# Exact Exponent for Atypicality of Random Quantum States

Eyuri Wakakuwa<EMAIL_ADDRESS>Department of Mathematical Informatics,
Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, 464-8601, Japan

###### Abstract

We study the properties of the random quantum states induced from the
uniformly random pure states on a bipartite quantum system by taking the
partial trace over the larger subsystem. Most of the previous studies have
adopted a viewpoint of “concentration of measure” and have focused on the
behavior of the states close to the average. In contrast, we investigate the
large deviation regime, where the states may be far from the average. We prove
the following results: First, the probability that the induced random state is
within a given set decreases no slower or faster than exponential in the
dimension of the subsystem traced out. Second, the exponent is equal to the
quantum relative entropy of the maximally mixed state and the given set,
multiplied by the dimension of the remaining subsystem. Third, the total
probability of a given set strongly concentrates around the element closest to
the maximally mixed state, a property that we call conditional concentration.
Along the same line, we also investigate an asymptotic behavior of coherence
of random pure states in a single system with large dimensions.

##  I Introduction

The concept of concentration of measure has played significant roles in
quantum information theory and the foundation of quantum statistical
mechanics. It asserts that functions on high-dimensional spaces are almost
constant if it is sufficiently continuous (see e.g. Ledoux (2001); Milman and
Schechtman (2009)). In the context of quantum information theory, Ref. Hayden
et al. (2006) applied this concept to analyzing entanglement properties of
multipartite quantum states. Ref. Gross et al. (2009) proved that almost all
pure states are useless for universal quantum computation. Ref. Aubrun et al.
(2011) provided a simple derivation of the counterexample Hastings (2009) to
the additivity conjecture of the classical capacity of quantum channels. As
for the foundation of quantum statistical mechanics, Ref. Popescu et al.
(2006a) applied the concept of concentration of measure to prove that a
physical system would be in the thermal state for almost all pure states of
the universe. Ref. Reimann (2015) generalized von Neumann’s approach to prove
thermalization of isolated quantum many-body systems, among many others.

Most of the previous researches along these lines are based on Levy’s lemma.
Ref. Hayden et al. (2006) proved that any function defined on the set of the
pure states takes values close to the average for the overwhelming majority of
the pure states if the dimension of the Hilbert space is sufficiently large
and the function is Lipschitz continuous. Mathematically, the concentration
inequalities obtained from Levy’s lemma are of the form

$\displaystyle{\rm
Pr}_{\Psi}\\{|f(\Psi)-{\mathbb{E}}_{\Psi}f(\Psi)|>\epsilon\\}\leq
2\exp{\left(-\frac{\kappa(d)\epsilon^{2}}{\eta^{2}}\right)}.$ (1)

Here, $\Psi$ is a pure state that is randomly chosen from the Hilbert space
according to the unitary invariant measure, $f$ is a function defined on the
set of the unit vectors therein, $\eta$ is the Lipschitz constant of $f$, $d$
is the dimension of the Hilbert space, and $\kappa(d)$ is a positive linear
function of $d$. The inequalities of the type (1) apply to any finite $d$ and
any Lipschitz function. Because of this generality, Levy’s lemma has been
widely applied to prove the concentration of functions of random quantum
states. However, the approaches from Levy’s lemma have limitations in that (i)
they provide us only upper bounds on the probability of the value of $f$
distant from the average, and (ii) they apply only to the properties that are
represented by the deviation of Lipschitz functions.

In this paper, we address the above problem by following the spirit of the
large deviation theory Dembo and Zeitouni (2009). Our primal focus is on the
random quantum states obtained from the uniformly random pure states in a
bipartite system by taking the partial trace over the larger subsystem. This
scenario was originally investigated in Lloyd and Pagels (1988); Zyczkowski
and Sommers (2001). To be concrete, we consider the random state ${\rm
Tr}_{E}[|\Psi\rangle\\!\langle\Psi|]$, where $|\Psi\rangle$ is the pure state
on a composite system $SE$ that obeys the unitary invariant measure. We focus
on an asymptotic limit of the dimension of $E$ going to infinity, while the
dimension of $S$ is assumed to be fixed. Our goal is to evaluate the
probability of the state ${\rm Tr}_{E}[|\Psi\rangle\\!\langle\Psi|]$ within a
given set $\Omega$.

The results are as follows: First, we prove that the probability of $\Omega$
converges to zero, no slower than or no faster than, exponential in the
dimension of $E$ unless $\Omega$ contains the maximally mixed state. Second,
we prove that the exponent is equal to the relative entropy of the maximally
mixed state $\pi$ against $\Omega$, multiplied by the dimension of $S$.
Namely, we prove the equation

$\displaystyle-\lim_{d_{E}\rightarrow\infty}\frac{1}{d_{E}}\ln{\rm
Pr}_{\Psi}\\{{\rm
Tr}_{E}[\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$}^{SE}]\in\Omega\\}=d_{S}\inf_{\rho\in\Omega}D(\pi\|\rho),$
(2)

where $\Psi$ is the uniformly random pure quantum state on $SE$. In the proof,
we exploit the fact that the unitary invariant measure is represented by the
multivariate normal distribution (see e.g. Section 7.2 in Watrous (2018)), to
which we apply Sanov’s theorem for continuous random variables in the version
of Groeneboom et al. (1979). Third, we prove a property that we call
conditional concentration: When we already know that the state is within a set
$\Omega$, the probability that it is not close to the optimal state (in terms
of the relative entropy from $\pi$) is exponentially small. Hence, the total
probability of $\Omega$ strongly concentrates around the element of $\Omega$
closest to the maximally mixed state. Finally, we show that a similar method
is applicable to analyzing the large deviation property of coherence of random
pure states in a single quantum system.

The main result, Eq. (2), is applicable to any subset of states that is
regular in a topological sense. Hence, it is possible to obtain a large
deviation bound in the case where the property of the states of our interest
is not necessarily represented by the deviation of Lipschitz functions.
Moreover, Eq. (2) provides not only an upper bound but also a lower bound on
the probability of the subset in the asymptotic limit. That is, Eq. (2) is
both a concentration bound and an anti-concentration bound. It is because of
this novelty that we can prove the property of conditional concentration. A
drawback of Eq. (2) from Eq. (1) is that it is valid only in the asymptotic
limit of the subsystem dimension to infinity. It is, therefore, left as an
open problem to obtain a large deviation bound that applies to finite $d$ and
is tight in the asymptotic limit.

This paper is organized as follows. In Section II, we present the formulation
of the problem and state the main results. Section III reviews a version of
Sanov’s theorem in general probability spaces and applies it to prove a
variant of Sanov’s theorem for continuous random variables. The proofs of the
main results are presented in Section IV. In Section V, we apply the main
results to several cases where the exponents can be calculated explicitly.
Section VI introduces the concept of conditional concentration and provides a
proof thereof. In Section VII, we investigate the large deviation property of
quantum coherence of random pure states in a single Hilbert space. Conclusions
are given in Section VIII.

### Notations

The sets of the linear operators, the positive semidefinite operators and the
normalized density operators on a Hilbert space ${\mathcal{H}}$ are denoted by
${\mathcal{L}}({\mathcal{H}})$, ${\mathcal{P}}({\mathcal{H}})$ and
${\mathcal{S}}({\mathcal{H}})$, respectively. The identity operator on
${\mathcal{H}}$ is denoted by $I$. $\ln{x}$ represents the natural logarithm
of $x$. For $p\geq 1$, the Schatten $p$-norm of
$W\in{\mathcal{L}}({\mathcal{H}})$ is defined by $\|W\|_{p}:=({\rm
Tr}[(W^{\dagger}W)^{p/2}])^{1/p}$ (see e.g. Section 1.1.3 in Watrous (2018)).
For any set $S$ in a topological space or a metric space, its closure,
interior and boundary are denoted by ${\rm cl}S$, ${\rm int}S$ and $\partial
S$, respectively.

For quantum states $\sigma,\rho\in{\mathcal{S}}({\mathcal{H}})$, the quantum
relative entropy is defined by

$\displaystyle D(\sigma\|\rho):={\rm Tr}[\sigma\ln{\sigma}-\sigma\ln{\rho}].$
(3)

For a finite alphabet ${\mathcal{X}}$ and probability distributions
$P\equiv\\{p(x)\\}_{x\in{\mathcal{X}}}$ and
$Q\equiv\\{q(x)\\}_{x\in{\mathcal{X}}}$, the classical relative entropy is
defined by

$\displaystyle D(P\|Q):=\sum_{x\in{\mathcal{X}}}p(x)\ln{\frac{p(x)}{q(x)}}.$
(4)

For binary probability distributions $\\{\alpha,1-\alpha\\}$ and
$\\{\beta,1-\beta\\}$, we define the binary relative entropy $\mathscr{D}$ by

$\displaystyle\mathscr{D}(\alpha\|\beta)$
$\displaystyle:=D(\\{\alpha,1-\alpha\\}\|\\{\beta,1-\beta\\}).$ (5)

For the properties of the quantum and classical relative entropies, see e.g.
Wilde (2013); Cover and Thomas (2005).

##  II Formulation and Main Result

Let ${\mathcal{H}}^{S}$ and ${\mathcal{H}}^{E}$ be finite-dimensional Hilbert
spaces such that

$\displaystyle\dim{{\mathcal{H}}^{S}}=m,\;\dim{{\mathcal{H}}^{E}}=n.$ (6)

Let the Hilbert space of the whole system be

$\displaystyle{\mathcal{H}}={\mathcal{H}}^{S}\otimes{\mathcal{H}}^{E}.$ (7)

For each $\mbox{$|\Psi\rangle$}\in{\mathcal{H}}$, we define

$\displaystyle\rho_{\Psi}:={\rm
Tr}_{E}[\mbox{$\mbox{$|\Psi\rangle$}\\!\mbox{$\langle\Psi|$}$}].$ (8)

We consider the unitarily invariant measure on ${\mathcal{H}}$. To clarify the
dependence on the dimension of ${\mathcal{H}}^{E}$, we denote it by $\mu_{n}$.
When $|\Psi\rangle$ is randomly chosen according to $\mu_{n}$, we represent it
as $\Psi\sim\mu_{n}$. The induced random state
$\rho_{\Psi}\in{\mathcal{S}}({\mathcal{H}}^{S})$ is referred to as the
rank-$n$ random state Hayden et al. (2006). We are interested in the
probability of the rank-$n$ random state $\rho_{\Psi}$ within a given subset
$\Omega\subseteq{\mathcal{S}}({\mathcal{H}}^{S})$, namely ${\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}$. We
focus on the asymptotic behavior of the probability in the limit of $n$ to
infinity. We regard ${\mathcal{S}}({\mathcal{H}}^{S})$ as a metric space
equipped with the trace distance. The following theorem is the main result of
this paper:

######  Theorem 1

For any subset $\Omega\subseteq{\mathcal{S}}({\mathcal{H}}^{S})$, it holds
that

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}$
$\displaystyle\geq m\inf_{\rho\in{\rm cl}\Omega}D(\pi\|\rho),$ (9)
$\displaystyle-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}$
$\displaystyle\leq m\inf_{\rho\in{\rm int}\Omega}D(\pi\|\rho),$ (10)

where $\pi\equiv I/m$ is the maximally mixed state on $S$. In particular, for
any subset $\Omega\subseteq{\mathcal{S}}({\mathcal{H}}^{S})$ satisfying ${\rm
cl}({\rm int}\Omega)={\rm cl}\Omega$, it holds that

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}=m\inf_{\rho\in\Omega}D(\pi\|\rho).$
(11)

In the terminology of the large deviation theory Dembo and Zeitouni (2009),
Theorem 1 asserts that the probability density of $\rho_{\Psi}$ satisfies the
large deviation principle with the rate function $mD(\pi\|\rho)$. A proof of
Theorem 1 will be provided in Section IV, based on the preliminary results
presented in Section III.

Figure 1: The example of a one-qubit system is depicted. Every given subset
$\Omega\subset{\mathcal{S}}({\mathbb{C}}^{2})$ is represented by a subset
$\tilde{\Omega}$ of the Bloch ball. Let $\vec{t}$ be the Bloch vector of the
state that is closest to the maximally mixed state $\pi$. Let $S$ be the plane
that is tangent to $\tilde{\Omega}$ at point $\vec{t}$. Let
$d(\tilde{\Omega})$ be the section of the Bloch sphere by $S$. The log of the
inverse radius of $d(\tilde{\Omega})$ is equal to the relative entropy
$D(\pi\|\rho(\vec{t}))$. The twice of it is the exponent for $\Omega$.

We present a few examples for which the exponent can be calculated concretely.
More applications will be provided in Section V.

Example 1 (qubit system): We consider a one-qubit system where $m=2$. Every
state $\rho$ is represented as

$\displaystyle\rho=\rho(\vec{t}):=\frac{1}{2}I+\frac{1}{2}\sum_{\kappa=x,y,z}t_{\kappa}\sigma_{\kappa},$
(12)

where $\vec{t}\equiv(t_{x},t_{y},t_{z})$ is a three dimensional real vector
satisfying $|\vec{t}|:=\sqrt{t_{1}^{2}+t_{2}^{2}+t_{3}^{2}}\leq 1$ and
$(\sigma_{x},\sigma_{y},\sigma_{z})$ is the Pauli operators defined by

$\displaystyle\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\>\sigma_{y}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\>\sigma_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ (13)

(see e.g. Nielsen and Chuang (2000)). The eigenvalues of $\rho(\vec{t})$ are
$(1\pm|\vec{t}|)/2$. Thus, we have

$\displaystyle
D(\pi\|\rho(\vec{t}))=\ln{\left(\frac{1}{\sqrt{1-|\vec{t}|^{2}}}\right)}.$
(14)

The exponent is equal to zero if and only if $\vec{t}=\vec{0}$ (i.e.,
$\rho(\vec{t})=\pi$), and goes to infinity when $\vec{t}$ approaches the
surface of the Bloch sphere (i.e., $|\vec{t}|\rightarrow 1$). Every subset
$\Omega\subset{\mathcal{S}}({\mathbb{C}}^{2})$ is mapped by the correspondence
(12) to a subset $\tilde{\Omega}$ of the unit ball in ${\mathbb{R}}^{3}$.
Thus, we have

$\displaystyle\inf_{\rho\in\Omega}D(\pi\|\rho)=\inf_{\vec{t}\in\tilde{\Omega}}D(\pi\|\rho(\vec{t}))=\inf_{\vec{t}\in\tilde{\Omega}}\ln{\left(\frac{1}{\sqrt{1-|\vec{t}|^{2}}}\right)},$
(15)

which is depicted in Figure 1. $\blacksquare$

Example 2 (Maximum Eigenvalue): Let $\lambda_{\rm max}(\rho)$ denote the
maximum eigenvalue of $\rho\in{\mathcal{S}}({\mathcal{H}}^{S})$. We consider
the set of states such that the maximum eigenvalue is no smaller than or equal
to $(1+(m-1)\epsilon)/m$, where $0<\epsilon<1$. The infimum of the relative
entropy is evaluated as

$\displaystyle\inf_{\rho:\lambda_{\rm
max}(\rho)\geq\frac{1+(m-1)\epsilon}{m}}D(\pi\|\rho)=\inf_{\rho:\lambda_{\rm
max}(\rho)=\frac{1+(m-1)\epsilon}{m}}D(\pi\|\rho)=\mathscr{D}_{\rm
ME}(\epsilon;m),$ (16)

where

$\displaystyle\mathscr{D}_{\rm
ME}(\epsilon;m):=-\frac{1}{m}\left[\ln{\left(1+(m-1)\epsilon\right)}+(m-1)\ln{\left(1-\epsilon\right)}\right].$
(17)

Indeed, the infimum is achieved by a state

$\displaystyle\tilde{\rho}_{\epsilon}=\frac{1+(m-1)\epsilon}{m}\mbox{$\mbox{$|1\rangle$}\\!\mbox{$\langle
1|$}$}+\frac{1-\epsilon}{m}\sum_{k=2}^{m}\mbox{$\mbox{$|k\rangle$}\\!\mbox{$\langle
k|$}$}.$ (18)

Note that every state satisfying $\lambda_{\rm
max}(\rho)\geq(1+(m-1)\epsilon)/m$ can be transformed to $\varrho$ by a random
unitary operation, under which $D(\pi\|\rho)$ is non-increasing. A graph of
the function $\mathscr{D}_{\rm ME}(\epsilon;m)$ for the case of $m=3$ is
presented in Figure 2. In the second order of $\epsilon$, the function (17) is
equal to $(m-1)\epsilon^{2}/2$. This is a constant factor improvement over the
large deviation bound obtained from Lemma III.4 in Hayden et al. (2006) (see
Appendix D for the detail). $\blacksquare$

Figure 2:  A graph of the function $\tilde{\mathscr{D}}_{\rm
ME}(\epsilon):=\mathscr{D}_{\rm ME}(\epsilon;m=3)$, which is given by (17), is
presented. It goes to infinity in the limit of $\epsilon\rightarrow 1$.

