# An information-theoretic treatment of quantum dichotomies

Francesco Buscemi  Graduate School of Informatics, Nagoya University, Nagoya,
Japan  David Sutter  Institute for Theoretical Physics, ETH Zurich,
Switzerland  Marco Tomamichel  Centre for Quantum Software and Information and
School of Computer Science,
University of Technology Sydney, Sydney<EMAIL_ADDRESS>
###### Abstract

Given two pairs of quantum states, we want to decide if there exists a quantum
channel that transforms one pair into the other. The theory of quantum
statistical comparison and quantum relative majorization provides necessary
and sufficient conditions for such a transformation to exist, but such
conditions are typically difficult to check in practice. Here, by building
upon work by Keiji Matsumoto, we relax the problem by allowing for small
errors in one of the transformations. In this way, a simple sufficient
condition can be formulated in terms of one-shot relative entropies of the two
pairs. In the asymptotic setting where we consider sequences of state pairs,
under some mild convergence conditions, this implies that the quantum relative
entropy is the only relevant quantity deciding when a pairwise state
transformation is possible. More precisely, if the relative entropy of the
initial state pair is strictly larger compared to the relative entropy of the
target state pair, then a transformation with exponentially vanishing error is
possible. On the other hand, if the relative entropy of the target state is
strictly larger, then any such transformation will have an error converging
exponentially to one. As an immediate consequence, we show that the rate at
which pairs of states can be transformed into each other is given by the ratio
of their relative entropies. We discuss applications to the resource theories
of athermality and coherence, where our results imply an exponential strong
converse for general state interconversion.

##  1 Introduction

Various pre- and partial orders have been the subject of extensive study both
in mathematical statistics [24, 5, 32, 56, 2, 55, 21] and in information
theory [50, 29, 20]. An example of paramount importance is that provided by
the majorization preorder [34]: a probability distribution $\vec{p}_{1}$ is
said to majorize another distribution $\vec{p}_{2}$, in formula
$\vec{p}_{1}\succeq\vec{p}_{2}$, whenever there exists a bistochastic111A
transformation is said to be bistochastic if it transforms probability
distributions to probability distributions, while keeping the uniform
distribution fixed. transformation $T$ such that $T\vec{p}_{1}=\vec{p}_{2}$.
The majorization preorder is particularly relevant and useful because of a
famous result by Hardy, Littlewood, and Pólya, according to which the relation
$\vec{p}_{1}\succeq\vec{p}_{2}$ can be expressed in terms of a finite set of
inequalities [24] of the form $f_{i}(\vec{p}_{1})\geq f_{i}(\vec{p}_{2})$,
intuitively capturing the idea that $\vec{p}_{1}$ is “less uniform” than
$\vec{p}_{2}$. Such inequalities can be conveniently visualized by plotting
the Lorenz curve of $\vec{p}_{1}$ versus that of $\vec{p}_{2}$ [34].

As it involves the comparison of two probability distributions relative to a
third one (i.e., the uniform distribution), the majorization preorder is
naturally generalized by considering two pairs of probability distributions,
that is, two dichotomies $(\vec{p}_{1},\vec{q}_{1})$ and
$(\vec{p}_{2},\vec{q}_{2})$, where now $\vec{q}_{1}$ and $\vec{q}_{2}$ are
arbitrary distributions. One then writes
$(\vec{p}_{1},\vec{q}_{1})\succeq(\vec{p}_{2},\vec{q}_{2})$ whenever there
exists a stochastic transformation simultaneously mapping $\vec{p}_{1}$ to
$\vec{p}_{2}$ and $\vec{q}_{1}$ to $\vec{q}_{2}$. As a consequence of
Blackwell’s equivalence theorem [5], also the more general case of dichotomies
is completely characterized by a finite set of simple inequalities, which
directly reduce to those of Hardy, Littlewood and Polya if $\vec{q}_{1}$ and
$\vec{q}_{2}$ are both taken to be uniform. Also in this more general
scenario, a relative Lorenz curve can be associated to each dichotomy, and the
preorder $\succeq$ visualized accordingly [47].

In relation to quantum information sciences, while the preorder of
majorization has found early applications in entanglement theory [41], the
notion of relative majorization have started being employed only more
recently, especially due to its applications in quantum thermodynamics [26, 6,
7, 47, 10, 23] and generalized resource theories [16]. In the quantum setting,
the objects of comparison are quantum dichotomies, namely, pairs of density
matrices.

Let us consider two arbitrary finite-dimensional quantum dichotomies
$(\rho_{1},\sigma_{1})$ and $(\rho_{2},\sigma_{2})$. In complete analogy with
the classical case à la Blackwell, the relative majorization preorder
$\succeq$ can be extended to the quantum setting by writing
$(\rho_{1},\sigma_{1})\succeq(\rho_{2},\sigma_{2})$ whenever there exists a
completely positive trace-preserving map $\mathcal{E}$ such that
$\mathcal{E}(\rho_{1})=\rho_{2}$ and $\mathcal{E}(\sigma_{1})=\sigma_{2}$
simultaneously. However, in the quantum case222With the notable exception of
the qubit case [1, 12] and pure states [11, 51]., there is no known simple set
of inequalities, analogously to the comparison of two relative Lorenz curves,
able to completely capture the relative majorization preorder [44, 27, 38,
10]: statistical conditions can be derived [8, 36, 28] but they typically
involve an infinite number of inequalities, thus becoming much more cumbersome
to verify.

In order to overcome such problems, in this paper we build upon an
information-theoretic approach for the comparison of quantum dichotomies first
considered by Matsumoto in [37]. This involves the relaxation of the order
$\succeq$ to allow for errors in the transformation, and the consideration of
an asymptotic regime, in which an increasing number of identical copies of one
dichotomy get transformed into something that resembles, up to an arbitrarily
high level of accuracy, many copies of the other dichotomy. More precisely,
while we allow for (small) errors in the transformation
$\mathcal{E}(\rho_{1})\approx\rho_{2}$, the condition
$\mathcal{E}(\sigma_{1})=\sigma_{2}$ must always be satisfied exactly, as if
it were a sort of “conservation rule.” We compute that the optimal rate at
which such transformation can happen is given by the ratio between the quantum
relative entropies $D(\rho_{1}\|\sigma_{1})$ and $D(\rho_{2}\|\sigma_{2})$.
Our result hence shows that quantum dichotomies, while enjoying a very rich
structure, are asymptotically characterized by a single number, that is, their
relative entropy.

The remainder of the paper is structured as follows. After introducing the
relevant one-shot divergences and their properties in Section 2, we establish
our main technical results in Section 3. First, in Section 3.1 we derive
sufficient conditions in terms of one-shot divergences for exact pairwise
state transformations. In Section 3.2 we relax this to allow for an error on
one of the states, and again find sufficient conditions in terms of smoothed
one-shot entropies. This then allows us to derive our main results in Section
3.3, Theorems 3.4 and 3.6, which together show that the relative entropy fully
characterizes when pairwise transformations are possible asymptotically.
Section 4 then takes an information-theoretic approach to the problem by
studying the maximal rate at which independent copies of states can be
transformed into each other in Theorem 4.1. We finally discuss applications of
our results to the resource theories of athermality and coherence in Section 5
and end with a conjecture characterizing the second-order asymptotic behavior
of resource transformations in Section 6.

_Note added._ Concurrent work [57, 49] also derives some of our results in the
setting of resource theories and thermodynamics. We explore these connections
in Section 5 for the convenience of the reader.

##  2 Preliminaries

###  2.1 Notation

Let $\mathrm{S}(\mathbb{C}^{d})$ denote the set of _quantum states_ on a
$d$-dimensional Hilbert space $\mathbb{C}^{d}$. A _quantum channel_
$\mathcal{E}:\mathrm{S}(\mathbb{C}^{d})\to\mathrm{S}(\mathbb{C}^{d^{\prime}})$
is a linear map that is _completely positive_ and _trace-preserving_ (CPTP).
We denote by $\leq$ the Löwner partial order, i.e., for two Hermitian matrices
$X$ and $Y$ the relation $X\geq Y$ means that $X-Y$ is positive semi-definite,
and the relation $X\gg Y$ means that the support of $Y$ is contained in the
support of $X$. Throughout this paper we denote by $\log$ the logarithm to
base $2$.

We denote the _trace distance_ between two states $\rho$ and $\sigma$ by
$T(\rho,\sigma):=\frac{1}{2}\left|\\!\left|{\rho-\sigma}\right|\\!\right|_{1}$,
where $\|\cdot\|_{1}$ denotes the Schatten 1-norm. The _fidelity_ is given as
$F(\rho,\sigma):=\|\sqrt{\rho}\sqrt{\sigma}\|_{1}^{2}$. We will also use the
_sine_ or _purified distance_ , $P(\rho,\sigma):=\sqrt{1-F(\rho,\sigma)}$.
Both $P$ and $T$ satisfy the triangle inequality and are non-increasing when a
CPTP map is applied to both states. They are related by the Fuchs-van de Graaf
inequalities stating that $T(\rho,\sigma)\leq
P(\rho,\sigma)\leq\sqrt{2T(\rho,\sigma)-T(\rho,\sigma)^{2}}$. When the choice
of metric is arbitrary, we use $\Delta$ to stand for either $T$ or $P$.

###  2.2 Some divergences and their properties

In this work we use several different non-commutative divergences. We will
introduce here only the measures and properties that are needed for this work
— an interested reader may consult [52] for a more comprehensive discussion
with references to all the original papers. For
$\rho,\sigma\in\mathrm{S}(\mathbb{C}^{d})$ the relative entropy is given by
$D(\rho\|\sigma):=\mathrm{tr}\,\rho(\log\rho-\log\sigma)$ if $\sigma\gg\rho$,
and $+\infty$ otherwise. To simplify the exposition in the following we assume
throughout that the states $\sigma$ always have full support and are thus
invertible, avoiding such infinities.

