# Moderate Deviations for C-Q Channels

1701.03195

## Communication Systems Concern

• Error Probability:
• Transmission Rate:
• Code Length:
\varepsilon
$\varepsilon$
R
$R$
n
$n$
\varepsilon
$\varepsilon$
R
$R$
C
$C$
1
$1$
n=1
$n=1$
n=\infty
$n=\infty$

Small Deviation

R\to C
$R\to C$
\varepsilon>0
$\varepsilon>0$

Large Deviation

\varepsilon\to 0
$\varepsilon\to 0$
R \neq C
$R \neq C$

Moderate Deviation

R\to C
$R\to C$
\varepsilon\to 0
$\varepsilon\to 0$

## Three Regimes:

Small Deviation

Large Deviation

Moderate Deviation

R\to C
$R\to C$
\varepsilon \neq0
$\varepsilon \neq0$
\varepsilon\to 0
$\varepsilon\to 0$
R \neq C
$R \neq C$
\varepsilon\to 0
$\varepsilon\to 0$
R\to C
$R\to C$

# Small Deviation:

## a.k.a. Second-Order Analysis

R_n(\varepsilon) = C + \sqrt{\frac{V}{n}}\Phi^{-1}(\varepsilon) + O(\frac{\log n}{n})
$R_n(\varepsilon) = C + \sqrt{\frac{V}{n}}\Phi^{-1}(\varepsilon) + O(\frac{\log n}{n})$

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)}
$\varepsilon_n(R) = e^{-\Theta(n)}$

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)}
$\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}$

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.
$\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.$
R_n= C- a_n
$R_n= C- a_n$

Cheng and Hsieh, arXiv: 1701.03195.

# Moderate Deviation:

\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
$\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}$
\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
$\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.$
R_n= C- a_n,
$R_n= C- a_n,$

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.

# 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\}
$\frac{1}{n a_n^2}\log\varepsilon_n(R_n)\leq \frac{4}{n a_n^2} - \frac{1}{a_n^2}\max_{s} \left\{ \tilde{E}_0(s,P)-sR\right\}$

# 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}}
$\tilde{E}_0(s,P) = s C_W - \frac{s^2}{2} V_W + \frac{s^3}{6} \frac{\partial^3 \tilde{E}_0(s,P)}{\partial s^3}|_{s=\bar{s}}$

# 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
$\tilde{E}_0(s,P):= -\log \sum_x P_x\text{Tr} W_x^{1-s}(PW)^s$

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

# 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)}
$\tilde{\alpha}_{e^{-nR_n}}(\rho^n\|\sigma^n) \geq \frac{A}{{s_n^\star} n^{-1/2}} e^{-n\phi_n(R_n'|\rho^n\|\sigma^n)}$

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

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
$\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \frac{A}{ n^{-t}} e^{-n\phi_n(R'|\rho^n\|\sigma^n)}, \quad t>1/2$