We prove the following uniqueness inequality for general QRTs in our
framework.

######  Proposition 20 (Uniqueness Inequality).

Let $\mathcal{H}$ be a quantum system. Suppose that there is no catalytically
replicable state; that is, $r_{\textup{conv}}\left(\phi\to\phi\right)=1$ for
any resource state
$\phi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$.
If a resource measure $R_{\mathcal{H}}$ satisfies the conventional
normalization, asymptotic continuity, and weak additivity, then for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$, $R_{\mathcal{H}}$ satisfies

$R_{\textup{D}}\left(\psi\right)\leqq R_{\mathcal{H}}\left(\psi\right)\leqq
R_{\textup{C}}\left(\psi\right).$ (177)

###### Proof.

First, we prove $R_{\textup{D}}\left(\psi\right)\leqq
R_{\mathcal{H}}\left(\psi\right)$ for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$. Let $\delta$ be an arbitrary
positive number. Due to Theorem 9, we take a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$R_{\textup{D}}\left(\psi\right)\leqq
r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}.$ (178)

Let $r\coloneqq r_{\textup{conv}}\left(\psi\to\phi\right)$. For any positive
integer $n$, it holds that

$rR_{\max}^{(\mathcal{H})}\leqq\frac{\left\lceil
rn\right\rceil}{n}R_{\max}^{(\mathcal{H})}.$ (179)

Then, by conventional normalization and weak additivity, it holds that

$\displaystyle\frac{\left\lceil rn\right\rceil}{n}R_{\max}^{(\mathcal{H})}$
$\displaystyle=\frac{\left\lceil
rn\right\rceil}{n}R_{\mathcal{H}}\left(\phi\right)$ (180)
$\displaystyle=\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\phi^{\otimes\left\lceil rn\right\rceil}\right)}{n}.$

By the definition of $r_{\textup{conv}}\left(\psi\to\phi\right)$ shown in
(38), there exist free operations
$\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}^{\otimes
n}\to\mathcal{H}^{\otimes\left\lceil rn\right\rceil}\right)\right)$ such that
for any $\epsilon$, it holds that

$\left\|\mathcal{N}_{n}\left(\psi^{\otimes n}\right)-\phi^{\otimes\left\lceil
rn\right\rceil}\right\|_{1}<\epsilon,$ (181)

for an infinitely large subset of $\mathbb{N}$. Because of asymptotic
continuity, there exists sufficiently large $n$ such that

$\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\phi^{\otimes\left\lceil
rn\right\rceil}\right)}{n}\leqq\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)\right)}{n}+\delta.$ (182)

By monotonicity and weak additivity, it holds that

$\displaystyle\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\mathcal{N}_{n}\left(\psi^{\otimes n}\right)\right)}{n}$
$\displaystyle\leqq\frac{R_{\mathcal{H}^{\otimes n}}\left(\psi^{\otimes
n}\right)}{n}$ (183)
$\displaystyle=\frac{nR_{\mathcal{H}}\left(\psi\right)}{n}$
$\displaystyle=R_{\mathcal{H}}\left(\psi\right).$

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

$R_{\textup{D}}\left(\psi\right)\leqq
R_{\mathcal{H}}\left(\psi\right)+\delta.$ (184)

Since we can take arbitrarily small $\delta$, it holds that
$R_{\textup{D}}\left(\psi\right)\leqq R_{\mathcal{H}}\left(\psi\right)$ holds.

Next, we prove $R_{\textup{C}}\left(\psi\right)\geqq
R_{\mathcal{H}}\left(\psi\right)$ for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$. Let $\delta$ be an arbitrary
positive number. Due to Theorem 9, we take a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$\displaystyle R_{\textup{C}}\left(\psi\right)+\delta$
$\displaystyle\geqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}$
(185)
$\displaystyle=r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)R_{\max}^{(\mathcal{H})}.$

Let $r\coloneqq r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$. For any
positive integer $n$, it holds that

$rR_{\max}^{(\mathcal{H})}\geqq\frac{\left\lfloor
rn\right\rfloor}{n}R_{\max}^{(\mathcal{H})}.$ (186)

By the conventional normalization, weak additivity, and monotonicity, it holds
that

$\displaystyle\frac{\left\lfloor rn\right\rfloor}{n}R_{\max}^{(\mathcal{H})}$
$\displaystyle=\frac{\left\lfloor
rn\right\rfloor}{n}R_{\mathcal{H}}\left(\phi\right)$ (187)
$\displaystyle=\frac{R_{\mathcal{H}^{\otimes\left\lfloor
rn\right\rfloor}}\left(\phi^{\otimes\left\lfloor rn\right\rfloor}\right)}{n}$
$\displaystyle\geqq\frac{R_{\mathcal{H}^{\otimes
n}}\left(\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\right)}{n},$

for any free operation $\mathcal{M}_{n}$. By the definition of
$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$ shown in (40), there
exist free operations
$\left(\mathcal{M}_{n}\in\mathcal{O}\left(\mathcal{H}^{\otimes\left\lfloor
rn\right\rfloor}\to\mathcal{H}^{\otimes n}\right)\right)$ such that for any
$\epsilon$, it holds that

$\left\|\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\psi^{\otimes n}\right\|_{1}<\epsilon,$ (188)

for an infinitely large subset of $\mathbb{N}$. Because of asymptotic
continuity, there exists sufficiently large $n$ such that

$\frac{R_{\mathcal{H}^{\otimes
n}}\left(\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\right)}{n}\geqq\frac{R_{\mathcal{H}^{\otimes
n}}\left(\psi^{\otimes n}\right)}{n}-\delta.$ (189)

Therefore, by (185), (186), (187), (189), and weak additivity, it holds that

$\displaystyle R_{\textup{C}}\left(\psi\right)+\delta$
$\displaystyle\geqq\frac{R_{\mathcal{H}^{\otimes n}}\left(\psi^{\otimes
n}\right)}{n}-\delta$ (190)
$\displaystyle=\frac{nR_{\mathcal{H}}\left(\psi\right)}{n}-\delta$
$\displaystyle=R_{\mathcal{H}}\left(\psi\right)-\delta.$

Since we can take arbitrarily small $\delta$, it holds that

$R_{\textup{C}}\left(\psi\right)\geqq R_{\mathcal{H}}\left(\psi\right).$ (191)

∎

######  Remark 6.

If we can assume strong superadditivity of the resource measure $R$, the
uniqueness inequality in Proposition 20 holds for $R$ satisfying the
conventional normalization, weak additivity, strong superadditivity, and lower
semi-continuity for each quantum system $\mathcal{H}$, instead of assuming
asymptotic continuity. A function
$R_{\mathcal{H}}:\mathcal{S}\left(\mathcal{H}\right)\to\mathbb{R}$ is lower
semi-continuous if for any fixed state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and any sequence of states
${\left(\psi_{n}\in\mathcal{S}\left(\mathcal{H}\right)\right)}_{n}$ converging
to $\psi$, i.e., $\lim_{n\to\infty}\left\|\psi-\psi_{n}\right\|_{1}=0$, it
holds that

$\liminf_{n\to\infty}R_{\mathcal{H}}\left(\psi_{n}\right)\geqq
R_{\mathcal{H}}\left(\psi\right).$ (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 either; that is, these
assumptions are independent. We prove the uniqueness inequality for $R$
satisfying the conventional normalization, weak additivity, strong
superadditivity, and lower semi-continuity in Appendix B. The combination of
strong superadditivity 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
normalization, asymptotic continuity, and weak additivity simultaneously. In
these QRTs, Proposition 20 is not applicable. Note that because of Proposition
7, the condition (194) in the following theorem is not satisfied for QRTs with
a unique maximally resourceful state, but may hold for QRTs with two or more
different equivalence classes of maximally resourceful states; that is,

$\left|\mathcal{G}\left(\mathcal{H}\right)/\sim\right|\geqq 2.$ (193)

######  Theorem 21 (Inconsistency of Conventional Properties of Resource
Measure).

Suppose that there exist maximally resourceful states
$\phi_{0},\phi_{1}\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$r_{\textup{conv}}\left(\phi_{0}\to\phi_{1}\right)>1.$ (194)

If $R_{\max}^{(\mathcal{H})}>0$, 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 continuity,
and weak additivity. Let
$\phi_{0},\phi_{1}\in\mathcal{G}\left(\mathcal{H}\right)$ be maximally
resourceful states such that