For $\alpha\in(0,1)\cup(1,2]$ the _Petz quantum Rényi divergence_ [43] is
defined as

$\displaystyle\bar{D}_{\alpha}(\rho\|\sigma):=\frac{1}{\alpha-1}\log\mathrm{tr}\,\rho^{\alpha}\sigma^{1-\alpha}\,.$
(1)

In the limit $\alpha\to 0$ the divergence converges to the _min-relative
entropy_ [48, 22], i.e.,

$\displaystyle D_{\min}(\rho\|\sigma)$
$\displaystyle:=-\log\mathrm{tr}\,\sigma\Pi_{\rho}=\lim_{\alpha\to
0}\bar{D}_{\alpha}(\rho\|\sigma)\,,$ (2)

where $\Pi_{\rho}$ is the projector onto the support of $\rho$.

Another non-commutative family of Rényi divergences is the _sandwiched quantum
Rényi divergence_ [40, 58], for $\alpha\in[\frac{1}{2},1)\cup(1,\infty)$
defined as

$\displaystyle\tilde{D}_{\alpha}(\rho\|\sigma):=\frac{1}{\alpha-1}\log\mathrm{tr}\big{(}\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\big{)}^{\alpha}\,.$
(3)

For $\alpha=\frac{1}{2}$ the sandwiched quantum Rényi divergence becomes
$\tilde{D}_{\nicefrac{{1}}{{2}}}(\rho\|\sigma)=-\log F(\rho,\sigma)$. In the
limit $\alpha\to\infty$ the sandwiched quantum Rényi divergence converges to
the so-called _max-divergence_ [48, 22], i.e.,

$\displaystyle D_{\max}(\rho\|\sigma)$
$\displaystyle:=\inf\big{\\{}\lambda\in\mathbb{R}:\rho\leq
2^{\lambda}\sigma\big{\\}}\,.$ (4)

Both non-commutative families of Rényi divergences introduced above satisfy
many desirable properties: they are monotonically increasing in $\alpha$, they
satisfiy the data-processing inequality (i.e. they are non-increasing when the
same CPTP map is applied to both states), and in the limit $\alpha\to 1$ they
both converge to the relative entropy.

Smooth variants of the max and min-divergence will be useful to treat problems
with finite errors. The _$\varepsilon$ -smooth max-divergence_ is defined as

$\displaystyle
D^{\varepsilon,\Delta}_{\max}(\rho\|\sigma):=\inf_{\tilde{\rho}\in\mathcal{B}_{\varepsilon}^{\Delta}(\rho)}D_{\max}(\tilde{\rho}\|\sigma)\,,$
(5)

where
$\mathcal{B}_{\varepsilon}^{\Delta}(\rho):=\\{\tilde{\rho}\in\mathrm{S}(\mathbb{C}^{d}):\Delta(\rho,\tilde{\rho})\leq\varepsilon\\}$
for $\varepsilon\in(0,1)$. We will use this definition for both trace distance
and purified distance, but note that in contrast to some other works the
optimization here only goes over close quantum states, not sub-normalized
states. As the smooth variant of $D_{\min}$ we introduce the so-called
_hypothesis testing divergence_.333In some earlier works (see, e.g., [22]) an
optimization over a ball of $\varepsilon$-close states is used instead to
define a smooth variant of $D_{\min}$, in analogy with
$D_{\max}^{\varepsilon}$ defined above. However, the quantity we use here,
first introduced in [9] as a generalization of $D_{\min}$, is more natural for
our problem here. For any $\varepsilon\in(0,1)$ it is defined as

$\displaystyle
D_{h}^{\varepsilon}(\rho\|\sigma):=-\log\inf\big{\\{}\mathrm{tr}\,\sigma
Q:0\leq Q\leq\mathds{1}\ \land\ \mathrm{tr}\,\rho\,Q\geq
1-\varepsilon\big{\\}}\,.$ (6)

In the limit $\varepsilon\to 0$ we recover $D_{\min}(\rho\|\sigma)$. Finally,
we note that both of these divergences satisfy the data-processing inequality,
namely

$\displaystyle D^{\varepsilon,\Delta}_{\max}(\rho\|\sigma)\geq
D^{\varepsilon,\Delta}_{\max}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))\qquad\textrm{and}\qquad
D^{\varepsilon}_{h}(\rho\|\sigma)\geq
D^{\varepsilon}_{h}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))$ (7)

for any CPTP map $\mathcal{E}$.

The smooth max-divergence and the hypothesis testing divergence are closely
related, as shown in [3, Theorem 4].

######  Proposition 2.1.

Let $\rho,\sigma\in\mathrm{S}(\mathbb{C}^{d})$ and $\varepsilon\in(0,1)$ and
$\nu\in(0,1-\varepsilon)$. It holds that

$\displaystyle D_{h}^{1-\varepsilon}(\rho\|\sigma)\geq
D_{\max}^{\sqrt{\varepsilon},P}(\rho\|\sigma)-\log\frac{1}{1-\varepsilon}\geq
D_{h}^{1-\varepsilon-\nu}(\rho\|\sigma)-\log\frac{4}{\nu^{2}}\,.$ (8)

We next recall two inequalities that relate the hypothesis testing relative
entropy and the max-relative entropy to the Rényi divergences.

######  Proposition 2.2.

Let $\varepsilon\in(0,1)$ and let $\rho,\sigma\in\mathrm{S}(\mathbb{C}^{d})$.
Then,

$\displaystyle D_{h}^{\varepsilon}(\rho\|\sigma)$
$\displaystyle\geq\bar{D}_{\alpha}(\rho\|\sigma)-\frac{\alpha}{1-\alpha}\log\frac{1}{\varepsilon}$
$\displaystyle\textrm{for }\alpha\in[0,1)\,,\quad\textrm{and}$ (9)
$\displaystyle D_{\max}^{\varepsilon,\Delta}(\rho\|\sigma)$
$\displaystyle\leq\tilde{D}_{\alpha}(\rho\|\sigma)+\frac{1}{\alpha-1}\log\frac{1}{\varepsilon^{2}}+\log\frac{1}{1-\varepsilon^{2}}$
$\displaystyle\textrm{for }\alpha\in(1,\infty]\,.$ (10)

We can interchange $\bar{D}_{\alpha}$ and $\tilde{D}_{\alpha}$ in the above
inequalities.

Note that we can interchange the Rényi relative entropies in both inequalities
since $\bar{D}_{\alpha}(\rho\|\sigma)\geq\tilde{D}_{\alpha}(\rho\|\sigma)$ for
all states due to the Araki-Lieb-Thirring inequality. Inequality (9) follows
immediately from [4, Proposition 3.2]. Inequality (10) is shown in [3, Theorem
3], tightening earlier bounds that were established as part of the fully
quantum asymptotic equipartition property (QAEP) [53]. The QAEP states that,
for all $\varepsilon\in(0,1)$, the regularized smooth entropies converge to
the relative entropy

$\displaystyle\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\varepsilon,\Delta}\left(\rho^{\otimes
n}\middle\|\sigma^{\otimes n}\right)=D(\rho\|\sigma)\,.$ (11)

The analogous statement for $D_{h}^{\varepsilon}$ is an immediate consequence
from quantum Stein’s lemma and its converse [25, 42].

##  3 Conditions for pairwise state transformation

In this section we derive sufficient conditions for the existence of a channel
that transforms $(\rho_{1},\sigma_{1})$ to $(\rho_{2},\sigma_{2})$, where the
first state is transformed either exactly or approximately, and the second
state always has to be transformed exactly.

###  3.1 Conditions for exact state transformation

We start by considering the case of exact transformations. We note that
conditions for an exact transformation derived in this section are very
restrictive. If we allow for approximate transformations as done in Section
3.2 we will find conditions that are considerably easier to fulfill.

For what follows, we can restrict ourselves to a very special class of
transformations, namely, test-and-prepare channels of the form

$\displaystyle\mathcal{E}(\rho)=\gamma_{1}\;\mathrm{tr}\,E\rho+\gamma_{2}\;\mathrm{tr}\,(\mathds{1}-E)\rho\;,$
(12)

where the $\gamma_{i}$’s are density matrices and $0\leq E\leq\mathds{1}$.
Hence test-and-prepare channels constitute a subset of measure-and-prepare
channels, in which the measurement is a simple binary test. Ref. [10] provides
a complete characterization of this case. The following lemma can be obtained
as a consequence of the results in [10], but we provide an independent proof
here for the sake of the reader (see also Ref. [39]).

######  Lemma 3.1.

Let $\rho_{1},\sigma_{1}\in\mathrm{S}(\mathbb{C}^{2})$ be commuting qubit
quantum states and let $\rho_{2},\sigma_{2}\in\mathrm{S}(\mathbb{C}^{d})$. The
following two conditions are equivalent:

  1. (i)

there exists a CPTP map
$\mathcal{E}:\mathrm{S}(\mathbb{C}^{2})\to\mathrm{S}(\mathbb{C}^{d})$ such
that $\mathcal{E}(\rho_{1})=\rho_{2}$ and
$\mathcal{E}(\sigma_{1})=\sigma_{2}$;

  2. (ii)

$D_{\max}(\rho_{1}\|\sigma_{1})\geq D_{\max}(\rho_{2}\|\sigma_{2})$ and
$D_{\max}(\sigma_{1}\|\rho_{1})\geq D_{\max}(\sigma_{2}\|\rho_{2})$.

###### Proof.

Since the implication (i)$\implies$(ii) is just the data-processing
inequality, we only need to prove the reverse implication (ii)$\implies$(i).

