theoremTheorem

#  Optimal Second-Order Rates for
Quantum Soft Covering and Privacy Amplification

Yu-Chen Shen1, Li Gao5, and Hao-Chung Cheng1,2,3,4 1Department of Electrical
Engineering and Graduate Institute of Communication Engineering,
National Taiwan University, Taipei 106, Taiwan (R.O.C.)
2Department of Mathematics, National Taiwan University
3Center for Quantum Science and Engineering, National Taiwan University
4Hon Hai (Foxconn) Quantum Computing Center, New Taipei City 236, Taiwan
5Department of Mathematics, University of Houston, Houston, TX 77204, USA
<EMAIL_ADDRESS>
###### Abstract.

We study quantum soft covering and privacy amplification against quantum side
information. The former task aims to approximate a quantum state by sampling
from a prior distribution and querying a quantum channel. The latter task aims
to extract uniform and independent randomness against quantum adversaries. For
both tasks, we use trace distance to measure the closeness between the
processed state and the ideal target state. We show that the minimal amount of
samples for achieving an $\varepsilon$-covering is given by the
$(1-\varepsilon)$-hypothesis testing information (with additional logarithmic
additive terms), while the maximal extractable randomness for an
$\varepsilon$-secret extractor is characterized by the conditional
$(1-\varepsilon)$-hypothesis testing entropy.

When performing independent and identical repetitions of the tasks, our one-
shot characterizations lead to tight asymptotic expansions of the above-
mentioned operational quantities. We establish their second-order rates given
by the quantum mutual information variance and the quantum conditional
information variance, respectively. Moreover, our results extend to the
moderate deviation regime, which are the optimal asymptotic rates when the
trace distances vanish at sub-exponential speed. Our proof technique is direct
analysis of trace distance without smoothing.

##  1\. Introduction

Questing the optimal rates for information-processing tasks is a core problem
in classical and quantum information theory [1, 2, 3, 4, 5, 6, 7, 8, 9].
Nowadays, considerable research focus has shifted from the first-order
characterization of the optimal rates to the _second-order_ quantities in the
asymptotic expansions of the optimal rates in coding blocklengths [10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20]. Such second-order terms quantifying how much
extra cost one has to pay in non-asymptotic scenarios are of significant
importance both in theory and practice.

Indeed, much progress has been made in deriving the exact second-order rates
for numerous quantum information-theoretic protocols. Nevertheless, this
problem remains open for certain tasks such as _privacy amplification against
quantum side information_ (or called _randomness extraction_) [21, 15], where
the operational quantities usually being used as a security criterion is the
trace distance [22, 21, 23, 24, 19, 25, 26, 27]. This manifests the fact that
existing analysis on operational quantities in terms of the trace distance as
in various quantum information-theoretic tasks [28, 29, 19, 30, 31, 32] still
has room for improvement. Hence, this problem will be the main focus of this
work. Moreover, we hope the proposed analysis on the trace distance would shed
new lights on the _one-shot quantum information theory_ [19].

In this paper, we study two tasks. The first task is privacy amplification
against quantum side information [21, 33, 34, 35, 36, 37, 19, 26]. Suppose
that a classical source $X$ at Alice’s disposal may be correlated with a
quantum adversary Eve, which can be modelled as a joint classical-quantum
(c-q) state $\rho_{XE}$. The goal of Alice is to extract from $X$ as much
uniform randomness as possible that is independent of Eve. Due to operational
motivation of composability [22, 21, 24, 34, 19], the _trace distance_ is
usually adopted as the security criterion to measure how far the extracted
state is from the perfectly uniform and independent randomness, i.e.

$\displaystyle\Delta(X{\,|\,}E)_{\rho}:=\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}\left(\rho_{XE}\right)-\frac{\mathds{1}_{Z}}{|\mathcal{Z}|}\otimes\rho_{E}\right\|_{1},$
(1.1)

where $\mathcal{R}^{h}$ denotes the random hash function applied by Alice, and
$\|\cdot\|_{1}$ is the Schatten $1$-norm. A randomness extractor satisfying
$\Delta(X{\,|\,}E)_{\rho}\leq\varepsilon$ is then said to be _$\varepsilon$
-secret_. We define $\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$ as the _maximal
extractable randomness_ $|\mathcal{Z}|$ for $\varepsilon$-secret randomness
extractors.

The second task studied in this work is _quantum soft covering_ [38]. Consider
a c-q state $\rho_{XB}:=\sum_{x\in\mathcal{X}}p_{X}(x)|x\rangle\langle
x|\otimes\rho_{B}^{x}$. The goal of quantum soft covering is to approximate
the marginal state $\rho_{B}$ using a random codebook $\mathcal{C}$ with
certain size $|\mathcal{C}|$. Here, the codewords in $\mathcal{C}$ are
independently sampled from the distribution $p_{X}$; through the c-q channel
$x\mapsto\rho_{B}^{x}$, one may construct an approximation state
$\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{B}^{x}$ to accomplish the
goal. Again, the trace distance is used as the figure of merit to measure
closeness, i.e.

$\displaystyle\Delta(X{\,:\,}B)_{\rho}:=\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{B}^{x}-\rho_{B}\right\|_{1}.$
(1.2)

We say that the random codebook $\mathcal{C}$ achieves an _$\varepsilon$
-covering_ if it satisfies $\Delta(X{\,:\,}B)_{\rho}\leq\varepsilon$. We then
define the _$\varepsilon$ -covering number_
$M^{\varepsilon}(X{\,:\,}E)_{\rho}$ as the _minimum random codebook size_
$|\mathcal{C}|$ to realize an _$\varepsilon$ -covering_.

Our main result is to provide _one-shot characterizations_ for both the
operational quantities $\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$ and
$M^{\varepsilon}(X{\,:\,}E)_{\rho}$. For privacy amplification, we show that
for every $0<\varepsilon<1$, the maximal extractable randomness using strongly
$2$-universal hash functions is characterized by the _$(1-\varepsilon)$
-conditional hypothesis testing entropy_
$H_{\textnormal{h}}^{1-\varepsilon}(X{\,|\,}E)_{\rho}$ [15]:

$\displaystyle\log\ell^{\varepsilon}(X{\,|\,}E)_{\rho}\approx
H_{\textnormal{h}}^{1-\varepsilon\pm\delta}(X{\,|\,}E)_{\rho}\,.$ (1.3)

Here, “$\approx$” means equality up to some logarithmic additive terms, and
$\delta$ is a parameter that be chosen for optimization (see Theorem 10 for
the precise statement, and see Section 2 for detailed definitions). With a
similar flavor, we prove that for every $0<\varepsilon<1$, the minimal random
codebook size for quantum soft covering is characterized by the
_$(1-\varepsilon)$ -conditional hypothesis testing information_:
$I_{\textnormal{h}}^{1-\varepsilon}(X{\,:\,}B)_{\rho}$ [39]:

$\displaystyle\log M^{\varepsilon}(X{\,:\,}B)_{\rho}\approx
I_{\textnormal{h}}^{1-\varepsilon\pm\delta}(X{\,:\,}B)_{\rho}$ (1.4)

(see Theorem 13 for the precise statement). Contrary to the previous studies,
the established one-shot entropic characterizations established in this work
are not smoothed min- and max-entropies [21, 40, 41, 42, 15, 43].

In the scenario that the underlying states are identical and independently
distributed, the established one-shot characterizations lead to the following
second-order asymptotic expansions of the optimal rates, respectively
(Propositions 11 and 14):

$\displaystyle\log\ell^{\varepsilon}(X^{n}{\,|\,}E^{n})_{\rho^{\otimes n}}$
$\displaystyle=nH(X{\,|\,}E)_{\rho}+\sqrt{nV(X{\,|\,}E)_{\rho}}\,\Phi^{-1}(\varepsilon)+O\left(\log
n\right);$ (1.5) $\displaystyle\log
M^{\varepsilon}(X^{n}{\,:\,}B^{n})_{\rho^{\otimes n}}$
$\displaystyle=nI(X{\,:\,}B)_{\rho}-\sqrt{nV(X{\,:\,}B)}\,\Phi^{-1}(\varepsilon)+O\left(\log
n\right).$ (1.6)

Here, the first-order terms are the conditional quantum entropy
$H(X{\,|\,}E)_{\rho}$ and the quantum mutual information
$I(X{\,:\,}B)_{\rho}$, whereas the second-order rates that can be expressed as
the quantum conditional information variance $V(X{\,|\,}E)$ and the quantum
mutual information variance $V(X{\,:\,}B)$; and $\Phi^{-1}$ is the inverse of
the cumulative normal distribution.

Furthermore, our results extend to the _moderate deviation regime_ [44, 45].
Namely, we derive both the optimal rates when the trace distances approach
zero sub-exponentially (Propositions 12 and 15):

$\displaystyle\frac{1}{n}\log\ell^{\varepsilon_{n}}(X^{n}{\,|\,}E^{n})_{\rho^{\otimes
n}}$
$\displaystyle=H(X{\,|\,}E)_{\rho}-\sqrt{2V(X{\,|\,}E)}\,a_{n}+o\left(a_{n}\right);$
(1.7) $\displaystyle\frac{1}{n}\log
M^{\varepsilon_{n}}(X^{n}{\,:\,}B^{n})_{\rho^{\otimes n}}$
$\displaystyle=I(X{\,:\,}B)_{\rho}+\sqrt{2V(X{\,:\,}B)}\,a_{n}+o\left(a_{n}\right).$
(1.8)

Here, $(a_{n})_{n\in\mathds{N}}$ is any moderate sequence satisfying $a_{n}\to
0$ and $na_{n}^{2}\to\infty$; and
$\varepsilon_{n}:=\mathrm{e}^{-na_{n}^{2}}\to 0$ .

The rest of the paper is organized as follows. In Section 2, we introduce
notations and auxiliary lemmas that will be used in our derivations. Section 3
presents the one-shot and second-order characterizations of privacy
amplification against quantum side information and Section 4 is devoted to
quantum soft covering and its one-shot and second-order characterizations. We
conclude our paper in Section 5.

##  2\. Notation and Auxiliary Lemmas

For an integer $M\in\mathds{N}$, we denote $[M]:=\\{1,\ldots,M\\}$. We use
‘$\wedge$’ to indicate ‘minimum value’ between two scalars or the conjunction
‘and’ between two statements. We use $\mathds{1}_{A}$ to denote the indicator
function for a condition $A$. The density operators considered in this paper
are positive semi-definite operators with unit trace. For a trace-class
operator $H$, the trace class norm (also called Schatten-$1$ norm) is defined
by

$\displaystyle\|H\|_{1}:=\operatorname{Tr}\left[\sqrt{H^{\dagger}H}\right].$
(2.1)

For positive semi-definite operator $K$ and positive operator $L$, we use the
following short notation for the _noncommutative quotient_.

$\displaystyle\frac{K}{L}:=L^{-\frac{1}{2}}KL^{-\frac{1}{2}}\,.$ (2.2)

We use $\mathds{E}_{x\sim p_{X}}$ to stand for taking expectation where the
underlying random variable is $x$ with probability distribution $p_{X}$ (with
finite support), e.g.
$\rho_{XB}=\sum_{x\in\mathcal{X}}p_{X}(x)|x\rangle\langle
x|\otimes\rho_{B}^{x}\equiv\mathds{E}_{x\sim p_{X}}\left(|x\rangle\langle
x|\otimes\rho_{B}^{x}\right)$.

