General Quantum Resource Theories: Distillation, Forma-
tion and Consistent Resource Measures
Kohdai Kuroiwa
1,2
and Hayata Yamasaki
3,4
1
Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West,
2
Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656,
Japan
3
Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
4
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 [24], coherence [511], athermal-
ity [1216], magic states [17, 18], asymmetry [19
23], purity [24], non-Gaussianity [2528], 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 [3140].
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 [2528] 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
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 [2528],
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
B
H
:
= {|ki}
k
. (1)
In cases where H represents a D-dimensional system,
B
H
is a ﬁnite set with cardinality D, while B
H
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
H
: |vi H, k|vik
H
5 1}
is bounded, where k·k
H
denotes the norm induced by
the inner product of H.
To deﬁne trace-class operators, we use the trace de-
ﬁned as Tr T
:
=
P
|ki∈B
H
hk |T |ki, where B
H
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
H
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
(a
n
: n N) is indexed by a natural number, that
is, a countably inﬁnite and totally ordered set, a net
a
n
is indexed by n in a directed set, a generalization
of totally ordered sets, while this directed set can be
uncountable. In the deﬁnition of the trace, the net is
indexed by any ﬁnite subset B
0
H
B
H
and deﬁned
as
P
|ki∈B
0
H
hk |T |ki, which approaches to the limit
Tr T as B
0
H
gets larger. Note that Tr T is independent
of the choice of B
H
. 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
T , and T
denotes the conjugation of T .
For any T T (H), the trace norm of T is deﬁned as
kT k
1
:
= 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
1
, A
2
, . . . B(H), or more generally, any net A
i
,
converges to A if and only if Tr [ T A
i
] 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
ψ
: M C that is positive semidef-
inite f
ψ
M
M
= 0, M M, satisﬁes the normal-
ization condition f
ψ
( ) = 1, and is also normal, i.e.,
continuous in terms of the ultraweak operator topol-
ogy. We require this continuity in order for f
ψ
to be
in the predual M
of M, i.e., the space whose dual
(M
)
equals (can be identiﬁed with) M. Given this
duality and under the condition of M = B (H), we
identify f
ψ
with the operator ψ T (H) that satisﬁes
for any M M
Tr [ψM ] = f
ψ
(M) . (3)
Note that we have one-to-one correspondence between
f
ψ
and ψ if M = B(H). This operator ψ is the
density operator representing the state f
ψ
, 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
f
ψ
.
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
M
(out)
representing the spaces of the input and the
output respectively, a channel
˜
E : M
(out)
M
(in)
in
the Heisenberg picture is deﬁned as a linear map that
is completely positive
n1
X
j,k=0
M
(in)
j
˜
E
M
(out)
j
M
(out)
k
M
(in)
k
= 0,
M
(in)
0
, . . . , M
(in)
n1
M
(in)
,
M
(out)
0
, . . . , M
(out)
n1
M
(out)
,
n N, (4)
unital
˜
E
(out)
=
(in)
, and normal, i.e., continu-
ous in terms of the ultraweak operator topologies of
M
(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
ψ
of a state with the density operator ψ of the state, un-
der the conditions of M
(in)
= B
H
(in)
and M
(out)
=
B
H
(out)
, we identify a channel
˜
E : M
(out)
M
(in)
in the Heisenberg picture with the channel
E : T
H
(in)
T
H
(out)
in the Schrödinger pic-
ture that satisﬁes for any M
(out)
M
(out)
Tr
h
E (ψ) M
(out)
i
=
f
ψ
˜
E
M
(out)
, (5)
where ψ and f
ψ
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
H
(in)
and
M
(out)
= B
H
(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
H
(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
H
(in)
and
M
(out)
= B
H
(out)
; that is, C
H
(in)
H
(out)
is
the set of the CPTP linear maps, while it would be
possible to use the Heisenberg picture otherwise. We
represent quantum operations as channels, while it is
possible to include measurements in our formulation
as channels from a quantum input system to a 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
i
in T (H) converges to T if and only if
Tr [T
i
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
i
in T (H) converges to T if and only if kT
i
T k
1
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
C
H
(in)
H
(out)
, and we use the bounded weak
(BW) topology. The bounded weak topology of
C
H
(in)
H
(out)
is the weakest topology such
that for any f
ψ
M
(in)
and M
(out)
M
(out)
,
a map S
ψ,M
(out)
: C
H
(in)
H
(out)
C given
by S
ψ,M
(out)
(E) =
f
ψ
˜
E
M
(out)
is continuous,
where E and
˜
E are related as (5). Note that if H
(in)
and H
(out)
are ﬁnite-dimensional, C
H
(in)
H
(out)
is compact, while C
H
(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
H
(in)
to S
H
(out)
. This set
is denoted by
O
H
(in)
H
(out)
C
H
(in)
H
(out)
. (7)
A map contained in O
H
(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
1
, H
2
and H
3
be arbitrary quantum
systems. For any M O(H
1
H
2
) and
N O(H
2
H
3
), it holds that N M
O(H
1
H
3
), where represents the composi-
tion.