$r\coloneqq r_{\textup{conv}}\left(\phi_{0}\to\phi_{1}\right)>1.$ (195)

By the definition of $r_{\textup{conv}}\left(\phi_{0}\to\phi_{1}\right)$ shown
in (38), there exist free operations
$\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}^{\otimes
n}\to\mathcal{H}^{\otimes\left\lceil rn\right\rceil}\right)\right)$ such that
for any $\epsilon$, it holds that

$\left\|\mathcal{N}_{n}\left({\phi_{0}}^{\otimes
n}\right)-{\phi_{1}}^{\otimes\left\lceil
rn\right\rceil}\right\|_{1}<\epsilon,$ (196)

for an infinitely large subset of $\mathbb{N}$. Then, by asymptotic
continuity, for a fixed $\epsilon>0$, there exists sufficiently large $n$ such
that

$\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\mathcal{N}_{n}\left(\phi_{0}^{\otimes
n}\right)\right)}{n}\geqq\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\phi_{1}^{\otimes\left\lceil
rn\right\rceil}\right)}{n}-\epsilon.$ (197)

By monotonicity, it holds that

$\frac{R_{\mathcal{H}^{\otimes n}}\left(\phi_{0}^{\otimes
n}\right)}{n}\geqq\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\phi_{1}^{\otimes\left\lceil
rn\right\rceil}\right)}{n}-\epsilon.$ (198)

Then, from weak additivity, it follows that

$R_{\mathcal{H}}\left(\phi_{0}\right)\geqq\frac{\left\lceil
rn\right\rceil}{n}R_{\mathcal{H}}\left(\phi_{1}\right)-\epsilon.$ (199)

Therefore, due to the conventional normalization, it is necessary that for any
$\epsilon$ and $n$ such that (197) holds, it holds that

$R_{\max}^{(\mathcal{H})}-\frac{\left\lceil
rn\right\rceil}{n}R_{\max}^{(\mathcal{H})}\geqq-\epsilon.$ (200)

Since $R_{\max}^{(\mathcal{H})}>0$, we have

$1-\frac{\left\lceil
rn\right\rceil}{n}\geqq-\frac{\epsilon}{R_{\max}^{(\mathcal{H})}}.$ (201)

Since $r>1$, this inequality does not hold for sufficiently small $\epsilon$
or sufficiently large $n$, which implies that there is no such $R$. ∎

The following example shows that the QRT of magic has no measure with the
conventional normalization, asymptotic continuity, and weak additivity.

######  Example 8 (QRT without Measure with Conventional 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 asymptotic 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 continuity and weak additivity simultaneously in
this QRT.

###  6.3 Consistency of Resource Measures

In this section, we introduce consistent resource measures in place of
resource measures in the previous axiomatic approach in Secs. 6.1 and 6.2. As
Theorem 21 and Example 8 suggest, the conventional normalization, asymptotic
continuity and weak additivity 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 normalization
given by (173) and (174).

First, we introduce a definition of a consistent resource measure. A
consistent resource measure quantifies the amount of a resource without
contradicting the state conversion rate.

######  Definition 22 (Consistent Resource Measure).

For quantum systems $\mathcal{H}$ and $\mathcal{H}^{\prime}$, a resource
measure $R$ is called a consistent resource measure if for any states
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and
$\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$, it holds that

$R_{\mathcal{H}}\left(\psi\right)r_{\textup{conv}}\left(\phi\to\psi\right)\leqq
R_{\mathcal{H}^{\prime}}\left(\phi\right).$ (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 measure. Then, for any resource state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$
that is not catalytically replicable and for any positive integer $n$, it
holds that

$R\left(\psi^{\otimes n}\right)=nR\left(\psi\right).$ (203)

###### Proof.

By (53), it holds that

$\displaystyle r_{\textup{conv}}\left(\psi\to\psi^{\otimes
n}\right)r_{\textup{conv}}\left(\psi^{\otimes n}\to\psi\right)$ (204)
$\displaystyle\leqq r_{\textup{conv}}\left(\psi\to\psi\right)$
$\displaystyle=1.$

Since the identity map is a free operation, we have

$\displaystyle r_{\textup{conv}}\left(\psi\to\psi^{\otimes n}\right)$
$\displaystyle\geqq{\frac{1}{n}},$ (205) $\displaystyle
r_{\textup{conv}}\left(\psi^{\otimes n}\to\psi\right)$
$\displaystyle\geqq{n}.$ (206)

Thus, we have

$\displaystyle r_{\textup{conv}}\left(\psi\to\psi^{\otimes n}\right)$
$\displaystyle=\frac{1}{n},$ (207) $\displaystyle
r_{\textup{conv}}\left(\psi^{\otimes n}\to\psi\right)$ $\displaystyle=n.$
(208)

By the definition of a consistent resource measure combined with the equations
above, it holds that

$\displaystyle nR\left(\psi\right)$ $\displaystyle\leqq R\left(\psi^{\otimes
n}\right),$ (209) $\displaystyle\frac{1}{n}R\left(\psi^{\otimes n}\right)$
$\displaystyle\leqq R\left(\psi\right).$ (210)

Therefore, it holds that

$R\left(\psi^{\otimes n}\right)=nR\left(\psi\right).$ (211)

∎

We prove that the uniqueness inequality holds for a consistent resource
measure that satisfies normalizations in the following propositions.

######  Proposition 24.

Let $R_{\mathcal{H}}$ be a consistent resource measure. Suppose that $0\leqq
R_{\mathcal{H}}\left(\psi\right)\leqq R_{\max}^{(\mathcal{H})}$ for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$. Then, $R_{\mathcal{H}}$
satisfies

$R_{\mathcal{H}}\left(\psi\right)\leqq R_{\textup{C}}\left(\psi\right).$ (212)

###### Proof.

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

$R_{\textup{C}}\left(\psi\right)+\epsilon\geqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}.$
(213)

Then, it holds that

$\displaystyle R_{\textup{C}}\left(\psi\right)+\epsilon$
$\displaystyle\geqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}$
(214)
$\displaystyle\geqq\frac{R_{\mathcal{H}}\left(\phi\right)}{r_{\textup{conv}}\left(\phi\to\psi\right)}$
(215) $\displaystyle\geqq R_{\mathcal{H}}\left(\psi\right),$ (216)

where the second inequality follows from the definition of consistent resource
measures. As we can take an arbitrarily small $\epsilon$,
$R_{\mathcal{H}}\left(\psi\right)\leqq R_{\textup{C}}\left(\psi\right)$ holds.
∎

######  Proposition 25.

Let $R_{\mathcal{H}}$ be a consistent resource measure. Suppose that there
exists a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that
$R_{\mathcal{H}}\left(\phi\right)=R_{\max}^{(\mathcal{H})}$. Then,
$R_{\mathcal{H}}$ satisfies

$R_{\mathcal{H}}\left(\psi\right)\geqq R_{\textup{D}}\left(\psi\right)$ (217)

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

###### Proof.

By the definition of the distillable resource, it holds that

$R_{\textup{D}}\left(\psi\right)\leqq
r_{\textup{conv}}\left(\psi\to{\phi}\right)R_{\max}^{(\mathcal{H})}.$ (218)

Then, it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle\leqq
r_{\textup{conv}}\left(\psi\to{\phi}\right)R_{\max}^{(\mathcal{H})}$ (219)
$\displaystyle=r_{\textup{conv}}\left(\psi\to{\phi}\right)R_{\mathcal{H}}\left({\phi}\right)$
(220) $\displaystyle\leqq R_{\mathcal{H}}\left(\psi\right).$ (221)

Therefore, $R_{\textup{D}}\left(\psi\right)\leqq
R_{\mathcal{H}}\left(\psi\right)$ holds. ∎

From Proposition 24 and Proposition 25, we obtain the following corollary.

######  Corollary 26.

Let $R_{\mathcal{H}}$ be a consistent resource measure. Suppose that
$R_{\mathcal{H}}$ satisfies the following assumptions:

  * •

For any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$, $0\leqq
R_{\mathcal{H}}\left(\psi\right)\leqq R_{\max}^{(\mathcal{H})}$;

  * •

There exists $\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that
$R_{\mathcal{H}}\left(\phi\right)=R_{\max}^{(\mathcal{H})}$.