By assumption, $\rho_{1}$ and $\sigma_{1}$ commute. Hence, we can see them as
classical binary probability distributions, namely,
$\rho_{1}\leftrightarrow\vec{p}_{1}=(p,1-p)$ and
$\sigma_{1}\leftrightarrow\vec{q}_{1}=(q,1-q)$. Moreover, we can assume that
$\frac{p}{q}\geq\frac{1-p}{1-q}$; otherwise, the first step is to map
$(p,1-p)$ and $(q,1-q)$ into $(1-p,p)$ and $(1-q,q)$, respectively. Notice
that this is equivalent to saying that, without loss of generality, we can
always assume $p(1-q)\geq q(1-p)$, that is, $p\geq q$.

Let us now define $M:=\frac{p}{q}$ and $m:=\frac{1-p}{1-q}$. Notice that $\log
M=D_{\max}(\rho_{1}\|\sigma_{1})$ and $\log
m=-D_{\max}(\sigma_{1}\|\rho_{1})$. Problems with such definitions occur when
$q=0$ or $q=1$. If $q=1$, since (as noticed above) we can assume that $p\geq
q$, then also $p=1$; that is, $\rho_{1}=\sigma_{1}$ and
$D_{\max}(\rho_{1}\|\sigma_{1})=-D_{\max}(\sigma_{1}\|\rho_{1})=0$ (or,
equivalently, $M=m=1$). By Condition (ii), this implies that also
$D_{\max}(\rho_{2}\|\sigma_{2})=0$, namely, $\rho_{2}=\sigma_{2}$, and there
is nothing to prove. The case $q=0$ will be addressed separately at the end of
the proof.

For the moment, we thus assume that $\infty>M>1>m\geq 0$, that is, $p>q>0$.
Condition (ii) then guarantees that $M\sigma_{2}-\rho_{2}\geq 0$ and
$\rho_{2}-m\sigma_{2}\geq 0$. We thus define a linear map
$\mathcal{E}:\mathrm{S}(\mathbb{C}^{2})\to\mathrm{S}(\mathbb{C}^{d})$ as
follows:

$\displaystyle\mathcal{E}(\cdot):=\langle
0|\cdot|0\rangle\frac{\rho_{2}-m\sigma_{2}}{1-m}+\langle
1|\cdot|1\rangle\frac{M\sigma_{2}-\rho_{2}}{M-1}\;.$ (13)

By construction, $\mathcal{E}$ is clearly CPTP. On the other hand, by direct
inspection, $\mathcal{E}(\rho_{1})=\rho_{2}$ and
$\mathcal{E}(\sigma_{1})=\sigma_{2}$.

In order to conclude the proof, we are left to address the case in which
$q=0$. In such a case, we consider a variant of the map in (13), that is,

$\displaystyle\mathcal{F}(\cdot):=\langle
0|\cdot|0\rangle\frac{\rho_{2}-m\sigma_{2}}{1-m}+\langle
1|\cdot|1\rangle\sigma_{2}\;.$

Again, by direct inspection it is easy to check that $\mathcal{F}$ is CPTP and
that $\mathcal{E}(\rho_{1})=\rho_{2}$ and
$\mathcal{E}(\sigma_{1})=\sigma_{2}$, as claimed. ∎

From the above, we obtain a sufficient condition for the existence of a
transformation, more precisely a test-and-prepare channel, for arbitrary pairs
of states.

######  Corollary 3.2.

Let $\rho_{1},\sigma_{1}\in\mathrm{S}(\mathbb{C}^{d_{1}})$ and
$\rho_{2},\sigma_{2}\in\mathrm{S}(\mathbb{C}^{d_{2}})$. Then, if either

$\displaystyle D_{\min}(\rho_{1}\|\sigma_{1})\geq
D_{\max}(\rho_{2}\|\sigma_{2})\qquad\textrm{or}\qquad
D_{\min}(\sigma_{1}\|\rho_{1})\geq D_{\max}(\sigma_{2}\|\rho_{2})\,,$ (14)

then there exists a test-and-prepare channel $\mathcal{E}$ such that
$\mathcal{E}(\rho_{1})=\rho_{2}$ and $\mathcal{E}(\sigma_{1})=\sigma_{2}$.

###### Proof.

It suffices to show the statement under the first condition; the second then
follows by symmetry. Let us consider the measurement channel

$\mathcal{M}(\cdot):=\operatorname{tr}[\cdot\;\Pi]|0\rangle\langle
0|+\operatorname{tr}[\cdot\;(\mathds{1}-\Pi)]|1\rangle\langle 1|\;,$

where $\Pi$ is the projector onto the support of $\rho_{1}$. Then, the binary
classical probability distributions obtained from the pair
$(\rho_{1},\sigma_{1})$ are
$\mathcal{M}(\rho_{1})\leftrightarrow\vec{p}_{1}=(p,1-p)=(1,0)$ and
$\mathcal{M}(\sigma_{1})\leftrightarrow\vec{q}_{1}=(q,1-q)=(\operatorname{tr}[\Pi\;\sigma_{1}],\operatorname{tr}[(\mathds{1}-\Pi)\;\sigma_{1}])$.

We only need to show that, if Eq. (14) holds, then $\vec{p}_{1}$ and
$\vec{q}_{1}$ satisfy condition (ii) in Lemma 3.1. That is indeed the case,
since, on the one hand,

$\frac{p}{q}=\frac{1}{\operatorname{tr}[\Pi\;\sigma_{1}]}=2^{D_{\min}(\rho_{1}\|\sigma_{1})}\;,$

that is, $D_{\max}(\vec{p}_{1}\|\vec{q}_{1})=D_{\min}(\rho_{1}\|\sigma_{1})$,
so that Eq. (14) guarantees that $D_{\max}(\vec{p}_{1}\|\vec{q}_{1})\geq
D_{\max}(\rho_{2}\|\sigma_{2})$. Notice that, in the above equation, the case
$q=0$ can be excluded, as it is equivalent to $\rho_{1}$ and $\sigma_{1}$
being perfectly distinguishable, and thus exactly transformable into any other
pair of states.

On the other hand, because

$2^{-D_{\max}(\vec{q}_{1}\|\vec{p}_{1})}=\frac{1-p}{1-q}=0\leq
2^{-D_{\max}(\sigma_{2}\|\rho_{2})}\;.$

the remaining condition, $D_{\max}(\vec{q}_{1}\|\vec{p}_{1})\geq
D_{\max}(\sigma_{2}\|\rho_{2})$, is guaranteed by definition. In the above
equation, the case $q=1$ can be excluded, as it is equivalent to $\rho_{1}$
and $\sigma_{1}$ having the same support, that is,
$D_{\min}(\rho_{1}\|\sigma_{1})=D_{\min}(\sigma_{1}\|\rho_{1})=0$, which, by
virtue of assumption (14), implies that $\rho_{2}=\sigma_{2}$, so that there
is nothing to prove. ∎

We note that condition (14) is very strong in general. To see this, it is
enough to simply consider two states with coinciding support, which is what
happens generically (in the generic case, states are invertible with
probability one). Then, since in such a case
$D_{\min}(\rho_{1}\|\sigma_{1})=D_{\min}(\sigma_{1}\|\rho_{1})=0$, the only
dichotomies satisfying Eq. (14) are the trivial ones, i.e., those with
$\rho_{2}=\sigma_{2}$, as a consequence of the fact that max-relative entropy
vanishes if and only if the two arguments coincide.

However, when errors are allowed in condition (14), in what follows we show
that a much more flexible framework can be constructed, which is in fact
sufficient to derive strong asymptotic results.

###  3.2 Sufficient condition for approximate state transformation

In this section we are interested in approximate state transformation, i.e., a
transformation from $(\rho_{1},\sigma_{1})$ to $(\rho_{2},\sigma_{2})$ where
we allow for a (small) error in the transformation $\rho_{1}\to\rho_{2}$,
while the transformation $\sigma_{1}\to\sigma_{2}$ is required to be exact.

######  Lemma 3.3.

Let $\varepsilon_{1},\varepsilon_{2}\in(0,1)$,
$\rho_{1},\sigma_{1}\in\mathrm{S}(\mathbb{C}^{d_{1}})$ and
$\rho_{2},\sigma_{2}\in\mathrm{S}(\mathbb{C}^{d_{2}})$. If either

$\displaystyle D_{h}^{\varepsilon_{1}}(\rho_{1}\|\sigma_{1})\geq
D_{\max}^{\varepsilon_{2},T}(\rho_{2}\|\sigma_{2})\qquad\textrm{or}\qquad
D_{h}^{\varepsilon_{1}}(\rho_{1}\|\sigma_{1})\geq
D_{\max}^{\varepsilon_{2},P}(\rho_{2}\|\sigma_{2})\,,$ (15)

then there exists a test-and-prepare quantum channel
$\mathcal{E}:\mathrm{S}(\mathbb{C}^{d_{1}})\to\mathrm{S}(\mathbb{C}^{d_{2}})$
satisfying $\mathcal{E}(\sigma_{1})=\sigma_{2}$ and
$T(\mathcal{E}(\rho_{1}),\rho_{2})\leq\varepsilon_{1}+\varepsilon_{2}$ or
$P(\mathcal{E}(\rho_{1}),\rho_{2})\leq\sqrt{\varepsilon_{1}}+\varepsilon_{2}$,
respectively.