For density operators $\rho$ and $\sigma$, we define the
$\varepsilon$-_information spectrum divergence_ [9, 46] as

$\displaystyle D_{\text{s}}^{\varepsilon}(\rho\parallel\sigma)$
$\displaystyle:=\sup_{c\in\mathds{R}}\left\\{\log
c:\operatorname{Tr}\left[\rho\left\\{\rho\leq
c\sigma\right\\}\right]\leq\varepsilon\right\\},$ (2.3)

and the $\varepsilon$-_hypothesis testing divergence_ [15, 47, 14] as

$\displaystyle D_{\text{h}}^{\varepsilon}(\rho\,\|\,\sigma):=\sup_{0\leq
T\leq\mathds{1}}\left\\{-\log\operatorname{Tr}[\sigma
T]:\operatorname{Tr}[\rho T]\geq 1-\varepsilon\right\\}.$ (2.4)

For positive semi-definite operator $\rho$ and positive definite operator
$\sigma$, we define the _collision divergence_ 111Note that for unnormalized
$\rho$, the collision divergence is usually defined as
$\log\operatorname{Tr}[(\sigma^{-\nicefrac{{1}}{{4}}}\rho\sigma^{-\nicefrac{{1}}{{4}}})^{2}]-\log\operatorname{Tr}[\rho]$.
However, we do not use such a definition for notational convenience. Indeed,
our derivations will rely on the joint convexity of $\exp D_{2}^{*}$ (see
Lemma 4) which holds for positive semi-definite operator $\rho$ and the
definition given in (2.5) as well. [21] as

$\displaystyle
D^{*}_{2}(\rho\,\|\,\sigma):=\log\operatorname{Tr}\left[\left(\sigma^{-\frac{1}{4}}\rho\sigma^{-\frac{1}{4}}\right)^{2}\right].$
(2.5)

The _quantum relative entropy_ [48, 49] and _quantum relative entropy
variance_ [15, 14] for density operator $\rho$ and positive definite operator
$\sigma$ are defined as

$\displaystyle D(\rho\,\|\,\sigma)$
$\displaystyle:=\operatorname{Tr}\left[\rho\left(\log\rho-\log\sigma\right)\right];$
(2.6) $\displaystyle V(\rho\,\|\,\sigma)$
$\displaystyle:=\operatorname{Tr}\left[\rho\left(\log\rho-\log\sigma\right)^{2}\right]-\left(D(\rho\,\|\,\sigma)\right)^{2}.$
(2.7)

By a classical-quantum state
$\rho_{XB}=\sum_{x\in\mathcal{X}}p_{X}(x)|x\rangle\langle
x|\otimes\rho_{B}^{x}$, we mean $p_{X}$ is a probability mass function on
$\mathcal{X}$ and each $\rho_{B}^{x}$ is a density operator on system $B$. We
define the $\varepsilon$-_hypothesis testing information_ [39] and the
_conditional $\varepsilon$-hypothesis testing entropy_ [15]222Note that we did
not optimize the second argument but instead just put $\rho_{B}$; hence our
definitions are slightly different from that proposed in Refs. [15, 39],
respectively, as

$\displaystyle I_{\text{h}}^{\varepsilon}(X{\,:\,}B)_{\rho}$
$\displaystyle:=D_{\text{h}}^{\varepsilon}\left(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B}\right);\quad
H_{\text{h}}^{\varepsilon}(X{\,|\,}B)_{\rho}:=-D_{\text{h}}^{\varepsilon}\left(\rho_{XB}\,\|\,\mathds{1}_{X}\otimes\rho_{B}\right).$
(2.8)

We note that both the quantities
$I_{\text{h}}^{\varepsilon}(X{\,:\,}B)_{\rho}$ and
$H_{\text{h}}^{\varepsilon}(X{\,|\,}B)_{\rho}$ can be formulated as a semi-
definite optimization problem [15], [31, §1]. The _quantum mutual information_
, _quantum conditional entropy_ , _quantum mutual information variance_ , and
the _quantum conditional information variance_ of $\rho_{XB}$ are defined,
respectively as

$\displaystyle I(X{\,:\,}B)_{\rho}$
$\displaystyle:=D(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})\,;\quad
H(X{\,|\,}B)_{\rho}:=-D(\rho_{XB}\,\|\,\mathds{1}_{X}\otimes\rho_{B})\,;$
(2.9) $\displaystyle V(X{\,:\,}B)_{\rho}$
$\displaystyle:=V(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})\,;\quad
V(X{\,|\,}B)_{\rho}:=V(\rho_{XB}\,\|\,\mathds{1}_{X}\otimes\rho_{B})\,.$
(2.10)

For any self-adjoint operator $H$ with eigenvalue decomposition
$H=\sum_{i}\lambda_{i}|e_{i}\rangle\langle e_{i}|$, we define the set
$\textnormal{{spec}}(H):=\\{\lambda_{i}\\}_{i}$ to be the eigenvalues of $H$,
and $|\textnormal{{spec}}(H)|$ to be the number of distinct eigenvalues of
$H$. We define the _pinching map_ with respect to $H$ as

$\displaystyle\mathscr{P}_{H}[L]:L\mapsto\sum_{i=1}e_{i}Le_{i}\,,$ (2.11)

where $e_{i}$ is the spectrum projection onto $i$th distinct eigenvalues.

######  Lemma 1 (Pinching inequality [50]).

For every $d$-dimensional self-adjoint operator $H$ and positive semi-definite
operator $L$,

$\displaystyle\mathscr{P}_{H}[L]\geq\frac{1}{|\textnormal{{spec}}(H)|}L\,.$
(2.12)

Moreover, it holds that for every $n\in\mathds{N}$,

$\displaystyle\left|\textnormal{{spec}}\left(H^{\otimes
n}\right)\right|\leq(n+1)^{d-1}\,.$ (2.13)

We have the following relation between divergences that will be used in our
proofs.

######  Lemma 2 (Relation between divergences [15, Lemma 12, Proposition 13,
Theorem 14]).

For every density operator $\rho$, positive semi-definite operator $\sigma$,
$0<\varepsilon<1$, and $0<\delta<1-\varepsilon$, we have

$\displaystyle
D_{\textnormal{s}}^{\varepsilon-\delta}\left(\mathscr{P}_{\sigma}[\rho]\,\|\,\sigma\right)$
$\displaystyle\leq
D_{\textnormal{h}}^{\varepsilon+\delta}(\rho\,\|\,\sigma)+\log|\textnormal{{spec}}(\sigma)|+2\log\delta\,;$
(2.14) $\displaystyle D_{\textnormal{h}}^{\varepsilon}(\rho\,\|\,\sigma)$
$\displaystyle\leq
D_{\textnormal{s}}^{\varepsilon+\delta}(\rho\,\|\,\sigma)-\log\delta\,.$
(2.15)

######  Lemma 3 (Lower bound on the collision divergence [51, Theorem 3]).

For every $0<\eta<1$ and $\lambda_{1},\lambda_{2}>0$, density operator $\rho$,
and positive semi-definite $\sigma$, we have333Ref. [51, Thm. 3] is stated for
$\lambda_{1}=\lambda\in(0,1)$ and $\lambda_{2}=1-\lambda$. We remark that the
result applies to the case $\lambda_{1},\lambda_{2}>0$ by following the same
proof.

$\displaystyle\exp
D_{2}^{*}\left(\rho\,\|\,\lambda_{1}\rho+\lambda_{2}\sigma\right)\geq\frac{1-\eta}{\lambda_{1}+\lambda_{2}\cdot\mathrm{e}^{-D_{\text{s}}^{\eta}(\rho\,\|\,\sigma)}}\,.$
(2.16)

######  Lemma 4 (Joint convexity [52, Proposition 3], [53, 16]).

The map

$\displaystyle(\rho,\sigma)\mapsto\exp D_{2}^{*}(\rho\,\|\,\sigma)$ (2.17)

is jointly convex on all positive semi-definite operators $\rho$ and positive
definite operators $\sigma$.

######  Lemma 5 (Variational formula of the trace distance [54, 55], [28,
§9]).

For density operators $\rho$ and $\sigma$,

$\displaystyle\frac{1}{2}\left\|\rho-\sigma\right\|_{1}=\sup_{0\leq\Pi\leq\mathds{1}}\operatorname{Tr}\left[\Pi(\rho-\sigma)\right].$
(2.18)

We list some basic properties of the noncommutative quotient introduced in
(2.2) as follows. Since they follow straightforwardly from basic matrix
theory, we will use them in our derivations without proofs.

######  Lemma 6 (Properties of the noncommutative quotient).

The noncommutative quotient defined in (2.2) satisfies the following:

  1. (a)

$\displaystyle\frac{A}{B}\geq 0$ for all $A\geq 0$ and $B>0$;

  2. (b)

$\displaystyle\frac{A+B}{C}=\frac{A}{C}+\frac{B}{C}$ for all $A,B\geq 0$ and
$C>0$;

  3. (c)

$\displaystyle\frac{A}{A+B}\leq\mathds{1}$ for all $A\geq 0$ and $B>0$;

  4. (d)

$\displaystyle\operatorname{Tr}\left[A\frac{B}{C}\right]=\operatorname{Tr}\left[B\frac{A}{C}\right]$
for all $A,B\geq 0$ and $C>0$.

  5. (e)

$\displaystyle\operatorname{Tr}\left[A\frac{A}{B}\right]=\exp
D_{2}^{*}(A\,\|\,B)$ for all $A\geq 0$ and $B>0$.

######  Lemma 7 (Jensen’s inequalities [56],[57]).

Let $\Phi$ be a unital positive linear map between two spaces of bounded
operators (possibly with different dimensions), and let $f$ be an operator
concave function. Then for every self-adjoint operator $H$, one has

$\displaystyle\Phi(f(H))\leq f(\Phi(H))\,.$ (2.19)

For examples, the unital positive linear map $\Phi$ considered in this paper
include (i) taking expectation $\mathds{E}$ on (possibly matrix-valued) random
variables; (ii) the pinching map $\mathscr{P}$; and (iii) the functional
$H\mapsto\operatorname{Tr}[\rho H]\in[0,1]$ for some density operator $\rho$.

######  Lemma 8 (Second-order expansion [15, 14]).

For every density operator $\rho$, positive definite operator $\sigma$,
$0<\varepsilon<1$, and $\delta=O(\nicefrac{{1}}{{\sqrt{n}}})$, we have the
following expansion444 We note that Lemmas 8 and 9 were originally stated for
normalized $\sigma$ [15, 14, 45]. They hold for positive definite operator
$\sigma$ as well by observing that
$D_{\text{h}}^{\varepsilon}(\rho\,\|\,\lambda\sigma)=D_{\text{h}}^{\varepsilon}(\rho\,\|\,\sigma)-\log\lambda$,
$D(\rho\,\|\,\lambda\sigma)=D(\rho\,\|\,\sigma)-\log\lambda$, and
$V(\rho\,\|\,\lambda\sigma)=V(\rho\,\|\,\sigma)$ for all $\lambda>0$. :

$\displaystyle D_{\textnormal{h}}^{\varepsilon\pm\delta}\left(\rho^{\otimes
n}\,\|\,\sigma^{\otimes
n}\right)=nD\left(\rho\,\|\,\sigma\right)+\sqrt{nV\left(\rho\,\|\,\sigma\right)}\Phi^{-1}(\varepsilon)+O\left(\log
n\right),$ (2.20)

where $\Phi$ is the cumulative normal distribution
$\Phi(u):=\int_{-\infty}^{u}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{1}{2}t^{2}}\mathrm{d}t$,
and its inverse
$\Phi^{-1}(\varepsilon):=\sup\\{u\mid\Phi(u)\leq\varepsilon\\}$.