Example 3 (Binary Projective Measurement): Consider a binary projective
measurement on $S$, which is represented by a pair $\\{\Pi_{0},\Pi_{1}\\}$ of
orthogonal projectors on ${\mathcal{H}}^{S}$ such that $\Pi_{0}+\Pi_{1}=I$.
The probability of obtaining an outcome $s=0,1$ when the above measurement is
performed on a state $\rho$ is described by the probability distribution
$P_{\rho}\equiv\\{p_{\rho},1-p_{\rho}\\}$, where $p_{\rho}={\rm
Tr}[\Pi_{0}\rho]$. Let $Q\equiv\\{q,1-q\\}$ be a binary probability
distribution. The variational distance (see e.g. Cover and Thomas (2005))
between $P_{\rho}$ and $Q$ is calculated to be $|p_{\rho}-q|$. We consider the
set of states such that $|p_{\rho}-q|$ is no greater than $\delta$. We have
that

$\displaystyle\inf_{\rho:|p_{\rho}-q|\leq\delta}D(\pi\|\rho)=\inf_{P^{\prime}:|p^{\prime}-q|\leq\delta}\inf_{\rho:P_{\rho}=P^{\prime}}D(\pi\|\rho),$
(19)

where $P^{\prime}\equiv\\{p^{\prime},1-p^{\prime}\\}$. The monotonicity of the
relative entropy implies

$\displaystyle\inf_{\rho:P_{\rho}=P^{\prime}}D(\pi\|\rho)\geq\inf_{\rho:P_{\rho}=P^{\prime}}D(P_{\pi}\|P_{\rho})=\mathscr{D}\left(\left.\frac{m_{0}}{m}\right\|p^{\prime}\right),$
(20)

where $m_{0}:={\rm rank}\Pi_{0}$ and $\mathscr{D}$ is the binary relative
entropy defined by (5). The equality holds for the state
$\rho_{P^{\prime}}:=p^{\prime}\pi_{0}+(1-p^{\prime})\pi_{1}$, where
$\pi_{s}:=\Pi_{s}/{\rm Tr}[\Pi_{s}]$ for $s=0,1$. Hence,

$\displaystyle\inf_{\rho:|p_{\rho}-q|\leq\delta}D(\pi\|\rho)=\inf_{P^{\prime}:|p^{\prime}-q|\leq\delta}\mathscr{D}\left(\left.\frac{m_{0}}{m}\right\|p^{\prime}\right).$
(21)

Due to the continuity of the binary relative entropy, we obtain

$\displaystyle\lim_{\delta\rightarrow
0}\inf_{\rho:|p_{\rho}-q|\leq\delta}D(\pi\|\rho)=\mathscr{D}\left(\left.\frac{m_{0}}{m}\right\|q\right).$
(22)

A graph of this function for the case of $m_{0}/m=1/2$ is depicted in Figure
3. $\blacksquare$

Figure 3:  A graph of the function $\tilde{\mathscr{D}}_{\rm
PM}(q):=\mathscr{D}(m_{0}/m=1/2\|q)$ is presented. It goes to infinity in the
limit of $q\rightarrow 0,1$.

We will use the following properties of the quantum relative entropy in the
subsequent sections:

######  Lemma 2

The relative entropy $D(\pi\|\rho)$ satisfies the following properties:

  1. 1.

Lower Bound: $D(\pi\|\rho)\geq 0$ with the equality if and only if $\rho=\pi$.

  2. 2.

Upper Bound: $D(\pi\|\rho)<\infty$ if and only if $\rho$ is a full-rank state.

  3. 3.

Convexity: For any $\rho,\sigma\in{\mathcal{S}}({\mathcal{H}}^{S})$ and any
$0\leq\lambda\leq 1$, it holds that

$\displaystyle D(\pi\|\lambda\rho+(1-\lambda)\sigma)\leq\lambda
D(\pi\|\rho)+(1-\lambda)D(\pi\|\sigma).$ (23)

  4. 4.

Monotonicity: For any unital map ${\mathcal{T}}$, it holds that

$\displaystyle D(\pi\|\rho)\geq D(\pi\|{\mathcal{T}}(\rho)).$ (24)

  5. 5.

Continuity: $D(\pi\|\rho)$ is a continuous function of $\rho$ in the interior
of ${\mathcal{S}}({\mathcal{H}}^{S})$.

  6. 6.

Trace-Log Representation: For any $\rho\in{\mathcal{S}}({\mathcal{H}}^{S})$,
it holds that

$\displaystyle D(\pi\|\rho)=-\frac{1}{m}{\rm Tr}\ln(m\rho).$ (25)

######  Proof:

Properties 1$\sim$5 immediately follow from standard properties of the quantum
relative entropy. Property 6 follows by $-{\rm Tr}\ln(m\rho)=-{\rm
Tr}\left[\ln(mI)+\ln{\rho}\right]={\rm
Tr}\left[\ln{\pi}-\ln{\rho}\right]=m{\rm
Tr}\left[\pi\ln{\pi}-\pi\ln{\rho}\right]=mD(\pi\|\rho)$. $\blacksquare$

##  III Sanov’s theorem for continuous variables

Let $({\mathcal{X}},{\mathcal{F}})$ be a measurable space and let $\Lambda$ be
the set of all probability measures on $({\mathcal{X}},{\mathcal{F}})$. A
probability measure $P\in\Lambda$ is said to be absolutely continuous with
respect to $Q\in\Lambda$ if $P({\mathcal{B}})=0$ for any
${\mathcal{B}}\in{\mathcal{F}}$ satisfying $Q({\mathcal{B}})=0$, which is
represented as $P\ll Q$. For $P,\>Q\in\Lambda$, the relative entropy
(Kullback-Leibler divergence) $D(P\|Q)$ is defined by

$\displaystyle
D(P\|Q):=\begin{cases}\left.\int_{{\mathcal{X}}}\ln{\left(\frac{dP}{dQ}\right)}dP\right.&(P\ll
Q)\\\ +\infty&otherwise.\end{cases}$ (26)

For any subset $\Sigma\subseteq\Lambda$ and any $Q\in\Lambda$, the relative
entropy $D(\Sigma\|Q)$ is defined as

$\displaystyle
D(\Sigma\|Q):=\begin{cases}\inf_{P\in\Sigma}D(P\|Q)&(\Sigma\neq\emptyset)\\\
+\infty&(\Sigma=\emptyset).\end{cases}$ (27)

The following theorem provides us with a way to calculate $D(\Sigma\|Q)$ when
$\Sigma$ is defined in terms of the expectation values of measurable
functions:

######  Theorem 3

[Corollary of Theorem 2 in Csiszár (1984); see also Jupp and Mardia (1983)]
Fix $w\in{\mathbb{N}}$ and let
$\bm{g}:{\mathcal{X}}\rightarrow{\mathbb{R}}^{w}$ be a measurable function on
$({\mathcal{X}},{\mathcal{F}})$. Let $\Sigma\subset\Lambda$ be a convex subset
of probability measures on $({\mathcal{X}},{\mathcal{F}})$ such that

$\displaystyle\Sigma=\left\\{P\in\Lambda\left|\int\bm{g}dP=\bm{0}\right.\right\\}.$
(28)

Let $Q\in\Lambda$ be a probability measure such that $D(\Sigma\|Q)<\infty$.
Suppose that there exists a $P\in\Sigma$ that is mutually absolutely
continuous with $Q$. It holds that

$\displaystyle
D(\Sigma\|Q)=\max_{\bm{\theta}\in{\mathbb{R}}^{w}}\left[-\ln{\left\\{\int{\exp{\left(\bm{\theta}\cdot\bm{g}\right)}dQ}\right\\}}\right].$
(29)

Let $E$ be a real Hausdorff topological space. The notion of $\tau$-continuity
of a function of probability measures to $E$ was introduced in Groeneboom et
al. (1979)(see Section 2 thereof):

######  Definition 4

A function $T:\Lambda\rightarrow E$ is said to be $\tau$-continuous at
$P\in\Lambda$ if, for any neighborhood $V\subset E$ of $T(P)$, there exist
$\delta>0$ and a partition ${\mathfrak{X}}=(\chi_{1},\cdots,\chi_{K})$ of
${\mathcal{X}}$ into sets $\chi_{\kappa}\in{\mathcal{F}}$, such that $T(Q)\in
V$ for any $Q\in\Lambda$ satisfying $\max_{1\leq\kappa\leq
K}|Q(\chi_{\kappa})-P(\chi_{\kappa})|<\delta$.

Let $X$ be a random variable taking values in ${\mathcal{X}}$ according to a
probability measure $Q\in\Lambda$. The empirical distribution
$\hat{P}_{n}^{X^{n}}$ of an $n$-tuple $X^{n}=(X_{1},\cdots,X_{n})$ of
${\mathcal{X}}$-valued random variables is the random probability measure
defined by

$\displaystyle\hat{P}_{n}^{X^{n}}({\mathcal{B}})=\frac{1}{n}|\\{i:X_{i}\in{\mathcal{B}}\\}|,\quad{\mathcal{B}}\in{\mathcal{F}}.$
(30)

The following theorem is one of the extensions of Sanov’s theorem Sanov (1957)
to general probability spaces:

######  Theorem 5

[Theorem 4.1 in Groeneboom et al. (1979)] Let $Q\in\Lambda$ be a probability
measure and let $\\{{\mathcal{B}}_{m}\in{\mathcal{F}}\\}_{m=1}^{\infty}$ be a
sequence of Borel sets such that
${\mathcal{B}}_{m}\subset{\mathcal{B}}_{m+1}\>(m=1,2,\cdots)$ and
$\lim_{m\rightarrow\infty}Q({\mathcal{B}}_{m})=1$. Define
$\Lambda_{m}:=\\{P\in\Lambda|P({\mathcal{B}}_{m})=1\\}$ for $m\in{\mathbb{N}}$
and $\Lambda^{*}:=\cup_{m=1}^{\infty}\Lambda_{m}$. Let $E$ be a real Hausdorff
topological space and let $T:\Lambda^{*}\rightarrow E$ be a function. Suppose
that the restriction $T|_{\Lambda_{m}}$ is linear and $\tau$-continuous at
each $P\in\Lambda_{m}$ satisfying $D(P\|Q)<+\infty$ for each
$m\in{\mathbb{N}}$. Let $A$ be a convex subset of $E$ such that $D(T^{-1}({\rm
int}A)\|Q)<+\infty$. It holds that

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{T(\hat{P}_{n}^{X^{n}})\in A\\}}=D(T^{-1}(A)\|Q).$ (31)

We apply Theorem 3 and 5 to the case where ${\mathcal{X}}$ and $E$ is finite-
dimensional real vector spaces equipped with the standard topology. Fix
$v,w\in{\mathbb{N}}$. Let $\Lambda({\mathbb{R}}^{v})$ be the set of all
probability measures on ${\mathbb{R}}^{v}$. For
$P,\>Q\in\Lambda({\mathbb{R}}^{v})$, the relative entropy $D(P\|Q)$ is defined
as

$\displaystyle
D(P\|Q)=\int_{z\in{\mathbb{R}}^{v}}P(z)\ln{\frac{P(z)}{Q(z)}}dz.$ (32)

The following theorem is immediately obtained as a corollary of Theorem 3:

######  Theorem 6

Let $\tilde{\bm{g}}:{\mathbb{R}}^{v}\rightarrow{\mathbb{R}}^{w}$ be a
continuous function. For $\bm{t}\in{\mathbb{R}}^{w}$, define a subset
$\Sigma_{\bm{t}}\subset\Lambda({\mathbb{R}}^{v})$ by

$\displaystyle\Sigma_{\bm{t}}=\left\\{P\in\Lambda({\mathbb{R}}^{v})\left|\int_{z\in{\mathbb{R}}^{v}}P(z)\tilde{\bm{g}}(z)dz=\bm{t}\right.\right\\}.$
(33)

Let $Q\in\Lambda$ be a probability measure such that
$D(\Sigma_{\bm{t}}\|Q)<+\infty$. Suppose that there exists a probability
measure $P\in\Sigma_{\bm{t}}$ that is mutually absolutely continuous with $Q$.
Then, it holds that

$\displaystyle
D(\Sigma_{\bm{t}}\|Q)=\max_{\bm{\theta}\in{\mathbb{R}}^{w}}\left[-\ln{\left\\{\int_{z\in{\mathbb{R}}^{v}}{Q(z)e^{\bm{\theta}\cdot(\tilde{\bm{g}}(z)-\bm{t})}dz}\right\\}}\right].$
(34)

Let $Z$ be a random variable taking values in ${\mathbb{R}}^{v}$ according to
a probability measure $Q\in\Lambda$. In the same way as (30), the empirical
distribution $\hat{P}_{n}^{Z^{n}}$ of an $n$-tuple
$Z^{n}=(Z_{1},\cdots,Z_{n})$ of ${\mathbb{R}}^{v}$-valued random variables is
the random probability measure defined by

$\displaystyle\hat{P}_{n}^{Z^{n}}({\mathcal{B}})=\frac{1}{n}|\\{i:Z_{i}\in{\mathcal{B}}\\}|,\quad{\mathcal{B}}\subset{\mathbb{R}}^{v}.$
(35)

The following theorem is an extension of Sanov’s theorem to continuous random
variables:

######  Theorem 7

Let $\bm{f}:{\mathbb{R}}^{v}\rightarrow{\mathbb{R}}^{w}$ be a continuous
function. Define a function
$\bm{F}:\Lambda({\mathbb{R}}^{v})\rightarrow{\mathbb{R}}^{w}$ by

$\displaystyle\bm{F}(P)=\int_{z\in{\mathbb{R}}^{v}}P(z)\bm{f}(z)dz$ (36)

and ${\mathfrak{D}}:{\mathbb{R}}^{w}\rightarrow{\mathbb{R}}$ by

$\displaystyle{\mathfrak{D}}(\bm{t})=D(\bm{F}^{-1}(\\{\bm{t}\\})\|Q).$ (37)

Let $\Xi$ be a subset of ${\mathbb{R}}^{w}$ such that ${\mathfrak{D}}$ is
continuous in ${\rm int}\Xi$ and ${\mathfrak{D}}(\bm{t^{\prime}})=+\infty$ for
any $\bm{t^{\prime}}\notin{\rm int}\Xi$. Let $\Theta$ be a subset of $\Xi$
such that $D(\bm{F}^{-1}({\rm int}\Theta)\|Q)<+\infty$. It holds that

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\hat{P}_{n}^{Z^{n}}\in\bm{F}^{-1}(\Theta)\\}}$
$\displaystyle\geq\inf_{\bm{t}\in{\rm cl}\Theta}{\mathfrak{D}}(\bm{t}),$ (38)
$\displaystyle-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\hat{P}_{n}^{Z^{n}}\in\bm{F}^{-1}(\Theta)\\}}$
$\displaystyle\leq\inf_{\bm{t}\in{\rm int}\Theta}{\mathfrak{D}}(\bm{t}).$ (39)

######  Proof:

Follows from Theorem 5. See Appendix A for a detailed proof. $\blacksquare$

##  IV Proof of The Main Result

This section provides a proof of Theorem 1. First, we introduce a few
definitions and notations that will be used in the proof. Second, we prove
Theorem 1 based on two lemmas and Theorem 7 in Section III. A proof of one of
the lemmas regarding the calculation of the exponent will be provided at the
end of this section and is based on Theorem 6 in Section III.

