
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
Accepted in Quantum 2020-10-21, click title to verify. Published under CC-BY 4.0. 2