Then, it holds that

$R_{\textup{D}}\left(\psi\right)\leqq R_{\mathcal{H}}\left(\psi\right)\leqq
R_{\textup{C}}\left(\psi\right),$ (222)

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

######  Remark 7.

The second condition of Corollary 26 can be replaced by the existence of a
state $\rho\in\mathcal{S}\left(\mathcal{H}\right)$ (which is not necessarily
maximally resourceful) such that
$R_{\mathcal{H}}\left(\rho\right)=R_{\max}^{(\mathcal{H})}$. Suppose a
resource measure $R_{\mathcal{H}}$ satisfies the following conditions:

  * •

The measure $R_{\mathcal{H}}$ is normalized in such a way that there exists a
positive real number $C$ such that $0\leqq
R_{\mathcal{H}}\left(\psi\right)\leqq C$ for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$;

  * •

The upper-bound of $R_{\mathcal{H}}$ is achieved by some state; that is, there
exists $\rho\in\mathcal{S}\left(\mathcal{H}\right)$ such that
$R_{\mathcal{H}}\left(\rho\right)=C$.

Then, there exists a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that
$R_{\mathcal{H}}\left(\phi\right)=C$ by monotonicity of the resource measure.

To observe whether a consistent resource measure satisfies asymptotic
continuity, take any two states
$\phi,\psi\in\mathcal{S}\left(\mathcal{H}\right)$ such that
$r_{\textup{conv}}\left(\phi\to\psi\right)\leqq 1$. Consider a consistent
resource measure $R$ satisfying $0\leqq R_{\mathcal{H}}\left(\psi\right)\leqq
R_{\max}^{(\mathcal{H})}$ for any $\psi\in\mathcal{S}(\mathcal{H})$. By the
definition of a consistent resource measure, it holds that

$\frac{R_{\mathcal{H}}\left(\psi\right)-R_{\mathcal{H}}\left(\phi\right)}{R_{\max}^{(\mathcal{H})}}\leqq
1-r_{\textup{conv}}\left(\phi\to\psi\right).$ (223)

Assume that the quantum system $\mathcal{H}$ is finite-dimensional, and take
$R_{\max}^{(\mathcal{H})}=\log_{2}(\dim\mathcal{H})$ as stated in (88).
Inequality (223) suggests that a consistent resource measure has asymptotic
continuity if $1-r_{\textup{conv}}\left(\phi_{n_{i}}\to\psi_{n_{i}}\right)$
converges to zero for any sequence of positive integers
${(n_{i})}_{i\in\mathbb{N}}$ and any sequences of states
$\left(\phi_{n_{i}}\in\mathcal{S}(\mathcal{H}^{\otimes n_{i}})\right)$ and
$\left(\psi_{n_{i}}\in\mathcal{S}(\mathcal{H}^{\otimes n_{i}})\right)$
satisfying $\lim_{i\to\infty}\|\phi_{n_{i}}-\psi_{n_{i}}\|_{1}=0$. However, in
general, $1-r_{\textup{conv}}\left(\phi_{n_{i}}\to\psi_{n_{i}}\right)$ is not
necessarily small even if $\|\phi_{n_{i}}-\psi_{n_{i}}\|_{1}$ is small,
because the convertibility of two states under free operations is not related
to the distance between the two states. Therefore, a consistent resource
measure is not necessarily asymptotically 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 regularized relative entropy of resource and widely used in known
QRTs such as bipartite entanglement [78], coherence [8] and magic states [17].
We give the definition of the relative entropy of resource $R_{\textup{R}}$ in
our framework.

######  Definition 27 (Relative Entropy of Resource).

The relative entropy of resource $R_{\textup{R}}$ is defined as

$R_{\textup{R}}\left(\psi\right)\coloneqq\inf_{\phi\in\mathcal{F}\left(\mathcal{H}\right)}D\left(\psi\|\phi\right),$
(224)

where $D\left(\cdot\|\cdot\right)$ is the quantum relative entropy defined as
$D\left(\psi\|\phi\right)=\operatorname{Tr}\psi\log_{2}\psi-\operatorname{Tr}\psi\log_{2}\phi$.

The relative entropy of resource $R_{\textup{R}}$ is subadditive since the set
of free operations is closed under tensor product. Therefore, by subadditivity
of $R_{\textup{R}}$, the regularized relative entropy of resource defined as

$R^{\infty}_{\textup{R}}\left(\rho\right)\coloneqq\lim_{n\to\infty}\frac{R_{\textup{R}}\left(\rho^{\otimes
n}\right)}{n}$ (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
$\mathcal{F}\left(\mathcal{H}\right)$ for each $\mathcal{H}$ 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
$\mathcal{F}\left(\mathcal{H}\right)$ for each $\mathcal{H}$ contains at least
one full-rank state, the relative entropy of resource is asymptotically
continuous [76, 79]. On the other hand, in Refs. [80, 1], it is shown that

$r_{\textup{conv}}(\phi\to\psi)\leqq\frac{f^{\infty}(\phi)}{f^{\infty}(\psi)}$
(226)

holds for an asymptotically continuous resource measure $f$ and its
regularization $f^{\infty}$. Therefore, the regularized relative entropy of
resource $R^{\infty}_{\textup{R}}$ is a consistent resource measure for a
convex QRT that has a full-rank free state in each dimension. The QRT of
bipartite 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 resource 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 discontinuous [81]. However, as mentioned
in the last paragraph 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 consistent resource measure that is not asymptotically
continuous. 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
quantum resources. Our framework is based on minimal assumptions, and hence
covers a broad range of QRTs including those with non-unique maximally
resourceful states, non-convexity, and infinite dimension. In our general
framework, the existence of maximally resourceful 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 manipulation, we investigated one-
shot and asymptotic state conversions, which are central tasks in QRTs. We
discovered the existence of catalytically replicable states, which are
resources that are infinitely replicable by free operations. In addition, we
introduced the distillable resource and the resource cost in our framework
without assuming uniqueness of maximally resourceful states. We showed that
the distillable resource and the resource cost are weakly subadditive.
Furthermore, 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, asymptotic continuity, and weak additivity are incompatible
with each other in general QRTs with non-unique maximally resourceful states.
Motivated by this incompatibility, 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, generalizing the previous
work on the uniqueness inequality in the entanglement theory to general QRTs.

Owing to the generality, our formulations and results broaden potential
applications of QRTs in the following future research directions. Since we
formulated 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 systems, our results provide a
foundation for applying QRTs to quantum field theory. Since we discovered a
counter-intuitive phenomenon of catalytically replicable resources, it is
interesting to find more situations where catalytically replicable states
arise. Furthermore, while we showed that the regularized relative entropy
serves as a consistent resource measure in convex finite-dimensional QRTs that
have full-rank free states, construction of a consistent resource measure for
all the QRTs in our framework including non-convex or infinite-dimensional
QRTs is still open. Finally, extension of our framework to dynamic resource
[84, 85, 86, 87, 88, 89, 90] would also be an interesting future direction.

We established general and fruitful structures of QRTs disclosing universal
properties of quantum resources. Owing to the broad applicability of our
formulations, 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 formulation of QRTs.

We acknowledge Debbie Leung, Yui Kuramochi, Toshihiko Sasaki, and Masato
Koashi for the helpful advice and discussion. K. K. was supported by Mike and
Ophelia Lazaridis, and research grants by NSERC. H. Y. was supported by CREST
(Japan Science and Technology Agency) JPMJCR1671, Cross-ministerial Strategic
Innovation Promotion Program (SIP) (Council for Science, Technology and
Innovation (CSTI)), and JSPS Overseas Research Fellowships.

##  Appendix 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 topology 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
$K\subset\mathcal{D}\left(\mathcal{H}\right)$, $K$ 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 operator
topology to show the “only if” part. To show the compactness in the trace norm
topology, take an arbitrary sequence ${(\psi_{n})}_{n\in\mathbb{N}}$ 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
${(\psi_{n(k)})}_{k\in\mathbb{N}}$ converging to some $\psi\in
K\subset\mathcal{D}\left(\mathcal{H}\right)$ in terms of the weak opertor
topology. Moreover, according to [92] (Lemma 2), a sequence in
$\mathcal{D}\left(\mathcal{H}\right)$ convergent to a density operator in
terms of the weak operator topology is in fact convergent to that density
operator in terms of the trace norm topology. Thus,
${(\psi_{n(k)})}_{k\in\mathbb{N}}$ is convergent to $\psi$ in terms of the
trace norm topology. Therefore $K$ is sequentially compact, hence compact, in
the trace norm topology. ∎

##  Appendix 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
continuity 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
general QRTs by completing proofs of both bounds, that is, the bounds for the
distillable resource as well as the resource cost.

