus, we do not assume asymptotic continuity in the

deﬁnition of a consistent resource measure.

6.4 Example of Consistent Resource Measures

In this section, we show an example of a consistent

resource measure, which is known as the regular-

ized relative entropy of resource and widely used in

known QRTs such as bipartite entanglement [78], co-

herence [8] and magic states [17]. We give the def-

inition of the relative entropy of resource R

in our

framework.

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Deﬁnition 27 (Relative Entropy of Resource). The

relative entropy of resource R

is deﬁned as

(ψ)

= inf

φ∈F(H)

D (ψkφ) , (224)

where D (·k·) is the quantum relative entropy deﬁned

as D (ψkφ) = Tr ψ log

ψ − Tr ψ log

φ.

The relative entropy of resource R

is subadditive

since the set of free operations is closed under ten-

sor product. Therefore, by subadditivity of R

, the

regularized relative entropy of resource deﬁned as

∞

(ρ)

= lim

n→∞

(ρ

⊗n

)

(225)

exists [49].

We show that the regularized relative entropy of

resource serves as a consistent resource measure for

a ﬁnite-dimensional convex QRT in which F (H) for

each H contains at least one full-rank state. Consider

a convex QRT that is deﬁned for ﬁnite-dimensional

systems and satisﬁes the axioms of QRTs given in

Sec. 3. It has been shown that if the set of free states

F (H) for each H contains at least one full-rank state,

the relative entropy of resource is asymptotically con-

tinuous [76, 79]. On the other hand, in Refs. [1, 80],

it is shown that

conv

(φ → ψ) 5

∞

(φ)

∞

(ψ)

(226)

holds for an asymptotically continuous resource mea-

sure f and its regularization f

∞

. Therefore, the regu-

larized relative entropy of resource R

∞

is a consistent

resource measure for a convex QRT that has a full-

rank free state in each dimension. The QRT of bipar-

tite entanglement [55], coherence [48] and magic [17]

are known as convex QRTs with full-rank free states.

In these QRTs, the regularized relative entropy of re-

source works as a consistent resource measure.

We remark that this proof of the existence of a

consistent measure is not applicable to non-convex

QRTs because the relative entropy of resource for

a non-convex set of free states can be discontinu-

ous [81]. However, as mentioned in the last para-

graph of Sec. 6.3, asymptotic continuity is a suﬃcient

but not necessary condition for a resource measure

to be consistent. Therefore, there may be a consis-

tent resource measure that is not asymptotically con-

tinuous. Thus, there may be a consistent resource

measure even in QRTs that are not convex, not ﬁnite-

dimensional, or do not contain full-rank free states

while further research is needed to explicitly construct

consistent resource measures in these QRTs.

7 Conclusion

We have formulated and investigated quantum state

conversion and resource measures in a framework of

general QRTs to ﬁgure out general properties of quan-

tum resources. Our framework is based on minimal

assumptions, and hence covers a broad range of QRTs

including those with non-unique maximally resource-

ful states, non-convexity, and inﬁnite dimension. In

our general framework, the existence of maximally re-

sourceful states is no longer trivial, but we proved

that there always exists a maximally resourceful state

in the general QRTs.

To clarify general properties of resource manipula-

tion, we investigated one-shot and asymptotic state

conversions, which are central tasks in QRTs. We dis-

covered the existence of catalytically replicable states,

which are resources that are inﬁnitely replicable by

free operations. In addition, we introduced the distil-

lable resource and the resource cost in our framework

without assuming uniqueness of maximally resource-

ful states. We showed that the distillable resource

and the resource cost are weakly subadditive. Fur-

thermore, we showed that the distillable resource is

always smaller or equal to the resource cost if there is

no catalytically replicable state.

As for quantiﬁcation of quantum resources, we

proved that the conventional normalization, asymp-

totic continuity, and weak additivity are incompati-

ble with each other in general QRTs with non-unique

maximally resourceful states. Motivated by this in-

compatibility, we introduced a consistent resource

measure, which is consistent with the asymptotic

state conversion rate. Moreover, we proved that a

normalized consistent resource measure is bounded by

the distillable resource and the resource cost, general-

izing the previous work on the uniqueness inequality

in the entanglement theory to general QRTs.

