11affiliationtext: Institute for Quantum Computing, Baidu Research, Beijing
100193, China22affiliationtext: Department of Mathematics and Statistics,
Institute for Quantum Science and Technology,
University of Calgary, AB, T2N 1N4, Canada

# Towards the ultimate limits of quantum channel discrimination

Kun Fang<EMAIL_ADDRESS>Gilad Gour<EMAIL_ADDRESS>Xin Wang
<EMAIL_ADDRESS>
###### Abstract

This note studies the difficulty of discriminating quantum channels under
operational regimes. First, we make a conjecture on the exponentially strong
converse of quantum channel hypothesis testing under coherent strategies,
meaning that any strategy to make the Type II error decays with an exponent
larger than the regularized channel relative entropy will unavoidably result
in the Type I error converging to one exponentially fast in the asymptotic
limit. This conjecture will imply the desirable quantum channel Stein’s Lemma
and the continuity of the regularized (amortized) Sandwiched Rényi channel
divergence at $\alpha=1$. We also remark that there was a gap in the proof of
the above conjecture in our previous arXiv version. Such gap exists since a
lemma basically comes from [Brandao and Plenio, 2010] was found to be false.
Second, we develop a framework to show the interplay between the strategies of
channel discrimination, the operational regimes, and variants of channel
divergences. This framework systematically underlies the operational meaning
of quantum channel divergences in quantum channel discrimination. Our work
makes an attempt towards understanding the ultimate limit of quantum channel
discrimination, as well as its connection to quantum channel divergences in
the asymptotic regime.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Exponentially Strong Converse of Channel Hypothesis Testing
  4. 4 Limits of Channel Divergence
  5. 5 Channel Discrimination and Channel Divergence
  6. 6 Discussions
  7. A Technical Lemmas
  8. B Some Useful Results
  9. C Towards the AEP of Max-relative Entropy under Channel Smoothing

##  1 Introduction

Distinguishability is central in information technology from both theoretical
and practical perspectives. A fundamental setting for distinguishability is
asymmetric hypothesis testing. Given a source generates sample $x$ from one of
two probability distributions $p\equiv\\{p(x)\\}_{x\in\mathcal{X}}$ or
$q\equiv\\{q(x)\\}_{x\in\mathcal{X}}$, asymmetric hypothesis testing seeks to
minimize the Type II error (decides $p$ when the fact is $q$) while keeping
the Type I error (decides $q$ when the fact is $p$) within a certain
threshold. The celebrated Chernoff-Stein’s Lemma [Cov99, Che52] states that,
for any constant bound on the Type I error, the optimal Type II error decays
exponentially fast in the number of copies, and the decay rate is exactly the
relative entropy

$D(p\|q)=\sum_{x\in\mathcal{X}}p(x)\log_{2}[p(x)/q(x)].$ (1)

In particular this lemma also states the “strong converse property”: any
possible scheme with Type II error decaying to zero with an exponent larger
than the relative entropy will result in the Type I error converging to one in
the asymptotic limit. Such strong converse property indicates no trade-off for
Type I error and Type II error exponent in the asymptotic limit, which is a
fundamental topic in information processing [Wol78]. The Chernoff-Stein’s
Lemma gives a rigorous operational interpretation for the relative entropy and
establishes a vital bridge between hypothesis testing and information theory
[Bla74].

A natural question is whether the above result generalizes to the quantum
case. Substantial efforts have been made to answer this fundamental question
in quantum information community (see, e.g., [HP91, ON00, Hay02, ANSV08a,
Hay07, BP10, CMW16, MO15, WW19a, WW19b]). The basic task is quantum state
discrimination, where we have an independent and identically distributed
(i.i.d.) quantum state which could be either $\rho^{\otimes n}$ or
$\sigma^{\otimes n}$. We set that $\rho^{\otimes n}$ is the null hypothesis
and $\sigma^{\otimes n}$ is the alternative hypothesis. The task is to apply a
binary measurement $\\{\Pi_{n},I-\Pi_{n}\\}$ on the state to determine which
one it is and the corresponding errors are defined similarly to the classical
case as follows:

$\displaystyle\text{(Type
I)}\quad\alpha_{n}(\Pi_{n})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[(I-\Pi_{n})\rho^{\otimes n}],\qquad\text{(Type
II)}\quad\beta_{n}(\Pi_{n})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[\Pi_{n}\sigma^{\otimes n}].$ (2)

The quantum state version of the Chernoff-Stein’s Lemma [HP91, ON00] (aka.
quantum Stein’s lemma) states that

$\lim_{n\rightarrow\infty}-\frac{1}{n}\log
D_{H}^{\varepsilon}\left(\rho^{\otimes n}\|\sigma^{\otimes
n}\right)=D(\rho\|\sigma),\quad\forall\varepsilon\in(0,1),$ (3)

where $D_{H}^{\varepsilon}(\rho\|\sigma)\mathrel{\mathop{\mathchar
58\relax}}=-\log\inf\\{\operatorname{Tr}[\Pi\sigma]\mathrel{\mathop{\mathchar
58\relax}}0\leqslant\Pi\leqslant I,\operatorname{Tr}[\Pi\rho]\geqslant
1-\varepsilon\\}$ denotes the quantum hypothesis testing relative entropy that
characterizes the decay rate of the optimal Type II error and
$D(\rho\|\sigma)=\operatorname{Tr}[\rho(\log\rho-\log\sigma)]$ denotes the
quantum relative entropy. This quantum Stein’s lemma delivers a rigorous
operational interpretation for the quantum relative entropy. Extended research
on quantum Stein’s lemma are presented in [Hay02, NH07, ANSV08a, Hay07, BP10,
WR12].

Although research in quantum hypothesis testing has largely focused on quantum
states, there are various situations in which quantum channels play the role
of the main objects of study. The task of channel discrimination is very
similar to that of state discrimination. Here, we are given black-box access
to $n$ uses of a channel ${\cal G}$ with the aim to identify it from
candidates ${\cal N}$ and ${\cal M}$. However, quantum channel discrimination
has more diversities in terms of discrimination strategies (e.g., product
strategy, coherent strategy, sequential strategy) due to its nature as dynamic
resources [CDP08, Hay09, PW09, CMW16, YF17, WBHK20, WW19b, FFRS20, PLLP19],
which leads to several variants of the potential quantum channel Stein’s
lemma. In particular, for the coherent strategies (also known as parallel
strategies in some literatures), the black box can be can be used $n$ times in
parallel to any state with a reference system before performing a measurement
at the final output to identify the channel. Based on the recently developed
resource theory of asymmetric distinguishability for quantum channels, the
state-of-the-art result [WW19b] arrives at

$\displaystyle\lim_{\varepsilon\rightarrow
0}\lim_{n\rightarrow\infty}\frac{1}{n}D_{H}^{\varepsilon}(\mathcal{N}^{\otimes
n}\|\mathcal{M}^{\otimes n})=D^{{\rm reg}}({\cal N}\|{\cal M}),$ (4)

with $D_{H}^{\varepsilon}(\mathcal{N}\|\mathcal{M})$ denotes the hypothesis
testing relative entropy of quantum channels and $D^{{\rm reg}}({\cal
N}\|{\cal M})$ denotes the regularized quantum relative entropy. That is, for
Type I error bounded by $\varepsilon$, the asymptotic optimal rate of the Type
II error exponent is given by $D^{{\rm reg}}({\cal N}\|{\cal M})$ when
$\varepsilon$ goes to $0$.

However, the condition of vanishing $\varepsilon$ lefts a notable gap to
achieve the quantum channel version of Stein’s lemma. Unlike state
discrimination, the dynamic feature of quantum channels raises challenging
difficulties in determining the optimal discrimination scheme as we have to
handle the additional optimization of the input states and the non-i.i.d.
structure of the testing states. To fill the gap, it requires a deeper
understanding and analysis on the error exponent in hypothesis testing of
quantum channels. The solution could promptly advance our understanding of
quantum channel discrimination, the resource theory of asymmetric
distinguishability for quantum channels, and the related field of quantum
metrology [PBG+18, BAB+18, DRC17]. Beyond the quantum channel Stein’s lemma,
there is an emergence of various channel divergences for analyzing different
regimes of quantum channel discrimination. How to unify the channel
divergences and the discrimination regimes in a consistent framework is
desirable for better understanding the manipulation of quantum channels.

#### Main contributions

In this note, we first make a conjecture on the exponentially strong converse
of quantum channel hypothesis testing under coherent strategies. This
conjecture directly implies the desirable quantum channel Stein’s Lemma that
the asymptotically vanishing $\varepsilon$ in Eq. (4) could be removed. The
conjecture also implies the continuity of the regularized (amortized)
Sandwiched Rényi channel divergence at $\alpha=1$, relating to the open
question in [FF21]. Based on the conjecture, we develop a framework to show
the operational meaning of quantum channel divergences in quantum channel
discrimination. In particular, we systematically show the interplay between
the strategies of channel discrimination (e.g., sequential, coherent,
product), the operational regimes (e.g., error exponent, Stein exponent,
strong converse exponent), and three variants of channel divergences (e.g.,
Petz, Umegaki, Sandwiched). Some results are built upon the conjecture we
make, and do not solve all the cases. But we hope this framework would
stimulate further investigations on quantum channel discrimination.

#### Important remark

In a previous version of this arXiv post (arXiv:2110.14842v1), we gave a proof
of the exponentially strong converse conjecture. However, a gap was found in
our previous proof (cf. Lemma 16 in the previous version). This lemma is
essentially borrowed from [BP10] and now found to be false. See more details
in Section 3. This gap in [BP10] indeed makes the generalized quantum Stein’s
lemma and our exponentially strong converse conjecture both open again. Since
there are plenty of follow-up works of [BP10], we believe that it is necessary
to double-check the correctness of relevant results, especially those directly
rely on [BP10].

##  2 Preliminaries

In this section we set the notations first. Then we review the mathematical
tool of quantum divergences and the operational task of quantum channel
discrimination.

###  2.1 Notation

In the table below, we summarize some of the notations used throughout the
note:

Symbol | Definition
---|---
$A,B,C,...$ | Quantum systems and their associated Hilbert spaces
$|A|,|B|,|C|,...$ | Dimension of the Hilbert spaces
$A^{n}$ |  Shorthand for $A_{1}A_{2}\cdots A_{n}$
$I$ | Identity operator on a suitable space
${\cal I}$ | Identity map
$\mathfrak{L}(A)$ |  Set of linear operators on system $A$
$\mathfrak{D}(A)$ |  Set of density matrices on system $A$
$\text{\rm CPTP}(A\to B)$ |  Set of completely positive trace-preserving maps from $\mathfrak{L}(A)$ to $\mathfrak{L}(B)$
$\log$ |  Logarithm in base $2$
$[n]$ |  Shorthand for $\\{1,\cdots,n\\}$
${\operatorname{supp}}(X)$ |  Support of operator $X$
$X_{+}$ |  The positive part of operator $X$
$\\{X\leqslant Y\\}$ |  The projector onto the positive part of operator $Y-X$
$\lambda_{\min}(X)$ |  Minimum eigenvalue of $X$
$\lambda_{\max}(X)$ |  Maximum eigenvalue of $X$
${\rm Sym}^{n}(A)$ |  Symmetric subspace of $A^{n}$
Table 1: Overview of notational conventions.

Throughout the note, we restrict ourselves to finite-dimensional Hilbert
spaces. Let $A$ and $A^{\prime}$ be two isomorphic Hilbert spaces with
orthonormal bases $\\{|i\rangle_{A}\\}$ and $\\{|i\rangle_{A^{\prime}}\\}$ and
define $|\Phi\rangle=\sum_{i}|i\rangle_{A}\otimes|i\rangle_{A^{\prime}}$ as
the unnormalized maximally entangled state. Then the Choi matrix of a linear
map ${\cal E}$ from $\mathfrak{L}(A)$ to $\mathfrak{L}(B)$ is defined by
$J_{{\cal E}}={\cal E}_{A\to B}(\Phi_{AA^{\prime}})$. We say a map ${\cal
N}\in\text{\rm CPTP}(A\to B)$ is positive definite, denoted by ${\cal N}>0$,
if its Choi matrix is positive definite. For any two maps ${\cal N}$ an ${\cal
M}$, denote ${\cal N}\geqslant{\cal M}$ if the Choi matrix of ${\cal N}-{\cal
M}$ is positive semidefinite.

###  2.2 Quantum divergences

A divergence between two quantum states is a real-valued function
$\boldsymbol{D}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{D}\times\mathfrak{D}\to{{\mathbb{R}}}\cup\\{\infty\\}$
that satisfies the data processing inequality $\boldsymbol{D}({\cal
E}(\rho)\|{\cal E}(\sigma))\leqslant\boldsymbol{D}(\rho\|\sigma)$ for all
${\cal E}\in\text{\rm CPTP}(A\to B),\rho,\sigma\in\mathfrak{D}(A)$.
Divergences are widely used to measure the distinguishability of physical
objects. Of particular relevance are the following quantum divergences that
will be frequently used in our discussion.

###### Definition 1

Let $\rho,\sigma\in\mathfrak{D}(A)$. The Umegaki relative entropy (also called
quantum relative entropy) is defined by [Ume54]

$\displaystyle D(\rho\|\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[\rho(\log\rho-\log\sigma)],$ (5)

if ${\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma)$ and
$+\infty$ otherwise.

###### Definition 2

Let $\rho,\sigma\in\mathfrak{D}(A)$. The Petz Rényi divergence is defined by
[Pet86]

$\displaystyle\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)\coloneqq\frac{1}{\alpha-1}\log\operatorname{Tr}\left[\rho^{\alpha}\sigma^{1-\alpha}\right]$

if $\alpha\in(0,1)$ or $\alpha\in(1,+\infty)$ with
${\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma)$, and
$+\infty$ otherwise.

###### Definition 3

Let $\rho,\sigma\in\mathfrak{D}(A)$. The Sandwiched Rényi divergence is
defined by [MLDS+13, WWY14]

$\displaystyle\widetilde{D}_{\alpha}(\rho\|\sigma)\coloneqq\frac{1}{\alpha-1}\log\operatorname{Tr}\left[\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right]^{\alpha},$
(6)

if $\alpha\in(0,1)$ or $\alpha\in(1,+\infty)$ with
${\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma)$, and
$+\infty$ otherwise.

###### Definition 4

Let $\rho,\sigma\in\mathfrak{D}(A)$, $\varepsilon\in[0,1]$. The quantum
hypothesis testing relative entropy is defined by

$\displaystyle D_{H}^{\varepsilon}(\rho\|\sigma)\mathrel{\mathop{\mathchar
58\relax}}=-\log\inf\\{\operatorname{Tr}[\Pi\sigma]\mathrel{\mathop{\mathchar
58\relax}}0\leqslant\Pi\leqslant I,\operatorname{Tr}[\Pi\rho]\geqslant
1-\varepsilon\\}.$ (7)

The following result establishes an inequality relating hypothesis testing
relative entropy and the Sandwiched Rényi divergence [CMW16, Lemma 5].