######  Proposition 29.

Let $\mathcal{H}$ be a quantum system. Suppose that there is no catalytically
replicable state; that is, $r_{\textup{conv}}\left(\phi\to\phi\right)=1$ for
any resource state
$\phi\in\mathcal{S}\left(\mathcal{H}\right)\setminus\mathcal{F}\left(\mathcal{H}\right)$.
If a resource measure $R_{\mathcal{H}}$ satisfies the conventional
normalization, weak additivity, strong superadditivity, and lower semi-
continuity, then for any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$,
$R_{\mathcal{H}}$ satisfies

$R_{\textup{D}}\left(\psi\right)\leqq R_{\mathcal{H}}\left(\psi\right)\leqq
R_{\textup{C}}\left(\psi\right).$ (227)

###### Proof.

First, we prove $R_{\textup{D}}\left(\psi\right)\leqq
R_{\mathcal{H}}\left(\psi\right)$ for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$. Let $\delta$ be an arbitrary
positive number. Due to Theorem 9, we take a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$R_{\textup{D}}\left(\psi\right)\leqq
r_{\textup{conv}}\left(\psi\to\phi\right)R_{\max}^{(\mathcal{H})}\leqq
R_{\textup{D}}\left(\psi\right)+\delta.$ (228)

Let $r\coloneqq r_{\textup{conv}}\left(\psi\to\phi\right)$. For any positive
integer $n$, it holds that

$rR_{\max}^{(\mathcal{H})}\leqq\frac{\left\lceil
rn\right\rceil}{n}R_{\max}^{(\mathcal{H})}.$ (229)

Then, by conventional normalization and weak additivity, it holds that

$\displaystyle\frac{\left\lceil rn\right\rceil}{n}R_{\max}^{(\mathcal{H})}$
$\displaystyle=\frac{\left\lceil
rn\right\rceil}{n}R_{\mathcal{H}}\left(\phi\right).$ (230)

By the definition of $r_{\textup{conv}}\left(\psi\to\phi\right)$ shown in
(38), there exist free operations
$\left(\mathcal{N}_{n}\in\mathcal{O}\left(\mathcal{H}^{\otimes
n}\to\mathcal{H}^{\otimes\left\lceil rn\right\rceil}\right)\right)$ such that
for any $\epsilon$, it holds that

$\left\|\mathcal{N}_{n}\left(\psi^{\otimes n}\right)-\phi^{\otimes\left\lceil
rn\right\rceil}\right\|_{1}<\epsilon,$ (231)

for an infinitely large subset of $\mathbb{N}$. Note that
$\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)\in\mathcal{S}\left(\mathcal{H}^{\otimes\lceil rn\rceil}\right)$ is a
state on a system composed of $\lceil rn\rceil$ subsystems. For
$k\in\left\\{1,\ldots,\lceil rn\rceil\right\\}$, let
$\tilde{\phi}_{\epsilon}^{(k)}\in\mathcal{S}\left(\mathcal{H}\right)$ denote a
state obtained by tracing out $(\lceil rn\rceil-1)$ subsystems except the
$k$th one for $\mathcal{N}_{n}\left(\psi^{\otimes n}\right)$. Strong
superadditivity yields

$R_{\mathcal{H}^{\otimes\lceil
rn\rceil}}\left(\mathcal{N}_{n}\left(\psi^{\otimes
n}\right)\right)\geqq\sum_{k=1}^{\lceil
rn\rceil}R_{\mathcal{H}}\left(\tilde{\phi}_{\epsilon}^{(k)}\right).$ (232)

Then defining

$\tilde{\phi}_{\epsilon}\coloneqq\operatorname*{argmin}_{\tilde{\phi}\in\left\\{\tilde{\phi}_{\epsilon}^{(k)}:k\in\left\\{1,\ldots,\lceil
rn\rceil\right\\}\right\\}}R_{\mathcal{H}}\left(\tilde{\phi}\right),$ (233)

we have

$R_{\mathcal{H}^{\otimes\lceil
rn\rceil}}\left(\mathcal{N}_{n}\left(\psi^{\otimes n}\right)\right)\geqq\lceil
rn\rceil R\left(\tilde{\phi}_{\epsilon}\right).$ (234)

From (231), monotonicity of the trace distance implies

$\left\|\tilde{\phi}_{\epsilon}-\phi\right\|_{1}<\epsilon.$ (235)

Because of lower semi-continuity, we can take a sufficiently small
$\epsilon>0$ such that

$R_{\mathcal{H}}\left(\phi\right)\leqq
R_{\mathcal{H}}\left(\phi_{\epsilon}\right)+\delta.$ (236)

Using (234), we obtain

$\frac{\left\lceil
rn\right\rceil}{n}R_{\mathcal{H}}\left(\phi_{\epsilon}\right)\leqq\frac{R_{\mathcal{H}^{\otimes\lceil
rn\rceil}}\left(\mathcal{N}_{n}\left(\psi^{\otimes n}\right)\right)}{n}.$
(237)

By monotonicity and weak additivity, it holds that

$\displaystyle\frac{R_{\mathcal{H}^{\otimes\left\lceil
rn\right\rceil}}\left(\mathcal{N}_{n}\left(\psi^{\otimes n}\right)\right)}{n}$
$\displaystyle\leqq\frac{R_{\mathcal{H}^{\otimes n}}\left(\psi^{\otimes
n}\right)}{n}$ (238)
$\displaystyle=\frac{nR_{\mathcal{H}}\left(\psi\right)}{n}$
$\displaystyle=R_{\mathcal{H}}\left(\psi\right).$

Therefore, by (228), (229), (230), (236), (237), and (238), it holds that

$\displaystyle R_{\textup{D}}\left(\psi\right)$ $\displaystyle\leqq
R_{\mathcal{H}}\left(\psi\right)+\frac{\lceil rn\rceil}{n}\delta$ (239)
$\displaystyle\leqq
R_{\mathcal{H}}\left(\psi\right)+\left(\frac{R_{\textup{D}}\left(\psi\right)+\delta}{R_{\max}^{(\mathcal{H})}}+\frac{1}{n}\right)\delta$
(240) $\displaystyle\leqq
R_{\mathcal{H}}\left(\psi\right)+\left(1+\frac{\delta}{R_{\max}^{(\mathcal{H})}}+1\right)\delta,$
(241)

where the last inequality follows from Proposition 8 and $\frac{1}{n}\leqq 1$.
Since we can take arbitrarily small $\delta$, it holds that
$R_{\textup{D}}\left(\psi\right)\leqq R_{\mathcal{H}}\left(\psi\right)$.

Next, we prove $R_{\textup{C}}\left(\psi\right)\geqq
R_{\mathcal{H}}\left(\psi\right)$ for any state
$\psi\in\mathcal{S}\left(\mathcal{H}\right)$. Let $\delta$ be an arbitrary
positive number. Due to Theorem 9, we take a maximally resourceful state
$\phi\in\mathcal{G}\left(\mathcal{H}\right)$ such that

$\displaystyle R_{\textup{C}}\left(\psi\right)+\delta$
$\displaystyle\geqq\frac{R_{\max}^{(\mathcal{H})}}{r_{\textup{conv}}\left(\phi\to\psi\right)}$
(242)
$\displaystyle=r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)R_{\max}^{(\mathcal{H})}$
$\displaystyle\geqq R_{\textup{C}}\left(\psi\right).$

Let $r\coloneqq r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$. For any
positive integer $n$, it holds that

$rR_{\max}^{(\mathcal{H})}\geqq\frac{\left\lfloor
rn\right\rfloor}{n}R_{\max}^{(\mathcal{H})}.$ (243)

By the conventional normalization, weak additivity, and monotonicity, it holds
that