Owing to the generality, our formulations and re-

sults broaden potential applications of QRTs in the

following future research directions. Since we for-

mulated a framework of QRTs applicable to non-

convex QRTs where randomness can be regarded as

a resource, it would be interesting to ﬁnd further

applications of non-convex QRTs, such as analyses

of random-number generation [82] and quantum t-

design [83]. In addition, since our framework forms

a basis of QRTs on inﬁnite-dimensional quantum sys-

tems, our results provide a foundation for applying

QRTs to quantum ﬁeld theory. Since we discovered

a counter-intuitive phenomenon of catalytically repli-

cable resources, it is interesting to ﬁnd more situa-

tions where catalytically replicable states arise. Fur-

thermore, while we showed that the regularized rela-

tive entropy serves as a consistent resource measure

in convex ﬁnite-dimensional QRTs that have full-rank

free states, construction of a consistent resource mea-

sure for all the QRTs in our framework including

non-convex or inﬁnite-dimensional QRTs is still open.

Finally, extension of our framework to dynamic re-

source [84–90] would also be an interesting future di-

rection.

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We established general and fruitful structures of

QRTs disclosing universal properties of quantum re-

sources. Owing to the broad applicability of our for-

mulations, our results open a way to quantitative

understandings of complicated quantum-mechanical

phenomena that are sometimes hard to analyze,

through a uniﬁed approach using our general formu-

lation of QRTs.

Acknowledgments

We acknowledge Debbie Leung, Yui Kuramochi,

Toshihiko Sasaki, and Masato Koashi for the help-

ful advice and discussion. K. K. was supported by

Mike and Ophelia Lazaridis, and research grants by

NSERC. H. Y. was supported by CREST (Japan Sci-

ence and Technology Agency) JPMJCR1671, Cross-

ministerial Strategic Innovation Promotion Program

(SIP) (Council for Science, Technology and Innova-

tion (CSTI)), and JSPS Overseas Research Fellow-

ships.

A Equivalence of Compactness in

Weak Operator Topology and Trace

Norm Topology

In this section, we prove the following lemma, which

shows that compactness in the weak operator topol-

ogy is equivalent to that in the trace norm topology

on a set of density operators. We exploit this lemma

in the proof of Theorem 1 in Sec. 3.1.

Lemma 28. For any set of density operators K ⊂

D(H), K is compact in terms of the weak operator

topology if and only if K is compact in the trace norm

topology.

Proof. Since the trace norm topology is stronger than

the weak operator topology, the “if” part is obvious.

Assume that K is compact in terms of the weak op-

erator topology to show the “only if” part. To show

the compactness in the trace norm topology, take an

arbitrary sequence (ψ

)

n∈N

in K. According to the

Eberlein-Šmulian theorem (e.g. [91], Theorem V.6.1),

in the weak operator topology, the condition of the

compactness coincides with that of the sequential

compactness. Therefore, there exists a subsequence

(ψ

n(k)

)

k∈N

converging to some ψ ∈ K ⊂ D(H) in

terms of the weak opertor topology. Moreover, ac-

cording to [92] (Lemma 2), a sequence in D(H) con-

vergent to a density operator in terms of the weak op-

erator topology is in fact convergent to that density

operator in terms of the trace norm topology. Thus,

(ψ

n(k)

)

k∈N

is convergent to ψ in terms of the trace

norm topology. Therefore K is sequentially compact,

hence compact, in the trace norm topology. Q.E.D.

B Uniqueness Inequality Based on

Strong Superadditivity and Lower Semi-

Continuity

In this section, we prove the uniqueness inequality

under the assumptions of strong superadditivity and

lower semi-continuity, instead of asymptotic continu-

ity as mentioned in Remark 6. In Ref. [77], strong

superadditivity and lower semi-continuity are used to

bound a resource cost in the context of a QRT of

coherence, and the extension for general resources

was mentioned. Our proof further generalizes this

statement to the full uniqueness inequality for gen-

eral QRTs by completing proofs of both bounds, that

is, the bounds for the distillable resource as well as

the resource cost.

Proposition 29. Let H be a quantum system. Sup-

pose that there is no catalytically replicable state;

that is, r

conv

(φ → φ) = 1 for any resource state

φ ∈ S (H) \ F (H). If a resource measure R

sat-

isﬁes the conventional normalization, weak additiv-

ity, strong superadditivity, and lower semi-continuity,

then for any state ψ ∈ S (H), R

satisﬁes

(ψ) 5 R

(ψ) . (227)

Proof. First, we prove R

(ψ) 5 R

(ψ) for any state

ψ ∈ S (H). Let δ be an arbitrary positive number.

