Proof of Proposition 9.6

I_α⁽²⁾(P,W) − R

. (9.164)

Item(c) in Proposition3.2then shows that ^1−α_α Iα⁽²⁾(P,W) is strictly concave on (0, 1), which in turn implies that FR,P(·, σ^?)is also strictly concave on (0, 1). Hence, the maximizer of Eq. (9.163) is unique.

(9.5-(c)) As shown in the proof of item (b), α^?= 1 is not a saddle point of FR,P(·, ·) for any R > R∞ and P ∈PR(X ). We assume (α^?, σ^?) is a saddle-point of FR,P(·, ·)with α^? ∈ (0, 1), it holds that

F_R,P(α^?, σ^?) = min

σ∈S(H)F_R,P(α^?, σ) = α^?− 1

α^? R +1 − α^? α^? min

σ∈S(H)D_α^?(Wkσ|P ). (9.165) Then, it is clear from Proposition3.2-(c)in Section3.3that

σ^?  W_x, ∀x ∈supp(P ), (9.166)

and thus item(c) is proved.

9.3.2 Proof of Proposition 9.6

(9.6-(a)) Fix any arbitrary P ∈ P(X ). Item (b)in Proposition 3.2shows that the map α 7→ Iα⁽²⁾(P,W) is

E_sp⁽²⁾(R, P ) = +∞ for all R ∈ (0, I₀⁽²⁾(P,W)); nite for all R > I₀⁽²⁾(P,W); and Esp⁽²⁾(R, P ) = 0, for all R ≥ I₁⁽²⁾(P,W).

For every α ∈ (0, 1], the function ^1−α_α (Iα⁽²⁾(P,W) − R) in Eq. (9.13) is an non-increasing, convex, and continuous function in R ∈ R>0. Since Esp⁽²⁾(R, P ) is the pointwise supremum of the above function, Esp⁽²⁾(R, P )is non-increasing, convex, and lower semi-continuous function for all R ≥ 0.

Furthermore, since a convex function is continuous on the interior of the interval if it is nite [121, Corollary 6.3.3], thus Esp⁽²⁾(R, P ) is continuous for all R > I₀⁽²⁾(P,W), and continuous from the right at R = I₀⁽²⁾(P,W).

To establish the continuity of Esp⁽²⁾(R, P ) in P ∈ P(X ), we rst claim that there exists some

¯

αR∈ (0, 1] such that for every P ∈ P(X ), sup

α∈(0,1]

1 − α α



I_α⁽²⁾(P,W) − R

= sup

α∈[ ¯αR,1]

1 − α α



I_α⁽²⁾(P,W) − R

. (9.167)

Recall that R > R∞= max_{P ∈P(X )}I₀⁽²⁾(P,W). The continuity, item(a)in Proposition3.2, implies that there is an ¯αR> 0 such that

R ≥ I_α⁽²⁾_¯R(P,W), ∀P ∈ P(X ). (9.168) Then, Eq. (9.168) and the monotone increases of the map α 7→ Iα⁽²⁾(P,W) yield that,

1 − α α



I_α⁽²⁾(P,W) − R

< 0, ∀P ∈P(X ), and α ∈ (0, ¯αR). (9.169) The non-negativity of E⁽²⁾sp(R, P ) ≥ 0 ensures that the maximizer α^? will not happen in the region (0, ¯αR), and thus Eq. (9.167) is evident. Finally, Berge's maximum theorem [109, Section IV.3], [110, Lemma 3.1] coupled with the compactness of [¯αR, 1]and item (a)in Proposition3.2 complete our claim:

P 7→ E_sp⁽²⁾(R, P ) = sup

α∈[ ¯αR,1]

1 − α α



I_α⁽²⁾(P,W) − R

is continuous on P(X ). (9.170)

(9.6-(b)) The statement follows since item (a) holds for any P ∈ P(X ).

(9.6-(c)) For any R ∈ (R∞, C_W) and P ∈ PR(X ), item (b) in Proposition 9.5 shows that the optimizer α^?_R,P is unique. Moreover, Eq. (9.148) follows from in Lemma 2.14-(d)in Section2.2.