2. Let H
(in)
1
, H
(out)
1
, H
(in)
2
and H
(out)
2
be arbitrary
quantum systems. For any M O(H
(in)
1
H
(out)
1
) and N O(H
(in)
2
H
(out)
2
), it holds that
M N O(H
(in)
1
H
(in)
2
H
(out)
1
H
(out)
2
).
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:
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
H
(in)
H
out
be the set
of LOCC from H
(in)
to H
(out)
. It is known
that O
LOCC
C
4
C
4
is not closed; that is,
O
LOCC
(C
4
C
4
) 6= O
LOCC
C
4
C
4
[55]. In this
case, we use O
C
4
C
4
=
O
LOCC
(C
4
C
4
) as
the set of free operation because for any CPTP
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map N O
LOCC
(C
4
C
4
) \O
LOCC
C
4
C
4
and
any > 0, we can construct a CPTP map
˜
N
O
LOCC
C
4
C
4
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
0
H
(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
O
0
H
(in)
H
(out)
is dense in the set of all the chan-
nels C
H
(in)
H
(out)
, but O
0
H
(in)
H
(out)
is
diﬀerent from C
H
(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
H
(in)
H
(out)
= C
H
(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
U
x
:
= {y X|x y} is closed for any x X [62]
(Proposition VI-1.6.(i)). Thus, it suﬃces to show
U
ψ
:
= {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
)
nN
in U
ψ
convergent to φ S (H) in terms of the trace norm
topology and prove φ U
ψ
. By the deﬁnition of the
preorder , for each n N, there exists a free op-
eration N
n
O(H) such that ψ = N
n
(φ
n
). Since
O(H) is compact in terms of the bounded weak (BW)
topology, there exists a subnet
N
n(i)
iI
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
: k {1, . . . , N
}} of S (H)
such that for any χ S(H)
min
k∈{1,...,N
}
kχ χ
k
k
1
<
kAk
. (18)
By deﬁnition of the convergence N
n(i)
BW
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
}
Tr
N
n(i)
(χ
k
) A
Tr (N (χ
k
) A)
< .
(19)
Thus, for any i = i
,A
and any χ S(H), we have
Tr
N
n(i)
(χ) A
Tr (N (χ) A)
=
Tr
N
n(i)
(χ χ
k
) A
+ Tr
N
n(i)
(χ
k
) A
Tr (N (χ χ
k
) A) Tr (N (χ
k
) A)|
5
Tr
N
n(i)
(χ χ
k
) A
+ |Tr (N (χ χ
k
) A)|
+
Tr
N
n(i)
(χ
k
) A
Tr (N (χ
k
) A)
<
Tr
N
n(i)
(χ χ
k
) A
+ |Tr (N (χ χ
k
) A)| + ,
(20)
where χ
k
is an element in the ﬁnite subset {χ
k
} of
S (H) in (18) satisfying
kχ χ
k
k
1
<
kAk
, (21)
and we use (19) in the last line. With
N
n(i)
:
= sup
N
n(i)
(T )
1
: T T (H), kT k
1
5 1
(22)
denoting the operator norm of the linear map N
n(i)
:
T (H) T (H), we have
Tr
N
n(i)
(χ χ
k
) A
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5
N
n(i)
(χ χ
k
)
1
· kAk
5
N
n(i)
· kχ χ
k
k
1
· kAk
< 1 ·
kAk
· kAk
= , (23)
where the last inequality follows from the fact that
any CPTP map N
n(i)
satisﬁes
N
n(i)
= sup
N
n(i)
(T )
1
5 sup kT k
1
5 1. (24)
In the same way as (23) by substituting N
n(i)
with
N, it holds that
|Tr (N (χ χ
k
) A)| < . (25)
Therefore, applying (23) and (25) to (20), for any
i = i
,A
and any χ S(H), we have
Tr
N
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)|
=
Tr
N
n(i)
φ
n(i)
A
Tr (N (φ) A)
5
Tr
N
n(i)
φ
n(i)
A
Tr
N
φ
n(i)
A
+
Tr
N
φ
n(i)
A
Tr (N (φ) A)
< 3 + kNk
·
φ
n(i)
φ
1
· 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)
Tr
N
φ
n(i)
A
Tr (N (φ) A)
5 kNk
·
φ
n(i)
φ
1
· kAk
, (28)
and the limit in the last line in terms of i yields
φ
n(i)
φ
1
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)
:
=
n
ψ S (H) : ∀H
0
, φ S (H
0
) ,
∃N O(H
0
H) s.t. ψ = N (φ)
o
.