###### Proof.

Let $Q^{*}$ denote the optimizer in (6) for
$D_{h}^{\varepsilon_{1}}(\rho_{1}\|\sigma_{1})$ that satisfies

$\displaystyle
2^{-D_{h}^{\varepsilon_{1}}(\rho_{1}\|\sigma_{1})}=\mathrm{tr}\,\sigma_{1}Q^{*}\qquad\textrm{and}\qquad\mathrm{tr}\sigma_{1}Q^{*}\leq\mathrm{tr}\,\rho_{1}Q^{*}=1-\varepsilon_{1}<1\,.$
(16)

By definition of the smooth max-relative entropy, there furthermore exists a
state $\tilde{\rho}_{2}\in\mathcal{B}_{\varepsilon_{2}}^{\Delta}(\rho_{2})$
for $\Delta\in\\{T,P\\}$, such that

$\displaystyle
D_{\max}^{\varepsilon_{2},\Delta}(\rho_{2}\|\sigma_{2})=D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})\,.$
(17)

Consider now the mapping

$\displaystyle\mathcal{E}\,:\,X\mapsto\mathcal{E}(X)$
$\displaystyle=\tilde{\rho}_{2}\mathrm{tr}(XQ^{*})+\mathrm{tr}\big{(}X(\mathds{1}-Q^{*})\big{)}\frac{\sigma_{2}-\tilde{\rho}_{2}\mathrm{tr}(\sigma_{1}Q^{*})}{\mathrm{tr}\big{(}\sigma_{1}(\mathds{1}-Q^{*})\big{)}}$
(18)
$\displaystyle=\left(\mathrm{tr}X-\frac{\mathrm{tr}\big{(}X(\mathds{1}-Q^{*})\big{)}}{\mathrm{tr}\big{(}\sigma_{1}(\mathds{1}-Q^{*})\big{)}}\right)\tilde{\rho}_{2}+\left(\frac{\mathrm{tr}\big{(}X(\mathds{1}-Q^{*})\big{)}}{\mathrm{tr}\big{(}\sigma_{1}(\mathds{1}-Q^{*})\big{)}}\right)\sigma_{2}\,.$
(19)

We start by proving that $\mathcal{E}$ is a quantum channel, i.e., a trace-
preserving completely positive map. To see that $\mathcal{E}$ is trace-
preserving is straightforward since $\sigma_{2}$ and $\tilde{\rho}_{2}$ are
density operators. We note that because $0\leq Q^{*}\leq\mathds{1}$ it
suffices to show that
$\sigma_{2}\geq\tilde{\rho}_{2}\mathrm{tr}(\sigma_{1}Q^{*})$ in order prove
that $\mathcal{E}$ is completely positive. By definition of the smooth max-
relative entropy and by using (15) and (16) we find

$\displaystyle
2^{-D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})}=2^{-D_{\max}^{\varepsilon_{2},\Delta}(\rho_{2}\|\sigma_{2})}\geq
2^{-D_{h}^{\varepsilon_{1}}(\rho_{1}\|\sigma_{1})}=\mathrm{tr}\sigma_{1}Q^{*}\,.$
(20)

By definition of the max-relative entropy we thus have

$\displaystyle\sigma_{2}\geq\tilde{\rho}_{2}\,2^{-D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})}\geq\tilde{\rho}_{2}\,\mathrm{tr}(\sigma_{1}Q^{*})\,.$
(21)

We have seen that $\mathcal{E}$ is indeed a quantum channel. It thus remains
to show that $\mathcal{E}(\sigma_{1})=\sigma_{2}$ and
$\Delta(\mathcal{E}(\rho_{1}),\rho_{2})\leq\varepsilon_{1}+\varepsilon_{2}$.
The first property is straightforward to verify. The second property requires
some more work. Define $\bar{\rho}_{2}:=\mathcal{E}(\rho_{1})$. The triangle
inequality immediately yields

$\displaystyle\Delta(\rho_{2},\bar{\rho}_{2})\leq\varepsilon_{2}+\Delta(\tilde{\rho}_{2},\bar{\rho}_{2})\,,$
(22)

and it thus remains to bound the second term.

Let us first consider the case where
$\Delta(\rho,\tau)=T(\rho,\tau)=\frac{1}{2}\|\rho-\tau\|_{1}$ is the trace
distance. Substituting the expression in (19), we find

$\displaystyle\left|\\!\left|{\tilde{\rho}_{2}-\bar{\rho}_{2}}\right|\\!\right|_{1}=\frac{1-\mathrm{tr}\rho_{1}Q^{*}}{1-\mathrm{tr}\sigma_{1}Q^{*}}\|\tilde{\rho}_{2}-\sigma_{2}\|_{1}\leq\varepsilon_{1}\frac{\|\tilde{\rho}_{2}-\sigma_{2}\|_{1}}{1-2^{-D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})}}\,,$
(23)

where we used (16) and (20) in the final step. Using (21) we find for
$\Pi_{+}$ denoting the projector on to the positive support of
$\rho_{2}-\sigma_{2}$

$\displaystyle\left|\\!\left|{\tilde{\rho}_{2}-\sigma_{2}}\right|\\!\right|_{1}=2\,\mathrm{tr}\,\Pi_{+}(\tilde{\rho}_{2}-\sigma_{2})$
$\displaystyle\leq
2\Big{(}1-2^{-D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})}\Big{)}\mathrm{tr}\,\Pi_{+}\tilde{\rho}_{2}$
(24) $\displaystyle\leq
2\Big{(}1-2^{-D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})}\Big{)}\,.$ (25)

Combining this with (22) finally gives
$\Delta\big{(}\mathcal{E}(\rho_{1}),\rho_{2}\big{)}\leq\varepsilon_{1}+\varepsilon_{2}$,
concluding the proof for the trace distance.

Let us now consider the case where
$\Delta(\rho,\tau)=P(\rho,\tau)=\sqrt{1-F(\rho,\tau)}$ is the purified
distance. First, we recall that
$\mathrm{tr}\rho_{1}Q^{*}\geq\mathrm{tr}\sigma_{1}Q^{*}$ by definition of
$Q^{*}$ as the optimizer for the hypothesis testing divergence. We then use
the concavity of fidelity (see, e.g., [52, Lemma 3.4]) and the expression in
(19) to bound

$\displaystyle F(\tilde{\rho}_{2},\bar{\rho}_{2})$
$\displaystyle\geq\left(1-\frac{1-\mathrm{tr}\rho_{1}Q^{*}}{1-\mathrm{tr}\sigma_{1}Q^{*}}\right)+\left(\frac{1-\mathrm{tr}\rho_{1}Q^{*}}{1-\mathrm{tr}\sigma_{1}Q^{*}}\right)F(\tilde{\rho}_{2},\sigma_{2})$
(26)
$\displaystyle=1-\frac{1-\mathrm{tr}\rho_{1}Q^{*}}{1-\mathrm{tr}\sigma_{1}Q^{*}}\left(1-F(\tilde{\rho}_{2},\sigma_{2})\right)$
(27) $\displaystyle\geq
1-\varepsilon_{1}\frac{1-F(\tilde{\rho}_{2},\sigma_{2})}{1-2^{-D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})}}\,.$
(28)

Using the monotonicity of sandwiched Rényi relative entropy we find
$D_{\max}(\tilde{\rho}_{2}\|\sigma_{2})\geq-\log
F(\tilde{\rho}_{2},\sigma_{2})$ and $F(\tilde{\rho}_{2},\bar{\rho}_{2})\geq
1-\varepsilon_{1}$, concluding that
$\Delta\big{(}\mathcal{E}(\rho_{1}),\rho_{2}\big{)}\leq\sqrt{\varepsilon_{1}}+\varepsilon_{2}$.
∎

###  3.3 Conditions for asymptotic state transformation

In the following we will consider an asymptotic setting given by four
sequences of states $\vec{\rho}_{1}=\\{\rho_{1}^{n}\\}_{n}$,
${\vec{\sigma}}_{1}=\\{\sigma_{1}^{n}\\}_{n}$,
${\vec{\rho}}_{2}=\\{\rho_{2}^{n}\\}_{n}$, and
${\vec{\sigma}}_{2}=\\{\sigma_{2}^{n}\\}_{n}$ for $n\in\mathbb{N}$. We assume
no specific structure for these states and the underlying Hilbert spaces, i.e.
in general we have
$\rho_{1}^{n},\sigma_{1}^{n}\in\mathrm{S}(\mathbb{C}^{d_{1}^{n}})$ and
$\rho_{2}^{n},\sigma_{2}^{n}\in\mathrm{S}(\mathbb{C}^{d_{2}^{n}})$ for
arbitrary dimensions $\\{d_{1}^{n}\\}_{n}$ and $\\{d_{2}^{n}\\}_{n}$. The only
requirement that we impose is that, for $i\in\\{1,2\\}$, the limits

$\displaystyle\tilde{D}_{\alpha}(\vec{\rho}_{i}\|\vec{\sigma}_{i}):=\lim_{n\to\infty}\frac{1}{n}\tilde{D}_{\alpha}(\rho_{i}^{n}\|\sigma_{i}^{n})$
(29)

exist and are continuous in $\alpha$ at $\alpha=1$. To simplify notation we
write $D(\vec{\rho}_{i}\|\vec{\sigma}_{i})$ :=
$\tilde{D}_{1}(\vec{\rho}_{i}\|\vec{\sigma}_{i})$.

A simple sequence satisfying this condition is given by independent and
identically distributed (iid) states, i.e. the sequences determined by
$\rho_{i}^{n}=\rho_{i}^{\otimes n}$ and $\sigma_{i}^{n}=\sigma_{i}^{\otimes
n}$ for $i\in\\{1,2\\}$. In this case the expressions simplify due to the
additivity of the Rényi divergence for product states and we have
$\tilde{D}_{\alpha}(\vec{\rho}_{i}\|\vec{\sigma}_{i})=\tilde{D}_{\alpha}(\rho_{i}\|\sigma_{i})$
as well as $D(\vec{\rho}_{i}\|\vec{\sigma}_{i})=D(\rho_{i}\|\sigma_{i})$. A
slightly more involved example is constructed by taking tensor products of
states, each randomly chosen from some finite sets. The resulting sequences of
product states satisfy
$\tilde{D}_{\alpha}(\vec{\rho}_{i}\|\vec{\sigma}_{i})=\mathbb{E}[\tilde{D}_{\alpha}(\rho_{i}\|\sigma_{i})]$
by the law of large numbers, where the expectation is taken over the joint
distribution of $\rho_{1},\sigma_{1},\rho_{2}$ and $\sigma_{2}$.

