Moderate Deviations for C-Q Channels

1701.03195



 

Communication Systems Concern

  • Error Probability:
  • Transmission Rate:
  • Code Length:
\varepsilon
ε\varepsilon
R
RR
n
nn
\varepsilon
ε\varepsilon
R
RR
C
CC
1
11
n=1
n=1n=1
n=\infty
n=∞n=\infty

Small Deviation

R\to C
R→CR\to C
\varepsilon>0
ε>0\varepsilon>0

Large Deviation

\varepsilon\to 0
ε→0\varepsilon\to 0
R \neq C
R≠CR \neq C

Performance Trade-offs

Moderate Deviation

R\to C
R→CR\to C
\varepsilon\to 0
ε→0\varepsilon\to 0

Three Regimes:

Small Deviation

Large Deviation

Moderate Deviation

R\to C
R→CR\to C
\varepsilon \neq0
ε≠0\varepsilon \neq0
\varepsilon\to 0
ε→0\varepsilon\to 0
R \neq C
R≠CR \neq C
\varepsilon\to 0
ε→0\varepsilon\to 0
R\to C
R→CR\to C

Small Deviation:

a.k.a. Second-Order Analysis

R_n(\varepsilon) = C + \sqrt{\frac{V}{n}}\Phi^{-1}(\varepsilon) + O(\frac{\log n}{n})
Rn(ε)=C+VnΦ−1(ε)+O(lognn)R_n(\varepsilon) = C + \sqrt{\frac{V}{n}}\Phi^{-1}(\varepsilon) + O(\frac{\log n}{n})

Strassen, Transactions of the Third Prague Conference on Information Theory, pp. 689–723, 1962.

Tomamichel and Tan, CMP 338(1):103–137, 2015.

Large Deviation:

a.k.a. Error Exponent Analysis

\varepsilon_n(R) = e^{-\Theta(n)}
εn(R)=e−Θ(n)\varepsilon_n(R) = e^{-\Theta(n)}

Shannon, Bell System Technical Journal, 38(3):611–656, 1959.

Burnashev and Holevo, Problems of information transmission, 34(2):97–107, 1998.

Moderate Deviation:

\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
εn(Rn)=e−Θ(nan2)\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}

Altug and Wagner. IEEE TIT 60(8):4417–4426, 2014.

Chubb, Tomamichel and Tan, arXiv: 1701.03114.

\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
{an}:an→0, ann→∞.\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
R_n= C- a_n
Rn=C−anR_n= C- a_n

Cheng and Hsieh, arXiv: 1701.03195.

Moderate Deviation:

\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
εn(Rn)=e−Θ(nan2)\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
{an}:an→0, ann→∞.\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
R_n= C- a_n,
Rn=C−an,R_n= C- a_n,

Cheng and Hsieh, arXiv: 1701.03195.

[Achievability] \(\limsup_{n\to\infty}\frac{1}{na_n^2}\log\varepsilon_n(R_n)\leq -\frac{1}{2V_W}\)

[Converse] \(\liminf_{n\to\infty}\frac{1}{na_n^2}\log\varepsilon_n(R_n)\geq -\frac{1}{2V_W}\)

Chubb, Tomamichel and Tan, arXiv: 1701.03114.

Achievability:

Step 1:

$$\varepsilon_n(R_n) \leq 4\exp\left(-n\left[\max_{0\leq s\leq 1} \tilde{E}_0(s,P)-sR_n\right]\right)$$

Hayashi, PRA  76(6): 06230,12007.

\frac{1}{n a_n^2}\log\varepsilon_n(R_n)\leq \frac{4}{n a_n^2} - \frac{1}{a_n^2}\max_{s} \left\{ \tilde{E}_0(s,P)-sR\right\}
1nan2logεn(Rn)≤4nan2−1an2maxs{E~0(s,P)−sR}\frac{1}{n a_n^2}\log\varepsilon_n(R_n)\leq \frac{4}{n a_n^2} - \frac{1}{a_n^2}\max_{s} \left\{ \tilde{E}_0(s,P)-sR\right\}

Achievability:

Step 2:

Apply Taylor Expansion to \(\tilde{E}_0(s,P)\) at \(s=0\). 

\tilde{E}_0(s,P) = s C_W - \frac{s^2}{2} V_W + \frac{s^3}{6} \frac{\partial^3 \tilde{E}_0(s,P)}{\partial s^3}|_{s=\bar{s}}
E~0(s,P)=sCW−s22VW+s36∂3E~0(s,P)∂s3∣s=s¯\tilde{E}_0(s,P) = s C_W - \frac{s^2}{2} V_W + \frac{s^3}{6} \frac{\partial^3 \tilde{E}_0(s,P)}{\partial s^3}|_{s=\bar{s}}

Property of \(\tilde{E}_0(s,P)\):

\tilde{E}_0(s,P):= -\log \sum_x P_x\text{Tr} W_x^{1-s}(PW)^s
E~0(s,P):=−log∑xPxTrWx1−s(PW)s\tilde{E}_0(s,P):= -\log \sum_x P_x\text{Tr} W_x^{1-s}(PW)^s

(a) Partial derivatives of \(\tilde{E}_0\) are continuous.

(b) \(\tilde{E}_0\) is concave in \(s\geq 0 \).

\text{(c)}\left.\frac{\partial}{\partial s}\tilde{E}_0(s,P)\right|_{s=0} = I(P,W).
(c)∂∂sE~0(s,P)∣s=0=I(P,W).\text{(c)}\left.\frac{\partial}{\partial s}\tilde{E}_0(s,P)\right|_{s=0} = I(P,W).
\text{(d)}\left.\frac{\partial^2}{\partial s^2}\tilde{E}_0(s,P)\right|_{s=0} = V(P,W).
(d)∂2∂s2E~0(s,P)∣s=0=V(P,W).\text{(d)}\left.\frac{\partial^2}{\partial s^2}\tilde{E}_0(s,P)\right|_{s=0} = V(P,W).

Converse

Similar to Quantum SP Bounds

1. A New Sharp Hoeffiding Bound.

2. Weak Hoeffiding Bound needs special attention.

New Sharp Converse Hoeffding Bound

\tilde{\alpha}_{e^{-nR_n}}(\rho^n\|\sigma^n) \geq \frac{A}{{s_n^\star} n^{-1/2}} e^{-n\phi_n(R_n'|\rho^n\|\sigma^n)}
α~e−nRn(ρn∥σn)≥Asn⋆n−1/2e−nϕn(Rn′∣ρn∥σn)\tilde{\alpha}_{e^{-nR_n}}(\rho^n\|\sigma^n) \geq \frac{A}{{s_n^\star} n^{-1/2}} e^{-n\phi_n(R_n'|\rho^n\|\sigma^n)}

\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\),  \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)

Chaganty-Sethuraman , The Annals of Probability, 21(3):1671–1690, 1993.

\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \frac{A}{ n^{-t}} e^{-n\phi_n(R'|\rho^n\|\sigma^n)}, \quad t>1/2
α~e−nR(ρn∥σn)≥An−te−nϕn(R′∣ρn∥σn),t>1/2\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \frac{A}{ n^{-t}} e^{-n\phi_n(R'|\rho^n\|\sigma^n)}, \quad t>1/2

Summary

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