$\displaystyle\frac{\left\lfloor rn\right\rfloor}{n}R_{\max}^{(\mathcal{H})}$
$\displaystyle=\frac{\left\lfloor
rn\right\rfloor}{n}R_{\mathcal{H}}\left(\phi\right)$ (244)
$\displaystyle=\frac{R_{\mathcal{H}^{\otimes\left\lfloor
rn\right\rfloor}}\left(\phi^{\otimes\left\lfloor rn\right\rfloor}\right)}{n}$
$\displaystyle\geqq\frac{R_{\mathcal{H}^{\otimes
n}}\left(\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\right)}{n},$

for any free operation $\mathcal{M}_{n}$. By the definition of
$r^{\prime}_{\textup{conv}}\left(\phi\to\psi\right)$ shown in (40), there
exist free operations
$\left(\mathcal{M}_{n}\in\mathcal{O}\left(\mathcal{H}^{\otimes\left\lfloor
rn\right\rfloor}\to\mathcal{H}^{\otimes n}\right)\right)$ such that for any
$\epsilon$, it holds that

$\left\|\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)-\psi^{\otimes n}\right\|_{1}<\epsilon,$ (245)

for an infinitely large subset of $\mathbb{N}$. Note that
$\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\in\mathcal{S}\left(\mathcal{H}^{\otimes n}\right)$ is
a state on a system composed of $n$ subsystems. For each
$k\in\left\\{1,\ldots,n\right\\}$, let
$\tilde{\psi}_{\epsilon}^{(k)}\in\mathcal{S}\left(\mathcal{H}\right)$ denote a
state obtained by tracing out $(n-1)$ subsystems except the $k$th one for
$\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor rn\right\rfloor}\right)$.
Strong superadditivity yields

$R_{\mathcal{H}^{\otimes
n}}\left(\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\right)\geqq\sum_{k=1}^{n}R_{\mathcal{H}}\left(\tilde{\psi}_{\epsilon}^{(k)}\right).$
(246)

Then defining

$\tilde{\psi}_{\epsilon}\coloneqq\operatorname*{argmin}_{\tilde{\psi}\in\left\\{\tilde{\psi}_{\epsilon}^{(k)}:k\in\left\\{1,\ldots,\lceil
rn\rceil\right\\}\right\\}}R_{\mathcal{H}}\left(\tilde{\psi}\right),$ (247)

we have

$R_{\mathcal{H}^{\otimes
n}}\left(\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\right)\geqq nR\left(\tilde{\psi}_{\epsilon}\right).$
(248)

Thus, we obtain

$\frac{R_{\mathcal{H}^{\otimes
n}}\left(\mathcal{M}_{n}\left(\phi^{\otimes\left\lfloor
rn\right\rfloor}\right)\right)}{n}\geqq
R\left(\tilde{\psi}_{\epsilon}\right).$ (249)

From (245), monotonicity of the trace distance implies

$\left\|\tilde{\psi}_{\epsilon}-\psi\right\|_{1}<\epsilon.$ (250)

Because of lower semi-continuity, we can take a sufficiently small
$\epsilon>0$ such that

$R_{\mathcal{H}}\left(\psi_{\epsilon}\right)\geqq
R_{\mathcal{H}}\left(\psi\right)-\delta.$ (251)

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

$\displaystyle R_{\textup{C}}\left(\psi\right)+\delta$ $\displaystyle\geqq
R_{\mathcal{H}}\left(\psi\right)-\delta.$ (252)

Since we can take arbitrarily small $\delta$, it holds that

$R_{\textup{C}}\left(\psi\right)\geqq R_{\mathcal{H}}\left(\psi\right).$ (253)