###### Lemma 5

For any $\alpha\in(1,+\infty)$ and $\varepsilon\in(0,1)$, it holds [CMW16]

$D_{H}^{\varepsilon}(\rho\|\sigma)\leqslant\widetilde{D}_{\alpha}(\rho\|\sigma)+\frac{\alpha}{\alpha-1}\log\frac{1}{1-\varepsilon}.$
(8)

###### Definition 6

Let $\rho,\sigma\in\mathfrak{D}(A)$. The max-relative entropy is defined by
[Dat09, Ren05]

$\displaystyle
D_{\max}(\rho\|\sigma)\coloneqq\log\inf\big{\\{}t\in{{\mathbb{R}}}\;\mathrel{\mathop{\mathchar
58\relax}}\;\rho\leqslant t\sigma\big{\\}}\;,$ (9)

if ${\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma)$ and
$+\infty$ otherwise. Let $F(\rho,\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\|\sqrt{\rho}\sqrt{\sigma}\|_{1}+\sqrt{(1-\operatorname{Tr}\rho)(1-\operatorname{Tr}\sigma)}$
be the generalized fidelity and $P(\rho,\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\sqrt{1-F^{2}(\rho,\sigma)}$ be the purified distance. Let
$\varepsilon\in(0,1)$. Then the smooth max-relative entropy is defined by

$\displaystyle D_{\max}^{\varepsilon}(\rho\|\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\inf_{\rho^{\prime}\mathrel{\mathop{\mathchar
58\relax}}P(\rho^{\prime},\rho)\leqslant\varepsilon}D_{\max}(\rho^{\prime}\|\sigma),$
(10)

where the infimum is taken over all subnormalized states that are
$\varepsilon$ close to the state $\rho$.

###### Definition 7

Let $\rho,\sigma\in\mathfrak{D}(A)$ and $\varepsilon\in[0,1]$. Three variants
of information spectrum relative entropy are respectively defined by [TH13,
DL14]

$\displaystyle D_{s}^{\varepsilon}(\rho\|\sigma)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\sup\\{\gamma\in{{\mathbb{R}}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Tr}(\rho\\{\rho\leqslant
2^{\gamma}\sigma\\})\leqslant\varepsilon\\}$ (11)
$\displaystyle\underline{D}_{s}^{\varepsilon}(\rho\|\sigma)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\sup\\{\gamma\in{{\mathbb{R}}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Tr}(\rho-2^{\gamma}\sigma)_{+}\geqslant
1-\varepsilon\\}$ (12)
$\displaystyle\overline{D}_{s}^{\varepsilon}(\rho\|\sigma)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\inf\\{\gamma\in{{\mathbb{R}}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Tr}(\rho-2^{\gamma}\sigma)_{+}\leqslant\varepsilon\\}.$
(13)

The divergence between quantum states can be naturally extended to quantum
channels. The main idea is to quantify the maximum divergence between the
outputs from the channels.

###### Definition 8

Let $\boldsymbol{D}$ be a divergence between quantum states. Let ${\cal
N},{\cal M}\in\text{\rm CPTP}(A\to B)$. Then the channel divergence induced by
$\boldsymbol{D}$ is defined in the worst-case scenario by [LKDW18]

$\displaystyle\boldsymbol{D}({\cal N}\|{\cal M})\mathrel{\mathop{\mathchar
58\relax}}=\sup_{\psi\in\mathfrak{D}(RA)}\boldsymbol{D}\left(\mathcal{N}_{A\to
B}(\psi_{RA})\big{\|}\mathcal{M}_{A\to B}(\psi_{RA})\right),$ (14)

where the supremum is taken over all quantum states $\psi$ on systems
$R\otimes A$ and $R$ is of arbitrary dimension.

Note that as a consequence of purification, data processing, and the Schmidt
decomposition, the supremum can be restricted to be with respect to pure
states and the reference system $R$ is isomorphic to the channel input system
$A$.

Some properties for state divergence can be naturally extended to the channel
divergences, for example, the following continuity holds.

###### Lemma 9

For any $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$, it holds [CMW16]

$\lim_{\alpha\to 1}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}\|\mathcal{M})=\lim_{\alpha\to
1}\widetilde{D}_{\alpha}(\mathcal{N}\|\mathcal{M})=D(\mathcal{N}\|\mathcal{M})\;.$
(15)

Moreover, if the state divergence $\boldsymbol{D}$ is superadditive under
tensor product of states, then the channel divergences $\boldsymbol{D}$
defined above is also superadditive under tensor product of channels. That is,
$\boldsymbol{D}({\cal N}_{1}\otimes{\cal N}_{2}\|{\cal M}_{1}\otimes{\cal
M}_{2})\geqslant\boldsymbol{D}({\cal N}_{1}\|{\cal
M}_{1})+\boldsymbol{D}({\cal N}_{2}\|{\cal M}_{2})$ for any ${\cal
N}_{1},{\cal M}_{1}\in\text{\rm CPTP}(A_{1}\to B_{1})$ and ${\cal N}_{2},{\cal
M}_{2}\in\text{\rm CPTP}(A_{2}\to B_{2})$. However, different than state
divergences, channel divergences are not additive in general [FFRS20],
motivating the definition of their regularization.

###### Definition 10

Let $\boldsymbol{D}$ be a divergence between quantum states. Let ${\cal
N},{\cal M}\in\text{\rm CPTP}(A\to B)$. Then the regularized channel
divergence induced by $\boldsymbol{D}$ is defined by

$\displaystyle\boldsymbol{D}^{\rm
reg}(\mathcal{N}\|\mathcal{M})\mathrel{\mathop{\mathchar
58\relax}}=\sup_{n\in{{\mathbb{N}}}}\frac{1}{n}\,\boldsymbol{D}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right).$ (16)

If $\boldsymbol{D}$ is superadditive for states, then it is superadditive for
channels and we have

$\displaystyle\boldsymbol{D}^{\rm
reg}(\mathcal{N}\|\mathcal{M})=\lim_{n\to\infty}\frac{1}{n}\,\boldsymbol{D}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right).$ (17)

Another way to induce a channel divergence is to use the idea of amortization.

###### Definition 11

Let $\boldsymbol{D}$ be a divergence between quantum states. Let ${\cal
N},{\cal M}\in\text{\rm CPTP}(A\to B)$. The amortized channel divergence
induced by $\boldsymbol{D}$ is defined by [BHKW18]

$\displaystyle\boldsymbol{D}^{A}({\cal N}\|{\cal M})\mathrel{\mathop{\mathchar
58\relax}}=\sup_{\psi,\varphi\in\mathfrak{D}(RA)}\Big{[}\boldsymbol{D}\left({\cal
N}_{A\to B}(\psi_{RA})\|{\cal M}_{A\to
B}(\varphi_{RA})\right)-\boldsymbol{D}\left(\psi_{RA}\|\varphi_{RA}\right)\Big{]},$
(18)

where the supremum is taken over all quantum states $\psi,\varphi$ on
$R\otimes A$ and $R$ is of arbitrary dimension.

It is worth noting that channel divergences have been used as vital tools to
many fundamental areas including the resource theory of quantum channels
[CG19, GW19, WW19b, FL20, RT21], quantum communication [WFD19, BHKW18, WFT19,
FFRS20, FWTB20], quantum coherence [DFW+18, SCG20], fault-tolerant quantum
computing [WWS19] and quantum thermodynamics [FBB19].

###  2.3 Quantum channel discrimination

The task of channel discrimination is very similar to that of state
discrimination. For a given unknown quantum channel ${\cal G}$, one aims to
identify it from the possible candidates ${\cal N}$ or ${\cal M}$. The
standard approach of discrimination is to perform hypothesis testing that
distinguishes between the null hypothesis and the alternative hypothesis. What
makes the channel discrimination different is the diverse choice of
discrimination strategies.

There are at least three different classes of available strategies illustrated
in Figure 1.

Figure 1: Three different classes of strategies for quantum channel
discrimination.

Each class of strategies is composed with two parts $(S_{n},\Pi_{n})$, where
$S_{n}$ is a method to generate a testing state and $\Pi_{n}$
($0\leqslant\Pi_{n}\leqslant I$) defines is a quantum test (a binary quantum
measurement) $\\{\Pi_{n},I-\Pi_{n}\\}$ performed on this state.

#### Product strategy

In a product strategy (Figure 1(a)) the testing state is created by choosing a
sequence of input states $\varphi_{i}\in\mathfrak{D}(R_{i}A_{i})$ and sending
them to the unknown channel ${\cal G}$ individually. The generated testing
state is given by ${\cal G}^{\otimes
n}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i})$.
The class of all product strategies is denoted as PRO. Note that the input
states considered here are not restricted to have the i.i.d. structure (e.g.
$\varphi^{\otimes n}$) but general tensor product states. That is, we allow
choosing different input states for different instances of ${\cal G}$, which
makes a difference with the product strategy discussed in [CMW16].

#### Coherent strategy

In a coherent strategy (Figure 1(b)) the testing state is created by choosing
an input state $\psi_{n}\in\mathfrak{D}(RA^{n})$ on the joint quantum systems
and sending the corresponding register to each copy of the channel. The
generated testing state is given by ${\cal G}^{\otimes n}(\psi_{n})$. The
class of all product strategies is denoted as COH. It is clear that if our
choice of $\psi_{n}$ is of a tensor product structure
$\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i}$
with $\varphi_{i}\in\mathfrak{D}(R_{i}A_{i})$ we get effectively a product
strategy. Thus we have the set inclusion $\text{\rm PRO}\subseteq\text{\rm
COH}$.

#### Sequential strategy

In a sequential strategy (Figure 1(c)) the testing state is created in an
adaptive way. That is, we first choose an initial state
$\psi_{n}\in\mathfrak{D}(R_{1}A_{1})$ and send it to one copy of the channel
${\cal G}$ followed by applying an update channel ${\cal P}_{1}$. After this,
apply another copy of the channel ${\cal G}$ and then an update channel ${\cal
P}_{2}$. Repeat his process $n$ times. We get the final testing state ${\cal
G}\circ{\cal P}_{n-1}\circ\cdots\circ{\cal P}_{2}\circ{\cal G}\circ{\cal
P}_{1}\circ{\cal G}(\psi_{n})$ with ${\cal P}_{i}\in\text{\rm
CPTP}(R_{i}B_{i}\to R_{i+1}A_{i+1})$. The class of all sequential strategies
is denoted as SEQ. It is clear that if all update channels ${\cal P}_{i}$ are
chosen as identity maps the sequential strategy reduces to a coherent
strategy. Thus we have $\text{\rm COH}\subseteq\text{\rm SEQ}$.

For a given strategy $(S_{n},\Pi_{n})$, let $\rho_{n}(S_{n})$ and
$\sigma_{n}(S_{n})$ be the testing state generated by $n$ uses of the channel
depending on whether it is ${\cal N}$ or ${\cal M}$. One can define the Type I
and Type II errors as

$\displaystyle\text{(Type
I)}\quad\alpha_{n}(S_{n},\Pi_{n})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[(I-\Pi_{n})\rho_{n}(S_{n})],\qquad\text{(Type
II)}\quad\beta_{n}(S_{n},\Pi_{n})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[\Pi_{n}\sigma_{n}(S_{n})].$ (19)

Since it is not always possible to perform perfect discrimination (i.e. both
errors are equal to zero simultaneously), one instead is interested in the
asymptotic behavior of $\alpha_{n}$ and $\beta_{n}$ as $n\to\infty$, expecting
there to be a tradeoff between minimizing $\alpha_{n}$ and minimizing
$\beta_{n}$.

##  3 Exponentially Strong Converse of Channel Hypothesis Testing

In this section we make a conjecture on the strong converse property of
hypothesis testing of quantum channels under coherent strategies. That is, any
strategy to make the Type II error decays with an exponent larger than $D^{\rm
reg}({\cal N}\|{\cal M})$ will unavoidably result in the Type I error
converging to one in the asymptotic limit. The conjecture we make is stronger
than a standard strong converse statement, as we have required not only the
convergence but also the behavior of _exponential_ convergence. Since the
convergence happens exponentially fast, we call this an _exponentially strong
converse property_.

###### Conjecture 12

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and ${\cal N}>0$. Consider
any sequence of coherent channel discrimination strategies with Type I error
$\\{\alpha_{n}\\}_{n\in{{\mathbb{N}}}}$ and Type II error
$\\{\beta_{n}\\}_{n\in{{\mathbb{N}}}}$. If the Type II error has

$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\log\beta_{n}=\mathrel{\mathop{\mathchar
58\relax}}r>D^{\rm reg}({\cal N}\|{\cal M}),$ (20)

then there exists $c>0$ such that the Type I error $1-\alpha_{n}\leqslant
2^{-cn}$ for sufficiently large $n$.

An attempt to prove Conjecture 12 can be found in a previous arXiv post
(arXiv:2110.14842v1). Here we aim to point out at which step our strategy
falls apart. Some primitive results of our previous proof still hold and may
be of independent interest. So we move them to Appendix B.

In our previous proof, we employ the following bounds on the state variance in
Problem 1, which is directly taken from [BP10, Eqs.(145-155) in Lemma III.9].
However, a flaw exists around [BP10, Eq.(147)] where the authors claim that a
maximizer of the optimization happens on the boundary by ruling out the
possibility of a maximizer on the critical points. Their argument is not valid
because the optimization is taken on the hyperplane with a normalized
condition and then a Lagrange multiplier is missing in the calculation of the
critical point in [BP10, Eq.(147)].

###### Problem 1

Let $\sigma\in\mathfrak{D}(A)$ and $t\in(0,1/|A|]$. For each
$\tau\in\mathfrak{D}(A)$ such that $\tau_{A}\geqslant tI_{A}$ define

$f_{\sigma}(\tau)\coloneqq\operatorname{Tr}\left[\sigma(\log\tau)^{2}\right]-\left(\operatorname{Tr}\left[\sigma\log\tau\right]\right)^{2}\;.$
(21)

Then,

$\max_{\begin{subarray}{c}\tau\in\mathfrak{D}(A)\\\ \tau\geqslant
tI\end{subarray}}f_{\sigma}(\tau)\leqslant\|\sigma\|_{\infty}\log^{2}\left(1+t^{-1}-|A|\right)\;.$
(22)

Actually, there exists a counter-example to Problem 1 given by Marco
Tomamichel. Consider $\sigma=I/d$ and $t=1/(2d)$ where $|A|=d$. Then let
$\tau=tP+3t(I-P)$ for some projector $P$ with $\operatorname{Tr}P=d/2$.
Therefore, the r.h.s. is $[\log(1+d)]^{2}/d$ which is vanishing as
$d\to\infty$. However, a direct calculation gives $f_{\sigma}(\tau)=[\log
3]^{2}/4$ which is independent of $d$.

The remaining part of this note proceeds on top of Conjecture 12.

##  4 Limits of Channel Divergence

In this section we show that the exponentially strong converse of channel
hypothesis testing in Conjecture 12 directly implies some foundational results
on the limits of channel divergences. In particular, the first implication of
Conjecture 12 is a complete analog of Stein’s lemma for quantum channels.

######  Theorem 13 (Channel Stein’s Lemma)

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ and $\mathcal{N}>0$.
Then, for any $\varepsilon\in(0,1)$, Conjecture 12 implies

$\lim_{n\to\infty}\frac{1}{n}D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)=D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})\;.$ (23)

###### Proof.

We prove the achievable direction first. The proof of this direction relies on
the relation of the hypothesis testing relative entropy and the Petz Rényi
divergence (see Lemma 27 in the Appendix). Applying Lemma 27 to channel
divergence gives

$\displaystyle D_{H}^{\varepsilon}({\cal N}\|{\cal
M})\geqslant\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}({\cal N}\|{\cal
M})+\frac{\alpha}{1-\alpha}\left(\frac{h(\alpha)}{\alpha}-\log\left(\frac{1}{\varepsilon}\right)\right),$
(24)

for any $\alpha\in(0,1)$ and $\varepsilon\in(0,1)$. Therefore,

$\displaystyle\liminf_{n\to\infty}\frac{1}{n}D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)$
$\displaystyle\geqslant\lim_{n\to\infty}\frac{1}{n}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\mathcal{N}^{\otimes n}\big{\|}\mathcal{M}^{\otimes
n}\right)=\sup_{n\in\mathbb{N}}\frac{1}{n}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\mathcal{N}^{\otimes n}\big{\|}\mathcal{M}^{\otimes
n}\right)\;.$ (25)

Hence,

$\displaystyle\liminf_{n\to\infty}\frac{1}{n}D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)$
$\displaystyle\geqslant\sup_{\alpha\in(0,1)}\sup_{n\in\mathbb{N}}\frac{1}{n}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\mathcal{N}^{\otimes n}\big{\|}\mathcal{M}^{\otimes
n}\right)$ (26)
$\displaystyle=\sup_{n\in\mathbb{N}}\sup_{\alpha\in(0,1)}\frac{1}{n}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\mathcal{N}^{\otimes n}\big{\|}\mathcal{M}^{\otimes
n}\right)$
$\displaystyle=\sup_{n\in\mathbb{N}}\frac{1}{n}D\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)$ $\displaystyle=D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})\;,$

where the first equality follows by swapping the two suprema, the second
equality follows by the fact that $\sup_{\alpha\in(0,1)}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}({\cal N}\|{\cal M})=\lim_{\alpha\to 1^{-}}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}({\cal N}\|{\cal M})=D({\cal N}\|{\cal M})$ (see Lemma 9), the
last equality follows by definition.