(9.6-(d)) The proof of this item is similar to [91, Proposition 3.4]. Fix any Po ∈ PR(X ) and consider arbitrary {Pk}_k∈N such that Pk ∈ PR(X ), ∀k ∈ N, and limn→+∞P_k = Po. Following from Eq. (9.148), we have

s^?_R,Pk = − ∂E_sp⁽²⁾(r, Pk)

∂r     r=R

. (9.171)

Given any R ∈ (R∞, C_W), the continuity of Esp⁽²⁾(R, ·)(see item (a)) implies that

k→+∞lim E_sp⁽²⁾(R, P_k) = E_sp⁽²⁾(R, P_o). (9.172) Then, continuity of the rst-order derivative in [128, Corollary VI.6.2.8], we have

k→+∞lim s^?_R,Pk = lim

k→+∞− ∂E_sp⁽²⁾(r, P_k)

∂r     r=R

= − ∂E_sp⁽²⁾(r, Po)

∂r     r=R

= s^?_R,Po, (9.173)

which completes the proof.

Achievability (Channel Coding)

In the error exponent regime (i.e. large deviation regime), the achievability for information transmis-sions means that one has to construct a coding strategy and show the probability of error achieves the desired upper bound given a xed transmission rate. The nite blocklength achievability bound for classical-quantum channel exponent was rst studied by Burnashev, Holevo [34,35], and Winter [37].

Specically, Burnashev and Holevo [34] introduced the following random coding exponent E_r(R) and the auxiliary function E0(s, P ) (see also Eqs. (9.1) and (9.5)):

E_r(R) = sup

0≤s≤1

sup

P ∈P(X ){E₀(s, P ) − sR} ; (10.1)

E0(s, P ) = − log Tr



 X

x∈X

P (x)W

1

x1+s

!1+s

. (10.2)

By quantum Sibson's identity given in Lemma3.3, it is easy to show that the random coding exponent can be expressed the Rényi capacity with Petz's version (see Eqs. (3.63) and (3.5)):

E_r(R) = sup

1 2≤α≤1

1 − α

α C_α,W− R
. (10.3)

Further, they showed that [34,35] for pure-state c-q channels (i.e. the channel outputs are all rank-one density operators), there exists a random coding strategy and some decoder (POVM) such that the average error probability over the ensemble, denoted by Pe(n, R), can be upper bounded as

P_e(n, R) ≤ 4 exp{−nE_r(R)}, ∀R < C_W, n ∈N. (10.4) However, for general c-q channels (i.e. the channel outputs are possibly non-rank-one density operators), the random coding bound by the exponent function in Eq. (10.3) is still open.

The slightly weaker and the best to date achievability bound was later proven by Hayashi [87,88, 129]:

P_e(n, R) ≤ 4 exp{−nE_r^↓(R)}, ∀R < C_W, n ∈N. (10.5)

The above bound holds for all c-q channels. However, it can be shown that

E_r^↓(R) ≤ E_r(R), ∀R < C_W. (10.6) Recently, Dalai [130] proposed a method to prove Eqs. (10.3) and (10.5). For the sake of complete-ness, we provide the proof below.

Theorem 10.1 (Dalai [130]). Given any classical-quantum channels W : X → S(H), and any random codes with size M and distribution P ∈ P(X ), we have the one-shot bound:

Pe(1, log M ) ≤ 6(M − 1)^sexp

For pure-state classical-quantum channels,

P_e(1, log M ) ≤ 6(M − 1)^sexp {−E₀(s, P )} , ∀s ∈ [0, 1]. (10.9)

Using Hayashi-Nagaoka inequality, Lemma2.9, we have

1 − Πxi ≤ 2(1 − πxi) + 4X

j6=i

π_xj. (10.12)

Hence, the average probability of error given realizations (x1, . . . , x_M) can be upper bounded as

Pr {error|(x1, . . . , x_M)} = 1

For 0 < α ≤ 1 and 0 < s ≤ 1, using2.10to bound the rst term in Eq. (10.14) as

Recalling the operator concavity of u 7→ u^s, we take expectation of the random code to obtain

2E Tr For the second term in Eq. (10.14), we re-index it to have

4 1

Again, using Lemma 2.10yields

Tr

Taking expectation and combining with Eq. (10.18), we have P_e(n, R) ≤ 6(M − 1)^sTr

ExW_x^1−αs

Ex[W_x^α]^s . (10.21) Invoking the denition of Er^↓(R) and choosing α = 1/(1 + s), we obtain Eq. (10.7).

For pure-state c-q channels, Eq. (10.21) can be rewritten as

P_e(n, R) ≤ 6(M − 1)^sTr [Ex[W_x]]^1+s, (10.22) because Wx^p = W_x for p ≥ 0 for pure-state c-q channels. The above expression equals to Eq. (10.9), which completes the proof.

Remark 10.1. To obtain the Eq. (10.9) for general c-q channels, one possible way of the above method is to employ the inequality



which in turn implies

X

j6=i

Wxj ≤



 X

j6=i

W_x^α_j





1 α

. (10.24)

Unfortunately, the operator inequality in Eq. (10.23) does not hold for general density operators Wxi.

