Table Of ContentSphere-Packing Bound for Symmetric Classical-Quantum Channels
Hao-Chung Cheng1,2, Min-Hsiu Hsieh2, and Marco Tomamichel2,3
1
Graduate Institute Communication Engineering, National Taiwan University, Taiwan (R.O.C.)
2Centre for Quantum Software and Information,
Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
7
1 3School of Physics, The University of Sydney, Australia
0
2
n
a Abstract. Weprovideasphere-packinglowerboundfortheoptimalerrorprobabilityinfiniteblocklengths
J
when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be
5 significantly improvedfrom theorderofthesubexponentialtothepolynomial. Thisestablished pre-factor
1
is essentially optimal because it matches the best known random coding upper bound in the classical
case. Ourapproachesrely onasharpconcentration inequality instronglarge deviationtheory andcrucial
]
h properties of theerror-exponent function.
p
t- 1. Introduction
n
a The probability of decoding error is one of the fundamental criteria for evaluating the performance
u
of a communication system. In Shannon’s seminal work [1], he pioneered the study of the noisy coding
q
[ theorem, which states that the error probability can be made arbitrarily small as the coding blocklength
grows when the coding rate R is below the channel capacity C. Later, Shannon [2] made a further step
2
v in exploring the exponential dependency of the optimal error probability ǫ∗(n,R) on the blocklength n
7 and rate R, and defined the reliability function as follows: given a fixed coding rate R < C, E(R) :=
5 limsup 1 logǫ (n,R). The quantity E(R) then provides a measure of how rapidly the error
9 probabinl→ity+∞ap−prnoache∗s zero with an increase in blocklength. This asymptotic characterization of the
2
0 optimal error probability under a fixed rate is hence called the error exponent analysis. For a classical
1. channel, the upper bounds of the optimal error can be established using a random coding argument [3].
0 On the other hand, the lower bound was first developed by Shannon, Gallager, and Berlekamp [4] and
7
was called the sphere-packing bound. Alternative approaches by Haroutunian [5] and Blahut [6] were
1
subsequently proposed.
:
v
In recent years, much attention has been paid to the finiteblocklength regime [7,8]. Altu˘g and Wagner
i
X employed strong large deviation techniques [9] to prove a sphere-packing bound with a finite blocklength
r n. Moreover, the pre-factor of the bound was significantly refined from the order of the subexponential
a
exp O(√n) [4] to the polynomial [10, 11]. This refinement is substantial especially at rates near
{− }
capacity, where the error-exponent function is close to zero; hence, the pre-factor dominants the bound
[12, 13].
Error exponent analysis in classical-quantum (c-q) channels is much more difficult because of the
noncommutative nature of quantum mechanics. Burnashev and Holevo [15, 16] investigated reliability
functions in c-q channels andproved therandom codingupperboundfor pure-state channels. Winter [17]
adopted Haroutunian’s method to derive a sphere-packing bound for c-q channels in the form of relative
entropy functions [5]. Dalai [18] employed Shannon-Gallager-Berlekamp’s approach to establish a sphere-
packing bound with Gallager’s expression [4]. It was later pointed out that these two sphere-packing
exponents are not equal for general c-q channels [19]. In this work, we initiate the study of the refined
sphere-packing bound in the quantum scenario. In particular, we consider a “symmetric c-q channel”
(see Section 2 for a detailed definition), which is an important class of covariant channels (e.g. [20]),
and establish a sphere-packing bound with the pre-factor improved from the order of the subexponential
E-mail address: [email protected], [email protected], [email protected].
1
in Dalai’s result [18] to the polynomial. Our result recovers Altu˘g and Wagner’s work [10] for classical
symmetricchannelsincludingthebinarysymmetricchannelandbinaryerasurechannel. Furthermore,the
proved pre-factor matches that of the best known random coding upper bound [21] in the classical case.
Hence, our result yields the exact asymptotics for the sphere-packing bound in symmetric c-q channels.
The main ingredients in our proof are a tight concentration inequality from Bahadur and Ranga Rao
[9], [13] (see Appendix A) and the major properties of the sphere-packing exponent [22]. We remark
that the result obtained in this paper might enable analysis in the medium error probability regime of a
classical-quantum channel [12, 13, 14]. We leave the case for general c-q channels as future work [23].
This paper is organized as follows. We introduce the necessary notation and state our main result in
Section 2. Section 3 includes the crucial properties of the error-exponent function. We provide the proof
of the main result in Section 4. Section 5 concludes this paper.