######  Lemma 9 (Moderate deviations [45, Theorem 1]).

Let $(a_{n})_{n\in\mathds{N}}$ be a moderate sequence satisfying

$\displaystyle\lim_{n\to\infty}a_{n}=0\ ,\
\lim_{n\to\infty}na_{n}^{2}=\infty\hskip 2.84544pt.$ (2.21)

and let $\varepsilon_{n}:=\mathrm{e}^{-na_{n}^{2}}$. For any density operator
$\rho$ and positive definite operator $\sigma$, the following asymptotic
expansions hold 555 We note that the $\varepsilon$-hypothesis testing
divergence $D_{\textnormal{h}}^{\varepsilon}$ used in [45] has an additional
$\log(1-\varepsilon)$ term than our definition in (2.4). Nonetheless, this
does not affect the moderate deviation results since the additional terms are
$\frac{1}{n}\log(1-\varepsilon_{n})=o(a_{n})$ and
$\frac{1}{n}\log\varepsilon_{n}=-a_{n}^{2}=o(a_{n})$ for any moderate sequence
$(a_{n})_{n\in\mathds{N}}$. :

$\displaystyle\begin{dcases}\frac{1}{n}D_{\textnormal{h}}^{1-\varepsilon_{n}}\left(\rho^{\otimes
n}\,\|\,\sigma^{\otimes
n}\right)=D\left(\rho\,\|\,\sigma\right)+\sqrt{2V\left(\rho\,\|\,\sigma\right)}a_{n}+o\left(a_{n}\right);\\\
\frac{1}{n}D_{\textnormal{h}}^{\varepsilon_{n}}\left(\rho^{\otimes
n}\,\|\,\sigma^{\otimes
n}\right)=D\left(\rho\,\|\,\sigma\right)-\sqrt{2V\left(\rho\,\|\,\sigma\right)}a_{n}+o\left(a_{n}\right).\\\
\end{dcases}$ (2.22)

##  3\. Privacy Amplification against Quantum Side Information

Let $\mathcal{H}_{E}$ is finite dimensional be a finite dimensional Hilbert
space. Consider a classical-quantum state
$\rho_{XE}=\sum_{x\in\mathcal{X}}p_{X}(x)|x\rangle\langle
x|\otimes\rho_{E}^{x}$. Without loss of generality, we assume that the
marginal density $\rho_{E}$ is invertible.

In this work, we consider a _strongly $2$-universal random hash function_
$h:\mathcal{X}\to\mathcal{Z}$ is a random function which satisfies for all
$x,x^{\prime}\in\mathcal{X}$ with $x\neq x^{\prime}$ and
$z,z^{\prime}\in\mathcal{Z}$,

$\displaystyle\Pr_{h}\left\\{h(x)=z\;\wedge\;h(x^{\prime})=z^{\prime}\right\\}=\frac{1}{|\mathcal{Z}|^{2}}\,.$
(3.1)

Namely, the output $h(x)$ for each input $x$ is uniform and pairwise
independent. Alice applies the linear operation $\mathcal{R}^{h}_{X\to Z}$ on
her system $X$ by the following:

$\displaystyle\mathcal{R}^{h}(\rho_{XE})$
$\displaystyle:=\sum_{x\in\mathcal{X}}p_{X}(x)|h(x)\rangle\langle
h(x)|\otimes\rho_{E}^{x}\,.$ (3.2)

A perfectly randomizing channel $\mathcal{U}_{X\to Z}$ from $\mathcal{X}$ to
$\mathcal{Z}$ is defined as

$\displaystyle\mathcal{U}(\theta_{X})=\frac{\mathds{1}_{Z}}{|\mathcal{Z}|}\left(\sum_{x}\theta_{X}(x)\right).$
(3.3)

We define the _maximal extractable randomness_ for an $\varepsilon$-secret
extractor [19, §7] as

$\displaystyle\ell^{\varepsilon}(X{\,|\,}E)_{\rho}:=\sup\left\\{\ell\in\mathds{N}:|\mathcal{Z}|\geq\ell\wedge\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}(\rho_{XE})-\mathcal{U}(\rho_{XE})\right\|_{1}\leq\varepsilon\right\\}.$
(3.4)

The main result of this section is to prove the following one-shot
characterization of the operational quantity
$\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$.

######  Theorem 10.

Let $\rho_{XE}$ be a classical-quantum state, and let
$h(x):\mathcal{X}\to\mathcal{Z}$ be a strongly $2$-universal hash function.
Then, for every $0<\varepsilon<1$ and
$0<c<\delta<\frac{\varepsilon}{3}\wedge\frac{(1-\varepsilon)}{2}$, we have

$\displaystyle
H_{\textnormal{h}}^{1-\varepsilon+3\delta}(X{\,|\,}E)_{\rho}-\log\frac{\nu^{2}}{\delta^{4}}\leq\log\ell^{\varepsilon}(X{\,|\,}E)_{\rho}\leq
H_{\textnormal{h}}^{1-\varepsilon-2\delta}(X{\,|\,}E)_{\rho}+\log\left(\frac{1+c}{c\delta}\right)+\log\left(\frac{\varepsilon+c}{\delta-c}\right).$
(3.5)

Here, $H_{\textnormal{h}}^{\varepsilon}$ is the conditional
$\varepsilon$-hypothesis testing entropy defined in (2.8) and
$\nu=|\textnormal{{spec}}(\rho_{E})|$.

In the i.i.d setting, the one-shot characterization from Theorem 10 combined
with the second-order expansion of $H_{\textnormal{h}}^{\varepsilon}$ leads to
the following second-order asymptotics of
$\log\ell^{\varepsilon}(X^{n}{\,|\,}E^{n})_{\rho^{\otimes n}}$.

######  Proposition 11.

Let $\epsilon\in(0,1)$. For any strongly $2$-universal hash function
$h^{n}:\mathcal{X}^{n}\to\mathcal{Z}^{n}$ and
$\frac{1}{2}\mathds{E}_{h^{n}}\left\|({\mathcal{R}^{h^{n}}}-\mathcal{U}^{\otimes
n})(\rho_{XE}^{\otimes n})\right\|_{1}\leq\epsilon$, we have the asymptotic
lower bound:

$\displaystyle\log\ell^{\varepsilon}(X^{n}{\,|\,}E^{n})_{\rho^{\otimes
n}}=nH(X{\,|\,}E)_{\rho}+\sqrt{nV(X{\,|\,}E)_{\rho}}\Phi^{-1}(\varepsilon)+O\left(\log
n\right).$ (3.6)

###### Proof.

Since $|\textnormal{{spec}}(\rho_{E})|\leq(n+1)^{|\mathcal{H}_{E}|-1}$ for
$|\mathcal{H}_{E}|$ being the rank of $\rho_{E}$, the additive terms grow of
order $O(\log n)$. We apply the established one-shot characterization, Theorem
10, with $\delta=n^{-\nicefrac{{1}}{{2}}}$ and $c=\frac{1}{2}\delta$, together
with the second-order expansion of the conditional hypothesis testing entropy,
Lemma 8, to arrive at the claim. ∎

Moreover, the one-shot characterization, can be extended to the moderate
deviation regime [45, 44]; namely, we derive the optimal rate of the maximal
required output dimension when the error approaches zero moderately quickly.

######  Proposition 12 (Moderate deviations for privacy amplification).

For every classical-quantum state $\rho_{XB}$ and any moderate sequence
$(a_{n})_{n\in\mathds{N}}$ satisfying (2.21) and
$\varepsilon_{n}:=\mathrm{e}^{-na_{n}^{2}}$, we have

$\displaystyle\begin{dcases}\frac{1}{n}\log\ell^{\varepsilon_{n}}(X^{n}{\,|\,}E^{n})_{\rho^{\otimes
n}}=H(X{\,|\,}E)_{\rho}-\sqrt{2V(X{\,|\,}E)}a_{n}+o\left(a_{n}\right);\\\
\frac{1}{n}\log\ell^{1-\varepsilon_{n}}(X^{n}{\,|\,}E^{n})_{\rho^{\otimes
n}}=H(X{\,|\,}E)_{\rho}+\sqrt{2V(X{\,|\,}E)}a_{n}+o\left(a_{n}\right).\end{dcases}$
(3.7)

###### Proof.

We prove the expansion for $\ell^{\varepsilon_{n}}(X^{n}{\,|\,}E^{n})$. For
every moderate deviation sequence $(a_{n})_{n\in\mathds{N}}$, we let
$\delta_{n}=\frac{1}{4}\operatorname{\mathrm{e}}^{-na_{n}^{2}}$ satisfying
(2.21) and let $c_{n}=\frac{1}{2}\delta_{n}$. Then,
$\varepsilon_{n}+2\delta_{n}$ or $\varepsilon_{n}-3\delta_{n}$ can be viewed
as another $e^{-nb_{n}^{2}}$ for another moderate deviation sequence
$(b_{n})_{n\in\mathds{N}}$; we have $b_{n}=a_{n}+(b_{n}-a_{n})=o(a_{n})$ (see
e.g. [58, §7.2]), $\frac{1}{n}\log\delta_{n}=-a_{n}^{2}=o(a_{n})$ and
$\frac{1}{n}\log|\textnormal{{spec}}(\rho_{E}^{\otimes n})|=O\left(\frac{\log
n}{n}\right)=o(a_{n})$. Then, applying Theorem 10 with Lemma 9 leads to our
first claim of the moderate derivation for privacy amplification. The second
line follows similarly. ∎

We prove the lower bound and upper bound of
$\log\ell^{\varepsilon}(X:E)_{\rho}$ in Section 3.1 and Section 3.2,
respectively.

###  3.1. Direct Bound

We prove the lower bound on $\log\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$ here.

###### Proof of achievability in Theorem 10.

We first claim that for any $c>0$ and strongly 2-universal hash function
$h:\mathcal{X}^{n}\to\mathcal{Z}^{n}$,

$\displaystyle\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}(\rho_{XE})-\mathcal{U}(\rho_{XE})\right\|_{1}\leq\operatorname{Tr}\left[\rho_{XE}\left\\{\mathscr{P}_{\rho_{E}}[\rho_{XE}]>c\mathds{1}_{X}\otimes\rho_{E}\right\\}\right]+\sqrt{c|\textnormal{{spec}}(\rho_{E})||\mathcal{Z}|}\,.$
(3.8)

Let $\delta\in(0,\varepsilon)$ and let

$\displaystyle c$
$\displaystyle=\exp\left\\{D_{\text{s}}^{1-\varepsilon+\delta}(\mathscr{P}_{\rho_{E}}[\rho_{XE}]\,\|\,\mathds{1}_{X}\otimes\rho_{E})+\xi\right\\},$
(3.9)

for some small $\xi>0$. Then, by definition of the information spectrum
divergence, (2.3), we have

$\displaystyle\operatorname{Tr}\left[\rho_{XE}\left\\{\mathscr{P}_{\rho_{E}}[\rho_{XE}]>c\mathds{1}_{X}\otimes\rho_{E}\right\\}\right]<\varepsilon-\delta\,.$
(3.10)

Choose
$|\mathcal{Z}|=\left\lfloor{\frac{\delta^{2}}{c|\texttt{spec}(\rho_{E})|}}\right\rfloor$.
Then, by (3.8) and (3.10), the $\varepsilon$-secret criterion is satisfied,
i.e.

$\displaystyle\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}(\rho_{XE})-\mathcal{U}(\rho_{XE})\right\|_{1}\leq\varepsilon\,,$
(3.11)

By the definition of (3.4), we have the following lower bound on
$\log\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$:

$\displaystyle\log\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$ $\displaystyle\geq-
D_{\text{s}}^{1-\varepsilon+\delta}(\mathscr{P}_{\rho_{E}}[\rho_{XE}]\,\|\,\mathds{1}_{X}\otimes\rho_{E})-\xi-\log|\texttt{spec}(\rho_{E})|+2\log\delta$
(3.12) $\displaystyle\geq
H_{h}^{1-\varepsilon+3\delta}(X{\,|\,}B)_{\rho}-\xi-2\log|\texttt{spec}(\rho_{E})|+4\log\delta,$
(3.13)

where we have used Lemma 2 in the last inequality. Since $\xi>0$ is arbitrary,
taking $\xi\to 0$ gives our claim of lower bound in Theorem 10.