As for the converse direction, suppose
$\limsup_{n\to\infty}\frac{1}{n}D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)>D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})$, then there exists a subsequence $n_{k}$ such
that
$\lim_{n_{k}\to\infty}\frac{1}{n_{k}}D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n_{k}}\big{\|}\mathcal{M}^{\otimes n_{k}}\right)>D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})$. This implies a sequence of strategies such
that the Type I error $\alpha_{n_{k}}\leqslant\varepsilon$ and the Type II
error $\lim_{n_{k}\to\infty}-\frac{1}{n_{k}}\log\beta_{n_{k}}>D^{\rm
reg}({\cal N}\|{\cal M})$. By Conjecture 12, we know that the second condition
implies $1-\alpha_{n_{k}}\leqslant 2^{-cn_{k}}$ for some $c>0$, which
contradicts to the first condition $\alpha_{n_{k}}\leqslant\varepsilon$. This
completes the proof.  $\sqcap$$\sqcup$

Besides the hypothesis testing relative entropy, quantum divergences such as
the max-relative entropy and information spectrum relative entropies also play
a crucial role in information processing such as data compression, randomness
extraction and source coding [TH13, DL14]. Of particular importance are their
asymptotic equipartition properties (AEP). Here we establish the AEP of their
channel divergences.

###### Theorem 14

Let
$\boldsymbol{D}^{\varepsilon}\in\\{D_{\max}^{\varepsilon},\underline{D}_{s}^{\varepsilon},\overline{D}_{s}^{\varepsilon},D_{s}^{\varepsilon}\\}$
be one of the quantum divergences. Let $\varepsilon\in(0,1)$ and
$\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ with ${\cal N}>0$. Then
Conjecture 12 implies

$\lim_{n\to\infty}\frac{1}{n}\boldsymbol{D}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)=D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M}).$ (27)

###### Proof.

This result can be seen as a direct consequence of the channel Stein’s lemma
in Theorem 13. As the proof works similarly for all
$\boldsymbol{D}^{\varepsilon}\in\\{D_{\max}^{\varepsilon},\underline{D}_{s}^{\varepsilon},\overline{D}_{s}^{\varepsilon},D_{s}^{\varepsilon}\\}$,
we only showcase the proof for $D_{\max}^{\varepsilon}$ without loss of
generality.

For any two quantum states $\rho,\sigma\in\mathfrak{D}(A)$, any
$\varepsilon\in(0,1)$, and any $\varepsilon^{\prime}\in(0,1-\varepsilon)$ it
is known that [DKF+14, Proposition 4.1] [DMHB13, Theorem 11],

$D_{H}^{\varepsilon^{\prime}}(\rho\|\sigma)+\log\left(1-\varepsilon-\varepsilon^{\prime}\right)\leqslant
D_{\max}^{\varepsilon}\left(\rho\|\sigma\right)\leqslant
D_{H}^{1-\frac{1}{2}\varepsilon^{2}}(\rho\|\sigma)+\log\left(\frac{2}{\varepsilon^{2}}\right)\;.$
(28)

Applying this to channel divergence gives

$D_{H}^{\varepsilon^{\prime}}(\mathcal{N}\|\mathcal{M})+\log\left(1-\varepsilon-\varepsilon^{\prime}\right)\leqslant
D_{\max}^{\varepsilon}\left(\mathcal{N}\|\mathcal{M}\right)\leqslant
D_{H}^{1-\frac{1}{2}\varepsilon^{2}}(\mathcal{N}\|\mathcal{M})+\log\left(\frac{2}{\varepsilon^{2}}\right)$
(29)

Taking $n$ copies of $\mathcal{N}$ and $\mathcal{M}$ we get

$\displaystyle\frac{1}{n}D_{H}^{\varepsilon^{\prime}}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)$
$\displaystyle+\frac{1}{n}\log\left(1-\varepsilon-\varepsilon^{\prime}\right)$
$\displaystyle\leqslant\frac{1}{n}D_{\max}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes
n}\right)\leqslant\frac{1}{n}D_{H}^{1-\frac{1}{2}\varepsilon^{2}}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes
n}\right)+\frac{1}{n}\log\left(\frac{2}{\varepsilon^{2}}\right)\;.$ (30)

From the channel Stein’s Lemma in Theorem 13 we get that for $\mathcal{N}>0$
we have

$\lim_{n\to\infty}\frac{1}{n}D_{H}^{\varepsilon^{\prime}}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes
n}\right)=\lim_{n\to\infty}\frac{1}{n}D_{H}^{1-\frac{1}{2}\varepsilon^{2}}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)=D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})\;.$ (31)

Therefore, taking the limit $n\to\infty$ in (30) we get that

$\lim_{n\to\infty}\frac{1}{n}D_{\max}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)=D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})\;.$ (32)

Finally, the proofs for
$\underline{D}_{s}^{\varepsilon},\overline{D}_{s}^{\varepsilon}$ and
$D_{s}^{\varepsilon}$ work similarly by using their one-shot equivalent
relations with $D_{H}$ in [TH13, Lemma 12] and [DL14, Proposition 2].
$\sqcap$$\sqcup$

Note that the above results are stronger than the usual AEPs as they hold for
any fixed $\varepsilon\in(0,1)$ instead of vanishing $\varepsilon$. In this
sense, we will call such a result by the _strong asymptotic equipartition
property_.

As $D_{H}$ and $D_{\max}$ are the two extreme cases of one-shot quantum
divergences, Theorems 13 and 14 implies that the strong asymptotic
equipartition property holds for any divergence between $D_{H}$ and $D_{\max}$
after appropriate smoothing.

Besides the above strong asymptotic equipartition properties, the
exponentially strong converse in Conjecture 12 also implies the continuity of
the regularized (amortized) Sandwiched Rényi channel divergence at $\alpha=1$,
relating to the open question in [FF21, Remark 5.6]. Moreover, this continuity
implies that the exponentially strong converse also holds for channel
discrimination under sequential strategies.

###### Theorem 15

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and ${\cal N}>0$. Then
Conjecture 12 implies

$\displaystyle\lim_{\alpha\to 1^{+}}\widetilde{D}_{\alpha}^{{\rm reg}}({\cal
N}\|{\cal M})=\inf_{\alpha>1}\widetilde{D}_{\alpha}^{{\rm reg}}({\cal
N}\|{\cal M})=D^{{\rm reg}}({\cal N}\|{\cal M}).$ (33)

As a consequence,

$\displaystyle\lim_{\alpha\to 1^{+}}\widetilde{D}_{\alpha}^{A}({\cal N}\|{\cal
M})=\inf_{\alpha>1}\widetilde{D}_{\alpha}^{A}({\cal N}\|{\cal M})=D^{A}({\cal
N}\|{\cal M}).$ (34)

###### Proof.

Note that the Sandwiched Rényi divergence between states is monotone-
increasing in $\alpha$. So this directly implies the monotonicity of
$\widetilde{D}_{\alpha}^{{\rm reg}}$ and thus the first equality in (33). It
remains to show the second equality. For any $\alpha>1$, we have
$\widetilde{D}_{\alpha}^{{\rm reg}}({\cal N}\|{\cal M})\geqslant D^{{\rm
reg}}({\cal N}\|{\cal M})$. Thus it is clear that
$\inf_{\alpha>1}\widetilde{D}_{\alpha}^{{\rm reg}}({\cal N}\|{\cal
M})\geqslant D^{{\rm reg}}({\cal N}\|{\cal M})$. We now prove the other
direction. If $\inf_{\alpha>1}\widetilde{D}_{\alpha}^{{\rm reg}}({\cal
N}\|{\cal M})>D^{{\rm reg}}({\cal N}\|{\cal M})$, we can find a real number
$r$ such that $\inf_{\alpha>1}\widetilde{D}_{\alpha}^{{\rm reg}}({\cal
N}\|{\cal M})>r>D^{{\rm reg}}({\cal N}\|{\cal M})$. Consider a sequence of
coherent channel discrimination strategies such that the Type II error
converges at an exponential rate $r$. By the result [FF21, Theorem 5.5 and
Remark 5.6], we know that the strong converse exponent is zero since
$r<\inf_{\alpha>1}\widetilde{D}_{\alpha}^{{\rm reg}}({\cal N}\|{\cal M})$,
which means the Type I error does not exponentially converge to one. However,
by Conjecture 12, the condition $r>D^{{\rm reg}}({\cal N}\|{\cal M})$ implies
that the Type I error has to converge exponentially to one, which forms a
contradiction and concludes that $\inf_{\alpha>1}\widetilde{D}_{\alpha}^{{\rm
reg}}({\cal N}\|{\cal M})\leqslant D^{{\rm reg}}({\cal N}\|{\cal M})$.
Finally, the assertion for the amortized channel divergence directly follows
from $\widetilde{D}_{\alpha}^{{\rm reg}}({\cal N}\|{\cal
M})=\widetilde{D}_{\alpha}^{A}({\cal N}\|{\cal M})$ [FF21, Theorem 5.4] and
$D^{{\rm reg}}({\cal N}\|{\cal M})=D^{A}({\cal N}\|{\cal M})$ [FFRS20,
Corollary 3].  $\sqcap$$\sqcup$

###### Corollary 16

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and ${\cal N}>0$. Consider
any sequence of sequential channel discrimination strategies with Type I error
$\\{\alpha_{n}\\}_{n\in{{\mathbb{N}}}}$ and Type II error
$\\{\beta_{n}\\}_{n\in{{\mathbb{N}}}}$. If the Type II error has

$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\log\beta_{n}=\mathrel{\mathop{\mathchar
58\relax}}r>D^{A}({\cal N}\|{\cal M}),$ (35)

then Conjecture 12 implies that there exists $c>0$ such that the Type I error
$1-\alpha_{n}\leqslant 2^{-cn}$ for sufficiently large $n$.

###### Proof.

By the result in [WBHK20, Proposition 20], for any sequential strategies and
$\alpha>1$ it holds

$\displaystyle-\frac{1}{n}\log(1-\alpha_{n})\geqslant\frac{\alpha-1}{\alpha}\left(-\frac{1}{n}\log\beta_{n}-\widetilde{D}_{\alpha}^{A}({\cal
N}\|{\cal M})\right).$ (36)

By the assumption (35), there exists $\delta>0$ such that
$-\frac{1}{n}\log\beta_{n}\geqslant r-\delta>D^{A}({\cal N}\|{\cal M})$ for
sufficiently large $n$. Since $r>D^{A}({\cal N}\|{\cal M})$ and
$\lim_{\alpha\to 1^{+}}\widetilde{D}_{\alpha}^{A}({\cal N}\|{\cal
M})=D^{A}({\cal N}\|{\cal M})$ from Theorem 15, then there exists
$\alpha_{0}>1$ such that $r-\delta>\widetilde{D}^{A}_{\alpha_{0}}({\cal
N}\|{\cal M})$. Then we have

$\displaystyle-\frac{1}{n}\log\beta_{n}-\widetilde{D}^{A}_{\alpha_{0}}({\cal
N}\|{\cal M})\geqslant r-\delta-\widetilde{D}^{A}_{\alpha_{0}}({\cal N}\|{\cal
M})=\mathrel{\mathop{\mathchar 58\relax}}b>0,$ (37)

and

$\displaystyle-\frac{1}{n}\log(1-\alpha_{n})\geqslant\frac{\alpha_{0}-1}{\alpha_{0}}b=\mathrel{\mathop{\mathchar
58\relax}}c>0,$ (38)

which is equivalent to $1-\alpha_{n}\leqslant 2^{-cn}$. This completes the
proof.  $\sqcap$$\sqcup$

##  5 Channel Discrimination and Channel Divergence

The task of channel discrimination aims to distinguish a quantum channel from
the other under a given type of strategy. A standard approach for
discrimination is to perform hypothesis testing and make a decision based on
the testing result. However, two types of error (Type I error and Type II
error) arise. In the same spirit of state discrimination, one can study the
asymptotic behavior of these errors in different operational regimes,
particularly,

  * •

Stein exponent regime that studies the exponent of the exponential decay of
the Type II error given that the Type I error is within a constant threshold;

  * •

strong converse exponent regime that studies the exponent of the exponential
convergence of the Type I error given that the Type II error exponentially
decays;

  * •

error exponent regime that studies the exponent of the exponential convergence
of the Type I error given that the Type II error exponentially decays.

The discrimination between quantum states in different operational regimes has
been well-studied. In particular, there is a nice correspondence between the
regime studied and the divergence of quantum states. More precisely, the Stein
exponent is given by the Umegaki relative entropy [HP91, ON00], the strong
converse exponent is determined by the Sandwiched Rényi divergence [MO15], and
the error exponent is determined by the Petz Rényi divergence [Hay07, Nag06,
ANSV08b]. However, when it comes to channel discrimination, the situation
becomes much involved due to the diverse range of discrimination strategies
and different extensions of channel divergence.

In this section we give a systematic study of the interplay between the
strategies of channel discrimination (e.g., sequential, coherent, product),
the operational regimes (e.g., error exponent, Stein exponent, strong converse
exponent), and three variants of channel divergences (e.g., Petz, Umegaki,
Sandwiched). We find a nice correspondence summarized in Table 2, which shows
that the proper divergences to use (Petz, Umegaki, Sandwiched) are determined
by the operational regime of interest, while the types of channel extension
(one-shot, regularized, amortized) are determined by the discrimination
strategies. The solution of each particular case includes both direct and
converse parts. Existing results are either for specific channels (e.g
replacer channels [CMW16], cq channels [BHKW18]) or providing partial
solutions (e.g. weak converse [WW19b], strong converse bounds [BHKW18]). Our
main contributions (highlighted in yellow) are mostly on the strong converse
parts, contributing to a complete picture of channel discrimination in this
unified framework (note that some results are built upon Conjecture 12).

$\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{D}_{\alpha}^{A}$ ? |  |  $D^{A}$ ✔ |  [WW19b] [FFRS20] [*,Thm. 20] |  $\widetilde{D}_{\alpha}^{A}$ ✔ |  [BHKW18] [FF21] [*,Thm. 15]
---|---|---|---|---|---
$\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{D}_{\alpha}^{\rm reg}$ ✔ – |  [*,Thm. 25] |  $D^{\rm reg}$ ✔ |  [WW19b] [FFRS20] [*,Thm. 19] |  $\widetilde{D}_{\alpha}^{\rm reg}$ ✔ |  [FF21] [*,Thm. 15]
$\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{D}_{\alpha}$ ✔ – |  [CMW16] [*,Thm. 24] |  $D$ ✔ |  [CMW16] [*,Thm. 18] |  $\widetilde{D}_{\alpha}$ ✔ |  [CMW16] [*,Thm. 22]
Figure 2: A table of relations between the operational regimes, channel
discrimination strategies and channel divergences. The symbol [*] represents
contributions in this note. “✔”,“✔ – ” and “?” represent that the case is
solved (may depend on Conjecture 12), partially solved and not solved,
respectively. Some key references are also included.