###  IV.1 Definitions and Notations

Let $\\{A_{r}\\}_{r=0}^{m^{2}-1}$ be a set of Hermitian operators on
${\mathcal{H}}^{S}$ that forms an orthonormal basis of the operator Hilbert
space ${\mathcal{L}}({\mathcal{H}}^{S})$ with respect to the Hilbert-Schmidt
inner product, that is, $\frac{1}{m}{\rm
Tr}[A_{r}A_{r^{\prime}}]=\delta_{rr^{\prime}}$. For convenience, we assume
that $A_{0}=I$. We introduce a notation

$\displaystyle{\bm{A}}\equiv(A_{0},\cdots,A_{m^{2}-1}).$ (40)

Define

$\displaystyle{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}:=\\{\bm{t}\in{\mathbb{R}}^{m^{2}}|\bm{t}\cdot\bm{A}\geq
0\\},$ (41)

where $\bm{t}\cdot\bm{A}\equiv\sum_{r=0}^{m^{2}-1}t_{r}A_{r}$ for
$\bm{t}\equiv(t_{0},\cdots,t_{m^{2}-1})\in{\mathbb{R}}^{m^{2}}$. The elements
of the set ${\mathcal{P}}({\mathcal{H}}^{S})$ have a one-to-one correspondence
with those of ${\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$.
Indeed, every element $\varrho$ of ${\mathcal{P}}({\mathcal{H}}^{S})$ is
mapped to an element of
${\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$ by
$\bm{t}(\varrho)=(1/m){\rm Tr}[\bm{A}\varrho]$, the inverse map of which is
given by $\varrho(\bm{t})=\bm{t}\cdot\bm{A}$. For any subset
$\Omega\subseteq{\mathcal{S}}({\mathcal{H}}^{S})$, we define

$\displaystyle{\mathbb{R}}^{m^{2}}_{\Omega}$
$\displaystyle:=\left\\{\bm{t}\in{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}\backslash\\{\bm{0}\\}\left|\frac{\bm{t}\cdot\bm{A}}{{\rm
Tr}[\bm{t}\cdot\bm{A}]}\in\Omega\right.\right\\}.$ (42)

It is straightforward to verify that
${\mathbb{R}}^{m^{2}}_{\Omega}=\lambda{\mathbb{R}}^{m^{2}}_{\Omega}$ for any
$\lambda>0$, thus ${\mathbb{R}}^{m^{2}}_{\Omega}$ is a linear cone. In
addition, we have

$\displaystyle{\rm
cl}{\mathbb{R}}^{m^{2}}_{\Omega}\backslash\\{\bm{0}\\}={\mathbb{R}}^{m^{2}}_{{\rm
cl}\Omega},\quad{\rm
int}{\mathbb{R}}^{m^{2}}_{\Omega}={\mathbb{R}}^{m^{2}}_{{\rm int}\Omega}.$
(43)

Let $\alpha_{r}$ and $\beta_{r}$ be the real part and the imaginary part of
$A_{r}$, i.e.,

$\displaystyle\alpha_{r}:=\frac{A_{r}+A_{r}^{*}}{2},\quad\beta_{r}:=\frac{A_{r}-A_{r}^{*}}{2i}$
(44)

with the superscript $*$ denoting complex conjugate. Define
$\alpha_{r,jk}:={\mathfrak{R}}{\mathfrak{e}}\mbox{$\langle
j|$}A_{r}\mbox{$|k\rangle$}$ and
$\beta_{r,jk}:={\mathfrak{I}}{\mathfrak{m}}\mbox{$\langle
j|$}A_{r}\mbox{$|k\rangle$}$. Let ${\mathcal{A}}_{r}\;(r=0,\cdots,m^{2}-1)$ be
a $2m$-dimensional real symmetric matrix defined by

$\displaystyle{\mathcal{A}}_{r}:=\begin{pmatrix}\alpha_{r}&\beta_{r}\\\
-\beta_{r}&\alpha_{r}\end{pmatrix}=\left(\begin{array}[]{c|c}\left[\alpha_{r,jk}\right]_{jk}&\left[\beta_{r,jk}\right]_{jk}\\\
\hline\cr-\left[\beta_{r,jk}\right]_{jk}&\left[\alpha_{r,jk}\right]_{jk}\end{array}\right).$
(47)

We introduce a notation

$\displaystyle\bm{{\mathcal{A}}}\equiv({\mathcal{A}}_{0},\cdots,{\mathcal{A}}_{m^{2}-1}).$
(48)

A simple calculation yields

$\displaystyle{\mathcal{A}}_{r}$
$\displaystyle=I\otimes\alpha_{r}+i\sigma_{y}\otimes\beta_{r}$ (49)
$\displaystyle=\frac{I+\sigma_{y}}{2}\otimes(\alpha_{r}+i\beta_{r})+\frac{I-\sigma_{y}}{2}\otimes(\alpha_{r}-i\beta_{r})$
(50)
$\displaystyle=\mbox{$\mbox{$|\varphi_{+}\rangle$}\\!\mbox{$\langle\varphi_{+}|$}$}\otimes
A_{r}+\mbox{$\mbox{$|\varphi_{-}\rangle$}\\!\mbox{$\langle\varphi_{-}|$}$}\otimes
A_{r}^{*},$ (51)

where $\varphi_{\pm}$ are the eigenvectors of $\sigma_{y}$. It follows that

$\displaystyle\bm{{\mathcal{A}}}=\mbox{$\mbox{$|\varphi_{+}\rangle$}\\!\mbox{$\langle\varphi_{+}|$}$}\otimes\bm{A}+\mbox{$\mbox{$|\varphi_{-}\rangle$}\\!\mbox{$\langle\varphi_{-}|$}$}\otimes\bm{A}^{*}.$
(52)

Let $\Lambda({\mathbb{R}}^{2m})$ be the set of all probability measures on
${\mathbb{R}}^{2m}$. Define functionals
$\Phi_{r}:\Lambda({\mathbb{R}}^{2m})\rightarrow{\mathbb{R}}\>(r=0,\cdots,m^{2}-1)$
and $\bm{\Phi}:\Lambda({\mathbb{R}}^{2m})\rightarrow{\mathbb{R}}^{m^{2}}$ by

$\displaystyle\Phi_{r}(P):=\int_{{\mathbb{R}}^{2m}}P(z)z^{T}{\mathcal{A}}_{r}zdz$
(53)

and

$\displaystyle\bm{\Phi}(P):=(\Phi_{0}(P),\cdots,\Phi_{m^{2}-1}(P)).$ (54)

It is straightforward to verify that $\bm{\Phi}$ satisfies the condition of
$\bm{F}$ in Theorem 7.

The one-dimensional standard normal distribution is defined by the probability
density

$\displaystyle
G_{1}(z):=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}z^{2}}\quad(z\in{\mathbb{R}}).$
(55)

Likewise, the standard normal distribution on the $D$-dimensional real vector
space is defined by the probability density

$\displaystyle
G_{D}(z):=\frac{1}{\sqrt{(2\pi)^{D}}}e^{-\frac{1}{2}z^{T}z}\quad(z\in{\mathbb{R}}^{D}),$
(56)

which is equivalent to $G_{D}(z)=\prod_{d=1}^{D}G_{1}(z_{d})$ for
$z\equiv(z_{1},\cdots,z_{D})$.

###  IV.2 Proof of Theorem 1

We prove Theorem 1 based on the two lemmas presented below. The first one is
that the unitary invariant measure on the set of normalized state vectors are
generated by the multivariate standard normal distribution:

######  Lemma 8

Let $\\{X_{d}\\}_{d=1}^{D}$ and $\\{Y_{d}\\}_{d=1}^{D}$ be a collection of
independent and identically distributed random variables that obey the
standard normal distribution. Let $\\{|e_{d}\rangle\\}_{d=1}^{D}$ be a fixed
orthonormal basis of ${\mathbb{C}}^{D}$, and let
${\mathcal{S}}({\mathbb{C}}^{D})\subset{\mathbb{C}}^{D}$ be the set of unit
vectors in ${\mathbb{C}}^{D}$. Define a vector-valued random variable $Z$,
taking values in ${\mathbb{C}}^{D}$, as
$Z=\sum_{d=1}^{D}(X_{d}+iY_{d})\mbox{$|e_{d}\rangle$}.$ Let $\mu:{\rm
Borel}({\mathcal{S}}({\mathbb{C}}^{D}))\rightarrow[0,1]$ be the Borel
probability measure on ${\mathcal{S}}({\mathbb{C}}^{D})$ defined by
$\mu({\mathcal{B}})={\rm Pr}\\{\lambda Z\in{\mathcal{B}}\text{ for some
}\lambda>0\\}$. The measure $\mu$ is unitarily invariant, i.e.,
$\mu(U{\mathcal{B}})=\mu({\mathcal{B}})$ for any ${\mathcal{B}}\in{\rm
Borel}({\mathcal{S}}({\mathbb{C}}^{D}))$ and any unitary matrix $U$ on
${\mathbb{C}}^{D}$.

######  Proof:

See e.g. Section 7.2 in Watrous (2018). $\blacksquare$

The second one is that the infimum of the relative entropy between the set
$\Phi^{-1}(\\{\bm{t}\\})$ and the standard normal distribution is represented
by the unnormalized density matrix corresponding to $\bm{t}$:

######  Lemma 9

For any $\bm{t}\in{\mathbb{R}}^{m^{2}}$, define

$\displaystyle{\mathfrak{D}}(\bm{t}):=D(\bm{\Phi}^{-1}(\\{\bm{t}\\})\|G_{2m}).$
(57)

It holds that

$\displaystyle{\mathfrak{D}}(\bm{t})=\begin{cases}{\rm
Tr}\left[\varsigma(\bm{t})-I-\ln{(\varsigma(\bm{t}))}\right]&(\bm{t}\in{\rm
int}{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})})\\\
+\infty&(\bm{t}\notin{\rm
int}{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})})\end{cases},$ (58)

where $\varsigma(\bm{t}):=\bm{t}\cdot\bm{A}/2m$. Thus,
${\mathfrak{D}}(\bm{t})$ is continuous in
${\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$.

######  Proof:

See Section IV.3. $\blacksquare$

Proof of Theorem 1: Let $\\{\mbox{$|k\rangle$}\\}_{k=1}^{m}$ and
$\\{\mbox{$|l\rangle$}\\}_{l=1}^{n}$ be a fixed orthonormal basis of
${\mathcal{H}}^{S}$ and ${\mathcal{H}}^{E}$, respectively. Let the orthonormal
basis of the whole space
${\mathcal{H}}={\mathcal{H}}^{S}\otimes{\mathcal{H}}^{E}$ be
$\\{\mbox{$|e_{kl}\rangle$}\\}_{1\leq k\leq m,1\leq l\leq n}$, where
$\mbox{$|e_{kl}\rangle$}=\mbox{$|k\rangle$}\mbox{$|l\rangle$}$. Let $X_{kl}$
and $Y_{kl}$ ($1\leq k\leq m,1\leq l\leq n$) be real random variables that are
independently and identically distributed with respect to the standard normal
distribution (SND). Let $|\tilde{\Psi}\rangle$ be a random unnormalized vector
defined by

$\displaystyle\mbox{$|\tilde{\Psi}\rangle$}=\sum_{\begin{subarray}{c}1\leq
k\leq m\\\ 1\leq l\leq
n\end{subarray}}(X_{kl}+iY_{kl})\mbox{$|e_{kl}\rangle$}.$ (59)

With a slight abuse of notation, we denote $\tilde{\Psi}\sim{\rm SND}(n)$. Due
to Lemma 8, the random state $|\Psi\rangle$ distributed with respect to the
unitarily invariant measure on ${\mathcal{H}}$ is equivalent to the random
state $\mbox{$|\tilde{\Psi}\rangle$}/\|\mbox{$|\tilde{\Psi}\rangle$}\|$. With
$\varrho_{\tilde{\Psi}}$ denoting ${\rm
Tr}_{E}[|\tilde{\Psi}\rangle\\!\langle\tilde{\Psi}|]$, we have the following
equivalence relations:

$\displaystyle{\rm
Tr}_{E}\left[\frac{\mbox{$\mbox{$|\tilde{\Psi}\rangle$}\\!\mbox{$\langle\tilde{\Psi}|$}$}}{\|\mbox{$|\tilde{\Psi}\rangle$}\|^{2}}\right]\in\Omega\;\Leftrightarrow\;\frac{\varrho_{\tilde{\Psi}}}{{\rm
Tr}[\varrho_{\tilde{\Psi}}]}\in\Omega\;\Leftrightarrow\;{\rm
Tr}[\bm{A}\varrho_{\tilde{\Psi}}]\in{\mathbb{R}}^{m^{2}}_{\Omega}.$ (60)

Thus, noting that ${\rm
Tr}[{\bm{A}}\varrho_{\tilde{\Psi}}]=\langle\tilde{\Psi}|({\bm{A}}^{S}\otimes
I^{E})|\tilde{\Psi}\rangle$, we have

$\displaystyle{\rm
Pr}\left\\{\rho_{\Psi}\in\Omega|\Psi\sim\mu_{n}\right\\}={\rm
Pr}\left\\{\left.\mbox{$\langle\tilde{\Psi}|$}({\bm{A}}^{S}\otimes
I^{E})\mbox{$|\tilde{\Psi}\rangle$}\in{\mathbb{R}}^{m^{2}}_{\Omega}\right|\tilde{\Psi}\sim\text{SND}(n)\right\\}.$
(61)

To evaluate the probability (61), we introduce a $2m$-dimensional real vector
random variable

$\displaystyle
Z=(X_{1},\cdots,X_{k},\cdots,X_{m};Y_{1},\cdots,Y_{k^{\prime}},\cdots,Y_{m})^{T}.$
(62)

Here, $X_{k}$ and $Y_{k^{\prime}}$ are distributed independently with respect
to the standard normal distribution. I.e., $Z$ obeys a probability
distribution

$\displaystyle G_{2m}(Z)=\prod_{k=1}^{m}G_{1}(X_{k})G_{1}(Y_{k}).$ (63)

Let $Z^{n}\equiv(Z_{1},\cdots,Z_{l},\cdots,Z_{n})$ be an independently and
identically distributed sequence of the random variable $Z$, where

$\displaystyle
Z_{l}\equiv(X_{1l},\cdots,X_{kl},\cdots,X_{ml};Y_{1l},\cdots,Y_{k^{\prime}l},\cdots,Y_{ml})^{T}.$
(64)

Noting that $\beta_{r,jk}=-\beta_{r,kj}$, we have

$\displaystyle\mbox{$\langle\tilde{\Psi}|$}(A_{r}^{S}\otimes
I^{E})\mbox{$|\tilde{\Psi}\rangle$}$ $\displaystyle=\sum_{1\leq l\leq
n}\left(\sum_{\begin{subarray}{c}1\leq j\leq m\\\ 1\leq k\leq
m\end{subarray}}\alpha_{r,jk}(X_{jl}X_{kl}+Y_{jl}Y_{kl})\right)+\sum_{1\leq
l\leq n}\left(\sum_{\begin{subarray}{c}1\leq j\leq m\\\ 1\leq k\leq
m\end{subarray}}\beta_{r,jk}(X_{jl}Y_{kl}-Y_{jl}X_{kl})\right)$ (65)
$\displaystyle=\sum_{1\leq l\leq n}Z_{l}^{T}{\mathcal{A}}_{r}Z_{l}.$ (66)

Thus, the probability (61) is equal to

$\displaystyle{\rm Pr}\left\\{\left.\frac{1}{n}\sum_{1\leq l\leq
n}Z_{l}^{T}\bm{{\mathcal{A}}}Z_{l}\in{\mathbb{R}}^{m^{2}}_{\Omega}\right|Z_{l}\sim\text{SND}\right\\}={\rm
Pr}\left\\{\hat{P}_{n}^{Z^{n}}\in\bm{\Phi}^{-1}({\mathbb{R}}^{m^{2}}_{\Omega})\right\\},$
(67)

where $\hat{P}_{n}^{Z^{n}}$ denotes the empirical distribution of the
sequence.

We apply Theorem 7, which is possible due to Lemma 9. It follows that

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\hat{P}_{n}^{Z^{n}}\in\bm{\Phi}^{-1}({\mathbb{R}}^{m^{2}}_{\Omega})\right\\}}$
$\displaystyle\geq\inf_{{\bm{t}}\in{\rm
cl}{\mathbb{R}}^{m^{2}}_{\Omega}}{\mathfrak{D}}({\bm{t}}),$ (68)
$\displaystyle-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\hat{P}_{n}^{Z^{n}}\in\bm{\Phi}^{-1}({\mathbb{R}}^{m^{2}}_{\Omega})\right\\}}$
$\displaystyle\leq\inf_{{\bm{t}}\in{\rm
int}{\mathbb{R}}^{m^{2}}_{\Omega}}{\mathfrak{D}}({\bm{t}}).$ (69)

For both $\ast={\rm cl},{\rm int}$, the infimum is calculated as

$\displaystyle\inf_{{\bm{t}}\in\ast{\mathbb{R}}^{m^{2}}_{\Omega}}{\mathfrak{D}}({\bm{t}})$
$\displaystyle=\inf_{\bm{t}\in\ast{\mathbb{R}}^{m^{2}}_{\Omega}}{\rm
Tr}\left[\frac{\bm{t}\cdot\bm{A}}{2m}-I-\ln{\left(\frac{\bm{t}\cdot\bm{A}}{2m}\right)}\right]$
(70)
$\displaystyle=\inf_{\bm{t}\in\ast{\mathbb{R}}^{m^{2}}_{\Omega}}\left[m\left(\frac{t_{0}}{2m}-1-\ln{\frac{t_{0}}{2m}}\right)-{\rm
Tr}\ln{\left(\frac{\bm{t}\cdot\bm{A}}{t_{0}}\right)}\right]$ (71)
$\displaystyle=\inf_{\bm{t}\in\ast{\mathbb{R}}^{m^{2}}_{\Omega}}-{\rm
Tr}\ln{\left(\frac{\bm{t}\cdot\bm{A}}{t_{0}}\right)}$ (72)
$\displaystyle=\inf_{\rho\in\ast\Omega}-{\rm Tr}\ln{(m\rho)}$ (73)
$\displaystyle=m\inf_{\rho\in\ast\Omega}D(\pi\|\rho).$ (74)

Here, (72) follows from $\inf_{t_{0}>0}\\{t_{0}/2m-1-\ln{(t_{0}/2m)}\\}=0$ and
the fact that $\ast{\mathbb{R}}^{m^{2}}_{\Omega}$ is a linear cone; (73) from
(43) the fact that the condition $\bm{t}\in\ast{\mathbb{R}}^{m^{2}}_{\Omega}$
is equivalent to $\bm{t}\cdot\bm{A}/mt_{0}\in\ast\Omega$ because ${\rm
Tr}[\bm{t}\cdot\bm{A}]=mt_{0}$; and the last line due to (25). All in all, we
obtain (9) and (10).