∎

## References

  * Chitambar and Gour [2019] Eric Chitambar and Gilad Gour.  Quantum resource theories.  _Rev. Mod. Phys._ , 91:025001, Apr 2019.  doi: 10.1103/RevModPhys.91.025001.  URL https://link.aps.org/doi/10.1103/RevModPhys.91.025001.
  * Horodecki et al. [2009] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki.  Quantum entanglement.  _Rev. Mod. Phys._ , 81:865–942, Jun 2009.  doi: 10.1103/RevModPhys.81.865.  URL https://link.aps.org/doi/10.1103/RevModPhys.81.865.
  * Rains [2001] E. M. Rains.  A semidefinite program for distillable entanglement.  _IEEE Trans. Inf. Theory_ , 47(7):2921–2933, Nov 2001.  ISSN 1557-9654.  doi: 10.1109/18.959270.  URL https://doi.org/10.1109/18.959270.
  * Brandão and Datta [2011] Fernando G. S. L. Brandão and N. Datta.  One-shot rates for entanglement manipulation under non-entangling maps.  _IEEE Trans. Inf. Theory_ , 57(3):1754–1760, March 2011.  ISSN 1557-9654.  doi: 10.1109/TIT.2011.2104531.  URL https://doi.org/10.1109/TIT.2011.2104531.
  * Streltsov et al. [2017] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio.  Colloquium: Quantum coherence as a resource.  _Rev. Mod. Phys._ , 89:041003, Oct 2017.  doi: 10.1103/RevModPhys.89.041003.  URL https://link.aps.org/doi/10.1103/RevModPhys.89.041003.
  * Marvian and Spekkens [2016] Iman Marvian and Robert W. Spekkens.  How to quantify coherence: Distinguishing speakable and unspeakable notions.  _Phys. Rev. A_ , 94:052324, Nov 2016.  doi: 10.1103/PhysRevA.94.052324.  URL https://link.aps.org/doi/10.1103/PhysRevA.94.052324.
  * Aberg [2006] Johan Aberg.  Quantifying superposition.  _arXiv preprintarXiv:quant-ph/0612146_, 2006.  URL https://arxiv.org/abs/quant-ph/0612146.
  * Baumgratz et al. [2014] T. Baumgratz, M. Cramer, and M. B. Plenio.  Quantifying coherence.  _Phys. Rev. Lett._ , 113:140401, Sep 2014.  doi: 10.1103/PhysRevLett.113.140401.  URL https://link.aps.org/doi/10.1103/PhysRevLett.113.140401.
  * Yadin et al. [2016] Benjamin Yadin, Jiajun Ma, Davide Girolami, Mile Gu, and Vlatko Vedral.  Quantum processes which do not use coherence.  _Phys. Rev. X_ , 6:041028, Nov 2016.  doi: 10.1103/PhysRevX.6.041028.  URL https://link.aps.org/doi/10.1103/PhysRevX.6.041028.
  * Chitambar and Gour [2016a] Eric Chitambar and Gilad Gour.  Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence.  _Phys. Rev. Lett._ , 117:030401, Jul 2016a.  doi: 10.1103/PhysRevLett.117.030401.  URL https://link.aps.org/doi/10.1103/PhysRevLett.117.030401.
  * Chitambar and Gour [2016b] Eric Chitambar and Gilad Gour.  Comparison of incoherent operations and measures of coherence.  _Phys. Rev. A_ , 94:052336, Nov 2016b.  doi: 10.1103/PhysRevA.94.052336.  URL https://link.aps.org/doi/10.1103/PhysRevA.94.052336.
  * Janzing et al. [2000] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and Th. Beth.  Thermodynamic cost of reliability and low temperatures: Tightening Landauer’s principle and the second law.  _Int. J. Theor. Phys._ , 39:2717, 2000.  doi: 10.1023/A:1026422630734.  URL https://doi.org/10.1023/A:1026422630734.
  * Brandão et al. [2013] Fernando G. S. L. Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens.  Resource theory of quantum states out of thermal equilibrium.  _Phys. Rev. Lett._ , 111:250404, Dec 2013.  doi: 10.1103/PhysRevLett.111.250404.  URL https://link.aps.org/doi/10.1103/PhysRevLett.111.250404.
  * Horodecki and Oppenheim [2013a] Michal Horodecki and Jonathan Oppenheim.  Fundamental limitations for quantum and nanoscale thermodynamics.  _Nat. Commun._ , 4:2059, 2013a.  doi: 10.1038/ncomms3059.  URL https://doi.org/10.1038/ncomms3059.
  * Halpern [2017] Nicole Yunger Halpern.  Toward physical realizations of thermodynamic resource theories.  In _Information and Interaction_ , pages 135–166. Springer, 2017\.  doi: 10.1007/978-3-319-43760-6_8.  URL https://doi.org/10.1007/978-3-319-43760-6_8.
  * Yunger Halpern and Limmer [2020] Nicole Yunger Halpern and David T. Limmer.  Fundamental limitations on photoisomerization from thermodynamic resource theories.  _Phys. Rev. A_ , 101:042116, Apr 2020.  doi: 10.1103/PhysRevA.101.042116.  URL https://link.aps.org/doi/10.1103/PhysRevA.101.042116.
  * Veitch et al. [2014] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson.  The resource theory of stabilizer quantum computation.  _New J. Phys._ , 16(1):013009, jan 2014.  doi: 10.1088/1367-2630/16/1/013009.  URL https://doi.org/10.1088%2F1367-2630%2F16%2F1%2F013009.
  * Howard and Campbell [2017] Mark Howard and Earl Campbell.  Application of a resource theory for magic states to fault-tolerant quantum computing.  _Phys. Rev. Lett._ , 118:090501, Mar 2017.  doi: 10.1103/PhysRevLett.118.090501.  URL https://link.aps.org/doi/10.1103/PhysRevLett.118.090501.
  * Gour and Spekkens [2008] Gilad Gour and Robert W Spekkens.  The resource theory of quantum reference frames: manipulations and monotones.  _New J. Phys._ , 10(3):033023, 2008.  doi: 10.1088/1367-2630/10/3/033023.  URL https://doi.org/10.1088/1367-2630/10/3/033023.
  * Gour et al. [2009] Gilad Gour, Iman Marvian, and Robert W. Spekkens.  Measuring the quality of a quantum reference frame: The relative entropy of frameness.  _Phys. Rev. A_ , 80:012307, Jul 2009.  doi: 10.1103/PhysRevA.80.012307.  URL https://link.aps.org/doi/10.1103/PhysRevA.80.012307.
  * Marvian and Spekkens [2013] Iman Marvian and Robert W Spekkens.  The theory of manipulations of pure state asymmetry: I. basic tools, equivalence classes and single copy transformations.  _New J. Phys._ , 15(3):033001, mar 2013.  doi: 10.1088/1367-2630/15/3/033001.  URL https://doi.org/10.1088%2F1367-2630%2F15%2F3%2F033001.
  * Marvian and Spekkens [2014a] Iman Marvian and Robert W. Spekkens.  Asymmetry properties of pure quantum states.  _Phys. Rev. A_ , 90:014102, Jul 2014a.  doi: 10.1103/PhysRevA.90.014102.  URL https://link.aps.org/doi/10.1103/PhysRevA.90.014102.
  * Marvian and Spekkens [2014b] Iman Marvian and Robert W Spekkens.  Extending Noether’s theorem by quantifying the asymmetry of quantum states.  _Nat. Commun._ , 5(1):1–8, 2014b.  doi: 10.1038/ncomms4821.  URL https://doi.org/10.1038/ncomms4821.
  * Gour et al. [2015] Gilad Gour, Markus P. Müller, Varun Narasimhachar, Robert W. Spekkens, and Nicole Yunger Halpern.  The resource theory of informational nonequilibrium in thermodynamics.  _Phys. Rep._ , 583:1 – 58, 2015.  ISSN 0370-1573.  doi: https://doi.org/10.1016/j.physrep.2015.04.003.  URL http://www.sciencedirect.com/science/article/pii/S037015731500229X.
  * Takagi and Zhuang [2018] Ryuji Takagi and Quntao Zhuang.  Convex resource theory of non-Gaussianity.  _Phys. Rev. A_ , 97:062337, Jun 2018.  doi: 10.1103/PhysRevA.97.062337.  URL https://link.aps.org/doi/10.1103/PhysRevA.97.062337.
  * Lami et al. [2018] Ludovico Lami, Bartosz Regula, Xin Wang, Rosanna Nichols, Andreas Winter, and Gerardo Adesso.  Gaussian quantum resource theories.  _Phys. Rev. A_ , 98:022335, Aug 2018.  doi: 10.1103/PhysRevA.98.022335.  URL https://link.aps.org/doi/10.1103/PhysRevA.98.022335.
  * Albarelli et al. [2018] Francesco Albarelli, Marco G. Genoni, Matteo G. A. Paris, and Alessandro Ferraro.  Resource theory of quantum non-Gaussianity and wigner negativity.  _Phys. Rev. A_ , 98:052350, Nov 2018.  doi: 10.1103/PhysRevA.98.052350.  URL https://link.aps.org/doi/10.1103/PhysRevA.98.052350.
  * Yamasaki et al. [2020] Hayata Yamasaki, Takaya Matsuura, and Masato Koashi.  Cost-reduced all-gaussian universality with the gottesman-kitaev-preskill code: Resource-theoretic approach to cost analysis.  _Phys. Rev. Research_ , 2:023270, Jun 2020.  doi: 10.1103/PhysRevResearch.2.023270.  URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.023270.
  * Wakakuwa [2017] Eyuri Wakakuwa.  Operational resource theory of non-markovianity.  _arXiv preprintarXiv:1709.07248 [quant-ph]_, 2017.  URL https://arxiv.org/abs/1709.07248.