###  5.1 Stein exponent

In this subsection we consider minimizing the Type II error probability, under
the constraint that the Type I error probability does not exceed a constant
threshold $\varepsilon\in(0,1)$. We characterize the exact exponent, named
_Stein exponent_ , with which the Type II error exponentially decays.

######  Definition 17 (Stein exponent)

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and $\varepsilon\in(0,1)$.
The Stein exponents of quantum channel discrimination by the strategy class
$\Omega\in\\{\text{\rm PRO},\text{\rm COH},\text{\rm SEQ}\\}$ are defined by

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\Omega,\sup}(\varepsilon|{\cal
N}\|{\cal M})$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\limsup_{n\to\infty}\frac{1}{n}D_{H,\Omega}^{\varepsilon}({\cal
N}^{\otimes n}\|{\cal M}^{\otimes n}),$ (39) $\displaystyle\text{\rm
E}^{\text{\rm st}}_{\Omega,\inf}(\varepsilon|{\cal N}\|{\cal M})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\liminf_{n\to\infty}\frac{1}{n}D_{H,\Omega}^{\varepsilon}({\cal
N}^{\otimes n}\|{\cal M}^{\otimes n}),$ (40)

where

$\displaystyle D_{H,\Omega}^{\varepsilon}({\cal N}^{\otimes n}\|{\cal
M}^{\otimes n})\mathrel{\mathop{\mathchar
58\relax}}=\sup_{(S_{n},\Pi_{n})\in\Omega}\left\\{-\frac{1}{n}\log\beta_{n}(S_{n},\Pi_{n})\mathrel{\mathop{\mathchar
58\relax}}\alpha_{n}(S_{n},\Pi_{n})\leqslant\varepsilon\right\\}$ (41)

the supremum is taken over all possible stategies $(S_{n},\Pi_{n})\in\Omega$
satisfying the condition.

The non-asymptotic quantity in (41) can also be written as a notion of
hypothesis testing relative entropy between the testing states,

$\displaystyle D_{H,\Omega}^{\varepsilon}({\cal N}^{\otimes n}\|{\cal
M}^{\otimes
n})=\sup_{S_{n}\in\Omega}D_{H}^{\varepsilon}(\rho_{n}(S_{n})\|\sigma_{n}(S_{n})),$
(42)

where the hypothesis testing relative entropy on the r.h.s. is between two
quantum states which is defined in (7) and the supremum is taken over all
strategies $S_{n}\in\Omega$ that generate the testing states $\rho_{n}(S_{n})$
and $\sigma_{n}(S_{n})$. More explicitly, when $\Omega=\text{\rm PRO}$, we
have $\rho_{n}(S_{n})={\cal N}^{\otimes
n}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i})$,
$\sigma_{n}(S_{n})={\cal M}^{\otimes
n}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i})$
and the supremum is taken over all $\varphi_{i}\in\mathfrak{D}(R_{i}A_{i})$.
When $\Omega=\text{\rm COH}$, we have $\rho_{n}(S_{n})={\cal N}^{\otimes
n}(\psi_{n})$, $\sigma_{n}(S_{n})={\cal M}^{\otimes n}(\psi_{n})$ and the
supremum is taken over all $\psi_{n}\in\mathfrak{D}(RA^{n})$. When
$\Omega=\text{\rm SEQ}$, we have $\rho_{n}(S_{n})={\cal N}\circ{\cal
P}_{n-1}\circ\cdots\circ{\cal P}_{2}\circ{\cal N}\circ{\cal P}_{1}\circ{\cal
N}(\psi_{n})$, $\sigma_{n}(S_{n})={\cal M}\circ{\cal
P}_{n-1}\circ\cdots\circ{\cal P}_{2}\circ{\cal M}\circ{\cal P}_{1}\circ{\cal
M}(\psi_{n})$ and the supremum is taken over all
$\psi_{n}\in\mathfrak{D}(R_{1}A_{1})$ and ${\cal P}_{i}\in\text{\rm
CPTP}(R_{i}B_{i}\to R_{i+1}A_{i+1})$.

###### Theorem 18

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and $\varepsilon\in(0,1)$.
It holds

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
PRO},\sup}(\varepsilon|{\cal N}\|{\cal M})=\text{\rm E}^{\text{\rm
st}}_{\text{\rm PRO},\inf}(\varepsilon|{\cal N}\|{\cal M})=D({\cal N}\|{\cal
M}).$ (43)

###### Proof.

It suffices to show that

$\displaystyle\lim_{n\to\infty}\frac{1}{n}D_{H,\text{\rm
PRO}}^{\varepsilon}({\cal N}^{\otimes n}\|{\cal M}^{\otimes n})=D({\cal
N}\|{\cal M}).$ (44)

For the achievable part, let $\varphi\in\mathfrak{D}(RA)$ be an optimal input
state for $D({\cal N}\|{\cal M})$, i.e., $D({\cal N}(\varphi)\|{\cal
M}(\varphi))=D({\cal N}\|{\cal M})$. Using $\varphi^{\otimes n}$ as the input
state in the product strategy, we have

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
PRO},\inf}(\varepsilon|{\cal N}\|{\cal
M})\geqslant\liminf_{n\to\infty}\frac{1}{n}D_{H}^{\varepsilon}([{\cal
N}(\varphi)]^{\otimes n}\|[{\cal M}(\varphi)]^{\otimes n})=D({\cal
N}(\varphi)\|{\cal M}(\varphi))=D({\cal N}\|{\cal M}),$ (45)

where the first equality follows from the quantum Stein’s lemma [HP91, ON00]
and the second equality follows from the optimality assumption of $\varphi$.
For the converse part, consider any input states
$\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i}$
with $\varphi_{i}\in\mathfrak{D}(R_{i}A_{i})$ and $\alpha>1$ we have

$\displaystyle\frac{1}{n}D_{H}^{\varepsilon}\left(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}{\cal
N}(\varphi_{i})\Big{\|}\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}{\cal
M}(\varphi_{i})\right)$
$\displaystyle\leqslant\frac{1}{n}\widetilde{D}_{\alpha}\left(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}{\cal
N}(\varphi_{i})\Big{\|}\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}{\cal
M}(\varphi_{i})\right)+\frac{1}{n}\frac{\alpha}{\alpha-1}\log\frac{1}{1-\varepsilon}$
(46) $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\widetilde{D}_{\alpha}\left({\cal
N}(\varphi_{i})\big{\|}{\cal
M}(\varphi_{i})\right)+\frac{1}{n}\frac{\alpha}{\alpha-1}\log\frac{1}{1-\varepsilon}$
(47) $\displaystyle\leqslant\widetilde{D}_{\alpha}({\cal N}\|{\cal
M})+\frac{1}{n}\frac{\alpha}{\alpha-1}\log\frac{1}{1-\varepsilon},$ (48)

where the first inequality follows from Lemma 5, the first equality follows
from the additivity of Sandwiched Rényi divergence under tensor product
states, the second inequality follows from the definition of channel
divergence. Taking the supremum of all input states
$\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i}$
and taking the limit of $n\to\infty$, we have

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
PRO},\sup}(\varepsilon|{\cal N}\|{\cal
M})=\limsup_{n\to\infty}\frac{1}{n}D_{H,\text{\rm PRO}}^{\varepsilon}({\cal
N}^{\otimes n}\|{\cal M}^{\otimes n})\leqslant\widetilde{D}_{\alpha}({\cal
N}\|{\cal M}).$ (49)

Finally, taking $\alpha\to 1$ and applying Lemma 9 we have the converse part.
$\sqcap$$\sqcup$

Note that we can actually extend the input choices of product strategy to
convex combination of tensor product states
$\sum_{j=1}^{m}p_{j}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i,j})$.
In this case Theorem 18 still holds by adding an extra step in the proof of
the converse part and using the joint quasi-convexity of the Sandwiched Rényi
divergence (e.g. [MO15, Corollary 3.16]). This indicates that shared
randomness between the input states for each uses of the channel will not help
to get a faster convergence rate of the type II error for channel
discrimination.

###### Theorem 19

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ with ${\cal N}>0$ and
$\varepsilon\in(0,1)$. Then, Conjecture 12 implies

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
COH},\sup}(\varepsilon|{\cal N}\|{\cal M})=\text{\rm E}^{\text{\rm
st}}_{\text{\rm COH},\inf}(\varepsilon|{\cal N}\|{\cal M})=D^{\rm reg}({\cal
N}\|{\cal M}).$ (50)

###### Proof.

The assertion is basically a restatement of Theorem 13 by noting that
$D_{H,\text{\rm COH}}^{\varepsilon}(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n})=D_{H}^{\varepsilon}(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n})$.  $\sqcap$$\sqcup$

###### Theorem 20

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ with ${\cal N}>0$ and
$\varepsilon\in(0,1)$. Then, Conjecture 12 implies

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
SEQ},\sup}(\varepsilon|{\cal N}\|{\cal M})=\text{\rm E}^{\text{\rm
st}}_{\text{\rm SEQ},\inf}(\varepsilon|{\cal N}\|{\cal M})=D^{A}({\cal
N}\|{\cal M}).$ (51)

###### Proof.

By definition it is clear that $\text{\rm E}^{\text{\rm st}}_{\text{\rm
SEQ},\inf}(\varepsilon|{\cal N}\|{\cal M})$ is monotone increasing in
$\varepsilon$. Thus for any fixed $\varepsilon\in(0,1)$ we have

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
SEQ},\inf}(\varepsilon|{\cal N}\|{\cal M})\geqslant\lim_{\varepsilon\to
0}\text{\rm E}^{\text{\rm st}}_{\text{\rm SEQ},\inf}(\varepsilon|{\cal
N}\|{\cal M})=D^{A}({\cal N}\|{\cal M}),$ (52)

where the equality follows from [WW19b, Theorem 6].

Next we prove the converse part. For any $\psi_{n}\in\mathfrak{D}(RA^{n})$,
${\cal P}_{i}\in\text{\rm CPTP}(R_{i}B_{i}\to R_{i+1}A_{i+1})$, denote

$\displaystyle\rho_{n}$ $\displaystyle={\cal N}\circ{\cal
P}_{n-1}\circ\cdots\circ{\cal P}_{2}\circ{\cal N}\circ{\cal P}_{1}\circ{\cal
N}(\psi_{n})$ (53) $\displaystyle\sigma_{n}$ $\displaystyle={\cal M}\circ{\cal
P}_{n-1}\circ\cdots\circ{\cal P}_{2}\circ{\cal M}\circ{\cal P}_{1}\circ{\cal
M}(\psi_{n}).$ (54)

Due to Lemma 5, it holds for any $\alpha>1$ that

$\displaystyle\frac{1}{n}D_{H}^{\varepsilon}(\rho_{n}\|\sigma_{n})\leqslant\frac{1}{n}\widetilde{D}_{\alpha}(\rho_{n}\|\sigma_{n})+\frac{1}{n}\frac{\alpha}{\alpha-1}\log\frac{1}{1-\varepsilon}.$
(55)

Note that for any quantum state $\rho,\sigma$ and quantum channels ${\cal
E},{\cal F}$, we have by definition

$\displaystyle\widetilde{D}_{\alpha}({\cal E}(\rho)\|{\cal
F}(\sigma))\leqslant\widetilde{D}_{\alpha}^{A}({\cal E}\|{\cal
F})+\widetilde{D}_{\alpha}(\rho\|\sigma).$ (56)

By using this relation and the data-processing inequality of
$\widetilde{D}_{\alpha}$ iteratively, we have
$\widetilde{D}_{\alpha}(\rho_{n}\|\sigma_{n})\leqslant
n\widetilde{D}_{\alpha}^{A}({\cal N}\|{\cal M})$. This gives

$\displaystyle\frac{1}{n}D_{H}^{\varepsilon}(\rho_{n}\|\sigma_{n})\leqslant\widetilde{D}_{\alpha}^{A}({\cal
N}\|{\cal M})+\frac{1}{n}\frac{\alpha}{\alpha-1}\log\frac{1}{1-\varepsilon}.$
(57)

Taking on both sides the supremum over all sequential strategies following by
the limit $n\to\infty$ gives

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
SEQ},\sup}(\varepsilon|{\cal N}\|{\cal
M})=\limsup_{n\to\infty}\frac{1}{n}D_{H,\text{\rm
SEQ}}^{\varepsilon}\left(\mathcal{N}^{\otimes n}\big{\|}\mathcal{M}^{\otimes
n}\right)\leqslant\widetilde{D}^{A}_{\alpha}(\mathcal{N}\|\mathcal{M})\;.$
(58)

Since the above inequality holds for all $\alpha>1$, by taking $\alpha\to
1^{+}$ and using Theorem 15 we have

$\displaystyle\text{\rm E}^{\text{\rm st}}_{\text{\rm
SEQ},\sup}(\varepsilon|{\cal N}\|{\cal M})\leqslant
D^{A}(\mathcal{N}\|\mathcal{M}).$ (59)

This completes the proof.  $\sqcap$$\sqcup$

Note that Theorem 19 and 20 have been proved in [WW19b, Theorem 3 and Theorem
6] for vanishing $\varepsilon$. But the above results are stronger as they
hold for any fixed $\varepsilon\in(0,1)$ without the need to take
${\varepsilon\to 0}$. This corresponds to the strong converse property of
channel discrimination under coherent or sequential strategies.

###  5.2 Strong converse exponent

In the task of state discrimination, the strong converse exponent is defined
by

$\displaystyle\text{\rm E}^{\text{\rm
sc}}(r|\rho\|\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\inf_{\\{\Pi_{n}\\}}\left\\{-\liminf_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\rho^{\otimes
n}\Pi_{n}\mathrel{\mathop{\mathchar
58\relax}}\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\sigma^{\otimes
n}\Pi_{n}\leqslant-r\right\\},$ (60)

where the infimum is taken over all possible sequences of quantum tests
$\\{\Pi_{n}\\}_{n\in{{\mathbb{N}}}}$ satisfying the condition. It has been
shown in [MO15, Theorem 4.10] that this exponent is precisely characterized
by:

$\displaystyle\text{\rm E}^{\text{\rm
sc}}(r|\rho\|\sigma)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-\widetilde{D}_{\alpha}(\rho\|\sigma)\right].$
(61)

We aim to extend this result to the channel case.

Let us start by defining the strong converse exponent of channel
discrimination.

######  Definition 21 (Strong converse exponent)

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and $r>0$. The strong
converse exponent of channel discrimination by the strategy class
$\Omega\in\\{\text{\rm PRO},\text{\rm COH},\text{\rm SEQ}\\}$ is defined by

$\displaystyle\text{\rm E}^{\text{\rm
sc}}_{\Omega}(r|\mathcal{N}\|\mathcal{M})\mathrel{\mathop{\mathchar
58\relax}}=\inf_{(S_{n},\Pi_{n})\in\Omega}\left\\{-\liminf_{n\to+\infty}\frac{1}{n}\log(1-\alpha_{n}(S_{n},\Pi_{n}))\mathrel{\mathop{\mathchar
58\relax}}\limsup_{n\to+\infty}\frac{1}{n}\log\beta_{n}(S_{n},\Pi_{n})\leqslant-r\right\\},$
(62)

where the infimum is taken over all possible strategies
$(S_{n},\Pi_{n})\in\Omega$ satisfying the condition.