Now we move on to prove (3.8). We first consider case where $\rho^{x}_{E}$ is
invertible for each $x\in\mathcal{X}$ and later argue that the general case
follows from approximation. Shorthand $p\equiv p_{X}$ and
$\rho_{x}\equiv\rho_{E}^{x}$. For every $x\in\mathcal{X}$, we take the
projection

$\displaystyle\Pi_{x}$
$\displaystyle=\left\\{\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]\leq
c\rho_{E}\right\\};$ (3.14) $\displaystyle\Pi^{\mathrm{c}}_{x}$
$\displaystyle:=\mathds{1}_{E}-\Pi_{x}.$ (3.15)

Observe that for every $z\in\mathcal{Z}$, we have

$\displaystyle\mathds{E}_{h}\left[\sum_{x:h(x)=z}p(x)\rho_{x}\right]=\frac{1}{|\mathcal{Z}|}\rho_{E}.$
(3.16)

We then use the fact that the Schatten $1$-norm $\|\cdot\|_{1}$ is additive
for direct sums to calculate

$\displaystyle\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}(\rho_{XE})-\mathcal{U}(\rho_{XE})\right\|_{1}$
(3.17)
$\displaystyle=\frac{1}{2}\mathds{E}_{h}\left\|\sum_{z}\ket{z}\bra{z}\otimes\left(\sum_{x:h(x)=z}p(x)\rho_{x}\right)-\frac{1}{|\mathcal{Z}|}\sum_{z}\ket{z}\bra{z}\otimes\rho_{E}\right\|_{1}$
(3.18)
$\displaystyle=\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}p(x)\rho_{x}-\frac{1}{|\mathcal{Z}|}\rho_{E}\right\|_{1}$
(3.19)
$\displaystyle=\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}p(x)\rho_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}p(x)\rho_{x}\right]\right\|_{1}$
(3.20)
$\displaystyle=\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}p(x)\rho_{x}(\Pi^{\mathrm{c}}_{x}+\Pi_{x})-\mathds{E}_{h}\left[\sum_{x:h(x)=z}p(x)\rho_{x}(\Pi^{\mathrm{c}}_{x}+\Pi_{x})\right]\right\|_{1}$
(3.21)
$\displaystyle\overset{(a)}{\leq}\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}\right]\right\|_{1}$
(3.22)
$\displaystyle\quad+\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}p(x)\rho_{x}\Pi_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}p(x)\rho_{x}\Pi_{x}\right]\right\|_{1}$
(3.23)
$\displaystyle\begin{split}&\leq\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}\right]\right\|_{1}\\\
&\quad+\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}\Pi_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}\Pi_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}\right]\right\|_{1}\\\
&\quad+\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}p(x)\rho_{x}\Pi_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}p(x)\rho_{x}\Pi_{x}\right]\right\|_{1},\end{split}$
(3.24)

where (a) follows from the triangle inequality of the Schatten $1$-norm
$\|\cdot\|_{1}$, and we then use triangle inequality again to decompose the
first term at (a) to arrive at (3.24).

We bound the three terms of (3.24) as follows. The first term of (3.24) is
bounded by

$\displaystyle\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}\right]\right\|_{1}$
(3.25)
$\displaystyle\leq\sum_{z\in\mathcal{Z}}\mathds{E}_{h}\left\|\sum_{x:h(x)=z}\Pi^{\mathrm{c}}_{x}\left(p(x)\rho_{x}\right)\Pi^{\mathrm{c}}_{x}\right\|_{1}$
(3.26)
$\displaystyle=\sum_{z\in\mathcal{Z}}\mathds{E}_{h}\operatorname{Tr}\left[\sum_{x\in\mathcal{X}}\mathbf{1}_{h(x)=z}\ket{x}\bra{x}\otimes\left(p(x)\rho_{x}\\{\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]>c\rho_{E}\\}\right)\right]$
(3.27)
$\displaystyle=\mathds{E}_{h}\operatorname{Tr}\left[\sum_{x\in\mathcal{X}}\ket{x}\bra{x}\otimes\left(p(x)\rho_{x}\\{\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]>c\rho_{E}\\}\right)\right]$
(3.28)
$\displaystyle=\operatorname{Tr}\left[\left(\sum_{x\in\mathcal{X}}\ket{x}\bra{x}\otimes(p(x)\rho_{x})\right)\left(\sum_{x\in\mathcal{X}}\ket{x}\bra{x}\otimes\\{\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]>c\rho_{E}\\}\right)\right]$
(3.29)
$\displaystyle=\operatorname{Tr}\left[\rho_{XE}\\{\mathscr{P}_{\rho_{E}}[\rho_{XE}]>c\mathds{1}_{X}\otimes\rho_{E}\\}\right].$
(3.30)

Let us shorthand for notational convenience

$\displaystyle
H_{h,z}:=\sum_{x:h(x)=z}\Pi_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}-\mathds{E}_{h}\left[\sum_{x:h(x)=z}\Pi_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}\right].$
(3.31)

The second term of (3.24) can be bounded by:

$\displaystyle\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|H_{h,z}\right\|_{1}$
$\displaystyle=\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\operatorname{Tr}\sqrt{H_{h,z}^{\dagger}H_{h,z}}$
(3.32)
$\displaystyle\overset{(a)}{\leq}\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\mathds{E}_{h}\left[H_{h,z}^{\dagger}H_{h,z}\right]}$
(3.33)
$\displaystyle\overset{(b)}{=}\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\mathrm{Var}_{h}\left[\sum_{x\in\mathcal{X}}\mathbf{1}_{\\{h(x)=z\\}}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi_{x}\right]}$
(3.34)
$\displaystyle\overset{(c)}{=}\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\sum_{x\in\mathcal{X}}\mathrm{Var}_{h}\left[\mathbf{1}_{\\{h(x)=z\\}}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi_{x}\right]}$
(3.35)
$\displaystyle\overset{(d)}{\leq}\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\sum_{x\in\mathcal{X}}\mathds{E}_{h}\left[\mathbf{1}_{\\{h(x)=z\\}}\Pi_{x}p(x)\rho_{x}\Pi^{\mathrm{c}}_{x}p(x)\rho_{x}\Pi_{x}\right]}$
(3.36)
$\displaystyle\overset{(e)}{\leq}\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\sum_{x\in\mathcal{X}}\mathds{E}_{h}\left[\mathbf{1}_{\\{h(x)=z\\}}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}\right]}$
(3.37)
$\displaystyle\overset{(f)}{\leq}\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}}.$
(3.38)

Here, in (a) we applied Jensen’s inequality, Lemma 7, with expectation
$\mathds{E}_{h}$ and the operator concavity of square-root; in (b) we denoted
a _matrix-valued variance_ for a random matrix $H_{h}$ as
$\mathrm{Var}[H_{h}]:=\mathds{E}[H_{h}^{\dagger}H_{h}]-(\mathds{E}[H_{h}])^{\dagger}\mathds{E}[H_{h}]$
(see e.g. [59, §2]); in (c) we applied pairwise independent property of the
strongly $2$-universal hash function (on each $x\in\mathcal{X}$); in (d) we
used the operator monotone of square-root and
$\mathrm{Var}[H_{h}]\leq\mathds{E}[H_{h}^{\dagger}H_{h}]$; (e) follows from
$\Pi_{x}^{\mathrm{c}}\leq\mathds{1}_{E}$ and the operator monotonicity of
square-root; in (f) we used the uniformity of the random hash function $h$.

The third term of (3.24) can be bounded similar to the second term as:

$\displaystyle\sum_{z\in\mathcal{Z}}\frac{1}{2}\mathds{E}_{h}\left\|\sum_{x\in\mathcal{X}}\mathbf{1}_{h(x)=z}p(x)\rho_{x}\Pi_{x}-\mathds{E}_{h}\left[\sum_{x\in\mathcal{X}}\mathbf{1}_{h(x)=z}p(x)\rho_{x}\Pi_{x}\right]\right\|_{1}$
(3.39)
$\displaystyle\leq\sum_{z\in\mathcal{Z}}\frac{1}{2}\operatorname{Tr}\sqrt{\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}}.$
(3.40)

Now by (3.38) and (3.40), we bound the sum of second and third term of (3.24)
as:

$\displaystyle\sum_{z\in\mathcal{Z}}\operatorname{Tr}\sqrt{\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}}$
(3.41)
$\displaystyle=\sum_{z\in\mathcal{Z}}\operatorname{Tr}\mathscr{P}_{\rho_{E}}\sqrt{\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}}$
(3.42)
$\displaystyle\overset{(a)}{\leq}\sum_{z\in\mathcal{Z}}\operatorname{Tr}\sqrt{\mathscr{P}_{\rho_{E}}\left[\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}\right]}$
(3.43)
$\displaystyle\overset{(b)}{=}\sum_{z\in\mathcal{Z}}\operatorname{Tr}\left[\rho_{E}\sqrt{\mathscr{P}_{\rho_{E}}\left[\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}\right]\rho_{E}^{-2}}\right]$
(3.44)
$\displaystyle\overset{(c)}{\leq}\sum_{z\in\mathcal{Z}}\sqrt{\operatorname{Tr}\left[\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\Pi_{x}(p(x)\rho_{x})^{2}\Pi_{x}\rho_{E}^{-1}\right]}$
(3.45)
$\displaystyle=\sum_{z\in\mathcal{Z}}\sqrt{\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\operatorname{Tr}\left[(p(x)\rho_{x})^{2}\Pi_{x}\rho_{E}^{-1}\Pi_{x}\right]}$
(3.46)
$\displaystyle\overset{(d)}{\leq}\sum_{z\in\mathcal{Z}}\sqrt{\sum_{x\in\mathcal{X}}\frac{1}{|\mathcal{Z}|}\operatorname{Tr}\left[(p(x)\rho_{x})^{2}c|\textnormal{{spec}}(\rho_{E})|(p(x)\rho_{x})^{-1}\right]}$
(3.47)
$\displaystyle=|\mathcal{Z}|\sqrt{\frac{c}{|\mathcal{Z}|}|\textnormal{{spec}}(\rho_{E})|}$
(3.48) $\displaystyle=\sqrt{c|\mathcal{Z}||\textnormal{{spec}}(\rho_{E})|}.$
(3.49)

where (a) follows from Jensen’s inequality, Lemma 7, with the pinching map
$\mathscr{P}_{\rho_{E}}$ and the operator concavity of square-root; (b) holds
because now every term is commuting with $\rho_{E}$; (c) follows from Jensen’s
inequality, Lemma 7, with the functional
$\operatorname{Tr}[\rho_{E}(\,\cdot\,)]$ and the operator concavity of square-
root; in (d) we use the fact that

$\displaystyle\Pi_{x}\rho_{E}^{-1}\Pi_{x}$ $\displaystyle\leq
c\,\Pi_{x}\left(\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]\right)^{-1}\Pi_{x}$
(3.50) $\displaystyle\leq
c\left(\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]\right)^{-1}$ (3.51)
$\displaystyle\leq
c\left|\textnormal{{spec}}(\rho_{E})\right|(p(x)\rho_{x})^{-1},$ (3.52)

since

$\displaystyle\Pi_{x}=\left\\{\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]\leq
c\rho_{E}\right\\}=\left\\{\rho_{E}^{-1}\leq
c\left(\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]\right)^{-1}\right\\},$ (3.53)

and we invoke the operator monotonicity of matrix inversion together with the
pinching inequality (Lemma 1), i.e.

$\displaystyle\mathscr{P}_{\rho_{E}}[p(x)\rho_{x}]\geq\frac{p(x)\rho_{x}}{|\texttt{spec}(\rho_{E})|}.$
(3.54)

The statement (3.8) is proved by combining (3.24), (3.30), (3.38), (3.40), and
(3.49).