When $\Omega$ satisfies ${\rm cl}({\rm int}\Omega)={\rm cl}\Omega$, it holds
that

$\displaystyle\inf_{\rho\in{\rm cl}\Omega}D(\pi\|\rho)=\inf_{\rho\in{\rm
int}\Omega}D(\pi\|\rho)$ (75)

due to the continuity of $D(\pi\|\rho)$. This implies (11) and complete the
proof of Theorem 1. $\blacksquare$

###  IV.3 Proof of Lemma 9

Let $z\equiv(z_{1},\cdots,z_{2m})^{T}$ denote the concrete values that $Z$
takes. By definition, the condition $P\in\bm{\Phi}^{-1}(\\{\bm{t}\\})$ is
equivalent to

$\displaystyle\int_{z\in{\mathbb{R}}^{2m}}P(z)z^{T}\bm{{\mathcal{A}}}zdz=\bm{t}.$
(76)

We first consider the case where $\bm{t}\in{\rm
int}{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$, which is
equivalent to $\bm{t}\cdot\bm{A}>0$. Define
$\bm{\theta}^{*}(\bm{t})\in{\mathbb{R}}^{m^{2}}$ by

$\displaystyle\bm{\theta}^{*}(\bm{t}):=\frac{{\rm Tr}{\bm{A}}}{2m}-{\rm
Tr}[{\bm{A}}(\bm{t}\cdot{\bm{A}})^{-1}].$ (77)

Due to the complete orthonormality of $\\{A_{r}\\}_{r=0}^{m^{2}-1}$, the
expression (77) is equivalent to

$\displaystyle
I-2\bm{\theta}^{*}(\bm{t})\cdot\bm{A}=\left(\frac{\bm{t}\cdot\bm{A}}{2m}\right)^{-1},$
(78)

which in turn is equivalent to

$\displaystyle\frac{\partial}{\partial\bm{\theta}}\left.\left[\bm{\theta}\cdot\bm{t}+{\rm
Tr}\ln{\left(I-2\bm{\theta}\cdot\bm{A}\right)}\right]\right|_{\bm{\theta}=\bm{\theta}^{*}(\bm{t})}=\bm{0}.$
(79)

Define a function
${\mathfrak{L}}:{\mathbb{R}}^{m^{2}}\times{\mathbb{R}}^{m^{2}}\rightarrow{\mathbb{R}}$
by

$\displaystyle{\mathfrak{L}}(\bm{\theta},\bm{t})=\int_{z\in{\mathbb{R}}^{2m}}{G_{2m}(z)\exp{\left(\bm{\theta}\cdot(z^{T}\bm{{\mathcal{A}}}z-\bm{t})\right)}dz}.$
(80)

With ${\mathcal{I}}$ denoting the $2m\times 2m$ identity matrix, we have

$\displaystyle
G_{2m}(z)\exp{\left(\bm{\theta}\cdot(z^{T}\bm{{\mathcal{A}}}z-\bm{t})\right)}=\frac{1}{(2\pi)^{m}}\exp{\left(-\bm{\theta}\cdot\bm{t}-\frac{1}{2}z^{T}({\mathcal{I}}-2\bm{\theta}\cdot\bm{{\mathcal{A}}})z\right)}.$
(81)

Thus, by taking the Gaussian integral, we obtain

$\displaystyle{\mathfrak{L}}(\bm{\theta},\bm{t})=\begin{cases}\exp{(-\bm{\theta}\cdot\bm{t})}({\rm
det}({\mathcal{I}}-2\bm{\theta}\cdot\bm{{\mathcal{A}}}))^{-1/2}&({\mathcal{I}}-2\bm{\theta}\cdot\bm{{\mathcal{A}}}>0)\\\
+\infty&\text{otherwise}.\end{cases}$ (82)

From (52), it holds that

$\displaystyle{\mathcal{I}}-2\bm{\theta}\cdot\bm{{\mathcal{A}}}=\mbox{$\mbox{$|\varphi_{+}\rangle$}\\!\mbox{$\langle\varphi_{+}|$}$}\otimes(I-2\bm{\theta}\cdot\bm{A})+\mbox{$\mbox{$|\varphi_{-}\rangle$}\\!\mbox{$\langle\varphi_{-}|$}$}\otimes(I-2\bm{\theta}\cdot\bm{A}^{*}).$
(83)

Hence, (82) yields

$\displaystyle-\ln{\mathfrak{L}}(\bm{\theta},\bm{t})=\begin{cases}\bm{\theta}\cdot\bm{t}+{\rm
Tr}\ln(I-2\bm{\theta}\cdot\bm{A})&(I-2\bm{\theta}\cdot\bm{A}>0)\\\
-\infty&\text{otherwise}.\end{cases}$ (84)

Due to (79) and the concavity of the log function, we obtain

$\displaystyle\bm{\theta}^{*}(\bm{t})=\mathop{\rm
arg~{}max}\limits_{\bm{\theta}\in{\mathbb{R}}^{m^{2}}}\left[-\ln{{\mathfrak{L}}(\bm{\theta},\bm{t})}\right].$
(85)

From (77), we have

$\displaystyle\bm{\theta}^{*}(\bm{t})\cdot\bm{t}={\rm
Tr}\left[\frac{\bm{t}\cdot\bm{A}}{2m}-I\right].$ (86)

Substituting this and (78) to (84), and noting that we have
$I-2\bm{\theta}^{*}(\bm{t})\cdot\bm{A}>0$ due to (78), we arrive at

$\displaystyle\max_{\bm{\theta}\in{\mathbb{R}}^{m^{2}}}\left[-\ln{{\mathfrak{L}}(\bm{\theta},\bm{t})}\right]=-\ln{{\mathfrak{L}}(\bm{\theta}^{*}(\bm{t}),\bm{t})}={\rm
Tr}\left[\frac{\bm{t}\cdot\bm{A}}{2m}-I-\ln{\left(\frac{\bm{t}\cdot\bm{A}}{2m}\right)}\right].$
(87)

Define a probability measure $P^{*}$ on ${\mathbb{R}}^{2m}$ by

$\displaystyle
P^{*}(z)=\frac{G_{2m}(z)\exp{\left[\bm{\theta}^{*}(\bm{t})\cdot(z^{T}\bm{{\mathcal{A}}}z-\bm{t})\right]}}{{\mathfrak{L}}(\bm{\theta}^{*}(\bm{t}),\bm{t})}.$
(88)

It is obvious that $P^{*}(z)>0$ for any $z\in{\mathbb{R}}^{2m}$. In addition,
we have

$\displaystyle\int_{z\in{\mathbb{R}}^{2m}}P^{*}(z)(z^{T}\bm{{\mathcal{A}}}z-\bm{t})dz=\left.\frac{\partial}{\partial\bm{\theta}}\ln{\mathfrak{L}}(\bm{\theta},\bm{t})\right|_{\bm{\theta}=\bm{\theta}^{*}(\bm{t})}=\bm{0},$
(89)

which implies $P^{*}\in\bm{\Phi}^{-1}(\\{\bm{t}\\})$. Furthermore, a simple
calculation using (89) yields

$\displaystyle
D(P^{*}\|G_{2m})=-\ln{{\mathfrak{L}}(\bm{\theta}^{*}(\bm{t}),\bm{t})}<+\infty.$
(90)

Thus, it follows from Theorem 6 and (87) that

$\displaystyle D(\bm{\Phi}^{-1}(\\{\bm{t}\\})\|G_{2m})={\rm
Tr}\left[\varsigma(\bm{t})-I-\ln(\varsigma(\bm{t}))\right].$ (91)

It remains to consider the case where $\bm{t}\notin{\rm
int}{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$. For each
$z\in{\mathbb{R}}^{2m}$, define an unnormalized vector
$|z\rangle\in{\mathcal{H}}^{S}$ by

$\displaystyle|z\rangle:=\sum_{l=1}^{m}(z_{l}-iz_{m+l})|e_{l}\rangle.$ (92)

Similarly to (51), we have
$z=|\varphi_{+}\rangle|z\rangle+|\varphi_{-}\rangle|z^{*}\rangle$. Thus, the
condition (76) is equivalent to

$\displaystyle\int_{z\in{\mathbb{R}}^{2m}}P(z)\mbox{$\mbox{$|z\rangle$}\\!\mbox{$\langle
z|$}$}dz=\frac{\bm{t}\cdot\bm{A}}{m}.$ (93)

For $\bm{t}\notin{\rm
cl}{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$, at least one of
the eigenvalues of $\bm{t}\cdot\bm{A}$ is negative. Thus, no probability
measure $P$ satisfies (93), which implies ${\mathfrak{D}}(\bm{t})=+\infty$.
For
$\bm{t}\in\partial{\mathbb{R}}^{m^{2}}_{{\mathcal{P}}({\mathcal{H}}^{S})}$, at
least one of the eigenvalues of $\bm{t}\cdot\bm{A}$ is equal to zero. Thus,
there exists a linear subspace
${\mathbb{V}}_{\bm{t}}\subsetneq{\mathbb{R}}^{2m}$ such that ${\rm
supp}P\subset{\mathbb{V}}_{\bm{t}}$ for any $P$ satisfying (93). This implies
${\mathfrak{D}}(\bm{t})=+\infty$ and completes the proof. $\blacksquare$

##  V Applications

We apply Theorem 1 to a few cases for which the exponent can be calculated
concretely.

###  V.1 Trace Distance

We consider the set of states such that the trace distance with the maximally
mixed state $\pi$ is no smaller than $t$, where $0<t<1$. We prove that

$\displaystyle\inf_{\rho:\frac{1}{2}\|\pi-\rho\|_{1}=t}D(\pi\|\rho)=\mathscr{D}_{\rm
TD}(t;m):=\min_{m^{\prime}}\mathscr{D}\left(\left.\frac{m^{\prime}}{m}\right\|\frac{m^{\prime}}{m}+t\right),$
(94)

where the minimum is taken over $m^{\prime}\in\\{1,\cdots,\lfloor
m(1-t)\rfloor\\}$. Since $\mathscr{D}_{\rm TD}(t;m)$ is monotonically non-
decreasing in $t$, it follows that

$\displaystyle\inf_{\rho:\frac{1}{2}\|\pi-\rho\|_{1}\geq
t}D(\pi\|\rho)=\mathscr{D}_{\rm TD}(t;m).$ (95)

First, we prove that $D(\pi\|\rho)\geq\mathscr{D}_{\rm TD}(t;m)$ for any
$\rho$ satisfying $\|\pi-\rho\|_{1}/2=t$. Let $\Pi_{\rho}$ be the projection
of the subspace of ${\mathcal{H}}^{S}$ spanned by the eigenvectors of
$\rho-\pi$ with nonnegative eigenvalues. Define probability distributions
$P\equiv\\{p(0),p(1)\\}$ and $Q\equiv\\{q(0),q(1)\\}$ by

$\displaystyle p(0)={\rm Tr}[\Pi_{\rho}\rho],\;p(1)={\rm
Tr}[(I-\Pi_{\rho})\rho],\;q(0)={\rm Tr}[\Pi_{\rho}\pi],\;q(1)={\rm
Tr}[(I-\Pi_{\rho})\pi].$ (96)

Due to the property of the trace distance (see e.g. Section 9.1.3 of Wilde
(2013)), we have

$\displaystyle p(0)-q(0)=q(1)-p(1)=t.$ (97)

Letting $m^{\prime}\equiv{\rm rank}\Pi_{\rho}$, it follows that

$\displaystyle
q(0)=\frac{m^{\prime}}{m},\;q(1)=1-\frac{m^{\prime}}{m},\;p(0)=\frac{m^{\prime}}{m}+t,\;p(1)=1-\frac{m^{\prime}}{m}-t.$
(98)

Thus, due to the monotonicity of the quantum relative entropy, we obtain
$D(\pi\|\rho)\geq D(P\|Q)\geq\mathscr{D}_{\rm TD}(t;m)$.

Second, we prove that there exists a state $\varrho$ such that
$\|\pi-\varrho\|_{1}/2=t$ and $D(\pi\|\varrho)=\mathscr{D}_{\rm TD}(t;m)$.
Define

$\displaystyle m^{*}_{t}:=\mathop{\rm
arg~{}min}\limits_{m^{\prime}}\mathscr{D}\left(\left.\frac{m^{\prime}}{m}\right\|\frac{m^{\prime}}{m}+t\right),$
(99)

where $m^{\prime}$ runs over $\\{1,\cdots,\lfloor m(1-t)\rfloor\\}$. Let
$\Pi_{m^{*}_{t}}$ be an arbitrary projection of rank $m^{*}_{t}$. Consider a
state

$\displaystyle\varrho_{t}^{*}=\left(\frac{m^{*}_{t}}{m}+t\right)\frac{\Pi_{m^{*}_{t}}}{m^{*}_{t}}+\left(1-\frac{m^{*}_{t}}{m}-t\right)\frac{I-\Pi_{m^{*}_{t}}}{m-m^{*}_{t}}.$
(100)

It is straightforward to verify that $D(\pi\|\varrho_{t}^{*})=\mathscr{D}_{\rm
TD}(t;m)$.

We consider the case where $m\gg 1$ and $t\ll 1$. Due to the properties of the
binary relative entropy (see Appendix B), it follows from (94) and (95) that

$\displaystyle\inf_{\rho:\frac{1}{2}\|\pi-\rho\|_{1}=t}D(\pi\|\rho)\approx\min_{\alpha}\mathscr{D}\left(\left.\alpha\right\|\alpha+t\right)\approx\frac{t^{2}}{2}.$
(101)

###  V.2 Entropy

Figure 4:  This graph shows the way how
$\mathscr{D}\left(\left.\mu/m\right\|r^{*}(\mu,\eta)\right)$ is obtained for a
fixed $m$, $\mu$ and $\eta$. The solid line and the dashed line represents the
functions $\mathscr{D}\left(\left.\mu/m\right\|r\right)$ and
$\mathscr{D}\left(r\left\|\mu/m\right.\right)$, respectively.

The von Neumann entropy of a state $\rho\in{\mathcal{S}}({\mathcal{H}}^{S})$
is defined by

$\displaystyle H(\rho):=-{\rm Tr}[\rho\ln{\rho}].$ (102)

We prove that

$\displaystyle\inf_{\rho:H(\rho)=\eta\ln{m}}D(\pi\|\rho)=\mathscr{D}_{\rm
vN}(\eta;m):=\min_{\mu}\mathscr{D}\left(\left.\frac{\mu}{m}\right\|r^{*}(\mu,\eta)\right)$
(103)

for any $0<\eta<1$, where the minimum is taken over
$\mu\in\\{1,\cdots,m-1\\}$. Here, $r^{*}(\mu,\eta)$ is the solution of an
equation

$\displaystyle\mathscr{D}\left(r\left\|\frac{\mu}{m}\right.\right)=(1-\eta)\ln{m}.$
(104)

A graphical representation of the function
$\mathscr{D}\left(\left.\frac{\mu}{m}\right\|r^{*}(\mu,\eta)\right)$ is
presented in Figure 4. Due to the convexity of the relative entropy, the
equation (104) has at most two solutions. Since (103) and (104) are both
invariant under the simultaneous exchange of $\mu$ with $m-\mu$ and $r$ with
$1-r$, we may assume $r^{*}(\mu,\eta)$ to be the smaller one of the solutions.
Due to the property of the binary relative entropy (see Appendix B),
$\mathscr{D}_{\rm vN}(\eta;m)$ is monotonically non-increasing in $\eta$.
Thus, it holds that

$\displaystyle\inf_{\rho:H(\rho)\leq\eta\ln{m}}D(\pi\|\rho)=\mathscr{D}_{\rm
vN}(\eta;m).$ (105)

To prove (103), note that both $H(\rho)$ and $D(\pi\|\rho)$ depend only on the
spectrum of $\rho$. Thus, it is sufficient to consider the case where $\pi$
and $\rho$ are classical probability distributions. That is, we only need to
prove that

$\displaystyle\inf_{P:H(P)=\eta\ln{m}}D(U\|P)=\mathscr{D}_{\rm vN}(\eta;m),$
(106)

where the infimum is taken over all probability distributions $P$ on the set
$\\{1,\cdots,m\\}$ and $U$ is the uniform distribution thereon. Since the set
of probability distributions satisfying $H(P)=\eta\ln{m}$ is closed, the
infimum in (106) can be replaced by the minimum. Hence, denoting
$P\equiv\\{p(1),\cdots,p(m)\\}$, we are to minimize the function

$\displaystyle D(U\|P)=-\ln{m}-\frac{1}{m}\sum_{k=1}^{m}\ln{p(k)},$ (107)

under the condition that

$\displaystyle\sum_{k=1}^{m}p(k)=1,\quad-\sum_{k=1}^{m}p(k)\ln{p(k)}=\eta\ln{m}.$
(108)

We apply the Lagrange multiplier method. Let
$P_{*}\equiv\\{p_{*}(1),\cdots,p_{*}(m)\\}$ be

$\displaystyle P_{*}:=\mathop{\rm
arg~{}min}\limits_{P:H(P)=\eta\ln{m}}D(U\|P).$ (109)

With $u$ and $v$ being the Lagrange multipliers, the condition for the
extremal value is given by

$\displaystyle\frac{1}{p_{*}(k)}-u(1+\ln{p_{*}(k)})-v=0\;(k=1,\cdots,m).$
(110)

Due to the concavity of the logarithm function, every equation of the form

$\displaystyle x+C_{1}\ln{x}+C_{2}=0\quad(C_{1},C_{2}\in{\mathbb{R}})$ (111)

has at most two solutions. Thus, $P_{*}$ is of the form

$\displaystyle p(k)=\begin{cases}q_{0}&(1\leq k\leq\mu)\\\ q_{1}&(\mu+1\leq
k\leq m)\end{cases},$ (112)

where $\mu\in\\{1,\cdots,m\\}$ and $q_{0}$ and $q_{1}$ satisfy the following
conditions that follow from (108):

$\displaystyle\mu q_{0}+(m-\mu)q_{1}$ $\displaystyle=1,$ (113)
$\displaystyle-\mu q_{0}\ln{q_{0}}-(m-\mu)q_{1}\ln{q_{1}}$
$\displaystyle=\eta\ln{m}.$ (114)

Solving (113) with respect to $q_{1}$ and substituting it to (114), we obtain

$\displaystyle-\mu q_{0}\ln{\left(\frac{\mu q_{0}}{\mu}\right)}-(1-\mu
q_{0})\ln{\left(\frac{1-\mu q_{0}}{m-\mu}\right)}=\eta\ln{m}.$ (115)

Letting $r\equiv\mu q_{0}$, this is equivalent to (104). From (112), the
relative entropy (107) is calculated as

$\displaystyle D(U\|P)=\frac{\mu}{m}\ln{\left(\frac{\mu/m}{\mu
q_{0}}\right)}+\left(1-\frac{\mu}{m}\right)\ln{\left(\frac{1-\mu/m}{1-\mu
q_{0}}\right)},$ (116)

which is equal to $\mathscr{D}(\mu/m\|r)$. Minimizing over $\mu$, we obtain
(103).

