# Moderate and Large Deviation Analysis for C-Q Channels

1701.03195 1704.05703

## Communication Systems Concern

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

Small Deviation

R\to C
\varepsilon>0

Large Deviation

\varepsilon\to 0
R \neq C

Moderate Deviation

R\to C
\varepsilon\to 0

## Three Regimes:

Small Deviation

Large Deviation

Moderate Deviation

R\to C
\varepsilon \neq0
\varepsilon\to 0
R \neq C
\varepsilon\to 0
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})

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)}

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)}

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.
R_n= C- a_n

Cheng and Hsieh, arXiv: 1701.03195.

# Classical-Quantum Channels

W: \mathcal{X} \to \mathbb{C}^{d\times d}

# Large Deviation

\lim_{n\to\infty} \frac{-1}{n} \log\varepsilon_n(R) = E(R)

# Error Exponent Analysis

E_{\text{rc}}(R) \leq E(R) \leq E_{\text{sp}}(R)

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

# Claissical Sphere-Packing Bounds

E_{\text{sp}}(R):=\sup_{s\geq 0} \left\{ \max_P E_0(s,P) -sR\right\}
\tilde{E}_{\text{sp}}(R):=\max_P\min_V\{D(V\|W|P):I(P,V)\leq R\}

Shannon, Gallager, and Berlekamp.  Information and Control, 10(1):65–103, 1967.

Haroutunian. Problemy Peredachi Informatsii, 4(4):37–48, 1968, (in Russian).

Blahut. IEEE TIT, 20(4):405–417, 1974.

{E}_{\text{sp}}(R)= \tilde{E}_{\text{sp}}(R)

# Quantum Sphere-Packing Bounds

E_{\text{sp}}(R):=\sup_{s\geq 0} \left\{ \max_P E_0(s,P) -sR\right\}
\tilde{E}_{\text{sp}}(R):=\max_P\min_V\{D(V\|W|P):I(P,V)\leq R\}

Dalai. IEEE TIT, 59(12):8027–8056, 2013.

Winter. PhD Thesis, Universitate Bielefeld, 1999.

Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703

{E}_{\text{sp}}(R)\leq \tilde{E}_{\text{sp}}(R)
E_0(s,P) := -\log {\rm Tr} \left[ \left( \sum_{x\in\mathcal{X}} P(x) W_x^{\frac1{1+s}}\right)^{1+s} \right]

### $$E_0(s,P)$$  is concave in $$s\geq 0$$

Cheng, MH. IEEE Transactions on Information Theory,  vol. 62, no. 10, pp. 5960–5965 (2016).

\tilde{E}_{\text{sp}}(R,P)=\sup_{0<\alpha\leq 1}\min_\sigma\{\frac{1-\alpha}{\alpha}\left(D^\flat_\alpha(W\|\sigma|P)-R\right)\}
{E}_{\text{sp}}(R,P)\leq\sup_{0<\alpha\leq 1}\min_\sigma\{\frac{1-\alpha}{\alpha}\left(D_\alpha(W\|\sigma|P)-R\right)\}

# Theorem:

{E}_{\text{sp}}(R)\leq \tilde{E}_{\text{sp}}(R)

# Proof:

D_\alpha(\rho\|\sigma):= \frac{1}{\alpha-1}\log \text{Tr}\rho^\alpha \sigma^{(1-\alpha)} \leq D^\flat_\alpha(\rho\|\sigma):=\frac{1}{\alpha-1}\text{Tr}[e^{\alpha\log\rho+(1-\alpha)\log\sigma}]

Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703

# Dalai's Sphere-Packing Bound

\log \frac{1}{\varepsilon_n(R)} \leq n E_{\text{sp}}(R) + O(\sqrt{n})

Dalai. IEEE TIT, 59(12):8027–8056, 2013.

Shannon, Gallager, and Berlekamp.  Information and Control, 10(1):65–103, 1967.