For the general non-invertible states $\\{\rho_{x}\\}_{x}$ we define the
approximation

$\displaystyle\rho_{x}^{\epsilon}:=(1-\epsilon)\rho_{x}+\epsilon\frac{\mathds{1}_{E}}{|\mathcal{H}_{E}|},$
(3.55)

and then

$\rho_{E}^{\epsilon}:=(1-\epsilon)\rho_{E}+\epsilon\frac{\mathds{1}_{E}}{|\mathcal{H}_{E}|}\
,\
\rho_{XE}^{\epsilon}=(1-\epsilon)\rho^{\epsilon}_{XE}+\epsilon\rho_{X}\otimes\frac{\mathds{1}_{E}}{|\mathcal{H}_{E}|}.$

Moreover, since $\mathscr{P}_{\rho_{E}}$ is unital completely positive and
trace-preserving,

$\displaystyle\mathscr{P}_{\rho_{E}}\left[\rho_{XE}^{\epsilon}\right]:=(1-\epsilon)\mathscr{P}_{\rho_{E}}\left[\rho_{XE}\right]+\epsilon\rho_{X}\otimes\frac{\mathds{1}_{E}}{|\mathcal{H}_{E}|}.$
(3.56)

It is clear that

$\displaystyle\lim_{\epsilon\to
0}\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}(\rho_{XE}^{\epsilon})-\mathcal{U}(\rho_{XE}^{\epsilon})\right\|_{1}=\frac{1}{2}\mathds{E}_{h}\left\|\mathcal{R}^{h}(\rho_{XE})-\mathcal{U}(\rho_{XE})\right\|_{1}.$
(3.57)

For the right-hand side of (3.8), we have

$\displaystyle\left\|\mathscr{P}_{\rho_{E}}\left[\rho_{XE}^{\epsilon}\right]-\mathscr{P}_{\rho_{E}}[\rho_{XE}]\right\|_{1}\leq
2\epsilon$ (3.58)

and for small enough $\epsilon$, the projection

$\displaystyle\left\\{\mathscr{P}_{\rho_{E}}[(\rho_{XE})^{\epsilon}]>c\mathds{1}_{X}\otimes(\rho_{E})^{\epsilon}\right\\}=\left\\{\mathscr{P}_{\rho_{E}}[\rho_{XE}]>c\mathds{1}_{X}\otimes\rho_{E}\right\\}.$
(3.59)

In fact, because $\mathscr{P}_{\rho_{E}}[(\rho_{XE})]$ and
$\mathds{1}_{X}\otimes\rho_{E}$ are commutative, they can be viewed as
functions on the finite set of spectrum. Therefore, we have

$\displaystyle\lim_{\epsilon\to
0}\operatorname{Tr}\left[(\rho_{XE})^{\epsilon}\left\\{\mathscr{P}_{\rho_{E}}[(\rho_{XE})^{\epsilon}]>c\mathds{1}_{X}\otimes(\rho_{E})^{\epsilon}\right\\}\right]=\operatorname{Tr}\left[\rho_{XE}\left\\{\mathscr{P}_{\rho_{E}}[\rho_{XE}]>c\mathds{1}_{X}\otimes\rho_{E}\right\\}\right],$
(3.60)

which proves (3.8) by approximation. ∎

###  3.2. Converse Bound

We prove the upper bound on $\log\ell^{\varepsilon}(X{\,|\,}E)_{\rho}$ here.

###### Proof of converse of Theorem 10.

We denote $p\equiv p_{X}$. For every $0<c<\delta<\frac{1-\varepsilon}{2}$, and
for any realization of the random hash function $h$, we choose the
noncommutative quotient

$\displaystyle\Pi=\frac{\mathcal{R}^{h}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\cdot\mathcal{U}(\rho_{XE})}.$
(3.61)

For every $\varepsilon$-secret randomness extractor, we calculate

$\displaystyle\varepsilon$
$\displaystyle\geq\frac{1}{2}\mathds{E}_{h}\left\|(\mathcal{R}^{h}-\mathcal{U})(\rho_{XE})\right\|_{1}$
(3.62)
$\displaystyle\overset{(a)}{\geq}\mathds{E}_{h}\operatorname{Tr}\left[(\mathcal{R}^{h}-\mathcal{U})(\rho_{XE})\frac{\mathcal{R}^{h}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
(3.63)
$\displaystyle\overset{(b)}{\geq}\mathds{E}_{h}\operatorname{Tr}\left[\mathcal{R}^{h}(\rho_{XE})\frac{\mathcal{R}^{h}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]-c,$
(3.64)

where in (a) we used the lower bound on trace distance, Lemma 5, with $\Pi$,
and in (b) we applied Lemma 6-d to obtain the following estimation:

$\displaystyle\operatorname{Tr}\left[\mathcal{U}(\rho_{XE})\frac{\mathcal{R}^{h}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
$\displaystyle=c\operatorname{Tr}\left[\mathcal{R}^{h}(\rho_{XE})\frac{c^{-1}\mathcal{U}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
$\displaystyle\leq c\operatorname{Tr}\left[\mathcal{R}^{h}(\rho_{XE})\right]$
$\displaystyle=c\hskip 2.84544pt.$

For every $z\in\mathcal{Z}$, we define

$\displaystyle\sigma_{zXE}:=$
$\displaystyle\ket{z}\bra{z}\otimes\left(\sum_{x:\,h(x)=z}\ket{x}\bra{x}\otimes
p(x)\rho^{x}_{E}\right);$ (3.65) $\displaystyle\tau_{zXE}:=$
$\displaystyle\left(\sum_{x:\,h(x)=z}\ket{x}\bra{x}\right)\otimes\left(\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE}))\right).$
(3.66)

Since $\mathcal{R}^{h}(\rho_{XE})=\sum_{z\in\mathcal{Z}}\sigma_{zE}$ for
$\sigma_{zE}=\operatorname{Tr}_{X}[\sigma_{zXE}]:=|z\rangle\langle
z|\otimes\sum_{x:\,h(x)=z}p(x)\rho_{E}^{x}$, we calculate

$\displaystyle\mathds{E}_{h}\operatorname{Tr}\left[\mathcal{R}^{h}(\rho_{XE})\frac{\mathcal{R}^{h}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
$\displaystyle=\sum_{z\in\mathcal{Z}}\mathds{E}_{h}\operatorname{Tr}\left[\sigma_{zE}\frac{\mathcal{R}^{h}(\rho_{XE})}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
(3.67)
$\displaystyle\overset{(a)}{\geq}\sum_{z\in\mathcal{Z}}\mathds{E}_{h}\operatorname{Tr}\left[\sigma_{zE}\frac{\sigma_{zE}}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
(3.68)
$\displaystyle=\sum_{z\in\mathcal{Z}}\sum_{x:h(x)=z}\mathds{E}_{h}\operatorname{Tr}\left[\ket{z}\bra{z}\otimes
p(x)\rho^{x}_{E}\frac{\sigma_{zE}}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
(3.69)
$\displaystyle\overset{(b)}{\geq}\sum_{z\in\mathcal{Z}}\sum_{x:h(x)=z}\mathds{E}_{h}\operatorname{Tr}\left[\ket{z}\bra{z}\otimes
p(x)\rho^{x}_{E}\frac{\ket{z}\bra{z}\otimes
p(x)\rho^{x}_{E}}{\mathcal{R}^{h}(\rho_{XE})+c^{-1}\mathcal{U}(\rho_{XE})}\right]$
(3.70)
$\displaystyle=\sum_{z\in\mathcal{Z}}\mathds{E}_{h}\exp{D_{2}^{*}(\sigma_{zXE}\,\|\,\tau_{zXE})}$
(3.71)
$\displaystyle\overset{(c)}{\geq}\sum_{z\in\mathcal{Z}}\exp{D_{2}^{*}\left(\mathds{E}_{h}\left[\sigma_{zXE}\right]\|\,\mathds{E}_{h}\left[\tau_{zXE}\right]\right)},$
(3.72)

where in (a) we used $\mathcal{R}^{h}(\rho_{XE})\geq\sigma_{zE}$ for all
$z\in\mathcal{Z}$ and Lemma 6-a & b; in (b) we used that for every $x,z$ s.t.
$h(x)=z$, $\sigma_{zE}\geq\ket{z}\bra{z}\otimes p(x)\rho^{x}_{E}$ and Lemma 6
again; and in (c) we used the joint convexity of $\exp
D_{2}^{*}(\,\cdot\|\,\cdot)$, Lemma 4. Then, we exploit the uniformity and
independence of the random hash function to calculate the expectations:

$\displaystyle\mathds{E}_{h}\left[\sigma_{zXE}\right]$
$\displaystyle=\frac{1}{|\mathcal{Z}|}\ket{z}\bra{z}\otimes\rho_{XE};$ (3.73)
$\displaystyle\mathds{E}_{h}\left[\left(\sum_{x:\,h(x)=z}\ket{x}\bra{x}\right)\otimes\mathcal{R}^{h}(\rho_{XE})\right]$
$\displaystyle=\mathds{E}_{h}\sum_{x,\bar{x},\bar{z}}|x\rangle\langle
x|\otimes\mathbf{1}_{\\{h(x)=z\\}}\mathbf{1}_{\\{h(\bar{x})=\bar{z}\\}}|\bar{z}\rangle\langle\bar{z}|\otimes
p(\bar{x})\rho_{E}^{\bar{x}}$ (3.74)
$\displaystyle=\sum_{x,\bar{x},\bar{z}}|x\rangle\langle
x|\otimes\left[\frac{1}{|\mathcal{Z}|}\mathbf{1}_{\\{x=\bar{x}\\}}\mathbf{1}_{\\{z=\bar{z}\\}}+\frac{1}{|\mathcal{Z}|^{2}}\mathbf{1}_{\\{x\neq\bar{x}\\}}\right]|\bar{z}\rangle\langle\bar{z}|\otimes
p(\bar{x})\rho_{E}^{\bar{x}}$ (3.75)
$\displaystyle=\frac{1}{|\mathcal{Z}|}\ket{z}\bra{z}\otimes\rho_{XE}+\frac{1}{|\mathcal{Z}|^{2}}\mathds{1}_{Z}\otimes(\mathds{1}_{X}\otimes\rho_{E}-\rho_{XE});$
(3.76)
$\displaystyle\mathds{E}_{h}\left[\left(\sum_{x:\,h(x)=z}\ket{x}\bra{x}\right)\otimes
c^{-1}\mathcal{U}(\rho_{XE})\right]$
$\displaystyle=c^{-1}\cdot\frac{\mathds{1}_{Z}}{|\mathcal{Z}|}\otimes\frac{\mathds{1}_{X}}{|\mathcal{Z}|}\otimes\rho_{E}.$
(3.77)

Here, a crucial observation is that we can employ the direct-sum structure of
$\mathds{1}_{Z}$ together with the definition of the collision diversion,
(2.5), to rewrite (3.72) as follows, i.e. for every $z\in\mathcal{Z}$,

$\displaystyle\exp{D_{2}^{*}\left(\mathds{E}_{h}\left[\sigma_{zXE}\right]\,\|\,\mathds{E}_{h}\left[\tau_{zXE}\right]\right)}$
(3.78) $\displaystyle=$ $\displaystyle\exp
D_{2}^{*}\left(\left.\frac{|z\rangle\langle
z|}{|\mathcal{Z}|}\otimes\rho_{XE}\,\right\|\frac{|z\rangle\langle
z|}{|\mathcal{Z}|}\otimes\rho_{XE}+\frac{\mathds{1}_{Z}}{|\mathcal{Z}|^{2}}\otimes\left((1+c^{-1})\mathds{1}_{X}\otimes\rho_{E}-\rho_{XE}\right)\right)$
(3.79) $\displaystyle=$ $\displaystyle\exp
D_{2}^{*}\left(\left.\frac{|z\rangle\langle
z|}{|\mathcal{Z}|}\otimes\rho_{XE}\,\right\|\frac{|z\rangle\langle
z|}{|\mathcal{Z}|}\otimes\rho_{XE}+\frac{|z\rangle\langle
z|}{|\mathcal{Z}|^{2}}\otimes\left((1+c^{-1})\mathds{1}_{X}\otimes\rho_{E}-\rho_{XE}\right)\right)$
(3.80) $\displaystyle=$ $\displaystyle\exp
D_{2}^{*}\left(\left.|\mathcal{Z}|^{-1}\rho_{XE}\,\right\||\mathcal{Z}|^{-1}\rho_{XE}+|\mathcal{Z}|^{-2}\left((1+c^{-1})\mathds{1}_{X}\otimes\rho_{E}-\rho_{XE}\right)\right).$
(3.81)