###### Theorem 22

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and $r>0$. Then it holds

$\displaystyle\text{\rm E}^{\text{\rm sc}}_{\text{\rm
PRO}}(r|\mathcal{N}\|\mathcal{M})=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-\widetilde{D}_{\alpha}(\mathcal{N}\|\mathcal{M})\right].$
(63)

###### Proof.

We first prove the converse part which closely follows the proof of its state
analog in [MO15, Lemma 4.7]. For any product strategy
$(\\{\varphi_{i}\\}_{i=1}^{n},\Pi_{n})$ with input states
$\varphi_{i}\in\mathfrak{D}(R_{i}A_{i})$ and measurement opeartor
$0\leqslant\Pi_{n}\leqslant I$. Let $\rho_{n}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{N}^{\otimes
n}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i})$,
$\sigma_{n}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{M}^{\otimes
n}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i})$
be the output states and $p_{n}\mathrel{\mathop{\mathchar
58\relax}}=(\operatorname{Tr}\rho_{n}\Pi_{n},\operatorname{Tr}\rho_{n}(I-\Pi_{n}))$
and $q_{n}\mathrel{\mathop{\mathchar
58\relax}}=(\operatorname{Tr}\sigma_{n}\Pi_{n},\operatorname{Tr}\sigma_{n}(I-\Pi_{n}))$
be the post-measurement states. Then the Type I error is
$\alpha_{n}=\operatorname{Tr}[(I-\Pi_{n})\rho_{n}]$ and the Type II error is
$\beta_{n}=\operatorname{Tr}[\Pi_{n}\sigma_{n}]$. By definition it suffices to
consider sequences $(\\{\varphi_{i}\\}_{i=1}^{n},\Pi_{n})$ such that
$\limsup_{n\to+\infty}\frac{1}{n}\log\beta_{n}\leqslant-r$. From the data-
processing of the Sandwiched Rényi divergence, we have for any $\alpha>1$ that

$\displaystyle\widetilde{D}_{\alpha}(\rho_{n}\|\sigma_{n})$
$\displaystyle\geqslant\widetilde{D}_{\alpha}(p_{n}\|q_{n})$
$\displaystyle\geqslant\frac{1}{\alpha-1}\log\left[(\operatorname{Tr}\rho_{n}\Pi_{n})^{\alpha}(\operatorname{Tr}\sigma_{n}\Pi_{n})^{1-\alpha}\right]=\frac{\alpha}{\alpha-1}\log(1-\alpha_{n})-\log\beta_{n}.$
(64)

This can be equivalently written as

$\displaystyle-\frac{1}{n}\log(1-\alpha_{n})\geqslant\frac{\alpha-1}{\alpha}\left[-\frac{1}{n}\log\beta_{n}-\frac{1}{n}\widetilde{D}_{\alpha}(\rho_{n}\|\sigma_{n})\right].$
(65)

By the assumption of $(\\{\varphi_{i}\\}_{i=1}^{n},\Pi_{n})$ and taking
$\limsup_{n\to\infty}$ on both sides, we have

$\displaystyle-\liminf_{n\to+\infty}\frac{1}{n}\log(1-\alpha_{n})\geqslant\frac{\alpha-1}{\alpha}\left[r-\liminf_{n\to+\infty}\frac{1}{n}\widetilde{D}_{\alpha}(\rho_{n}\|\sigma_{n})\right].$
(66)

By the additivity of Sandwiched Rényi divergence under tensor product states
and the definition of channel divergence, we have
$\widetilde{D}_{\alpha}(\rho_{n}\|\sigma_{n})=\sum_{i=1}^{n}\widetilde{D}_{\alpha}({\cal
N}(\varphi_{i})\|{\cal M}(\varphi_{i}))\leqslant n\widetilde{D}_{\alpha}({\cal
N}\|{\cal M})$. Thus

$\displaystyle-\liminf_{n\to+\infty}\frac{1}{n}\log(1-\alpha_{n})\geqslant\frac{\alpha-1}{\alpha}\left[r-\widetilde{D}_{\alpha}({\cal
N}\|{\cal M})\right].$ (67)

Finally taking the infimum over all product strategies and the supremum over
all $\alpha>1$ on both sides, we can conclude the converse part.

We then proceed to show the achievable part. Let $\varphi\in\mathfrak{D}(RA)$
be an optimal quantum state such that
$\widetilde{D}_{\alpha}(\mathcal{N}\|\mathcal{M})=\widetilde{D}_{\alpha}(\mathcal{N}(\varphi)\|\mathcal{M}(\varphi))$.
Consider the task of distinguishing quantum states $\mathcal{N}(\varphi)$ and
$\mathcal{M}(\varphi)$. Suppose the optimal test in $\text{\rm E}^{\text{\rm
sc}}(r|\mathcal{N}(\varphi)\|\mathcal{M}(\varphi))$ is given by the sequence
$\\{\Pi_{n}\\}_{n\in{{\mathbb{N}}}}$. Then by the quantum converse Hoeffiding
theorem (see (61)) we have

$\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}[\mathcal{M}(\varphi)]^{\otimes
n}\Pi_{n}$ $\displaystyle\leqslant-r\qquad\text{and}$ (68)
$\displaystyle-\liminf_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}[\mathcal{N}(\varphi)]^{\otimes
n}\Pi_{n}$
$\displaystyle=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-\widetilde{D}_{\alpha}({\cal
N}(\varphi)\|{\cal M}(\varphi))\right].$ (69)

Note that $(\\{\varphi\\}_{i=1}^{n},\Pi_{n})$ is a product stategy for the
task of channel discrimination between $\mathcal{N}^{\otimes n}$ and
$\mathcal{M}^{\otimes n}$. We have

$\displaystyle\text{\rm E}^{\text{\rm sc}}_{\text{\rm
PRO}}\left(r|\mathcal{N}\|\mathcal{M}\right)$
$\displaystyle\leqslant-\liminf_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}\mathcal{N}^{\otimes
n}(\varphi^{\otimes n})\Pi_{n}$ (70)
$\displaystyle=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-\widetilde{D}_{\alpha}(\mathcal{N}(\varphi)\|\mathcal{M}(\varphi))\right]$
(71)
$\displaystyle=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-\widetilde{D}_{\alpha}(\mathcal{N}\|\mathcal{M})\right]$
(72)

where the first equality follows from (69), the second equality follows from
the optimality assumption of $\varphi$. This completes the proof.
$\sqcap$$\sqcup$

Note here that one can extend the input choices of product strategy to convex
combination of tensor product states
$\sum_{j=1}^{m}p_{j}(\operatorname*{\text{\raisebox{0.96873pt}{\scalebox{0.6}{$\bigotimes$}}}}_{i=1}^{n}\varphi_{i,j})$.
In this case Theorem 22 still holds by adding an additional step in the proof
of the converse part and using the joint quasi-convexity of the Sandwiched
Rényi divergence (e.g. [MO15, Corollary 3.16]). This indicates that shared
randomness between the input states for each use of the channel will provide
no advantage in reducing the convergence rate of the Type I error.

Also note that the strong converse exponents under coherent and sequential
strategies have been settled in [FF21, Theorem 5.5]. However, in terms of the
exact threshold for the exponential convergence, their result only implies the
threshold to be $\inf_{\alpha>1}\widetilde{D}^{\rm reg}({\cal N}\|{\cal M})$.
The continuity result in Theorem 15 would help to completely determine this
threshold to be $D^{\rm reg}({\cal N}\|{\cal M})$.

###  5.3 Error exponent

In the task of state discrimination, the error exponent is defined by

$\displaystyle\text{\rm E}^{\text{\rm
er}}(r|\rho\|\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\sup_{\\{\Pi_{n}\\}}\left\\{-\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}[(I-\Pi_{n})\rho^{\otimes
n}]\mathrel{\mathop{\mathchar
58\relax}}\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}[\Pi_{n}\sigma^{\otimes
n}]\leqslant-r\right\\},$ (73)

where the supremum is taken over all possible sequences of quantum tests
$\\{\Pi_{n}\\}_{n\in{{\mathbb{N}}}}$ satisfying the condition. It has been
shown in [Hay07, Nag06, ANSV08b] that the error exponent is precisely given
by:

$\displaystyle\text{\rm E}^{\text{\rm
er}}(r|\rho\|\sigma)=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)\right].$ (74)

We aim to extend this result to the channel case.

######  Definition 23 (Error exponent)

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$ and $r>0$. The error
exponent of quantum channel discrimination by the strategy class
$\Omega\in\\{\text{\rm PRO},\text{\rm COH},\text{\rm SEQ}\\}$ is defined by

$\displaystyle\text{\rm E}^{\text{\rm
er}}_{\Omega}(r|\mathcal{N}\|\mathcal{M})\mathrel{\mathop{\mathchar
58\relax}}=\sup_{(S_{n},\Pi_{n})\in\Omega}\left\\{-\limsup_{n\to+\infty}\frac{1}{n}\log\alpha_{n}(S_{n},\Pi_{n})\mathrel{\mathop{\mathchar
58\relax}}\limsup_{n\to+\infty}\frac{1}{n}\log\beta_{n}(S_{n},\Pi_{n})\leqslant-r\right\\},$
(75)

where the supremum is taken over all possible strategies
$(S_{n},\Pi_{n})\in\Omega$ satisfying the condition.

###### Theorem 24

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ and $r>0$. Then it
holds

$\displaystyle\text{\rm E}^{\text{\rm er}}_{\text{\rm
PRO}}(r|\mathcal{N}\|\mathcal{M})\geqslant\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}({\cal N}\|{\cal M})\right].$ (76)

###### Proof.

Let $\varphi\in\mathfrak{D}(RA)$ an optimal input state such that
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}\|\mathcal{M})=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}(\varphi)\|\mathcal{M}(\varphi))$. Consider the
task of distinguishing quantum states $\mathcal{N}(\varphi)$ and
$\mathcal{M}(\varphi)$. Suppose the optimal test in $\text{\rm E}^{\text{\rm
er}}(r|\mathcal{N}(\varphi)\|\mathcal{M}(\varphi))$ is given by the sequence
$\\{\Pi_{n}\\}_{n\in{{\mathbb{N}}}}$. Then by the quantum Hoeffding theorem
(74) we have

$\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}[\mathcal{M}(\varphi)]^{\otimes
n}\Pi_{n}$ $\displaystyle\leqslant-r\qquad\text{and}$ (77)
$\displaystyle-\limsup_{n\to+\infty}\frac{1}{n}\log(1-\operatorname{Tr}[\mathcal{N}(\varphi)]^{\otimes
n}\Pi_{n})$
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}({\cal N}(\varphi)\|{\cal M}(\varphi))\right].$ (78)

Note that $(\\{\varphi\\}_{i=1}^{n},\Pi_{n})$ is a product strategy for the
task of channel discrimination between $\mathcal{N}^{\otimes n}$ and
$\mathcal{M}^{\otimes n}$. Then we have

$\displaystyle\text{\rm E}^{\text{\rm er}}_{\text{\rm
PRO}}\left(r|\mathcal{N}\|\mathcal{M}\right)$
$\displaystyle\geqslant-\limsup_{n\to+\infty}\frac{1}{n}\log(1-\operatorname{Tr}\mathcal{N}^{\otimes
n}(\varphi^{\otimes n})\Pi_{n})$ (79)
$\displaystyle=-\limsup_{n\to+\infty}\frac{1}{n}\log(1-\operatorname{Tr}[\mathcal{N}(\varphi)]^{\otimes
n}\Pi_{n})$ (80)
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}(\varphi)\|\mathcal{M}(\varphi))\right]$ (81)
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}\|\mathcal{M})\right]$ (82)

where the second equality follows from (78), the third equality follows from
the optimality assumption of $\varphi$. This completes the proof.
$\sqcap$$\sqcup$

###### Theorem 25

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ and $r>0$. Then it
holds

$\displaystyle\text{\rm E}^{\text{\rm er}}_{\text{\rm
COH}}(r|\mathcal{N}\|\mathcal{M})\geqslant\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}^{\rm reg}({\cal N}\|{\cal M})\right].$ (83)

###### Proof.

For any given $m\in{{\mathbb{N}}}$, let $\psi_{m}\in\mathfrak{D}(RA^{m})$ an
optimal input state such that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes
m})=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}(\psi_{m})\|\mathcal{M}^{\otimes
m}(\psi_{m}))$. Denote $\rho_{m}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{N}^{\otimes m}(\psi_{m})$ and
$\sigma_{m}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{M}^{\otimes
m}(\psi_{m})$. Consider the task of distinguishing quantum states $\rho_{m}$
and $\sigma_{m}$. Suppose the optimal test in $\text{\rm E}^{\text{\rm
er}}(r|\rho_{m}\|\sigma_{m})$ is given by the sequence
$\\{\Pi_{m,n}\\}_{n\in{{\mathbb{N}}}}$. Then by the quantum Hoeffding theorem
(see (74)) we have

$\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log\operatorname{Tr}[\sigma_{m}]^{\otimes
n}\Pi_{m,n}$ $\displaystyle\leqslant-r\qquad\text{and}$ (84)
$\displaystyle-\limsup_{n\to+\infty}\frac{1}{n}\log(1-\operatorname{Tr}[\rho_{m}]^{\otimes
n}\Pi_{m,n})$
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho_{m}\|\sigma_{m})\right].$ (85)

Note that $(\psi_{m}^{\otimes n},\Pi_{m,n})$ is a coherent strategy for the
task of channel discrimination between $\mathcal{N}^{\otimes mn}$ and
$\mathcal{M}^{\otimes mn}$, satisfying

$\displaystyle\limsup_{n\to+\infty}\frac{1}{mn}\log\operatorname{Tr}\mathcal{M}^{\otimes
mn}(\psi_{m}^{\otimes n})\Pi_{m,n}$ $\displaystyle\leqslant-\frac{r}{m}.$ (86)

Then we have

$\displaystyle\text{\rm E}^{\text{\rm er}}_{\text{\rm
COH}}\left(\frac{r}{m}\Big{|}\mathcal{N}\Big{\|}\mathcal{M}\right)$
$\displaystyle\geqslant-\limsup_{n\to+\infty}\frac{1}{mn}\log(1-\operatorname{Tr}\mathcal{N}^{\otimes
mn}(\psi_{m}^{\otimes n})\Pi_{m,n})$ (87)
$\displaystyle=-\limsup_{n\to+\infty}\frac{1}{mn}\log(1-\operatorname{Tr}[\rho_{m}]^{\otimes
n}\Pi_{m,n})$ (88)
$\displaystyle=\frac{1}{m}\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho_{m}\|\sigma_{m})\right]$ (89)
$\displaystyle=\frac{1}{m}\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes m})\right]$ (90)
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[\frac{r}{m}-\frac{1}{m}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes m})\right],$
(91)

where the second equality follows from (85), the third equality follows from
the optimality assumption of $\psi_{m}$. Replacing $r/m$ as $r$, we have

$\displaystyle\text{\rm E}^{\text{\rm er}}_{\text{\rm
COH}}\left(r|\mathcal{N}\|\mathcal{M}\right)\geqslant\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\frac{1}{m}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes m})\right].$
(92)

Since (92) holds for any integer $m\in{{\mathbb{N}}}$, we have

$\displaystyle\text{\rm E}^{\text{\rm er}}_{\text{\rm
COH}}\left(r|\mathcal{N}\|\mathcal{M}\right)$
$\displaystyle\geqslant\sup_{m\in{{\mathbb{N}}}}\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\frac{1}{m}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes m})\right]$ (93)
$\displaystyle=\sup_{0<\alpha<1}\sup_{m\in{{\mathbb{N}}}}\frac{\alpha-1}{\alpha}\left[r-\frac{1}{m}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes m})\right]$ (94)
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\sup_{m\in{{\mathbb{N}}}}\frac{1}{m}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\mathcal{N}^{\otimes m}\|\mathcal{M}^{\otimes m})\right]$ (95)
$\displaystyle=\sup_{0<\alpha<1}\frac{\alpha-1}{\alpha}\left[r-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}^{\rm reg}(\mathcal{N}\|\mathcal{M})\right].$ (96)

This completes the proof.  $\sqcap$$\sqcup$

##  6 Discussions

In this note, we make a conjecture on the exponentially strong converse of
quantum channel hypothesis testing under coherent strategies. This conjecture
directly implies a complete analog of Stein’s lemma for quantum channels. It
also directly leads to the establishment of strong asymptotic equipartition
property for equivalent channel divergences such as the max-relative entropy
and variants of information spectrum relative entropies. Together with the
known results on the strong converse exponent of channel discrimination, we
show that the conjecture also implies the continuity of the regularized
(amortized) Sandwiched Rényi channel divergence. We also give a systematic
study on the interplay between the strategies of channel discrimination, the
operational regimes, and variants of channel divergences.

