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

Performance Trade-offs

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.

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\}

\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}}

\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).

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)}

\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

Moderate Deviations for C-Q Channels

Communication Systems Concern

Performance Trade-offs

Three Regimes:

Small Deviation:

a.k.a. Second-Order Analysis

Large Deviation:

a.k.a. Error Exponent Analysis

Moderate Deviation:

Moderate Deviation:

Achievability:

Step 1:

Achievability:

Step 2:

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

Converse

Similar to Quantum SP Bounds

1. A New Sharp Hoeffiding Bound.

2. Weak Hoeffiding Bound needs special attention.

New Sharp Converse Hoeffding Bound

Summary

Thank you!