We consider the case where $\eta=1-\delta$, $\delta\ll 1$ and $m\gg 1$. As in
(103) and (104), we are to minimize the function $\mathscr{D}(\alpha\|r)$ over
$0<\alpha<1$ and $0<r<1$, under the condition that

$\displaystyle\mathscr{D}(r\|\alpha)=\delta\ln{m}.$ (117)

Due to the properties of the binary relative entropy (see Appendix B), we have
$\mathscr{D}(\alpha\|\alpha+\epsilon)\approx\mathscr{D}(\alpha+\epsilon\|\alpha)$
in the second order of $\epsilon\ll 0$. Substituting $r=\alpha+\epsilon$ to
(117) and $\mathscr{D}(\alpha\|r)$, we arrive at

$\displaystyle\left.\mathscr{D}_{\rm
vN}(\eta;m)\right|_{\eta=\delta\ln{m}}\approx\min_{\begin{subarray}{c}0<\alpha<1\\\
0<r<1\end{subarray}}\mathscr{D}(\alpha\|r)\approx\delta\ln{m}$ (118)

in the first order of $\delta$. As an upper bound on the probability of large
deviation, this is a quadratic improvement with respect to $\delta$ and an
improvement by factor $8\pi^{2}\ln{m}$ over the exponent obtained from Theorem
III.3 in Hayden et al. (2006) (see Appendix D for the detail).

###  V.3 Expectation Value

Let $W$ be a Hermitian operator on ${\mathcal{H}}^{S}$. For
$w\in{\mathbb{R}}$, we calculate the exponent for the set of states such that
the expectation value of $W$ is equal to $w$. Without loss of generality, we
may assume that ${\rm Tr}[W]=0$ and $0<w<\|W\|_{\infty}$. We prove that

$\displaystyle\inf_{\rho:{\rm Tr}[W\rho]=w}D(\pi\|\rho)=\mathscr{D}_{\rm
exp}(w;W,m):=\frac{1}{m}{\rm
Tr}\ln\left[\left(1-w\nu^{*}(w)\right)I+\nu^{*}(w)W\right],$ (119)

where $\nu^{*}(w)$ is the solution of

$\displaystyle\frac{1}{m}{\rm Tr}[\\{(1-w\nu)I+\nu W\\}^{-1}]=1$ (120)

under the condition that $\nu\neq 0$ and

$\displaystyle\frac{1}{m}\\{(1-w\nu)I+\nu W\\}^{-1}\geq 0.$ (121)

Let $\\{A_{r}\\}_{r=1}^{m^{2}-1}$ be the set of traceless Hermitian operators
on ${\mathcal{H}}$ that are orthonormal with respect to the Hilbert-Schmidt
inner product. Every state $\rho\in{\mathcal{S}}({\mathcal{H}})$ is
represented by a $(m^{2}-1)$-dimensional real vector
$\vec{t}\equiv(t_{1},\cdots,t_{m^{2}-1})\in{\mathbb{R}}^{m^{2}-1}$ as

$\displaystyle\rho=\rho(\vec{t}):=\frac{1}{m}(I+\vec{t}\cdot\vec{A})=\frac{1}{m}\left(I+\sum_{r=1}^{m^{2}-1}t_{r}A_{r}\right).$
(122)

Using the trace-log representation of the relative entropy (see Property 6 in
Lemma 2), we have

$\displaystyle-m\frac{\partial}{\partial
t_{r}}D(\pi\|\rho(\vec{t}))=\frac{\partial}{\partial t_{r}}{\rm
Tr}\ln{(m\rho(\vec{t}))}=\frac{1}{m}{\rm Tr}[A_{r}\rho(\vec{t})^{-1}].$ (123)

Multiplying $A_{r}$ and taking the summation over $r$, we obtain

$\displaystyle-m\vec{A}\cdot{\rm
grad}_{\vec{t}}D(\pi\|\rho(\vec{t}))=\sum_{r=1}^{m^{2}-1}\frac{1}{m}{\rm
Tr}[A_{r}\rho(\vec{t})^{-1}]A_{r}=\rho(\vec{t})^{-1}-\frac{{\rm
Tr}[\rho(\vec{t})^{-1}]}{m}I.$ (124)

Due to the convexity of the relative entropy (see Property 3 in Lemma 2), the
infimum in (119) is achieved by the state $\rho(\vec{t})$ such that ${\rm
grad}_{\vec{t}}D(\pi\|\rho(\vec{t}))\propto{\rm grad}_{\vec{t}}{\rm
Tr}[W\rho(\vec{t})]$, which is equivalent to $\vec{A}\cdot{\rm
grad}_{\vec{t}}D(\pi\|\rho(\vec{t}))\propto W$. Thus, due to (124), it follows
that $\rho(\vec{t})^{-1}=\mu I+m\nu W$ for some
$\mu,\nu\in{\mathbb{R}}\backslash\\{0\\}$. Multiplying $\rho(\vec{t})$ on both
sides, and substituting the condition that ${\rm Tr}[W\rho(\vec{t})]=w$, we
have that $\mu=m(1-w\nu)$. From the normalization condition and the positive
semidefiniteness of $\rho(\vec{t})$, we obtain (120) and (121).

We consider the case where $w$ is sufficiently small so that
$m|w|\ll\|W\|_{2}$. Suppose now that the solution of (120) and (121) satisfies

$\displaystyle|\nu^{*}(w)|\cdot\|wI-W\|_{\infty}\ll 1.$ (125)

In this case, the expression (119) is equivalent to

$\displaystyle\mathscr{D}_{\rm
exp}(w;W,m)=-\frac{1}{m}\sum_{\ell=1}^{\infty}\frac{\nu^{*}(w)^{\ell}}{\ell}{\rm
Tr}[(wI-W)^{\ell}]$ (126)

and the condition (120) is represented as

$\displaystyle\frac{1}{m}\sum_{\ell=0}^{\infty}\nu^{\ell}{\rm
Tr}[(wI-W)^{\ell}]=1.$ (127)

The expressions (127) and (126) can be well approximated by the summations up
to the order $\ell=2$. Noting that we have $mw^{2}\ll{\rm Tr}[W^{2}]$ from the
assumption, we obtain from (127) that

$\displaystyle\nu^{*}(w)\approx-\frac{mw}{{\rm Tr}[W^{2}]}.$ (128)

The condition (125) is indeed satisfied, because we have
$|\nu^{*}(w)|\cdot\|wI-W\|_{\infty}\leq 2|\nu^{*}(w)|\cdot\|W\|_{\infty}\leq
2|\nu^{*}(w)|\cdot\|W\|_{2}\approx 2mw/\|W\|_{2}\ll 1$. Substituting (128) to
(126), we obtain

$\displaystyle\mathscr{D}_{\rm exp}(w;W,m)\approx\frac{mw^{2}}{2{\rm
Tr}[W^{2}]}.$ (129)

This shows an improvement over the exponent that can be deduced from the
results of Popescu et al. (2006b); Reimann (2015) by a constant factor (see
Appendix D for the detail).

Figure 5:  A graph of the function $\tilde{\mathscr{D}}_{\rm
exp}(w):=\mathscr{D}_{\rm exp}(w;W=\mbox{$\mbox{$|1\rangle$}\\!\mbox{$\langle
1|$}$}-\mbox{$\mbox{$|3\rangle$}\\!\mbox{$\langle 3|$}$},m=3)$, which is given
by (133), is presented. It goes to infinity in the limit of $w\rightarrow\pm
1$.

Example: We consider the case where $m=3$ and
$W=\mbox{$\mbox{$|1\rangle$}\\!\mbox{$\langle
1|$}$}-\mbox{$\mbox{$|3\rangle$}\\!\mbox{$\langle 3|$}$}$ (see Appendix C for
a detailed calculation). The conditions (120) and (121) yield

$\displaystyle 1-(w-k)\nu\geq 0\;(k=-1,0,1)$ (130)

and

$\displaystyle\frac{1}{3}\sum_{k=-1,0,1}\frac{1}{1-w\nu+k\nu}=1.$ (131)

It follows that

$\displaystyle\nu^{*}(w)=\frac{1-3w^{2}-\sqrt{1+3w^{2}}}{3w(1-w^{2})}.$ (132)

The infimum of the relative entropy is calculated by (119) to be

$\displaystyle\mathscr{D}_{\rm
exp}(w;W=\mbox{$\mbox{$|1\rangle$}\\!\mbox{$\langle
1|$}$}-\mbox{$\mbox{$|3\rangle$}\\!\mbox{$\langle
3|$}$},m=3)=\frac{1}{3}\sum_{k=-1,0,1}\ln{[1-(w-k)\nu^{*}(w)]}.$ (133)

A graph of this function is depicted in Figure 5.

##  VI Conditional Concentration

Theorem 1 implies that the probability of the set
$\Omega\in{\mathcal{S}}({\mathcal{H}}^{S})$ is essentially the same as the
probability of the state in $\Omega$ that is closest to the maximally mixed
state in terms of the quantum relative entropy. In this section, we prove that
the total probability of the states in $\Omega$ that is $\epsilon$-away from
the closest state $\rho^{*}$, conditioned by $\Omega$, is exponentially small
with respect to $n$ for any $\epsilon>0$. Thus, the total probability of
$\Omega$ strongly concentrates around $\rho^{*}$. We refer to this property as
conditional concentration. (As similar concepts, see e.g. Section 11.6 in
Cover and Thomas (2005) for the “conditional limit theorem” and La Cour and
Schieve (2015) for the “conditional large deviation property”.) To be precise:

######  Theorem 10

Let $\Omega\subseteq{\mathcal{S}}({\mathcal{H}}^{S})$ be such that ${\rm
cl}({\rm int}\Omega)={\rm cl}\Omega$. Define $\Omega_{*}\subseteq{\rm
cl}\Omega$ by

$\displaystyle\Omega_{*}:=\left\\{\rho\in{\rm
cl}\Omega\left|D(\pi\|\rho)=\min_{\rho^{\prime}\in{\rm
cl}\Omega}D(\pi\|\rho^{\prime})\right.\right\\}.$ (134)

Let ${\mathcal{B}}_{\epsilon}(\Omega_{*})$ be the set of the
$\epsilon$-neighbourhoods of $\Omega_{*}$, i.e.,

$\displaystyle{\mathcal{B}}_{\epsilon}(\Omega_{*}):=\left\\{\rho\in{\mathcal{S}}({\mathcal{H}}^{S})\left|\inf_{\sigma\in\Omega_{*}}\frac{1}{2}\|\rho-\sigma\|_{1}\leq\epsilon\right.\right\\}.$
(135)

For any $\epsilon>0$, there exists $\delta>0$ and it holds that

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\frac{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash{\mathcal{B}}_{\epsilon}(\Omega_{*})\right|\Psi\sim\mu_{n}\right\\}}{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}}\geq\delta.$
(136)

######  Proof:

Define nonnegative functions $f,g:{\rm cl}\Omega\rightarrow{\mathbb{R}}$ by

$\displaystyle f(\rho)=D(\pi\|\rho)-\min_{\rho^{\prime}\in{\rm
cl}\Omega}D(\pi\|\rho^{\prime}),\quad
g(\rho)=\inf_{\sigma\in\Omega_{*}}\frac{1}{2}\|\rho-\sigma\|_{1}.$ (137)

By definition, we have $f(\rho)\geq 0$, $g(\rho)=0$ and

$\displaystyle f(\rho)=0\;\Leftrightarrow\;g(\rho)=0.$ (138)

Define

$\displaystyle\mathfrak{F}_{\leq}(\delta):=\\{\rho\in{\rm
cl}\Omega|f(\rho)\leq\delta\\}$ (139)

and

$\displaystyle\zeta(\delta):=\max_{\rho\in\mathfrak{F}_{\leq}(\delta)}g(\rho).$
(140)

This is a monotonically nondecreasing function of $\delta$ by definition. It
satisfies $\lim_{\delta\rightarrow 0}\zeta(\delta)=0$ as we prove later. Thus,
for any $\epsilon>0$, there exists $\delta>0$ such that
$\mathfrak{F}_{\leq}(\delta)\subseteq{\mathcal{B}}_{\epsilon}(\Omega_{*})$. It
follows that

$\displaystyle{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash{\mathcal{B}}_{\epsilon}(\Omega_{*})\right|\Psi\sim\mu_{n}\right\\}\leq{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash\mathfrak{F}_{\leq}(\delta)\right|\Psi\sim\mu_{n}\right\\}.$
(141)

Hence, we have

$\displaystyle\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\frac{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash{\mathcal{B}}_{\epsilon}(\Omega_{*})\right|\Psi\sim\mu_{n}\right\\}}{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}}$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\frac{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash\mathfrak{F}_{\leq}(\delta)\right|\Psi\sim\mu_{n}\right\\}}{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}}$ (142)
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash\mathfrak{F}_{\leq}(\delta)\right|\Psi\sim\mu_{n}\right\\}}-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}$ (143)
$\displaystyle=-\inf_{\rho\in\Omega\backslash\mathfrak{F}_{\leq}(\delta)}D(\pi\|\rho)+\inf_{\rho\in\Omega}D(\pi\|\rho)$
(144)
$\displaystyle=-\inf_{\rho\in\Omega\backslash\mathfrak{F}_{\leq}(\delta)}D(\pi\|\rho)+\min_{\rho\in{\rm
cl}\Omega}D(\pi\|\rho)$ (145) $\displaystyle\leq\delta,$ (146)

where (144) follows from Theorem 1, (145) from the continuity of
$D(\pi\|\rho)$ with respect to $\rho$ and (146) from (139). For any
sufficiently small $\delta>0$, (146) is equal to $\delta$, which implies
(136).