# Theorem:

\log \frac{1}{\varepsilon_n(R)} \leq n E_{\text{sp}}(R) + O(\log{n})

Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703, 2017.

Altug and Wagner, IEEE TIT, 60(3): 1592–1614, 2014.

# Proof:

## Step 1：

$$\varepsilon_{\max}(\mathcal{C}_n) \geq \max_\sigma\min_{\mathbb{x}^n\in \mathcal{C}_n} \tilde{\alpha}_{\frac{1}{|\mathcal{C}_n|}}(W_{\mathbb{x}^n}||\sigma).$$

## Step 2：

Two one-shot converse Hoeffding bounds for $$\tilde{\alpha}_\mu(\cdot|\cdot)$$.

\tilde{\alpha}_\mu(\rho\|\sigma) = \min_\Pi\{\alpha(\Pi,\rho): \beta(\Pi,\sigma)\leq \mu\}

# For Bad Codewords, Use Weak Converse Hoeffding Bound.

\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \kappa_1 e^{-\kappa_2\sqrt{n}-n\phi_n(R'|\rho^n\|\sigma^n)}
\phi_n(r|\rho^n\|\sigma^n):= \sup_{\alpha\in(0,1]} \left\{\frac{1-\alpha}{\alpha}(\frac{1}{n}D_\alpha(\rho^n\|\sigma^n)-r)\right\}

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

Blahut. IEEE TIT, 20(4):405–417, 1974.

Audenaert et. al., PRL 98:160501, 2007.

# For Good Codewords, Use Sharp Converse Hoeffding Bound.

\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \frac{A}{n^{-t}} e^{-n\phi_n(R'|\rho^n\|\sigma^n)}

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

Bahadur and Rao, The Annals of Mathematical Statistics, 31(4):1015–1027, 1960.

Altug and Wagner, IEEE TIT, 60(3): 1592–1614, 2014.

t> 1/2

# If Channel is Symmetric

W_x = V^{x-1}W_1 (V^\dagger)^{x-1},
\log \frac{1}{\varepsilon_n(R)} \leq n E_{\text{sp}}(R) + \frac{1}{2}(1+|E'_{\text{sp}}(R)|)\log{n}+o(1)

# Property of $$E(R)$$:

I_{\alpha}^{(1)}(P,W):= \inf_\sigma D_\alpha(P\circ W\|P\otimes\sigma)
I_{\alpha}^{(2)}(P,W):= \inf_\sigma D_\alpha(W\|\sigma|P)

(a) The map $$(\alpha,P)\to I_\alpha$$ is continuous on $$[0,1]\times\mathcal{P}(\mathcal{X})$$.

(b) The map $$\alpha\to I_\alpha$$ is monotone increasing on $$[0,1]$$.

(c) The map $$\alpha\to \frac{1-\alpha}{\alpha}I_\alpha$$ is strictly concave on $$(0,1]$$.

# Property of $$E(R)$$:

E^{(1)}_{\text{sp}}(R,P):= \sup_{0<\alpha\leq 1} \frac{1-\alpha}{\alpha} \left(I_{\alpha}^{(1)}(P,W)-R\right)

(a) The map $$R\to E^{(\cdot)}_{\text{sp}}$$ is convex, continuous, and non-increasing.

(b) $$E^{(\cdot)}_{\text{sp}}$$ is differentiable w.r.t. $$R$$.

(c) $${E'}^{(\cdot)}_{\text{sp}}$$ is continuous.

E^{(2)}_{\text{sp}}(R,P):= \sup_{0<\alpha\leq 1} \frac{1-\alpha}{\alpha} \left(I_{\alpha}^{(2)}(P,W)-R\right)
E_{\text{sp}}(R,P):= \sup_{s>0} \left(E_{0}(s,P)-sR\right)

# Moderate Deviation:

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

# 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}}

# 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

(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{(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)}

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

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