Hence, (3.72) and (3.81) show that

$\displaystyle\sum_{z\in\mathcal{Z}}\exp{D_{2}^{*}\left(\mathds{E}_{h}\left[\sigma_{zXE}\right]\|\,\mathds{E}_{h}\left[\tau_{zXE}\right]\right)}$
(3.82) $\displaystyle=$ $\displaystyle\sum_{z\in\mathcal{Z}}\exp
D_{2}^{*}\left(\left.|\mathcal{Z}|^{-1}\rho_{XE}\,\right\||\mathcal{Z}|^{-1}\rho_{XE}+|\mathcal{Z}|^{-2}\left((1+c^{-1})\mathds{1}_{X}\otimes\rho_{E}-\rho_{XE}\right)\right)$
(3.83) $\displaystyle=$ $\displaystyle\exp
D^{*}_{2}\left(\rho_{XE}\,\|\left(1-|\mathcal{Z}|^{-1}\right)\rho_{XE}+|\mathcal{Z}|^{-1}(1+c^{-1})\mathds{1}_{X}\otimes\rho_{E}\right)$
(3.84) $\displaystyle{\geq}$
$\displaystyle(\delta+\varepsilon)\left(1-\frac{1}{|\mathcal{Z}|}+\frac{1+c^{-1}}{|\mathcal{Z}|}\operatorname{\mathrm{e}}^{-D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XE}\,\|\,\mathds{1}_{X}\otimes\rho_{E})}\right)^{-1},$
(3.85)

where we apply Lemma 3 with $\eta=1-\varepsilon-\delta$,
$\lambda_{1}=1-\frac{1}{|\mathcal{Z}|}$, and
$\lambda_{2}=\frac{1}{|\mathcal{Z}|}(1+c^{-1})$ in the last inequality.

Combining (3.64), (3.72), and (3.85) gives

$\displaystyle\varepsilon\geq(\delta+\varepsilon)\left(1-\frac{1}{|\mathcal{Z}|}+\frac{1+c^{-1}}{|\mathcal{Z}|}\operatorname{\mathrm{e}}^{-D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XE}\,\|\,\mathds{1}_{X}\otimes\rho_{E})}\right)^{-1}-c,$
(3.86)

which can be translated to

$\displaystyle\log|\mathcal{Z}|$ $\displaystyle\leq-
D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XE}\,\|\,\mathds{1}_{X}\otimes\rho_{E})+\log(1+c^{-1})-\log\left(\frac{\delta-c}{\varepsilon+c}+\frac{1}{|\mathcal{Z}|}\right)$
(3.87) $\displaystyle\leq-
D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XE}\,\|\,\mathds{1}_{X}\otimes\rho_{E})+\log(1+c^{-1})-\log\left(\frac{\delta-c}{\varepsilon+c}\right)$
(3.88)
$\displaystyle\overset{(a)}{\leq}H_{\text{h}}^{1-\varepsilon-2\delta}(X{\,|\,}E)_{\rho}+\log(1+c^{-1})-\log\left(\frac{\delta-c}{\varepsilon+c}\right)-\log\delta$
(3.89)
$\displaystyle=H_{\text{h}}^{1-\varepsilon-2\delta}(X{\,|\,}E)_{\rho}+\log\left(\frac{1+c}{c\delta}\right)+\log\left(\frac{\varepsilon+c}{\delta-c}\right),$
(3.90)

where we applied Lemma 2 in (a). That completes the proof. ∎

##  4\. Quantum Soft Covering

In this section, we consider a classical-quantum state
$\rho_{XB}=\sum_{x\in\mathcal{X}}p_{X}(x)|x\rangle\langle
x|\otimes\rho_{B}^{x}$ be a classical-quantum state. We assume that $\rho_{B}$
is invertible and the Hilbert space $\mathcal{H}_{B}$ is finite dimensional.
Let $\mathcal{C}$ be a random codebook where each codeword $x\in\mathcal{X}$
is drawn independently according to distribution $p_{X}$. The goal of quantum
soft covering is to approximate the state $\rho_{B}$ using the (random)
codebook-induced state
$\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{B}^{x}$. We define the
_minimal random codebook size_ for an $\varepsilon$-covering as

$\displaystyle
M^{\varepsilon}(X{\,:\,}B)_{\rho}:=\inf\left\\{M\in\mathds{N}:|\mathcal{C}|\leq
M\wedge\frac{1}{2}\mathds{E}_{\mathcal{C}\sim p_{X}^{\otimes
M}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{B}^{x}-\rho_{B}\right\|_{1}\leq\varepsilon\right\\}.$
(4.1)

The main result of this section is to prove the following one-shot
characterization of the operational quantity
$M^{\varepsilon}(X{\,:\,}B)_{\rho}$.

######  Theorem 13 (One-shot characterization for quantum soft covering).

Given a classical-quantum state $\rho_{XB}$, for every $0<\varepsilon<1$ and
$0<c<\delta<\frac{\varepsilon}{3}\wedge\frac{(1-\varepsilon)}{2}$, we have

$\displaystyle
I_{\textnormal{h}}^{1-\varepsilon-2\delta}\left(X{\,:\,}B\right)_{\rho}-\log\frac{1+c}{c\delta}-\log\frac{\varepsilon+c}{\delta-c}\leq\log
M^{\varepsilon}(X{\,:\,}B)_{\rho}$ $\displaystyle\leq
I_{\textnormal{h}}^{1-\varepsilon+3\delta}\left(X{\,:\,}B\right)_{\rho}+\log\frac{\nu^{2}}{\delta^{4}}.$
(4.2)

Here, $I_{\textnormal{h}}^{\varepsilon}$ is the $\varepsilon$-hypothesis
testing information defined in (2.8) and
$\nu=|\textnormal{{spec}}(\rho_{B})|$.

######  Remark 4.1.

In Theorem 13, we express the operational quantity $\log
M^{\varepsilon}(X{\,:\,}B)_{\rho}$ in terms of the
$(1-\varepsilon)$-hypothesis testing information. However, the lower bound can
be improved to
$D_{\textnormal{s}}^{1-\varepsilon-\delta}(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})$
and the upper bound can be improved to
${D_{\textnormal{s}}^{1-\varepsilon+\delta}\left(\mathscr{P}_{\rho_{B}}\left[\rho_{XB}\right]\,\|\,\rho_{X}\otimes\rho_{B}\right)}$
(both with additional additive logarithmic terms).

In the scenario where the underlying state is identical and independently
prepared, i.e. $\rho_{XB}^{\otimes n}$, the established one-shot
characterization, Theorem 13, gives the following second-order asymptotics of
the logarithmic random codebook size, $\log
M^{\varepsilon}(X^{n}{\,:\,}B^{n})_{\rho^{\otimes n}}$, as a function of
blocklength $n$, in which the optimal second-order rate is obtained.

######  Proposition 14 (Second-order rate for quantum soft covering).

For every classical-quantum state $\rho_{XB}$ and $0<\varepsilon<1$, we have

$\displaystyle\log M^{\varepsilon}(X^{n}{\,:\,}B^{n})_{\rho^{\otimes
n}}=nI(X{\,:\,}B)_{\rho}-\sqrt{nV(X{\,:\,}B)}\Phi^{-1}(\varepsilon)+O\left(\log
n\right).$ (4.3)

###### Proof.

Since $|\textnormal{{spec}}(\rho_{B}^{\otimes
n})|\leq(n+1)^{|\mathcal{H}_{B}|-1}$ for $|\mathcal{H}_{B}|$ being the rank of
$\rho_{B}$, the additive terms grow of order $O(\log n)$. Then applying
Theorem 13, with $\delta_{n}=n^{-\nicefrac{{1}}{{2}}}$ and
$c_{n}=\frac{1}{2}n^{-\nicefrac{{1}}{{2}}}$, together with the second-order
expansion of the hypothesis testing information, Lemma 8, proves our claim. ∎

Moreover, the one-shot characterization, can be extended to the moderate
deviation regime [45, 44]; namely, we derive the optimal rates of the minimal
required random codebook size when the error approaches $0$ or $1$ moderately.

######  Proposition 15 (Moderate deviations for quantum soft covering).

For every classical-quantum state $\rho_{XB}$ and every moderate sequence
$(a_{n})_{n\in\mathds{N}}$ satisfying (2.21) and
$\varepsilon_{n}:=\mathrm{e}^{-na_{n}^{2}}$, we have

$\displaystyle\begin{dcases}\frac{1}{n}\log
M^{\varepsilon_{n}}(X^{n}{\,:\,}B^{n})_{\rho^{\otimes
n}}=I(X{\,:\,}B)_{\rho}+\sqrt{2V(X{\,:\,}B)}a_{n}+o\left(a_{n}\right);\\\
\frac{1}{n}\log M^{1-\varepsilon_{n}}(X^{n}{\,:\,}B^{n})_{\rho^{\otimes
n}}=I(X{\,:\,}B)_{\rho}-\sqrt{2V(X{\,:\,}B)}a_{n}+o\left(a_{n}\right).\\\
\end{dcases}$ (4.4)

###### Proof.

We prove the first assertion. For every moderate deviation sequence
$(a_{n})_{n\in\mathds{N}}$, we let
$\delta_{n}=\frac{1}{4}\mathrm{e}^{-na_{n}^{2}}$ satisfying (2.21) and let
$c_{n}=\frac{1}{2}\delta_{n}$. Then, $\varepsilon_{n}-2\delta_{n}$ or
$\varepsilon+3\delta_{n}$ can be viewed as $\mathrm{e}^{-nb_{n}^{2}}$ for
another moderate deviation sequence $(b_{n})_{n\in\mathds{N}}$; we have
$b_{n}=a_{n}+(b_{n}-a_{n})=o(a_{n})$ (see e.g. [58, §7.2]),
$\frac{1}{n}\log\delta_{n}=-a_{n}^{2}=o(a_{n})$ and
$\frac{1}{n}\log|\textnormal{{spec}}(\rho_{B}^{\otimes n})|=O\left(\frac{\log
n}{n}\right)=o(a_{n})$. Then, applying Theorem 13 together with Lemma 9 leads
to our first claim of the moderate derivation for quantum soft covering. The
second line follows similarly. ∎

######  Remark 4.2.

We remark that the upper bound may be viewed as a quantum generalization of a
classical result by Hayashi [60, Lemma 2]. However, such a generalization is
non-trivial due to difficulties of non-commutativity.

The proofs of the one-shot achievability (i.e. upper bound) and converse (i.e.
lower bound) of Theorem 13 are presented in Section 4.1 and 4.2, respectively.

###  4.1. Direct Bound

We prove the upper bound on $\log M^{\varepsilon}(X:B)_{\rho}$ here.

###### Proof of achievability of Theorem 13.

Throughout the proof, we use the short notation: $\rho_{x}\equiv\rho_{B}^{x}$
and $M\equiv|\mathcal{C}|$. For every $x\in\mathcal{C}$, we define a
projection $\Pi_{x}:=\left\\{\mathscr{P}_{\rho_{B}}\left[\rho_{x}\right]\leq
c\rho_{B}\right\\}$ and its complement
$\Pi_{x}^{\mathrm{c}}:=\mathds{1}_{B}-\Pi_{x}$.