2. Notation and Main Result
2.1. Notation. Throughout this paper, we consider a finite-dimensional Hilbert space . The set of
H
densityoperators(i.e.positivesemi-definiteoperatorswithunittrace)on aredefinedas ( ). Forρ,σ
H S H ∈
( ),wewriteρ σifsupp(ρ) supp(σ),wheresupp(ρ)denotesthesupportofρ. Theidentityoperator
oSnH isdenotedb≪y1 . Whenth⊂ereisnopossibilityofconfusion,weskipthesubscript . WeuseTr[ ]as
theHtracefunction. LHetN,R,andR denotethesetofintegers, realnumbers,andposHitiverealnumb·ers,,
>0
respectively. Define [n] := 1,2,...,n for n N. Given a pair of positive semi-definite operators
ρ,σ ( ), we define the ({quantum) r}elative e∈ntropy as D(ρ σ) := Tr[ρ(logρ logσ)], when ρ σ,
∈ S H k − ≪
and + otherwise. For every α [0,1), we define the (Petz) quantum R´enyi divergences D (ρ σ) :=
α
1 log∞Tr ρασ1 α . For α = 1,∈D (ρ σ) := lim D (ρ σ) = D(ρ σ). Let = 1,2,..., k be
aα−fi1nite alphabet−, and let P( ) be t1hekset of probαa→b1ilityα dikstributionskon . InXparti{cular, we|Xde|}note
(cid:2) (cid:3) X X
by U the uniform distribution on . A classical-quantum (c-q) channel W maps elements of the
X X
finite set to the density operators in ( ), i.e., W : ( ). Let be a finite alphabetical
X S H X → S H M
set with size M = . An (n-block) encoder is a map f : n that encodes each message
n
m to a codewor|Md x|n(m) := x (m)...x (m) n. The codeMwor→d xXn(m) is then mapped to a state
1 n
W∈nM = W W ( n). T∈heXdecoder is described by a positive operator-valued
x⊗n(m) x1(m) ⊗ ··· ⊗ xn(m) ∈ S H⊗
measurement (POVM) Π = Π ,...,Π on n, where Π 0 and M Π = 1. The pair
n { n,1 n,M} H⊗ n,i ≥ i=1 n,i
(f ,Π ) =: is called a code with rate R = 1 log . The error probability of sending a message
n n Cn n |M| P
m with the code n is ǫm(W, n) := 1 Tr Πn,mWxn(m) . We use ǫmax(W, n) = maxm ǫm(W, n)
C C − C ∈M C
and ǫ¯(W, ) = 1 ǫ (W, ) to denote the maximal error probability and the average error
Cn M m m Cn (cid:0) (cid:1)
pemropbiarbicialiltyd,isrtersibpuectitoiPvneloyf.∈xMGn.iven a sequence xn ∈ Xn, we denote by Pxn(x) := n1 ni=11{x = xi} the
P
Throughout this paper, we consider a symmetric c-q channel defined as
W := Vx 1W (V )x 1, x , (1)
x − 1 † −
∀ ∈ X
where W ( ) is an arbitrary density operator, and V satisfies V V = VV = V = 1 . We
1 † † |X|
define the f∈ollSowHing conditional entropic quantities for the channel W with P P( ): D (W σHP) :=
α
∈ X k |
P(x)D (W σ). Themutual information ofthec-qchannelW : ( )withpriordistribution
P x∈XP( ) isαdefinxked as I(P,W) := D(W PW P), where PWα := X → SPH(x)Wα, α (0,1]. The
P(cla∈ssicaXl) capacity of the channel W : k ( )|is denoted by C := maxxP∈XP( )I(P,xW). ∈Let
X → S H P ∈ X
E(1)(R,P) := sup E (s,P) sR
sp { 0 − }
s 0
≥
α 1
E(2)(R,P) := sup min − (R D (W σ P)),
sp 0<α 1σ ( ) α − α k |
≤ ∈S H
where we denote by E (s,P) := logTr PW1/(1+s) 1+s an auxiliary function [16, 22]. The sphere-
0
−
packing exponent is defined by h i
(cid:0) (cid:1)
E (R):= max E(1)(R,P) = max E(2)(R,P), (2)
sp sp sp
P P( ) P P( )
∈ X ∈ X
2
where the last equality follows from [24, Proposition IV.2]. Further, we define a rate [25, p. 152], [18]:
R := lim max min P(x)D (W σ)
1 x
∞ s + P P( )σ ( ) 1+s k
→ ∞ ∈ X ∈S H x
X∈X
= max min P(x)Tr W0σ . (3)
x
P P( )σ ( )
∈ X ∈S H x
X∈X (cid:2) (cid:3)
It follows that E (R)= + for any R R (see also [4, p. 69] and [3, Eq. (5.8.5)]).
sp
∞ ≤ ∞
Consider a binary hypothesis whose null and alternative hypotheses are ρ ( ) and σ ( ),
respectively. The type-I error and type-II error of the hypothesis testing, for a∈nSopHerator 0 ∈QS H1,
are defined as α(Q;ρ) := Tr[(1 Q)ρ], and β(Q;σ) := Tr[Qσ]. There is a trade-off between≤these≤two
−
errors. Thus, we can define the minimum type-I error, when the type-II error is below µ (0,1), as
∈
α (ρ σ) := min α(Q;ρ) :β(Q;σ) µ . (4)
µ
k 0 Q 1 ≤
≤ ≤
2.2. Main Result. Let us now consider any sym(cid:8)metric c-q channel wit(cid:9)h capacity C.
b
Theorem 1 (Exact Sphere-Packing Bound). For any rate R [0,C), there exist an N N such that
0
∈ ∈
for all codes of length n N , we have
n 0
C ≥
1 o(1)
ǫ ( ) − exp nE (R) , (5)
max Cn ≥ n21(1+|Es′p(R)|) {− sp }
(1)
where Es′p(R) := ∂maxP∈P(X)Esp (r,P)/∂r|r=R.