In the following we show that sufficient and necessary conditions for
asymptotic state transformations are determined by
$\lambda_{1}:=D(\vec{\rho}_{1}\|\vec{\sigma}_{1})$ and
$\lambda_{2}:=D(\vec{\rho}_{2}\|\vec{\sigma}_{2})$.

  * •

If $\lambda_{1}>\lambda_{2}$ we show the existence of a sequence of channels
that transform $(\rho_{1}^{n},\sigma_{1}^{n})$ to
$(\rho_{2}^{n},\sigma_{2}^{n})$ where the transformation
$\rho_{1}^{n}\to\rho_{2}^{n}$ has an error that is vanishing exponentially as
$n\to\infty$ and the transformation $\sigma_{1}^{n}\to\sigma_{2}^{n}$ is
exact.

  * •

On the other hand, if $\lambda_{1}<\lambda_{2}$ we show that any
transformation for which $\sigma_{1}^{n}\to\sigma_{2}^{n}$ is exact leads to
an error exponentially approaching one as $n\to\infty$ in the transformation
$\rho_{1}^{n}\to\rho_{2}^{n}$.

We note that the case where $\lambda_{1}=\lambda_{2}$ is left as an open
question. For example, in case of four states $\rho_{1},\rho_{2},\sigma_{1},$
and $\sigma_{2}$ such that $D(\rho_{1}\|\sigma_{1})=D(\rho_{2}\|\sigma_{2})$
it is unknown if there exists a sequence of channels that take
$\sigma_{1}^{\otimes n}$ to $\sigma_{2}^{\otimes n}$ for each $n$ and that
take $\rho_{1}^{\otimes n}$ to $\rho_{2}^{\otimes n}$ up to asymptotically
vanishing error.

Our main technical results are formally presented in the next two theorems.

######  Theorem 3.4 (Achievability with exponentially small error).

Let $\vec{\rho}_{1}$, $\vec{\sigma}_{1}$, $\vec{\rho}_{2}$ and
$\vec{\sigma}_{2}$ for $n\in\mathbb{N}$ be sequences satisfying the condition
in (29) and furthermore

$\displaystyle
D(\vec{\rho}_{1}\|\vec{\sigma}_{1})>D(\vec{\rho}_{2}\|\vec{\sigma}_{2})\,.$
(30)

Then there exists $\gamma>0$, $n_{0}\in\mathbb{N}$, and sequence
$\\{\mathcal{E}_{n}\\}_{n\in\mathbb{N}}$ of (test-and-prepare) quantum
channels such that

$\displaystyle\mathcal{E}_{n}(\sigma_{1}^{n})=\sigma_{2}^{n}\quad\forall
n\in\mathbb{N}\qquad\textnormal{and}\qquad\Delta\big{(}\mathcal{E}_{n}(\rho_{1}^{n}),\rho_{2}^{n}\big{)}\leq
2^{-\gamma n}\quad\forall n\geq n_{0}\,.$ (31)

######  Remark 3.5.

The above theorem strengthens [37, Theorem 2.7] in that we are also able to
show the exponential decay of the error in the transformation. Furthermore, we
extend the result [37, Theorem 2.7] to hold for states that do not necessarily
satisfy an iid assumption.

###### Proof.

We show the statement for trace distance, and the statement for purified
distance then follows by the Fuchs-van de Graaf inequality.

By the assumption of the theorem and the continuity guaranteed by the
assumption at the beginning of the section, there exists a $\delta>0$ and
$\kappa>0$ such that

$\displaystyle\tilde{D}_{1-\delta}(\vec{\rho}_{1}\|\vec{\sigma}_{1})\geq\tilde{D}_{1+\delta}(\vec{\rho}_{2}\|\vec{\sigma}_{2})+\kappa\,.$
(32)

Hence, by their definition as limits, there exists a $n_{0}\in\mathbb{N}$ such
that for all $n\geq n_{0}$,

$\displaystyle\tilde{D}_{1-\delta}(\rho_{1}^{n}\|\sigma_{1}^{n})\geq\tilde{D}_{1+\delta}(\rho_{2}^{n}\|\sigma_{2}^{n})+\frac{n\kappa}{2}.$
(33)

Let us now set $\varepsilon_{n}=\frac{1}{2}2^{-\gamma n}$ for some $\gamma>0$
to be determined later. From Lemma 3.3 we learn that the maps
$\mathcal{E}_{n}$ with the desired properties exist if

$\displaystyle D_{h}^{\varepsilon_{n}}(\rho_{1}^{n}\|\sigma_{1}^{n})\geq
D_{\max}^{\varepsilon_{n},T}(\rho_{2}^{n}\|\sigma_{2}^{n})\,.$ (34)

Indeed, Proposition 2.2 together with (33) imply

$\displaystyle
D_{h}^{\varepsilon_{n}}(\rho_{1}^{n}\|\sigma_{1}^{n})-D_{\max}^{\varepsilon_{n},T}(\rho_{2}^{n}\|\sigma_{2}^{n})$
$\displaystyle\geq\frac{n\kappa}{2}-\frac{1-\delta}{\delta}\log\frac{1}{\varepsilon_{n}}-\frac{1}{\delta}\log\frac{1}{\varepsilon_{n}^{2}}-\log\frac{1}{1-\varepsilon_{n}^{2}}$
(35)
$\displaystyle\geq\frac{n\kappa}{2}-\frac{3}{\delta}\log\frac{1}{\varepsilon_{n}}$
(36) $\displaystyle\geq\frac{n\kappa}{2}-\frac{4\gamma n}{\delta}\,,$ (37)

where to reach (36) we used that $1-\varepsilon_{n}^{2}\geq\varepsilon_{n}$
since $\varepsilon_{n}\leq\frac{1}{2}$, and in the last step we assumed
$\gamma n\geq 3$. We conclude that the choice $\gamma=\frac{\kappa\delta}{8}$
ensures that (34) holds. ∎

######  Theorem 3.6 (Exponential strong converse).

Let $\vec{\rho}_{1}$, $\vec{\sigma}_{1}$, $\vec{\rho}_{2}$ and
$\vec{\sigma}_{2}$ for $n\in\mathbb{N}$ be sequences satisfying the condition
in (29) and furthermore

$\displaystyle
D(\vec{\rho}_{1}\|\vec{\sigma}_{1})<D(\vec{\rho}_{2}\|\vec{\sigma}_{2})\,.$
(38)

Then there exists $\gamma>0$ such that for all sequences of quantum channels
$\\{\mathcal{E}_{n}\\}_{n\in\mathbb{N}}$ that satisfy
$\mathcal{E}_{n}(\sigma_{1}^{n})=\sigma_{2}^{n}$ there exists an
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$ we have

$\displaystyle\Delta\big{(}\mathcal{E}_{n}(\rho_{1}^{n}),\rho_{2}^{n}\big{)}\geq
1-2^{-\gamma n}\,.$ (39)

###### Proof.

We show the statement for purified distance, and the statement for trace
distance then follows by the Fuchs-van de Graaf inequality.

We again start by observing that the assumption of the theorem and the
continuity guaranteed by the assumption at the beginning of the section imply
the existence of a $\delta>0$ and $\kappa>0$ such that
$\tilde{D}_{1+\delta}(\vec{\rho}_{1}\|\vec{\sigma}_{1})\leq\tilde{D}_{1-\delta}(\vec{\rho}_{2}\|\vec{\sigma}_{2})-\kappa$,
and thus, there exists a $n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$,
we have

$\displaystyle\tilde{D}_{1+\delta}(\rho_{1}^{n}\|\sigma_{1}^{n})-\tilde{D}_{1-\delta}(\rho_{2}^{n}\|\sigma_{2}^{n})\leq-\frac{n\kappa}{2}.$
(40)

It thus suffices to prove that (40) implies the desired property for all
sequences of quantum channels. In the following we prove the contrapositive.
Suppose that for all $\gamma>0$, there exists a family of channels
$\\{\mathcal{E}_{n}\\}_{n\in\mathbb{N}}$ such that for some $n\geq n_{0}$ we
have both $\mathcal{E}_{n}(\sigma_{1}^{n})=\sigma_{2}^{n}$ and
$P(\mathcal{E}_{n}(\rho_{1}^{n}),\rho_{2}^{n})<1-2^{-\gamma n}$. Let us then
fix a $\gamma$, to be determined later, and set
$\varepsilon_{n}=\frac{1}{2}2^{-\gamma n}$. By Proposition 2.2 we have

$\displaystyle\tilde{D}_{1+\delta}(\rho_{1}^{n}\|\sigma_{1}^{n})$
$\displaystyle\geq
D_{\max}^{\varepsilon_{n},P}(\rho_{1}^{n}\|\sigma_{1}^{n})-\frac{1}{\delta}\log\frac{1}{\varepsilon_{n}^{2}}-\log\frac{1}{1-\varepsilon_{n}^{2}}$
(41) $\displaystyle\geq
D_{\max}^{\varepsilon_{n},P}\big{(}\mathcal{E}_{n}(\rho_{1}^{n})\big{\|}\sigma_{2}^{n}\big{)}-\frac{1}{\delta}\log\frac{1}{\varepsilon_{n}^{2}}-\log\frac{1}{\varepsilon_{n}}$
(42) $\displaystyle\geq
D_{\max}^{1-\varepsilon_{n},P}(\rho_{2}^{n}\|\sigma_{2}^{n})-\frac{1}{\delta}\log\frac{1}{\varepsilon_{n}^{2}}-\log\frac{1}{\varepsilon_{n}}$
(43)

where the penultimate step uses the data-processing inequality for the smooth
max-divergence and the fact that $1-\varepsilon_{n}^{2}\geq\varepsilon_{n}$
since $\varepsilon_{n}\leq\frac{1}{2}$. The final step follows from the
definition of the smooth max-relative entropy as an optimization over a ball
of close states and the triangle inequality of the purified distance.