In a previous version of this arXiv post, we provided a proof of the
exponentially strong converse conjecture. However, a gap was founded in a
technical lemma borrowed from [BP10] which turned out to be false. This gap
makes the generalized quantum Stein’s lemma in [BP10] and our exponentially
strong converse conjecture both open again. As we have shown that our
conjecture would imply many desirable results and would contribute to a
complete understanding of channel discrimination, we encourage interested
readers to further investigate on this. More importantly, we believe that it
is necessary to double-check the correctness of relevant results of [BP10] by
either fixing the gap or finding alternative proofs.

#### Acknowledgements.

The authors thank the organization of the 23rd Annual Conference on Quantum
Information Processing (QIP 2020), held in Shenzhen, China during January
2020, at which this work was initiated. GG would like to acknowledge support
from the Natural Sciences and Engineering Research Council of Canada (NSERC).
The authors also thank Marco Tomamichel for pointing out the gap in the proof
of Lemma 16 in our previous arXiv version that basically comes from [BP10].

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##  Appendix A Technical Lemmas

###### Lemma 26

For any two matrices $M,N\in{\rm Herm}(A)$ we have

$\displaystyle\operatorname{Tr}(M-N)_{+}\leqslant\operatorname{Tr}M_{+}+\operatorname{Tr}N_{+}-\operatorname{Tr}N\;.$
(97)

###### Proof.

Observe that $\operatorname{Tr}(X)_{+}=(\|X\|_{1}+\operatorname{Tr}X)/2$. Then
it holds

$\displaystyle\operatorname{Tr}(M-N)_{+}$
$\displaystyle=\frac{1}{2}\operatorname{Tr}(M-N)+\frac{1}{2}\|M-N\|_{1}\leqslant\frac{1}{2}\operatorname{Tr}(M-N)+\frac{1}{2}\|M\|_{1}+\frac{1}{2}\|N\|_{1}\;.$
(98)

By using the observation again to the r.h.s., we have the asserted result.
$\sqcap$$\sqcup$

Recall that the hypothesis testing relative entropy can be lower bounded by
the Petz Rényi divergence [QWW18, Proposition 3],

$\displaystyle D_{H}^{\varepsilon}(\rho\|\sigma)\geqslant\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)-\frac{\alpha}{1-\alpha}\log\frac{1}{\varepsilon}.$
(99)

Here we provide a tighter lower bound with a simple proof. This result can be
of independent interest.

###### Lemma 27

Let $\varepsilon\in(0,1)$ and $\rho,\sigma\in\mathfrak{D}(A)$. For all
$\alpha\in(0,1)$,

$D_{H}^{\varepsilon}(\rho\|\sigma)\geqslant\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)+\frac{\alpha}{1-\alpha}\left(\frac{h(\alpha)}{\alpha}-\log\left(\frac{1}{\varepsilon}\right)\right)\;,$
(100)

where $h(\alpha)=-\alpha\log\alpha-(1-\alpha)\log(1-\alpha)$ is the binary
entropy.

###### Proof.

Let $\alpha\in(0,1)$ and recall that

$2^{-D^{\varepsilon}_{H}\left(\rho\|\sigma\right)}=\max_{t\geqslant
0}\Big{\\{}t(1-\varepsilon)-\operatorname{Tr}(t\rho-\sigma)_{+}\Big{\\}}\;.$
(101)

To bound the term $\operatorname{Tr}(t\rho-\sigma)_{+}$ we use the quantum
weighted geometric-mean inequality; i.e. for any two positive semidefinite
matrices $M,N$ and any $\alpha\in[0,1]$

$\frac{1}{2}\operatorname{Tr}\Big{[}M+N-\big{|}M-N\big{|}\Big{]}\leqslant\operatorname{Tr}\big{[}M^{\alpha}N^{1-\alpha}\big{]}\;.$
(102)

Since the term $|M-N|$ can be expressed as $|M-N|=2(M-N)_{+}-(M-N)$, the above
inequality is equivalent to

$\operatorname{Tr}(M-N)_{+}\geqslant\operatorname{Tr}[M]-\operatorname{Tr}\big{[}M^{\alpha}N^{1-\alpha}\big{]}$
(103)

Taking $M=t\rho$ and $N=\sigma$ we have

$\displaystyle\operatorname{Tr}(t\rho-\sigma)_{+}\geqslant
t-t^{\alpha}\operatorname{Tr}\big{[}\rho^{\alpha}\sigma^{1-\alpha}\big{]}=t-t^{\alpha}2^{(\alpha-1)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)}\;.$ (104)

Substituting this into (101) gives

$\displaystyle 2^{-D^{\varepsilon}_{H}\left(\rho\|\sigma\right)}$
$\displaystyle=\max_{t\geqslant
0}\Big{\\{}t(1-\varepsilon)-\operatorname{Tr}(t\rho-\sigma)_{+}\Big{\\}}\leqslant\max_{t\geqslant
0}\Big{\\{}-t\varepsilon+t^{\alpha}2^{(\alpha-1)D_{\alpha}(\rho\|\sigma)}\Big{\\}}\;.$
(105)

It is straightforward to check that for fixed
$\alpha,\rho,\sigma,\varepsilon$, the function
$t\mapsto-t\varepsilon+t^{\alpha}2^{(\alpha-1)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)}$ obtains its maximal value at

$t=\left(\frac{\alpha}{\varepsilon}\right)^{\frac{1}{1-\alpha}}2^{-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}(\rho\|\sigma)}\;.$ (106)

Substituting this value into the optimization in (105) gives

$2^{-D^{\varepsilon}_{H}\left(\rho\|\sigma\right)}\leqslant(1-\alpha)\left(\frac{\alpha}{\varepsilon}\right)^{\frac{\alpha}{1-\alpha}}2^{-D_{\alpha}(\rho\|\sigma)}\;.$
(107)

By taking $-\log$ on both sides we get (100). This concludes the proof.
$\sqcap$$\sqcup$

In the following we prove several properties of channel divergences that will
be frequently used.

###### Lemma 28

Let $\boldsymbol{D}$ be a quantum divergence for states. Let
$\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$. If $\boldsymbol{D}$ is
superadditive under tensor product of states, then

$\displaystyle\boldsymbol{D}^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})\geqslant\frac{1}{n}\boldsymbol{D}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right),\quad\forall n\in{{\mathbb{N}}}.$
(108)

###### Proof.

Denote $a_{n}\coloneqq\frac{1}{n}\boldsymbol{D}(\mathcal{N}^{\otimes
n}\|\mathcal{M}^{\otimes n})$ and observe that

$\displaystyle a_{2n}=\frac{1}{2n}\boldsymbol{D}({\cal N}^{\otimes 2n}\|{\cal
M}^{\otimes 2n})\geqslant\frac{1}{2n}(\boldsymbol{D}({\cal N}^{\otimes
n}\|{\cal M}^{\otimes n})+\boldsymbol{D}({\cal N}^{\otimes n}\|{\cal
M}^{\otimes n}))=a_{n}\;,$ (109)

where the inequality follows by the superadditivity of the channel divergence
$\boldsymbol{D}$ (directly follows from the superadditivity for states).
Hence, since $a_{2n}\geqslant a_{n}$ for all $n\in\mathbb{N}$, we also have
$a_{4n}\geqslant a_{2n}\geqslant a_{n}$, and more generally, for any
$k\in\mathbb{N}$, $a_{n2^{k}}\geqslant a_{n}$. By taking $k\to\infty$ and
using the fact that the limit of $\\{a_{n}\\}_{n\in{{\mathbb{N}}}}$ exists,
the assertion follows.  $\sqcap$$\sqcup$

A desirable property of quantum divergence is the additivity, which holds for
most of the divergences between states (e.g. Umegaki, Petz, Sandwiched).
However, such a property does not trivially extend to the channel divergences
(see e.g [FFRS20]). Nevertheless, the next lemma shows that the additivity of
amortized channel divergence follows from the additivity of the associated
state divergence.

###### Lemma 29

Let $\boldsymbol{D}$ be a quantum divergence for states. Let ${\cal
N}_{1},{\cal M}_{1}\in\text{\rm CPTP}(A_{1}\to B_{1})$ and ${\cal N}_{2},{\cal
M}_{2}\in\text{\rm CPTP}(A_{2}\to B_{2})$. If $\boldsymbol{D}$ is additive
under tensor product of states, then

$\displaystyle\boldsymbol{D}^{A}({\cal N}_{1}\otimes{\cal N}_{2}\|{\cal
M}_{1}\otimes{\cal M}_{2})=\boldsymbol{D}^{A}({\cal N}_{1}\|{\cal
M}_{1})+\boldsymbol{D}^{A}({\cal M}_{1}\|{\cal M}_{2}).$ (110)

###### Proof.

For any quantum state $\psi,\varphi\in\mathfrak{D}(RA_{1}A_{2})$, it holds

$\displaystyle\boldsymbol{D}({\cal N}_{1}\otimes{\cal N}_{2}(\psi)\|{\cal
M}_{1}\otimes{\cal M}_{2}(\varphi))$
$\displaystyle\leqslant\boldsymbol{D}^{A}({\cal N}_{1}\|{\cal
M}_{1})+\boldsymbol{D}({\cal N}_{2}(\psi)\|{\cal M}_{2}(\varphi))$ (111)
$\displaystyle\leqslant\boldsymbol{D}^{A}({\cal N}_{1}\|{\cal
M}_{1})+\boldsymbol{D}^{A}({\cal N}_{2}\|{\cal
M}_{2})+\boldsymbol{D}(\psi\|\varphi),$ (112)

where the two inequalities follow by using the definition of the amortized
channel divergence twice. Then move $\boldsymbol{D}(\psi\|\varphi)$ to the
l.h.s. and supremum over all input states $\psi,\varphi$, we have one
direction of the stated result. On the other hand, for any input states
$\rho_{1},\sigma_{1}\in\mathfrak{D}(R_{1}A_{1})$ and
$\rho_{2},\sigma_{2}\in\mathfrak{D}(R_{2}A_{2})$, we have

$\displaystyle\boldsymbol{D}^{A}({\cal N}_{1}\otimes{\cal N}_{2}\|{\cal
M}_{1}\otimes{\cal M}_{2})$ (113)
$\displaystyle\geqslant\sup_{\rho_{1},\rho_{2},\sigma_{1},\sigma_{2}}\Big{[}\boldsymbol{D}({\cal
N}_{1}\otimes{\cal N}_{2}(\rho_{1}\otimes\rho_{2})\|{\cal M}_{1}\otimes{\cal
M}_{2}(\sigma_{1}\otimes\sigma_{2}))-\boldsymbol{D}(\rho_{1}\otimes\rho_{2}\|\sigma_{1}\otimes\sigma_{2})\Big{]}$
(114)
$\displaystyle=\sup_{\rho_{1},\rho_{2},\sigma_{1},\sigma_{2}}\Big{[}\boldsymbol{D}({\cal
N}_{1}(\rho_{1})\|{\cal
M}_{1}(\sigma_{1})-\boldsymbol{D}(\rho_{1}\|\sigma_{1})\Big{]}+\Big{[}\boldsymbol{D}({\cal
N}_{2}(\rho_{2})\|{\cal
M}_{2}(\sigma_{2})-\boldsymbol{D}(\rho_{2}\|\sigma_{2})\Big{]}$ (115)
$\displaystyle=\sup_{\rho_{1},\sigma_{1}}\Big{[}\boldsymbol{D}({\cal
N}_{1}(\rho_{1})\|{\cal
M}_{1}(\sigma_{1})-\boldsymbol{D}(\rho_{1}\|\sigma_{1})\Big{]}+\sup_{\rho_{1},\sigma_{1}}\Big{[}\boldsymbol{D}({\cal
N}_{2}(\rho_{2})\|{\cal
M}_{2}(\sigma_{2})-\boldsymbol{D}(\rho_{2}\|\sigma_{2})\Big{]}$ (116)
$\displaystyle=\boldsymbol{D}^{A}({\cal N}_{1}\|{\cal
M}_{1})+\boldsymbol{D}^{A}({\cal N}_{2}\|{\cal M}_{2}),$ (117)

where the inequality follows as tensor product states are particular choices
of input states for $\boldsymbol{D}^{A}({\cal N}_{1}\otimes{\cal N}_{2}\|{\cal
M}_{1}\otimes{\cal M}_{2})$, the first equality follows by the additivity
assumption of $\boldsymbol{D}$.  $\sqcap$$\sqcup$

By the chain rules of Umegaki relative entropy [FFRS20] and Sandwiched Rényi
divergence [FF21], we know from Lemma 29 that $D^{\rm reg}$ and
$\widetilde{D}^{\rm reg}_{\alpha}$ ($\alpha>1$) are also additive under tensor
product of channels. The additivity of these divergences can be hard to prove
solely from their definitions.

The above result directly leads to the following order of relation:

###### Lemma 30

Let $\boldsymbol{D}$ be a quantum divergence for states. Let
$\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$. If $\boldsymbol{D}$ is
superadditive under tensor product of states, then

$\displaystyle\boldsymbol{D}({\cal N}\|{\cal M})\leqslant\boldsymbol{D}^{\rm
reg}({\cal N}\|{\cal M})\leqslant\boldsymbol{D}^{A}({\cal N}\|{\cal M}).$
(118)

###### Proof.

The first inequality follows from Lemma 28 by choosing $n=1$. We also have
that

$\displaystyle\frac{1}{n}\boldsymbol{D}({\cal N}^{\otimes n}\|{\cal
M}^{\otimes n})\leqslant\frac{1}{n}\boldsymbol{D}^{A}({\cal N}^{\otimes
n}\|{\cal M}^{\otimes n})\leqslant\boldsymbol{D}^{A}({\cal N}\|{\cal M})$
(119)

where the first inequality follows by definition and the second inequality
follows from (112). Taking $n\to\infty$, we have $\boldsymbol{D}^{\rm
reg}({\cal N}\|{\cal M})\leqslant\boldsymbol{D}^{A}({\cal N}\|{\cal M})$.
$\sqcap$$\sqcup$

##  Appendix B Some Useful Results

The first lemma is an analog of a result of Ogawa and Nagaoka [ON00] that was
originally used to establish the strong converse of quantum Stein’s lemma. A
similar result was proved by Brandão and Plenio for tensor product states
[BP10]. Here we extend it further to permutation-symmetric states.