3. Properties of the Sphere-Packing Exponent
Lemma 2 (Optimal Input Distribution). For any R > R , the distribution U is a maximizer of
(1) (2) ∞ X
E (R, ) and E (R, ).
sp sp
· ·
Proof. We first prove that U attains maxP P( )E0(s,P). From Eq. (1), it is not hard to verify that
U Wα = VU WαV for all αX (0,1]. Hence∈, itXfollows that
†
X X ∈
Tr[Wxα(U Wα)1−αα] = Tr[Vx−1W1αV†x−1(U Wα)1−αα] (6)
X X
= Tr[W1αV†x−1(U Wα)1−ααVx−1] (7)
X
= Tr[W1α(U Wα)1−αα] (8)
X
= Tr[(U Wα)α1] (9)
X
for all α (0,1]. The above equation shows that the distribution U that maximizes E (s,P), s 0
0
∈ X ∀ ≥
[16, Eq. (38)]. Then we have
E(1)(R,U ) = sup max E (s,P) sR = E (R).
sp X s≥0(cid:26)P∈P(X) 0 − (cid:27) sp
(2) (1)
Further, Jensen’s inequality implies that E (R,U ) E (R,U ) = E (R), which completes the
sp sp sp
proof. X ≥ X (cid:3)
Lemma 3 (Saddle-Point Property). Consider any R (R ,C) and P P( ). Let ( ) :=
P,W
∈ ∞ ∈ X S H
σ ( ) : x supp(P), supp(W ) supp(σ) = . We define
x
{ ∈ S H ∀ ∈ ∩ 6 ∅}
α 1
F (α,σ) := − (R D (W σ P)), (10)
R,P α
α − k |
on (0,1] ( ), and let P ( ) := P P(X) : min sup F (α,σ) R . The follow-
P,W R σ ( ) 0<α 1 R,P >0
ing holds×S H X ∈ ∈S H ≤ ∈
(cid:8) (cid:9)
(i) For any P P( ), F (, ) has a saddle-point with the saddle-value:
R,P
∈ X · ·
min sup F (α,σ) = sup min F (α,σ) = E(2)(R,W,P). (11)
R,P R,P sp
σ ( )0<α 1 0<α 1σ ( )
∈S H ≤ ≤ ∈S H
3
(ii) The saddle-point is unique for P P ( ).
R
(iii) Let P P ( ). The unique sadd∈le-poinXt (α,σ) of F (, ) satisfies α (0,1) and
R R,P
∈ X · · ∈
P(x)Wαe(1 α)Dα(Wx σ) 1/α
σ = x x − k W , x supp(P). (12)
∈X x
Tr(cid:0)P x P(x)Wxαe(1−α)Dα(Wxkσ(cid:1)) 1/α ≫ ∀ ∈
∈X
h i
The proof is provided in A(cid:0)pPpendix B.1. (cid:1)
Lemma 4 (Representation). For any R (R ,C), let (α⋆,σ⋆) be the saddle-point of F (, ). It
∈ ∞ R R R,UX · ·
follows that
U Wα⋆R 1/α⋆R
(α⋆R,σR⋆) = −Es′p(R), Tr(cid:0) UX Wα⋆R(cid:1) 1/α⋆R . (13)
X
h i
(2) (cid:0) (cid:1) (2)
Proof. Since Lemma 2 implies that U attains E (R, ), one observes from the definition of E that
sp sp
all the quantities Dα⋆R(WxkσR⋆), x ∈ XXare equal. By ite·m (iii) of Lemma 3, we obtain a representation
of σ⋆ in Eq. (13). The optimal α⋆ = ∂E (r,U )/∂r follows from [22, Eq. (42)]. (cid:3)
R R − sp X |r=R
Lemma 5 (Invariance). For any R (R ,C), we have
∈ ∞
F (α⋆,σ⋆)= E (R)> 0, P P( ), (14)
R,P R R sp ∀ ∈ X
where α⋆ and σ⋆ are defined in Eq. (13).
R R
Proof. Following the argument in Lemma 2 and recalling Eq. (13) in Lemma 4, one can verify that
sup F (α,σ⋆) = sup E (s,U ) sR = E (R) for all P P( ). Further, we obtain
E α(R∈()0,>1] 0Rf,oPr R R(R ,C) fsr≥o0m{th0e resuXlt−in [22}, Propospsition 10]. ∈ X (cid:3)
sp
∈ ∞
4. Proof of the Main Result
For rates in the range R R , we have E (R)= + . The bound in Eq. (5) obviously holds. Hence,
sp
≤ ∞ ∞
we consider the case of R (R ,C) and fix the rate throughout the proof.
∈ ∞
We first pose the channel coding problem into a binary hypothesis testing through Lemma 6, which
originates from Blahut [6] for the classical case.