The inequality only holds under the weak majorization. 3

Lastly, the following Conjecture 10.1 was posed by Holevo [35]. Note that to achieve Eq. (10.26), the right-hand side of Eq. (10.25) allows to have any sub-exponential prefactors exp{o(n)}.

Conjecture 10.1 (Random Coding Bound for Classical-Quantum Channels). Given any classical-quantum channels W : X → S(H), transmission rate R < C_W, and random codes with distribution P ∈P(X ), one has

P_e(n, R) ≤ exp{−nE_r(R, P )}, ∀n ∈N. (10.25) In particular,

ε^∗(n, R) ≤ exp{−nE_r(R)}, ∀n ∈N. (10.26)

Optimality (Channel Coding)

In this chapter, we present the weak and strong sphere-packing bounds for c-q channels. In Section11.1, we rst review existing approaches of proving classical sphere-packing bound. In Section 11.2, we provide the proof of a weak sphere-packing bound by using Wolfowitz strong converse. This bound is new in the quantum scenario and will be used in the moderate deviation analysis in Section 12.

In Section 11.3, we prove our main result of a nite blocklength strong sphere-packing bound for c-q channels, see Theorem 11.1 below, which improve Dalai's prefactor [38, 39] from the order of subexponential e^O(√n) to polynomial. Lastly, in Section 11.4, we obtain exact asymptotics (i.e. exact prefactors) of the strong sphere-packing bound for a symmetric c-q channels, which can be seen as a generalization of classical symmetric channels [21].

Theorem 11.1 (Finite Blocklength Strong Sphere-Packing Bound of Constant Composition Codes).

Consider a classical-quantum channel W : X → S(H) and R ∈ (R∞, C_W). For every γ > 0, there exist an N0 ∈ N and a constant A > 0 such that for all constant composition codes Cn of length n ≥ N0

with message size |Cn| ≥ exp{nR}, we have

¯

ε (C_n) ≥ A

n¹²(^1+|Esp⁰ (R)|+γ) exp {−nE_sp(R)} . (11.1) The following corollary generalizes the rened sphere-packing bound for constant composition codes to arbitrary codes by using the standard argument [30, p. 95]. We delay the proof to the end of Section 11.3.5.

Corollary 11.1 (Finite Blocklength Strong Sphere-Packing Bound of General Codes). Consider a classical-quantum channel W : X → S(H) and R ∈ (R∞, C_W). There exist some t > 1/2 and N0 ∈N such that for all codes of length n ≥ N0, we have

ε^∗(n, R) ≥ n^−texp {−nE_sp(R)} . (11.2) Theorem 11.1yields

log 1

¯

ε(C_n) ≤ nE_sp(R) + 1 2 1 +

E_sp⁰ (R)


log n + o(log n), (11.3)

where the term ¹₂ 1 +

E_sp⁰ (R) 

can be viewed as a second-order term (see the discussions in [18,

Section 4.4]). On the other hand, for the case of classical non-singular channels¹, it was shown that [131, Theorem 3.6], for all constant composition codes Cn and rate R ∈ (C1/2,W, C_W),

log 1

¯

ε(Cn) ≥ nE_r(R) +1 2 1 +

E_r⁰(R)


log n + Ω(1), (11.4)

where E_r(R) is the random coding exponent dened in Eq. (9.1), and note that E_r(R) = E_sp(R) for all R ≥ C_1/2,W [21, p. 160], [36]. Hence our result, Theorem11.1, matches the achievability up to the logarithmic order. We note that whether the third order o(log n) in Eq. (11.3) can be improved to O(1)is still unknown even for the classical case.

11.1 Literature Review of Classical Sphere-Packing Bound

This section reviews existing proof approaches of classical sphere-packing bounds:

ε^∗(n, R) ≥ f (n) exp {−n [E_sp(R − g(n))]} , (11.5) ε^∗(n, R) ≥ f (n) exp

n

−nh

Ee_sp(R − g(n)) io

, (11.6)

where f(n) is the pre-factor of the bound, and g(n) is the back-o from the rate. We remark that E_sp coincides withEe_sp in the classical case. The reason why we distinguish the notation Esp andEe_sp here is because of their possible quantum generalizations (recalling that they are not equal in the quantum case, i.e. Theorem9.1in Section9.1). Table11.1below summarizes the comparisons of existing results.