Instantiating Proposition 2.1 with $\varepsilon=(1-\varepsilon_{n})^{2}$ and
$\nu=\varepsilon_{n}-\varepsilon_{n}^{2}$ further yields

$\displaystyle D_{\max}^{1-\varepsilon_{n},P}(\rho_{2}^{n}\|\sigma_{2}^{n})$
$\displaystyle\geq
D_{h}^{\varepsilon_{n}}(\rho_{2}^{n}\|\sigma_{2}^{n})+\log\frac{1}{2\varepsilon_{n}-\varepsilon_{n}^{2}}-2\log\frac{2}{\varepsilon_{n}-\varepsilon_{n}^{2}}$
(44) $\displaystyle\geq
D_{h}^{\varepsilon_{n}}(\rho_{2}^{n}\|\sigma_{2}^{n})-2\log\frac{4}{\varepsilon_{n}}$
(45)
$\displaystyle\geq\tilde{D}_{1-\delta}(\rho_{2}^{n}\|\sigma_{2}^{n})-\frac{1-\delta}{\delta}\log\frac{1}{\varepsilon_{n}}-2\log\frac{4}{\varepsilon_{n}},$
(46)

where for the second inequality we used that
$\varepsilon_{n}^{2}\leq\frac{1}{2}\varepsilon_{n}$ since
$\varepsilon_{n}\leq\frac{1}{2}$ and the last inequality is a consequence of
Proposition 2.2. Combining (43) and (46) we find

$\displaystyle\tilde{D}_{1+\delta}(\rho_{1}^{n}\|\sigma_{1}^{n})-\tilde{D}_{1-\delta}(\rho_{2}^{n}\|\sigma_{2}^{n})$
$\displaystyle\geq-\frac{3}{\delta}\log\frac{1}{\varepsilon_{n}}-2\log\frac{4}{\varepsilon_{n}}$
(47)
$\displaystyle=-n\gamma\left(\frac{3}{\delta}+2\right)-\frac{3}{\delta}-6\,.$
(48)

However, this contradicts (40) for sufficiently large $n_{0}$ (and thus $n$)
as long as $\gamma$ chosen small enough so that
$\gamma\left(\frac{3}{\delta}+2\right)<\frac{\kappa}{2}$. ∎

It is worth noting that we made no attempts to characterize the exact error
exponents and strong converse exponents here. This is because finding the
exact error exponent for this problem is still open even in simple commutative
cases where, for example, $\sigma_{1}$ and $\sigma_{2}$ are proportional to
the identity and thus commute with $\rho_{1}$ and $\rho_{2}$.

##  4 Rates for pairwise state transformations

From an information theoretic perspective, a natural question to ask is at
what rate we can transform between pairs of states asymptotically. The main
result of this section has been suggested already in the work of Matsumoto
[37], and is a consequence of Theorems 3.4 and 3.6 of the previous section. It
has been obtained independently in concurrent work [57]. Hence, in this
section we ask at what rate we can transform a pair of states
$(\rho_{1},\sigma_{1})$ into $(\rho_{2},\sigma_{2})$ up to some asymptotically
vanishing error in the first transformation. To make this precise, let us
first say that a triplet $(n,m,\varepsilon)$ for $m,n\in\mathbb{N}$ and
$\varepsilon\in[0,1]$ is achievable by a transformation if and only if there
exists a channel $\mathcal{E}$ such that

$\displaystyle\mathcal{E}\big{(}\sigma_{1}^{\otimes
n}\big{)}=\sigma_{2}^{\otimes
m}\quad\text{and}\quad\Delta\big{(}\mathcal{E}(\rho_{1}^{\otimes
n}),\rho_{2}^{\otimes m}\big{)}\leq\varepsilon\,.$ (49)

We can now define the maximal achievable pairwise state transformation rate
with error $\varepsilon$ on input block length $n$ as

$\displaystyle\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\,\varepsilon,\Delta}(n):=\max\left\\{\frac{m}{n}\,:\,(n,m,\varepsilon)\text{
is achievable}\right\\}\,.$ (50)

Our goal is to understand the asymptotics of this quantity for $n\to\infty$
when $\varepsilon$ is constant. We can determine the first order asymptotics
of
$\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\,\varepsilon,\Delta}(n)$,
which turns out to be independent of the metric $\Delta\in\\{T,P\\}$ and
$\varepsilon$.

######  Theorem 4.1.

Let $\rho_{1},\sigma_{1}\in\mathrm{S}(\mathbb{C}^{d_{1}})$ and
$\rho_{2},\sigma_{2}\in\mathrm{S}(\mathbb{C}^{d_{2}})$ be two pairs of states.
For all $\varepsilon\in(0,1)$, the pairwise state transformation rate
satisfies

$\displaystyle\lim_{n\to\infty}\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\,\varepsilon,\Delta}(n)=\frac{D(\rho_{1}\|\sigma_{1})}{D(\rho_{2}\|\sigma_{2})}\,.$
(51)

######  Remark 4.2.

Another viewpoint on this question can be taken by fixing a rate $R$ and
asking how the minimal achievable error $\varepsilon$ behaves as a function of
$n$. Strong qualitative statements for when we transform above and below the
critical rate $D(\rho_{1}\|\sigma_{1})/D(\rho_{2}\|\sigma_{2})$ are immediate
from Theorems 3.4 and 3.6. Namely, for transformations below the critical rate
the error will drop exponentially as $n\to\infty$, and for transformations
above the critical rate the error will approach one exponentially fast as
$n\to\infty$.

###### Proof.

First consider the case $D(\rho_{2}\|\sigma_{2})=0$. This implies that
$\rho_{2}=\sigma_{2}$ and we can employ a sequence of constant output
channels, $\mathcal{E}_{n}(\cdot)=\sigma_{2}^{\otimes m}$, for any
$m\in\mathbb{N}$, which implies
$\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\,\varepsilon,\Delta}(n)=\infty$.

In case $D(\rho_{2}\|\sigma_{2})>0$ we first show that the rate (51) can be
achieved. Let $R=\frac{p}{q}>0$ with $p,q\in\mathbb{N}$ with
$D(\rho_{1}\|\sigma_{1})>RD(\rho_{2}\|\sigma_{2})$. The additivity of the
relative entropy for tensor product states yields

$\displaystyle D\big{(}\rho_{1}^{\otimes q}\big{\|}\sigma_{1}^{\otimes
q}\big{)}>D\big{(}\rho_{2}^{\otimes p}\big{\|}\sigma_{2}^{\otimes
p}\big{)}\,.$ (52)

Hence, Theorem 3.4 implies the existence of a sequence of channels
$\\{\mathcal{E}_{n}\\}_{n\in\mathbb{N}}$ such that

$\displaystyle\mathcal{E}_{n}(\sigma_{1}^{\otimes nq})=\sigma_{2}^{\otimes
Rnq}\qquad\text{and}\qquad\Delta\big{(}\mathcal{E}_{n}(\rho_{1}^{\otimes
nq}),\rho_{2}^{\otimes Rnq}\big{)}\leq\varepsilon\quad\text{ for large enough
$n$ \, . }$ (53)

We thus showed that $(nq,Rnq,\varepsilon)$ is achievable. Moreover, by just
throwing away $s\in\\{1,2,\ldots,q-1\\}$ systems, we also know that
$(nq+s,Rnq,\varepsilon)$ is achievable. Hence,

$\displaystyle\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\,\varepsilon,\Delta}(nq+s)\geq
R\cdot\frac{nq}{nq+s}\,,$

and, thus,
$\liminf_{N\to\infty}\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\,\varepsilon,\Delta}(N)\geq
R$. Hence the achievability statement follows because $\mathbb{Q}$ is dense in
$\mathbb{R}$.

We next prove a strong converse, i.e., we show that if $\varepsilon$ is
bounded away from $1$ then for large $n$ we must have
$D(\rho_{1}\|\sigma_{1})\geq RD(\rho_{2}\|\sigma_{2})$. Let
$\\{\mathcal{E}_{n}\\}_{n\in\mathbb{N}}$ be a sequence of channels satisfying
(49). The fully quantum asymptotic equipartition property [53, 52] states that
for any $\delta\in(0,1-\varepsilon)$ we have

$\displaystyle D(\rho_{1}\|\sigma_{1})$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\delta,\Delta}(\rho_{1}^{\otimes
n}\|\sigma_{1}^{\otimes n})$ (54)
$\displaystyle\geq\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\delta,\Delta}\big{(}\mathcal{E}_{n}(\rho_{1}^{\otimes
n})\|\mathcal{E}_{n}(\sigma_{1}^{\otimes n})\big{)}$ (55)
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\delta,\Delta}\big{(}\mathcal{E}_{n}(\rho_{1}^{\otimes
n})\|\sigma_{2}^{\otimes\lceil Rn\rceil}\big{)}\,,$ (56)

where the penultimate step follows from the data-processing inequality for the
smooth max-relative entropy [22]. Because
$\Delta(\mathcal{E}_{n}(\rho_{1}^{\otimes n}),\rho_{2}^{\otimes\lceil
Rn\rceil})\leq\varepsilon$, we further find

$\displaystyle\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\delta,\Delta}\big{(}\mathcal{E}_{n}(\rho_{1}^{\otimes
n})\|\sigma_{2}^{\otimes\lceil
Rn\rceil}\big{)}\geq\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\delta+\varepsilon,\Delta}\big{(}\rho_{2}^{\otimes\lceil
Rn\rceil}\|\sigma_{2}^{\otimes\lceil
Rn\rceil}\big{)}=RD(\rho_{2}\|\sigma_{2})\,,$ (57)

where the final step again uses the asymptotic equipartition property.
Combining all of this gives $D(\rho_{1}\|\sigma_{1})\geq
RD(\rho_{2}\|\sigma_{2})$, concluding the proof. ∎

##  5 Applications to resource theories

The above results have some immediate consequences in resource theories: in
what follows we consider in particular the resource theory of athermality and
the resource theory of coherence. In concurrent work, Wang and Wilde [57] also
derived some of our results, interpreting them in terms of a resource theory
of “asymmetric distinguishability”. Our perspective is different insofar as we
interpret Theorems 3.4, 3.6 and 4.1 as building blocks that have applications
in different resource theories.