Lemma 6(HypothesisTestingReduction). For anycode withmessage sizeenr, there existsanxn
n n
C ∈ C
such that
ǫ ( ) max α W n σ n . (15)
max n exp nr x⊗n ⊗
C ≥ σ (H) {− } k
∈S
(cid:0) (cid:1)
The proof is provided in Appendix B.2.
b
Let us now commence with the proof of Theorem 1. Fix arbitrary γ,ξ > 0. Let γ := 1 +γ logn and
n 2 n
R := R γ . The choice of the rate back-off term γ will become evident later. Choose N N such
n n n 1
− (cid:0) (cid:1)∈
that R R ξ > R . Let σ⋆ be defined in Eq. (13), and from Lemma 6, we have
n ≥ − ∞ R
ǫ ( ) α W n σ⋆ n . (16)
max Cn ≥ exp{−nRn} x⊗n k R⊗
Inn tpheafonldloqwnin:=g, wenprqovi,dwehaerleo(wper,bqou)nadbrefoNrutshsbeatuy(cid:0)mpe-S-Izkeor lraordiαs(cid:1)terxipb{u−tniRonn}s[2W6]x⊗nonfk(σWR⋆⊗n,σ.⋆L)feotrpenve:r=y
i=1 xi i=1 xi xi xi (cid:0) xi (cid:1) R
wNi ∈he[rne].PSxinncisetDhαe(eWmNxpikirσicR⋆a)l=disDtrαi(bpuxtiikoqnxio)f,fxonr.aMllαor∈eov(0e,r1,]i,tewmes(hiioi)rtihnbanLdemφnm(aRn3)im:=psliuepsαt∈h(a0t,1]tFheRns,tPaxtne(ασ,R⋆σR⋆),
dominates all the channel outputs: σR⋆ ≫ Wx, for all x ∈ supp(Pxn), Hence, we have pn ≪ qn. Subse-
quently, for every i [n], we let q (ω) = 0, for all ω supp(p ). We apply Nagaoka’s argument [27] by
choosing δ = exp nR∈ nφ (R )xito yield, for any 06∈ Q x1i,
n n n n
{ − } ≤ ≤
α(U;pn)+δβ(U;qn)
α Q ;W n +δβ Q ;σ⋆ n , (17)
n x⊗n n R⊗ ≥ 2
where α(U;pn) := p(cid:0)n(ω), β(U(cid:1);qn) :=(cid:0) qn(cid:1)(ω), and U := ω : pn(ω)enφn(Rn) > qn(ω)enRn .
ω Uc ω U
∈ ∈
4
P P (cid:8) (cid:9)
Next, we employ Bahadur-Ranga Rao’s concentration inequality, Theorem 9 in Appendix A, to further
lower bound α(U;pn) and β(U;qn). Before proceeding, we need to introduce some notation. We define
the tilted distributions, for every i [n], ω supp(p ), and t [0,1] by
∈ ∈ xi ∈
p (ω)1 tq (ω)t
qˆ (ω) := xi − xi . (18)
xi,t p (ω)1 tq (ω)t
ω∈supp(pxi) xi − xi
Let P
Λ0,xi(t) := logEpxi etlogpqxxii ;
(cid:20) (cid:21) (19)
Λ1,xi(t) := logEqxi etlogpqxxii .
(cid:20) (cid:21)
Since pn and qn are mutually absolutely continuous, the maps t Λ (t), j 0,1 are differentiable
7→ j,xi ∈ { }
for all t [0,1]. One can immediately verify the following partial derivatives with respect to t:
∈
q q
Λ (t) = E log xi , Λ (t) = Var log xi ,
′0,xi qˆxi,t p ′0′,xi qˆxi,t p
(cid:20) xi(cid:21) (cid:20) xi(cid:21) (20)
q p
Λ (t) = Var log xi , Λ (t)= E log xi .
′0′,xi qˆxi,t p ′1,xi qˆxi,1−t q
(cid:20) xi(cid:21) (cid:20) xi(cid:21)
With Λ (t) in Eq. (19), we can define
j,xi
Λj,Pxn(t) := Pxn(x)Λj,x(t), j ∈{0,1}; (21)
x
X∈X
Λ (z) := sup tz Λ (t) , j 0,1 , (22)
∗j,Pxn t R{ − j,Pxn } ∈ { }
∈
whereΛ (z)inEq.(22)istheFenchel-Legendre transform ofΛ (t). Thequantities Λ (z)would
∗j,Pxn j,Pxn ∗j,Pxn
appear in the lower bounds of α(U;pn) and β(U;qn) obtained by Bahadur-Randga Rao’s inequality as
shown later.
In the following, we relate the Fenchel-Legendre transform Λ (z) to the desired error-exponent
∗j,Pxn
function φ (R ). Such a relationship is stated in Lemma 7; the proof is provided in Appendix B.3.
n n
Lemma 7. Under the prevailing assumptions and for all R (R ,C), the following holds:
n
∈ ∞
(i) Λ (φ (R ) R )= φ (R );
∗0,Pxn n n − n n n
(ii) Λ (R φ (R )) = R ;
∗1,Pxn n− n n n
(iii) There exists a unique t⋆ = s⋆ (0,1), such that Λ (t⋆) = φ (R ) R , where s⋆ :=
1+s⋆ ∈ ′0,Pxn n n − n
∂φn(r) .