We prove that $\zeta(\delta)$ defined by (140) satisfies
$\lim_{\delta\rightarrow 0}\zeta(\delta)=0$. The proof is along the same line
as Appendix B-I of Wakakuwa et al. (2017), which was based on an idea used in
Koashi and Imoto (2001). By definition, we have that

$\displaystyle\forall\delta>0,\>\exists\rho\in{\rm
cl}\Omega\;:\;g(\rho)=\zeta(\delta)\wedge f(\rho)\leq\delta.$ (147)

Define $\zeta_{0}:=\lim_{\delta\rightarrow 0}\zeta(\delta)$. Due to the
monotonicity, we have $\zeta(\delta)\geq\zeta_{0}$ for all $\delta>0$.
Consequently, we have that

$\displaystyle\forall\delta>0,\>\exists\rho\in{\rm
cl}\Omega\;:\;g(\rho)\geq\zeta_{0}\wedge f(\rho)\leq\delta.$ (148)

Define ${\mathfrak{F}}_{\geq}(\zeta_{0}):=\\{\rho\in{\rm
cl}\Omega\>|\>g(\rho)\geq\zeta_{0}\\}$. Due to the continuity of $g$,
${\mathfrak{F}}_{\geq}(\zeta_{0})$ is a closed subset of ${\rm cl}\Omega$.
Hence $\beta:=\min_{\rho\in{\mathfrak{F}}_{\geq}(\zeta_{0})}f(\rho)$ exists
due to the continuity of $f$. By definition, we have that

$\displaystyle\forall\rho\in{\rm cl}\Omega:\>g(\rho)\geq\zeta_{0}\Rightarrow
f(\rho)\geq\beta.$ (149)

Suppose now that $\zeta_{0}>0$. We have $f(\rho)>0$ for all
$\rho\in{\mathfrak{F}}_{\geq}(\zeta_{0})$ due to (138). Thus we have
$\beta>0$, in which case (149) contradicts with (148) because $\delta$ can be
arbitrarily small. $\blacksquare$

The conditional concentration property can be generalized to any continuous
function of states as follows:

######  Theorem 11

Let $v\in{\mathbb{N}}$ and let
$\gamma:{\mathcal{S}}({\mathcal{H}}^{S})\rightarrow{\mathbb{R}}^{v}$ be a
uniformly continuous function. Define a function
$D_{\gamma}:{\mathbb{R}}^{v}\rightarrow{\mathbb{R}}$ by

$\displaystyle D_{\gamma}(x)=\inf_{\rho\in\gamma^{-1}(\\{x\\})}D(\pi\|\rho).$
(150)

Let $\Gamma\subseteq{\mathbb{R}}^{v}$ be an open set and define
$\Gamma_{*}\subset{\rm cl}\Gamma$ by

$\displaystyle\Gamma_{*}:=\left\\{x\in{\rm
cl}\Gamma\left|D_{\gamma}(x)=\min_{x^{\prime}\in{\rm
cl}\Gamma}D_{\gamma}(x^{\prime})\right.\right\\}.$ (151)

Furthermore, let ${\mathfrak{B}}_{\iota}(\Gamma_{*})$ be the set of the
$\iota$-neighbourhoods of $\Gamma_{*}$, i.e.,

$\displaystyle{\mathcal{B}}_{\iota}(\Gamma_{*}):=\left\\{x\in{\mathbb{R}}^{v}\left|\min_{x_{*}\in\Gamma_{*}}\|x-x_{*}\|\leq\iota\right.\right\\}.$
(152)

For any $\iota>0$, there exists $\delta>0$ and it holds that

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\frac{{\rm
Pr}\left\\{\left.\gamma(\rho_{\Psi})\in\Gamma\backslash\mathfrak{B}_{\iota}(\Gamma_{*})\right|\Psi\sim\mu_{n}\right\\}}{{\rm
Pr}\left\\{\left.\gamma(\rho_{\Psi})\in\Gamma\right|\Psi\sim\mu_{n}\right\\}}}\geq\delta.$
(153)

######  Proof:

Define $\Omega:=\gamma^{-1}(\Gamma)$. Since $\gamma$ is a continuous function,
$\Omega$ is an open set and thus ${\rm cl}({\rm int}\Omega)={\rm cl}\Omega$.
Define $\Omega_{*}\subset{\mathcal{S}}({\mathcal{H}}^{S})$ and
${\mathcal{B}}_{\epsilon}(\Omega_{*})\subset{\mathcal{S}}({\mathcal{H}}^{S})$
in the same way as in Theorem 10. Due to the uniform continuity of $\gamma$,
for any $\iota>0$, there exists $\epsilon>0$ such that
$\gamma({\mathcal{B}}_{\epsilon}(\Omega_{*})\cap\Omega)\subseteq\mathfrak{B}_{\iota}(\Gamma_{*})$.
Thus, $\gamma(\rho_{\Psi})\in\Gamma\backslash\mathfrak{B}_{\iota}(\Gamma_{*})$
implies
$\gamma(\rho_{\Psi})\in\Gamma\backslash\gamma({\mathcal{B}}_{\epsilon}(\Omega_{*})\cap\Omega)$,
which in turn implies
$\rho_{\Psi}\in\Omega\backslash{\mathcal{B}}_{\epsilon}(\Omega_{*})$.
Consequently, we have

$\displaystyle{\rm
Pr}\left\\{\left.\gamma(\rho_{\Psi})\in\Gamma\backslash\mathfrak{B}_{\iota}(\Gamma_{*})\right|\Psi\sim\mu_{n}\right\\}\leq{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash{\mathcal{B}}_{\epsilon}(\Omega_{*})\right|\Psi\sim\mu_{n}\right\\}.$
(154)

By definition, we have

$\displaystyle{\rm
Pr}\left\\{\left.\gamma(\rho_{\Psi})\in\Gamma\right|\Psi\sim\mu_{n}\right\\}={\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}.$ (155)

Thus, from Theorem 10, we obtain

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\frac{{\rm
Pr}\left\\{\left.\gamma(\rho_{\Psi})\in\Gamma\backslash\mathfrak{B}_{\iota}(\Gamma_{*})\right|\Psi\sim\mu_{n}\right\\}}{{\rm
Pr}\left\\{\left.\gamma(\rho_{\Psi})\in\Gamma\right|\Psi\sim\mu_{n}\right\\}}}$
$\displaystyle\geq-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\frac{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\backslash{\mathcal{B}}_{\epsilon}(\Omega_{*})\right|\Psi\sim\mu_{n}\right\\}}{{\rm
Pr}\left\\{\left.\rho_{\Psi}\in\Omega\right|\Psi\sim\mu_{n}\right\\}}}\geq\delta$
(156)

and complete the proof. $\blacksquare$

##  VII Coherence of Random Pure States

We apply approachs presented in the previous sections to investigate the large
deviation property of coherence of random pure states in a single quantum
system. Ref. Singh et al. (2016) proved that coherence of random pure states
in a system with large dimension strongly concentrates around a near-maximal
value. To be precise, let ${\mathcal{H}}_{n}$ be a $n$-dimensional Hilbert
space with a fixed orthonormal basis $\\{|e_{l}\rangle\\}_{l=1}^{n}$ for each
$n\in{\mathbb{N}}$. For each $\mbox{$|\Psi\rangle$}\in{\mathcal{H}}_{n}$, the
relative entropy of coherence Baumgratz et al. (2014) is calculated to be

$\displaystyle
C_{r}(\Psi):=H(\\{p_{\Psi}(l)\\}_{l=1}^{n})=-\sum_{l=1}^{n}p_{\Psi}(l)\ln{p_{\Psi}(l)},$
(157)

where $p_{\Psi}(l):=|\langle e_{l}|\Psi\rangle|^{2}$. It was proved in Ref.
Singh et al. (2016) (see Theorem 1 therein) that, for any $n\geq 3$ and
$\omega>0$, it holds that

$\displaystyle{\rm
Pr}\left\\{\left.\left|\frac{C_{r}(\Psi)}{\ln{n}}-\frac{H_{n}-1}{\ln{n}}\right|>\omega\right|\Psi\sim\nu_{n}\right\\}\leq
2\exp\left(-\frac{n\omega^{2}}{36\pi^{3}\ln{2}}\right),$ (158)

where $\nu_{n}$ is the unitarily invariant measure on ${\mathcal{H}}_{n}$ and
$H_{n}=\sum_{k=1}^{n}(1/k)$. The proof in Ref. Singh et al. (2016) is based on
evaluation of the Lipschitz constant of $C_{r}$ and application of Levy’s
lemma, which followed the lines of Section III in Hayden et al. (2006) to
prove concentration of entanglement. Noting that
$\lim_{n\rightarrow\infty}(H_{n}-1)/\ln{n}=1$, we obtain an asymptotic bound
from (158) as

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.1-\frac{C_{r}(\Psi)}{\ln{n}}>\omega\right|\Psi\sim\nu_{n}\right\\}\geq\frac{\omega^{2}}{36\pi^{3}\ln{2}},$
(159)

which provides a lower bound on the exponent of coherence of random pure
states. In the following, we provide a large deviation bound that is
complementary to (159), namely an upper bound on the exponent:

######  Theorem 12

For any $0<\omega<1$, it holds that

$\displaystyle-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.1-\frac{C_{r}(\Psi)}{\ln{n}}\geq\omega\right|\Psi\sim\nu_{n}\right\\}}\leq\ln{\left(\frac{1}{1-\omega}\right)}.$
(160)

######  Proof:

Fix arbitrary $\delta$ satisfying $0<\delta<1-\omega$ and define
$p^{*}(\Psi):=\max_{1\leq l\leq n}{p_{\Psi}(l)}$. A simple calculation yields

$\displaystyle H(\\{p_{\Psi}(l)\\}_{l=1}^{n})\leq
h(p^{*}(\Psi))+(1-p^{*}(\Psi))\ln{n}\leq 1+(1-p^{*}(\Psi))\ln{n},$ (161)

where $h$ is the binary entropy defined by $h(x)=-x\ln{x}-(1-x)\ln{(1-x)}$.
Thus, for any $n\geq e^{1/\delta}$, we have

$\displaystyle\frac{C_{r}(\\{\Psi\\})}{\ln{n}}\leq\delta+(1-p^{*}(\Psi)),$
(162)

which implies

$\displaystyle{\rm
Pr}\left\\{\left.1-\frac{C_{r}(\Psi)}{\ln{n}}\geq\omega\right|\Psi\sim\nu_{n}\right\\}\geq{\rm
Pr}\left\\{\left.p^{*}(\Psi)\geq\omega+\delta\right|\Psi\sim\nu_{n}\right\\}.$
(163)

It follows that

$\displaystyle-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.1-\frac{C_{r}(\Psi)}{\ln{n}}\geq\omega\right|\Psi\sim\nu_{n}\right\\}\leq-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.p^{*}(\Psi)\geq\omega+\delta\right|\Psi\sim\nu_{n}\right\\}.$
(164)

As we prove below, we have

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.p^{*}(\Psi)\geq\kappa\right|\Psi\sim\nu_{n}\right\\}=\ln{\left(\frac{1}{1-\kappa}\right)}.$
(165)

for any $0<\kappa<1$. Substituting $\omega+\delta$ to $\kappa$ and taking the
limit of $\delta\rightarrow 0$, we arrive at (160).

To prove (165), note that the condition $p^{*}(\Psi)\geq\kappa$ is equivalent
to the condition that there exists $l\>(1\leq l\leq n)$ satisfying
$p_{\Psi}(l)\geq\kappa$. Thus, we have that

$\displaystyle{\rm Pr}\left\\{\left.|\langle
e_{1}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}$
$\displaystyle\leq{\rm
Pr}\left\\{\left.p^{*}(\Psi)\geq\kappa\right|\Psi\sim\nu_{n}\right\\}\leq\sum_{l=1}^{n}{\rm
Pr}\left\\{\left.|\langle
e_{l}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}.$ (166)

Define a normalized state vector
$|+_{n}\rangle:=(1/\sqrt{n})\sum_{l=1}^{n}|e_{l}\rangle$. Due to the unitary
invariance of $\nu_{n}$, we have

$\displaystyle{\rm Pr}\left\\{\left.|\langle
e_{l}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}={\rm
Pr}\left\\{\left.|\langle+_{n}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}$
(167)

for any $l$. It follows from (166) that

$\displaystyle{\rm
Pr}\left\\{\left.|\langle+_{n}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}\leq{\rm
Pr}\left\\{\left.p^{*}(\Psi)\geq\kappa\right|\Psi\sim\nu_{n}\right\\}\leq
n{\rm
Pr}\left\\{\left.|\langle+_{n}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}.$
(168)

Hence, to prove (165), it suffices to prove that

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\left.|\langle+_{n}|\Psi\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}}=\ln{\left(\frac{1}{1-\kappa}\right)}$
(169)

for any $0<\kappa<1$.

To prove (169), let $Z=(X,Y)^{T}$ be a $2$-dimensional real vector random
variable that obeys the standard normal distribution (SND), i.e., $Z$ obeys a
probability distribution $G_{2}(Z)=G_{1}(X)G_{1}(Y)$, where $G_{1}$ is defined
by (55). Let $Z_{1},\cdots,Z_{l},\cdots,Z_{n}$ be an i.i.d. sequence of the
random variable $Z$, where $Z_{l}\equiv(X_{l},Y_{l})^{T}$. Define a random
unnormalized vector $|\tilde{\Psi}\rangle$ by

$\displaystyle\mbox{$|\tilde{\Psi}\rangle$}=\sum_{1\leq l\leq
n}(X_{l}+iY_{l})\mbox{$|e_{l}\rangle$},$ (170)

and a normalized one by
$|\bar{\Psi}\rangle\equiv\mbox{$|\tilde{\Psi}\rangle$}/\|\mbox{$|\tilde{\Psi}\rangle$}\|$.
With a slight abuse of notation, we denote $\bar{\Psi}\sim{\rm SND}(n)$. Due
to Lemma 8 in Section IV.2, the random state $|\Psi\rangle$ distributed with
respect to the unitarily invariant measure on ${\mathcal{H}}_{n}$ is
equivalent to the random state $|\bar{\Psi}\rangle$. Thus, the probability in
the L.H.S. of (169) is equal to ${\rm
Pr}\left\\{\left.|\langle+_{n}|\bar{\Psi}\rangle|^{2}\geq\kappa\right|\Psi\sim\nu_{n}\right\\}$.

Define $\Xi\subset{\mathbb{R}}^{3}$ and $\Theta(\kappa)\subset\Xi$ for
$0<\kappa<1$ by

$\displaystyle\Xi$
$\displaystyle:=\left\\{(s,t)\in{\mathbb{R}}\times{\mathbb{R}}^{2}\left|s>0,\frac{\|t\|^{2}}{s}<1\right.\right\\},$
(171) $\displaystyle\Theta(\kappa)$
$\displaystyle:=\left\\{(s,t)\in\Xi\left|\kappa\leq\frac{\|t\|^{2}}{s}\right.\right\\}.$
(172)

Define functionals $\eta:\Lambda({\mathbb{R}}^{2})\rightarrow{\mathbb{R}}$ and
$\zeta:\Lambda({\mathbb{R}}^{2})\rightarrow{\mathbb{R}}^{2}$ by

$\displaystyle\eta(P):=\int_{z\in{\mathbb{R}}^{2}}\|z\|^{2}P(z)dz,\quad\zeta(P):=\int_{z\in{\mathbb{R}}^{2}}zP(z)dz,$
(173)

in addition to $\xi:\Lambda({\mathbb{R}}^{2})\rightarrow{\mathbb{R}}$ and
$\Phi:\Lambda({\mathbb{R}}^{2})\rightarrow{\mathbb{R}}^{3}$ by

$\displaystyle\xi(P):=\frac{\|\zeta(P)\|^{2}}{\eta(P)},\quad\Phi(P):=(\eta(P),\zeta(P)).$
(174)

A simple calculation yields

$\displaystyle|\langle+_{n}|\bar{\Psi}\rangle|^{2}=\frac{\left\|\frac{1}{n}\sum_{l=1}^{n}Z_{l}\right\|^{2}}{\frac{1}{n}\sum_{1\leq
l\leq n}\|Z_{l}\|^{2}}=\xi(\hat{P}_{n}^{Z^{n}}),$ (175)

where $\hat{P}_{n}^{Z^{n}}$ denotes the empirical distribution of the
sequence. Thus, the condition $|\langle+_{n}|\bar{\Psi}\rangle|^{2}\geq\kappa$
is equivalent to the condition that $\xi(\hat{P}_{n}^{Z^{n}})\geq\kappa$,
which in turn is equivalent to $\Phi(\hat{P}_{n}^{Z^{n}})\in\Theta(\kappa)$.
Hence, it suffices to prove that

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\left\\{\hat{P}_{n}^{Z^{n}}\in\Phi^{-1}(\Theta(\kappa))\right\\}}=\ln{\left(\frac{1}{1-\kappa}\right)}$
(176)

for any $0<\kappa<1$.

To prove (176), define a function
$D^{*}:{\mathbb{R}}\times{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ by

$\displaystyle D^{*}(s,t):=D(\Phi^{-1}\\{(s,t)\\}\|G_{2}).$ (177)

As we prove below, it holds that

$\displaystyle
D^{*}(s,t)=\begin{cases}\frac{s}{2}-\ln{\left(\frac{s-\|t\|^{2}}{2}\right)}-1&(s,t)\in\Xi,\\\
+\infty&(s,t)\notin\Xi.\end{cases}$ (178)

It is obvious that $D^{*}(s,t)$ is continuous in $\Xi$, and that ${\rm
int}\Xi=\Xi$ from (171). Thus, we can apply Theorem 7. A simple calculation
yields

$\displaystyle\inf_{(s,t)\in{\rm
cl}\Theta(\kappa)}D^{*}(s,t)=\inf_{(s,t)\in{\rm
int}\Theta(\kappa)}D^{*}(s,t)=-\ln{(1-\kappa)},$ (179)

which leads to (176).