We claim that for any random codebook $\mathcal{C}$ with its codeword
independently drawn according to distribution $p_{X}$ and for any $c>0$, we
have

$\displaystyle\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{B}^{x}-\rho_{B}\right\|_{1}\leq\operatorname{Tr}\left[\rho_{XB}\left\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}]>c\rho_{X}\otimes\rho_{B}\right\\}\right]+\sqrt{\frac{|\textnormal{{spec}}(\rho_{B})|c}{|\mathcal{C}|}}.$
(4.5)

Then, let $\delta\in(0,\varepsilon)$ and choose

$\displaystyle
c=\exp\left\\{D_{\text{s}}^{1-\varepsilon+\delta}\left(\mathscr{P}_{\rho_{B}}[\rho_{XB}]\,\|\,\rho_{X}\otimes\rho_{B}\right)+\xi\right\\}$
(4.6)

for some small $\xi>0$. By definition of the $\varepsilon$-information
spectrum divergence (2.3),

$\displaystyle\operatorname{Tr}\left[\rho_{XB}\left\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}]>c\rho_{X}\otimes\rho_{B}\right\\}\right]=\operatorname{Tr}\left[\mathscr{P}_{\rho_{B}}[\rho_{XB}]\left\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}]>c\rho_{X}\otimes\rho_{B}\right\\}\right]<\varepsilon-\delta.$
(4.7)

Letting

$\displaystyle|\mathcal{C}|=\Big{\lceil}{|\texttt{spec}(\rho_{B})|c}{\delta^{-2}}\Big{\rceil},$
(4.8)

we obtain the $\varepsilon$-covering, i.e.

$\displaystyle\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{B}^{x}-\rho_{B}\right\|_{1}\leq\varepsilon,$
(4.9)

and then together (4.5) we have the following upper bound on $\log
M^{\varepsilon}(X:B)_{\rho}$:

$\displaystyle\log M^{\varepsilon}(X{\,:\,}B)_{\rho}$ $\displaystyle\leq
D_{\text{s}}^{1-\varepsilon+\delta}\left(\mathscr{P}_{\rho_{B}}[\rho_{XB}]\,\|\,\rho_{X}\otimes\rho_{B}\right)+\xi+\log|\texttt{spec}(\rho_{B})|-2\log\delta$
(4.10) $\displaystyle\leq
I_{\text{h}}^{1-\varepsilon+3\delta}\left(X{\,:\,}B\right)_{\rho}+\xi+2\log|\texttt{spec}(\rho_{B})|-4\log\delta,$
(4.11)

where we have used Lemma 2 in the last inequality. Since $\xi>0$ is arbitrary,
we take $\xi\to 0$ to obtain the upper bound in (4.2).

Now, we move on to prove (4.5). We first prove the case where all
$\\{\rho_{x}\\}_{x}$ are invertible. Use triangle inequality of the norm
$\|\cdot\|_{1}$, we obtain

$\displaystyle\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}-\rho_{B}\right\|_{1}$
$\displaystyle=\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}(\Pi_{x}^{\mathrm{c}}+\Pi_{x})-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}(\Pi_{x}^{\mathrm{c}}+\Pi_{x})\right]\right\|_{1}$
(4.12)
$\displaystyle\leq\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}^{\mathrm{c}}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right]\right\|_{1}$
$\displaystyle\quad+\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}\right]\right\|_{1}$
(4.13)
$\displaystyle\begin{split}&\leq\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right]\right\|_{1}\\\
&\quad+\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}\rho_{x}\Pi_{x}^{\mathrm{c}}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}\rho_{x}\Pi_{x}^{\mathrm{c}}\right]\right\|_{1}\\\
&\quad+\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}\right]\right\|_{1}.\end{split}$
(4.14)

The first term in (LABEL:eq:one-shot_second-order1) can be further bounded
using triangle inequality again as:

$\displaystyle\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right]\right\|_{1}$
$\displaystyle\leq\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right\|_{1}+\frac{1}{2}\left\|\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right]\right\|_{1}$
(4.15)
$\displaystyle\leq\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right\|_{1}$
(4.16) $\displaystyle=\mathds{E}_{x\sim
p_{X}}\operatorname{Tr}\left[\rho_{x}\left\\{\mathscr{P}_{\rho_{B}}\left[\rho_{x}\right]>c\rho_{B}\right\\}\right]$
(4.17)
$\displaystyle=\operatorname{Tr}\left[\rho_{XB}\left\\{\mathscr{P}_{\rho_{B}}\left[\rho_{XB}\right]>c\rho_{X}\otimes\rho_{B}\right\\}\right].$
(4.18)

Next, we bound the second term in (LABEL:eq:one-shot_second-order1):

$\displaystyle\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}\rho_{x}\Pi_{x}^{\mathrm{c}}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}\Pi_{x}^{\mathrm{c}}\right]\right\|_{1}=\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]\right\|_{1}$
(4.19)
$\displaystyle=\frac{1}{2}\mathds{E}_{\mathcal{C}}\operatorname{Tr}\sqrt{\left(\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]\right)^{\dagger}\left(\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]\right)}$
(4.20)
$\displaystyle\overset{(a)}{\leq}\frac{1}{2}\operatorname{Tr}\sqrt{\mathds{E}_{\mathcal{C}}\left(\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]\right)^{\dagger}\left(\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}-\mathds{E}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]\right)}$
(4.21)
$\displaystyle\overset{(b)}{=}\frac{1}{2}\operatorname{Tr}\sqrt{\mathrm{Var}_{\mathcal{C}}\left[\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]}$
(4.22)
$\displaystyle\overset{(c)}{=}\frac{1}{2}\operatorname{Tr}\sqrt{\frac{1}{|\mathcal{C}|}\mathrm{Var}_{x\sim
p_{X}}\left[\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]}$ (4.23)
$\displaystyle\overset{(d)}{\leq}\frac{1}{2}\operatorname{Tr}\sqrt{\frac{1}{|\mathcal{C}|}\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}\Pi_{x}^{\mathrm{c}}\rho_{x}\Pi_{x}\right]}$ (4.24)
$\displaystyle\overset{(e)}{\leq}\frac{1}{2}\operatorname{Tr}\sqrt{\frac{1}{|\mathcal{C}|}\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]}.$ (4.25)

Here, in (a) we applied Jensen’s inequality, Lemma 7, with expectation
$\mathds{E}_{\mathcal{C}}$ and the operator concavity of square-root; in (b)
we denoted a matrix-valued variance for a random matrix $H$ as
$\mathrm{Var}[H]:=\mathds{E}[H^{\dagger}H]-(\mathds{E}[H])^{\dagger}\mathds{E}[H]$
(see e.g. [59, §2]); in (c) we applied the mutual independence of the
codewords in the random codebook; in (d) we used the operator monotone of
square-root and $\mathrm{Var}[H]\leq\mathds{E}[H^{\dagger}H]$; and (e) follows
from $\Pi_{x}^{\mathrm{c}}\leq\mathds{1}_{B}$ and the operator monotonicity of
square-root.

Applying the same reasoning on the third term of (LABEL:eq:one-shot_second-
order1), we thus upper bound the sum of the second and the third term of
(LABEL:eq:one-shot_second-order1) by

$\displaystyle\operatorname{Tr}\sqrt{\frac{1}{|\mathcal{C}|}\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]}.$ (4.26)

To further upper bound this term, we use Jensen’s inequality, Lemma 7, with
the pinching map $\mathscr{P}_{\rho_{B}}$ and the operator concavity of
square-root to have

$\displaystyle\operatorname{Tr}\sqrt{\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]}$
$\displaystyle=\operatorname{Tr}\mathscr{P}_{\rho_{B}}\sqrt{\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]}$ (4.27)
$\displaystyle\leq\operatorname{Tr}\sqrt{\mathscr{P}_{\rho_{B}}\left[\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]\right]}$ (4.28)
$\displaystyle=\operatorname{Tr}\left[\rho_{B}\sqrt{\mathscr{P}_{\rho_{B}}\left[\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]\right]\rho_{B}^{-2}}\right]$
(4.29)
$\displaystyle\overset{(a)}{\leq}\sqrt{\operatorname{Tr}\left[\mathds{E}_{x\sim
p_{X}}\left[\Pi_{x}\rho_{x}^{2}\Pi_{x}\right]\right]\rho_{B}^{-1}}$ (4.30)
$\displaystyle=\sqrt{\mathds{E}_{x\sim
p_{X}}\operatorname{Tr}\left[\rho_{x}^{2}\Pi_{x}\rho_{B}^{-1}\Pi_{x}\right]},$
(4.31)

where (a) follows from Jensen’s inequality, Lemma 7, with the functional
$\operatorname{Tr}[\rho_{B}(\,\cdot\,)]$ and the operator concavity of square-
root.

Now, since

$\displaystyle\Pi_{x}=\left\\{\mathscr{P}_{\rho_{B}}[\rho_{x}]\leq
c\rho_{B}\right\\}=\left\\{\rho_{B}^{-1}\leq
c\left(\mathscr{P}_{\rho_{B}}[\rho_{x}]\right)^{-1}\right\\},$ (4.32)

we obtain

$\displaystyle\Pi_{x}\rho_{B}^{-1}\Pi_{x}$ $\displaystyle\leq
c\Pi_{x}\left(\mathscr{P}_{\rho_{B}}[\rho_{x}]\right)^{-1}\Pi_{x}$ (4.33)
$\displaystyle=c\left(\mathscr{P}_{\rho_{B}}[\rho_{x}]\right)^{-1/2}\Pi_{x}\left(\mathscr{P}_{\rho_{B}}[\rho_{x}]\right)^{-1/2},$
(4.34)

where we used the fact that $\Pi_{x}$ commutes with
$\mathscr{P}_{\rho_{B}}[\rho_{x}]$. Then for each $x$,

$\displaystyle\operatorname{Tr}\left[\rho_{x}^{2}\Pi_{x}\rho_{B}^{-1}\Pi_{x}\right]\leq$
$\displaystyle
c\operatorname{Tr}\left[\rho_{x}^{2}\mathscr{P}_{\rho_{B}}(\rho_{x})^{-1/2}\Pi_{x}\mathscr{P}_{\rho_{B}}(\rho_{x})^{-1/2}\right]$
(4.36) $\displaystyle\leq$ $\displaystyle
c\operatorname{Tr}\left[\rho_{x}^{2}\mathscr{P}_{\rho_{B}}(\rho_{x})^{-1}\right]$
(4.37) $\displaystyle\overset{(a)}{\leq}$ $\displaystyle
c|\textnormal{{spec}}(\rho_{B})|\operatorname{Tr}\left[\rho_{x}^{2}\rho_{x}^{-1}\right]$
(4.38) $\displaystyle=$ $\displaystyle c|\textnormal{{spec}}(\rho_{B})|,$
(4.39)

where in (a) we used the the pinching inequality (Lemma 1), i.e.

$\rho\leq|\textnormal{{spec}}(\rho_{B})|\mathscr{P}_{\rho_{B}}[\rho_{x}],$

and the operator monotonicity of inversion. Combining (LABEL:eq:one-
shot_second-order1), (4.26), (4.31), and (4.39) arrives at the desired (4.5).

For the general non-invertible state $\rho_{x}$, we define the approximation

$\rho_{x}^{\epsilon}:=(1-\epsilon)\rho_{x}+\varepsilon\frac{\mathds{1}_{B}}{|\mathcal{H}_{B}|},$

and then

$\rho_{B}^{\epsilon}:=(1-\epsilon)\rho_{B}+\epsilon\frac{\mathds{1}_{B}}{|\mathcal{H}_{B}|}\
,\
\rho_{XB}^{\epsilon}=(1-\epsilon)\rho^{\epsilon}_{XB}+\epsilon\rho_{X}\otimes\frac{\mathds{1}_{B}}{|\mathcal{H}_{B}|}.$

Moreover, since the pinching map $\mathscr{P}_{\rho_{B}}$ is unital completely
positive and trace-preserving,

$\mathscr{P}_{\rho_{B}}(\rho_{XB}^{\epsilon})=(1-\epsilon)\mathscr{P}_{\rho_{B}}(\rho_{XB})+\epsilon\rho_{X}\otimes\frac{\mathds{1}_{B}}{|\mathcal{H}_{B}|}\hskip
2.84544pt.$

It is clear that

$\lim_{\epsilon\to
0}\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}^{\epsilon}-\rho_{B}^{\epsilon}\right\|_{1}=\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\rho_{x}-\rho_{B}\right\|_{1}.$

For the right-hand side of the desired inequality (4.5), we have

$\left\|\mathscr{P}_{\rho_{B}}(\rho_{XB}^{\epsilon})-\mathscr{P}_{\rho_{B}}(\rho_{XB})\right\|_{1}\leq
2\epsilon$

and for small enough $\epsilon$, the projection

$\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}^{\epsilon}]>c\rho_{X}\otimes\rho_{B}^{\epsilon}\\}=\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}]>c\rho_{X}\otimes\rho_{B}\\}.$

Indeed, because $\mathscr{P}_{\rho_{B}}[\rho_{XB}]$ and
$\rho_{X}\otimes\rho_{B}$ are commutative, they can be viewed as functions on
the finite sets of spectrum. Therefore, we have

$\lim_{\epsilon\to
0}\operatorname{Tr}\left[\rho_{XB}^{\epsilon}\left\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}^{\epsilon}]>c\rho_{X}\otimes\rho_{B}^{\epsilon}\right\\}\right]=\operatorname{Tr}\left[\rho_{XB}\left\\{\mathscr{P}_{\rho_{B}}[\rho_{XB}]>c\rho_{X}\otimes\rho_{B}\right\\}\right]\
,$

which proves (4.5) by approximation. ∎

###  4.2. Converse Bound

We prove the lower bound on on $\log M^{\varepsilon}(X{\,:\,}B)_{\rho}$ here.