∂r |r=Rn
Item(iii)inLemma7showsthattheoptimizertinEq.(22)alwaysliesinthecompactset[0,1]. Further,
Eqs. (19) and (20) ensure that Λ (t) = Λ (1 t), Λ (t)= Λ (1 t), Λ (t) = Λ (1 t). We
0,xi 1,xi − ′0,xi − ′1,xi − ′0′,xi ′1′,xi −
define the following quantities:
V := max Λ (t); (23)
max ′0′,x
t [0,1],x
∈ ∈X
V := min Λ (t); (24)
min ′0′,x
t [0,1],x
∈ ∈X
T := max T (t); (25)
max 0,x
t [0,1],x
∈ ∈X
3
q
T (t):= E log x Λ (t) ; (26)
0,x qˆx,t p − ′0,x
"(cid:12) x (cid:12) #
(cid:12) (cid:12)
T (t) := T (1 t); and K := 15√2πT /V(cid:12) . Note that(cid:12)for every x , Λ () and T ()
1,x 0,x − max max (cid:12)min (cid:12) ∈ X ′0′,x · 0,x ·
are continuous functions on [0,1] from the definitions in Eqs. (20), (26) (see also [10, Lemma 9]). The
maximization and minimization in the above definitions are well-defined and finite. Moreover, Lemma 8
guarantees that V is bounded away from zero.
min
5
Lemma 8 (Positivity). For any Rn ∈ (R∞,C) and Pxn ∈ P(X), Λ′0′,Pxn(t) > 0, for all t ∈ [0,1].
Proof. Assume Λ (t) is zero for some t [0,1]. This is equivalent to
′0′,Pxn ∈
Λ′ (t)
pxi(ω) = qxi(ω)·e− 0,xi , ∀ω ∈ pxi, ∀i ∈[n]. (27)
Summing the right-hand side of Eq. (27) over ω ∈ pxi gives 1 = Tr p0xiqxi e−Λ′0,xi(t), ∀i ∈ [n]. Then,
Eqs. (27) and the above equation imply that
(cid:2) (cid:3)
α 1
φn(Rn)= sup α− Rn+ Pxn(x)logTr p0xqx
0<α 1 !
≤ xX∈X (cid:2) (cid:3)
= 0,
wimhpelrieeswtehautseφth(Re fa)c=t tEhat(RRn)>> 0R,∞wh=ich−leaxd∈sXtoPxanc(xon)tloragdTicrtipo0xnq.x ; see Eq. (3)). However, Lemma(cid:3)5
n n sp n
P (cid:2) (cid:3)
Now, we are ready to derive the lower bounds to α(U;pn) and β(U;qn). Let N N be sufficiently
2
∈
large such that for all n N ,
2
≥
1+(1+K )2
√n max . (28)
≥ √V
min
Applying Bahadur-Randga Rao’s inequality (Theorem 9) to Z = logq logp with the probability
i i i
−
measure λ = p , and z = R φ (R ) gives
i i n n n
−
n
1
α(U;pn) = Pr Z R φ (R ) (29)
i n n n
n ≥ −
( )
i=1
X
2A
exp nΛ (φ (R ) R ) (30)
≥ √n − ∗0,Pxn n n − n
(cid:8) (cid:9)
where A := e−Kmax . Similarly, applying Theorem 9 to Z = logp logq with the probability measure
√4πVmax i i − i
λ = q , and z = φ (R ) R yields
i i n n n
−
n
1
β(U;qn) = Pr Z φ (R ) R (31)
i n n n
n ≥ −
( )
i=1
X
2A
exp nΛ (R φ (R )) . (32)
≥ √n − ∗1,Pxn n− n n
(cid:8) (cid:9)
Continuing from Eq. (30) and item (i) in Lemma 7 gives
2A
α(U;pn) exp nφ (R ) . (33)
n n
≥ √n {− }
Eq. (32) together with item (iii) in Lemma 7 yields
2A
β(U;qn) exp nR = 2Anγexp nR . (34)
n
≥ √n {− } {− }
Let N N such that Anγ > 1, for all n N . Then Eq. (34) implies that β(U;qn) > 2exp nR .
3 3
∈ ≥ {− }
Thus, we can bound the left-hand side of Eq. (17) from below by A e nφn(Rn). For any test 0 Q 1
√n − ≤ n ≤
such that β(Q ;σ⋆ n) exp nR , we have
n R⊗ ≤ {− }
αexp{−nRn} Wx⊗nnkσR⋆⊗n = α(Qn;ρn)
A A
exp(cid:0) nφ (R ) (cid:1)= exp nE (R ) , (35)
b≥ √n {− n n } √n {− sp n }
where the last equality follows from Lemma 5.
6
Finally, it remains to remove the back-off term R = R γ in Eq. (35). By Taylor’s theorem, we have
n n
−
γ2
E (R γ ) = E (R) γ E (R)+ nE (R¯), (36)
sp − n sp − n s′p 2 s′′p
for some R¯ (R ξ,R) and E (R¯) := ∂2Es(p1)(r,UX) . Further, one can calculate that
∈ − s′′p ∂r2
(cid:12)r=R¯
(cid:12)
Es′′p(R¯) = − ∂2E0∂(cid:12)(cid:12)(ss2,UX) −1 (37)
(cid:18) (cid:12)s=s¯(cid:19)
(1+s¯)3 ((cid:12)(cid:12)1+s¯)3
= (cid:12) =:Υ, (38)
Λ s¯ ≤ Vmin
′0′,UX 1+s¯
(cid:16) (cid:17)
1 α⋆
where s¯= − R¯. From item (iii) in Lemma 3, it follows that both s¯and E (R) = s⋆ are both positive
α⋆R¯ | s′p |
and finite for R¯ (R ,C) and R (R ,C). Together with the fact that V > 0, we have Υ R .
min >0
We apply Taylor∈’s exp∞ansion on th∈e fun∞ction n () again to yield ∈
− ·
logn
n−21(1+|Es′p(R)|)−γnΥ = n−21(1+|Es′p(R)|)· 1− nx¯Γ γnΥ
(cid:18) (cid:19)
= n−21(1+|Es′p(R)|) (1 o(1)), (39)
· −
where the first equality holds for some x¯ (0,γ ), and the last line follows from the definition γ =
n n
∈
(1 +γ)log. Finally, by combining Eqs. (16), (35), and (39), we obtain the desired Eq. (5) for sufficiently
2 n
large n N := max N ,N ,N .