Let us first consider the resource theory of athermality under Gibbs-
preserving maps. There, we are given a Hamiltonian $E$, an inverse temperature
$\beta$ and a Gibbs state $\gamma=\frac{1}{Z}e^{-\beta E}$, where $Z$ is the
normalization factor. One then asks whether there exists a quantum channel
that has the Gibbs state $\gamma$ of a quantum system as a fixed point and
transforms $\rho_{1}$ to $\rho_{2}$. Theorem 3.4 reveals that there exists a
sequence of Gibbs-preserving maps from $\rho_{1}^{\otimes n}$ to
$\rho_{2}^{\otimes n}$, with exponentially vanishing error as $n\to\infty$, if

$\displaystyle D(\rho_{1}\|\gamma)>D(\rho_{2}\|\gamma)\qquad\textrm{or,
equivalently,}\qquad F_{H}(\rho_{1})>F_{H}(\rho_{2})\,,$ (58)

where we used that the Helmholtz free energy $F_{H}=U-TH$, where $U$ is the
internal energy, $T$ is the temperature and $H$ is the entropy, that is,

$\displaystyle F_{H}(\rho)=\mathrm{tr}\rho
E+\frac{1}{\beta}\mathrm{tr}\rho\log\rho=\frac{1}{\beta}\left(D(\rho\|\gamma)-\log
Z\right)\,.$ (59)

Furthermore, as shown in Theorem 3.6, no such sequence of Gibbs-preserving
maps can exist if the inequality in (58) is strictly reversed. In fact, any
sequence of Gibbs-preserving maps would incur an error approaching one
exponentially fast as $n\to\infty$. In concurrent work by Sagawa _et al._ [49]
similar results are discussed for the resource theory of athermality with
thermal operations (which is more restrictive than the Gibbs-preserving maps
discussed here) and for sequences of states beyond the iid case. Since thermal
operations are Gibbs-preserving, our strong converse results also apply to
their setting, showing that there is no asymptotic advantage in allowing
Gibbs-preserving maps.

Another resource theory in which Theorem 4.1 above plays a role is the
resource theory of coherence, in particular, the resource theory of coherence
based on dephasing-covariant incoherent operations (DIO) [15, 35]. A DIO
operation $\mathcal{E}$ is such that its action commutes with the completely
dephasing channel $\mathsf{diag}$, that is

$\displaystyle\mathcal{E}\circ\mathsf{diag}=\mathsf{diag}\circ\mathcal{E}\;.$
(60)

In this framework, the rate at which coherence can be distilled from an
initial resource state $\rho$ is defined, as usual, as the optimal rate at
which the transformation

$\displaystyle\rho^{\otimes
n}\xrightarrow{\operatorname{DIO}}|+\rangle\langle+|^{\otimes m}\;,$ (61)

where $|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ is one unit of
coherence, can be achieved with asymptotically vanishing error. Such a rate is
known to be equal to $D(\rho\|\mathsf{diag}(\rho))$ [45, 14]. Recently, a
relaxation of the DIO paradigm has been proposed and motivated [46]: instead
of requiring that the transformation $\mathcal{E}$ and the completely
dephasing channel $\mathsf{diag}$ commute on all states, the commutation
relation is enforced only on the initial resource state $\rho$. In other
words, one considers the $\rho$-DIO condition

$\displaystyle(\mathcal{E}\circ\mathsf{diag})(\rho)=(\mathsf{diag}\circ\mathcal{E})(\rho)\;.$
(62)

The existence of a $\rho$-DIO channel transforming $\rho$ into $\sigma$ can
then be easily reformulated as the existence of a channel $\mathcal{E}$
achieving the following mapping of quantum dichotomies:

$\displaystyle(\rho,\mathsf{diag}(\rho))\xrightarrow{\mathcal{E}}(\sigma,\mathsf{diag}(\sigma))\;.$
(63)

Once formulated in this form, our Theorem 4.1 implies that the rate at which
coherence can be distilled from $\rho$ by means of $\rho$-DIO operations is
given by $D(\rho\|\mathsf{diag}(\rho))$. The asymptotic distillation rate
under $\rho$-DIO has been independently computed in Ref. [46]. Beyond that,
Theorem 3.6 establishes an exponential strong converse which implies that if
we try to distill at a rate exceeding $D(\rho\|\mathsf{diag}(\rho))$ then the
error will go to one exponentially fast. Interestingly, since the asymptotic
distillation rate is the same for both DIO and $\rho$-DIO operations, and
since $\rho$-DIO operations constitute a larger set than DIO operations, we
have that the above mentioned exponential strong converse property holds for
DIO state distillation too. Beyond that, our results also yield an exponential
strong converse for general DIO transformations between arbitrary states
$\rho$ and $\sigma$ for any rate exceeding the ratio of
$D(\rho\|\mathsf{diag}(\rho))$ and $D(\sigma\|\mathsf{diag}(\sigma))$.

##  6 Discussion

Given the result of Theorem 4.1, we are left to wonder how quickly
$\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\varepsilon,\Delta}(n)$
approaches the asymptotic limit, or, in other words: what are the higher order
terms in the expansion of
$\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\varepsilon,\Delta}(n)$
for large $n$? Such questions have recently attracted a lot of interest in
both classical and quantum information theory. In the quantum setting, the
first tight results in this direction were achieved for hypothesis testing
[33, 54]:

$\displaystyle D_{h}^{\varepsilon}(\rho^{\otimes n}\|\sigma^{\otimes
n})=nD(\rho\|\sigma)+\sqrt{nV(\rho\|\sigma)}\Phi^{-1}(\varepsilon)+O(\log n),$
(64)

where
$V(\rho\|\sigma):=\mathrm{tr}\rho\left(\log\rho-\log\sigma\right)^{2}-D(\rho\|\sigma)^{2}$
is the _relative entropy variance_ , i.e. a quantum generalization of the
variance of the log-likelihood ratio, and $\Phi(\cdot)$ is the cumulative
normal distribution function. This is the constant error regime of quantum
hypothesis testing. Similar second-order expansions can be derived for the
case where $\varepsilon$ is not constant but approaches $0$ and $1$ slower
than exponentially, the so-called moderate deviation regime [17, 13].

Our goal is to establish similar asymptotic expansions for
$\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\varepsilon,\Delta}(n)$.
While this is ultimately beyond the scope of this work, we will justify the
following conjecture.

######  Conjecture 6.1.

Let $\rho_{1},\sigma_{1}\in\mathrm{S}(\mathbb{C}^{d_{1}})$ and
$\rho_{2},\sigma_{2}\in\mathrm{S}(\mathbb{C}^{d_{2}})$ such that

$\displaystyle\nu:=\frac{D(\rho_{1}\|\sigma_{1})}{V(\rho_{1}\|\sigma_{1})}\cdot\frac{V(\rho_{2}\|\sigma_{2})}{D(\rho_{2}\|\sigma_{2})}\,.$
(65)

is finite, i.e. $V(\rho_{1}\|\sigma_{1})>0$ and $D(\rho_{2}\|\sigma_{2})>0$.
Then, for any $\varepsilon\in(0,1)$, we have

$\displaystyle\lim_{n\to\infty}\sqrt{n}\left(\hat{R}_{\rho_{1},\sigma_{1}\to\rho_{2},\sigma_{2}}^{\varepsilon,P}(n)-\frac{D(\rho_{1}\|\sigma_{1})}{D(\rho_{2}\|\sigma_{2})}\right)=\frac{\sqrt{V(\rho_{1}\|\sigma_{1})}}{D(\rho_{2}\|\sigma_{2})}Z_{\nu}^{-1}(\varepsilon)\,,$
(66)

where $Z_{\nu}(\cdot)$ is the cumulative Rayleigh-normal distribution function
[31].

This form is of interest since it shows a resonance behavior when $\nu=1$,
where the contribution in the second-order term turns positive even for
arbitrarily small $\varepsilon$. This means that there exist pairs of states
that can be transformed into each other without loss due to finite size
effects (up to second order). This effect has been observed both analytically
and numerically in the commutative case [30], and its applications to fully
quantum resource theories remain to be explored.

The limit expression in (66) was shown to hold for the case where $\sigma_{1}$
and $\sigma_{2}$ are both proportional to the identity in the work of Kumagai
and Hayashi [31], and the above conjecture thus constitutes a natural fully
quantum generalization of their result. Building on that and an embedding
technique from quantum thermodynamics, the equality was also shown for general
$\sigma_{1}$ and $\sigma_{2}$ as long as $\rho_{1}$ and $\rho_{2}$ commute
with $\sigma_{1}$ and $\sigma_{2}$, respectively [18].444To be more precise,
the cited work [18] only shows this for the case where
$\sigma_{1}=\sigma_{2}$, but the required extension follows after a close
inspection of the proof. The same special case can also be solved in the
moderate error regime by adapting the results in [19].

Finally, note that since we are now concerned with higher order contribution
that are a function of the error threshold $\varepsilon$, the limit does in
general depend on how exactly we measure the error. It is thus not obvious how
an appropriate conjecture for the trace distance would look like, for example.

#### Acknowledgements.

FB acknowledges partial support from the Japan Society for the Promotion of
Science (JSPS) KAKENHI, Grant No.19H04066, and the program for FRIAS-Nagoya
IAR Joint Project Group. DS acknowledges support from the Swiss National
Science Foundation via the NCCR QSIT as well as project No. 200020_165843.