(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) =
n
ψ S (H) : ∃N O(C H) s.t. ψ = N (1)
o
.
(31)
Proof. By the deﬁnition (30) of F (H), it trivially
holds that
F (H)
n
ψ S (H) : ∃N O(C H) s.t. ψ = N (1)
o
.
(32)
To show the converse inclusion, assume that
ψ
n
ψ S (H) : ∃N O(C H) s.t. ψ = N (1)
o
.
(33)
Let N O(C H) be a free operation such that
ψ = N (1). Consider an arbitrary quantum system
H
0
and an arbitrary state φ S (H
0
). Since Tr
O(H
0
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
S
C
2
. 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
C
2
is the set of logical
states in the logical 2-dimensional space of the GKP
code. Take the quantum operations on S
C
2
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
1
and H
2
, and two
quantum states φ S (H
1
) and ψ S (H
2
), 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
1
, N
2
, . . . 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
r
0
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
0
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)
:
=
n
r = 0 :
N
n
O
H
n
1
H
⊗drne
2
: n N
,
lim inf
n→∞
N
n
φ
n
ψ
⊗drne
1
= 0
o
,
(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
r
conv
(φ ψ) is deﬁned as
r
conv
(φ ψ)
:
= sup R(φ ψ) . (38)
Similarly, we can consider the other deﬁnition of a
state conversion rate r
0
conv
(φ ψ). Here, we deﬁne
another set of asymptotic achievable rates
R
0
(φ ψ)
:
=
n
r = 0 :
N
0
n
O
H
⊗brnc
1
H
n
2
: n N
,
lim inf
n→∞
N
0
n
φ
⊗brnc
ψ
n
1
= 0
o
.
(39)
Here, achievable rate r
0
= 0 is a positive number for
which we can generate n copies of ψ from r
0
n copies of
φ. However, r
0
n is not necessarily an integer in gen-
eral; therefore, in (39), we regard r
0
as an achievable
rate when we can generate n copies of ψ from br
0
nc
copies of φ, which guarantees that r
0
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
0
conv
(φ ψ) is deﬁned as
r
0
conv
(φ ψ)
:
= inf R
0
(φ ψ) , (40)
where r
0
conv
(φ ψ) is inﬁnity if the set on the right-
hand side is empty.
These two conversion rates r
conv
(φ ψ) and
r
0
conv
(φ ψ) are related to each other as shown
in the following theorem. Hereafter, we will use
r
conv
(φ ψ) as the asymptotic states conversion rate
rather than r
0
conv
(φ ψ).
Theorem 4 (Relation Between Two Conversion
Rates). Let H and H
0
be quantum systems. For any
states φ S (H) and ψ S (H
0
), it holds that,
r
conv
(φ ψ) =
1
r
0
conv
(φ ψ)
, (41)
where we regard 1/0 = .
Proof. It suﬃces to show that
r R(φ ψ)
1
r
R
0
(φ ψ) , (42)
and that
r R
0
(φ ψ)
1
r
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
n
φ
n
ψ
⊗drne
1
< . (44)
Let n
0
= drne. Because n 5 bn
0
/rc, we can deﬁne a
free operation M
n
0
as the partial trace over bn
0
/rcn
systems so that
M
n
0
φ
b
n
0
/r
c
= φ
n
. (45)
From (44) and (45), it holds that
N
n
M
n
0
φ
b
n
0
/r
c
ψ
n
0
1
< , (46)
and 1/r R
0
(φ ψ) follows.
On the other hand, assume that r R
0
(φ ψ) to
show (43). Choose a ﬁxed positive real number > 0.
Let n be an arbitrary positive integer such that
N
0
n
φ
⊗brnc
ψ
n
1
< (47)
Let n
0
= brnc. Because n = dn
0
/re, we can deﬁne a
free operation M
0
n
0
as the partial trace over ndn
0
/re
systems so that
M
0
n
0
ψ
n
= ψ
d
n
0
/r
e
. (48)
From (47) and (48), it holds that
M
0
n
0
N
0
n
φ
n
0
ψ
d
n
0
/r
e
1
=
M
0
n
0
N
0
n
φ
n
0
M
0
n
0
ψ
n
1
5
N
0
n
φ
n
0
ψ
n
1
< ,
(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)
:
=
n
r = 0 :
N
n
O
H
n
1
H
⊗brnc
2
: n N
,
lim inf
n→∞
N
n
φ
n
ψ
⊗brnc
1
= 0
o
,
(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
1
n
n
(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
1
, H
2
and H
3
be
quantum systems. Let ρ S (H
1
), σ S (H
2
) and
ω S (H
3
) 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
r
conv
(ρ ω) = r
conv
(ρ σ) r
conv
(σ ω) . (53)
Accepted in Quantum 2020-10-21, click title to verify. Published under CC-BY 4.0. 10
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 pro<