0 1 2 3
≥ { }
5. Discussion
In this work, we establish a sphere-packing bound with a refined polynomial pre-factor that coincides
with the best classical results [10, Theorem 1] to date. As discussed by Altu˘g and Wagner [10, Sec. VII],
the pre-factor is correct for binary symmetric channels but slightly worse for binary erasure channels (in
the order of 1/√n). On the other hand, our pre-factor matches the recent result of the random coding
upper bound [21, Theorem 2], where the pre-factor has been shown to be exact. Hence, we conjecture
that the established result is optimal for general symmetric c-q channels.
Thiswork admitsvariety ofpotential extensions. First, thesymmetricc-qchannelstudiedinthis paper
is a covariant channel with a cyclic group:
W = (g)W (g) , g,x , (40)
Uin(g)xUin(g)† Uout xUout † ∀ ∈ X
where and are the unitary representations on and ( ) such that (g)x (g) = (x +
in out in in †
U U X S H U U
g) mod and (g) = Vg. It would be interesting to investigate whether the refined sphere-packing
out
|X| U
bound can be extended to covariant quantum channels : ( ) ( ) with arbitrary compact
in out
N S H → S H
groups. Second, the random coding bound in the quantum case has been proved only for pure-state
channels [16]. It is promisingto prove the boundfor this class of c-q channels by employing the symmetry
property. Finally, therefinementprovidesanewpossibilityformoderatedeviationanalysisinc-qchannels
[13], which is left as future work.
Acknowledgements
MH is supported by an ARC Future Fellowship under Grant FT140100574. MT is funded by an ARC
Discovery Early Career Researcher Award (DECRA) fellowship and acknowledges supportfrom the ARC
Centre of Excellence for Engineered Quantum Systems (EQUS).
7
Appendix A. A Tight Concentration Inequality
Let (Z )n bea sequence of independent, real-valued random variables whose probability measures are
i i=1
λ . Let Λ (t) := logE etZi and define the Fenchel-Legendre transform of 1 n Λ () to be: Λ (z) :=
i i n i=1 i · ∗n
sup zt 1 n Λ (t) , z R. Thenb there exists a real number t⋆ (0,1] for every z R
sucht∈Rthat z−=n 1 i=1n (cid:2)iΛ (t(cid:3)⋆) a∀nd ∈Λ (z) = zt⋆ 1 n Λ (t⋆). Define the Ppr∈obability measure λ˜∈via
(cid:8) Pn i=1 ′i(cid:9) ∗n − n i=1 i i
dλ˜i(z ) := et⋆zi Λi(t⋆), and let Z¯ := Z E [Z ]. Furthermore, define m := n Var Z¯ , m :=
dλi i − P i i − λ˜i i P 2,n i=1 λ˜i i 3,n
ni=1Eλ˜i Z¯i 3 , and Kn(t⋆):= 15√m2π2,mn3,n. With these definitions, we can nowPstate the fo(cid:2)llow(cid:3)ing sharp
Pconcentrathi(cid:12)on(cid:12)iniequality for n1 ni=1Zi:
(cid:12) (cid:12)
Theorem 9 (Bahadur-RangaPRao’s Concentration Inequality [11, Proposition 5], [28]). Given
√m2,n 1+(1+Kn(t⋆))2, (41)
≥
it follows that
1 n e Kn(t⋆)
Pr Zi z e−nΛ∗n(z) − . (42)
(n ≥ ) ≥ 2 2πm2,n
i=1
X
Appendix B. Proofs of Miscellapneous Lemmas
B.1. Proof of Lemma 3. Let R > R and P P( ) be arbitrary. It is convenient to reparameterize
the function F by the substitution α∞= 1 : ∈ X
R,P 1+s
F (α,σ) = sR+sD (W σ P) =:K (s,σ). (43)
R,P |α=1+1s − 1+1s k | R,P
In the following, we prove the existence of a saddle-point of K (, ) on R ( ), where R :=
R,P 0 P,W 0
[0, ). By Ref. [29, Lemma 36.2], (s⋆,σ⋆) is a saddle point of K ·(·, ) if a≥nd×onSly if tHhe supremum≥ in
R,P
∞ · ·
sup inf K (s,σ) (44)
R,P
s∈R≥0σ∈SP,W(H)
is attained at s⋆, the infimum in
inf sup K (s,σ) (45)
R,P
σ∈SP,W(H)s∈R≥0
is attained at σ⋆, and the two extrema in Eqs. (44), (45) are equal and finite. We first claim that