## References

  * [1] P. Alberti and A. Uhlmann.  “A problem relating to positive linear maps on matrix algebras”.  Reports on Mathematical Physics 18(2): 163–176 (1980).
  * [2] P. M. Alberti and A. Uhlmann.  Stochasticity and partial order.  volume 9 of Mathematics and Its Applications, Deutscher Verlag der Wissenschaften (1982).
  * [3] A. Anshu, M. Berta, R. Jain, and M. Tomamichel.  “A minimax approach to one-shot entropy inequalities”.  Preprint, arXiv: 1906.00333 (2019).
  * [4] K. M. R. Audenaert, M. Mosonyi, and F. Verstraete.  “Quantum State Discrimination Bounds for Finite Sample Size”.  Journal of Mathematical Physics 53(12): 122205 (2012).
  * [5] D. Blackwell.  “Equivalent Comparisons of Experiments”.  The Annals of Mathematical Statistics 24(2): 265–272 (1953).
  * [6] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner.  “The Second Laws of Quantum Thermodynamics”.  Proceedings of the National Academy of Sciences USA 112(11): 3275–3279 (2014).
  * [7] F. Buscemi.  “Fully quantum second-law–like statements from the theory of statistical comparisons”.  Preprint, arXiv: 1505.00535v1 .
  * [8] F. Buscemi.  “Comparison of Quantum Statistical Models: Equivalent Conditions for Sufficiency”.  Communications in Mathematical Physics 310(3): 625–647 (2012).
  * [9] F. Buscemi and N. Datta.  “The Quantum Capacity of Channels With Arbitrarily Correlated Noise”.  IEEE Transactions on Information Theory 56(3): 1447–1460 (2010).
  * [10] F. Buscemi and G. Gour.  “Quantum relative Lorenz curves”.  Physical Review A 95(1) (2017).
  * [11] A. Chefles.  “Deterministic Quantum State Transformations”.  Physics Letters A 270(1-2): 14–19 (2000).
  * [12] A. Chefles, R. Jozsa, and A. Winter.  “On the Existence of Physical Transformations between Sets of Quantum States”.  International Journal of Quantum Information 02(01): 11–21 (2004).
  * [13] H.-C. Cheng and M.-H. Hsieh.  “Moderate deviation analysis for classical-quantum channels and quantum hypothesis testing”.  IEEE Transactions on Information Theory 64(2): 1385–1403 (2018).
  * [14] E. Chitambar.  “Dephasing-covariant operations enable asymptotic reversibility of quantum resources”.  Physical Review A 97(5) (2018).
  * [15] E. Chitambar and G. Gour.  “Critical Examination of Incoherent Operations and a Physically Consistent Resource Theory of Quantum Coherence”.  Physical Review Letters 117(3) (2016).
  * [16] E. Chitambar and G. Gour.  “Quantum resource theories”.  Reviews of Modern Physics 91(2) (2019).
  * [17] C. T. Chubb, V. Y. F. Tan, and M. Tomamichel.  “Moderate Deviation Analysis for Classical Communication over Quantum Channels”.  Communications in Mathematical Physics 355(3): 1283–1315 (2017).
  * [18] C. T. Chubb, M. Tomamichel, and K. Korzekwa.  “Beyond the thermodynamic limit: finite-size corrections to state interconversion rates”.  Quantum 2: 108 (2018).
  * [19] C. T. Chubb, M. Tomamichel, and K. Korzekwa.  “Moderate deviation analysis of majorization-based resource interconversion”.  Physical Review A 99(3): 032332 (2019).
  * [20] J. E. Cohen, J. H. B. Kempermann, and G. Zbaganu.  Comparisons of Stochastic Matrices with Applications in Information Theory, Statistics, Economics and Population.  Birkhäuser (1998).
  * [21] G. Dahl.  “Matrix majorization”.  Linear Algebra and its Applications 288: 53 – 73 (1999).
  * [22] N. Datta.  “Min- and Max- Relative Entropies and a New Entanglement Monotone”.  IEEE Transactions on Information Theory 55(6): 2816–2826 (2009).
  * [23] G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian.  “Quantum majorization and a complete set of entropic conditions for quantum thermodynamics”.  Nature Communications 9(1) (2018).
  * [24] G. H. Hardy, J. E. Littlewood, and G. Pólya.  Inequalities.  Cambridge University Press (1934).
  * [25] F. Hiai and D. Petz.  “The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability”.  Communications in Mathematical Physics 143(1): 99–114 (1991).
  * [26] M. Horodecki and J. Oppenheim.  “Fundamental limitations for quantum and nanoscale thermodynamics”.  Nature Communications 4(1) (2013).
  * [27] A. Jenčová.  “Comparison of Quantum Binary Experiments”.  Reports on Mathematical Physics 70(2): 237–249 (2012).
  * [28] A. Jenčová.  “Comparison of quantum channels and statistical experiments”.  In Proc. IEEE ISIT 2016, pages 2249–2253, (2016).
  * [29] J. Körner and K. Marton.  “Comparison of two noisy channels”.  Colloquia Mathematica Societatis Janos Bolyai, Topics in Information Theory 16: 411–424, (1977).
  * [30] K. Korzekwa, C. T. Chubb, and M. Tomamichel.  “Avoiding Irreversibility: Engineering Resonant Conversions of Quantum Resources”.  Physical Review Letters 122(11): 110403 (2019).
  * [31] W. Kumagai and M. Hayashi.  “Second Order Asymptotics of Optimal Approximate Conversion for Probability Distributions and Entangled States and Its Application to LOCC Cloning”.  Preprint, arXiv: 1306.4166 (2013).
  * [32] L. Le Cam.  “Sufficiency and Approximate Sufficiency”.  The Annals of Mathematical Statistics 35(4): 1419–1455 (1964).
  * [33] K. Li.  “Second-Order Asymptotics for Quantum Hypothesis Testing”.  Annals of Statistics 42(1): 171–189 (2014).
  * [34] A. W. Marshall, I. Olkin, and B. C. Arnold.  Inequalities: Theory of Majorization and Its Applications.  Springer (2011).
  * [35] I. Marvian and R. W. Spekkens.  “How to quantify coherence: Distinguishing speakable and unspeakable notions”.  Physical Review A 94(5) (2016).
  * [36] K. Matsumoto.  “A quantum version of randomization criterion”.  Preprint, arXiv: 1012.2650 (2010).
  * [37] K. Matsumoto.  “Reverse Test and Characterization of Quantum Relative Entropy”.  Preprint, arXiv: 1010.1030 (2010).
  * [38] K. Matsumoto.  “An example of a quantum statistical model which cannot be mapped to a less informative one by any trace preserving positive map”.  Preprint, arXiv: 1409.5658 (2014).
  * [39] K. Matsumoto.  “On the condition of conversion of classical probability distribution families into quantum families”.  Preprint, arXiv: 1412.3680 (2014).
  * [40] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel.  “On Quantum Rényi Entropies: A New Generalization and Some Properties”.  Journal of Mathematical Physics 54(12): 122203 (2013).
  * [41] M. A. Nielsen.  “Conditions for a Class of Entanglement Transformations”.  Physical Review Letters 83(2): 436–439 (1999).
  * [42] T. Ogawa and H. Nagaoka.  “Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing”.  IEEE Transactions on Information Theory 46(7): 2428–2433 (2000).
  * [43] D. Petz.  “Quasi-entropies for Finite Quantum Systems”.  Reports on Mathematical Physics 23(1): 57–65 (1986).
  * [44] D. Reeb, M. J. Kastoryano, and M. M. Wolf.  “Hilbert’s projective metric in quantum information theory”.  Journal of Mathematical Physics 52(8): 082201 (2011).
  * [45] B. Regula, K. Fang, X. Wang, and G. Adesso.  “One-Shot Coherence Distillation”.  Physical Review Letters 121(1) (2018).
  * [46] B. Regula, V. Narasimhachar, F. Buscemi, and M. Gu.  “Coherence manipulation with dephasing-covariant operations”.  Preprint, arXiv: 1907.08606 (2019).
  * [47] J. M. Renes.  “Relative submajorization and its use in quantum resource theories”.  Journal of Mathematical Physics 57(12): 122202 (2016).
  * [48] R. Renner.  Security of Quantum Key Distribution.  PhD thesis, ETH Zurich, (2005).  Available at arXiv: quant-ph/0512258.
  * [49] T. Sagawa, P. Faist, K. Kato, K. Matsumoto, H. Nagaoka, and F. G. S. L. Brandao.  “Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein’s Lemma”.  Preprint, arXiv: 1907.05650 .
  * [50] C. E. Shannon.  “A note on a partial ordering for communication channels”.  Information and control 1(4): 390–397 (1958).
  * [51] V. Siddhu and R. B. Griffiths.  “Degradable Quantum Channels using Pure-State to Product-of-Pure-State Isometries”.  Physical Review A 94(5): 052331 (2016).
  * [52] M. Tomamichel.  Quantum Information Processing with Finite Resources — Mathematical Foundations.  volume 5 of SpringerBriefs in Mathematical Physics, Springer International Publishing (2016).
  * [53] M. Tomamichel, R. Colbeck, and R. Renner.  “A Fully Quantum Asymptotic Equipartition Property”.  IEEE Transactions on Information Theory 55(12): 5840–5847 (2009).
  * [54] M. Tomamichel and M. Hayashi.  “A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks”.  IEEE Transactions on Information Theory 59(11): 7693–7710 (2013).
  * [55] E. Torgersen.  Comparison of statistical experiments.  volume 36 of Encyclopedia of Mathematics and its Applications, Cambridge University Press (1991).
  * [56] E. N. Torgersen.  “Comparison of experiments when the parameter space is finite”.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 16(3): 219–249 (1970).
  * [57] X. Wang and M. M. Wilde.  “Resource theory of asymmetric distinguishability”.  Preprint, arXiv: 1905.11629 (2019).
  * [58] M. M. Wilde, A. Winter, and D. Yang.  “Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy”.  Communications in Mathematical Physics 331(2): 593–622 (2014).