Finite Composition Pre-factor Rate back-o Classical-quantum Tightness

Bounds\Settings blocklength dependent f (n) g(n) channels

Shannon-Gallager- No Yes e^−O(√n) O_{log n}

n

 Dalai [38] Strong

(a) Berlekamp [30]

Haroutunian [31]

No Yes e^−o(n) o(1) Winter [37] Weak

Omura [133]

(b)

Csisár-Korner [25]

(c) Blahut [32] No No e^−O(

√n) O

n⁻¹²

Eqs. (11.148) & (11.153) Strong Yes Yes n⁻¹²(1+|E⁰_sp(R)|+o(1)) 0 Theorem11.1 Strong (d) Altu§-Wagner [91]

(e) Elkayam-Feder [134] Yes Yes O n^−t

O_{log n}

n

 Unknown Unknown

Agustin-Nakibo§lu Yes No O n^−t

0 Unknown Unknown

(f) [135,106,105,136]

Table 11.1: Dierent sphere-packing bounds are compared by (i) the bound is nite blocklength or asymptotical; (ii) whether or not they are dependent on the constant composition codes; (iii) & (iv) the asymptotics of f(n) and g(n); (v) the corresponding c-q generalizations. The parameter t in rows (e) and (f) is some value in the range t > 1/2; and (vi) whether their error exponent expressions for c-q channels are in the strong form (Eq. (1.4)) or weak form (Eq. (12.51)).

(a) Shannon, Gallager and Berlekamp obtained the rst classical sphere-packing bound Eq. (11.5),

1For classical singular channels, one has log_ε(C¯¹_n) ≥ nE_r(R) +¹₂log n + Ω(1)[131]. Further, it was conjectured that [132] that log_ε(C¯¹_n) ≤ nE_sp(R)+¹₂log n+o(log n),for all asymmetric classical singular channels and constant composition codes. However, such a result remains open.

where [30, Theorem 5]

f (n) = e^−O(

√n); g(n) = O log n n



. (11.7)

Their method is based on distinguishing two codewords, followed by Chebyshev's inequality. The works [137] and [138] further improved the coecients in f(n) and g(n) for short to moderate blocklengths.

Remarkably, Shannon-Gallager-Berlekamp's result can be extended to c-q channels with almost the same asymptotics in Eq. (11.7) [38]. See also the result by Dalai and Winter for constant composition codes [39].

(b) Haroutunian [31], Omura [133], Csiszár and Körner [25], Ahlswede [139] subsequently proposed a sphere-packing exponent using discrimination functions (i.e. the relative entropy function in Eq. (1.5)), and obtained the following classical sphere-packing bound for constant composition codes Cn:

¯

ε(W, Cn) ≥ 1 2expn

−n eE_sp(R − δ)(1 + δ)o

, (11.8)

for all δ > 0 and all suciently large n ∈ N, and ¯ε(W, Cn) denotes the average error of the code C_n. The idea is to apply strong converse bounds [140, 141, 142, 133, 25] to a dummy channel, and then use a data-processing inequality for the discrimination function between the dummy and true channels. Recently, Altu§ and Wagner employed a particular strong converse result, Wolfowitz's strong converse result [143], and obtained a form of Eq. (11.6) with [43, Lemma 3]:

g(n) = O

 1

√n



. (11.9)

Following the arguments in [139, Theorem 49], Winter proved a weak sphere-packing bound Eq. (11.8) for constant composition codes in c-q channels [37, Theorem II.20]. We remark that Altu§ and Wagner's result [43] can also be extended to a weak sphere-packing bound for c-q channels when combining Winter's approach [37] with Sharma and Warsi's strong converse result [125, Theorem 3].

(c) Blahut related the channel coding problem to hypothesis testing [32, Theorem 20] (see also [23, Theorem 10.2.1]) and independently obtained a weak sphere-packing bound Eq. (11.6) with

f (n) = e^−O(

√n); g(n) = O

 1

√n



. (11.10)

In Section11.3, we generalize Blahut's result to a strong sphere-packing bound for c-q channels.

(d) In Ref. [48], Altu§ and Wagner applied a sharp concentration inequality to rene the sphere-packing bound Eq. (11.7) with

f (n) = e^−O(

√n); g(n) = O log n n



, (11.11)

for some t > 1/2 and all suciently large n ∈ N.

(e) Elkayam and Feder [134] established a general expression for the error probability in terms of the cumulative distribution function [144, Theorem 6]. Combined with the method of types and Polyanskiy's minimax meta-converse [145, Theorem 3], they proved a classical sphere-packing bound for constant composition codes with

f (n) = Θ n^−t
; g(n) = O log n n



, (11.12)

for some t > 1/2. This sphere-packing bound also had a polynomial pre-factor; however, it is unknown whether this method can be extended to c-q channels.

在文檔中 量子資訊理論中的錯誤率分析 (頁 109-119)