s R , inf K (s,σ) = inf K (s,σ). (46)
0 R,P R,P
∀ ∈ ≥ σ P,W( ) σ ( )
∈S H ∈S H
To see that, observe that for any s R , the definition of the α-R´enyi divergence yields
0
∈ ≥
σ ( ) ( ), D (W σ P) = + , (47)
P,W 1
∀ ∈S H \S H 1+s k | ∞
which, in turn, implies
σ ( ) ( ), K (s,σ) = + . (48)
P,W R,P
∀ ∈ S H \S H ∞
Hence, Eq. (46) yields
sup inf K (s,σ) = sup inf K (s,σ) = sup min K (s,σ),
R,P R,P R,P (49)
s∈R≥0σ∈SP,W(H) s∈R≥0σ∈S(H) s∈R≥0σ∈S(H)
wherethelastequalityinEq.(49)followsfromthelowersemi-continuityofthemapσ D (W σ P)
1/(1+s)
7→ k |
[24, Corollary III.25] and the compactness of ( ). Further, by the fact R > R and the definition of
(2) S H ∞
E , we have
sp
E(2)(R,P) = sup min K (s,σ) < + , (50)
sp s∈R≥0σ∈S(H) R,P ∞
which guarantees the supremum in the right-hand side of Eq. (49) is attained at some s R , i.e.,
0
∈ ≥
sup inf K (s,σ) = max min K (s,σ) < + . (51)
R,P R,P
s∈R≥0σ∈SP,W(H) s∈R≥0σ∈S(H) ∞
8
Thus, we complete our claim in Eq. (44). It remains to show that the infimum in Eq.(45) is attained
at some σ⋆ ( ) and the supremum and infimum are exchangeable. To achieve this, we will show
P,W
that (R , ∈ S ( )H,K ) is a closed saddle-element (see Definition 10 below) and apply Rockafellar’s
0 P,W R,P
≥ S H
saddle-point result, Theorem 11, to conclude our claim.
Definition 10 (Closed Saddle-Element [29]). The triple ( , ,F) is called a closed saddle-element if for
A B
any1 x ri( ) (resp. y ri( )):
∈ A ∈ B
(a) (resp. ) is convex;
B A
(b) F(x, ) (resp. F(,y)) is convex (resp. concave) and lower (resp. upper) semi-continuous; and
· ·
(c) any accumulation point of (resp. ) that does not belong to (resp. ), say y (resp. x )
o o
B A B A
satisfies lim F(x,y) = + (resp. lim F(x,y) = ).
y→yo ∞ x→xo −∞
Theorem 11 (The Existence of Saddle-Points [29, Theorem 8], [30, Theorem 37.3]). Let ( , ,F) be any
closed saddle-element on Rm Rn. A B
×
(I) Nonon-zerox hastheproperty that, forallx ri( )andy ri( ), thehalf-line x+tx : t 0
0 0
∈ A ∈ B { ≥ }
is contained in and F(x+tx ,y) is a non-zero and non-decreasing function for t 0.
0
A ≥
(II) Nonon-zeroy hastheproperty that, forallx ri( )andy ri( ), thehalf-line y+ty : t 0
0 0
∈ A ∈ B { ≥ }
is contained in and F(x,y+ty ) is a non-increasing function for t 0.
0
B ≥
If condition (I) is satisfied, then
max inf F(x,y) = inf supF(x,y) <+ . (52)
x y y x ∞
∈A ∈B ∈B ∈A
If condition (II) is satisfied, then
< sup inf F(x,y) = minsupF(x,y). (53)
−∞ x y y x
∈A ∈B ∈B ∈A
If (I) and (II) are both satisfied, then F has a saddle-point on .
A×B
Fix an arbitrary s ri(R ) = R . We check that ( ( ),K (s, )) fulfills the three items
0 >0 P,W R,P
∈ ≥ S H ·
in Definition 10. (a) The set ( ) is clearly convex. (b) Since the map σ D (W σ P)
P,W 1/(1+s)
S H 7→ k |
is convex (owing to Lieb’s concavity theorem [31]) and lower semi-continuous on ( ) [24, Corollary
L H +
III.25], by Eq. (43), σ K (α,σ) is also convex and lower semi-continuous on ( ). (c) Due to
R,P P,W
7→ S H
the compactness of ( ), any accumulation point of ( ) that does not belong to ( ), say σ ,
P,W P,W o
S H S H S H
satisfies σ ( ) ( ). By Eqs. (47), (48), one finds K (α,σ ) = + .
o P,W R,P o
∈ S H \S H ∞
Next,fixanarbitraryσ ri( ( )). Owingtotheconvexityof ( ),itfollowsthatri( ( ))
P,W P,W P,W
∈ S H S H S H
= ri(cl( ( ))) (see e.g. [30, Theorem 6.3]). We first claim cl( ( )) = ( ). To see this, ob-
P,W P,W
S H S H S H
serve that ( ) ( ) since a full-rank density operator is not orthogonal with every W , x .