Due to Theorem 9, we take a maximally resourceful

state φ ∈ G (H) such that

(ψ) 5 r

conv

(ψ → φ) R

(H)

max

5 R

(ψ) + δ. (228)

Let r

= r

conv

(ψ → φ). For any positive integer n, it

holds that

(H)

max

drne

(H)

max

. (229)

Then, by conventional normalization and weak addi-

tivity, it holds that

drne

(H)

max

drne

(φ) .

(230)

By the deﬁnition of r

conv

(ψ → φ) shown in (38), there

exist free operations



∈ O



⊗n

→ H

⊗drne



such

that for any , it holds that





⊗n



− φ

⊗drne



< , (231)

for an inﬁnitely large subset of N. Note that

(ψ

⊗n

) ∈ S



⊗drne



is a state on a system com-

posed of drne subsystems. For k ∈ {1, . . . , drne},

let

(k)



∈ S (H) denote a state obtained by trac-

ing out (drne − 1) subsystems except the kth one for

(ψ

⊗n

). Strong superadditivity yields

⊗drne



⊗n



drne

k=1



(k)





. (232)

Accepted in Quantum 2020-10-21, click title to verify. Published under CC-BY 4.0. 25

Then deﬁning



= argmin

φ∈



(k)



:k∈{1,...,drne}







, (233)

we have

⊗drne



⊗n



= drneR







. (234)

From (231), monotonicity of the trace distance implies





− φ



< . (235)

Because of lower semi-continuity, we can take a suﬃ-

ciently small  > 0 such that

(φ) 5 R

(φ



) + δ. (236)

Using (234), we obtain

drne

(φ



) 5

⊗drne

(ψ

⊗n

))

. (237)

By monotonicity and weak additivity, it holds that

⊗drne

(ψ

⊗n

))

⊗n

(ψ

⊗n

)

(ψ)

= R

(ψ) .

(238)

Therefore, by (228), (229), (230), (236), (237),

and (238), it holds that

(ψ) 5 R

(ψ) +

drne

δ (239)

5 R

(ψ) +



(ψ) + δ

(H)

max



δ (240)

5 R

(ψ) +



1 +

(H)

max

+ 1



δ, (241)

where the last inequality follows from Proposition 8

and

5 1. Since we can take arbitrarily small δ, it

holds that R

(ψ) 5 R

(ψ).

Next, we prove R

(ψ) = R

(ψ) for any state ψ ∈

S (H). Let δ be an arbitrary positive number. Due

to Theorem 9, we take a maximally resourceful state

φ ∈ G (H) such that

(ψ) + δ =

(H)

max

conv

(φ → ψ)

= r

conv

(φ → ψ) R

(H)

max

= R

(ψ) .

(242)

Let r

= r

conv

(φ → ψ). For any positive integer n, it

holds that

(H)

max

brnc

(H)

max

. (243)

By the conventional normalization, weak additivity,

and monotonicity, it holds that

brnc

(H)

max

brnc

(φ)

⊗brnc



⊗brnc



⊗n



⊗brnc



(244)

for any free operation M

. By the deﬁnition of

conv

(φ → ψ) shown in (40), there exist free opera-

tions



∈ O



⊗brnc

→ H

⊗n



such that for any

, it holds that





⊗brnc



− ψ

⊗n



< , (245)

for an inﬁnitely large subset of N. Note that



⊗brnc



∈ S (H

⊗n

) is a state on a system com-

posed of n subsystems. For each k ∈ {1, . . . , n},

let

(k)



∈ S (H) denote a state obtained by trac-

ing out (n − 1) subsystems except the kth one for



⊗brnc



. Strong superadditivity yields

⊗n



⊗brnc



k=1



(k)





. (246)

Then deﬁning



= argmin

ψ∈



(k)



:k∈{1,...,drne}







, (247)

we have

⊗n



⊗brnc



= nR







. (248)

Thus, we obtain

⊗n



⊗brnc



= R







. (249)

From (245), monotonicity of the trace distance implies





− ψ



< . (250)

Because of lower semi-continuity, we can take a suﬃ-

ciently small  > 0 such that

(ψ



) = R

(ψ) − δ. (251)

Therefore, by (242), (243), (244), (249), and (251) it

holds that

(ψ) + δ = R

(ψ) − δ.

(252)

Since we can take arbitrarily small δ, it holds that

(ψ) = R

(ψ) . (253)

Q.E.D.

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