###### Proof of converse of Theorem 13.

Throughout this proof, we write $\rho_{x}\equiv\rho_{B}^{x}$ and
$M:=|\mathcal{C}|$. For every $0<c<\delta<\frac{1-\varepsilon}{2}$ and for any
realization of the random codebook $\mathcal{C}$, we choose the noncommutative
quotient

$\displaystyle\Pi=\frac{\frac{1}{M}\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}}{\frac{1}{M}\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}+c^{-1}\rho_{B}}.$
(4.40)

Lemma 5 then implies that

$\displaystyle\frac{1}{2}\left\|\frac{1}{M}\sum_{x\in\mathcal{C}}\rho_{B}^{x}-\rho_{B}\right\|_{1}\geq\operatorname{Tr}\left[\frac{1}{M}\sum_{x\in\mathcal{C}}\rho_{B}^{x}\Pi\right]-\operatorname{Tr}\left[\rho_{B}\Pi\right]$
(4.41)

Using Lemma 6-d, the second term in (4.41) can be lower bounded as

$\displaystyle-\operatorname{Tr}\left[\rho_{B}\Pi\right]$
$\displaystyle=-c\operatorname{Tr}\left[\frac{1}{M}\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}\cdot\frac{c^{-1}\rho_{B}}{\frac{1}{M}\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}+c^{-1}\rho_{B}}\right]$
(4.42)
$\displaystyle\geq-c\operatorname{Tr}\left[\frac{1}{M}\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}\right]$
(4.43) $\displaystyle=-c\,,$ (4.44)

since
$\frac{c^{-1}\rho_{B}}{\frac{1}{M}\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}+c^{-1}\rho_{B}}\leq\mathds{1}_{B}$
by Lemma 6-c.

For each $x\in\mathcal{C}$, we use (by recalling Lemma 6-a & b)

$\displaystyle\Pi\geq\frac{\rho_{x}}{\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}+c^{-1}M\rho_{B}}$
(4.45)

to lower bound the first term in (4.41) as

$\displaystyle\operatorname{Tr}\left[\frac{1}{M}\sum_{x\in\mathcal{C}}\rho_{B}^{x}\Pi\right]$
$\displaystyle\geq\frac{1}{M}\sum_{x\in\mathcal{C}}\operatorname{Tr}\left[\rho_{x}\frac{\rho_{x}}{\sum_{\bar{x}\in\mathcal{C}}\rho_{\bar{x}}+c^{-1}M\rho_{B}}\right]$
(4.46) $\displaystyle=\frac{1}{M}\exp
D_{2}^{*}\left(\rho_{XB}^{\mathcal{C}}\,\|\,\rho_{X}^{\mathcal{C}}\otimes\rho_{B}^{\mathcal{C}}+c^{-1}\rho_{X}^{\mathcal{C}}\otimes\rho_{B}\right),$
(4.47)

where we recall the definition of the sandwiched Rényi divergence, (2.5), and

$\displaystyle\rho_{XB}^{\mathcal{C}}$
$\displaystyle:=\sum_{x\in\mathcal{C}}\frac{1}{M}|x\rangle\langle
x|\otimes\rho_{x}.$ (4.48)

With (4.41), (4.44), and (4.47), we take expectation over the random codebook
and use the joint convexity of $\exp D_{2}^{*}(\,\cdot\|\,\cdot)$, Lemma 4, to
obtain

$\displaystyle\frac{1}{2}\mathds{E}_{\mathcal{C}}\left\|\frac{1}{M}\sum_{x\in\mathcal{C}}\rho_{B}^{x}-\rho_{B}\right\|_{1}\geq\frac{1}{M}\exp
D_{2}^{*}\left(\left.\mathds{E}_{\mathcal{C}}\left[\rho_{XB}^{\mathcal{C}}\right]\right\|\mathds{E}_{\mathcal{C}}\left[\rho_{X}^{\mathcal{C}}\otimes\rho_{B}^{\mathcal{C}}+c^{-1}\rho_{X}^{\mathcal{C}}\otimes\rho_{B}\right]\right)-c.$
(4.49)

Now, we apply the mutual independence between the codeword to have

$\displaystyle\mathds{E}_{\mathcal{C}}\left[\rho_{XB}^{\mathcal{C}}\right]$
$\displaystyle=\rho_{XB};$ (4.50)
$\displaystyle\mathds{E}_{\mathcal{C}}\left[\rho_{X}^{\mathcal{C}}\otimes\rho_{B}^{\mathcal{C}}\right]$
$\displaystyle=\mathds{E}_{\mathcal{C}}\left[\frac{1}{M^{2}}\sum_{m,\bar{m}\in[M]}|x_{m}\rangle\langle
x_{m}|\otimes\rho_{{x}_{\bar{m}}}\right]$ (4.51)
$\displaystyle=\mathds{E}_{\mathcal{C}}\left[\frac{1}{M^{2}}\sum_{m\in[M]}|x_{m}\rangle\langle
x_{m}|\otimes\rho_{{x}_{m}}\right]+\mathds{E}_{\mathcal{C}}\left[\frac{1}{M^{2}}\sum_{m\neq\bar{m}}|x_{m}\rangle\langle
x_{m}|\otimes\rho_{{x}_{\bar{m}}}\right]$ (4.52)
$\displaystyle=\frac{1}{M}\rho_{XB}+\left(1-\frac{1}{M}\right)\rho_{X}\otimes\rho_{B};$
(4.53) $\displaystyle
c^{-1}\mathds{E}_{\mathcal{C}}\left[\rho_{X}^{\mathcal{C}}\otimes\rho_{B}\right]$
$\displaystyle=c^{-1}\rho_{X}\otimes\rho_{B}.$ (4.54)

By putting them together, we obtain

$\displaystyle\varepsilon$ $\displaystyle\geq\frac{1}{M}\exp
D_{2}^{*}\left(\rho_{XB}\,\|\,M^{-1}\rho_{XB}+\left(1+c^{-1}-M^{-1}\right)\rho_{X}\otimes\rho_{B}\right)-c$
(4.55)
$\displaystyle\overset{(a)}{\geq}\frac{\varepsilon+\delta}{M}\left[M^{-1}+\left(1+c^{-1}-M^{-1}\right)\operatorname{\mathrm{e}}^{-D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})}\right]^{-1}-c,$
(4.56)

where we apply Lemma 3 in (a) with $\eta=1-\varepsilon-\delta$,
$\lambda_{1}=M^{-1}$, and $\lambda_{2}=1+c^{-1}-M^{-1}$. In other words, we
get an lower bound on $\log M$ as

$\displaystyle\log M$ $\displaystyle\geq
D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})-\log\left(1+c^{-1}+M^{-1}\right)+\log\frac{\delta-c}{\varepsilon+c}$
(4.57) $\displaystyle\geq
D_{\text{s}}^{1-\varepsilon-\delta}(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})-\log\left(1+c^{-1}\right)+\log\frac{\delta-c}{\varepsilon+c}$
(4.58) $\displaystyle\geq
D_{\text{h}}^{1-\varepsilon-2\delta}(\rho_{XB}\,\|\,\rho_{X}\otimes\rho_{B})-\log\left(1+c^{-1}\right)+\log\frac{\delta-c}{\varepsilon+c}+\log\delta$
(4.59) $\displaystyle\geq
I_{\text{h}}^{1-\varepsilon-2\delta}(X{\,:\,}B)_{\rho}-\log\frac{1+c}{c\delta}-\log\frac{\varepsilon+c}{\delta-c},$
(4.60)

completing the proof. ∎

##  5\. Conclusions

The _large deviation analysis_ [61, 62, 63, 16, 64, 65, 66, 67, 68, 69, 70,
71] of privacy amplification against quantum side information and quantum soft
covering has been investigated in previous literature [21, 26, 72, 73, 38,
27], wherein one fixes the rate or the size of $|\mathcal{Z}|$ and
$|\mathcal{C}|$ and studies the optimal errors in terms of the trace distance.
Also, some _moderate deviation analysis_ [44, 45] were studied for
characterizing the minimal trace distance while the rates approach the first-
order limits with certain speed [38, 27]. In this paper, we took another
perspective—what are the optimal rates when the trace distances are upper
bounded by a constant $\varepsilon\in(0,1)$. This corresponds to the so-called
_small error regime_ [10, 11, 12] or the _non-vanishing error regime_ [13]. We
establish the second-order rates for fixed $\varepsilon\in(0,1)$ and establish
the optimal rates when trace distances vanishes no faster than
$O(\nicefrac{{1}}{{\sqrt{n}}})$.

In light of the duality between smooth min- and max-entropies, the purified
distance has been recognized as an appropriate distance measure [42, 15, 43,
25, 73]. Our work suggests that if one considers the trace distance as the
performance benchmark without going into the smooth entropy framework [74, 34,
15, 36, 75, 25], the conditional hypothesis testing entropy and the hypothesis
testing information666In Ref. [15], it was shown that up to second-order
terms, $D_{\text{h}}^{1-\varepsilon}$ scales as the relative entropy version
of the smooth min-entropy $D_{\text{max}}^{\sqrt{\varepsilon}}$. Although the
two quantities are asymptotically equivalent, they arise in very different
proof methodologies.  are the natural one-shot characterizations777 In Ref.
[26], Dupuis asked is it possible to use the Rényi-type entropies/information
to characterize operational quantities in one-shot information theory. We
would like to point out that although there are essentially no differences
between the three deviation regimes in the one-shot setting, there are at
least two different types of operational quantities of interest; their
characterizations might be different. As observed in [26] and our previous
works [27, 38], indeed, the Rényi-type quantities are more favorable in
characterizing the optimal error given a fixed size or cardinality such as
$|\mathcal{Z}|$ and $|\mathcal{C}|$ considered in this paper. On the other
hand, if one concerns the size or cardinality given a fixed error, the
hypothesis-testing-type quantities or the information-spectrum-type quantities
might be more direct for characterizations.  . An interesting open question is
comparison between the conditional hypothesis testing entropy with the
partially trace-distance-smoothed min-entropy [25].

## Acknowledgement

H.-C. Cheng would like to thank Kai-Min Chung for his insightful discussions,
and also thank Marco Tomamichel for helpful comments and pointing out relevant
references. Y.-C. Shen and H.-C. Cheng are supported by the Young Scholar
Fellowship (Einstein Program) of the Ministry of Science and Technology in
Taiwan (R.O.C.) under Grant MOST 110-2636-E-002-009, and are supported by the
Yushan Young Scholar Program of the Ministry of Education in Taiwan (R.O.C.)
under Grant NTU-110V0904, Grant NTU-CC-111L894605, and Grand NTU-111L3401.

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