S H ++ ⊆ SP,W H x ∈ X
Hence,
( )= cl ( ) cl( ( )). (54)
S H S H ++ ⊆ SP,W H
On the other hand, the fact P,W( ) ( (cid:0)) leads to(cid:1)
S H ⊆ S H
cl( ( )) cl( ( )) = ( ). (55)
P,W
S H ⊆ S H S H
By Eqs. (54) and (55), we deduce that
ri( ( )) = ri(cl( ( )))= ri( ( )) = ( ) , (56)
SP,W H SP,W H S H S H ++
where the last equality in Eq. (56) follows from [32, Proposition 2.9]. Hence, we obtain
σ ri( ( )) and x , σ W . (57)
P,W x
∀ ∈ S H ∀ ∈ X ≫
Now,weverifythat(R ,K (,σ))satisfiesthethreeitemsinDefinition10. (a)ThesetR isobviously
0 R,P 0
≥ · ≥
convex. (b) From Eqs. (57) and the definition of the R´enyi divergence , the map s D (W σ P) is
1/(1+s)
7→ k |
continuousonR . Further,s sD (W σ P)isconcave onR [24,AppendixB].ByEq.(43),the
0 1/(1+s) 0
map s K (s≥,σ) is concave7→and continuouskon| R . (c) Since R≥ is closed, there is no accumulation
R,P 0 0
point o7→f R that does not belong to R . ≥ ≥
0 0
≥ ≥
1Wedenote byri and cl therelative interior and theclosure of a set, respectively.
9
We are now in a position to prove item (i) of this Proposition. Since the set ( ) is bounded,
P,W
S H
condition (II) is satisfied. Equation (53) in Theorem 11 implies that
< sup inf K (s,σ) = min sup K (s,σ). (58)
R,P R,P
−∞ s∈R≥0σ∈SP,W(H) σ∈SP,W(H)s∈R≥0
Then Eqs. (51) and (58) lead to the existence of a saddle-point of K (, ) on R ( ). Note
R,P 0 P,W
· · ≥ ×S H
that K (s,σ) = F (1/(1+s),σ). We conclude the existence of a saddle-point of F (, ) on (0,1]
R,P R,P R,P
· · ×
( ). Hence, item (i) is proved.
P,W
S H
We postpone the proof of the uniqueness of the optimizer to later and now show item (iii). Given any
R (R ,C) and P P ( ), one finds
R
∈ ∞ ∈ X
min sup F (α,σ) (0,+ ). (59)
R,P
σ ( )0<α 1 ∈ ∞
∈S H ≤
If α⋆ = 1 and σ⋆ is a saddle point of F (, ), by Eq. (10) we deduce that F (1,σ⋆) = 0 for every
R,P R,P
· ·
possible σ⋆, which contradicts Eq. (59). Hence, α⋆ = 1 is not a saddle point of F (,σ⋆).
R,P
·
For any saddle-point (α⋆,σ⋆) of F (, ), it holds that
R,P
· ·
α⋆ 1 1 α⋆
FR,P(α⋆,σ⋆) = σmi(n)FR,P(α⋆,σ) = α−⋆ R+ −α⋆ σmi(n)Dα⋆(Wkσ|P). (60)
∈S H ∈S H
We claim the minimizer of Eq. (60) must satisfy
1
Wα⋆ α⋆
σ⋆ = (cid:18) x∈X P(x)Tr[Wxα⋆(xσ⋆)1−α⋆](cid:19) = x P(x)Wxα⋆e(1−α⋆)Dα⋆(Wxkσ) α1⋆ (61)
∈X
P 1 1
Tr P(x) Wxα⋆ α⋆ Tr(cid:0)P x P(x)Wxα⋆e(1−α⋆)Dα⋆(Wxkσ(cid:1)) α⋆
x Tr[Wα⋆(σ⋆)1−α⋆] ∈X
" ∈X x # h i
(cid:18) (cid:19) (cid:0)P (cid:1)
P
for every α⋆ (0,1). Our approach closely follows from Hayashi and Tomamichel [33, Lemma 5]. For
∈
two density operators σ,ω ( ) and a map G : ( ) ( ) (where ( ) denotes the self-adjoint
∈ S H S H → L H sa L H sa
operators on ), define the Fr´echet derivative (see e.g. [33, Appendix C], [34]2)
H
∂ G(σ) := DG(σ)[ω σ]. (62)
ω
−
By letting
g (σ) := P(x)logTr Wασ1 α , (63)
α x −
x
X∈X (cid:2) (cid:3)
it follows that
σ⋆ = argminD (W σ P) = argmaxg (σ), α (0,1). (64)
α α
k | ∀ ∈
σ ( ) σ ( )
∈S H ∈S H
Since the map σ g (σ) is strictly concave for every α (0,1) [31], a sufficient and necessary condition
α
7→ ∈
for σ to be an optimizer of Eq. (64) is ∂ g (σ) = 0 for all ω ( ). Direct calculation shows that
ω α
∈ S H
Wα
∂ g (σ) = Tr P(x) x ∂ σ1 α. (65)
ω α Tr[Wασ1 α] ω −
" x − #
x
X∈X
Next, we check that the fixed-points of the following map achieves the optimum:
1
P(x) Wxα α
x Tr[Wασ1−α]
σ ∈X x . (66)
7→ Tr(cid:16)P P(x) Wxα (cid:17) α1
x Tr[Wασ1−α]
∈X x
(cid:20) (cid:21)
(cid:16) (cid:17)
P
2Wenote that the Fr´echet derivativeof functions involving matrices has other applications in quantum information theory;
see e.g. [35, 36, 37].
10