  * Wakakuwa [2019] Eyuri Wakakuwa.  Communication cost for non-markovianity of tripartite quantum states: A resource theoretic approach.  _arXiv preprintarXiv:1904.08852 [quant-ph]_, 2019.  URL https://arxiv.org/abs/1904.08852.
  * Horodecki and Oppenheim [2013b] Michal Horodecki and Jonathan Oppenheim.  (Quantumness in the context of) resource theories.  _Int. J. Mod. Phys. B_ , 27(1-3):1345019, 2013b.  doi: 10.1142/S0217979213450197.  URL https://doi.org/10.1142/S0217979213450197.
  * Brandão and Gour [2015] Fernando G. S. L. Brandão and Gilad Gour.  Reversible framework for quantum resource theories.  _Phys. Rev. Lett._ , 115:070503, Aug 2015.  doi: 10.1103/PhysRevLett.115.070503.  URL https://link.aps.org/doi/10.1103/PhysRevLett.115.070503.
  * Korzekwa et al. [2019] Kamil Korzekwa, Christopher T. Chubb, and Marco Tomamichel.  Avoiding irreversibility: Engineering resonant conversions of quantum resources.  _Phys. Rev. Lett._ , 122:110403, Mar 2019.  doi: 10.1103/PhysRevLett.122.110403.  URL https://link.aps.org/doi/10.1103/PhysRevLett.122.110403.
  * Liu et al. [2019a] C. L. Liu, Xiao-Dong Yu, and D. M. Tong.  Flag additivity in quantum resource theories.  _Phys. Rev. A_ , 99:042322, Apr 2019a.  doi: 10.1103/PhysRevA.99.042322.  URL https://link.aps.org/doi/10.1103/PhysRevA.99.042322.
  * Liu et al. [2019b] Zi-Wen Liu, Kaifeng Bu, and Ryuji Takagi.  One-shot operational quantum resource theory.  _Phys. Rev. Lett._ , 123:020401, Jul 2019b.  doi: 10.1103/PhysRevLett.123.020401.  URL https://link.aps.org/doi/10.1103/PhysRevLett.123.020401.
  * Takagi et al. [2019] Ryuji Takagi, Bartosz Regula, Kaifeng Bu, Zi-Wen Liu, and Gerardo Adesso.  Operational advantage of quantum resources in subchannel discrimination.  _Phys. Rev. Lett._ , 122:140402, Apr 2019.  doi: 10.1103/PhysRevLett.122.140402.  URL https://link.aps.org/doi/10.1103/PhysRevLett.122.140402.
  * Takagi and Regula [2019] Ryuji Takagi and Bartosz Regula.  General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks.  _Phys. Rev. X_ , 9:031053, Sep 2019.  doi: 10.1103/PhysRevX.9.031053.  URL https://link.aps.org/doi/10.1103/PhysRevX.9.031053.
  * Fang and Liu [2020] Kun Fang and Zi-Wen Liu.  No-go theorems for quantum resource purification.  _Phys. Rev. Lett._ , 125:060405, Aug 2020.  doi: 10.1103/PhysRevLett.125.060405.  URL https://link.aps.org/doi/10.1103/PhysRevLett.125.060405.
  * Regula et al. [2020] Bartosz Regula, Kaifeng Bu, Ryuji Takagi, and Zi-Wen Liu.  Benchmarking one-shot distillation in general quantum resource theories.  _Phys. Rev. A_ , 101:062315, Jun 2020.  doi: 10.1103/PhysRevA.101.062315.  URL https://link.aps.org/doi/10.1103/PhysRevA.101.062315.
  * Vijayan et al. [2019] Madhav Krishnan Vijayan, Eric Chitambar, and Min-Hsiu Hsieh.  One-shot distillation in a general resource theory.  _arXiv preprintarXiv:1906.04959 [quant-ph]_, 2019.  URL https://arxiv.org/abs/1906.04959.
  * [41] J. Eisert and M. M. Wolf.  _Gaussian Quantum Channels_ , pages 23–42.  doi: 10.1142/9781860948169_0002.  URL https://www.worldscientific.com/doi/abs/10.1142/9781860948169_0002.
  * Holevo [2007] A. S. Holevo.  One-mode quantum Gaussian channels: Structure and quantum capacity.  _Probl. Inf. Transm._ , 43, Mar 2007.  doi: 10.1134/S0032946007010012.  URL https://doi.org/10.1134/S0032946007010012.
  * Bera et al. [2017] Anindita Bera, Tamoghna Das, Debasis Sadhukhan, Sudipto Singha Roy, Aditi Sen(De), and Ujjwal Sen.  Quantum discord and its allies: a review of recent progress.  _Rep. Prog. Phys._ , 81(2):024001, dec 2017.  doi: 10.1088/1361-6633/aa872f.  URL https://doi.org/10.1088%2F1361-6633%2Faa872f.
  * Gudder [2008] Stanley Gudder.  Quantum markov chains.  _J. Math. Phys._ , 49(7):072105, 2008.  doi: 10.1063/1.2953952.  URL https://doi.org/10.1063/1.2953952.
  * Jonathan and Plenio [1999] Daniel Jonathan and Martin B. Plenio.  Entanglement-assisted local manipulation of pure quantum states.  _Phys. Rev. Lett._ , 83:3566–3569, Oct 1999.  doi: 10.1103/PhysRevLett.83.3566.  URL https://link.aps.org/doi/10.1103/PhysRevLett.83.3566.
  * Bennett et al. [1996a] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters.  Mixed-state entanglement and quantum error correction.  _Phys. Rev. A_ , 54:3824–3851, Nov 1996a.  doi: 10.1103/PhysRevA.54.3824.  URL https://link.aps.org/doi/10.1103/PhysRevA.54.3824.
  * Hayden et al. [2001] Patrick M Hayden, Michal Horodecki, and Barbara M Terhal.  The asymptotic entanglement cost of preparing a quantum state.  _J. Phys. A: Math. Gen._ , 34(35):6891–6898, aug 2001.  doi: 10.1088/0305-4470/34/35/314.  URL https://doi.org/10.1088%2F0305-4470%2F34%2F35%2F314.
  * Winter and Yang [2016] Andreas Winter and Dong Yang.  Operational resource theory of coherence.  _Phys. Rev. Lett._ , 116:120404, Mar 2016.  doi: 10.1103/PhysRevLett.116.120404.  URL https://link.aps.org/doi/10.1103/PhysRevLett.116.120404.
  * Donald et al. [2002] Matthew J. Donald, Michał Horodecki, and Oliver Rudolph.  The uniqueness theorem for entanglement measures.  _J. Math. Phys._ , 43(9):4252–4272, 2002.  doi: 10.1063/1.1495917.  URL https://doi.org/10.1063/1.1495917.
  * Reed and Simon [1980] Michael Reed and Barry Simon.  _Methods of modern mathematical physics. vol. 1. Functional analysis_.  Academic New York, 1980.
  * Takesaki [2012] Masamichi Takesaki.  _Theory of Operator Algebras I_.  Springer, 2012.  ISBN 0387903917;9780387903910;.  doi: 10.1007/978-1-4612-6188-9.  URL https://doi.org/10.1007/978-1-4612-6188-9.
  * Kitaev [1997] A Yu Kitaev.  Quantum computations: algorithms and error correction.  _Russ. Math. Surv._ , 52(6):1191–1249, dec 1997\.  doi: 10.1070/rm1997v052n06abeh002155.  URL https://doi.org/10.1070%2Frm1997v052n06abeh002155.
  * Brandão and Plenio [2008] Fernando G. S. L. Brandão and Martin Plenio.  Entanglement theory and the second law of thermodynamics.  _Nat. Phys._ , 4:873–877, 2008.  doi: 10.1038/nphys1100.  URL https://doi.org/10.1038/nphys1100.
  * Contreras-Tejada et al. [2019] Patricia Contreras-Tejada, Carlos Palazuelos, and Julio I. de Vicente.  Resource theory of entanglement with a unique multipartite maximally entangled state.  _Phys. Rev. Lett._ , 122:120503, Mar 2019.  doi: 10.1103/PhysRevLett.122.120503.  URL https://link.aps.org/doi/10.1103/PhysRevLett.122.120503.
  * Chitambar et al. [2014] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter.  Everything you always wanted to know about locc (but were afraid to ask).  _Commun. Math. Phys._ , 328(1):303–326, May 2014\.  ISSN 1432-0916.  doi: 10.1007/s00220-014-1953-9.  URL https://doi.org/10.1007/s00220-014-1953-9.
  * Nielsen and Chuang [2010] Michael A. Nielsen and Isaac L. Chuang.  _Quantum Computation and Quantum Information: 10th Anniversary Edition_.  Cambridge University Press, 2010.  doi: 10.1017/CBO9780511976667.  URL https://doi.org/10.1017/CBO9780511976667.
  * Groisman et al. [2005] Berry Groisman, Sandu Popescu, and Andreas Winter.  Quantum, classical, and total amount of correlations in a quantum state.  _Phys. Rev. A_ , 72:032317, Sep 2005.  doi: 10.1103/PhysRevA.72.032317.  URL https://link.aps.org/doi/10.1103/PhysRevA.72.032317.
  * Anshu et al. [2018] Anurag Anshu, Min-Hsiu Hsieh, and Rahul Jain.  Quantifying resources in general resource theory with catalysts.  _Phys. Rev. Lett._ , 121:190504, Nov 2018.  doi: 10.1103/PhysRevLett.121.190504.  URL https://link.aps.org/doi/10.1103/PhysRevLett.121.190504.
  * Pirandola et al. [2019] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden.  Advances in quantum cryptography.  _arXiv preprintarXiv:1906.01645 [quant-ph]_, 2019.  URL https://arxiv.org/abs/1906.01645.
  * Preskill [2018] John Preskill.  Quantum computing in the NISQ era and beyond.  _Quantum_ , 2:79, Aug 2018.  ISSN 2521-327X.  doi: 10.22331/q-2018-08-06-79.  URL http://dx.doi.org/10.22331/q-2018-08-06-79.
  * Bravyi and Haah [2012] Sergey Bravyi and Jeongwan Haah.  Magic-state distillation with low overhead.  _Phys. Rev. A_ , 86:052329, Nov 2012.  doi: 10.1103/PhysRevA.86.052329.  URL https://link.aps.org/doi/10.1103/PhysRevA.86.052329.