###### Lemma 31

Let $\mu\in\mathbb{R}$ and $\rho_{n},\sigma_{n}\in\mathfrak{D}(A^{n})$ be
symmetric under permutations of the $n$ subsystems such that
${\operatorname{supp}}(\rho_{n})\subseteq{\operatorname{supp}}(\sigma_{n})$.
Then, for any $r\in\mathbb{R}$ and $s\in[0,1]$ the following relation holds

$\operatorname{Tr}\left(\rho_{n}-2^{\mu n}\sigma_{n}\right)_{+}\leqslant
2^{-nrs+\log\operatorname{Tr}[\rho^{1+s}_{n}]}+2^{-ns(\mu-r)+s|A|\log(1+n)+\log\operatorname{Tr}[\rho_{n}\sigma^{-s}_{n}]}\;.$
(120)

###### Proof.

Let $\Pi$ be the projection to the positive part of $\rho_{n}-2^{\mu
n}\sigma_{n}$ and $\Pi=\sum_{x=1}^{|A|^{n}}a_{x}\Pi_{x}$ be a decomposition of
$\Pi$ into orthogonal rank-one projectors, where $a_{x}\in\\{0,1\\}$ and
$\sum_{x}\Pi_{x}=I_{A^{n}}$ (i.e. the set $\\{\Pi_{x}\\}$ forms a von-Neumann
rank-one projective measurement). In general, this decomposition of $\Pi$ is
not unique, and the precise choice of $\\{\Pi_{x}\\}$ will be determined later
on in the proof. Finally, denote by
$p_{x}\coloneqq\operatorname{Tr}\left[\rho_{n}\Pi_{x}\right]$,
$q_{x}\coloneqq\operatorname{Tr}\left[\sigma_{n}\Pi_{x}\right]$ (note that
$p_{x}$ and $q_{x}$ depends on $n$), and let $\mathfrak{I}$ be the set of all
$x$ for which $p_{x}>2^{\mu n}q_{x}$. Since
$\left(\rho_{n}-2^{n\mu}\sigma_{n}\right)_{+}=\Pi\left(\rho_{n}-2^{n\mu}\sigma_{n}\right)\Pi$,
we have

$\displaystyle\operatorname{Tr}\left(\rho_{n}-2^{n\mu}\sigma_{n}\right)_{+}=\sum_{x}a_{x}\left(p_{x}-2^{\mu
n}q_{x}\right)\leqslant\sum_{x\in\mathfrak{I}}\left(p_{x}-2^{\mu
n}q_{x}\right)\leqslant\sum_{x\in\mathfrak{I}}p_{x}={\rm Pr}(\mathfrak{I})\;,$
(121)

where ${\rm Pr}(\mathfrak{I})$ is the probability of the set $\mathfrak{I}$
with respect to the probability distribution $\\{p_{x}\\}$. Note that the set
$\mathfrak{I}$ can be written as

$\mathfrak{I}=\left\\{x\;\mathrel{\mathop{\mathchar
58\relax}}\;\frac{1}{n}\log p_{x}>\mu+\frac{1}{n}\log q_{x}\right\\}\;.$ (122)

We would like to replace the set $\mathfrak{I}$ with two sets: one depends
solely on $p_{x}$, and the other only on $q_{x}$. This can be done in the
following way. For any $r\in\mathbb{R}$ define the two sets

$\mathfrak{I}^{(1)}\coloneqq\left\\{x\;\mathrel{\mathop{\mathchar
58\relax}}\;\frac{1}{n}\log p_{x}\geqslant
r\right\\}\quad\text{and}\quad\mathfrak{I}^{(2)}\coloneqq\left\\{x\;\mathrel{\mathop{\mathchar
58\relax}}\;\frac{1}{n}\log q_{x}\leqslant r-\mu\right\\}\;.$ (123)

Note that if $x\in\mathfrak{I}$ then either $x\in\mathfrak{I}^{(1)}$ or
$x\in\mathfrak{I}^{(2)}$. We therefore conclude that

$\operatorname{Tr}\left(\rho_{n}-2^{n\mu}\sigma_{n}\right)_{+}\leqslant{\rm
Pr}\left(\mathfrak{I}^{(1)}\right)+{\rm Pr}\left(\mathfrak{I}^{(2)}\right)\;.$
(124)

From Cramér’s theorem [DZ98] it follows that

$\displaystyle-\log{\rm
Pr}\left(\mathfrak{I}^{(1)}\right)\geqslant\sup_{s\in[0,1]}\left\\{nsr-\log\sum_{x}p_{x}^{1+s}\right\\}$
(125) $\displaystyle-\log{\rm
Pr}\left(\mathfrak{I}^{(2)}\right)\geqslant\sup_{s\in[0,1]}\left\\{ns(\mu-r)-\log\sum_{x}p_{x}q_{x}^{-s}\right\\}\;.$
(126)

We first bound (125) in terms of $\rho_{n}$. For this purpose, let
$\Delta\in\text{\rm CPTP}(A^{n}\to A^{n})$ be the completely dephasing map
(also a pinching map) $\Delta(\omega)\coloneqq\sum_{x}\Pi_{x}\omega\Pi_{x}$
defined on all $\omega\in\mathfrak{D}(A^{n})$. Then, the density matrix
$\Delta\left(\rho_{n}\right)$ is diagonal (in the basis that the operators
$\\{\Pi_{x}\\}$ project to) with components $\\{p_{x}\\}$ on its diagonal.
Hence, denoting by $\pi_{A^{n}}\mathrel{\mathop{\mathchar
58\relax}}=I_{A^{n}}/|A|^{n}$ the completely mixed state in
$\mathfrak{D}(A^{n})$, and by $\alpha\coloneqq 1+s$, we get by direct
calculation that

$\displaystyle-\log\sum_{x}p_{x}^{1+s}=n(\alpha-1)\log|A|-(\alpha-1)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\Delta(\rho_{n})\big{\|}\pi_{A^{n}}\right)\;.$ (127)

Since $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\Delta(\rho_{n})\big{\|}\pi_{A^{n}}\right)=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\Delta(\rho_{n})\big{\|}\Delta(\pi_{A^{n}})\right)\leqslant\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\rho_{n}\big{\|}\pi_{A^{n}}\right)$, we get

$\displaystyle-\log\sum_{x}p_{x}^{1+s}\geqslant
n(\alpha-1)\log|A|-(\alpha-1)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{D}_{\alpha}\left(\rho_{n}\big{\|}\pi_{A^{n}}\right)=-\log\operatorname{Tr}\left[\rho_{n}^{\alpha}\right]=-\log\operatorname{Tr}\left[\rho^{1+s}_{n}\right].$
(128)

Together with (125), this gives the first term on the r.h.s. of (120).

For the second term, observe that

$\displaystyle\sum_{x}p_{x}q_{x}^{-s}=\operatorname{Tr}\left[\Delta\left(\rho_{n}\right)\left(\Delta\left(\sigma_{n}\right)\right)^{-s}\right]=\operatorname{Tr}\left[\rho_{n}\left(\Delta\left(\sigma_{n}\right)\right)^{-s}\right]\;.$
(129)

We now estimate this term by utilizing the symmetry of $\rho_{n}$ and
$\sigma_{n}$. Since $\rho_{n}$ and $\sigma_{n}$ are symmetric under
permutations of the $n$ subsystems they can be expressed as

$\rho_{n}=\bigoplus_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}I^{B_{\lambda}}\otimes\rho^{C_{\lambda}}_{\lambda}\quad\text{and}\quad\sigma_{n}=\bigoplus_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}I^{B_{\lambda}}\otimes\sigma^{C_{\lambda}}_{\lambda}$
(130)

where $\lambda$ represents an irrep of the natural representation of the
permutation group $\mathcal{S}_{n}$ on $A^{n}$, and
$\rho_{\lambda},\sigma_{\lambda}\geqslant 0$. We therefore have

$\rho_{n}-2^{\mu n}\sigma_{n}=\bigoplus_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}I^{B_{\lambda}}\otimes\left(\rho^{C_{\lambda}}_{\lambda}-2^{\mu
n}\sigma_{\lambda}^{C_{\lambda}}\right)$ (131)

The condition
${\operatorname{supp}}(\rho_{n})\subseteq{\operatorname{supp}}(\sigma_{n})$
implies that without loss of generality we can assume that $\sigma_{n}>0$
(otherwise we can restrict our consideration to the subspace of
${\operatorname{supp}}(\sigma_{n})$ and embed $\rho_{n}$ in this space).
Therefore, under this assumption we have that each $\sigma_{\lambda}>0$. Let
$P_{\lambda}$ be the projector to the support of $\left(\rho_{\lambda}-2^{\mu
n}\sigma_{\lambda}\right)_{+}$, and let
$P_{\lambda}\coloneqq\sum_{j=1}^{|C_{\lambda}|^{n}}a_{\lambda,j}P_{\lambda,j}$
be a decomposition of $P_{\lambda}$ into orthogonal rank-one projectors, where
$a_{\lambda,j}\in\\{0,1\\}$ and $\sum_{j}P_{\lambda,j}=I^{C_{\lambda}}$.
Moreover, for each $\lambda\in{\rm Irr}(\mathcal{S}_{n})$ decompose
$I^{B_{\lambda}}\coloneqq\sum_{k=1}^{|B_{\lambda}|}|\psi_{\lambda,k}\rangle\langle\psi_{\lambda,k}|^{B_{\lambda}}$,
where $\\{|\psi_{\lambda,k}\rangle\\}_{k}$ forms an orthonormal basis of
$B_{\lambda}$. Finally, we denote by $x\coloneqq\\{\lambda,j,k\\}$ and take
$\Pi_{x}\coloneqq|\psi_{\lambda,k}\rangle\langle\psi_{\lambda,k}|^{B_{\lambda}}\otimes
P_{\lambda,j}^{C_{\lambda}}$. With this choice of $\Pi_{x}$ we get that

$\displaystyle\Delta(\sigma_{n})$ $\displaystyle=\bigoplus_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}\sum_{j,k}|\psi_{\lambda,k}\rangle\langle\psi_{\lambda,k}|^{B_{\lambda}}\otimes
P_{\lambda,j}^{C_{\lambda}}\sigma_{\lambda}^{C_{\lambda}}P_{\lambda,j}^{C_{\lambda}}$
(132) $\displaystyle=\bigoplus_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}I^{B_{\lambda}}\otimes\sum_{j}P_{\lambda,j}^{C_{\lambda}}\sigma_{\lambda}^{C_{\lambda}}P_{\lambda,j}^{C_{\lambda}}$
$\displaystyle=\bigoplus_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}I^{B_{\lambda}}\otimes\Delta_{\lambda}^{C_{\lambda}\to
C_{\lambda}}\left(\sigma_{\lambda}^{C_{\lambda}}\right)$

where each
$\Delta_{\lambda}(\cdot)\coloneqq\sum_{j}P_{\lambda,j}(\cdot)P_{\lambda,j}$ is
a completely dephasing map in $\text{\rm CPTP}(C_{\lambda}\to C_{\lambda})$.
Therefore,

$\displaystyle\sum_{x}p_{x}q_{x}^{-s}$
$\displaystyle=\operatorname{Tr}\left[\rho_{n}\left(\Delta\left(\sigma_{n}\right)\right)^{-s}\right]=\sum_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}|B_{\lambda}|\operatorname{Tr}\left[\rho_{\lambda}\big{(}\Delta_{\lambda}(\sigma_{\lambda})\big{)}^{-s}\right]\;.$
(133)

From the pinching inequality, for each $\lambda\in{\rm Irr}(\mathcal{S}_{n})$
we have
$\Delta_{\lambda}(\sigma_{\lambda})\geqslant\frac{1}{|C_{\lambda}|}\sigma_{\lambda}$.
Moreover, since the function $r\mapsto r^{\alpha}$ is operator anti-monotone
for $\alpha\in[-1,0]$ we get that

$\big{(}\Delta_{\lambda}(\sigma_{\lambda})\big{)}^{-s}\leqslant\left(\frac{1}{|C_{\lambda}|}\sigma_{\lambda}\right)^{-s}\;.$
(134)

Substituting this into (133) gives

$\sum_{x}p_{x}q_{x}^{-s}\leqslant\sum_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}|C_{\lambda}|^{s}|B_{\lambda}|\operatorname{Tr}\left[\rho_{\lambda}\sigma_{\lambda}^{-s}\right]\;.$
(135)

Now, since $C_{\lambda}$ can be viewed as a subspace of ${\rm Sym}^{n}(A)$,
its dimension cannot exceed that of ${\rm Sym}^{n}(A)$ which itself is bounded
by $(n+1)^{|A|}$. We therefore conclude that

$\displaystyle\sum_{x}p_{x}q_{x}^{-s}$
$\displaystyle\leqslant(n+1)^{s|A|}\sum_{\lambda\in{\rm
Irr}(\mathcal{S}_{n})}|B_{\lambda}|\operatorname{Tr}\left[\rho_{\lambda}\sigma_{\lambda}^{-s}\right]=(n+1)^{s|A|}\operatorname{Tr}\left[\rho_{n}\sigma_{n}^{-s}\right]\;.$
(136)

Together with (126), this gives the second term on the r.h.s. of (120).
$\sqcap$$\sqcup$

The next lemma shows that the eigenvalues of the output from $n$ use of a
positive definite channel are uniformly bounded by an exponential factor.

###### Lemma 32

Let $\mathcal{N}\in\text{\rm CPTP}(A\to B)$ and $\mathcal{N}>0$. Then, there
exists $b\in(0,1)$ such that for any $n\in\mathbb{N}$

$\max_{\rho\in\mathfrak{D}(R^{n}A^{n})}\left\|\mathcal{N}^{\otimes
n}\left(\rho_{R^{n}A^{n}}\right)\right\|_{\infty}\leqslant b^{n}\;.$ (137)

###### Proof.

Since $\mathcal{N}>0$ we have its Choi matrix $J_{{\cal N}}>0$. Then there
exists $\tau\in\mathfrak{D}(B)$ with $\|\tau\|_{\infty}<1$ (e.g. the maximally
mixed state) and its associated replacer channel ${\cal R}_{\tau}$ such that
$tJ_{{\cal N}}>J_{{\cal R}_{\tau}}$ for some $t\in(0,\infty)$. Equivalently,
we have $t{\cal N}>{\cal R}_{\tau}$. Set $\varepsilon\coloneqq 1/t$ and then
$\mathcal{M}\coloneqq(\mathcal{N}-\varepsilon\mathcal{R}_{\tau})/(1-\varepsilon)>0$;
in particular, $\mathcal{M}\in\text{\rm CPTP}(A\to B)$ and
$\mathcal{N}=(1-\varepsilon)\mathcal{M}+\varepsilon\mathcal{R}_{\tau}$.
Observe that

$\mathcal{N}^{\otimes n}=\sum_{k=0}^{n}{n\choose
k}(1-\varepsilon)^{k}\varepsilon^{n-k}\mathcal{F}_{n,k}$ (138)

where $\mathcal{F}_{n,k}\in\text{\rm CPTP}(A^{n}\to B^{n})$ is a uniform
convex combination of ${n\choose k}$ channels all having the form
$\mathcal{M}^{\otimes k}\otimes\mathcal{R}_{\tau}^{\otimes n-k}$ up to
permutations of the $n$ channels. Now, observe that

$\displaystyle\left\|\mathcal{M}^{\otimes k}\otimes\mathcal{R}_{\tau}^{\otimes
n-k}\left(\rho_{R^{n}A^{n}}\right)\right\|_{\infty}=\left\|\mathcal{M}^{\otimes
k}\left(\rho_{R^{n}A^{k}}\right)\otimes\tau^{\otimes
n-k}\right\|_{\infty}\leqslant\left\|\tau^{\otimes
n-k}\right\|_{\infty}=\|\tau\|_{\infty}^{n-k}\;.$ (139)

Note that the order that $\mathcal{N}$ and $\mathcal{R}_{\tau}$ appear in the
equation above does not effect this upper bound. Therefore, since
$\mathcal{F}_{n,k}$ is a convex combination of such channels we conclude that
also

$\left\|\mathcal{F}_{n,k}\left(\rho_{R^{n}A^{n}}\right)\right\|_{\infty}\leqslant\|\tau\|_{\infty}^{n-k}\;.$
(140)

Hence, for any $\rho\in\mathfrak{D}(R^{n}A^{n})$

$\displaystyle\left\|\mathcal{N}^{\otimes
n}\left(\rho_{R^{n}A^{n}}\right)\right\|_{\infty}$
$\displaystyle\leqslant\sum_{k=0}^{n}{n\choose
k}(1-\varepsilon)^{k}\varepsilon^{n-k}\left\|\mathcal{F}_{n,k}\left(\rho^{R^{n}A^{n}}\right)\right\|_{\infty}$
(141) $\displaystyle\leqslant\sum_{k=0}^{n}{n\choose
k}(1-\varepsilon)^{k}\varepsilon^{n-k}\|\tau\|_{\infty}^{n-k}$
$\displaystyle=\Big{(}1-\varepsilon+\|\tau\|_{\infty}\varepsilon\Big{)}^{n}\;.$

The proof is completed by taking $b\coloneqq
1-\varepsilon+\|\tau\|_{\infty}\varepsilon$ which is clearly in $(0,1)$.
$\sqcap$$\sqcup$

The next lemma shows that by utilizing the permutation symmetry of tensor
product channels we can restrict the optimal input states in the
discrimination strategies to be symmetric states. This reduces the problem
from the most general form to a particular one that can be tackled more
easily.