To prove (178), note that the condition $\Phi(P)=(s,t)$ is equivalent to

$\displaystyle\int_{z\in{\mathbb{R}}^{2}}\|z\|^{2}P(z)dz=s\wedge\int_{z\in{\mathbb{R}}^{2}}zP(z)dz=t.$
(180)

For $(s,t)\in\Xi$, suppose that there exists $P\in\Lambda({\mathbb{R}}^{2m})$
such that

$\displaystyle P\in\Phi^{-1}(s,t),\;D(P\|Q)<\infty,\;P(z)>0\>(\forall
z\in{\mathbb{R}}^{2}).$ (181)

From Theorem 6, it follows that

$\displaystyle D^{*}(s,t)$
$\displaystyle=\max_{\mu\in{\mathbb{R}},\nu\in{\mathbb{R}}^{2}}\left[-\ln{\left\\{\int_{z\in{\mathbb{R}}^{2}}{G_{2}(z)e^{-\mu(\|z\|^{2}-s)-\nu\cdot(z-t)}dz}\right\\}}\right]$
(182)
$\displaystyle=-\min_{\mu\in{\mathbb{R}},\nu\in{\mathbb{R}}^{2}}\left[\mu
s+\nu\cdot t+\frac{\|\nu\|^{2}}{2(1+2\mu)}-\ln{(1+2\mu)}\right],$ (183)

where the second line follows by taking the Gaussian integral. The minimum is
achieved by

$\displaystyle(\mu^{*},\nu^{*})=\left(\frac{1}{s-\|t\|^{2}}-\frac{1}{2}\>,\>\frac{-2t}{s-\|t\|^{2}}\right)$
(184)

which yields

$\displaystyle
D^{*}(s,t)=\frac{s}{2}-\ln{\left(\frac{s-\|t\|^{2}}{2}\right)}-1.$ (185)

It is straightforward to verify that a probability distribution $P^{*}$
defined by

$\displaystyle
P^{*}(z)=\frac{G_{2}(z)e^{-\mu^{*}(\|z\|^{2}-s)-\nu^{*}(z-t)}dz}{\int_{z^{\prime}\in{\mathbb{R}}^{2}}{G_{2}(z^{\prime})e^{-\mu^{*}(\|z^{\prime}\|^{2}-s)-\nu^{*}(z^{\prime}-t)}dz^{\prime}}}$
(186)

indeed satisfies the condition (181). For $(s,t)\notin{\rm cl}\Xi$, we have
$\Phi^{-1}(s,t)=\emptyset$ because $\eta(P)\geq 0$ and $\xi(P)\leq 1$ for any
$P\in\Lambda({\mathbb{R}}^{2m})$. For $(s,t)\in\partial{\Xi}$, any
$P\in\Phi^{-1}(s,t)$ satisfies $\eta(P)=\|\zeta(P)\|^{2}$. Thus, there exists
$\iota_{(s,t)}\in{\mathbb{R}}^{2}$ such that $P(z)=\delta(z-\iota_{(s,t)})$,
which implies $D(P\|G_{2})=+\infty$. In both cases, we have
$D^{*}(s,t)=+\infty$. All in all, we obtain (176) and complete the proof.
$\blacksquare$

Noting that
$\ln{(\frac{1}{1-\omega})}=\sum_{\ell=1}^{\infty}\ell^{-1}\omega^{\ell}$,
there remains a quadratic gap between the lower bound (159) and the upper
bound (160) with respect to $\omega$, in addition to the constant factor
$36\pi^{2}\ln{2}$. To close this gap and obtain the exact exponent for the
coherence of random pure states is left as a future work.

##  VIII Conclusion

We have investigated the properties of the random quantum states induced from
the uniformly random pure states on a bipartite system by taking the partial
trace over the larger subsystem. We have proved that the probability of the
induced random state within a given set decreases no slower or faster than
exponential in the dimension of the system traced out. The exponent is given
by the quantum relative entropy between the maximally mixed state and the set
of states to be considered. We have also shown that the total probability of a
given set strongly concentrates around the element closest to the maximally
mixed state. The same method was applied to investigating the large deviation
property of coherence of random quantum states in a single system.

The main result (Theorem 1) can be regarded as a “Sanov-type” theorem for
atypicality of random quantum states. Quantum extensions of Sanov’s theorem
have been proposed in the context of quantum hypothesis testing Bjelaković et
al. (2005); Audenaert et al. (2008); Nötzel (2014). However, it was pointed
out in Nagaoka and Hayashi (2007) (see Remark 8 therein) that finding a
meaningful and useful quantum extension of Sanov’s theorem remains open.
Theorem 1 provides some insights on this problem from the viewpoint of random
quantum states. However, it cannot be regarded exactly as a quantum version of
Sanov’s theorem. Note that, in Eq. (10), the first argument in the relative
entropy is fixed to the maximally mixed state, and the infimum is taken over
the second argument thereof, as opposed to the original Sanov’s theorem and
those presented in Bjelaković et al. (2005); Audenaert et al. (2008); Nötzel
(2014).

We conclude this paper with a list of problems that are left for future work.

As discussed in the introduction, the large deviation bound obtained in this
paper is valid only in the asymptotic limit, where the dimension of the
subsystem traced out goes to infinity. It tells us nothing in the case where
the dimension is finite. To obtain a large deviation bound that is valid in
the finite-dimensional case and is tight in the asymptotic limit is left as an
important open problem.

In the main result of this paper, we assumed that the pure state is chosen
randomly from the entire Hilbert space, which is represented as a tensor
product of two Hilbert spaces. The concept of measure concentration has also
been applied to investigating properties of random pure quantum states in a
single system Reimann (2015); Singh et al. (2016) or in a subspace of the
tensor product of two Hilbert spaces Hayden et al. (2006); Popescu et al.
(2006a, b). It would be fruitful to explore the applicability of the method
presented in this paper to those cases, besides the analysis of the coherence
property of random pure states presented in Section VII.

The Haar measure on the set of the unitary operators is generated from the
multivariate normal distribution, in the same way as the unitary invariant
measure on the set of the normalized pure states is (see e.g. Section 7.2 in
Watrous (2018)). Hence, it would be natural to expect that the approach
presented in this paper is extended to investigating the large deviation
property of random unitary operators Hiai and Petz (2000).

Ref. Low (2009) investigated the concentration bounds obtained in Hayden et
al. (2006) from the viewpoint of the unitary designs and the state designs. It
proved that a large deviation bound, similar to Eq. (1), holds when the state
$t$-designs or the unitary $t$-designs are used instead of the unitary
invariant measure or the Haar measure. In this direction, it would be natural
to consider an extension of Theorem 1 to the case where the unitary invariant
measure is replaced by the state $t$-designs.

Finally, it would be nice to apply the main result (Theorem 1) to concrete
examples besides those presented in Section II and Section V. One interesting
scenario would be the case where the system $S$ is composed of two subsystems.
Applying Theorem 1, we may investigate the large deviation property on the
probability of the random state induced on $S$ being entangled. This scenario
was addressed in Section V.B of Hayden et al. (2006).

Note added.— After the completion of this work, we became aware that Eq. (10)
and Eq. (169) can be proved much more simply from the result of Zyczkowski and
Sommers (2001) (see Eqs. (3.6) and (2.3) therein).

## Acknowledgement

EW acknowledges support from MEXT Quantum Leap Flagship Program (MEXT QLEAP)
Grant No. JPMXS0120319794.

##  Appendix A Proof of Theorem 7

We provide proof of Theorem 7. First, we consider the case where $\Theta$ is
convex and ${\rm cl}({\rm int}\Theta)={\rm cl}\Theta$. Second, we extend the
result to the case where these conditions are not necessarily satisfied.

###  A.1 Proof for Convex Regular Subsets

Assume that $\Theta$ is convex and ${\rm cl}({\rm int}\Theta)={\rm cl}\Theta$.
We prove that

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta\\}}=\inf_{\bm{t}\in\Theta}{\mathfrak{D}}(\bm{t}),$
(187)

which implies (38) and (39).

Define $B_{m}:=[-m,m)^{\times v}$, $\Lambda_{m}:=\\{P\in\Lambda|P(B_{m})=1\\}$
for each $m\in{\mathbb{N}}$ and let
$\Lambda^{*}:=\cup_{m=1}^{\infty}\Lambda_{m}$. It is obvious that
$\lim_{m\rightarrow\infty}Q(B_{m})=1$ and that the function
$\bm{F}|_{\Lambda_{m}}$ is linear. Due to Theorem 5, it suffices to prove that

$\displaystyle\inf_{P\in\bm{F}^{-1}(\Theta)\cap\Lambda^{*}}D(P\|Q)=\inf_{\bm{t}\in\Theta}{\mathfrak{D}}(\bm{t})$
(188)

and that $\bm{F}$ is $\tau$-continuous in $\Lambda_{m}$ for each
$m\in{\mathbb{N}}$.

To prove (188), fix an arbitrary $\epsilon,\delta>0$ and define an open convex
set

$\displaystyle{\mathcal{I}}^{\epsilon}(\Theta):=\left\\{\bm{t}\in\Theta\left|\inf_{\bm{\tau}\in\partial\Theta}\|\bm{t}-\bm{\tau}\|>\epsilon\right.\right\\}.$
(189)

For any $P\in\bm{F}^{-1}({\mathcal{I}}^{\epsilon}(\Theta))$ and
$m\in{\mathbb{N}}$, define $P_{m}\in\Lambda_{m}$ by

$\displaystyle P_{m}(z)=\begin{cases}\frac{P(z)}{P(B_{m})}&(z\in B_{m})\\\
0&(z\notin B_{m})\end{cases}.$ (190)

We have

$\displaystyle\bm{F}(P_{m})$ $\displaystyle=\frac{1}{P(B_{m})}\int_{z\in
B_{m}}P(z)\bm{f}(z)dz,$ (191) $\displaystyle D(P_{m}\|Q)$
$\displaystyle=\frac{1}{P(B_{m})}\int_{z\in
B_{m}}P(z)\ln\frac{P(z)}{Q(z)}dz-\ln P(B_{m})$ (192)

and $\lim_{m\rightarrow\infty}P(B_{m})=1$. Thus, for any sufficiently large
$m$, it holds that

$\displaystyle\|\bm{F}(P_{m})-\bm{F}(P)\|$ $\displaystyle<\epsilon,$ (193)
$\displaystyle D(P_{m}\|Q)-D(P\|Q)$ $\displaystyle\leq\delta.$ (194)

It follows from (193) that $P_{m}\in\bm{F}^{-1}(\Theta)\cap\Lambda^{*}$. Thus,
from (194), we have

$\displaystyle\inf_{P\in\bm{F}^{-1}({\mathcal{I}}^{\epsilon}(\Theta))}D(P\|Q)\geq\inf_{P\in\bm{F}^{-1}(\Theta)\cap\Lambda^{*}}D(P\|Q)-\delta.$
(195)

We take the limit of $\epsilon,\delta\rightarrow 0$. Due to the continuity of
${\mathfrak{D}}(\bm{t})$ and the assumption that ${\rm cl}({\rm
int}\Theta)={\rm cl}\Theta$, the L.H.S. of (195) is evaluated as

$\displaystyle\lim_{\epsilon\rightarrow
0}\inf_{P\in\bm{F}^{-1}({\mathcal{I}}^{\epsilon}(\Theta))}D(P\|Q)=\lim_{\epsilon\rightarrow
0}\inf_{\bm{t}\in{\mathcal{I}}^{\epsilon}(\Theta)}{\mathfrak{D}}(\bm{t})=\inf_{\bm{t}\in\Theta}{\mathfrak{D}}(\bm{t}).$
(196)

Thus, we arrive at

$\displaystyle\inf_{\bm{t}\in\Theta}{\mathfrak{D}}(\bm{t})\geq\inf_{P\in\bm{F}^{-1}(\Theta)\cap\Lambda^{*}}D(P\|Q).$
(197)

The inverse inequality is straightforward as

$\displaystyle\inf_{\bm{t}\in\Theta}{\mathfrak{D}}(\bm{t})=\inf_{P\in\bm{F}^{-1}(\Theta)}D(P\|Q)\leq\inf_{P\in\bm{F}^{-1}(\Theta)\cap\Lambda^{*}}D(P\|Q).$
(198)

Thus, we obtain (188).

To prove $\tau$-continuity of $\bm{F}$ in $\Lambda_{m}$, we first assume that
$w=1$. For $l\in{\mathbb{N}}$ and
$\mu\equiv(\mu_{1},\cdots,\mu_{v})\in{\mathbb{Z}}^{\times v}$, define
${\mathbb{R}}_{\mu,l,m}^{v}\subseteq{\mathbb{R}}^{v}$ by

$\displaystyle{\mathbb{R}}_{\mu,l,m}^{v}:=\bigtimes_{s=1}^{v}\left[\frac{m\mu_{s}}{l},\frac{m(\mu_{s}+1)}{l}\right).$
(199)

It is obvious that
$\\{{\mathbb{R}}_{\mu,l,m}^{v}\\}_{\mu\in\\{-l,\cdots,0,\cdots,l-1\\}^{\times
v}}$ is a partition of $B_{m}$. For any $P,Q\in\Lambda_{m}$, it holds that

$\displaystyle\int_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}P(z)f(z)dz$
$\displaystyle\leq\sup_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}f(z)\cdot\int_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}P(z)dz,$
(200) $\displaystyle\int_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}Q(z)f(z)dz$
$\displaystyle\geq\inf_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}f(z)\cdot\int_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}Q(z)dz.$
(201)

Thus, we have

$\displaystyle\int_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}(P(z)-Q(z))f(z)dz$
$\displaystyle\leq\sup_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}\\!\\!\\!f(z)\cdot\int_{z^{\prime}\in{\mathbb{R}}_{\mu,l,m}^{v}}\\!\\!\\!\\!\\!\\!(P(z^{\prime})-Q(z^{\prime}))dz^{\prime}+\left(\sup_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}\\!\\!\\!f(z)-\inf_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}\\!\\!\\!f(z)\right)\cdot\int_{z^{\prime}\in{\mathbb{R}}_{\mu,l,m}^{v}}\\!\\!\\!\\!\\!\\!Q(z^{\prime})dz^{\prime}$
(202) $\displaystyle\leq\delta\cdot f_{m}^{*}+\Delta_{f,l,m}^{*}\cdot
Q({\mathbb{R}}_{\mu,l,m}^{v}),$ (203)

where we defined

$\displaystyle\delta:=\max_{\mu}|Q({\mathbb{R}}_{\mu,l,m}^{v})-P({\mathbb{R}}_{\mu,l,m}^{v})|,\quad\Delta_{f,l,m}^{*}:=\max_{\mu}\left(\sup_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}f(z)-\inf_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}f(z)\right)$
(204)

and $f_{m}^{*}:=\sup_{z\in B_{m}}f(z)$. The maximizations in (204) are taken
over all $\mu\in\\{-l,\cdots,l-1\\}^{\times v}$. It follows that

$\displaystyle\bm{F}(P)-\bm{F}(Q)$ $\displaystyle=\int_{z\in[-m,m)^{\times
v}}(P(z)-Q(z))f(z)dz$ (205)
$\displaystyle=\sum_{\mu}\int_{z\in{\mathbb{R}}_{\mu,l,m}^{v}}(P(z)-Q(z))f(z)dz$
(206) $\displaystyle\leq(2l)^{v}\delta\cdot
f_{m}^{*}+\Delta_{f,l,m}^{*}\sum_{\mu}Q({\mathbb{R}}_{\mu,l,m}^{v})$ (207)
$\displaystyle=(2l)^{v}\delta\cdot f_{m}^{*}+\Delta_{f,l,m}^{*}.$ (208)

Exchanging the roles of $P$ and $Q$, we also have

$\displaystyle\bm{F}(Q)-\bm{F}(P)\leq(2l)^{v}\delta\cdot
f_{m}^{*}+\Delta_{f,l,m}^{*}.$ (209)

Thus, we arrive at

$\displaystyle|\bm{F}(P)-\bm{F}(Q)|\leq(2l)^{v}\delta\cdot
f_{m}^{*}+\Delta_{f,l,m}^{*}.$ (210)

Due to the continuity of $f$, we have $f_{m}^{*}<\infty$ and
$\lim_{l\rightarrow\infty}\Delta_{f,l,m}^{*}=0$. Thus, we conclude that
$\bm{F}$ is $\tau$-continuous in $\Lambda_{m}$. It is straightforward to
extend the above argument to the case where $w\geq 2$, which completes the
proof. $\blacksquare$

###  A.2 Extension to General Subsets

It remains to consider the case where $\Theta$ is not necessarily convex and
${\rm cl}({\rm int}\Theta)={\rm cl}\Theta$ does not necessarily hold. For
$l\in{\mathbb{N}}$ and
$\mu\equiv(\mu_{1},\cdots,\mu_{w})\in{\mathbb{Z}}^{\times w}$, define
${\mathbb{R}}_{\mu,l}^{w}\subseteq{\mathbb{R}}^{w}$ by

$\displaystyle{\mathbb{R}}_{\mu,l}^{w}:=\bigtimes_{r=1}^{w}\left[\frac{\mu_{r}}{l},\frac{\mu_{r}+1}{l}\right).$
(211)

Any ${\mathbb{R}}_{\mu,l}^{w}$ is a convex set and satisfies ${\rm cl}({\rm
int}{\mathbb{R}}_{\mu,l}^{w})={\rm cl}{\mathbb{R}}_{\mu,l}^{w}$. Thus, from
(187), it holds that

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in{\mathbb{R}}_{\mu,l}^{w}\\}}=\inf_{\bm{t}\in{\mathbb{R}}_{\mu,l}^{w}}{\mathfrak{D}}(\bm{t}).$
(212)