  * Berger et al. [2003] Ulrich Berger, G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott.  _Continuous Lattices and Domains_ , volume 93.  Cambridge University Press, 2003.  doi: 10.1017/CBO9780511542725.  URL https://doi.org/10.1017/CBO9780511542725.
  * Gottesman et al. [2001] Daniel Gottesman, Alexei Kitaev, and John Preskill.  Encoding a qubit in an oscillator.  _Phys. Rev. A_ , 64:012310, Jun 2001.  doi: 10.1103/PhysRevA.64.012310.  URL https://link.aps.org/doi/10.1103/PhysRevA.64.012310.
  * Pantaleoni et al. [2020] Giacomo Pantaleoni, Ben Q. Baragiola, and Nicolas C. Menicucci.  Modular bosonic subsystem codes.  _Phys. Rev. Lett._ , 125:040501, Jul 2020.  doi: 10.1103/PhysRevLett.125.040501.  URL https://link.aps.org/doi/10.1103/PhysRevLett.125.040501.
  * Weedbrook et al. [2012] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd.  Gaussian quantum information.  _Rev. Mod. Phys._ , 84:621–669, May 2012.  doi: 10.1103/RevModPhys.84.621.  URL https://link.aps.org/doi/10.1103/RevModPhys.84.621.
  * Horodecki et al. [2003] Michał Horodecki, Aditi Sen(De), and Ujjwal Sen.  Rates of asymptotic entanglement transformations for bipartite mixed states: Maximally entangled states are not special.  _Phys. Rev. A_ , 67:062314, Jun 2003.  doi: 10.1103/PhysRevA.67.062314.  URL https://link.aps.org/doi/10.1103/PhysRevA.67.062314.
  * Horodecki et al. [1998] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki.  Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature?  _Phys. Rev. Lett._ , 80:5239–5242, Jun 1998.  doi: 10.1103/PhysRevLett.80.5239.  URL https://link.aps.org/doi/10.1103/PhysRevLett.80.5239.
  * Bennett et al. [1996b] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher.  Concentrating partial entanglement by local operations.  _Phys. Rev. A_ , 53:2046–2052, Apr 1996b.  doi: 10.1103/PhysRevA.53.2046.  URL https://link.aps.org/doi/10.1103/PhysRevA.53.2046.
  * Wilde [2017] Mark M. Wilde.  _Entanglement Manipulation_ , page 517–536.  Cambridge University Press, 2 edition, 2017.  doi: 10.1017/9781316809976.022.  URL https://doi.org/10.1017/9781316809976.022.
  * Wootters and Zurek [1982] W. K. Wootters and W. H. Zurek.  A single quantum cannot be cloned.  _Nature_ , 299:802–803, Oct 1982.  doi: 10.1038/299802a0.  URL https://doi.org/10.1038/299802a0.
  * Wang et al. [2020] Xin Wang, Mark M. Wilde, and Yuan Su.  Efficiently computable bounds for magic state distillation.  _Phys. Rev. Lett._ , 124:090505, Mar 2020.  doi: 10.1103/PhysRevLett.124.090505.  URL https://link.aps.org/doi/10.1103/PhysRevLett.124.090505.
  * Plbnio and Virmani [2007] Martin B. Plbnio and Shashank Virmani.  An introduction to entanglement measures.  _Quantum Info. Comput._ , 7(1):1–51, jan 2007\.  ISSN 1533-7146.  doi: 10.5555/2011706.2011707.  URL https://dl.acm.org/doi/10.5555/2011706.2011707.
  * Bromley et al. [2018] Thomas R Bromley, Marco Cianciaruso, Sofoklis Vourekas, Bartosz Regula, and Gerardo Adesso.  Accessible bounds for general quantum resources.  _J. Phys. A: Math. Theor._ , 51(32):325303, jul 2018.  doi: 10.1088/1751-8121/aacb4a.  URL https://doi.org/10.1088%2F1751-8121%2Faacb4a.
  * Liu et al. [2017] Zi-Wen Liu, Xueyuan Hu, and Seth Lloyd.  Resource destroying maps.  _Phys. Rev. Lett._ , 118:060502, Feb 2017.  doi: 10.1103/PhysRevLett.118.060502.  URL https://link.aps.org/doi/10.1103/PhysRevLett.118.060502.
  * Regula [2017] Bartosz Regula.  Convex geometry of quantum resource quantification.  _J. Phys. A: Math. Theor._ , 51(4):045303, dec 2017.  doi: 10.1088/1751-8121/aa9100.  URL https://doi.org/10.1088%2F1751-8121%2Faa9100.
  * Synak-Radtke and Horodecki [2006] Barbara Synak-Radtke and Michał Horodecki.  On asymptotic continuity of functions of quantum states.  _J. Phys. A: Math. Gen._ , 39(26):L423–L437, jun 2006.  doi: 10.1088/0305-4470/39/26/l02.  URL https://doi.org/10.1088%2F0305-4470%2F39%2F26%2Fl02.
  * Lami [2019] L. Lami.  Completing the grand tour of asymptotic quantum coherence manipulation.  _IEEE Trans. Inf. Theory_ , pages 1–1, 2019.  ISSN 1557-9654.  doi: 10.1109/TIT.2019.2945798.  URL https://doi.org/10.1109/TIT.2019.2945798.
  * Vedral et al. [1997] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight.  Quantifying entanglement.  _Phys. Rev. Lett._ , 78:2275–2279, Mar 1997.  doi: 10.1103/PhysRevLett.78.2275.  URL https://link.aps.org/doi/10.1103/PhysRevLett.78.2275.
  * Brandão and Plenio [2010] Fernando G. S. L. Brandão and Martin B. Plenio.  A generalization of quantum Stein’s lemma.  _Commun. Math. Phys._ , 295(3):791–828, May 2010\.  ISSN 1432-0916.  doi: 10.1007/s00220-010-1005-z.  URL https://doi.org/10.1007/s00220-010-1005-z.
  * Horodecki et al. [2002] Michał Horodecki, Jonathan Oppenheim, and Ryszard Horodecki.  Are the laws of entanglement theory thermodynamical?  _Phys. Rev. Lett._ , 89:240403, Nov 2002.  doi: 10.1103/PhysRevLett.89.240403.  URL https://link.aps.org/doi/10.1103/PhysRevLett.89.240403.
  * Weis and Knauf [2012] Stephan Weis and Andreas Knauf.  Entropy distance: New quantum phenomena.  _J. Math. Phys._ , 53(10):102206, Oct 2012.  ISSN 1089-7658.  doi: 10.1063/1.4757652.  URL http://dx.doi.org/10.1063/1.4757652.
  * Herrero-Collantes and Garcia-Escartin [2017] Miguel Herrero-Collantes and Juan Carlos Garcia-Escartin.  Quantum random number generators.  _Rev. Mod. Phys._ , 89:015004, Feb 2017.  doi: 10.1103/RevModPhys.89.015004.  URL https://link.aps.org/doi/10.1103/RevModPhys.89.015004.
  * Dankert et al. [2009] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine.  Exact and approximate unitary 2-designs and their application to fidelity estimation.  _Phys. Rev. A_ , 80:012304, Jul 2009.  doi: 10.1103/PhysRevA.80.012304.  URL https://link.aps.org/doi/10.1103/PhysRevA.80.012304.
  * Theurer et al. [2019] Thomas Theurer, Dario Egloff, Lijian Zhang, and Martin B. Plenio.  Quantifying operations with an application to coherence.  _Phys. Rev. Lett._ , 122:190405, May 2019.  doi: 10.1103/PhysRevLett.122.190405.  URL https://link.aps.org/doi/10.1103/PhysRevLett.122.190405.
  * Gour and Winter [2019] Gilad Gour and Andreas Winter.  How to quantify a dynamical quantum resource.  _Phys. Rev. Lett._ , 123:150401, Oct 2019.  doi: 10.1103/PhysRevLett.123.150401.  URL https://link.aps.org/doi/10.1103/PhysRevLett.123.150401.
  * Liu and Yuan [2020] Yunchao Liu and Xiao Yuan.  Operational resource theory of quantum channels.  _Phys. Rev. Research_ , 2:012035, Feb 2020.  doi: 10.1103/PhysRevResearch.2.012035.  URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.012035.
  * Gour and Wilde [2018] Gilad Gour and Mark M. Wilde.  Entropy of a quantum channel.  _arXiv preprintarXiv:1808.06980 [quant-ph]_, 2018.  URL https://arxiv.org/abs/1808.06980.
  * Li et al. [2020] Lu Li, Kaifeng Bu, and Zi-Wen Liu.  Quantifying the resource content of quantum channels: An operational approach.  _Phys. Rev. A_ , 101:022335, Feb 2020.  doi: 10.1103/PhysRevA.101.022335.  URL https://link.aps.org/doi/10.1103/PhysRevA.101.022335.
  * Liu and Winter [2019] Zi-Wen Liu and Andreas Winter.  Resource theories of quantum channels and the universal role of resource erasure.  _arXiv preprintarXiv:1904.04201 [quant-ph]_, 2019.  URL https://arxiv.org/abs/1904.04201.
  * Takagi et al. [2020] Ryuji Takagi, Kun Wang, and Masahito Hayashi.  Application of the resource theory of channels to communication scenarios.  _Phys. Rev. Lett._ , 124:120502, Mar 2020.  doi: 10.1103/PhysRevLett.124.120502.  URL https://link.aps.org/doi/10.1103/PhysRevLett.124.120502.
  * Dunford and Schwartz [1958] Nelson Dunford and Jacob T Schwartz.  _Linear operators. Part I, General theory_.  New York : Interscience., 1958.
  * Dell’Antonio [1967] G. F. Dell’Antonio.  On the limits of sequences of normal states.  _Commun. Pure Appl. Math._ , 20(2):413–429, 1967\.  doi: 10.1002/cpa.3160200209.  URL https://doi.org/10.1002/cpa.3160200209.