###### Lemma 33

Let ${\cal N},{\cal M}\in\text{\rm CPTP}(A\to B)$. For any $n\in\mathbb{N}$
there exists a pure state $|\varphi\rangle\in{\rm Sym}^{n}(\tilde{R}A)$ such
that

$\max_{\psi\in\mathfrak{D}(R^{n}A^{n})}D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}(\psi_{R^{n}A^{n}})\big{\|}\mathcal{M}^{\otimes
n}(\psi_{R^{n}A^{n}})\right)=D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}(\varphi_{\tilde{R}^{n}A^{n}})\big{\|}\mathcal{M}^{\otimes
n}(\varphi_{\tilde{R}^{n}A^{n}})\right).$ (142)

###### Proof.

First recall a variational expression of the hypothesis testing relative
entropy [BG17, Eq.(2)]

$\displaystyle D_{H}^{\varepsilon}(\rho\|\sigma)=-\log\max_{t\geqslant
0}\big{\\{}(1-\varepsilon)t-\operatorname{Tr}(t\rho-\sigma)_{+}\big{\\}}\;.$
(143)

Therefore, we have

$\displaystyle{D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)}=-\log\max_{t\geqslant
0}\left\\{(1-\varepsilon)t-\operatorname{Tr}\big{(}t\mathcal{N}^{\otimes
n}(\psi_{R^{n}A^{n}})-\mathcal{M}^{\otimes
n}(\psi_{R^{n}A^{n}})\big{)}_{+}\right\\}\;,$ (144)

for some state $\psi_{R^{n}A^{n}}\in\mathfrak{D}(R^{n}A^{n})$. Let

$\omega_{XR^{n}A^{n}}=\frac{1}{n!}\sum_{\pi\in\mathcal{S}_{n}}|\pi\rangle\langle\pi|_{X}\otimes
P_{\pi}\psi_{R^{n}A^{n}}P_{\pi}^{*}$ (145)

where $X$ is a ‘flag’ system of dimension $|X|=n!$. By construction, the
marginal state $\omega_{R^{n}A^{n}}$ is symmetric under permutations (i.e. has
support on ${\rm Sym}^{n}(RA)$), so there exists a symmetric purification of
$\omega_{R^{n}A^{n}}$ which we denote by $\varphi_{C^{n}R^{n}A^{n}}$, where
$C\cong RA$ [Ren05, Lemma 4.2.2]. Let $\omega_{DXR^{n}A^{n}}$ be a
purification of $\omega_{XR^{n}A^{n}}$ and thus also a purification of
$\omega_{R^{n}A^{n}}$. Since all purifications of a density matrix are related
via isometries, there exists an isometry $V_{C^{n}\to DX}$ such that

$\displaystyle\omega_{DXR^{n}A^{n}}=(V_{C^{n}\to
DX})\varphi_{C^{n}R^{n}A^{n}}(V_{C^{n}\to DX})^{\dagger}.$ (146)

Taking a partial trace of the system $D$ on both sides gives

$\omega_{XR^{n}A^{n}}=\mathcal{E}_{C^{n}\to
X}\left(\varphi_{C^{n}R^{n}A^{n}}\right)\;,$ (147)

where $\mathcal{E}(\cdot)=\operatorname{Tr}_{D}V(\cdot)V^{\dagger}\in\text{\rm
CPTP}(C^{n}\to X)$. Let $\tilde{R}\coloneqq CR$, then
$|\varphi_{\tilde{R}^{n}A^{n}}\rangle\in{\rm Sym}^{n}(\tilde{R}A)$ and

$\displaystyle\operatorname{Tr}\Big{(}t\mathcal{N}^{\otimes
n}\big{(}\varphi_{\tilde{R}^{n}A^{n}}\big{)}-\mathcal{M}^{\otimes
n}\big{(}\varphi_{\tilde{R}^{n}A^{n}}\big{)}\Big{)}_{+}$
$\displaystyle\geqslant\operatorname{Tr}\Big{(}t\mathcal{N}^{\otimes
n}\big{(}\omega_{XR^{n}A^{n}}\big{)}-\mathcal{M}^{\otimes
n}\big{(}\omega_{XR^{n}A^{n}}\big{)}\Big{)}_{+}$ (148)
$\displaystyle=\frac{1}{n!}\sum_{\pi\in\mathcal{S}_{n}}\operatorname{Tr}\big{(}t\mathcal{N}^{\otimes
n}\left(P_{\pi}\psi_{R^{n}A^{n}}P_{\pi}^{*}\right)-\mathcal{M}^{\otimes
n}(P_{\pi}\psi_{R^{n}A^{n}}P_{\pi}^{*})\Big{)}_{+}$ (149)
$\displaystyle=\operatorname{Tr}\big{(}t\mathcal{N}^{\otimes
n}(\psi_{R^{n}A^{n}})-\mathcal{M}^{\otimes n}(\psi_{R^{n}A^{n}})\big{)}_{+}$
(150)

where the first inequality follows from the data processing inequality of
$\operatorname{Tr}(\cdot)_{+}$ 111This can be easily seen from the equation
$\operatorname{Tr}(X)_{+}=(\|X\|_{1}+\operatorname{Tr}X)/2$ and the data
processing inequality of trace norm., the first equality follows from the
block diagonal structure of $\mathcal{N}^{\otimes
n}\big{(}\omega_{XR^{n}A^{n}}\big{)}-t\mathcal{M}^{\otimes
n}\big{(}\omega_{XR^{n}A^{n}}\big{)}$, the second equality follows because
$\operatorname{Tr}(\cdot)_{+}$ is unitary invariant and ${\cal N}^{\otimes
n},{\cal M}^{\otimes n}$ commute with permutations. Together with (144), we
can conclude that

$\displaystyle{D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)}$
$\displaystyle\leqslant-\log\max_{t\in\mathbb{R}}\left\\{(1-\varepsilon)t-\operatorname{Tr}\big{(}t\mathcal{N}^{\otimes
n}(\varphi_{\tilde{R}^{n}A^{n}})-\mathcal{M}^{\otimes
n}(\varphi_{\tilde{R}^{n}A^{n}})\big{)}_{+}\right\\}$ (151)
$\displaystyle={D_{H}^{\varepsilon}\left(\mathcal{N}^{\otimes
n}\left(\varphi_{\tilde{R}^{n}A^{n}}\right)\big{\|}\mathcal{M}^{\otimes
n}\left(\varphi_{\tilde{R}^{n}A^{n}}\right)\right)}.$

This completes the proof.  $\sqcap$$\sqcup$

Remark 1 We say that a quantum divergence, $\boldsymbol{D}$, satisfies _the
direct sum property_ if there exists a one-to-one function
$f\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}_{+}\to\mathbb{R}_{+}$ such
that for any pair of cq-states $\rho,\sigma\in\mathfrak{D}(XA)$ of the form
$\rho_{XA}\coloneqq\sum_{x}p_{x}|x\rangle\langle x|^{X}\otimes\rho_{x}^{A}$
and $\sigma^{XA}\coloneqq\sum_{x}p_{x}|x\rangle\langle
x|^{X}\otimes\sigma_{x}^{A}$ where $\\{p_{x}\\}$ is a probability distribution
and $\rho_{x},\sigma_{x}\in\mathfrak{D}(A)$, we have
$f^{-1}\left(\boldsymbol{D}\left(\rho_{XA}\big{\|}\sigma_{XA}\right)\right)=\sum_{x}p_{x}f^{-1}\left(\boldsymbol{D}\left(\rho_{x}^{A}\big{\|}\sigma_{x}^{A}\right)\right)$.
The direct sum property is essentially equivalent to the _general mean
property_ used by Rényi and Müller-Lennert et al. for its generalization to
the quantum case and holds for almost all the quantum divergences studied in
the literature. Following a similar proof, we can show that Lemma 33 holds for
any quantum divergence with the direct sum property.

##  Appendix C Towards the AEP of Max-relative Entropy under Channel Smoothing

For any $\mathcal{N}\in\text{\rm CPTP}(A\to B)$, let

$\displaystyle\mathfrak{B}_{\text{cptp}}^{\varepsilon}(\mathcal{N})\coloneqq\Big{\\{}\mathcal{N}^{\prime}\in\text{\rm
CPTP}(A\to B)\;\mathrel{\mathop{\mathchar
58\relax}}\;\frac{1}{2}\left\|\mathcal{N}^{\prime}-\mathcal{N}\right\|_{\diamond}\leqslant\varepsilon\Big{\\}}$
(152)
$\displaystyle\mathfrak{B}_{\text{cp}}^{\varepsilon}(\mathcal{N})\coloneqq\Big{\\{}\mathcal{N}^{\prime}\in\text{\rm
CP}(A\to B)\;\mathrel{\mathop{\mathchar
58\relax}}\;\frac{1}{2}\left\|\mathcal{N}^{\prime}-\mathcal{N}\right\|_{\diamond}\leqslant\varepsilon\Big{\\}}\;.$
(153)

Note that the second “Ball” of channels contains trace increasing channels. We
define

$\displaystyle
D_{\max}^{\varepsilon,\diamond,\text{cptp}}(\mathcal{N}\|\mathcal{M})$
$\displaystyle\coloneqq\inf_{\mathcal{N}^{\prime}\in\mathfrak{B}_{\text{cptp}}^{\varepsilon}(\mathcal{N})}D_{\max}\left(\mathcal{N}^{\prime}{}\big{\|}\mathcal{M}\right),$
(154) $\displaystyle
D_{\max}^{\varepsilon,\diamond,\text{cp}}(\mathcal{N}\|\mathcal{M})$
$\displaystyle\coloneqq\inf_{\mathcal{N}^{\prime}\in\mathfrak{B}_{\text{cp}}^{\varepsilon}(\mathcal{N})}D_{\max}\left(\mathcal{N}^{\prime}\big{\|}\mathcal{M}\right)\;,$

where
$D_{\max}\left(\mathcal{N}\big{\|}\mathcal{M}\right)\mathrel{\mathop{\mathchar
58\relax}}=D_{\max}(J_{{\cal N}}\|J_{{\cal M}})$. Recall also the definitions

$\displaystyle
D_{\max}^{\varepsilon}(\mathcal{N}\|\mathcal{M})\coloneqq\sup_{\psi\in\mathfrak{D}(RA)}D_{\max}^{\varepsilon}\left(\mathcal{N}^{A\to
B}(\psi^{RA})\big{\|}\mathcal{M}^{A\to B}(\psi^{RA})\right).$ (155)

It is easy to see from the definitions that

$D_{\max}^{\varepsilon}(\mathcal{N}\|\mathcal{M})\leqslant
D_{\max}^{\varepsilon,\diamond,\text{cptp}}(\mathcal{N}\|\mathcal{M})\quad\text{and}\quad
D_{\max}^{\varepsilon,\diamond,\text{cp}}(\mathcal{N}\|\mathcal{M})\leqslant
D_{\max}^{\varepsilon,\diamond,\text{cptp}}(\mathcal{N}\|\mathcal{M})\;.$
(156)

###### Lemma 34

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ and
$\varepsilon\in(0,\frac{1}{2})$. Then,

$\displaystyle
D_{\max}^{\varepsilon,\diamond,\text{cp}}(\mathcal{N}\|\mathcal{M})\geqslant
D_{\max}^{2\varepsilon}(\mathcal{N}\|\mathcal{M})+\log(1-2\varepsilon).$ (157)

###### Proof.

Let $\psi\in\mathfrak{D}(RA)$ be such that
$D_{\max}^{2\varepsilon}(\mathcal{N}\|\mathcal{M})=D_{\max}^{2\varepsilon}\left(\mathcal{N}(\psi)\|\mathcal{M}(\psi)\right)$.
Set $\rho\coloneqq\mathcal{N}(\psi)$, $\sigma\coloneqq\mathcal{M}(\psi)$. Let
$\mathcal{N}^{\prime}\in\text{\rm CP}(A\to B)$ be such that
$D_{\max}^{\varepsilon,\diamond,\text{cp}}(\mathcal{N}\|\mathcal{M})=D_{\max}\left(\mathcal{N}^{\prime}\|\mathcal{M}\right)$.
Denote $\omega\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{N}^{\prime}(\psi)$,
$\tilde{\omega}\mathrel{\mathop{\mathchar
58\relax}}=\omega/\operatorname{Tr}[\omega]$ and
$s\coloneqq\operatorname{Tr}\left[\omega\right]$. Then,

$\displaystyle
D_{\max}^{\varepsilon,\diamond,\text{cp}}(\mathcal{N}\|\mathcal{M})=D_{\max}\left(\mathcal{N}^{\prime}\|\mathcal{M}\right)=D_{\max}\left(\omega\|\sigma\right)=D_{\max}\left(\tilde{\omega}\|\sigma\right)+\log
s.$ (158)

Observe that since
$\frac{1}{2}\left\|\mathcal{N}^{\prime}-\mathcal{N}\right\|_{\diamond}\leqslant\varepsilon$
we have in particular

$\frac{1}{2}\left\|s\tilde{\omega}-\rho\right\|_{1}\leqslant\varepsilon\;.$
(159)

By the data processing of trace norm, the above inequality also implies that
$\frac{1}{2}|s-1|\leqslant\varepsilon$. Therefore,

$\displaystyle\frac{1}{2}\left\|\tilde{\omega}-\rho\right\|_{1}$
$\displaystyle\leqslant\frac{1}{2}\left\|\tilde{\omega}-s\tilde{\omega}\right\|_{1}+\frac{1}{2}\left\|s\tilde{\omega}-\rho\right\|_{1}\leqslant
2\varepsilon\;.$ (160)

This implies

$\displaystyle D_{\max}\left(\tilde{\omega}\big{\|}\sigma\right)\geqslant
D_{\max}^{2\varepsilon}\left(\rho\|\sigma\right)=D_{\max}^{2\varepsilon}(\mathcal{N}\|\mathcal{M}).$
(161)

Continuing with (158), we get that

$\displaystyle
D_{\max}^{\varepsilon,\diamond,\text{cp}}(\mathcal{N}\|\mathcal{M})\geqslant
D_{\max}\left(\tilde{\omega}\big{\|}\sigma\right)+\log\left(1-2\varepsilon\right)\geqslant
D_{\max}^{2\varepsilon}(\mathcal{N}\|\mathcal{M})+\log\left(1-2\varepsilon\right)\;.$
(162)

This completes the proof.  $\sqcap$$\sqcup$

Combining Lemma 34 and Theorem 13, we get

###### Lemma 35

Let $\mathcal{N},\mathcal{M}\in\text{\rm CPTP}(A\to B)$ with ${\cal N}>0$ and
$\varepsilon\in(0,\frac{1}{2})$. Then Conjecture 12 implies

$\liminf_{n\to\infty}\frac{1}{n}D_{\max}^{\varepsilon,\diamond,\text{cp}}\left(\mathcal{N}^{\otimes
n}\big{\|}\mathcal{M}^{\otimes n}\right)\geqslant D^{{\rm
reg}}(\mathcal{N}\|\mathcal{M})\;.$ (163)

The other direction of this result remains an open question as well.