Define

$\displaystyle{\mathbb{Z}}_{\Theta,+}^{w,l}:=\\{\mu\in{\mathbb{Z}}^{w}:{\mathbb{R}}_{\mu,l}^{w}\cap\Theta\neq\emptyset\\},\quad{\mathbb{Z}}_{\Theta,-}^{w,l}:=\\{\mu\in{\mathbb{Z}}^{w}:{\mathbb{R}}_{\mu,l}^{w}\subset\Theta\\}$
(213)

and

$\displaystyle\Theta_{+}^{l}:=\bigcup_{\mu\in{\mathbb{Z}}_{\Theta,+}^{w,l}}{\mathbb{R}}_{\mu,l}^{w},\quad\Theta_{-}^{l}:=\bigcup_{{\mathbb{Z}}_{\Theta,-}^{w,l}}{\mathbb{R}}_{\mu,l}^{w}.$
(214)

It is obvious that

$\displaystyle{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta_{\pm}^{l}\\}=\sum_{\mu\in{\mathbb{Z}}_{\Theta,\pm}^{w,l}}{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in{\mathbb{R}}_{\mu,l}^{w}\\}.$ (215)

Hence, we have

$\displaystyle\max_{\mu\in{\mathbb{Z}}_{\Theta,\pm}^{w,l}}{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in{\mathbb{R}}_{\mu,l}^{w}\\}\leq{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta_{\pm}^{l}\\}\leq|{\mathbb{Z}}_{\Theta,\pm}^{w,l}|\cdot\max_{\mu\in{\mathbb{Z}}_{\Theta,\pm}^{w,l}}{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in{\mathbb{R}}_{\mu,l}^{w}\\},$ (216)

which implies

$\displaystyle-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta_{\pm}^{l}\\}}=-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{\max_{\mu\in{\mathbb{Z}}_{\Theta,\pm}^{w,l}}{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in{\mathbb{R}}_{\mu,l}^{w}\\}}.$ (217)

Using (212), the R.H.S. of the above equality is further calculated to be

$\displaystyle\min_{\mu\in{\mathbb{Z}}_{\Theta,\pm}^{w,l}}-\lim_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in{\mathbb{R}}_{\mu,l}^{w}\\}}=\min_{\mu\in{\mathbb{Z}}_{\Theta,\pm}^{w,l}}\inf_{\bm{t}\in{\mathbb{R}}_{\mu,l}^{w}}{\mathfrak{D}}(\bm{t})=\inf_{\bm{t}\in\Theta_{\pm}^{l}}{\mathfrak{D}}(\bm{t}).$
(218)

Thus, noting that $\Theta_{-}^{l}\subseteq\Theta\subseteq\Theta_{+}^{l}$, we
have

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta\\}}\geq-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta_{+}^{l}\\}}=\inf_{\bm{t}\in\Theta_{+}^{l}}{\mathfrak{D}}(\bm{t})$
(219)

and

$\displaystyle-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta\\}}\leq-\liminf_{n\rightarrow\infty}\frac{1}{n}\ln{{\rm
Pr}\\{\bm{F}(\hat{P}_{n}^{Z^{n}})\in\Theta_{-}^{l}\\}}=\inf_{\bm{t}\in\Theta_{-}^{l}}{\mathfrak{D}}(\bm{t}).$
(220)

From (214), we have that

$\displaystyle\lim_{l\rightarrow\infty}\Theta_{+}^{l}={\rm
cl}\Theta,\quad\lim_{l\rightarrow\infty}\Theta_{-}^{l}={\rm int}\Theta.$ (221)

Since ${\mathfrak{D}}(t)$ is continuous by assumption, it follows that

$\displaystyle\lim_{l\rightarrow\infty}\inf_{t\in\Theta_{+}^{l}}{\mathfrak{D}}(t)=\inf_{t\in{\rm
cl}\Theta}{\mathfrak{D}}(t),\quad\lim_{l\rightarrow\infty}\inf_{t\in\Theta_{-}^{l}}{\mathfrak{D}}(t)=\inf_{t\in{\rm
int}\Theta}{\mathfrak{D}}(t).$ (222)

All in all, we obtain (38), (39) and complete the proof. $\blacksquare$

##  Appendix B Properties of the binary relative entropy

We summarize the properties of the binary relative entropy. For binary
probability distributions $\\{\alpha,1-\alpha\\}$ and $\\{\beta,1-\beta\\}$,
the binary relative entropy $\mathscr{D}$ is defined by

$\displaystyle\mathscr{D}(\alpha\|\beta):=D(\\{\alpha,1-\alpha\\}\|\\{\beta,1-\beta\\})=\alpha\ln{\frac{\alpha}{\beta}}+(1-\alpha)\ln{\frac{1-\alpha}{1-\beta}}.$
(223)

It is nonnegative, and is equal to zero if and only if $\alpha=\beta$. It is a
convex function both in $\alpha$ and $\beta$. When $\alpha\neq\beta$, it is
finite if and only if $0<\beta<1$. The derivatives of the binary relative
entropy are calculated as

$\displaystyle\frac{\partial\mathscr{D}(\alpha\|\beta)}{\partial\alpha}=\ln{\left(\frac{\alpha}{1-\alpha}\cdot\frac{1-\beta}{\beta}\right)},\quad\frac{\partial\mathscr{D}(\alpha\|\beta)}{\partial\beta}=-\frac{\alpha}{\beta}+\frac{1-\alpha}{1-\beta}$
(224)

and

$\displaystyle\frac{\partial^{2}\mathscr{D}(\alpha\|\beta)}{\partial\alpha^{2}}=\frac{1}{\alpha}+\frac{1}{1-\alpha},\quad\frac{\partial^{2}\mathscr{D}(\alpha\|\beta)}{\partial\beta^{2}}=\frac{\alpha}{\beta^{2}}+\frac{1-\alpha}{(1-\beta)^{2}}.$
(225)

It follows from (224) that

$\displaystyle(\beta-\alpha)\frac{\partial\mathscr{D}(\alpha\|\beta)}{\partial\beta}\geq
0.$ (226)

In particular, we have

$\displaystyle\left.\frac{\partial\mathscr{D}(\alpha\|\beta)}{\partial\alpha}\right|_{\alpha=\beta}=\left.\frac{\partial\mathscr{D}(\alpha\|\beta)}{\partial\beta}\right|_{\alpha=\beta}=0$
(227)

and

$\displaystyle\left.\frac{\partial^{2}\mathscr{D}(\alpha\|\beta)}{\partial\alpha^{2}}\right|_{\alpha=\beta}=\left.\frac{\partial^{2}\mathscr{D}(\alpha\|\beta)}{\partial\beta^{2}}\right|_{\alpha=\beta}=\frac{1}{\alpha(1-\alpha)}.$
(228)

Thus, in the second order of $\epsilon\ll 1$, we have

$\displaystyle\mathscr{D}(\alpha\|\alpha+\epsilon)\approx\mathscr{D}(\alpha+\epsilon\|\alpha)\approx\frac{\epsilon^{2}}{2\alpha(1-\alpha)}.$
(229)

##  Appendix C Proof of Eq. (132)

We prove that the solution of an equation

$\displaystyle\frac{1}{3}\sum_{k=-1,0,1}\frac{1}{1-w\nu+k\nu}=1$ (230)

for $\nu$, under the condition $\nu\neq 0$ and

$\displaystyle 1-(w-k)\nu\geq 0\;(k=-1,0,1),$ (231)

is given by

$\displaystyle\nu^{*}(w)=\frac{1-3w^{2}-\sqrt{1+3w^{2}}}{3w(1-w^{2})}.$ (232)

From (230), we have

$\displaystyle 3w\nu^{3}-3w^{3}\nu^{3}+6w^{2}\nu^{2}-2\nu^{2}-3w\nu=0.$ (233)

Noting that $\nu\neq 0$, we obtain

$\displaystyle 3w(1-w^{2})\nu^{2}+2(3w^{2}-1)\nu-3w=0.$ (234)

Thus, we have the following two candidates for the solution:

$\displaystyle\nu_{\pm}(w)=\frac{1-3w^{2}\pm\sqrt{1+3w^{2}}}{3w(1-w^{2})}.$
(235)

We prove that $\nu_{-}(w)$ is the only solution that satisfies the condition
(231), regardless of the value of $w$ satisfying $-1\leq w\leq 1$. To be
explicit, we prove that

$\displaystyle 1-(w+1)\nu_{-}(w)$ $\displaystyle\geq 0,$ (236) $\displaystyle
1-w\nu_{-}(w)$ $\displaystyle\geq 0,$ (237) $\displaystyle 1-(w-1)\nu_{-}(w)$
$\displaystyle\geq 0$ (238)

for any $-1\leq w\leq 1$ and

$\displaystyle 1-(w+1)\nu_{+}(w)$ $\displaystyle\leq 0\quad(w\geq 0),$ (239)
$\displaystyle 1-(w-1)\nu_{+}(w)$ $\displaystyle\leq 0\quad(w\leq 0).$ (240)

We will use the fact that

$\displaystyle|1-3w|\leq\sqrt{1+3w^{2}}\leq|1+3w|\quad(w\geq 0),$ (241)
$\displaystyle|1+3w|\leq\sqrt{1+3w^{2}}\leq|1-3w|\quad(w\leq 0).$ (242)

For $\nu_{-}(w)$, we have

$\displaystyle 1-(w+1)\nu_{-}(w)$
$\displaystyle=\frac{-(1-3w)+\sqrt{1+3w^{2}}}{3w(1-w)},$ (243) $\displaystyle
1-w\nu_{-}(w)$ $\displaystyle=\frac{2+\sqrt{1+3w^{2}}}{3(1-w^{2})},$ (244)
$\displaystyle 1-(w-1)\nu_{-}(w)$
$\displaystyle=\frac{(1+3w)-\sqrt{1+3w^{2}}}{3w(1+w)},$ (245)

which yields (236)-(238). For $\nu_{+}(w)$, we have

$\displaystyle 1-(w+1)\nu_{+}(w)$
$\displaystyle=\frac{-(1-3w)-\sqrt{1+3w^{2}}}{3w(1-w)},$ (246) $\displaystyle
1-(w-1)\nu_{+}(w)$ $\displaystyle=\frac{(1+3w)+\sqrt{1+3w^{2}}}{3w(1+w)},$
(247)

which yields (239) and (240). $\blacksquare$

##  Appendix D Comparison with Previous Results

We compare the exponents obtained for a few examples in the main sections to
those obtained from the previous results based on Levy’s lemma. We compare the
exponents for the maximum eigenvalue (Example 2 in Section II) and entropy
(Section V.2) with those obtained from the results of Hayden et al. (2006),
and that for the expectation value (Section V.3) with those deduced from the
results of Popescu et al. (2006b); Reimann (2015). Note that the exp is taken
base $2$ in Hayden et al. (2006), while we adopt base $e$ here as in Popescu
et al. (2006b); Reimann (2015).

As for the maximum eigenvalue, Lemma III.4 in Hayden et al. (2006) asserts
that for any $0<\varepsilon\leq 1$, it holds that

$\displaystyle{\rm Pr}\left\\{\left.\lambda_{\rm
max}(\rho_{\Psi})>\frac{1+\varepsilon}{m}\right|\Psi\sim\mu_{n}\right\\}\leq\left(\frac{10m}{\varepsilon}\right)^{2m}\exp{\left(-\frac{n\varepsilon^{2}}{14}\right)}$
(248)

and

$\displaystyle{\rm Pr}\left\\{\left.\lambda_{\rm
min}(\rho_{\Psi})<\frac{1-\varepsilon}{m}\right|\Psi\sim\mu_{n}\right\\}\leq\left(\frac{10m}{\varepsilon}\right)^{2m}\exp{\left(-\frac{n\varepsilon^{2}}{14}\right)}.$
(249)

It is straightforward to verify that (248) yields

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.\lambda_{\rm
max}(\rho_{\Psi})>\frac{1+(m-1)\epsilon}{m}\right|\Psi\sim\mu_{n}\right\\}\geq\frac{(m-1)^{2}\epsilon^{2}}{14}.$
(250)

As shown in Example 2 in Section II, the exact exponent is given by
$m(m-1)\epsilon^{2}/2$. This is an improvement over (250) by factor
$7/(1-1/m)$, when viewed as an upper bound on the probability of large
deviation.

Theorem III.3 in Hayden et al. (2006) provides a large deviation bound for the
entropy. It asserts that for any $3\leq m\leq n$, it holds that

$\displaystyle{\rm
Pr}\left\\{\left.S(\rho_{\Psi})<\ln{m}-\frac{m}{n}-\epsilon\right|\Psi\sim\mu_{n}\right\\}\leq\exp{\left(-\frac{(mn-1)\epsilon^{2}}{8\pi^{2}(\ln{m})^{2}}\right)}.$
(251)

The equation (251) is equivalent to

$\displaystyle{\rm
Pr}\left\\{\left.1-\frac{S(\rho_{\Psi})}{\ln{m}}>\frac{m}{n\ln{m}}+\varepsilon\right|\Psi\sim\mu_{n}\right\\}\leq\exp{\left(-\frac{(mn-1)\varepsilon^{2}}{8\pi^{2}}\right)}.$
(252)

Hence, for any $\delta>\kappa>0$ and sufficiently large $n$, it holds that

$\displaystyle{\rm
Pr}\left\\{\left.1-\frac{S(\rho_{\Psi})}{\ln{m}}>\delta\right|\Psi\sim\mu_{n}\right\\}\leq\exp{\left(-\frac{(mn-1)(\delta-\kappa)^{2}}{8\pi^{2}}\right)}.$
(253)

It follows that

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.1-\frac{S(\rho_{\Psi})}{\ln{m}}>\delta\right|\Psi\sim\mu_{n}\right\\}\geq\frac{m(\delta-\kappa)^{2}}{8\pi^{2}}.$
(254)

Since this relation holds for any $\delta>0$, we arrive at

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\rm
Pr}\left\\{\left.1-\frac{S(\rho_{\Psi})}{\ln{m}}>\delta\right|\Psi\sim\mu_{n}\right\\}\geq\frac{m\delta^{2}}{8\pi^{2}}.$
(255)

We have shown in Section V.2 that the exact exponent is, when $\delta$ is
small, approximately equal to $\delta m\ln{m}$. As an upper bound on the
probability of large deviation, this is a quadratic improvement with respect
to $\delta$ and an improvement by factor $8\pi^{2}\ln{m}$ over (254).

A large deviation bound for the expectation value can be deduced from
Inequality (64) in Popescu et al. (2006b). Let
${\mathcal{H}}_{R}\subseteq{\mathcal{H}}_{S}\otimes{\mathcal{H}}_{E}$ be a
subspace with dimension $d_{R}$ and let $W$ be a Hermitian operator on
${\mathcal{H}}^{S}$. With $|\Psi\rangle\in{\mathcal{H}}_{R}$ randomly chosen
with respect to the unitary invariant measure $\mu$ on ${\mathcal{H}}_{R}$,
Ineq. (64) in Popescu et al. (2006b) reads

$\displaystyle{\rm Pr}\\{\left.|{\rm Tr}[\rho_{\Psi}W]-{\rm Tr}[\pi W]|\geq
w\right|\Psi\sim\mu\\}\leq
2\exp\left(-\frac{d_{R}w^{2}}{18\pi^{3}\|W\|_{\infty}^{2}}\right).$ (256)

Let ${\rm dim}{\mathcal{H}}_{S}=m$, ${\rm dim}{\mathcal{H}}_{E}=n$ and
${\mathcal{H}}_{R}={\mathcal{H}}_{S}\otimes{\mathcal{H}}_{E}$. We assume ${\rm
Tr}[W]=0$ and $w>0$. Then, (256) yields

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\rm Pr}\\{\left.{\rm
Tr}[\rho_{\Psi}W]\geq
w\right|\Psi\sim\mu_{n}\\}\geq\frac{mw^{2}}{18\pi^{3}\|W\|_{\infty}^{2}}.\quad\quad$
(257)

One can deduce a similar result from Equation (7) in Reimann (2015), in which
case we have

$\displaystyle-\limsup_{n\rightarrow\infty}\frac{1}{n}\ln{\rm Pr}\\{\left.{\rm
Tr}[\rho_{\Psi}W]\geq
w\right|\Psi\sim\mu_{n}\\}\geq\frac{2mw^{2}}{9\pi^{3}\Delta_{W}^{2}},\quad\quad$
(258)

where $\Delta_{W}$ is the difference between the minimum and the maximum
eigenvalues of $W$. Noting that $\Delta_{W}\leq 2\|W\|_{\infty}$, the bound
(258) is tighter than (257). We have shown in Section V.3 that the exact
exponent is, when $w$ is small, approximately equal to
$m^{2}w^{2}/2\|W\|_{2}^{2}$. As an upper bound on the probability of large
deviation, this is an improvement by factor
$9\pi^{3}m\Delta_{W}^{2}/4\|W\|_{2}^{2}$ over (258). Note that we have
$m\Delta_{W}^{2}\geq\|W\|_{2}^{2}$.

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